. . J < ~ s / / WILLIAMS $ ROGERS SEKIf!^,; : ,-.;; NEW COMMERCIAL ARITHMETIC BY JOHN H. MOORE NEW YORK-:. CINCINNATI-:. CHIC AGO AMERICAN BOOK COMPANY COPYRIGHT, 1904 AND 1907, BY L. L. WILLIAMS AND F. E. ROGERS. ENTERED AT STATIONERS' HALL, LONDON. MOORE'S COM. AB. EDUCJTTON DEFT* PREFACE A commercial arithmetic should be comprehensive in its scope, but should contain no complicated or obsolete subjects. It should furnish abundant material for drills in modern business problems, and, by natural and progressive steps in the methods of developing the subjects presented, should cultivate in the student those qualities of accuracy, rapidity, and self-reliance that will be so valuable to him later. With these objects in mind this book has been written. It is not intended for beginners, but for students pursuing a commercial course in business and secondary schools. While it may be assumed that these students have previously completed a more elementary arithmetic, yet experience has demonstrated that it is usually neces- sary for them to review the fundamental operations, and become familiar with the short methods which- are applicable to simple calcu- lations, before they can do effective work in commercial arithmetic. The underlying principles of arithmetic are, therefore, briefly re- viewed, and many practical counting-room methods having a direct bearing upon them are carefully illustrated and explained. Great care has been taken to make the methods of developing all the principles natural and businesslike. All of the operations given in connection with the illustrative problems are accompanied with solutions which enable the student to understand the principles involved. The student is taught to understand a process before he is taught to summarize it in a rule. Solutions and rules are omitted in all cases where it is thought the student can prepare them without assistance. The few rules given in the book all follow solutions, and are intended to aid the student to produce intelligent results. In no case are they intended to be committed to memory. Mental work has received due emphasis throughout the book. Oral exercises of a thoroughly practical nature accompany every sub- ject, and in many cases methods of computation are introduced and developed through a series of oral drills. 3 4 PREFACE An attempt has been made to make the treatment of the whole subject highly educative, but methods and topics distinctively utili- tarian in their value have received due attention. Arithmetical puzzles and improbable conditions have been studiously avoided, and a feature is made of concrete business problems from the outset. Particular attention has been devoted to the subject of Addition. The group method is carefully developed through a series of oral and written drills. The exercises on tabulation and all the exercises calling for vertical and horizontal additions are especially valuable. Only small common fractions are introduced ; they are the only ones used in ordinary business. In connection with this subject special care has been devoted to the topics Quantity, Price, and Cost, and Bills and Accounts. The methods developed and the forms illustrated in this part of the book are especially helpful and practi- cal. In the chapter on Denominate Numbers a feature is made of the subject Practical Measurements. In the preparation of this portion of the book the author consulted mechanics, contractors, and busi- ness men, 'thoroughly versed in their several departments, in order to get at current, practical usages. In the chapter on Percentage and its Applications, the subjects Commercial Discounts, Interest, Bank Discount, and Customhouse Business have been especially emphasized because they are so closely connected with modern busi- ness transactions. In the chapter on Sharing, the subject Partner- ship has been thoroughly covered. All the problems given in this work are treated from the accountant's standpoint, and are entirely free from all unusual conditions. In the preparation of all the sub- jects, business men have been consulted freely. In connection with many of the subjects a great deal of valuable information is given. Numerous business forms are also introduced, and made the basis of a series of problems. Some of the problems given have been taken from the Williams and Bogers's Commercial Arithmetic, by Oscar F. Williams ; but the majority of them are new. Acknowledgment is due to Professor C. D. Clarkson of the Depart- ment of Commerce in Drexel Institute, Philadelphia, for valuable assistance in perfecting the volume. CONTENTS SIMPLE NUMBERS PAGE Preliminary Definitions .7 Notation and Numeration . 8 Addition 12 Subtraction 25 Multiplication 36 Division 47 Properties of Numbers .50 UNITED STATES MONEY 58 METHODS FOR PROVING WORK 65 FRACTIONS Common Fractions 70 Decimal Fractions 93 Quantity, Price, and Cost . . . . . . . . 108 Bills and Accounts .......... 121 DENOMINATE NUMBERS Measures 134 Denominate Quantities 150 Practical Measurements 163 PERCENTAGE AND ITS APPLICATIONS Percentage ' 184 Commercial Discounts ......... 198 Gain and Loss 206 Marking Goods 214 Commission 219 Interest . 228 Present Worth and True Discount 256 Negotiable Paper 259 Bank Discount 263 Partial Payments 272 Equation of Accounts 277 Cash Balance 291 Savings-bank Accounts 296 Stocks 300 Bonds 310 5 6" CONTENTS PERCENTAGE AND ITS APPLICATIONS PAGE Insurance 314 Taxes 326 Customhouse Business 333 Exchange . 340 SHARING Proportional Parts 357 Partnership 359 Building and Loan Associations ....... 371 RATIO AND PROPORTION Ratio 377 Proportion 378 STORAGE Cash Storage 381 Credit or Average Storage 383 APPENDIX Metric System of Measures . . 385 Powers and Roots 389 Compound Interest Table for Annual Payments , . . 398 NEW COMMERCIAL ARITHMETIC SIMPLE NUMBERS PRELIMINARY DEFINITIONS 1. Arithmetic is the science of numbers and the art of com- puting by them. 2. A unit is a single thing, or a definite quantity regarded as a single thing. In selling cloth by the yard the unit is one yard of cloth ; in measuring lands by the acre the unit is one acre of land ; in counting the number of students in a class the unit is one student ; in buying bricks by the thousand the unit is one thousand bricks ; in selling "posts by the hundred the unit is one hundred posts. 3. An integral unit is one, or a whole thing. 4. A decimal unit is one of the parts obtained by dividing an integral unit into tenths, hundredths, and so on. 5. A fractional unit is one of the parts obtained by dividing an integral unit into any number of equal parts. 6. A number is a unit or two or more units. 7. An integer is an integral unit or two or more integral units. 8. An abstract number is a number not associated with any particular thing or quantity; as, 2, 7, 11. 9. A concrete number is a number associated with some particu- lar thing or quantity ; as, 11 men, 6 cords of wood. 10. A denominate number is a concrete number expressing standard money value, or standard measure or weight; as, 1 dollar; 5 gallons ; 6 pounds, 4 ounces. 11. Like numbers are numbers that have the same unit value ; as, 2, 6, 9 ; 3 houses, 7 houses, 5 houses ; 2 years, 9 years, 20 years. r. I "S ' ' SIMPLE NUMBERS [ 12-21 '121 TJnKke numbers are numbers that have different unit values ; as, 12, 16 days, 4 boys, 2 hours. 13. A simple number is a number consisting of a unit or a collec- tion of units of the same kind ; as 2, 12 men, 8 pounds. 14. A compound number is a number consisting of two or more denominations of the same unit; as, 7 bushels, 3 pecks, 1 quart; 6 pounds, 3 ounces. 15. A problem is a question to be solved. 16. A principle is a general law used as a basis for computations. 17. A rule is a concise outline of the steps to be taken in the performance of a computation. ORAL EXERCISE 1. All denominate numbers are concrete. Are all concrete numbers denominate ? Explain. 2. All the following numbers are concrete. Are they denomi- nate ? Explain. 16 pounds, 12 men, 4 rods, 7 dollars, 9 houses. 8. State clearly the difference between a concrete number and a denominate number. 4- Give an example of a compound number. d. Is there any difference between a compound number and a denominate number? Explain. 6. What is the unit of 16? of 75 barrels of molasses? of $7500? of ^ of a week? of 2\ dozen? 7. Give an example of a simple abstract number ; of a simple concrete number. 8. Name two like numbers ; two unlike numbers. NOTATION AND NUMERATION 18. Notation is the art of writing numbers. 19. Numbers are generally expressed by figures or letters, but they may also be expressed by words. 20. Numeration is the art of giving oral expression to numbers. 21. The two methods of notation in use are the Arabic and the 22-26] NOTATION AND NUMERATION 9 ARABIC NOTATION 22. The Arabic Method of Notation, first used by the Arabs, com- prises ten characters or figures, as follows : O 123J4567 8 9 Naught One Two Three Four Five Six Seven Eight Nine The figures 1, 2, 3, 4, 6, 6, 7, 8, 9 are called digits, and the figure is called eero, naught, or cipher. 23. The value of a digit is determined (1) by its name, and (2) by its position in a number. A digit, when standing alone, always equals the number of units which its name indicates; when com- bined with other digits its value is determined by the place which it occupies in the number. 24. The value of a digit in any given number increases from right to left, and decreases from left to right in a tenfold ratio. Thus, in the number eleven, expressed 11, the second 1 from the right has a value ten times as great as the first 1. 25. Orders of Units. The place which a figure occupies in a number is called its order. The ones of a number are called units of the first order; the tens, units of the second order; the hundreds, units of the third order; the thousands, units of the fourth order; and so on. Ten units of any given order are equal to one unit of the next higher order. 26c Periods. Numbers containing four figures or more are, for convenience, separated by the comma into periods of three figures each. Beginning at the right, the first group is the period of units; the second, the period of thousands; the third, the period of millions; the fourth, the period of billions ; and so on. One thousand units of any given period are equal to one unit of the next higher period. The left-hand period of any number may consist of one, two, or three figures. 10 SIMPLE NUMBERS [2d NUMERATION TABLE 1 PERIOD NUMBER NUMERATION ORDER H* Hundreds of quintillions 21st 7th Oi Tens of quintillions 20th CO Quintillions 19th 0) Hundreds of quadrillions 18th 6th -5 Tens of quadrillions 17th CD Quadrillions 16th H- 1 Hundreds of trillions 15th 5th 0> Tens of trillions 14th M Trillions 13th Hundreds of billions 12th 4th 0& Tens of billions llth CO Billions 10th CO Hundreds of millions 9th 3d M Tens of millions 8th ~J Millions 7th 00 Hundreds of thousands 6th 2d CO Tens of thousands 5th Oi Thousands 4th Hundreds 3d 1st CO Tens 2d ^ Units 1st ORAL EXERCISE 1. Read the following numbers: 5,005; 1,925; 3,036; 4,569; 260; 715. NOTE. In reading numbers, always express them in the shortest way possible. Thus, 1520 should be read fifteen hundred twenty, not one thousand five hundred twenty. This is important in writing amounts in checks, notes, and drafts, where the space is often limited. Do not read and between periods or between hundreds and units. Thus, 16,725 should be read sixteen thousand, seven hundred twenty-five, not sixteen thousand and seven hundred and twenty-five. This distinction is of the utmost importance in connection with the writing of decimals. 2. What is the name of the second period of notation? the third ? the fourth ? the fifth ? the seventh ? 26-29] NOTATION AND NUMERATION 11 8. What are the names of the successive periods expressed by seven figures ? by eleven figures ? 4- How many figures are required to write millions ? trillions ? 5. How many units of the first order in the second period of 9,321 ? 6. Eead the following numbers: 246,920,460; 750,861,432,120; 9,246,921,006 ; 1,269,247,268,490,621 ; 700,600,070,000,000,002. 7. Express in figures three units of the fourth order, three of the third, nine of the second, and seven of the first. 8. Distinguish between an order and a period as related to numbers. ROMAN NOTATION 27. The Roman Method of Notation is used extensively in number- ing volumes, chapters, sections, and the other important divisions of books; also in numbering dials and tabular outlines. It employs seven characters or letters, as follows : I V X L O D M 1 5 1O 50 100 500 100O 28. Roman Values. The value of Roman characters is twofold. 1. Each character when standing alone has a definite value, as above. 2. Each character also has a varying value when written in varying positions in combination with other Roman numerals. 29. General Principles. 1. Repeating a letter repeats its value. Thus, II represents two ; XX, twenty; CCC, three hundred. 2. When a letter of less value is placed before one of greater value the number indicated is the difference between the values of such numbers. Thus, IX represents nine ; XC, ninety. 3. When a letter of less value is placed after one of greater value, the number indicated is the sum of the values of such letters. Thus, CX represents one hundred ten; LXIV, sixty-four. 12 SIMPLE NUMBERS [ 29-34 4. A bar placed over a letter multiplies the value of the letter by one thousand. Thus, V represents five thousand ; C, one hundred thousand. 5. A letter should not be repeated more than three times in expressing numbers. 6. A bar is never placed over the letter I. TABLE OF ROMAN NUMERALS WITH ARABIC EQUIVALENTS I .... 1 XII ... 12 L .... 50 DCC . . 700 II .... 2 XIII ... 13 LX . . . 60 DCCC . 800 III. ... 3 XIV ... 14 LXX . . 70 CM . . 900 IV. ... 4 XV ... 15 LXXX . . 80 M. . . 1000 V .... 5 XVI ... 16 XC . . . 90 MM . . 2000 VI. ... 6 XVII . . .17 C . . . .100 V. . . 5000 VII ... 7 XVIII . . 18 CO ... 200 X. . . 10000 VIII ... 8 XIX ... 19 CCC ... 300 L . . . 50000 IX. ... 9 XX ... 20 CD ... 400 C . . . 100000 X . . . .10 XXX ... 30 D . . . .600 D . . . 500000 XI .... 11 XL ... 40 DC . . .600 M. . . 1000000 ORAL EXERCISE Bead the following expressions : XCII; XXVII; XXIX; CCXVII; DLXX; DCC; MDCCCLIII; MMDXLIV; MCDLXX. ADDITION 30. Addition is the process of combining several numbers into one equivalent number. 31. The sum or amount is the result obtained by addition. 32. The sign + signifies addition and is read plus. 33. The sign = signifies equality and is read equals. 34. General Principles. 1. Only the same orders of units of like numbers can be added. 2. The sum always expresses units of the same name as the several numbers to be added. _!_^_^_i 2 2 9 _? J? _J? _f 40. Horizontal Addition. Numbers, wfrem written in horizontal lines, as on invoices and other business forms, may be added with- out being rewritten in vertical columns. 41. In adding numbers horizontally, add from left to right and then verify all results by adding from right to left. The group method may be employed to advantage where numbers are written horizontally. The ability to add horizontally saves a great deal of time in making out bills and in performing other commercial opera- tions. ORAL EXERCISE Add from left to right and review from right to left the following : 1. 9, 9, 2, 5, 4, 3, 1, 6, 2. 6. 21, 32, 40, 82, 56, 30. 8. 42, 21, 46, 32, 14, 21. 7. 31, 18, 28, 36, 45, 21. 3. 52, 46, 35, 72, 68, 50. 8. 67, 61, 60, 63, 62, 65. 4. 21, 26, 32, 34, 81, 63, 45, 90, 31. 9. 51, 67, 34, 58, 56, 29. 5. 66, 31, 41, 18, 41, 62, 59, 35, 45. 10. 62, 60, 51, 28, 35, 62. 11. How many days in the summer months ? 12. Find the sum of the four numbers that may be expressed by the figures 2 and 3 ; 4 and 5 ; 6 and 7. 13. Find the sum of all the even numbers from 6 to 12 inclusive. NOTE. When figures to be added appear in consecutive order, and there is an odd number of them, the total may be found by multiplying the middle figure by the number of consecutive figures. Thus, 3 + 4 + 5 + 6 + 7 = 5x5 = 25. When any numbers appear in consecutive order their total may be found by multiplying one half the sum of the first and last numbers by the number of consecutive numbers. Thus, 14 + 15 + 16 + 17 + 18 = 16 x 6 = 80. 14- Find the sum of all the numbers from 7 to 19 inclusive. 15. Find the sum of all the numbers from 1 to 9 inclusive. 16. Find the sum of all the numbers from 3 to 19 inclusive ; of all the numbers from 5 to 13 inclusive. 41] ADDITION 19 17. Find the sum of 42, 42, 42, 42, 75. NOTE. When a number is repeated several times in any addition, the work may be shortened by multiplication. 18. A man who was born in 1853 died when he was forty-nine years old. In what" year did he die ? 19. What is the sum of 15, 23, 36, 18, 28, 92 ? of 21, 22, 23, 24, 25,26,27,28,29,30,31? WRITTEN EXERCISE Drill on the following and similar problems until correct results can be obtained in twenty seconds or less. 1. 2426^264 62462148 64292862 56259421 62462962 52462564 62469264 62462942 2. 47257386 52472164 83492752 26534721 23425625 32462813 42612542 78955473 8. 27452462 87950241 20724065 86957447 72757786 77777777 88888888 22222222 * 56319217 48263547 62519546 38641948 95722618 77554286 62496246 62462942 5. 72519218 67482153 72186349 39256258 78295416 87596357 21111016 20407030 ft 57264592 87492165 48576901 66875465 52163441 10205211 93758617 58759218 Drill on the fol owing and similar exercises until correct results may be obtained in twenty seconds or less. 7. 8 9. 10. 11. 12. 2714 4052 4032 3146 1487 1846 2652 6021 5061 4219 2116 1092 1493 1473 4728 2614 4574 1531 7510 6687 3214 9743 6589 1675 1126 7214 6010 6478 3752 1832 4251 9386 5271 2592 1678 1645 6859 7521 2642 7286 7593 1729 3114 4268 5537 4924 9164 1011 7996 7821 6214 6214 7386 4010 4216 5275 9146 7585 9552 6020 3114 3942 3910 2137 7829 5190 6996 4728 1120 7214 3687 1786 7245 3659 2110 2110 2014 2405 3865 2854 1640 1016 1730 7216 5125 7529 2114 4032 3019 4520 6219 2110 1431 2016 2170 7121 4346 1011 5214 6147 2590 2514 20 SIMPLE NUMBERS T 41-43 18. Show the totals of the following columns downwards and from left to right. Prove the results by adding the vertical and horizontal totals. 6249 2145 2592 6014 2172 4592 _ 4625 1687 1649 5019 1645 7126 1872 1421 3145 2041 1392 5218 4124 3652 1650 6215 1746 9041 3635 1926 1722 9013 7592 7592 4216 4521 1490 7016 6219 6218 3417 1725 7518 4110 5764 7527 1641 1686 2041 6211 2047 2692 4356 4035 4250 2140 6211 1420 14. Complete the following table by showing the totals of the columns vertically and horizontally. Prove the work by adding the vertical and horizontal totals. DEPARTMENTAL SALES FOR THE WEEK ENDING Nov. 15, 1903 DAYS CLOTHING DRY GOODS FURNISHINGS MILLINERY HOUSEHOLD UTENSILS TOTAL Monday $790.50 $988.40 $126.50 $256.85 $496.80 Tuesday 640.18 890.50 90.18 420.62 841.62 Wednesday 960.70 950.40 75.60 398.40 462.60 Thursday 490.18 960.80 214.90 425.60 521.90 Friday 930.50 720.60 126.70 396.80 762.80 Saturday 840.15 989.72 215.20 469.65 925.64 Total 42. The Two-column Method of Addition. Some accountants are very partial to the two-column method of addition, claiming that it is more rapid and accurate. 43. In adding two columns at once, combine first the tens of the numbers and then the units. Thus, in adding 75 and 32 think of 105 (75 + 30) and 2, or 107. 48] ADDITION 21 To illustrate this method of addition, take the accompanying example. Beginning with the number 46 at the top of the column, add first the ^Q tens and then the units of the successive numbers, as follows: 32 46-f30 = 76; 76 + 2= 78 65 78 + 60 = 138 ; 138 -f 6 = 143 51 143 + 60 = 193 ; 193 + 1 = 194 26 194 + 20 = 214 ; 214 + 6 = 220 220 In making computations in this manner name results only. Thus, beginning at the top of the accompanying example and adding downwards, read 76, 8, 138, 143, 193, 4, 214, 220. ORAL EXERCISE 1. Add the following by double columns as explained above : 45 24 52 39 28 57 62 27 33 41 12 13 19 57 92 39 26 58 56 52 31 34 48 45 37 43 56 38 14 12 2. Announce the totals of the above combinations at sight from left to right and from right to left. Thus, 84, 50, etc. NOTE. Require that this work be done rapidly. Drill on the above and similar combinations until the student can announce the totals as rapidly as he can count 1, 2, 3, etc. 3. Add the following by double columns, naming results only : 61 63 82 21 43 19 42 37 51 28 46 27 43 19 24 39 17 19 23 17 31 24 33 25 52 25 32 27 41 26 434140366444162872111021111621 4- Add the numbers in problem 3 horizontally by the two-column method. Add from left to right and verify the work by adding from right to left 5. Name results only in determining the totals of the following by the two-column method : 28 14 64 48 37 51 45 16 24 81 59 72 27 45 52 46 26 81 52 43 42 92 41 36 47 31 16 52 92 41 25 42 95 13 94 18 61 72 33 16 73 41 95 27 92 92 18 62 24 26 36 43 86 37 52 87 64 65 46 68 51 32 51 37 51 41 86 14 42 19 49 17 84 85 72 28 16 28 82 28 35 91 92 48 37 51 86 76 41 14 22 SIMPLE NUMBERS [44-47 44. Proving Addition. The simplest way to test the correct- ness of addition is to add the columns a second time in reverse order. 45. Accountants who have to add very long columns of figures frequently begin at the right-hand column and write on a piece of waste paper the full sum of each column added, and then, to verify the work, begin at the left side and add the columns in reverse order, again writing the full sums on a piece of waste paper. If the sum of the totals shown by the first addition is the same as the sum of the totals shown by the second addition, the work is assumed to be correct. 46. The accountant not infrequently has to perform his work amid more or less confusion. In adding long columns, if this method is employed, he can be interrupted or can leave his work for a time and resume it again without examining in detail the columns which have already been completed. 47. This method of proving addition is illustrated in the follow- ing example and solution : SOLUTION. Beginning with the right-hand column and adding downwards, the total is 28. Write 28 to the right of the numbers added, or on a piece of waste paper ; without carrying add the next 19 4225 28 column, and the total is 15, which should be 17 6248 15 written as shown in the accompanying illus- 17 tration ; add the next column without carry- 1 Q an( * tlie tota ^ * s ^ ' ad( * tne next c l umn and the total is 19. The sum of these totals 20878 2130 20878 is 20878. 20878 VERIFICATION. Beginning with the left- hand column and adding upwards, the total is 19. Write this to the left of the figures to be added or on a piece of waste paper, as shown in the accompanying illustration ; without carrying add the next column, and the total is 17, which write as shown in the illustration ; add the next column, and the total is 15 ; the next, and the total is 28. The sum of these totals is 20878, or the same as found by the first addition ; hence it is assumed that the work is correct. WRITTEN REVIEW 1. In the following statement add the columns downwards and from left to right, and then prove the work by adding the vertical and horizontal totals. 47] ADDITION 23 STATE ASSESSMENTS YEAR ARMORIES METROPOLI- TAN ABOLITION OP GRADE METROPOLITAN HIGHWAYS TOTAL SEWER CROSSINGS WATER 1895-1896 $21,498.29 $59,702.19 $25,811.94 $285,600.54 $161.67 1890-1897 28,056.27 119,321.10 45,583.53 211,901.92 100.45 1897-1898 28,056.27 95,421.14 62,677.92 199,900.41 571.94 1898-1899 34,223.15 75,753.81 56,854.31 258,990.00 153.23 1899-1900 34,223.15 129,773.27 71,662.03 411,861.54 101.82 1900-1901 34,223.15 12,625.73 131,074.00 578,696.96 68.78 Total 2. Complete the following sales sheet, ing the vertical and horizontal totals. Prove the work by add- SUMMARY OF DAILY SALES JULY 2 SHOES GLOVES HATS DRESS GOODS CLOTHING TOTAL A to D Ledger $237.31 $126.92 $132.16 $263.64 $423.09 E to H Ledger 228.80 140.75 110.25 357.18 387.75 I to L Ledger 238.84 231.78 106.35 676.83 627.71 M to P Ledger 143.54 157.57 161.69 382.55 641.23 Q to T Ledger 848.49 657.02 510.45 510.59 651.45 U to Z Ledger 556.51 213.19 388.54 811.82 680.29 Total 3. The sales of a dry goods house for the week ending Nov. 22, 1903, were as follows: Monday, domestics, $540.10; notions, $325.85; woolens, $864.98; dress goods, $325.78. Tuesday, do- mestics, $995.85; notions, $419.62; woolens, $919.10; dress goods, $146.84. Wednesday, domestics, $975.89; notions, $853.64; woolens, $ 1659.89 ; dress goods, $ 1259.89. Thursday, domestics, $856.74; notions, $459.13; woolens, $756.85; dress goods, $588.74. Friday, domestics, $862.47; notions, $817.39; woolens, $1249.86; dress goods, $1560.84. Saturday, domestics, $1529.84; notions, $915.62; woolens, $958.22; dress goods, $1079.54. Arrange these facts in tabular form, in six, columns, with proper headings. Show (a) the total sales for each department, (b) the total daily sales, and (c) the total sales for the week. SIMPLE NUMBERS [ 47 4. The records of a city post office show the following mail for one week : Monday, registered letters, 725 ; ordinary letters, 15,279 ; postal cards, 2147; book packets, 963; parcels, 184; newspapers, 26,419. Tuesday, registered letters, 461 ; ordinary letters, 12,365 ; postal cards, 2011; book packets, 395; parcels, 416; newspapers, 21,936. Wednesday, registered letters, 369 ; ordinary letters, 16,285 ; postal cards, 1989; book packets, 618; parcels, 365; newspapers, 23,162. Thursday, registered letters, 8490 ; ordinary letters, 14,317 ; postal cards, 416; book packets, 562; parcels, 213; newspapers, 23,164. Friday, registered letters, 959; ordinary letters, 25,162; postal cards, 2116; book packets," 475 ; parcels, 163; newspapers, 22,790. Saturday, registered letters, 416; ordinary letters, 11,259; postal cards, 659; book packets, 384; parcels, 175; newspapers, 21,218. .Arrange these facts in tabular form, in eight columns, with proper headings. Find (a) the total number of separate pieces of mail for each day, (6) the total number of pieces of each class, and (c) the total number of pieces for the week. Copy or write from dictation and find the sums of the following: 5. .*2x* 92451826 40159061 6. 24164290 72154031 7. 32169528 62169528 8. 12345678 28968457 52192165' 16941762 62195437 10475631 87965421 15304693 65954370 20047509 74926587 \ 59346599 ^ 31462845 32168492 65109011 52416011 33715586 88475*621 92657788 65945876 11141017 21411731 10401721 41627428 78991047 74839101 92517496 17283142 31426357 10108765 93479491 65493762 21407110 56461086 59627488 58911476 11169042 77562345 95178654 72491368 25172825 87693421 72958649 72159072 41627598 24683157 58721985 21311510 66901080 36912141 58759271 21411631 72164010 11354678 21864925 47293742 69957788 10019087 17264592 40171650 28521654 98798778 18259015., 21101670 29364124 76453111 48-54] SUBTRACTION 25 SUBTRACTION 48. Subtraction is the process of finding the difference between two numbers. 49. The subtrahend is the number to be subtracted. 50. The minuend is the number from which the subtrahend is to be subtracted. 51. The remainder or difference is the number obtained by sub- traction. 52. The sign signifies subtraction and is read minus or less. When the sign of subtraction is placed between two numbers, it indicates that the number written after it is to be taken from the one written before it 53. Numbers written within a parenthesis ( ), under a vinculum , or separated by the sign of multiplication x , are to be considered together. Thus, 16 - (4 + 2) or 16 - 4 + 2 signifies that the sum of 4 and 2 is to be subtracted from 16 ; 16 - 4 x 2 signifies that the product of 4 and 2 is to be subtracted from 16. 54. General Principles. 1. Only the same orders of units of like numbers can be subtracted. 2. The sum of the subtrahend and remainder is equal to the minuend. ORAL EXERCISE 1. Subtract by 4's from 44 to ; from 39 to 3. 2. Subtract by 6's from 49 to 1 ; from 78 to 0. 3. Subtract by 5's from 135 to ; from 121 to 1. 4. Subtract by 7's from 38 to 3 ; from 64 to 1 ; from 44 to 2. 5. Subtract by 8's from 91 to 3 ; from 55 to 7 ; from 37 to 5. 6. Subtract by 9's from 131 to 23; from 57 to 12; from 95 to 5. 7. Subtract by 15's from 90 to 0; from 120 to 0; from 76 to 1. 8. Subtract by 13's from 41 to 2; from 57 to 5; from 63 to 11. 9. Subtract by ll's from 88 to 0; from 72 to 6; from 91 to 3. 10 Subtract by 12's from 54 to 6; from 128 to 8; from 145 to t 26 SIMPLE NUMBERS [55 55. Example. Find the difference between 348 and 185. 348 SOLUTION. Since only units of the same order can be subtracted, 185 write units under units, tens under tens, and hundreds under hun- Jg3 dreds, and draw a line beneath. Beginning at the right, 5 units from 8 units leaves 3 units. Write 3 under the column of units. Since 8 tens cannot be taken from 4 tens, transform 1 of the 3 hundreds into 10 tens, and add it to the 4 tens, making 14 tens ; then, 8 tens from 14 tens leaves 6 tens, which write under the column of tens. Since 1 of the 3 hundreds has been taken, there are only 2 hundreds remaining ; then, 1 hundred from 2 hundreds leaves 1 hundred. The difference between the two numbers given is, therefore, 163. In practice think only of results and write them without hesitating. Thus, in the above problem write or think only 3, 6, 1. ORAL EXERCISE 1. From what number must we subtract $2.54 to have $7,46 remaining ? 2. If I pay $375 for a carriage and sell it at a loss of $73.75, how much do I receive for it ? S. The smaller of two numbers is 96; their difference is 46. What is the larger number ? 4- Pronounce at sight the difference between the numbers in each of the following groups. 79 62 56 85 67 78 89 98 67 34 27 47 19 21 49 fc !3 14 12 105 107 108 106 127 168 99 119 121 97 53 38 56 77 48 38 47 69 5. Pronounce at sight the difference between the numbers in each of the following groups. The subtrahend is placed above the minuend in order to give practice in find- ing the difference between numbers that are so arranged. If one is not able to subtract in this manner, he is frequently required to rearrange the numbers on separate paper to subtract them. This is a waste of time, since by a little practice one can readily subtract numbers that are not regularly arranged. 55-60] SUBTRACTION 27 79 58 72 29 47 57 62 49 26 92 164 149 132 98 124 126 93 78 18 123 93 47 244 169 158 137 214 54 246 186 94 488 239 248 267 264 SHORT METHODS 56. The complement of a number is the difference between such number and a unit of the next higher order. Thus, the complement of 6 is 4, since 4 is the difference between 6 and 10, or 1 ten, a unit of the next higher order than 6 ; the complement of 83 is 17, since 17 is the difference between 83 and 100, or 1 hundred, a unit of the next higher order than 83. 57. Two numbers whose sum is equal to a unit of the next higher order are called complementary numbers. Thus, 209 and 791 are complementary numbers, since their sum is equal to 1000 ; 2467 and 7633 are complementary numbers since their sum is equal to 10000. 58. If two numbers of more than one figure are complementary numbers, the sum of their units figures is 10, and of each of their corresponding higher orders, 9. Thus, 642 and 358 are complementary numbers ; the sum of the units figures is 10, and the sum of the figures in the corresponding higher orders is 9. 59. The foregoing principle may be applied to advantage in mak- ing change. Since we read numbers from left to right, it is gener- ally best in making change to begin at the left to subtract. In beginning at the left to subtract, take 1 from the number of units of the highest order in the minuend, and regard each of the lower orders as 9 except the last, which must be regarded as 10. 60. Example. A gave a twenty-dollar bill in payment for an account of $14.72. How much change should he receive? SOLUTION. Begin at the left to subtract. 1 from the highest order in the minuend leaves 1. 1 from 1 leaves 0. 4 from 9 leaves 5, which write in the units' column. 7 from 9 leaves 2, which write in the tenths' column. 2 from 10 leaves 8, which write in the hundredths' column. The result is $5.28. 2# SIMPLE NUMBERS [ 00-62 DRILL EXERCISE By inspection, find the difference between the following numbers : 400 600 300 700 900 1000 100 200 300 500 132 175 86 263 458 532 52 31 57 138 $1.00 $2.00 $3.00 $4.00 $5.00 $8.00 $7.00 $8.00 .39 1.15 1.17 1.58 2.21 2.39 5.36 1.37 $20.00 $20.00 $10.00 $30.00 $30.00 $40.00 $50.00 2.59 8.76 5.72 28.61 29.57 25.86 6.78 Subtract each of the following numbers from $2.00: 27^, 52^, 89 $1.52, $1.13, $1.41, $1.59, 85^ 41 37 56^, 18 97 If $100 is offered in payment for each of the following accounts, what amount of change should be returned ? $ 25.95, $ 85.67, $ 37.54, $92.18, $65.51, $87.75, $69.52, $18.75, $37.58, $88.13, $71.15, $41.30, $39.18, $25.72. Exercises similar to the above should be continued until correct results can be given without a moment's hesitation. 61. Frequently an accountant finds it desirable to take the sum of several numbers from the sum of several other numbers without transferring the totals from the books of record to separate paper. The following explanations will be found suggestive of short cuts which may be employed to advantage in such cases. 62. Examples. 1. From 24,794 subtract the sum of 4159, 6490, and 4462. 24794 SOLUTION. For convenience write the numbers under each , .. -Q other with the minuend set off from the subtrahend by a straight line. Beginning at the right and adding the units of the subtra- hend the sum is 11, which, subtracted from 14 (the next higher 4462 number ending with 4), leaves 3, the units of the required result. 9683 The sum of the figures in the tens' column plus 1 (the number of tens added to the minuend in the previous subtraction) is 21 , which, subtracted from 29 (the next higher number ending with 9), leaves J, the tens' figure of the required result. 62] SUBTRACTION 29 The sum of the figures in the hundreds' column plus 2 (the number of hun- dreds added to the minuend in the previous subtraction) is 11, which, subtracted from 17, leaves 6, the hundreds' figure of the required result. The sum of the figures in the thousands 1 column plus 1 (the number of thousands added to the minuend in the previous subtraction) is 15, which, subtracted from 24, leaves 9, or the thousands* figure of the required result. 2. The gross weights and tares of 5 barrels of sugar are as fol- lows : 319-19, 322-21, 311-17, 322-19, 329-21 pounds. Find the net weight. SOLUTION. In billing, the above numbers would be written horizontally as follows: 319-19, 322-21, 311-17, 322-19, 329-21. The minuend is the gross weight, and the subtrahend is the tare. Adding the units of the subtrahend horizontally the sum is 27, which, subtracted from the next higher order of units (30), leaves 3. 3 added to the units of the minu- end equals 26. Write 6 as the units of the net weight. Since the tens of the subtrahend are 1 more than the tens of the minuend, add 1 to the next higher order of the subtrahend, or subtract 1 from the next higher order of the minu- end. Add 1 to the tens of the subtrahend, and the result is 8, which, subtracted from the next higher order of units (10), leaves 2. Adding 2 to the tens of the minuend, the result is 10. Write as the tens of the net weight. Since the tens of the minuend are the same as the tens of the subtrahend, there is nothing to carry. Adding the hundreds of the minuend, the result is 15. Write 15 as the hundreds of the net weight. The net weight is then 1506 pounds. NOTE. In billing where items are listed as gross weight and tare the above process will be found especially helpful. Sufficient practice should be required to give the student facility in making the extensions properly. This principle may also be used to advantage hi finding the balances of ledger accounts. WRITTEN EXERCISE NOTE. In the first four problems below the gross weight in pounds is written to the left of the hyphen and the tare in pounds to the right of the hyphen. Find the net weight as explained in Example 2, above. 1. 10 casks of hams, 392-67, 412-71, 402-71, 411-67, 408-68, 425-71, 400-69, 399-70, 398-71, 426-68. & 6 baskets pork loins, 312-49, 301-56, 297-48, 415-43, 312-49, 314-56. 8. 4 tubs of lard, 71-14, 70-15, 69-14, 62-15. 4. 8 casks shoulders, 428-19, 322-21, 327-19, 311-17, 314-17, 315-18, 317-21, 342-24. SIMPLE NUMBERS [62 By either of the short methods explained in 62 find the balances of the following accounts. 6. *Dr. The Union Bank, in account 6. *Dr. The Union Bank, in account with 7. 1. _A* -. / /^ ^ tf "faL,, 260 / 2. ' ^ ^2-^X ^ 3 / &4JL &r 2-0 J^ 72/7 ^ / tMA n ^ / /!*/ A ?/ 00 7? 7? 720 KS 62-64] SUBTRACTION 8. 81 ,-fc What is the balance in Deposit Aug. 17, $65.98. What is Find the balances of the following bank accounts without using pen or pencil except to write the results. 9. Balance in bank June 1, $ 650.40. Checks from June 1 to July 1, $ 145.20, $ 14.90, $ 60.50, $ 20.40. \>auk July 1 ? .70. Balance in bank Aug. 15, $ 695.40. Checks from Aug. 15 to Sept. 1, $ 146.20, $ 90.50, $60.95. the balance in bank Sept. 1 ? 11. Balance in bank Jan. 15, $460.40. Deposit Jan. 20, $ 152.65. Checks from Jan. 15 to Feb. 1, $172.40, $ 14.90, $ 16.95, $ 40.65. What is the balance in bank Feb. 1 ? 63. Combining Addition and Subtraction in One Process. When a number, or the total of several numbers, is to be taken away from the total of several other numbers, the two processes may be combined as in 62, or as shown in the following examples. 64. Examples. 1. From the sum of 12 and 6 take 3. SOLUTION. Adding 12 to 6 we have 18. It is self-evident that 18 8 is equivalent to 18+ (10 - 3)- 10. When a number is both added to and subtracted from any quantity, the value of the quantity is not changed. Applying this principle in solving the above problem, mentally take 3 from 10, add the difference to 6 aid 12, and subtract 10 from the result. Thus, 7, 25, 15, the required result. 32 SIMPLE NUMBERS 2. From the sum of 827 and 534 subtract 356. [ C4-67 SOLUTION. Arranging the numbers horizontally and adding the units, by naming the results only, we have from the right 4 (10^6), 8, 15, 5 (15 10) to write as the first figure of the required result. Adding the tens, we have 5 (10 5), 8, 10, (10 10) to write as the tens of the required result. Adding the hundreds, we have 7 (10 - 3), 12, 20, 10 (20 - 10) to write as the hundreds of the required result. The final result is, therefore, 1005. 3. From the sum of 729 and 642 subtract 211. SOLUTION. 9, 11, 20, or to write and 1 to carry to the minuend. 10 (9 -f 1 carried), 14, 16, or 6 to write. 8, 14, 21, or 11 to write. The final result is, therefore, 1160. 4. From the sum of 321 and 811 subtract 369. SOLUTION. 1, 2, 3, or 3 to write and 1 to subtract from the tens of the minu- end. 3 (4 1), 4, 6, or 6 to write and 1 to subtract from the minuend. 6 (7 1), 14, 17, or 7 to write. The result is, therefore, 763. NOTE. In this class of work grouping may be used to advantage. To show every step in the process the results in the above solutions were determined with- out grouping. 65. Hence the following rule : Take each order of units in the subtrahend from 10, add the difference to the same order of units in the minuend, and deduct 10 from the result obtained. In adding any order of units if the result is less than 20,there is nothing to carry to the next higher order in the minuend ; if the sum is 20 or more, there is always something to carry to the next higher order in the minuend ; if the sum is less than 10, there is 1 to subtract from the next higher order in the minuend. Thus, if the sum of any order of units is a number from 10 to 19 inclusive, carry nothing ; a number from 20 to 29 inclusive, carry 1 ; a number from 1 to 9 inclusive, subtract 1. 66. Individual Ledger Balances. The above method is particularly helpful in making extensions on a banking individual ledger. 67. Example. Find the balance to the credit of D. Roe in the following bank account : NAMES BALANCES CHECKS DEPOSITS BALANCES D. Roe 692 85 146 25 625 42 67] SUBTRACTION 33 SOLUTION. The first column shows the balance on deposit, the second the amount withdrawn by checks, and the third the amount deposited. The sum of the old balance and the deposits is therefore the minuend, and the sum of the checks the subtrahend. Employing the principles just explained, the balance of the foregoing account may be determined mentally, as follows : 2, 7 (2 4- 10 - 5), 12. Write 2 in the new balance column. 4, 12 (4 -j- 10 - 2), 20. Write in the new balance column and carry 1. 6 (5 -f 1 carried), 10 (6 -f 10 6), 12. Write 2 in the new balance column. 2, 8 (2 -f- 10 - 4), 17. Write 7 in the new balance column. 6, 15 (6 -f- 10 - 1), 21. Write 11 in the new balance column. The new balance is, therefore, $1172.02. WRITTEN EXERCISE 1. Copy or write from dictation the following individual ledger accounts and find the new balances without using pen or pencil except to write the results. After extending the new balances add the old balances, checks, deposits, and new balances, respectively. Prove the work. The sum of the total old balances and the total deposits minus the total checks .should equal the total new balances. NAMES BALANCES CHECKS DEPOSITS BALANCES Allen, E. W. 962 59 421 65 875 90 Briggs, C. W. 725 42 126, 42 215. 95 Comer, L. M. 826 54 217 47 421 66. Day, O. D. 924 54 413 86 966 75 Emery, A. L. 592 87 436 58 297 62 Foley, B. E. 726 88 315 92 496 87 Good, J. I. 925 43 413 86 575 94 Hall, L. 0. 1426 88 613 92 726 48, Irwin, Chas. E. 1217 95 214 86 926 45 Jones, Chas. H. 725 77 216 54 818 72 ' t . In the following account find (a) the total checks, and (6) the new balances. Prove the work. - - 7 34 SIMPLE NUMBERS [67 NAMES BALANCES CHECKS IN DETAIL TOTAL CHECKS DEPOSITS BALANCES Ames, D. T. 9241 10 126 95 1400 00 Ballou, M. T. 6418 40 200 216 00 10 700 900 00 00 Collins, W. T. 1421 19 500 417 00 40 920 1240 00 10 Dorman & Co. 2146 11 200 711 00 40 1750 92 Evans & Son 1492 20 400 240 00 10 1120 00 Farley Bros. 1742 20 410 920 00 19 1750 00 Grant & Snow Co. 2114 90 750 00 . 8710 2500 00 00 Hall & Smith 6218 10 1200 00 1100 09 , 1460 41 Irwin, J. T. 1721 10 200 1140 00 80 1400 62 Jamison, M. I. 4216 91 600 00 1721 42 . ORAL REVIEW 1. From 100 take 15; 25; 42; 16; 73; 81; 19; 16; 14; 22; 33; 45; 55; 65; 72; 87; 64; 47; 35; 51; 17. 2. I gave a fifty dollar bill in payment for an account of $ 23.45. How much change should I receive ? NOTE. In making change it is always advisable to determine the amount by subtraction and then to verify the result by addition. Thus, if $ 10 is received in payment for a bill of $7.42, by inspection, determine the balance and prove the result by adding to the amount purchased the change counted out. If the amount of the payment is obtained by such addition, the result is assumed to be correct. 7. $30- $22.79 = ? 8. $15- $11.68=? 9. $25- $23.75= ? 10. $5-$1.03 + $2.17 S. $10- $4.17=? 4. $15- $2.77=? 5. $20-$1.75'+$2.16=? 6. $20- $2.95=? 11. From $3 take 89^; 74#; 67^; 83^; 74^; 18^; 24^; '; 68^; 38^; 27^; 52^. ' 12. In an account with Charles Spencer a payment of $ 17 which he makes is erroneously charged, instead of credited, to his account. What is the error in the balance of his account ? Explain. 67] SUBTRACTION 35 13. By inspection, find the difference between the following : 900 400 2000 800 5000 3000 9 52 172 24 57 127 $100.00 14.65 $200.00 15.65 $400.00 17.24 $50.00 5.21 $100.00 11.45 $200.00 15.65 $500.00 65.95 $300.00 11.42 $600.00 18.52 $700.00 17.25 $ 1000.00 127.50 $200.00 18.50 $150.00 1.92 $100.00 19.00 $7000.00 18.59 $100.00 7.92 $3000.00 15.49 $10.00 7.92 WRITTEN REVIEW 1. A has $950, which is $275 more than I have, and I have $ 300 more than B. How much have we together ? 2. A and B together owe me $9275; A owes me $3150. After paying me $ 1900 on account, how much does B still owe me ? 3. A produce dealer bought 200 barrels of apples for $415. Had he received $ 75 more in selling them his gain would have been equal to the amount originally paid for the apples. What amount was received from the sale of the apples ? 4- A furniture dealer bought a stock of goods amounting to $5216. After selling goods amounting to $4917, he took an ac- count of stock and found that he had furniture on hand amount- ing to $ 1937. Did he gain or lose, and how much ? 5. A retail hardware dealer bought merchandise amounting to $ 1249. After selling from this stock articles amounting to $ 842, he took an account of the stock remaining unsold and found that it was worth $ 311. Did he gain or lose, and how much ? 6. At the close of the business, July 1, a merchant had cash in the safe amounting to $314. July 2 he received from sales $ 526 ; on account, $435 ; the cash in the safe at the close of July 2 amounted to $ 219. What ^yere the total disbursements for July 2 ? 7. A father divided his farm, consisting of 675 acres, among his three sons, Harvey, William, and Albert. Harvey received 75 more acres than William, who received 225 acres, and to Albert was given the remainder. How many acres were given to Albert ? 36 SIMPLE NUMBERS f 68-76 MULTIPLICATION 68. Multiplication is the process of taking one of two numbers as many times as there are units in the other. 69. The multiplicand is the number that is to be taken a required number of times. 70. The multiplier is the number which indicates how many times the multiplicand is to be taken or multiplied. 71. The product is the number obtained by multiplication. 72. The sign x signifies multiplication, arid is read times or multiplied by. The sign x is read times when the multiplier precedes the multiplicand and multiplied by when the multiplicand precedes the multiplier. 73. General Principles. 1. The multiplier always signifies a num- ber of times, and is an abstract quantity. 2. The multiplicand may be either an abstract or a concrete number. 3. The product always has the same name as the multiplicand. 74. Factors are the numbers used in obtaining a product. 75. The numerical result of one number by another is the same whichever factor is regarded as the multiplier. The above general principles are to be recognized only in explanations of work done. For illustration, take the following example : If one barrel of apples cost $3, what will 125 barrels cost? SOLUTION: Since 1 barrel of apples cost $3, 125 barrels will cost 125 times $ 3, which is $ 375. We cannot multiply 125 by $3, but since 3 times 125 times is equal to 125 times 3, we may interchange the factors and have 3 tim $ 125. The product, it will be observed, is the same in either case. Hence, An interchange of the factors in any multiplication does not affect the product. 76. In multiplying one number by another, always use the smaller quantity as the multiplier. It should be remembered, however, that the product always has the same name as the true multiplicand. 70-78] MULTIPLICATION 37 MULTIPLICATION TABLE 1 a .'{ 4 5 7 8 10 11 12 13 14 15 10 17 IS 111 20 21 22 2:J 24 25 2 4 6 8 10 1-2 14 16 18 20 22 21 26 28 30 82 84 86 38 40 42 -14 46 48 50 8 6 9 12 ir. 18 21 24 27 8Q 88 86 89 42 46 48 51 51 5T M 68 66 69 72 75 4 8 12 16 so -24 28 32 36 40 41 IS 62 56 (id 64 68 72 T6 Sd S4 ss 82 96 100 5 Id 15 20 25 88 35 40 45 50 55 60 65 70 75 SO 86 90 95 loo 105 110 115 12o 125 6 12 18 24 80 86 4'2 48 64 80 66 w T8 84 90 96 102 10s 114 121' (26 182 188 144 150 7 14 21 28 35 42 48 56 88 To 77 s-l 91 98 105 112 119 1-26 168 (40 147 154 161 16s 175 8 Itl 24 88 40 4^ 56 64 T2 BO 88 96 104 112 120 12S 186 144 162 160 16s 176 1S4 192 200 9 [8 27 36 45 54 68 72 M 90 98 toe 111 126 185 144 158 1 02 171 1SI 189 19s 207 2 1C, 225 10 20 30 40 60 60 70 80 M LOO 110 120 180 140 150 160 170 180 190 200 210 220 280 240 250 11 22 88 44 55 06 77 88 99 no 121 182 148 154 165 176 181 19s 209 22 ii 281 242 268 264 275 1*2 24 86 4- (in 72 M 96 108 120 132 141 166 168 180 192 204 216 22- 241 252 264 276 23s 300 i:i 26 55 ;>2 DO rS 51 104 1 1 * 18d 143 156 168 18-2 195 208 221 454 247 26i 278 286 299 U2 325 U 2- 42 66 TO 54 98 112 126 140 15i'l68 182 196 210 224 28s 252 26(1 2 si 294 !()s m $86 350 15 30 45 60 75 M lor. 1-20 185 1 5i i 165180 1 1*5 210 225 240 255 270 286 801 815 ',80 146 -500 375 16 82 4s 64 80 1)6 112 1-28 144 160 176 192 20s 224 240 256 272 288 804 821 386 (62 (68 }S4 400 17 84 51 68 85 109 iisi 186 158 170 187'204 281 288 255 272 2-9 ;o6 828 8K 857 174 !91 408 425 18 86 64 7'2 90 108 126 144 162 180 198216 '284 252 270 288 o()6 524 842 8 ('.( 87- 596 414 432 450 19 88 57 76 95 114 1:53 152 171 190 209 228 24T 266 286 804 8-28 U2 861 8-1 :!99 41S 487 456 475 20 40 80 80 HIM 120 140 160 180 200 220 240 260 2-0 800 820 840 860 ;;su KM 120 440 460 4SO 500 ^T 42 68 >4 lo, 126 14, li>- [Sy 2io 231 2;V2 2,:; 21) I 81,> 88C> 85',' 51 899 42( 141 46-2 483 504 525 22 44 66 88 110 132 ir,4 176 19- 220 242 261 286 808 880 852 874 896 4ls 44( 462 4*4 506 528 550 28 46 96 M 115 188 161 184 2(i7 280 253 2T( 209 822 845 868 891 414 487 46( 488 506 529 552 575 24 48 72 M 120 144 16s 102 216 240 264 288 812 886 860 384 408 48'2 456 484 504 5-28 55-2 576 600 u so 75 10(1 128 160 175 200 225 250 275 30( 8-25 861 876 400 425 450 475 5<;( 5-25 550 575 (500 625 1 2 3 4 I 7 8 9 10 11 12 18 14 15 1C 17 18 lit 20 21 22 23 24 25 77. In the accompanying table take the multiplicand in the figures arranged horizontally at the top or the bottom, and the multi- plier in the column at the left ; the product is then the number under or above the multiplicand and opposite the multiplier. 78. Examples. 1. Find the product of 2418 x 7. 2418 SOLUTION. Write the multiplier 7 below the unit figure of 7 the multiplicand as shown in the margin, and begin at the 1fSQ2f right to multiply. 7 times 8 units equals 56 units, or 5 tens and 6 units. Write 6 units in the place of units, and reserve 5 tens to add to the product of tens. 7 times 1 ten equals 7 tens, and adding the 5 tens reserved gives 12 tens, or 1 hundred and 2 tens. Write 2 in the place of tens, and reserve 1 to add to the product of hundreds. 7 times 4 hundreds equals 28 hundreds, and adding the 1 hundred reserved gives 29 hundreds, or 2 thousands and 9 hundreds. Write 9 in the place of hundreds, and reserve 2 to add to the product of thousands. 7 times 2 thousands equals 14 thousands, and adding the 2 thousands reserved gives 16 thousands. Write this entire sum to the left of the figures already written, thus completing the multiplication and obtaining as a result 16,926. 38 SIMPLE NUMBERS [ 78-83 2. Find the product of 417 x 352. SOLUTION. Write the multiplier below t,ne multiplicand in the same unit order from the right. The multiplier is composed of 2 units, 6 tens, and 3 hun- dreds, or 2 -f 50 -f 300. Multiplying the -nultiplicand by 2, 60, and 300 respec- tively, and adding the products, the results are as follows : (a) FULL PARTIAL PRODUCTS (6) ABBREVIATED PARTIAL PRODUCTS 417 417 352 352 834 Partial Product by 2 834 20850 Partial Product by 60 2086 125100 Partial Product by 300 1251 _ 146784 Complete Product 146784 The ciphers at the right of the partial products are of no value in finding the complete product, and they should be omitted in practice as shown in (6). Observe that the first right-hand figure of each partial product is then always directly under the figure of the multiplier used. CONTRACTIONS IN MULTIPLICATION 79. To multiply any number by 10, 100, 1000, etc. 80. Every cipher that is annexed to a number moves each digit one place to the left, or converts units into tens, tens into hundreds, and so on. 81. Hence to multiply a number by 10, 100, 1000, etc., To one factor annex as many ciphers as there are ciphers in the other factor. 82. To multiply by any number of 10's, 100's, 1000's, etc. 83. Since 50 is 5 tens, 500 5 hundreds, etc., to multiply a number by 50, 1600, 7000, etc., Omit the ciphers on the right of the factors, multiply the remaining part of the multiplicand by the remaining part of the multiplier, and to the product thus obtained annex the ciphers omitted. ORAL EXERCISE By inspection, find the products of: 1. 34,000x600. 5. 710x6000. 9. 716x20. 9. 216,000x300. 6. 72x3000. 10. 160x20. 3. 5400x70. 7. 714,000x200. 11. 805x2000. 4. 5120x2000. 8. 28,000x20. 12. 43,007x100 84-86] MULTIPLICATION 39 84. To multiply any number by 11. 85. Examples. 1, Find the product of 72 x 11. (a; FULL OPERATION (6) CONTRACTED OPERATION 72 Write 2 in the place of the units of the 11 product. 2 + 7 = 9. Write 9 in the place of 72 the tens of the product. Bring down 7 for the 72 place of the hundreds of the product. 792 The completed product is therefore 792. SOLUTION. By glancing at (a) it will be observed that : 1. The partial product by the units of the multiplier contains the same figures as the partial product by the tens. 2. The first figure in the partial product by tens of the multiplier falls under the second figure of the partial product by units. Making practical use of these observations, we have the abbreviated opera- tion as in (6). 2. Find the product of 89 X 11. SOLUTION. Write 9 as the first figure of the product. 9 + 8 = 17. Write 7 as the second figure of the product, and carry 1. 8+1=9. Write 9 as the third figure of the product, thus completing the multiplication and obtaining as a re- sult 979. 3. Find the product of 195 X 11. SOLUTION. Write 5 as the first figure in the product. 6 +9 = 14. Write 4 as the second figure in the product, and carry 1. 9 + 1 + 1 = 11. Write 1 as the third figure in the product, and carry 1. 1 + 1 = 2. Write 2 as the fourth figure in the product, thus completing the multiplication and obtaining as a re- sult 2145. 86. Hence the rule : Write as the first figure of the product the first figure of the multiplicand. Beginning at the right of the multiplicand add the units and tens, the tens and hundreds, the hundreds and thousands, and so on. Finally, bring down tJie left-hand figure of the multipli- cand as the left-hand figure of the product. Carry when necessary. 40 SIMPLE NUMBERS [86-88 ORAL EXERCISE By inspection, find the products of : 1. 62 X 11. 11. 37 X 11. 21. 52 x 11. 81. 225 x 11. 2. 51 X 11. 12. 11 x 43. 22. 85 x 11. 82. 11 X 428. S. 11 X 71. 18. 59 x 11. 23. 11 X 93. S3. 11 x 927. 4^ 45 X 11. 14. 11 x 78. 24. 11 x 68. / 34- 11 x 728. 6. 11 X 29. 15. 81 x 11. 25. 121 X 11. 85. 726 x 11. 6. 11 X 75. 16. 21 x 11. 26. 132x11. 86. 487 x 11. 7. 11 X 84. 17. 24 x 11. 27. 11 X 141. 87. 1926 xll. 8. 11 X 91. 18. 11 x 26. 28. 11 X 164. 88. 11 X 1726. 9. 86 x 11. 19. 11 X 34. 29. 11 X 214. 89. 11 X 2814. 10. 32x11. m 11x47. SO. 216x11. 40. 5419x11. 87. To multiply by any number of ll's ; as, 22, 33, etc. 88. Examples. 1. Multiply 24 by 22. SOLUTION. 4x2=8. Write 8 as the first figure of the product. 6(4 + 2) x 2 = 12. Write 2 as the second figure of the product, and carry 1. 2x2 + 1 = 5. Write 5 as the third figure of the product, thus completing the multiplication and obtaining as a result 528. 2. Find the product of 121 x 77. SOLUTION. Write 7 as the first figure of the product. 3(1 + 2) x 7 = 21. Write 1 as the second figure of the product and carry 2. 3(2 + 1) x 7 + 2 = 23. Write 3 as the third figure of the product and carry 2. 1x7 + 2 = 9. Write 9 as the fourth figure of the product, thus completing the multiplication, and obtaining as a result 9317. It should be observed that this method is practically the same as the method of multiplying by 11. WRITTEN EXERCISE Find the products of : 1. 21x22. 5. 33x425. 9. 125x88. 13. 146x55. 2. 44x36. 6. 42x33. 10. 22x1214. 14. 22x216. 8. 12x22. 7. 66x71. 11. 44x74. 15. 151x44. 4. 66x215. 8. 214x33. 12. 25x88. 16. 66x125 89-00] MULTIPLICATION 41 89. To multiply by any number, one part of which is contained a certain number of times in another part. 90. Examples. L Find the product of 254 x 357. SOLUTION. Observe that the tens and hundreds (35) of the 357 multiplier make a number five times the units (7). Multiply 254 1778 by 7 in the usual manner. Multiply the resulting partial product 8890 b y 5 ( the Partial product by 7 multiplied by 6 equals the partial 90678 product by 35), and complete the multiplication by adding. 8. Find the product of 12,121 x 12,816. 12121 12816 SOLUTION. Multiply 12,121 by 16, obtaining 193,936. 1Q o Q qfi Multiply 193,936 by 8 (128 is just 8 times 16), and obtain 1,651,488. Add the two partial products and obtain 165,342,736, thus completing the multiplication. 155342736 WRITTEN EXERCISE 1. If a page contains 1864 ems, how many ems in a book of 794 pages ? 2. If 735 men can dig a canal in 9328 days, how many men would be required to complete the work in 1 day ? S. How many links in 639 chains, each chain having 8471 links ? 4. The Boston " Boot Maker " will enable a workman to make 324 pairs of boots daily. How many can be made with this machine in 328 days ? 5. What must be paid for grading a railroad 1809 miles long at $1288 a mile? 6. What will 248 acres of land cost at $ 217 per acre ? 7. If the circulation of the city library is 27,126 books daily, how much would it be in 168 days ? 8. A dray horse can draw 10 loads, of 1569 pounds each, per day. How many pounds can 749 horses draw in 1 day at the same rate ? 9. A barrel of flour weighs 196 pounds. What is the weight of 639 barrels ? 10. A railway is 1449 miles in length, and was completed at an average cost of $106,775 per mile. What was the total cost of constructing it ? 42 SIMPLE NUMBERS [ 91-93 91. Cross Multiplication. The possibilities of what is known as cross multiplication are almost without end. The method is par- ticularly helpful in making mental extensions on invoices. By it the product of any two numbers of two figures each may be ascer- tained mentally, and the product of any number by any other num- ber of two figures may be obtained by simply writing the completed product. By intelligent, persistent practice any number may be multiplied by any other number of three or four figures without writing any of the partial products that are ordinarily written in multiplying one number by another. 92. Examples. 1. Find the product of 74 x 23. SOLUTION. 4 x 3 = 12. Write 2 as the first figure of the product 74 and carry 1. 7x3 + 1 (carried) + 8 (4 x 2) = 30. Write as the 23 second figure of the product and carry 3. 7x2 + 3 (carried) = 17. 1702 Write 17 to the left of the figures already written in the product, thus completing the multiplication and obtaining a product of 1702. & Find the product of 124 x 62. . ^ , SOLUTION. 4x2 = 8. Write 8 as the first figure of the product. 2 x 2 + 24 (4 x 6) = 28. Write 8 as the second figure of the prod- uct and carry 2. 1 x 2 + 12 (2 x 6) + 2 (carried) = 16. Write 6 7688 as the third figure of the product and carry 1. 1x6 + 1 (car- ried) = 7. Write 7 as the fourth figure of the product, thus com- pleting the multiplication and obtaining a product of 7688. 3. Find the product of 2146 x 32. SOLUTION. 6 x 2 = 12. Write 2 and carry 1. 4x2 + 1 (carried) + 18(6 x 3) = 27. Write 7 and carry 2. 1x2 + 2 ^ (carried) + 12(4 x 3) = 16. Write 6 and carry 1. 2x2 + 1 68672 (carried) + 3(1 x 3) = 8. Write 8. 2x3=6. Write 6, thus completing the multiplication and obtaining a product of 68672. 4. Find the product of 214 x 236. SOLUTION. 4x6 = 24. Write 4 and carry 2. 1x6 + 2 + 12(4 x 3) = 20. Write and carry 2. 2 x 6 + 2 + 3(1 x 3) + 236 8 (4 x 2) = 25. Write 5 and carry 2. 2x3 + 2 + 2(1x2) = 10. 50504 Write and carry 1. 2x2+1=5. Write 5, thus completing the multiplication and obtaining a product of 50">04- 93. The method of cross multiplication can hardly be covered by a set rule, since it includes such a wide range of numbers. In attempting to make practical application of this method the "follow- ing principles are important. 93-95] MULTIPLICATION 43 1. Units multiplied by units equal the units of the product. 2. Tens multiplied by units, plus units multiplied by tens equal the tens of the product. 3. Hundreds multiplied by units plus tens multiplied by tens plus units multiplied by hundreds equal hundreds of the product. 4. Thousands multiplied by units plus hundreds multiplied by tens plus tens multiplied by hundreds plus units multiplied by thousands equal thousands of the product. WRITTEN EXERCISE Find the product in each of the following problems. Do not use pen or pencil except to write the product. 1. 2. 3. * 5. 6. 24 x 32. 41 x 35. 2115 x 32. 127 x 23. 53 x 42. 2144 x 36. 7. 8. 9. 10. 11. 12. 1121 x 42. 116 x 45. 47 x 26. 37 x 48. 1174 x 26. 181 x 59. 13. 14. 15. 16. 17. 18. 125 x 34. 36 x 58. 1215 x 57. 2125 x 64. 164 x 32. 172 x 27. 19. 20. 21. 22. c> KO. 24. 34 x 51. 1217 x 42. 241 x 36. 142 x 28. 45 x 91. 3215 x 42. 94. To multiply any number by the numbers from 101 to 109 inclusive. 95. Examples. 1. Find the product of 64 x 102. 64 .. ~o SOLUTION. 64 x 2 = 128. Write 28 and carry 1. 64 x 1 + 1 = 65. Write 65 to complete the product. 6528 2. Find the product of 215 x 102. 215 SOLUTION. 15 x 2 = 30. Write 30 for the first figures of the 102 product. 2x2 + 5 = 9. Write 9 as the third figure of the product. 21930 21 x 1 = 21. Write 21 to complete the product. S. Find the product of 2265 x 104. 2265 SOLUTION. 5x4 = 20. Write and carry 2. 6 x 4 + 2 = 26. 104 Write 6 and carry 2. 2 x 4 + 2 + 5 = 15. Write 5 and carry 1 235560 2x4 + 1+6 = 15. Write 5 and carry 1. 22 x 1 + 1 = 23. Write 23 to complete the product. NOTE. The above method of multiplication may be used to advantage in billing where the price is $1.02 and $1.03, etc. 44 SIMPLE .NUMBERS Cf 96-100 ORAL EXERCISE By inspection, find the product of : 1. 32 x 102. 4. 53 x 105. 7. 58 x 102. 10. 105 x 94. 2. 103 x 47. 5. 72 x 106. 8. 104 x 32. 11. 71 x 102. 3. 39 x 104. 6. 114 x 105. 9. 106 x 58. 12. 88 x 101. 96. To multiply by any number of three figures, the tens of which is a cipher. 97. Examples. 1. Find the product of 126 X 302. SOLUTION. 26 x 2 = 52. Write 52 as the first two figures of the product. 1x2 + 6x3 = 20. Write as the third figure ** of the product and carry 2. 12 x 3 + 2 = 38. Write 38 to com- 38052 pi ete t b e product. 2. Find the product of 1215 x 304 SOLUTION. 15 x 4 = 60. Write 60 as the first two figures 1215 of the product. 2x4 + 5x3 = 23. Write 3 as the third figure 304 of the product, and carry 2. 1x4 + 2 (carried) + 1x3 = 9. 369360 Write 9 as the fourth figure of the product. 12 x 3 = 36. Write 36 to complete the product. It will be observed from the above solutions that this method is practically the same as 94. WRITTEN EXERCISE Find the product in each of the following problems without using pen or pencil except to write the figures of the completed product. 1. 121 x 202. 5. 305 x 408. 9. 413 x 301. 13. 123 x 407 9. 116 x 403. 6. 431 x 309. 10. 365 x 308. 14. 218 x 905 3. 151 x 304. 7. 918 x 201. 11. 413 x 503. 15. 721 x 801 4. 165 x 405. 8. 725 x 402. 12. 936 x 405. 16. 718 x 203 98. To square any number of two figures. 99. To square any number of two figures the method of cross multiplication may be used, or the work may be further contracted as shown in the following example. 100. Example. Square 72. _rt SOLUTION. 2x2 = 4. Write 4 as the first figure of the product. 14 (7 + 7 ) x 2 = 28. Write 8 as the second figure of the product, and *" carry 2. 7x7 + 2 = 51. Write 51 to complete the product ; or, 5184 2x2 = 4. Write 4 as the first figure of the product. 4 (2 + 2} x 7 = 28. Write 8 as the second figure of the product, and carry 2 7 x 7 + 2 = 61. Write 51 to complete the product. 101] MULTIPLICATION 45 101. Therefore the following rule : Multiply the units of the multiplicand by the units of the multiplier and write the result in the product t carrying as usual. Add the tens in the multiplier to the tens of the multi- plicand and multiply by the units of the multiplier; or, add the units of the multiplier to the units of the multiplicand and multiply the sum by the tens of the multiplier. Write the result in the product, carrying as usual. Multiply the tens of the multiplicand by the tens of the multiplier, and write the full result in the product. WRITTEN EXERCISE 1. Find the sum of the squares of the following numbers. 24, 36, 32, 34, 67, 84, 92, 76, 89, 47, 39, 38, 43, 56, 75, 88, 95, 83, 94, 71, 29, 44, 59, 65, 73. 2. Square the numbers written below, writing the results hori- zontally ; then find the sum of -the squares. 37, 48, 68, 62, 98, 26, 27. 8 Square the numbers written below, and from the sum of the first five squares subtract the sum of the second five. 63, 61, 49, 76, 81, 41, 33, 23, 35, 37. ORAL REVIEW By inspection, find the cost of each of the following items : 1. 215 pounds of butter at 22^; 102 pounds at 28 2. 104 bushels of garden corn at $ 1.75; 204 bushels at 52 8. 125 bushels of wheat at $ 1.02 ; 115 bushels at $1.05. 4. 102 barrels of apples at $1.48; 103 barrels at $1.67. 5. 33 bushels of pears at $1.39; 55 bushels at $1.15. 6. 65 bushels of potatoes at 65 ^ ; 58 bushels at 52 #. 7. 44 baskets of peaches at $1.23; 52 baskets at 87 8. 64 barrels of flour at $5.15; 33 barrels at $6.50. 9. 26 bags of bran at $1.03; 32 bags at $1.05. 46 SIMPLE NUMBERS [ 101 10. 125 pounds of coffee at 22^; 164 pounds at 33^. 11. 56 gallons of molasses at 36^; 84 gallons at 34^. 12. 43 pounds of chocolate at 43 ^ ; 52 pounds at 52 ^. 18. 23 sacks of pancake flour at 42^ ; 45 sacks at 45^. 14. 62 boxes of ice cream salt at 62^; 53 boxes at 51^. WRITTEN REVIEW 7. A manufacturer sold 171 corn shellers at $23 each. How much did he receive for them ? 2. There are 5280 feet in a mile. How many feet are there in 104 miles ? 8. What will 462 barrels of petroleum cost at $ 1.08 per barrel ? 4. In freighting, lime and flour are each estimated to weigh 200 pounds per barrel ; pork and beef, each 32 pounds ; apples and pota- toes, each 150 pounds ; cider, whisky, and vinegar, each 150 pounds. What will be the weight of the freight in a car containing 22 barrels of each of these products ? 5. If a bushel of barley weighs 48 pounds, of clover seed 60 pounds, of flax seed 55 pounds, of beans 60 pounds, of buckwheat 48 pounds, of rye 56 pounds, of corn 56 pounds, of oats 32 pounds, of potatoes 60 pounds, of timothy seed 45 pounds, of wheat 60 pounds, what will be the total weight of 77 bushels of each product ? 6. A man rented a farm of 132 acres of grain land, 76 acres of pasture land, and 45 acres of meadow land ; paying for the grain land $7 per acre, for the pasture land $4 per acre, and for the meadow land $ 11 per acre. He produced 61 bushels of oats per acre on 45 acres, 32 bushels of barley per acre on 30 acres, 75 bushels of corn per acre on 15 acres, 150 bushels of potatoes per acre on 9 acres, 28 bushels of buckwheat per acre on 20 acres, and 24 bushels of beans per acre on the remainder of the grain ground. He relet the pasture land for $ 200, and on the meadow cut 2 tons per acre of hay, worth $13 per ton. If he paid $695 for labor and $467 for other ex- penses, and sold the oats at 26 ^ per bushel, the barley at 65 ^, the corn at 40^, the potatoes at 33^, the buckwheat at 60^, and the beans at $ 2, did he gain or lose, and how much ? 102-111] DIVISION 47 DIVISION 102. Division is the process of finding how many times one number is contained in another. 103. The dividend is the number to be divided. 104. The divisor is the number by which the dividend is to be divided. 105. The quotient is the result obtained by division. 106. Division is exact when all the dividend is divided and the quotient is an integer. 107. The remainder is the part left undivided when the division is not exact. 108. The sign -*- signifies division and is read divided by. Thus 24 -7- 8 = 3 is read 24 divided by 8 equals 3. The dividend in division corresponds to the product in multiplication, and the divisor and quotient to the multiplicand and multiplier, respectively. 109. General Principles. 1. Multiplying the dividend or divid- ing the divisor multiplies the quotient. 2. Dividing the dividend or multiplying the divisor divides the quotient. 3. Multiplying or dividing both the dividend and divisor by the same number does not change the quotient. 4. When the divisor and dividend are like numbers, the quotient is an abstract number. 5. When the divisor is an abstract number, the dividend and quotient are like numbers. 110. When the divisor is so small that the division may be per- formed mentally, the process is called short division. 111. Example. Divide 3713 by 8. SOLUTION. Write the divisor at the left of the dividend with a curved line between them. 8)3713 8 is not contained in 3 thousands, therefore divide 37 hun- dreds by 8. 8 is contained in 37 hundreds 4 times with a remainder 5. Write 4 in the quotient in the place of hundreds. Reducing 5 hundreds to tens and adding the one ten of the dividend, the result is 51. 8 is contained 6 times in 51 with a remainder 8. Write 6 in the quotient in the 48 SIMPLE NUMBERS [ 111-113 place of tens. Reducing the 3 tens to units and adding the 3 units of the divi- dend, the result is 33 units. 8 is contained in 33 units 4 times with a remainder 1. Write 4 in the place of units and place the remainder over the divisor with a line between ; thus, $. The complete quotient is 464. 112. When the divisor is so large that each step in the division must be Written, the process is called long division. 113. Example. Divide 5207 by 98. SOLUTION. Since 98 is more than 52, it is necessary to take __ 520 for the first partial dividend. The nearest number of tens 98)5207 represented by the divisor is 10 ; take 10, therefore, for the 490 trial divisor. The number of tens in the partial dividend is 307 52. 10 is contained 5 times in 52. Write 5 in the quotient 294 over the right-hand figure of the partial dividend as shown in ~~13 the margin. Multiplying the exact divisor by 5 and subtract- ing the product from the partial dividend, the remainder is 30, to which annex the 7 units of the dividend, and the second partial dividend is 307. The nearest number of tens represented by the second partial dividend is 31, which will contain 10 3 times. Write 3 in the quotient. Multiplying the exact divisor by 3 and subtracting the product, the remainder is 13, to be written in the form of a fraction and annexed to the quotient. The complete quotient is 53H- ORAL EXERCISE 1. The quotient is 61. If the dividend and divisor were each multiplied by 4, what would the quotient be ? 2. The quotient is 53. If the dividend and divisor were each divided by 3, what would the quotient be? 3. If the divisor were 4 times what it is, the quotient would be 1606. What is the quotient ? 4. The quotient of one number divided by another is 12. What would the quotient be if the divisor were multiplied by 3 ? divided by 3? 5. How many 15's must we add together to get 4590 ? WRITTEN EXERCISE 1. $21,735 was received from the sale of a farm at $35 an acre. How many acres did the farm contain ? 2. What number must be added to 21,786 that it may be exactly divisible by 168 ? 113-115] DIVISION 49 8. The remainder is 14, the quotient 5041, and the divisor 15, What is the dividend ? 4. The remainder is 7, the quotient 19,023, and the dividend 247,306. What is the divisor ? 5. If 8 men can do a piece of work in 24 days, in how many days can 12 men do the same work ? 6. I sell my village home for $ 3250, my store for $5000, my stock of goods for $11,250, receiving in part payment $8775, and for the remainder, Iowa prairie land at $15 per acre. How many acres should I receive ? 7. If there are 128 cubic feet in 1 cord, how many cords in 141,492 cubic feet? 8. If 93 be added to a certain number, it will contain 648 twenty-five times. What is the number? 9. What number must be subtracted from 3476 that it may be exactly divisible by 155 ? 10. A man bought 490 acres of land at $ 40 an acre and after paying $2900 for improvements sold it for $25,000. Did he gain or lose, and how much ? CONTRACTIONS IN DIVISION 114. To divide any number by 10, 100, 1000, etc. 115. By the decimal system of notation numbers increase in value from right to left and decrease from left to right in a tenfold ratio ; hence to divide a number by 10, 100, 1000, etc. : From, the right in the dividend point off as many places as the divisor contains ciphers. The figures so cut off express the remainder, to be written in fractional form. ORAL EXERCISE By inspection, find the complete quotients of: L 759 -=-10. 4. 4997-- 10000. 7. 297249-*- 10. 2. 7527929-^1000. 5. 75627 -j- 10. 8. 759^-10000. 3. 29-5-100. 6. 8967^100. 9. 4627490 -=- 1000. 50 SIMPLE NUMBERS [116-122 116. To divide by any number of 10's, 100's, 1000's, etc. 117. Example. Find the quotient of 14,131 -s- 4000. SOLUTION. Mark off as many figures in the dividend as there are ciphers in the divisor, thus dividing by 1000. The **Tffinr first quotient is then 14 with a first remainder 131. Dividing 4|000)14|131 14 by 4 gives 3 as the final quotient with a second remainder 2. 12 Multiplying the second remainder by 1000 to obtain its true ~2131 value and adding the first remainder, the result is the true 4QQQ remainder, 2131, to be written in fractional form and placed by the side of the integral quotient. The completed quotient is then Sfflft. ORAL EXERCISE By inspection, find the complete quotients of: 1. 1627 -f- 400. 8. 762179 -=-190000. 15. 7849-5-260. 2. 571119 -j- 19000. 9. 51295 -j- 17000. 16. 8479-5-280 5. 48887 --40. 10. 6439-- 160. 17. 9579-- 190. 4. 6427 --80. 11. 19279 -- 160. 18. 125265 -- 250. 6. 9687-- 120. 12. 15597-5-5000. 19. 16219 -v- 4000. 6. 7879-5-390. IS. 21259 -v- 700. 20. 39379-5-13000. 7. 19249-5-4000. 14. 72899-^-24000. 21. 4179-5-1200. PROPERTIES OF NUMBERS 118. Properties of numbers are those qualities which belong to and are inseparable from them. 119. All integral numbers are: (1) odd or even; (2) prime or composite. 120.. An odd number is a number that cannot be exactly divided by 2 ; as, 5, 9, 23. 121. An even number is a number that can be exactly divided by 2 ; as, 6, 8, 44. 122. Factors of numbers are those numbers the continued product of which will produce the number. ^ 23-131] PROPERTIES OF NUMBERS 51 123. A prime number is a number that cannot be resolved into two or more factors ; or, it is a number that has no integral factors except unity and itself. Thus, 23, 59, 11, and 13 are prime numbers. 2 is the only even number that is prime. 124. A composite number is a number that can be resolved into two or more factors ; or, it is a number which is the product of tw r o or more integral factors. 125. A prime factor is a prime number used as a factor. 126. A composite factor is a composite number used as a factor. 127. An exact divisor of a number is any integral factor of that number. 128. A common divisor of two or more numbers is any exact divisor of those numbers. 129. The greatest common divisor of two or more numbers is the greatest exact divisor common to those numbers. 130. Numbers having no common divisor or factor are said to be relatively prime. 131. Tests of Divisibility. * In arithmetical computations it is fre- quently necessary to determine whether one number is divisible by another or not. In dividing numbers the following tests of divisi- bility will be found helpful. 1. When a number is even, it is divisible by 2. 2. When the sum of the digits of any number is divisible by 3 or 9, the whole number is divisible by 3 or 9. 3. When the right-hand figure of any number is 5 or 0, the whole number is divisible by 5. 4. When the right-hand figure is 0, the whole number is divis- ible by 10. 5. When the number expressed by the two right-hand figures of a number is divisible by 4, the whole number is divisible by 4. 6. When a number is even and divisible by 3, it is also divisible by 6. 7. When the number expressed by the three right-hand figures of a number is divisible by 8, the whole number is divisible by 8. 52 SIMPLE NUMBERS [ 132-ldG 132. A multiple of a number is one or more times the number; or, it is that product of which the given number is an exact divisor. 133. A common multiple of two or more numbers is that product of which the given numbers is an exact divisor. 134. The least common multiple of two or more numbers is the least product of which each of the given numbers is an exact divisor. FACTORING 135. Factoring is the process of separating or dissolving a com- posite number into factors. 136. General Principles. 1. Any composite number is divisible by each of its several factors successively. 2. A composite number is equal to the product of all its prime factors. 137. To find the prime factors of a composite number. 138. Example. Find the prime factors of 4290. 4290 SOLUTION. The given number ends with a 0, hence is exactly 2 3 11 gpjg divisible by 5. Dividing by 5, the quotient 858 is obtained. 858, being an even number, is exactly divisible by 2. Dividing by 2, the quotient 429 is obtained. The sum of the digits in 429 is divisible by 3 ; there- fore the whole number is divisible by 3. Dividing by 3, the quotient 143 143 is obtained. 143 is exactly divisible by 11. Dividing byll, the quotient 13 is obtained. The several divisors, 5, 2, 3, 11, and the last quotient, 13, are the prime factors required. 139. Therefore the following rule : Divide the given number by any prime factor. Divide the successive quotients in tlw same manner until a quotient that is prime is obtained. The several divisors and the last quotient are the prime factors required. WRITTEN EXERCISE Find the prime factors of : 1. 144. 8. 924. 6. 135. 7. 1575. 9. 951. 2. 124. 4. 289. 6. 25785. 8. 252. 10. 1527. 140-144] PROPERTIES OF NUMBERS 53 GREATEST COMMON DIVISOR 140. To find the greatest common divisor of two or more numbers. 141. Example. Find the greatest common divisor of 42, 66, and 84. SOLUTION. Arrange the numbers as shown in Jtlie margin. The first prime factor that will divide all the numbers is 2. Divide by 2, obtaining 21, 33, and 42 7 11 14 as the quotients. The only prime factor that is common to these numbers is 3. Divide by 3, obtaining as the quotient 7, 11, and 14. There is no factor common to all of these numbers. Since 2 will divide all the given numbers, and 3 will divide the resulting quotients, the product of 2 x 3, or 6, is the greatest common divisor of 42, 66, and 84. 142. Hence the following rule : Write the numbers in a horizontal line, separating them by dashes. Divide by any prime number that will divide all the given numbers without a remainder, writing the quotients in a line below. Continue the process until the quotients have no common factor. Multiply together the several divisors, and the result is the greatest common divisor. 143. Sometimes the series of numbers of which the greatest common divisor is to be found cannot be factored by inspection, and a method similar to the following is employed. 144. Example. Find the greatest common divisor of 697 and 779. 697)779(1 (597 SOLUTION. Divide the greater number by the less, ~82^697(8 *^ e Divisor bv tne remainder, and so continue until ' ^ there is no remainder. The last divisor is the greatest - common divisor. Therefore the greatest common divi- 41)82(2 sor of 697 and 779 is 41. 82 WRITTEN EXERCISE Find the greatest common divisor of: 1. 22, 55, and 99. 8. 679 and 1869. 2. 24, 36, 60, and 96. 4- 32, 48, 80, 112, and 144. 54 SIMPLE NUMBERS [ 144-147 5. A farmer has a triangular piece of land which he wishes to inclose with a board fence so that the boards may be of the greatest length possible and no fractional lengths used. If the sides of the tract of land are 84, 96, and 108 feet, respectively, what is the length of the longest board that can be used in making the inclosure ? 6. How many boards will inclose, without waste, a rectangular garden, 98 feet long by 70 feet wide, the fence being straight and 5 boards high, if the boards be of equal length and the longest possible ? LEAST COMMON MULTIPLE 145. General Principles. 1. The product of two or more numbers, or any number of times their product, is a common multiple of those numbers. 2. Two or more numbers may have any number of common multiples but only one least common multiple. 3. A multiple of a number contains all the prime factors of that number. 4. A common multiple of two or more numbers contains all the prime factors of each of the numbers. 5. The least common multiple of two or more numbers is the least number that will contain all the prime factors of the given numbers. 146. To find the least common multiple of two or more numbers. 147. Example. Find the least common multiple of 12, 16, 63, and 90. / a \ SOLUTIONS, (a) Since no number less 12 2 2 3 than ^ can be divic * ed by 90, it is evident that the least common multiple cannot be less than that number; hence, it must 63 = 3 X 3 X 7 contain the prime factors 3, 3, 2, and 5 in 90 = 3x3x2x5 order to be divisible by 90 ; it must con- 90x7x2x2x2 = 5040 te * n 7 as a factor in order to be divisible by 63 ; and it must contain 2 as a factor three more times in order to be divisible by 16 ; hence, the product of the factors 3, 3, 2, 5, 7, 2, 2, and 2, or 6040, must be the least common multiple of the num- bers 12, 16, 63, and 90. Or, 147-148] PROPERTIES OF NUMBERS 55 (fo (6) First divide by 2 ; 63 not being divisible by .jo _ IQ _ gQ _ on 86345 2, bring it to the lower line and divide again by 2 ; neither 63 nor 45 being divisible by 2, bring both to the quotient line. Next divide by 3 ; 4 not being divisible by 3, bring it to the quotient line and 1 4 21 15 divide again by 3. The remaining numbers, 4, 7, 3 46345 1 4 J 5 and 5, being relatively prime, should be taken together with the prime divisors 2, 2, 3, and 3 as factors of the least common multiple. Their product is 6040, the same as found in 147. NOTE. The latter method of determining the least common multiple will, in the majority of cases, be found the most convenient for practical purposes. When one of the given numbers is a factor of another, reject the smaller one. 148. Therefore the following rule : Write the numbers in a horizontal line, separating them by dashes. Divide by any factor common to all the numbers, or by any prime factor common to two or more of them/. In the same manner divide the quotients obtained, and continue the process until the quotients are relatively prime. The product of the several divisors and the undivided numbers is the least common multiple. WRITTEN EXERCISE Find the least common multiple of : 1. 12, 20, and 32. 2. 25, 90, and 225. 3. 6, 16, and 26. 4. A, B, and C are traveling men. A makes a visit to Boston every four months, B every three months, and C every two months, tf they are all in Boston on January 1, 1903, when will they all be in that city together again for the first time ? 5. What is the least number of acres that a piece of land can contain to be exactly divided into lots of 12, 14, and 18 acres respec- tively ? 56 SIMPLE NUMBERS [ 149-152 CANCELLATION 149. Cancellation is the process of shortening the operation of division, or the combined operations of multiplication and division, by omitting or striking out equal factors from the dividend and divisor. 150. General Principles. 1. Canceling a factor from a number has the effect of dividing the number by that factor. 2. Canceling a factor from both dividend and divisor does not affect the value of the quotient. 151. Example. Divide 18 x 36 by 3 x 32. (6) SOLUTION. Indicate the pro- 9 cess by either of the methods jo . shown in the margin. 3 and 36 contain the common factor 3, which cancel and write 12 in the dividend. 12 and 32 contain the common factor 4, 27 = 6j which cancel from both and write the factors 3 and 8 in the divi- dend and divisor, respectively. 18 and 8 contain the common factor 2, which cancel from both and write the factors 9 and 4 in the dividend and divisor, respectively. The product of the remaining factors of the dividend, divided by the remaining factors of the divisor, is the required quotient, or 6f. 152. Hence the following rule : Indicate the operation in convenient form. Cancel from the dividend and divisor all factors common to both. Divide the product of the remaining factors of the divi- dend by the product of the remaining factors of the divisor. The result obtained is the required quotient. WRITTEN EXERCISE 1. Divide the product of 9, 8, 12, and 24 by the product of 2, 14, and 8. 2. What is the quotient of 35 X 75 -5- 7 X 5 x 3 ? 152] PROPERTIED I'F NUMBERS 57 S. Multiply together 18, 42, and 64, and divide the product by the product of 6, 16, and 32. 12x3x35x24 x 2 = ? 6x9x5x6 5. How many bushels of potatoes at 60^ per bushel will pay for 450 pounds of sugar at 6^ per pound ? 6. A farmer traded 4 hogs weighing 325 pounds at 6^ per pound for sugar at 5 ^ per pound. How many entire barrels of 312 pounds each should the farmer have received ? 7. How many yards of cloth at 15^ per yard should be given for 9 barrels of pork, each barrel containing 200 pounds, at 6^ per pound? 8. How many pieces of cloth containing 45 yards each should be received for 5 baskets of eggs, each basket containing 21 dozen at 18^ per dozen, if the cloth be valued at 8^ per yard ? 9. If 320 acres of land produce 25,600 bushels of wheat, how many bushels of wheat will 110 acres produce at the same rate ? 10. If a horse trots 5 miles in 30 minutes, how far can he trot in 21 minutes at the same rate ? 11. How many sections of Texas prairie land, each containing 640 acres, at $ 5 per acre, should be given for an Ohio farm of 400 acres, at $ 40 per acre ? 12. A farmer exchanged 196 loads of oats, each load containing 30 sacks of 2 bushels each, worth 30^ per bushel, for flour at 5^ per pound. At 196 pounds per barrel, how many barrels should he have received ? 18. A farmer exchanged 250 bushels of wheat at 80 ^ per bushel for cloth at 40^ per yard. How many yards of cloth should he have received? 14. If 9 men earn $ 108 in 6 days, how much will 15 men earn in 4 days at the same rate ? 15. 30 half chests of Japan tea containing 75 pounds each at 30^ per pound were exchanged for brown sugar at 2\$ per pound. If a barrel of brown sugar weighs 300 pounds, how many barrels should have been received ? UNITED STATES MOINEY 153. Money is a standard measure of value used as a medium of exchange. 154. Currency is the term applied to money or its equivalent. 155. A decimal currency is a currency whose denominations increase and decrease on a scale of ten. 156. United States money, commonly called Federal money, is the legal currency of the United States. It is a decimal currency and consists of coin and paper money. 157. The denominations and scale of United States money are shown in the following TABLE 10 mills (m.) =1 cent (j> or ct.) 10 cents = 1 dime (d.) 10 dimes or 100 cents = 1 dollar ($) 10 dollars = 1 eagle (E.) The dollar sign, $, is always written before the number. The mill is not a coin. It is used only as a decimal of a cent, which is the smallest money of the mint and the smallest recognized in business. The eagle and the dime are used only as names of coins and never in reading United States money. 158. Bullion is pure gold or silver in bars or ingots. 159. Coins are pieces of metal converted into money by being stamped by the authority of the government in such a way as to indicate the rate at which they shall pass in trade. 160. The coins of the United States are of two kinds, namely : 1. Those made by the authority of the government, in unlimited quantities, for private persons from metal deposited by them. The government provides that private persons may deposit metal with the United States mints or assay offices in unlimited quantities for the purpose of having it weighed, refined, assayed, and returned to them in the form of stand- ard coins or in ingots of standard fineness. 2. Those subsidiary coins made from silver, nickel, and copper by the authority of the government. 58 161-167] UNITED STATES MONEY 59 The government has the power to buy metal for the purpose of making it into subsidiary coins for itself. Since subsidiary coins may be sold to private individuals at more than cost, their quantity is restricted. 161. The coins of the United States are gold, silver, bronze, and nickel. COINS OF THE UNITED STATES COINS COMPOSITION WEIGHT VALUE Gold Quarter-eagle & pure gold and ^ alloy 64.5 Troy grains $2.50 Half-eagle ^ pure gold and ^ alloy 129 Troy grains 6.00 Eagle T % pure gold and ^ alloy 258 Troy grains 10.00 Double-eagle ^j pure gold and ^ alloy 516 Troy grains 20.00 Silver Dime r ? 5 pure silver and ^ alloy 38.58 Troy grains $0.10 Quarter-dollar T % pure silver and ^ alloy 96.45 Troy grains .25 Half-dollar T 9 ^ pure silver and ^ alloy 192.9 Troy grains .50 Dollar ^ pure silver and -J^ alloy 412.5 Troy grains 1.00 Bronze and Nickel 1-cont piece ^ copper and -fa tin and zinc 48 Troy grains $0.01 5-cent piece | copper and \ nickel 77.16 Troy grains .05 162. The standard unit of value in the United States is the gold dollar, which contains 23.22 Troy grains of pure gold and weighs 25.8 Troy grains. The gold dollar being so inconveniently small is not coined now. 163. The paper money of the United States at present consists of gold certificates, silver certificates. United States notes, treasury notes, and national bank bills. 164. Gold certificates are issued for gold deposited with the Treasurer of the United States. They represent values of $20 and up ward to $20,000. 165. Silver certificates are issued for silver deposited with the Treasurer of the United States in amounts not less than $10. They represent values of $ 1 and upward to $ 100. 166. United States notes (greenbacks) represent values of $10 and upward to $1000. 167. National bank bills are notes issued by national banks under the supervision of the national government. They are now issued in amounts of $5 and upward to $100. 60 UNITED STATES MONEY [ 1C8-172 168. Treasury notes of 1890 are now in the course of retirement. They were formerly issued in amounts of $ 1 and upward to $20. They cannot be reissued. 169. Legal tender is the term applied to such money as may be legally offered in payment of debts. 170. The following are legal tender in the United States as noted : 1. Gold coins, except when below the standard weight because of abrasion. 2. Silver dollars and treasury notes of 1890 in all cases where the contract does not expressly stipulate otherwise. 3. United States notes (greenbacks), except for interest on the public debts and for duties on imports. 4. National bank notes for any debt to a national bank, and for taxes and other dues to the United States, except duties on imports. 5. Silver coins less than one dollar in all cases where the amount does not exceed ten dollars in one payment. 6. Nickel and copper coins in all cases where the amount does not exceed twenty-five cents in one payment. NOTATION OP UNITED STATES MONEY 171. The dollars form the integral part of the number, and are written to the left of the dot, called the decimal point. The cents and mills form the fractional part of the number and are written to the right of the decimal point. 172. Since ten dimes make one dollar, the figures written in the first place to the right of the decimal point are tenths of a dollar, or dimes. Since one hundred cents make one dollar, the figures written in the second place to the right of the decimal point are hundredths of a dollar, or cents. Since one thomand mills make a dollar, the figures written in the third place to the right of the decimal point are thou- sandths of a dollar, or mills. Thus: Six dollars, forty-eight cents, two mills, is written $6.482. In writing cents less than ten, a cipher should occupy the first place to the right of the decimal point. In the final results all mills less than five are rejected, and all five or more are counted as a whole cent. 173-178] UNITED STATES MONEY 61 173. Whenever it is desirable to express United States money in written words, the cents should be written as hundredths of a dollar, and in fractional form. Thus : Twenty-live and j$$ dollars. REDUCTION OF UNITED STATES MONEY 174. Reduction is the process of changing the unit without changing the value of a number. 175. To reduce dollars to cents. 176. Since there are one hundred cents in a dollar, to reduce dollars to cents, multiply by 100 by annexing two ciphers to the numbers expressing a whole number of dollars, or by moving the decimal point two places to the right in numbers expressing cents, or dollars and cents. 177. To reduce cents to dollars, Divide by 100 by removing the decimal point two places to the left. ORAL EXERCISE By inspection, reduce : 1. $7.25 to cents. 3. 241 to dollars. 5. $ 157.32 to cents. %. $119 to cents. 4. 92,798^ to dollars. 6. 72,572^ to dollars. 7. Which is the heavier, a gold eagle or a silver dollar ? A gold eagle or a silver half-dollar ? 8. Which is the heavier, a five-dollar gold piece or a five-cent nickel piece ? a gold dollar or a bronze cent ? 9. Why are the gold and silver coins of the United States never made more than T % pure ? 10. Are national bank notes legal tender ? silver dollars ? pennies ? gold certificates ? silver certificates ? 11. Under what circumstances would gold coin not have full legal tender value? ADDITION AND SUBTRACTION OF UNITED STATES MONEY 178. To add or subtract United States money, Write dollars under dollars and cents under cents. Add or subtract as in simple numbers, placing the decimal point in the result directly under the points in the numbers added or subtracted. 62 UNITED STATES MONEY [178 *, S. 4- $47;198.76 $919,010.01 $876,311.40 2.91 1,889.76 40.32 1,487.59 33.44 21.56 101.11 221.34 2,197.77 321,876.34 2,345.66 140.40 8,198.99 1.06 999.88 2,345.98 66.87 278,811.33 1.11 221,198.32 9,435.23 231.59 44,859.83 1.06 22.81 321.54 6.05 45,728.67 2,378.95 200,400.58 3,221.55 34.40 3,211.59 19.88 70.87 20.87 987,111.23 1,100.58 12.549.15 WRITTEN EXERCISE Copy or write from dictation and find the sum of each of the following problems : * 1. $157,926.04 52.19 9,261,549.62 1,694,247.57 5,216.90 425.86 52.95 1,076.87 27,214.95 276,421.87 932.17 4.26 259,426.74 9,275.18 Copy or write from dictation the following numbers and find the sums by horizontal addition : f 5. 932 10 , 22,11s 03 , 81 21 , 967 s2 , 221 6. 346 50 , 291 75 , 100 31 , 269 11 , 80 93 . 7. 2165 84 , 72 43 , 90 20 , 117 65 , 600" 1127". 8. 15 16 , 2 s7 , II 46 , 107 90 , 9*, 81", 123" 2 , 6 01 , 15 80 , ll n . 9. A merchant bought cottons, for 3467 s25 ; linens, for 1326 15 ; woolens, for 4215 75 ; delaines, for 1025 48 ; brocades, for 1127 50 . If all were sold for 13256 28 , how much was gained ? * Addition is so interwoven with all arithmetical processes that proficiency in the subject should be insisted upon. If the principles of grouping have not been mastered, simple addition should be reviewed before any advance steps are taken. Accurate answers for problems similar to 1, 2, 3, and 4 should be obtained in from twenty to twenty-five seconds. t Under some circumstances it is desirable to write United States money expressed Jn dollars and cents without the dollar sign and the decimal point, with the decimal part placed slightly above that expressing the integers or dollars. Thus, $5.25 may be written S 26 . $13.08 may be written 138. This is advisable only where the sum of several items is to be found by horizontal addition. 179] UNITED STATES MONEY t>3 MULTIPLICATION OF UNITED STATES MONEY 179. To multiply United States money, Multiply as in simple numbers, and from the right in the product point off as many places as there are places to the right of the decimal point in the multiplicand. Money is a concrete expression. In a critical analysis of its multipli- cation, therefore, the money cost or price of the article is a concrete multiplicand. The number of things bought or sold is an abstract multiplier and their product is concrete and of the same name or denomination as the multiplicand. However, since the United States money scale is decimal, these terms may be interchanged for convenience. WRITTEN EXERCISE 1. Find the amount of the following bill. Make all extensions mentally as explained in 85-88. 28 Ib. lard at 11 112 Ib. butter at 22 46 bu. salt at 22^. 132 bu. onions at 44^. 117 bu. apples at 33^. 113 bu. potatoes at 66^. 2. Find the total cost of the following farm produce. Make all extensions mentally as explained in 95-97. 24 bu. wheat at $ 1.02. 103 bu. rye at 83 215 bu. barley at $1.04. 105 bu. peas at 72^. 108 bu. oats at 42^. 204 Ib. butter at 34 3. Find the amount of the following bill. Make all extensions mentally as explained in 91-93. 54 yd. jeans at 21 48 yd. print at 24^. 27 yd. delaine at 32 121 yd. ticking at 15 tf. 64 yd. gingham at 23 61 yd. sheeting at 13 42 yd. drilling at 21 f. 217 yd. cashmere at $1.13. 4- Find the total cost of the following items. Make the exten- sions mentally as explained in 95-97. 116 yd. moquette carpet at $ 3.02. 115 yd. border No. 1 at $ 3.06. 131 yd. Brussels carpet at $ 2.05. 64 yd. border No. 2 at $2.07. 103 yd. ingrain carpet at $1.04. 43 yd. border No. 3 at $1.08. 5. Find the amount of the following bill. Make all extensions X as explained in 90. 325 yd. tapestry Brussels at $ 2.17. 648 yd. Axminster at $ 2.55. 547 yd. 3-ply ingrain at $ 1.26. 328 yd. body Brussels at $ 2.48. 427 yd. moquette at $ 2.79. 255 yd. velvet at $ 2.79. 64 UNITED STATES MONEY [ 180-182 DIVISION OF UNITED STATES MONEY 180. To divide United States money. 181. Examples. 1. If 5 hats are worth $25.65, what is 1 hat worth ? g 13 SOLUTION. Since 5 hats are worth 2565 cents ($25.65), 1 -^ - hat is worth of 2565 cents, or 613 cents. 613 cents equals 2. If 4 boxes of oranges are worth $17, what is 1 box worth? 4 OK SOLUTION. Since 4 boxes of oranges cost 1700 cents ($17.00), one box will cost | of 1700 cents, or 425 cents. A\1 7 nrt ., 425 cents equals $4.25. 8. How many boxes of oranges at $ 4.25 per box can be bought for $17? . SOLUTION. Since 1 box of oranges cost 425 cents ($4.25), - as many times 1 box can be bought for 1700 cents as 425 is con- tained times in 1700, or 4 times. 182. Therefore the following rule : Divide as in simple numbers. From the right of the quotient point off as many decimal places as the number of places in the dividend exceed those in the divisor. When the dividend and divisor each contain the same number of decimal places, the numbers may be regarded as integers in performing the operation of division. When the divisor alone contains cents, both dividend and divisor may be reduced to cents and the division performed exactly as in whole numbers. WRITTEN EXERCISE 1. A dealer bought wheat at 95^, oats at 45^, and corn at 65^ per bushel. He paid $332.50 for the wheat, $191.25 for the oats, and $ 113.75 for the corn. How many bushels of each did he buy in all? 2. Having sold my mill for $17,250, and 316 barrels of flour in stock at $5.15 per barrel, I invested out of the proceeds $ 1185.85 in furnishing a house, $ 1260 in farming utensils, $ 1582.25 in live stock, and with the remainder paid in full for a farm of 163 acres. What was the cost of the farm per acre ? METHODS FOR PROVING WORK 183. Addition is generally verified as explained in 44. 184. A good way to test the accuracy of subtraction is to add the remainder and the subtrahend. If the work is correct, the result obtained should equal the minuend. 185. The work of multiplication may be verified by interchang- ing the multiplier and the multiplicand, and remultiplying. If the results obtained by both operations are the same, the work is assumed to be correct. 186. The operation of division may be verified by multiplying together the quotient and divisor and adding to the product the remainder, if any. CASTING OUT NINES AND ELEVENS 187. The basis of our numerical system being 10, every power * of 10 is 1 more than some multiple of 9 ; and 10 or any power of 10 multiplied by a single digit is some multiple of 9, plus that digit. Thus, 10 = 9 + 1 ; 100 = 11 x 9 + 1 ; and 60 = 6 x 9 + 6 ; 500 = 55 x 9 -f 5. 188. 4582 = 4000 + 500 + 80 + 2. Since the excess of nines (the remainder after the nines are cast out) in 4000 is 4, in 500, 5, in 80, 8, in 2, 2, it follows that the excess of nines in any num- ber is the same as the excess of nines in the sum of the digits of that number. Thus, the excess of nines in 4682 = 4 + 5 + 8 -f- 2, or 19. 19 = 10. 10=1. 189. In finding the excess of nines, it is always best to omit all nines and also to reject them as soon as they occur in the addition. Thus, in casting out the nines in 954,727, begin at the left and drop the nines as soon as they occur. The excess of nines in 954,727 is then found to be 7. * A power of a number is the product arising from multiplying a number by itself one or more times. 66 METHODS FOR PROVING WORK [ 190-197 190. Since 11 is just 1 more than the numerical basis 10, even powers of 10 are multiples of 11, plus 1, and odd powers of 10 are multiples of 11, minus 1. Thus, 10 x 10 or 100, an even power of 10, equals 11x9+1. 10 x 10 x 10 X 10 x 10, an odd power of 10, equals 9091 x 11 - 1. 191. Any even power of 10 multiplied by a single digit is some multiple of 11 plus that digit. Any odd power of 10 multiplied by a single digit is some multiple of 11 minus that digit. Thus, 600 = 54 x 11 + 6 ; and 6000 = 546 x 11 - 6. 192. 10 multiplied by any single digit is some multiple of 11 minus the difference between 11 and that digit. Thus, 30 = 2x114-8; 50 = 4x11+6; 80 = 7x11+3; 90 = 8x11+2; etc. 193. It therefore follows that : The digit in the odd place of any number of two figures, with eleven added whenever necessary, minus the digit in the even place, equals the excess of elevens in that number. Applying this principle to all numbers, the sum of the digits in the odd places, increased by eleven or a multiple of eleven whenever neces- sary, minus the sum of the digits in the even places, is equal to the excess of elevens in the entire number. 194. The elevens may be dropped from the partial additions in the same manner as explained for nines. 195. The properties of nine and eleven, as explained above, may be used in proving addition, subtraction, multiplication, and division. 196. To prove addition by casting out the nines and elevens. 197. Example. Add 375, 425, 623, and 412. Prove the work by casting out (a) the nines and (b) the elevens. (a) Excess of nines. 375= 6 425= 2 412= 7 SOLUTION. The excess of nines in 375 Is 6 ; In 425 is 2 ; in 623 is 2 ; in 412 is 7. The excess of nines in p v the sum of 6, 2, 2, and 7 is o. I he excess ( 1835 is also 8. Since the excess of nines in all the num- bers is equal to the excess of nines in the sum of the 1835 = 17 = 8 numbers, the work is assumed to be correct. 197-201] METHODS FOR PROVING WORK 67 (b) Excess of elevens. 375 s= 1 425 = 7 62S - 7 SOLUTION. 16 (11 + 5) - 7 + 3 = 12. 12 = 1, or tne excess of elevens in 376. 5 2 + 4 = 7, or the excess of elevens in 425. 3 2 + 6 = 7, or the excess of elevens in 623. 2 1 -f- 4 = 5, or the excess of _ elevens in 412. The sum of 1, 7, 7, and 5 equals 20, 1835 = 20 = 9 which, divided by 11, gives a remainder 9. (5-3) f (8 1) = 9, the excess of elevens in the sum. Since the excess of elevens in all the numbers is equal to the excess of elevens in the sum, the work is assumed to be correct. 198. To prove subtraction by casting out the nines and elevens. 199. The excess of nines or elevens in the minuend minus the excess of nines or elevens in the subtrahend should equal the excess of nines or elevens in the remainder; or, the excess of nines or elevens in the subtrahend plus the excess of nines or elevens in the remainder should equal the excess of nines or elevens in the minuend. 200. To prove multiplication by casting out the nines and elevens. 201. Example. Find the product of 512 x 324 and verify the result by casting out (a) the nines and (b) the elevens. (a) Excess of nines. 512 = 81 SOLUTION. Use the contracted method of multi- 324 I Paying explained in 90. The excess of nines in 512 _ = is 8 ; in 324, 0. 8x0 = 0. The excess of nines in 2048 the completed product is 0. Since the excess of nines 16384 to *he multiplicand multiplied by the excess of nines in the multiplier is equal to the excess of nines in the product, the work is assumed to be correct. Excess of elevens. _ g i SOLUTION. 2 1 -f 5 = 6, or the excess of elevens _ f 30 = 8 in 512. 4-2 + 3 = 5, or the excess of elevens in 324. (&) Excess of elevens. 512 = 6 1 SOLUTION. 2 1 -f 5 = 6, or the excess of elevens 324 - - 6 x 6 = 30, which, divided by 11, gives a remainder 8. 8-8 + 8 5-1-6 1 = 8, or the excess of elevens in the completed product. Since the excess of elevens "165888 = 8 in the multiplicand multiplied by the excess of elevens in the multiplier is equal to the excess of elevens in the completed product, the work is assumed to be correct. 68 METHODS FOR PROVING WORK [202-203 202. To prove division by casting out the nines and elevens. 203. Examples ID division may be proved by multiplying the excess of nines or elevens in the divisor by the excess of nines or elevens in the quotient. If the work is correct, the result should equal the excess of nines or elevens in the dividend, or the dividend minus the remainder when there is a remainder. On the whole, the proof of casting out the elevens is more reliable than tbe proof of casting out tbe nines, but neither proof is practiced very generally by accountants except in proving long multiplications and divisions. WRITTEN EXERCISE 1. Multiply 125,426 by 567 in two lines of partial products. Verify the work by casting out the elevens. 2. Multiply 112,121 by 12,816 in two lines of partial products. Verify the work by casting out the nines. 8. Multiply 121,214 by 112,568 in three lines of partial products. Verify the work by casting out the elevens. 4. Find the amount of the following bill, making the extensions mentally as explained in 90. Verify the work by casting out the elevens. 248 yd. black dress silk, $1.24. 248 yd. black wool crepon, $2.79. 576 yd. Amazon cloth, $ 2.17. 568 yd. English camel's hair, $ 2.17. 357 yd. cashmere, $1.55. 124 doz. cotton hose, $1.86. 5. Find the amount of the following bill, making the extensions mentally as explained in 85-88. Verify the work by casting out the nines. 116 gr. bone buttons, 22^. 141 yd. feather ticking, 11 112 pc. black Chantilly lace, 88 118 yd. fancy gingham, 11 53 yd. cotton surah lining, 66 . 54 yd. gunner's duck, 22^. 124 yd. wash silk, 44^. 83 yd. Scotch cheviot, 55^. WRITTEN REVIEW 1. Find the total of the products called for in the oral exercise, page 40. . 2. Find the total cost of the items in the oral review, pages 45 and 46. 203] METHODS FOR PROVING WORK 69 3. A man earned $312.50 during February. During March he earned $49.50 more than in February. In April he earned as much as he did during February and May. If he earned $200 in May, how much did he earn in the four months ? 4. A finds that in five months he spends as much as he earns in four. At that rate, how long will it take him to save $ 2400 if he earns $1200 a year? 5. Multiply 12,501 by 486, making only two lines of partial products. Verify the work by casting out the nines. 6. A man bought an equal number of barrels of flour and oat- meal for $260. If the flour cost $6.20 and the oatmeal $6.80 per barrel, how many barrels of each did he buy ? 7. A merchant failed in business, and the excess of his liabilities over resources was found to be $3000. If he could pay his creditors but 6C $ on a dollar, what were his total liabilities, and how much did A, whom he owed $ 1500, receive ? 8. A and B had $9245 divided between them. The difference between their shares was $ 245. What had each ? 9. A merchant's cash receipts for a week were as follows : Mon- day, $921.40; Tuesday, $525.44; Wednesday, $321.50; Thursday, $425.60; Friday, $926.80; Saturday, $120.40. What were his average daily sales ? 10. How many barrels of apples at $ 2.50 must be given for 1200 bushels of potatoes at 50 ^ ? 11. A merchant's gains for February amounted to $1600, or $1100 less than his gains for January. If his gains for January were $60 more than four times his gain for March, what were his total gains for the three months ? 12. Multiply 113,214 by 12,816 in two lines of partial products. 18. Multiply 21,213 by 96,486 in three lines of partial products. Verify the work by casting out the elevens. FRACTIONS COMMON FRACTIONS 204. Quantity is anything which may be measured or which may oe regarded as being made up of parts like the whole. Numbers are the expressions by which we measure quantities. The basis of all numbers is the unit. There are integral units (3) or whole things, and fractional units (5) or parts of things obtained by dividing integral units into any number of equal parts. 205. A fraction is one or more fractional units. 206. A common fraction is a fraction expressed by two numbers, one written above and the other below a horizontal line. 207. The terms of the fraction are the two integers, called numer- ator and denominator, used to express one or more fractional units. 208. The denominator is the term which indicates the number of parts into which a unit has been divided ; it is written below the line, and denominates or names the size or value of each part of the fraction. 209. The numerator is the term which indicates the number of equal parts taken to form the fraction; it is written above the line. 210. To read fractions, Pronounce the numerator first and then the denominator. Thus, } , f , and |, are read, one seventh, two thirds, and seven eighths. 211. All fractions express unperformed division. The denomi- nator is the divisor, the numerator the dividend, and the value of the number expressed by the fraction the quotient. Hence, When the numerator and denominator are equal,, the value expressed by tlie fraction is 1 ; when the numerator is less than the denominator, the value expressed by the fraction is 70 211-214] COMMON FRACTIONS 71 less than 1; when the numerator is more than the denomi- nator, tlie value expressed by the fraction is greater than 1 ; and Of two fractions Jiaving the same denominator, the one having the larger numerator expresses the greater value. Of two fractions having the same numerator, the one having the smaller denominator expresses the greater value. DRILL EXERCISE 1. What is the unit of 5 tons of coal ? of 5 thousand feet of lumber ? of 7 dozen of eggs ? of -| of a week ? of 2^- acres of land ? 2. What is the fractional unit of a number divided into 3 parts ? into 31 parts ? into 13 parts ? 8. What is the fractional unit of f ? of -& ? of ^ ? of f ? 4. A unit contains how many sevenths ? fifteenths ? elevenths ? twenty-fifths? eighths? sixths? 5. Which is the greater, 1 of a number or 1 of a number ? Why ? 6. How many times -fa of a number is 1 ? Why ? 7. A receives -J- of the profits of a business, and B i. If B's profits in one month are $400, what are A's profits for the same time? SOLUTION. When a number is divided into 3 equal parts, each part is twice as large as the pa"rts of the same unit divided into 6 equal parts. Hence, | of a number is twice of the same number ; and if of a number is $400, ^ of the same number is twice $400, or $800. Therefore, A's profits are $800. 8. A invested -fa in the capital stock of a certain business, and B f If A's investment was $ 9000, what was B's ? Why ? CLASSIFICATION OF FRACTIONS 212. For convenience fractions may be classified as simple, com- plex, and compound. 213. A simple fraction is a fraction which has one numerator and one denominator, each of which is an integer. Simple fractions are either proper or improper. 214. A proper fraction is a fraction whose numerator is less than its denominator, and whose value is less than 1 ; as, f, ^, f . 72 FRACTIONS [ 215-225 215. An improper fraction is a fraction whose numerator is equal to, or greater than, its denominator, and whose value is 1, or more than !;,> 216. A mixed number is a whole number and a fraction united ; as, 21 74, 2501 217. A complex fraction is a fraction having one or both of its terms fractional j as, I, -, 5, _X 9 I i A 218. A compound fraction is a fractional part of a whole number or mixed number, or another fraction ; as, | x ^, x 2 J, f of T 9 7 . 219. General Principles. 1. Multiplying the numerator or divid- ing the denominator multiplies the fraction. 2. Dividing the numerator or multiplying the denominator di- vides the fraction. 3. Multiplying or dividing both the numerator and denominator by the same number does not alter the value of the fraction. REDUCTION OF FRACTIONS 220. The process of changing the form without changing the value of fractions is called reduction of fractions. 221. To reduce a whole number to a fraction. 222. Example. Reduce 7 to a fraction whose denominator is 7. SOLUTION. Since in 1 unit there are 7 sevenths, in 7 units there must be 7 times 7 sevenths, or 49 sevenths. Hence, 7 equals -*-. 223. The value of a mixed number may be represented by an improper fraction. 224. To reduce a mixed number to an improper fraction. 225. Example. Keduce 17| to halves. 17^- SOLUTION. Since in 1 there are 2 halves, in 17 there must be 35 17 times 2 halves, or 34 halves. Thirty-four halves plus 1 half ~ equals 35 halves. Hence, 17 equals ^. 225-228] COMMON FRACTIONS 73 ORAL EXERCISE 1. How many fourths in 6 ? in 25 ? in 16 ? in 71 ? in 12J ? 2. How many thirds in 15 ? fourths ? fifths ? sixths ? ninths ? S. How many thirds in 12J? in 8? in!7? in42|? in27|? ^. How many sixths in 2J- ? in 1 ? in 4J ? in 51 ? in 12| ? . Nine equals how many times % ? how many times ? & How many more pieces in 5 melons cut into fourths than in 3 melons cut into fifths ? 7. A has 2J tons of coal which he proposes to distribute among some poor families. How many families will he aid if he gives each one of a ton ? WRITTEN EXERCISE Reduce the following mixed numbers to equivalent improper fractions : 1. 13}. S. 1040^. 6. 17|. 7. 14$. 9. 121f 11. 43f 2. 27J. 4. 186^. & 78 f- 8 - 26 A- 10 - 17 A- 12 - 425 A- 226. The value of an improper fraction represents a quantity which may be expressed by a whole or a mixed number. 227. To reduce an improper fraction to a whole or a mixed number. 228. Example. Eeduce if^ to a mixed number. o4 SOLUTION. Since 5 fifths equals 1, 134 fifths must equal as K ? many times 1 as there are times 6 in 134, which are 26| times. Hence, if* equals 26|. ORAL EXERCISE Convert into whole or mixed numbers : 1. - 2 . 3. *. 5. . 7. - **' - *& WRITTEN EXERCISE Convert into whole or mixed numbers : J. ^ft. 7. 6. j 8. ii. 10. 74 FRACTIONS [ 229-234 229. To reduce a fraction to its lowest terms. 230. Examples. 1. Reduce to its lowest terms. J as J. SOLUTION. Observing that the terms of the fraction T$& are divisible by 3, first divide by 3 and obtain as a result f$. Observing that the terms of the fraction f are divisible by 7, divide by 7 and obtain as a result . Since 3 and 5, the terms of the last frac- tion, are relatively prime, the reduction cannot be carried further. Hence, ffo reduced to its lowest terms equals f . Since both numerator and denominator have been divided by the same numbers, the value of the fraction remains unchanged (219). 8. Reduce fJ-J to its lowest terms. SOLUTION. Being unable to determine a common Or. 0. .LI. = 41. divisor of 697 and 779 by inspection, find their greatest 697 -f- 41 = 17 common divisor (144) and obtain 41. Dividing the terms 779 H- 41 = 19 of the fraction by 41 the result is }$. 17 and 19 being relatively prime, the reduction cannot be carried further, WRITTEN EXERCISE Reduce to their lowest terms : 1. . 3. ft. 6. ^ 7. {if* 9. s. ft 4- A* A HW- A Hf 10. 231. In obtaining final results, aU proper fractions should be reduced to their lowest terras and att improper fractions to whole or mixed numbers. 232. To reduce a fraction to higher terms. 233. Example. Reduce f to a fraction whose denominator is 63. SOLUTION. Since 9 is contained 7 times in 63, the given 63 -4- 9 = 7. fraction may be reduced to a fraction whose denominator is _ 05 63, by multiplying both of its terms by 7. Multiplying both terms of | by 7, the result is ||. || has the same value as | (219). In practice think only of results. Thus, for the above work: 03 -t- 9 = 7. 6x7 = 35. Jf. 234. Hence the following rule: Divide the required denominator by the denominator of the given fraction. 234-239] COMMON FRACTIONS 75 Multiply the numerator of the given fraction by the quo- tient thus obtained and write the product over the required denominator. The result is tJie fraction in higher terms. ORAL EXERCISE 1. How many ninths in f ? fifteenths ? thirtieths ? sixtieths ? 2. Change -| to an equivalent fraction having 45 for its denomi- nator. 8. Eeduce J to a fraction whose denominator is 32 ; 64 ; 128. 4> Eeduce -J to sixty-thirds; to one-hundred-eighths; to ninety- ninths. 5. If ^ of a number is 9, what is | of the same number ? 6. If ^ of a number is 3, what is ^ of the same number ? 7. How many thirty-seconds in 1 ? in \ ? in 1 ? in 2 ? in 4J ? 235. A common denominator of two or more fractions is any num- ber which will contain each of the given denominators an exact number of times. 236. The least common denominator of two or more fractions is the least number that will contain each of the given denominators an exact number of times. 237. The least common multiple (134) of all the denominators of the given fractions is the least common denominator. 238. To replace two or more fractions by two or more equivalent fractions having the least common denominator. 239. Example. Eeduce f , -J-, and ^ to equivalent fractions having a common denominator. ~ T5 "* SOLUTION. The least common multiple of the -R- ITS-- gi ven denominators is found to be 36. Using 36 2-1-3 -jig- = - 2 g-. as the least common denominator reduce each of vx Q ,x o v, Q Q the given fractions to thirty-sixths, as explained XoX^Xo = oo. . . . L.C.M. = 36. m233 ' 76 FRACTIONS [ 240-244 240. Hence the following rule : Find the least common multiple of each of the given de- nominators for the least common denominator, and proceed as in 234. ORAL EXERCISE By inspection, reduce to equivalent fractions having the least common denominator : 1- f> f * i f. 5. i i 1, i 7. i i, i T V 2. f |. 4- I, f e. i, i, ^ A- * i i T^ i- WRITTEN EXERCISE Reduce to equivalent fractions having the least common denomi- nator : ADDITION OF FRACTIONS 241. Similar fractions are fractions having the same denominator or unit value. 242. In order that fractions may be added, they must be similar and parts of like units. 243. The denominator is the namer of the fraction ; hence, simi- lar fractions are analogous to like numbers. 244. To add similar fractions, Add tfo numerators. Place the sum over the common denominator, and reduce the result to its simplest form. ORAL EXERCISE By inspection, find the sums of: i- i + i + f + | + i + i + i 4. tV + A + A + TV . * + f + 5 + 4 + * + i} + f 5 . A + A + A+A. * + + + + 4 + 6. . + , + + . 244-247] COMMON FRACTIONS 77 By horizontal addition, find the sums of: 7. 3 pieces of print containing 41 2 , 24 1 , 40 3 yards, respectively. NOTE, lii the dry goods business fourths (quarters) are very common frac- tions, and they are usually written without denominators by placing the numera- tors slightly above the integers. Thus, 41 1 = 41, 41 2 = 41| (41 fc), and 41 3 = 41|. 8. 3 pieces wash silk containing 42 1 , 45 1 , 43 1 yards, respectively. 9. 3 pieces duck containing 37 2 , 41 2 , 45 yards, respectively. 10. 4 pieces cashmere containing 42 1 , 45 1 , 46 1 , 42 1 yards, respec- tively. 245. To add fractions not having a common denominator. 246. Examples. 1. Find the sum of f and f . 5 3 40-4-21 SOLUTION. Since the given fractions are not ~f~ Q = f-jT* similar, reduce them to equivalent fractions having the least common denominator as in 240. Then =5 = l-g5_, add the numerators and place the sum, 61, over the 56 least common denominator, 66. $ = *& 8. Find the sum of 47J, 16|, and 17|. Eighths 47 J 4 SOLUTION. By inspection, the common denominator 16} 6 of the fractions is found to be 8. The sum of the frac- 17J- 5 tions is then 1$, which, added to the sum of the whole 81 J 15 numbers, gives 81 J, the required result T * 247. Hence the following rule : Reduce the given fractions to equivalent fractions having the least common denominator t and add as in 244. WRITTEN EXERCISE Find the sums of the following fractions : i. f , 4, f 4- f, A, * 7. A, j, A. 10. |, 4, |. * f . t A- s. &, I, A- * f f A' t A & f A, A- A. 1. 1- * A> A- ^- I. T, 1- 78 FRACTIONS [ 248-250 SHORT METHODS 248. When the numerators of any two fractions are alike, the work of addition may be performed as shown in the following examples. 249. Examples. 1. Find the sum of -J- and -fa. SOLUTION. The common denominator is 2 x 17, or 34. -|- -J^ = ^J. Since the numerator of each of the fractions is 1, the numerator of the first equivalent fraction having a common denominator 34 is equal to the denominator of the second fraction ; and the numerator of the second equivalent fraction having a common denominator 34 is equal to the denominator of the first fraction. 2 + 17 = 19; hence the sum of the given fractions is found to be f . 8. Find the sum of -f- and f . SOLUTION. The common denominator is 7 x 9, or 63. 2- -|- -3- = |-. If the numerator of each of the given fractions were 1, the sum of the numerators in the equivalent fractions having a common denominator 63 would be 16 (7 + 9). But the numerator of the given fractions is 2 ; hence the sum of the numerators of the equivalent fractions having a common denominator 63, is 32 (2 x 16). Therefore, $ + f = if. In practice think only of results. Thus, 7 + 9 x 2 = 32 ||. ORAL EXERCISE By inspection, find the sum of : 1. 1 + 1. 8 . A + f 16. f + f M. |-f f 9. 2. * + . 9. i-f-J. 18. f + f. 23. f + A SO. 3. i + i. 10. i + i- 17. A + f ** * + * 91. f + A- 4. A + * 11- i + i- 18. A + f ** 1 + 1- * t + f 5. A + t ^ i + J* ^ * + i ^ f + f- ^. A+^ ^ * + ^ A+f ^ l + f ^ f r. A + ^ * + f ^- A + f ^ A- + *- In the following problems add the first two fractions, and to the sum add the other fraction. ss. i + t + f as. t + t+TV 40- l + ? + rt- i+i-t-A- . i+i+A> # ?+f+tt- 250. The business man's fractions are usually of the simplest sort, and ability to add them rapidly is of the utmost importance. In a great many cases the least common denominator can be determined by inspection and the fractions added as rapidly as whole numbers. 251] COMMON FRACTIONS 79 251. Examples. 1. Find the sum of -J-, J, , and ^ SOLUTION. By inspection, find the least com- 1,1,1, _ L __ 1 5 mon denominator to be 16. Reducing each fraction r 1 6 T5* to sixteenths at sight, and adding, say or think, 1, 3, 7, 15, ft. 2. Find the sum of f , T ^, , and J. SOLUTION. By inspection, find the least yf =1^-. common denominator to be 16. 4, 6, 7, 19, tt. or IrV ORAL EXERCISE By inspection, find the sums of the following problems : 1. 2. 8. 4- 5. 6. 7. 8. 9. 10. 11. 12. iiifiViiilif* A t i i A i f A * i f * AAAAI I i t A V A f _lJLil_LAlJiJLA I The above may be used as an "open-book" exercise, different students being required to announce at sight the least common denominator, and then the successive steps necessary to arrive at a total. Students should be drilled on exercises similar to the above until they can add the fractious given as rapidly as they can whole numbers. WRITTEN EXERCISE Copy or write from dictation and add the following problems. Add the fractions as explained in 251. 1040J 1620J 1342| 1647| 1842^ 16211- 1831J 80 FRACTIONS [ 252-250 SUBTRACTION OF FRACTIONS 252. In order that fractions may be subtracted they must be similar and parts of like units. DRILL EXERCISE 0. 478 yards -28 yards =? 5. f-f =? 8. fi - if =? S. 195 acres - 88 acres = ? 6. J-f=?^ttJ- 253. To subtract similar fractions, Find the difference between the numerators of the given fractions write the result over the denominator. 254. To subtract fractions not having a common denominator. 255. Examples. 1. Subtract from . 7 _ _ o-i SOLUTION. Since only similar fractions and parts of like? " units may be subtracted, reduce the given fractions to equiva- ** _ lent fractions having a common denominator. | = f ; | = if. A tt-H = A- Hence, J-| = A- 0. From 16J take 12|. SOLUTION. Reduce the fractions to a common denomi- "i " l^rf nator. Since we cannot take ^ from ^, we take 1 from 16, 12| = 12^ change it to ^f, and add it to ^, making ^f . Then ^| - ^ = ^ ; and 15 - 12 = 3. Hence, 16| - 12| = 256. Therefore the following rule : Reduce the given fractions to equivalent fractions having the least common denominator and subtract as in 253. ORAL EXERCISE Find the values of the following : 1. lf-f S. 7f-lf 5. 12f-3f. 7. ll^-Sf & 3J-f 4. 25J-14f 6. 17|J-f 8. 120 - 56f. 256-258] COMMON TRACTIONS 81 WRITTEN EXERCISE Find the difference between : 1. 240f and 89. 4. ll^J and 21f 7. 1050f and 2020| 2. | J and |f. 5. 104^ and 84f 8. 79f and 49J. 5. 14|and21f 6. 9Jf and 21. 9. 541 1 and 29 J. 10. From 216J acres of land, lots of 21 acres, 16-f acres, 26|| acres, acres, and 63 J acres were sold. How many acres remained unsold ? SHORT METHODS 257. When the numerators of any two fractions are alike, the work of subtraction may be shortened, as shown in the following examples. 258. Examples. 1. From | take -3^. SOLUTION. The common denominator is 17 x 9, or J TT = ITS* 153. The numerator of the first equivalent fraction hav- ing a common denominator 153 is 1 x 17, or 17. The numerator of the second equivalent fraction having a common denominator 153 is 1x9, or 9. Hence 17, the denominator of the subtrahend, minus 9, the denominator of the minuend, written over the common denominator, 153, equals the required result, or jl^. 2. From | take T \. SOLUTION. The common denominator is 91. 13 7 fy -fa = -J-J. x 2 = 12, which write over the common denominator 91. ORAL EXERCISE By inspection, find the value of: IS. 2__2 T . 19f !_|. $5. 14. l- 20. f-^. 26. 5. _i, 9, -2^-. 15. A A- #* f fr # 7 - 124| 13f |-^. 10. |-f 16. f-tV 22. f-f. 28. 64|-52f 5. |-f 11. f &. 17. f-f. 23. f-J. 29. 83i-72. 0. -.. j& fft. 7<5. |-f ^. f-f. m 89|-72f. 82 FRACTIONS [ 258-260 MULTIPLICATION OF FRACTIONS DRILL EXERCISE 1. If 1 pound of sugar is worth 5^, how much are 5 pounds worth ? 2. If 1 pound of tea is worth $ ^, how much are 3 pounds worth ? 8. If 12 men earn $48, how many dollars does one man earn? 4. What is | of 48 ? 6. What is f of 15 ? 5. 48x^ = ? 7. 15 xf =? 259. To multiply a fraction by a whole number, a whole number by a fraction, or a fraction by a fraction. 260. Examples. 1. f x 4 = ? (a) SOLUTIONS. () Multiplying the numerator of 3 3x43 a f raction multiplies that fraction (219). Hence, ;; X 4 = P = = 1-J-. to multiply f by 4, multiply the numerator 3 by 4 and divide the result by the denominator 8 as shown in (a). Or, (P) (6) Dividing the denominator of any fraction 3 ^ __ 3 __ 3 __ ^ i multiplies that fraction (219). Hence, to multiply 8 8-5-42 f by 4 divide the denominator 8 by 4 and write the numerator over the quotient as shown in (6). Or, / c \ (c) Arranging (a) in another convenient form 3 3x43 * or canceuation we iiave the P rocess (c). X 4 = - 2 = - = 1-J-. Multiplying the numerator and leaving the P denominator unchanged as in (a), the number of parts taken has been multiplied, and the value of each part left the same. Hence, the whole fraction has been multiplied. Dividing the denominator and leaving the numerator unchanged as in (6), the size of each part has been multiplied and the number of parts left the same. Hence, the whole fraction has been multiplied. A whole number may be expressed in fractional form by writing 1 for its denominator. Hence, - x 4 = , and the process is indicated in convenient form for cancellation. 2. What will 5 dozen oranges cost at 9 f per dozen ? 3 SOLUTION. Since 1 dozen oranges cost $f, 6 dozen will cost x 1 = 5 times $|, which, by cancellation, as shown in the margin, is equal to $3. 260-261] COMMON FRACTIONS 83 8. If 1 mat of coffee cost $24, what will f of a mat cost ? (a) SOLUTIONS, (a) f of $24 = } of 3 x $24. 3 x $24 = $72. 24 x = 10 2 . $ of $72 ($72 -:- 7) = $ lOf. Hence, f of a mat of coffee at $24 a mat will cost $ 10 . Or, (&) (6) Since 1 mat of coffee cost $ 24, f of a mat will cost 24 -J- 7 = 3f f of $24. f of $24 is equal to 3 times } of $24. } of $24 3 8 X3 = 10 2 . ($ 24 -*- 7 ) = $ 3 f 3 times $3? = 10?. Solution (6) is shorter than solution (a) when the denomi- nator of the divisor is a factor of the whole number. 4. If 1 pound of tea cost $|, what will | of a pound cost ? 9 :. ^ K SOLUTIONS, (a) If 1 pound of tea cost $|, $ of a pound * x - = = ^H cost of $f- of $| is equal to 2 times $ of $f. 8 3 24 12 | of $ f equals $ . 2 x $ & = $ J$ = $ T \. Hence, if /JA 1 pound of tea cost $|, | of a pound will cost $-^j. Or, M n K (^) B y cancellation the operation is a little shorter X J = and the result appears in its lowest form, p o U ^. Find the product of 7J X 3j. j~ ^ ww SOLUTION. Reduce the mixed numbers to impropei 4f X = = 27 J. fractions and proceed as explained in Example 4. 2 p A 261. From the foregoing solutions the following general rule may be derived: Express tlw whole or mixed, numbers as improper frac- tions. Cancel all equivalent factors from the numerators and denominators. Find the product of the remaining numerators for the numerator of the resulting fraction, and the product of the remaining denominators for the denominator of the resulting fraction. NOTE. The same rule holds good for finding the product of more than two fractions. ORAL EXERCISE Find the value of : 1. JoflS. 8. fxa 5. }of27. 7. Hf45. 2. 18x|. 4- |x4. 6. |of6. 8. 84 FRACTIONS [ 261 P. Find the cost of 20 yards of cashmere at $f a yard. 10. If 1 yard of silk is worth $ T 7 , what are 8 yards worth ? 11. Required, the cost of 150 yards of muslin at $ a yard. 12. What will 2i pounds of sugar cost at 3^ per pound ? 18. What will 2 pounds of beef cost at 6|^ per pound ? 14. What will | of a pound of tea cost if 1 pound cost $ -f ? #?. John was given -J- of a farm and James f as much. What part had James ? 16. If \ of a stock were lost by fire and the remainder sold at f of its cost, what part of the first cost was received ? 17. Divide 21 into 2 parts, one of which shall be f of the other. 18. Divide 60 into 2 parts, one of which shall be { of the other. 19. So divide $ 150 between A and B that A's part may be J of B's. 20. Tea costing $ f per pound is sold for f of its cost. For what price per pound is the tea sold ? What is the loss per pound ? WRITTEN EXERCISE Fmd the product of : 1. ^x85. & ^ x!2. 6. A^xia 7. if*x40. *. |Jx8. ^. J^x9. 6. VxlL & ^f x28. P. xf xf xfxf xf xfxlOa? m xx5xAxxi-f If. What will be the cost of 7 tons of hay at f of $15 per ton ? 12. Find the cost of 7J pounds of beef at 8^ per pound. 18. At $2 per day, how much will a man earn in 17 days ? 14. Paid $^ for some stationery and | as much for some pens and ink. How much did I pay for both ? 15. A having $750 invested -J- of it in insurance and paid } of the remainder for a horse. How much did he have left ? 16. A owning f of a mill sold f of his share to B. What part of the whole mill did he still own ? 262-265] COMMON FRACTIONS 85 SHORT METHODS 262. To multiply together two mixed numbers when the integers are alike and the fractions are |- 263. Example. What will 8 yards of lining cost at 8^ per yard? 8^ SOLUTION. The sum of J of each of two like numbers is equal to gl either of the numbers. Since 8 multiplied by added to of 8 is 797^ equal to 8, in finding the product of 8 by 8|, we may say, 9x8 f $ of = 72, the required result. ORAL EXERCISE By inspection, find the value of : 1. 2% yards at $ 2|. 4. 3 yards at $ 3 J. 7. 6 pounds at 6J 2. 7-| barrels at $7f 5. 8J pounds at 8^. 8. 11| acres at $ 11 J. 5. 5|- yards at $ 5J. 5. 9J pounds at 9j P. 12 dozen at $ 12 J. 264. To multiply together any two mixed numbers when the frac- tions are -. 265. Examples. 1. Find the cost of 120J yards of lining at 6%tf per yard. SOLUTION, j of 120 -1- 6 times \ is equal to \ of 120 + 6. \ of 126 = 63, which write as shown in the margin. \ of \ - , 63J which write as shown in the margin. 6 times 120 = 720. Adding 720 ti* 6 Partial products, the result is 7.83. The required result is, /-r 00^ therefore, $7.83. 2. Find the cost of 87 J pounds of crackers at S^ per pound. _ SOLUTION. of 87 + 8 = 47, which, added to \ of \ = 47f . 87 x 8 = 696. Add, and the result is 7.43|, or $ 7.44. 7.43|' Observe that In dividing numbers by 2 there can never be a remainder of more than 1. Also that in multiplying together any two numbers ending in , the fraction in the resulting product must be either \ or J. 86 FRACTIONS [ 266-267 266. Therefore the following rule : If the sum of the integers is even, write f in the resulting product. If not, write \ in resulting product. Find one half of the sum of the integers and to the result add the product of the integers. ORAL EXERCISE By inspection, find the value of : 1. 9 yards at 6j 4. 161 yards at 10i 7. 351 yards at 2. 4|- pounds at 9i 5. 20 yards at 8j 8. 121 yards at S. 12J- pounds at 7^. 6. 321 yards at 2 J/ P. 211 pounds at 4j WRITTEN EXERCISE Find the value of : 1. 162$. x 16J. 123J- X 51. 7. 120J x 4. m 204| x 7|. & 144J x 3J, 5. 60| x 121 8 . H5J X 51 ^. 2151 x 8i x 4f (5. 90J x 3|. P. 16| x 5J. 12. 1101 x 8J. Mixed numbers ending in , |, etc., may be multiplied together in prac- tically the same manner as shown in the foregoing explanations. DIVISION OP FRACTIONS DRILL EXERCISE 1. If 8 hats cost $24, what will 1 hat cost? 2. If | of an article cost $ 24, what will \ of an article cost ? 8. If 5 barrels of apples cost $ 15, what will 7 barrels cost ? 4. If f of an acre of land cost $ 15, what will acre cost ? 5. If 3 pounds of tea cost $ 1.05, what will 5 pounds cost ? 6. If f of a pound of tea cost 21 what will f of a pound cost ? 7. If 5 dimes will purchase 1 pound of tea, how many pounds will $ 4 purchase ? 8. If i^ of a pound of tea cost 25^, what will 4 pounds cost? 9. If 4 pounds of sugar cost $ \, what will 1 pound cost ? 10. If | of a yard of cloth cost $ |, what will 1 yard cost? 267. The reciprocal of a whole number is 1 divided by that number. the reciprocal of 3 is J. 268-272] COMMON FRACTIONS 87 268. The reciprocal of a fraction is 1 divided by that fraction. 269. Example. Find the reciprocal of . SOLUTION. In 1 there are 6 fifths. \ is contained In 6 fifths 6 times, f Is 4 times . Therefore, | is contained in f \ as many times as $. of 6 is J ; the reciprocal of f equals f. Hence, I divided by any fraction is the fraction inverted. 270. \ is the reciprocal of 4, and 4 is the reciprocal of . Hence, 27&# dividend, multiplied by the reciprocal of the divisor, is equal to the quotient. 271. To divide a fraction by a whole number, a whole number by a fraction, or a fraction by a fraction. 272. Examples. 1. Divide f by 2. (a) SOLUTIONS, (a) Dividing the numerator (divi 4. 4 _._ 2 2 dend) divides the fraction (219). Hence, divide the K~*~ = - =- numerator of the fraction f by 2 and the quotient is. Or, (6) Multiplying the denominator (divisor) divides ^ . o_ ^ _ 4 _2 th e fraction (219). Hence, multiply the denomi- 5 5x2 10 5* nator of the fraction f and the quotient is . Or, (c) (c) The dividend multiplied by the reciprocal 2 of the divisor is equal to the quotient (270). The 4_ jL _2 == | x l == ?- reciprocal of 2 is . Therefore, divided by 2 is 5 ' 5 % 5 equal to | x }. | x * = f #. If 2 pounds of coffee cost $ f , what will 1 pound cost ? o ~| ., o SOLUTION. If 2 pounds of coffee cost $, 1 2 X - = -. pound wijl cost i of f , or $f. 5 jfi o 5. At $|- per yard, how many yards of cloth may be purchased forfU? SOLUTIONS, (a) Since the denominator names or tells the size or value of the parts taken, similar -. . _._ 7 _ 112 _._ 7 fractions may be divided as concrete whole numbers. 8 ~~ 8 8 In concrete whole numbers the operation of division _ H2 -s- 7 = 16 is performed without regard to the unit named, so in dividing similar fractions the denominator may be ignored. Reducing 14 to the same value as |, the result is ^p. H 2 + 1 bein & analogous to - -*. , divide 112 by 7 and obtain 16. dollar^ dollars' 88 FRACTIONS [ 272-273 In a critical analysis of the above problem say in general : Since 1 yard cost $, as many yards may be purchased for $14 as $ is contained times in 14. 14 = i|2, and is contained in ^ 16 times. Hence, 16 x 1 yard, or 16 yards, may be purchased for $ 14. Or, x,v (6) Since 1 yard cost $f, as many times 1 yard may be purchased for $ 1 as $ $ is contained times in ~. ~ $1, or f times (the reciprocal of ). f x 1 yard = f 14 -5- E X - = 16. yards. Since for $ 1 f yards may be purchased, for 1 / $14 14 times f yards, or 16 yards, may be purchased. 4. If 1 yard of cloth cost $ ^, how many yards may be purchased forff? SOLUTIONS, (a) Since 1 yard cost $, as many ( a ) yards may be bought for $f as ^ is contained times } -- - = -j^j- -J- -^j- = in f . | = T 9 s ; = ^j. 4 is contained in 9 2 times. 9 _._ 4 = 21 Hence, 2 x 1 yard, or 2 yards, may be purchased. Or, (6) Since $ is contained in $1 f times (the .,,. reciprocal of |), 3 yards of cloth may be purchased i 8 Vs 9 91 for $1. If 3 yards may be purchased for $1, f of 3 f ~*~7 = x T = f = ^* yards may be purchased for $|. Hence the simple 273. Therefore the following rule : Multiply the dividend ~by the reciprocal of the divisor. NOTE. Reduce mixed numbers to improper fractions before applying the rule. ORAL EXERCISE Find the value of : 1. 6-*- & 24-*-}. 6. 10 +f 7. }-*-6. 12-*-}. 4. 15-*-. ft 25 + !. * t^ 3 - P. If 4 pounds of coffee cost $ f , what will 1 pound cost ? 70. If |_i. O f a f arm b e grain land and evenly divided into 3 6 elds, what part of the farm will each field contain? 11 Divide 9 by |; 12 by 1J; \ by f ; 2| by ^. J& What part of 1 is |? of 2 ? of 3? of 4? of 5? of 10? 18. What part of 9 is 4? of5? of 6 ? of 7? of 11 ? of 17? 273] COMMON FRACTIONS 89 WRITTEN EXERCISE L If | of an acre of land sells for $45, what will 1 acre sell for at the same rate ? 2. A farm of 471 acres is divided into shares of 94J acres each. How many shares are there ? S. A church collection of $ 232 was divided among poor families, to each of which was given $5J. How many families shared the bounty ? 4. When potatoes are worth $ f per bushel and apples $ f per bushel, how many bushels of potatoes would pay for a load of apples measuring 30 bushels ? 5. A woman buys f of a cord of wood worth $ 6| per cord and pays for it in work at $ per day. How many days must she work to make full payment ? 6. A dealer paid f of $78f for f of 25 cords of wood. What was the cost per cord ? 7. if _^. of a farm of 67 acres be divided into 63 village lots, what part of an acre will each lot contain ? 8. 1760 bushels of wheat were put into sacks containing 2J bushels. How many sacks were there ? 9. At $ -J per day, how long would it take to earn $ 15 f ? 10. How many fields of 9| acres can be made from a farm con- taining 125 J- acres ? WRITTEN REVIEW 1. From the sum of % and 5J take the difference between and 21. 2, Divide into six equal parts the product of 11^ multiplied 8. An estate is so divided among A, B, and C that A gets f, B ^, and C the remainder, which was $4200. What is the amount of the estate ? 4. If 14 bushels of apples can be bought for $3|, how many bushels can be bought for f f ? 5. A woman having $ 1 gave f of it for coffee at 33^ per pound. How many pounds did she buy ? 6. Having bought | of a ship, I sold f of my share for $12,000. What was the value of the ship at that rate ? 90 FRACTIONS [273 7. What must be the amount of an estate it, when it is divided into three parts, the first part is double the second, the second double the third, and the difference between the second and the third is $7500? 8. Having paid $ 119 for a watch and chain, I discover that the cost of the chain was only -^ the cost of the watch. What was the cost of the watch ? 9. An estate valued at $ 120,000 was so distributed that A re- ceived ^ ; B ^ of the estate more than A ; C as much as A and B together, less $600; and 2 charities the remainder in equal parts. How much did each charity receive ? 10. A painter worked 17^ days, and after expending ^ of his wages for board had $ 15 left. How much did he earn per day ? 11. A mechanic worked 21f days, and after paying his board with | of his earnings had $ 66 j- left. How much did he earn per day? 12. If \ of the trees of an orchard are apple, J peach, J- pear, ^ plum, and the remaining 21 trees cherry, how many trees in all ? IS. A, B, and C rented a pasture for $37. A put in 3 cows for 4 months, B 5 cows for 6 months, and G 8 cows for 4 months. How much had each to pay ? 14- A farmer sold two cows for $ 75, receiving for one cow only { as much as for the other. What was the price of each ? 15. After selling 450 horses a dealer had -f- of his stock remain- ing. How many had he at first ? 16. If 8 horses consume 4 bushels of oats in 3 days, how many bushels will 12 horses consume in the same time ? 17. A and B can do a piece of work in 10 days which A alone can do in 18 days. In what time can B alone do the work ? 18. John and Calvin agree to build a wall for $ 86. If Calvin can work only |- as fast as John, how should the money be divided ? 19. What is the length of a pole that stands in the sand, -J in the water, and 25 \ feet above the water ? SO. A. colt and cow cost $124. If the colt cost $4 more than It times the cost of the cow, what was the cost of each ? 273] COMMON FRACTIONS 91 21. A tree 84 feet high was so broken in a storm that the part standing was f of the length of the part broken. How many feet were standing ? 22. A farmer has f of his sheep in one pasture, f in another, and the remainder of his flock, 72 sheep, in a third pasture. How many sheep has he ? 28. For a horse and carriage I paid $540. What was the cost of each, if the cost of the carriage was 1 J times the cost of the horse ? 24' Peter can do a piece of work in 12 days, and Charles in 15 days. How many days will it require for its completion if they both join in the work ? 25. A can do a piece of work in 21 days, B in 18 days, and C in 15 days. In how many days can the three working together per- form the work ? 26. John and his father have joint work which they can do, working together, in 25 days. If it requires 60 days for John work- ing alone to complete the work, how many days will it require for the father to complete it ? 27. A, B, and C together have $ 2520. C has twice as much as B, who has J as much as A. How much has each ? 28. A farmer bought 3 farms of 240 acres each at $llf an acre. He built three barns at a cost of $1245 each, spent $1275 in improving the houses, and put up 752 J rods of fence at $ 2} per rod. He then sold the farms for $ 35| per acre. Did he gain or lose, and how much ? 29. A and B joined in purchasing a farm costing $ 4500, A pay- ing $ 2000 and B the remainder. After owning the farm six months they sold it for $ 6300. Of this sum how much should each receive ? 80. By what number must $ be multiplied to produce 2401 ? 81. The difference between \ and of a number is 90 less than of the same number. What is the number? 82. If a man can dig 20 bushels of potatoes in a day, and can pick up 30 bushels in a day, how many bushels can he dig and pick up in 20 days ? 83. D can cut f of a cord of wood in | day. In 1 day E can cut | as much as D can cut in a whole day. If they work together, how long will it take them to cut 70 cords ? 92 FRACTIONS [273 34. If 3 J acres of land cost $ 65, what will 125^ acres cost at the same rate ? 35. A works at the rate of $2f a day, and B at the rate of $3 a day. How long will it take A to earn as much as B earns in 19 days ? 86. A tank has an inlet by which it may be filled in 10 hours, and an outlet by which, when filled, it may be emptied in 6 hours. If both inlet and outlet be opened when the tank is full, in what time will it be emptied ? 87. A arid B are engaged to perform a certain piece of work for $ 35.55. It is supposed that A does ^ more work than B, and they are to be paid proportionately. How much should each receive ? 88. If you buy 60 lemons at the rate of 6 for 10^, and twice as many more at the rate of 5 for 8 ^, and sell the entire lot at the rate of 3 for 4^, will you gain or lose, and how much ? 89. Henry bought a basket of oranges at the rate of 3 for 2^, and gained 50^ by selling them at the rate of 2 for 3^. How many oranges did he buy ? 40. There are 108 bushels of corn in 2 bins. In one of the bins there are 12 bushels less than one half as many bushels as in the other. How many bushels in each ? 41. Three brothers join in paying off the mortgage on their father's farm. The eldest pays - of it, and the others pay the remainder in equal shares. If the eldest brother pays $90 more than the amount paid by each of his younger brothers, what is the amount of the mortgage ? 42. A can dig 1\ bushels of potatoes in \ of a day, and B can dig 5J bushels in \ of a day. How many bushels can they both dig in 7f days ? 43. A dealer bought 250} bushels of corn at 60f ^ per bushel. If he sold the whole amount of his purchase at 65^ per bushel, what was his gain ? 44- Having bought 120| cords of wood at $ 5 per cord, I sold J of it at $ 6 per cord and the remainder for $ 340. Did I gain or lose, and how much ? 45. What is the value of 8 pieces of dress silk containing 48 1 , 42 2 , 45 2 , 40 1 , 43 s , 42 2 , 45 2 , and 42 1 yards at $1 per yard ? 273-275] DECIMAL FRACTIONS 93 46. I bought 240 J bushels of oats at 30}^ per bushel, 190| bushels of corn at 60 \$ per bushel, and 30 bushels of wheat at $ 1.12, and sold the whole for $ 320. Did I gain or lose, and how much? 47. Find the total cost of the items in the oral exercise, page 85; of the items in the oral exercise, page 86. 48. A man bought 5 bags of wheat, weighing respectively 120, 1L'4, 128J, 132|, and 131J pounds, at $11 per bushel. If each bag, independent of the wheat it contained, weighed 1 pound, and there are 60 pounds in a bushel of wheat, did he gain or lose by selling the whole purchase for $ 15 ? 49. A produce dealer's sales for a day are as follows: 3411 bushels of wheat at $11- per bushel, 410 \ bushels of barley at 80^ per bushel, 1120^ bushels of oats at 30J^ per bushel, 310^ bushels of buckwheat at $ f per bushel, 250 bushels of beans at $ 2-J- per bushel, 13861 bushels of potatoes at 501^ per bushel, 1050J bushels apples at $ i per bushel, and 6301 bushels of turnips at 70J ^ per bushel. Find the total sales for the day. 50. The six fields of a farm measure, respectively, 10, 12J, 19}, 26 T \, 301, and 2-f^ acres, and are valued at $ 250 per acre. How much is the farm worth ? DECIMAL FRACTIONS 274. A decimal fraction, or a decimal, is a fraction having for its denominator 10 or some power of 10 ; as, - If a unit be divided into ten equal parts, the parts are called tenths; if a tenth of a unit be divided into ten equal parts, the parts are called hundredths ; if a hundredth of a unit be divided into ten equal parts, the parts are called thousandths; and soon. To obviate the trouble of writing the denominators of decimal fractions, an abbreviated method of notation is used as shown in the following examples : 275. Compared with Common Fractions. Decimal fractions are in most respects quite similar to common fractions. The points of dif- ference may be stated as follows: 94 TRACTIONS [ 275-281 1. The denominator of a common fraction is always written, while that of a decimal fraction is only indicated. 2. The denominator of a common fraction may be any number, while that of a decimal fraction must be 10 or some power of 10. 276. The decimal point is a period (.). It is always placed at the left of tenths, and by its position indicates the denominator and determines the value of the decimal fraction ; as, .4, .47, .315. When the decimal point is used to separate the integral from the fractional part in a mixed decimal, or dollars and cents in a decimal currency, it is called a separatrix. The figures written at the right of the decimal point constitute the numerator of the fraction, and the number of figures written at the right of the decimal point indicates the power of 10 which constitutes the denomi- nator of the fraction. 277. A pure decimal corresponds to a proper fraction, the value being less than 1 ; as .5, .27, .207, .3241. 278. A mixed decimal corresponds to an improper fraction, the value being more than 1 ; as, 8.17, 17.8, 24.113. 279. A complex decimal corresponds to a complex fraction, and (OOJA T7m )> 10| )0y 100 280. General Principles. 1. Decimals increase in value from right to left, and decrease in value from left to right in a tenfold ratio. 2. A decimal should contain as many places as there would be ciphers in its denominator if written, the decimal point representing the unit 1 of such denominator. 3. The value of any decimal fraction depends upon its distance from the decimal point. 4. Prefixing a cipher to a decimal is equivalent to dividing it by 10. 5. Annexing one or more ciphers to a decimal does not alter its value. NUMERATION OF DECIMAL FRACTIONS 281. The abbreviated method used to indicate decimal fractions is nothing more than an extension of the method by which whole numbers are represented. & 281-283] DECIMAL FRACTIONS 95 The relation of orders in a mixed decimal fraction is clearly shown by the following NUMERATION TABLE H n i 127392416 72519275 THE INTEGRAL PART THE FRACTIONAL PART The above number Is read, one hundred twenty-seven million, three hundred ninety-two thousand, four hundred sixteen and seventy-two million, five hundred nineteen thousand, two hundred seventy-five hundred-millionths. 282. The order of a decimal fraction may be found by numerating either from right to left or from left to right, but it should be remembered that the decimal point stands in the position of the unit 1 in the decimal denominator. The order of a decimal may usually be determined by inspection if the fact to be drawn from the following illustration be observed. If .29 be numerated from the right as in integers, the point Is in the hun- dreds' place, hence read twenty-nine hundredths. In .1137 the point is in tbe ten-thousands' place, hence read eleven hundred thirty-seven ten-thousandths; in .031631 the point is in the millions' place, hence read thirty-one thousand, six hundred thirty-one millionths. 283. Hence the following rule : Numerate from the decimal point to determine the de- nominator. Read the decimal as a whole number, and give to it the denomination of the right-hand figure. In reading whole numbers never read and between periods or between hun- dreds and tens and units. Thus, in reading 615, say six hundred fifteen, and not six hundred and fifteen. In reading mixed decimals always connect the Integral and fractional parts by and; as 2.5, read two and five tenths; 17.016, read seventeen and sixteen thousandths. 96 FRACTIONS [ 283~28f ORAL EXERCISE Kead the following decimals : 1. .297. 7. .2. IS. .02. 19. 638.6|. 2. .1471. . .20. ^. .002. 80. 341.131^. 3. .2442. P. .200. 16. .0002. 7. 801.00801. 4. .105. m .2000. m .00002. 22. 6000.58302. 5. .963. 11. .214698. 77. .000002. #. 9001.00901. 6. .56007. ./. 4003755. 18. 136.251. jg 3000.00030003. NOTATION OF DECIMAL FRACTIONS 284. Example. Write as a decimal thirty-four hundredths. SOLUTION. Observe that in writing thirty-four hundredths as a common fraction the mental operation is as follows : After writing 34, the numerator, the question is, "34 what?" The answer is "34 hundredths," and 100 is written below as a denominator. The result is ^. In writing the decimal form of the same fraction reason in practically the same way. Write 34, the numerator, and make it hundredths by placing before 34 a decimal point, which represents 1 of the decimal denominator. The result is .34. Notice that 34 occupies two places corresponding to the two ciphers in the denominator. 285. Hence the following rule : Write the decimal the same as a whole number, prefixing ciphers when necessary to give each figure its true local value. Place the decimal point before the left-hand figure of the decimal. WRITTEN EXERCISE Express by figures the following decimals : 1. Twenty-six hundredths. 3. Six ten-thousandths. 9. Twenty-seven hundredths. 4. Four hundredths. 6. Five and seven tenths. 6. Five hundred and five hundredths. 7. Twenty-two hundred-thousandths. , 8. Five thousand and five thousandths. 9. One million and one millionth. 10. Five hundred thousandths. 11. Five hundred-thousandths. 285-288] DECIMAL FRACTIONS 97 18. Seventy-seven tenths. IS. Two thousand two thousandths. 14- Two thousand and two thousandths. 15. Eleven and one hundred seven millionths. 16. Eighty-three and five hundred four ten-thousandths. 17. Seven hundred ten and two hundred forty-three hundred- thousandths. 18. Fifty-four million fifty-four thousand fifty-four and fifty-four million fifty-four thousand fifty-four ten-billionths. 19. Write the following as decimal fractions : Io0> i oooo> -M oao"> REDUCTION OF DECIMAL FRACTIONS 286. To reduce a decimal to a common fraction. 287. Examples. 1. Keduce .035 to a common fraction. SOLUTION. The decimal .035 is read thirty- .035 = yjj-7F = -2-Shj-. five thousandths, which as a common fraction is writtpn 3 5 35 7 win/ben iggfl-. HT" K SOLUTION. .5 x .5 = .25. 9 x 8 = 72. If 1 acre cost $ 8.50, ;* 8.5 acres will cost 8.5 times $ 8.50, or $ 72.26. 72.25 2. Find the value of 17.5 acres of prairie land at $ 4.50 per acre. SOLUTION. The sum of the numbers to the left of the deci- mals is odd ; hence, the decimal fraction in the product is .75(|). ' (.5) of 17 +4 + 17 x 4 = 78.5. Rejecting the .5, which was 78.75 included in the .75 first written, the result is $78.75. 8. Find the cost of 15.5 dozen men's kid gloves at $ 5.50 per dozen. 15.5 SOLUTION. The sum of the numbers to the left of .5 is even ;" t$ $ hence, the decimal fraction in the product is .25(). |(.5) of ' 15 + 6 + 15 x 6 = 86. Hence, the required result is $ 85.26. ORAL EXERCISE Find the value of: 1. 21.5 A. at 9 5.50. 7. 125 Ib. at 35^* 13. 105 Ib. at 45 2. 17.5 doz. at $ 5.50. 8. 16.5 doz. at $ 3.50. 14. 255 Ib. at 45 4. 3. 16.5 doz. at $4.50. 9. 15.5 Ib. at 8j 15. 115 Ib. at 55 4. 25.5 yd. at $ 4.50. 10. 12.5 yd. at 7.5 16. 115 gal. at 35 f. 6. 19.5 yd. at $ 2.50. 11. 115 Ib. at 5.5 17. 205 gal. at 45 6. 15.5 yd. at $ 2.50. 12. 135 Ib. at 35 f. 18. 215 gal. at 55 DIVISION OP DECIMAL FRACTIONS 304. Division is the reverse of multiplication. Since in multi- plication the decimal places of the two factors are added to deter- mine the number of decimal places in the product, in division the number of decimal places in the quotient is found by subtracting the decimal places, if any, in the divisor from those in the dividend. * In this example and all similar cases, perform the multiplication just as if the decimal point were to the left of the 5's; then, in the product point off as many places as there are decimal places in the numbers multiplied. Thus, in this problem say, or think: odd number, therefore write 75. | of 12 + 3 -f 12 X 3 = 43^. Reject - which has been included in the 75 just tvritten and the product is 4375. Point off two places and the result is $43.75. 104 FRACTIONS [ 305-306 305. Example. .17).085(.5 Divide .085 by .17. 85 SOLUTION. The divisor has two decimal places, and the dividend has three decimal places, therefore point off one (21) place from the right in the quotient. 306. Hence the following rule : When needed, annex ciphers to the dividend to make its places equal in number to those of the divisor. Divide as in whole numbers, and from tfo right of the quotient point off as many places as the number of places in the dividend exceeds those in the divisor. Do not commence the division until the number of decimal places in the dividend is at least equal to the number of decimal places in the divisor. Supply any deficiency in the dividend by annexing ciphers. If the divisor and dividend have the same number of decimal places, the quotient obtained to the limit of the dividend as given will be a whole number. If the number of decimal places in the dividend be greater than the number of decimal places in the divisor, point off from the right of the quotient the number of places equal to such excess, prefixing ciphers to the quotient if necessary. If after division there be a remainder, ciphers may be annexed to it and the division continued to exactness or to two or three places ordinarily demanded in business computations. Such ciphers should be considered as parts of the dividend. ORAL EXERCISE By inspection, find the value of: 1. 1-1. 8. .In-.Ol. 15. .001-100. 22. .022-110. 2. l-s-.l. 9. .l-i-.OOl. 16. .0001-*-.!. 23. 2.2-.00011. 3. l-i-.Ol. 10. .1+10. 17. 100-.01. 24. 2200-.00011. 4. 10 -.1. 11. .1-100. 18. 10-10000. 25. .022-11000. 5. 10-.01. 12. 1-10. 19. .22-11. 26. 2200-.000022. 6. .1-1. 18. 1 + 100. 20. 2.2-.011. 27. .00001-10000. 7. .l-s-.l. 14. 1-1000. 21. 220-11000. 28. 10000 -.0001. 306-308] DECIMAL FRACTIONS 105 WRITTEN EXERCISE Find the sums oi the quotients in the following problems : 3. 64-4-16. .64-?- 16. 64 -*- .16. 640 ^ .16. 640 -5- 1600. 64 --.016. 64 -K. 00016. 6400 -*- .16. 6400 + .00016. 640 -s- 16000. 39 -*- 130, 3900 -4- .13. 390 -H 13000. 3900 -*- 130000. .039 -*- 13. .0039 -^ 130. .00039 -- 13000. .000039 -*- 13. .0000039 -5- 1.3. .039 + . 013. 1. 0. 9-1-9. 75-*- 250. 9-*- .9. 7.5 -?- 2500. 9-5- .09 75 -s- .25. 9 -s- .009. 7500 -i- .25. .009 -j- 9. .75 -- 2500. 900 -5- 900. 750 -f- 25000. 900 -T- .09. "7500 -i- .0025. 9000 -5- .0009. .075 -5- .025. 900 -?- 9000. 750 -j- .0025. .009 -*- 90000. 75 -*- .000025. * 6. 11 -*- 22. 150 -*-.3. 110 -*- .22. 150 -?- .03. 11 -5- .022. 150 -r- 3000. 1100 -*- .22. 150 -5- .003. 11000 -*- .022. 1500-*- .03. 110 -*- 2200. 15 -j- 30000. 1100 -f- 22000. 15 -*- .0003. 11 -*- .000022. 1500 -*- .003. 11000 -*- .22. 150 -*- .00003. 11 *- 2200. 15-*- 300. SHORT METHODS 307. To divide a decimal by 1 followed by any number of ciphers, Move the decimal point in the dividend to the left as many places as there are ciphers in the divisor. 308. To divide a number by .01, .001, or 1 preceded by any number of decimal ciphers, Move the decimal point in the dividend as many places to the right as there are places in the divisor. 106 FRACTIONS [ 308 ORAL EXERCISE 1. 897-1-10. 6. 357.16 -j- 1000. P. .113-1- .001. 2. 1.37-5-1000. 6. 14.27 -j- 100. 10. .171 -*- .0001. 8. 17.3 -f- 10. 7. .82 -f- 100. 11. .75-!- ,0001. 4. 2.47-f-lOOO. & .075-^.01. 12. 13.54 -*- .001. ORAL REVIEW 1. The sum of two numbers is .3. The smaller number is .05. What is the product of the two numbers ? 2. If .75 of a mill is worth $ 7500, what is .5 of it worth ? S. If .75 of a stock of goods is worth $ 225, what is three times the stock worth? 4. Five times a certain decimal is .4. What is the decimal ? 5. Three times a certain decimal is .15. What is twelve times the same decimal ? 6. How many thousandths in seven units ? 7. J-.5xi = ? 5. The product of two numbers is .0006. If one of the num- bers is .03, what is the other ? 9. The sum of two numbers is 15. If one of them is 6.5, what is the product of the numbers ? 10. $2.50 is how many hundredths times $75? 11. What will 7.5 thousand envelopes cost at $ 2.50 per thousand ? 12. Find the cost of 12.5 thousand feet of plank at $8.50 per thousand. WRITTEN REVIEW 1. The sum of three numbers is 4.5. If the smaller is .95 and the larger 2.05, what is the product of the three numbers ? 2. Multiply the sum of sixty-five hundred and sixty-five and one hundred seven millionths by the product of nine hundred millionths and one hundred twenty-seven and seventeen hundredths. 8. What is the cost of 6 barrels of sugar weighing 301, 314, 297, 309, 313, and 315 pounds, respectively, at 6J^ per pound? 308] DECIMAL FRACTIONS 107 4. If a wheelman travels 10.3 hours per day, how many days will be required for him to travel 558.0025 miles at the rate of 7.88 per hour ? 5. I sold a lumber man 381.25 pounds of butter at 28| ^ per pound, 2468.375 pounds of cheese at 11.4^ per pound, and 2356.5 pounds of dressed beef at 7f ^ per pound, and received pay in lumber at $ 23.12 per thousand feet. How many thousand feet of lumber should I have received ? 6. A man's salary is $2500 per year. If he spends $650.25 for board, $119.25 for books, $31.85 for other literature, $63.40 for charity, $209.75 for clothes, $ 109.90 for traveling expenses, $115.60 for incidental expenses, and saves the remainder, how long will it take him to pay for a piece of property valued at $ 8400 ? 7. A merchant had on hand Jan. 1, 1904, a stock of merchan- dise aggregating $11750.90. During the year he bought goods amounting to $7315.90 and sold goods amounting to $15364.85. If on Dec. 31, 1904, he has stock on hand valued at $9215.75, has he gained or lost for the year and how much ? 8. Having bought 25 gross of steel pens at $1.25 per gross, I sold them at 12^ each. If there are 144 pens in a gross, did I gain or lose, and how much ? 9. Find the total cost of the items in the oral exercise, page 103. 10. If a boy receives $1.25 a day, and a man $3.75 a day, how long will it take the boy to earn as much as the man can earn in 16 days ? 11. In a certain business school .5 of the students study book- keeping, .75 of the remainder study shorthand and typewriting, and the remainder, 125 pupils, study the English branches. How many students in each department, and in the entire school ? 12. C. W. Allen bought of J. E. Seel & Co., dealers in flour and feed, 135 barrels roller process flour at $6.75 per barrel, 135 barrels searchlight pastry flour at $ 5.75 per barrel, 375 sacks puritan pan- cake flour at 23 ^ per sack, 195 sacks chef pastry flour at 25 ^ per sack, 250 bags bran at $1.50 per bag, 1500 pounds corn meal at 2J^ per pound. Find the amount of the bill. 108 FRACTIONS [ 308-315 IS. A and B are in partnership. A is to receive .75 of the profits and B the remainder. At the end of one year B draws $ 1250 as his share of the profits. If the total losses for the year were $ 950, what was the total gain for the year ? the net gain ? 14> January 1, A and B join in the purchase of some real estate, A paying .4 of the purchase price, and B the remainder. They share the profits arising from the sale in proportion to their investments. The property is sold at a profit of $ 2500 and B receives $ 6000 as his share. How much did A and B pay for the real estate ? QUANTITY, PRICE, AND COST 309. The essential elements of every business transaction in- volving the money value of property or labor are quantity, price, and cost. 310. The fixed unit used in estimating the money value of com- modities is termed a commercial unit. A yard, a dozen, a bushel, and an acre are commercial units. 311. Quantity is the number of commercial units in any given commodity. 312. Price, is the value put upon a commercial unit. 313. Cost is the value of a quantity. 314. An aliquot part of a number is one of the even parts of that number. 20, 25, 33$, 50, etc., are aliquot parts of 100. 315. The unit of an aliquot part is the number which must be divided to obtain the part. $1 is the unit of the aliquot parts 20 ?, 25 ?, 83Jft etc. DRILL EXERCISE 1. Name three commodities of which a dozen is the commercial unit ; a yard ; a pound ; a ton. 2. Name three quantities of which the commercial unit is 1 rod ; 1 acre ; 1 barrel ; 1 bag ; 1 bale. 8. Name three aliquot parts of 1 yard; of 1 bushel; of 1 day; of ft 314-317] QUANTITY, PKICE, AND COST 109 4. Name three aliquot parts of 50 ; of 250 ; of J. 6- Name four aliquot parts of 1 ton ; of 2 yards ; of 30. 6 What aliquot part of $1 is 50^? 33^? 25^? 7. What aliquot part of 25^ is 12^? ? 1|^? 8. What aliquot part of 50^ is 2^? 316. The aliquot parts of $ 1 are especially useful in computa- tions where the quantity and price are given to find the cost. TABLE OP ALIQUOT PARTS PARTS OF $1.00 PABTS OF 50 j? PARTS OF 25 p PARTS MORB OR LESS THAN $ 1.00 112| ? = $ more 133J $ = \ more 110 ^ = ^ more 83^ = i less = f less 40 ^ = f less 37^ = f less 317. Many of the ordinary business computations may be mate- rially shortened by the use of aliquot parts. DRILL EXERCISE 1. Formulate a short method for finding the cost of a quantity when the price is 33 ^. SOLUTION. Since 33^ is of $1, to find the cost of a quantity when the price is 33 ^ consider the quantity as dollars and divide by 3. 2. Formulate a short method for finding the cost of a quantity when the price is 50^; 25^; 20/; 16f^; 12J^; 6J^; 8J 110 FRACTIONS [ 317-319 8. Formulate a short method for finding the cost of a quantity (a) when the price is 10 $ ; (6) when the price is 5 SOLUTIONS, (a) Since 10 p is fo of $ 1, to find the cost of a quantity when the price is Wp, point off from the right one place in the quantity considered as dollars. (6) Since 5j* is % of 10^, to find the cost of a quantity when the price is 5^, point off one place in the quantity considered as dollars and divide by 2. 4. Formulate a short method for finding the cost of a quantity when the price is 3^; 2J^; lf^; 1}^; lj 318. GENERAL RULE, find the cost of the total quantity by multiplying $ 1 by the given quantity; then take such part of the product thus obtained as the given price is a part 319. An abbreviation is a part of a word used to indicate an entire word. Many abbreviations are used in computations involving quantity, price, and cost. The most important of these are shown in the following list BUSINESS ABBREVIATIONS Al. . first quality doz. . . dozen No. . nnmber Apr. . April Dr. . . debtor Nov. November acct. . account ea. . . each Oct.. October amt. . amount E.&O.E. errors and omis- oz. . ounce Aug. . August "sions excepted p. . page bal. . balance Feb. . . February pp. . pages bbl. . barrel ft.. . . foot or feet pt. . pint B/L . bill of lading f rt. . . freight payt. payment bot. . . bought gaL .\ gallon Pd. . paid bu. . . bushel gro. . . gross pkg.. package bx. . . box hhd. . . hogshead pc. . piece cd. . . cord hr. . . hour pr. . pair ctg. . . cartage in. . . inch qt. . quart Co.. . Company Jan. . . January reed, received C.O.D. collect on delivery Jr. . . Junior R.R.. railroad Cr. . . creditor Ib. . . pound sec. . second cwt. . hundredweight Mar. . . March s. . . shilling da. . . day mem. . . memorandum Sept. September d. . . pence mo. . . month Sr. . Senior Dec. . December Messrs. . Gentlemen or Sirs wk. . week disc. . discount Mr. . . Mister yd. . yard do. . . ditto, or the same Mrs. . . Mistress yr. . year 31 9J QUANTITY, PRICE, AND COST BUSINESS CHARACTERS & number @ at a/c account c/o care of x by (in surface 8 dollars 1* ia 1 cents 1* If or 11 If per cent by the thousand C by the hundred ^ check mark ** ditto ' feet " inches pounds sterling ORAL EXERCISE By inspection, find the cost of: 1. 350 Ib. tea at 50 2. 870 Ib. coffee at 33^. 3. 124 Ib. raisins at 25 4. 24 Ib. raisins at 16f 5. 190 Ib. rice at 10 6. 160 Ib. seed at 7. 123 Ib. meal at 8. 855yd. prints at 20 9. 144 yd. gingham at 10. 180 yd. silesia at 16f 71. 192 yd. lining at 12. 1140 yd. prints at 18. 284 yd. lining at 1J. 960 yd. ticking at 15. 168 Ib. hamatl6f 70. 368 yd. plaids at 27. 88 yd. lace at 87| 18. 340 yd. mohair at 75 19. 390 yd. alpaca at 66f 20. 484 Ib. lard at 12} 21. 1680 doz. eggs at 16f f. 22. 240 Ib. pork at 28. 1152 yd. linen at 728 gal. cider at #5. Ill qt. berries at #0. 880 Ib. salt at lj #7. 164 yd. cotton at Z>"^ Model Receipted BUI (Dry Goods) In the model given above the number of yards in the different pieces of cloth is not uniform. Since the price is so much per yard, it is necessary to list the number of yards in each piece as shown. 40 1 , 39 2 , 40 3 , etc., in the first item on the bill are understood to mean 40, 89 f (i) 30f, etc. 429 2 equals the total number of yards in the 10 pieces. In finding the total number of yards in any number of pieces the various items should not be copied to another sheet, but should be added horizontally as they stand. The fractions should be added first, and then the integers. 124 FRACTIONS New York,. 19^' TO T. B. CUNNINGHAM, . Terms:. Model Receipted Bill (Groceries) In the above model the prices given are free on board cars New York city, and the shippers prepaid the freight charges to Boston, Mass. In all such cases the freight is part of the selling price, and is usually added to the bill as shown in the model. When goods are sold so that all transportation charges fall upon the buyers, the cost of cartage is also added to the amount of the bill. In certain lines of business a charge is also made for the crates used in packing. The above model shows the proper arrangement for all such additions. Had the above bill of goods been sold free on board cars Boston, Mass., and had the shippers not prepaid the freight charges, these charges would be deducted by the consignees from the amount of the bill, on the arrival of the goods. The freight bill would then be sent to the shippers for credit. Any conditions as to time of credit, manner of payment, or discount for prepayment should always be recorded on a bill. 344] BILLS AND ACCOUNTS 125 Please remit only by draft on New York, Boston, or Philadelphia, or by Post Office or Ezprcx Money Order, as the Clearing Home compels as to pay eolkeuonchaigeion local checks. Sold to "-^' -*^ TAYLOR, WOOD & CO, DEALERS IN PROVISIONS 69 and 71 Second Street, ina field, cMass., Terms, Shipped via Model Receipted Bill (Provisions) The second item in the above model shows how gross weights and tares are recorded in billing. The numbers to the left of the hyphen are the gross weights, and the numbers to the right of the hyphen, the tares. Thus, 74-14 is under- stood to mean that the gross weight of a tub is 74 Ib. and the tare 14 Ib. The 170 is the net weight of the three tubs. In finding the net weight the various items should not be copied to another sheet, but should be added horizontally as explained in 62. Bills on which commercial discounts are allowed should always be arranged as shown in the above model. Commercial discounts are fully explained on pages 198-206. In retail business, where running accounts are kept with customers, a tran- script of the charges, or of the charges and credits, is made, giving items and dates of purchases and payments. This transcript partakes of the nature of both statement and bill and is called an account current. 126 FRACTIONS 344-345 Boston, Mass.,. ir. 19 In account with L O HOLLIS -&, Model Statement The above statement is an abstract of W. L. Anderson's account. On Jan. 81 a statement was rendered showing a debit balance of $201.39. This amount is taken as a basis for the February statement, and to it are added the debit items of the ledger since Jan. 31, making a total of $1984.30. From this total is deducted the sum of the credits hi the ledger since Jan. 31, or $1250, leaving a debit balance of $734.80. This $734.30 will be the first item entered upon the March statement. 345. Statements are usually rendered the end of each month. By an exchange of statements errors are less likely to occur, and when made, are more readily detected. 346] BILLS AND ACCOUNTS 127 346. Wages are usually calculated on the basis of 8 or 10 hours to a day. In finding the amount due, in order to avoid fractions, it PAY ROLL. Weekending No. Names of Employees MOD. Tat. Wed. ITiur. Fri. Sat. Totals Rate Amount & -74 /7 It, /z-i Model Pay Roll is best to find the total time in hours, multiply by the rate per day and then divide by 8 or 10, carrying decimals to three places. Checks are sometimes used in paying off employees, but more generally the envelope system is used and each employee is paid in currency the amount due him. To pay off the employees in this manner the bookkeeper usually draws from the bank the exact amount of money and just the denominations wanted. To do this with absolute accuracy it is generally necessary for him to classify and record the denominations required for the payment of the amount due each name on the pay roll in a manner similar to the following : BILLS FRACTIONAL CURRENCY $20 $10 $$ $2 $1 W? 25? 10? 5? 1? 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 4 2 1 6 b 6 3 1 2 6 2 10 128 FRACTIONS [ 340 Seconfc National iSanfc CHELSEA, MASS. Pay Roll Memorandum E. W. FOWLE & require the following : CO. Pennies 10 3 12 25 60 20 10 10 60 50 50 Nickels 2 Dimes 6 Quarters 2 Halves 1 Dollars 3 2's . 6 5's 5 10's 6 20's 1 121 80 The foregoing model shows what should be done to ascertain the de- nominations required to pay off the amount shown in the model pay roll, page 127. Many times columns are provided for this memoranda on the right-hand side of the pay roll, im- mediately after the "Remarks" column. After the amount of the pay roll and the denominations required have been ascertained, they are written on the pay roll memorandum as shown in the accompanying model. This memorandum is attached to a check, payable to "Pay Roll," which is taken to the bank and cashed. WRITTEN EXERCISE Find the amount of each of the following bills : 1 SAGINAW, MICH., Sept. 1, 1904. Messrs. SAGE BROS. & Co., Tonawanda, N.Y. Bought of WESTON & BROWN- TERMS : Sight draft without notice after 30 da. 26416 ft. Clear Pine 2500 per M 146250 ft. Pine Plank 1250 per M 11670 Cedar Posts 1000 per C 81275 ft. Clapboards 2500 per M 71000 Shingles 410 per M 66200 ft. Pine Lumber 2500 per M 111224 Cedar R. R. Ties 333J per C 31000 ft. Pine Boards 1663 per M 346] BILLS AND ACCOUNTS 129 f WORCESTER, MASS., May 15, 1904. Messrs. R. E. BARNES & Co., Detroit, Mich. Bought of OSGOOD, TOWER & Co. TERMS : Cash Case No. of Yd. Price Items Amount #19 16 pcs. Bleached Cotton 412 453 411 452 44 441 471 45 3 42 423 431 433 47 44 442 6J? #6 12 pcs. Muslin 37 1 323 33 353 341 32 352 33 3 37 38i 38i 36 10? #31 9 pcs. Delaine 39 40 2 41i S9 3 382 40 423 441 42 16? #7 24 pcs. Windsor Prints 2F 2?3 253 28 26 222 24 25 82 312 28 241 25 27 2 22 281 24 1 22 26 24 312 32 22 212 G$? #21 21 pcs. Merrimac Prints 281 32 343 28 2 26 241 22 2 24 2 26 2 24 261 33 28 2 34 27 1 30 32 3 24 e #169 20 pcs. Simpson Mourning 6? Prints 40 38 34 2 40i 32 40 41 2 40 162 40 29 3 30 27 2 19 2 41i 88 2 30 43 42 413 6|? #173 15 pcs. Striped Denim 40 42 41 40 38 41 43 1 44 2 45 1 40 3 46 38 40 1 38 s 40 1 $$? 1 130 FRACTIONS s C346 BOSTON, MASS., Jan. 3, 1904. Messrs. MARTIN & WARREN, Milwaukee, Wis. Bought of HARRIS BROS. & Co. TERMS : Interest after 60 da. 8 pcs. F. A. Cambric 56 52 1 45 3 50 2 62 40 1 60 51 1 22 t 5 gro. Jet Buttons 112$ 8 pcs. P. D. Goods 35 45 3 55 2 50 3 51 62 46 1 60 26 ^ 4 pcs. G. Flannel 35 3 4040 2 403 33^ 8 pcs. V. Barege 20 1 25 24 2 27 26 3 22 24 2 22 16 ^ 6 pcs. E. Lining 40 62 2 64 55 1 45 2 50 2 3^ 3 pcs. B. Silk , 685866 98j* CLEVELAND, OHIO, Oct. 15, 1904. Messrs. BROWN, HORTON & Co., Springfield, Mass. Bought of EOBINSON, CAREY & Co. TERMS: Net cash. 10 bbl. Pork 1666 5 bbl. Mess Beef 1125 3 bbl. Hams - 275-56 281-60 287-62 12J j* 3 bbl. Shoulders 248-37 252-42 371-40 8$jZ 8 tubs Lard 71-14 70-15 69-14 11 ? 6 bkt. Pork Loins 314 301 294 312 302 315 8Jj* 346] BILLS AND ACCOUNTS 131 WRITTEN REVIEW Find the amount of each of the following invoices : 1. Thurston & Denton, Buffalo, N.Y., bought of Brown Bros. & Co., Boston, Mass., Jan. 27, the following : 8 pcs. M. shirting, 40 2 41 1 46* 51 2 45 1 50 3 43 34 1 , at 6f ^; 15 pcs. crash, 613 yd., at 6J^; 6 pcs. C. jeans, 50 2 45 50 55 1 51 1 46 2 , at 5^; 25 doz. M. L. thread at 59^; 10 pcs. R. print, 41 55 2 45 l 51 46 50 3 40 56 2 42 1 52 2 , at 6J^; 4 pcs. N. sateen, 55 1 55 2 60 3 50 2 , at 6|^; 5 gro. F. braid at $ 7.621; 16 doz. L, shirts at $7.25; 6 pcs. T. R. prints, 25 1 35 3 30 2 31 21 1 25 1 , at 4|^j 25 cases E. batts at $ 6 ; 20 gro. S. P. buttons at 49^. 2. I. F. Hoyt, Milwaukee, Wis., bought of Mann & Co., of the same city, Sept. 4: 10 pcs. N. sateen, 55 2 51 50 s 54 1 56 55 52 2 53 51 8 50, at 5^; 15 pcs. T. A. flannel, 62 3 65 1 61 58 2 55 63 1 65 s 62 60 2 63 56 3 60 1 58 62 2 65 1 , at 33^; 20 pcs. E. gingham, 50 52 1 51 51 2 55 60 3 62 1 61 2 58 55 2 56 1 53 3 51 55 3 61 2 61 58 1 56 54 2 51 1 , at 6JJ*; 10 pcs. B. checks, 45 52 1 41 2 40 55 3 50 2 45 51 1 42 50 3 , at 25 8. Brown Bros. & Co., Maiden, Mass., bought of W. D. Adams & Co., Boston, Mass., June 18 : 20 pcs. L. gingham, 58 2 46 1 41 3 38 1 46 2 45 3 51 2 55 38 2 35 37 3 49 3 40 2 51 s 44 44 2 40 37 1 33 s 46 2 , at 8^; 24 pcs. W. print, 44 1 46 3 51 2 39 3 41 2 45 48 3 51 34 s 37 2 35 36 2 41 s 34 8 49 1 37 2 34 36 2 42 3 48 43 2 53 1 38 1 42, at 6J^ ; 20 pcs. E. lining, 45 54 1 39 2 48 8 46 2 38 2 47 2 37 2 45 3 46 3 42 8 44 3 45 s 43 1 35 2 54 2 34 s 42 2 53 2 44 l , at 4j 4. Jan. 21, 1904, J. H. Palmer, Sons & Co., sold Morrison, Price & Long the goods shown below; terms, 2% 10 da., 1% 30 da., net 60 da. 10 doz. knives and forks : 3 doz. # 5 at $ 8.33 J, 3 doz. # 7 at $6.66}, 4 doz. #9 at $10; 9 doz. razors : 3 doz. #12 at $9, 3 doz. #13 at $12.50, 3 doz. #18 at $16.66f ; 12 doz. panel saws: 6 doz. #1 at $15, 4 doz. #4 at $21, 2 doz. #5 at $25; 4 doz. nutmeg graters: 2 doz. #1 at $2.25, 2 doz. #4 at $1.75; 6 doz. pocket knives : 2 doz. # 12 at $ 6, 2 doz. # 16 at $ 7.50, 2 doz. # 20 at $ 3.75 ; 5 doz. burnished teapots : 2 doz. # 1 at $ 6.75, 3 doz. # 2 at $7.25 ; 35 doz. dippers: 18 doz. #3 at $1.75, 10 doz. #6 at $1.35, 7 doz. #4 at $1.25; 2} doz. wash boilers at $37.75; f doz. #2 kettles at $5.87; 27 tea kettles at 97^; 8 doz. padlocks at $8.75; f doz. 3-qt. saucepans at $ 9.37 ; 11 doz. 2-qt. saucepans at $ 7.85 ; 2 doz. #44 dishpans at $2.47; 3f doz. #14 cups at 78f^; 59 faucets at $ 1.47 ; 2\ doz. carpet stretchers at $ 2.95 ; 6| doz. wrought wrenches at $12.75; 51 doz. cast steel axes at $12.50. 132 FRACTIONS [346 5. W. C. Blanchard, Hartford, Conn., bought of M. C. Woods, Utica, N.Y., July 15: 10 pcs. B. gingham, 60 61 s 50 1 60 3 51 61 8 61 1 50 55 51 s , at 8^; 10 doz. F. E. braid, at 23^; 10 pcs. B. checks, 45 41 55 1 42 52 40 2 50 55 51 8 45 2 , at 24^; 15 gro. G. buttons, at $1.124; 2 P cs - T - A - flannel, 65 60, at 30^; 6 pcs. E. lining, 40 55 l 45 2 52 41 50 1 , at 5^; 5 doz. L. L. gloves, at $3.05; 4 pcs. M. sateen, 55 3 55 50 60 s , at 5J^; 5 gro. T. braid, at $ 7.621; 3 doz. L. shirts, at $7.20; 6 pcs. T. B. print, 25 35 30 3 31 21 25 1 , at 4f ^; 10 cases E. batts, at $6; 20 gro. S. P. buttons, at 49^; 4 pcs. V. barege, 20 23 25 25, at 16f J*j 7 pcs. W. print, 45 8 51 45 50 46 2 55 50 3 , at 5j 6. Finu. the amount of Baker, Taylor & Co.'s inventory, Jan. 1, with items as follows : 8 pcs. F. A. cambric, 56 52 45 50 52 54 46 50, at 22^; 5 gro. J. buttons, at $1.12i; 15 pcs. P. D. goods, 55 45 8 55 2 50 8 51 52 46 1 50 52 1 54 48 2 50 3 o2 55 1 50, at 50^; 4 pcs. G. flannel, 35 8 40 40 2 40 8 , at 25^; 6 pcs. E. lining, 40 52 2 54 55 1 45 2 50 2 , at 31^; 10 pcs. V. barege, 20 1 25 23 2 27 26 8 22 24 2 22 26 8 28, at 16| t ; 10 pcs. B. H. checks, 45 52 55 41 40 2 51 s 51 1 53 50 2 46, at 24 ^ ; 5 pcs. W. prints, 25 2 31 8 30 28 2 27, at 5jtf ; 15 pcs. A. F. cashmere, 62 1 65 8 60 1 63 58 8 60 2 56 2 58 2 60 62 2 55 8 58 1 60 8 58 55 1 , at 19 X; 20 pcs. L. gingham, 45 48 1 46 1 44 8 45 s 44 8 46 44 48 46 42 50 2 51 s 46 2 47 1 46 1 48 49 45 1 48, at Bender the following statements : 7. On Feb. 28, 1904, the debits and credits of Mason & Hamlin's account with Lord & Taylor, Boston, Mass., were as follows : Debits : Jan. 1, To merchandise, $ 900.62 ; Jan. 27, To merchandise, $ 200.56 ; Feb. 18, To merchandise, $260.93. Credits: Feb. 1, By cash, $175; Feb. 15, By Cash, $200. 8. On May 31 the debits and credits of Burke, Fitzsimmons & Hone's account with C. D. Gray, Kochester, N. Y., were as follows : Debits : April 15, To merchandise, $ 900.46 ; April 30, To merchan- dise, $ 340.92 ; May 15, To merchandise, $ 135.40. Credits : April 30, By merchandise returned, $35.40; May 15, By cash, $300.90; May 20, By cash, $ 600. 9. The amount of the model pay roll, page 127, was determined on the basis of an 8-hour daj. Find the amount on the basis of a 10-hour day. 346] BILLS AND ACCOUNTS 133 10. Find the amount of the following pay roll (a) on the basis of an 8-hour day ; (b) on the basis of a 10-hour day : NAME MON. TUBS. WED. THCB. FRI. SAT. RA.TK 7 8 10 9 8 12 $2 00 8 8 8 8 9 11 3 00 5 8 5 1 10 9 3 50 Drowne, William .... 9 7 9 8 8 8 3.00 8 8 8 7 8 9 2 50 Keyser, Frederick . . . 8 7 7 8 9 8 1.75 Martin, Charles .... 9 10 8 7 9 8 3.00 Smith, Martin 7 8 8 9 9 5 3 00 Warren, William .... 9 9 9 9 9 10 3.50 Weeks, Thomas .... 8 9 10 9 8 9 2.50 11. Find the amount of the following pay roll (a) on the basis of an 8-hour day ; (b) on the basis of a 10-hour day : NAME MON. TlTES. WED. TlIUR. FRI. SAT. KATE Breen, Mildred .... 9 6 9 8 8 8 10 9 11 9 9 8 $3.50 2.75 Garret, Ellen 7 7 8 9 9 9 3.25 Cutter, James 11 9 8 8 8 9 4 10 Ernst, Harry . . . . , 9 9 7 6 7 9 4.00 Foley, Maude 9 5 8 8 7 10 3 25 Gordon, Ruth 8 7 8 8 8 8 2 75 Healey, Grace 7 8 6 8 6 5 1 75 8 8 7 9 9 9 4 00 Lang, James 8 8 9 10 8 11 4 50 Penny, George A Pratt, Helen 7 8 6 9 8 9 8 7 8 7 8 8 4.50 3 50 Schiller, Helen 7 9 8 8 7 7 6 00 Smith, Frank 8 7 8 8 9 8 4 20 Tuckerman, Leo .... Walker, Florence .... 8 8 5 9 9 9 8 9 7 8 9 9 5.10 5.50 12. Make pay roll memorandums for (a) and (b) in problem 10 ; for (a) arid (b) in problem 11. Assume that you are to draw the money from City National Bank. DENOMINATE NUMBERS MEASURES 347. Concrete numbers in which the unit has been established by law or custom are called denominate numbers. Numbers expressed in units of the same denomination are simple numbers. Simple numbers having denominate units are simple denominate numbers. Numbers expressed in units of two or more denominations are compound numbers, or compound denominate numbers. 348. A measure is a standard unit by which quantity is estimated. Quantity may be length, breadth, thickness, area, volume, capacity, weight, value, time, number, or amount. 349. The principal measures are those of Weight, Extension, Time, Capacity, Value, and Angles. 350. A standard unit of measure is a unit which has been estab- lished by law or custom as the one by which other units are to be adjusted. The Winchester bushel has been adopted by the United States as the standard unit for dry quantities, such as grain, seeds, etc. ; the gold dollar has been estab- lished as the standard unit of money value ; etc. 351. A quantity is measured by finding how many times it con- tains any standard unit of measure. The unit of extent is the yard; of weight, the Troy pound; etc. The num- ber of yards in a piece of cloth may be ascertained by applying the yard measure ; the weight of a body, by the use of the pound; etc. MEASURES OP WEIGHT 352. Weight is a quantity of matter expressed numerically with reference to some standard unit. 353. The standard unit of weight in the United States is the Troy pound. 354. There are three kinds of weight ID use: Troy Weight, Avoirdupois Weight, and Apothecaries' Weight 134 355-357] MEASURES 135 Troy Weight 355. Troy weight is used for weighing gold, silver, and jewels. TABLE 24 grains (gr.) = 1 pennyweight (pwt). 20 pennyweights = 1 ounce (oz.). 12 ounces = 1 pound (lb.). Ib. oz. pwt. gr. 1 = 12 = 240 = 5760. The grains of the Troy, Avoirdupois, and Apothecaries' weights are the same. The Troy pound is equal to 22.7944 cubic inches of pure water at its greatest density, and is identical with the Troy pound of Great Britain. In weighing diamonds and gems the unit generally employed is the carat, which is about 3.2 Troy grains. The term carat is also used to express the fineness of gold, 24 carats being pure. Thus, a carat means ^ part, and gold 18 carats fine contains 18 parts gold, or is | pure. Avoirdupois Weight 356. Avoirdupois weight is used for weighing all heavy articles, such as groceries, coal, provisions, grain, and the metals, except gold and silver. 357. The unit of Avoirdupois weight is the pound, which con- tains 7000 grains. TABLE 16 ounces (oz.) as 1 pound (lb.). 100 pounds as 1 hundredweight (cwt.). 20 hundredweight, or 2000 pounds = 1 ton (t.). t. cwt. lb. oz. 1 20 = 2000 = 82,000. "Hundredweight" and "pounds" may be read together as pounds, or pounds may be read as so many hundredths of a hundredweight. Thus, 17 hundredweight, 29 pounds, may be read "1729 pounds,' 1 or "17.29 hundred- weight" ; and 2 tons, 7 hundredweight, 31 pounds, may be read "2 tons, 7.31 hundredweight." 136 DENOMINATE NUMBERS (.' 358-359 358. In Great Britain the ton equals 2240 pounds. This in the United States is called the long or gross ton, and is used in the custom- houses and in wholesale transactions in coal and iron. LONG TON TABLE 112 pounds = 1 long hundredweight (1. cwt.). 2240 pounds = 1 long ton (1. t.). 359. A great many commodities are bought and sold by weight. The weight of the standard measure is, in some cases, uniform throughout the United States ; but in others it is regulated by State statutes. The following table shows the weights of the standards frequently used in buying and selling various commodities. TABLE or WEIGHTS OF PRODUCTS COMMODITIES STANDARD MEASURE WEIGHT IN AVOIRDUPOIS POUNDS EXCEPTIONS Barley bushel 48 Ala., Ga., Ky., Pa., 47; Cal., 50; La., 32. Beans bushel 60 Me., 62 ; Mass., 70. Beef barrel 200 Beets bushel 60 Butter firkin 100 Clover seed bushel 60 New Jersey, 64. Corn in the ear bushel 70 Miss., 72 ; Ohio, Ind., Ky., 68. Corn meal bushel 50 Ala., Ark., Ga., 111., Miss., N.C., Tenn., 48. Corn, shelled bushel 56 Cal., 62. Fish quintal 100 Flour barrel 196 Grain cental 100 Nails keg 100 Oats bushel 32 Ida., Ore., 36 ; Md., 26 ; N. J., Va., 30 Onions bushel 60 Peas bushel 60 Pork barrel 200 Md., Pa., Va., 56. Potatoes bushel CO Rye bushel 56 Cal., 54. Timothy seed bushel 45 Ark., 60 ; N. Dak., S. Dak., 42. Wheat bushel 60 | 360-365] MEASURES 1ST Apothecaries' Weight 360. Apothecaries' weight is used by physicians and druggists in compounding and prescribing medicines. TABLE 20 grains = 1 scruple (sc. or 3). 8 scruples = 1 dram (dr. or 3). 8 drains = 1 ounce (oz. or 5 ) 12 ounces = 1 pound (Ib. or ft)). Ib. oz. dr. sc. gr. 1 = 12 = 96 = 288 = 5760. The pound, ounce, and grain of this weight are identical with those of the Troy weight, but the ounce is differently divided. Drugs and medicines are bought and sold at wholesale by Avoirdupois weight. COMPARATIVE TABLE OP WEIGHTS 1 Troy pound = 5760 gr. ; 1 Troy ounce = 480 gr. 1 Apothecaries' pound = 5760 gr. ; 1 Apothecaries' ounce = 480 gr. 1 Avoirdupois pound = 7000 gr. ; 1 Avoirdupois ounce = 437 J gr. 176 Troy or Apothecaries' pounds = 144 Avoirdupois pounds. MEASURES OF EXTENSION 361. Extension is that property of a body by vvhich it occupies a portion of a space. It has one or more of the dimensions, length, breadth, and thickness, and may therefore be a line, a surface, or a solid. 362. Magnitude is the term applied to one or more of the dimen- sions, length, breadth, and thickness. 363. A line is a magnitude of only one dimension length. 364. A surface is a magnitude of two dimensions length and breadth. 365. A solid is a magnitude of three dimensions length breadth, and thickness. 138 DENOMINATE NUMBERS [366-370 366. The standard unit of extension in the United States is the yard. The standard yard prescribed at Washington has been fixed with the greatest precision. It is determined by a brass rod or pendulum, which vibrates seconds in a vacuum at the sea level at 62 Fahrenheit, in the latitude of London, Eng. This pendulum is divided into 391,393 equal parts, and 360,000 of these parts constitute a yard. A copy of the standard, which is identical with the present standard of Great Britain, is kept in each State capitol. Long Measure 367. Long measure is used in measuring lengths and distances. TABLE 12 inches (in.) = 1 foot (ft.). 8 feet = 1 yard (yd.). 6$ yards or 16 } feet = 1 rod (rd.). 820 rods or 6280 feet = 1 mile (ml). mL rd. yd. ft. In. 1 = 820 = 1760 = 5280 = 63,860. The terms pole and perch are sometimes used instead of rod. Formerly the mile was divided into 8 furlongs of 40 rods each. The furlong is now practically obsolete. The hand, used in measuring horses, is equal to four inches. 6280 feet is the legal mile in the United States and England, and hence is sometimes called the statute mile. The knot, used in navigation, is equal to 1.152$ statute miles, or 6086 feet. It is sometimes called a geographic mile. A league is equal to three knots or geographic miles. A pace is equal to three feet, and five paces approximate a rod. The fathom, used in measuring depths at sea, is equal to six feet Square Measure 368. Square measure is iised in computing surfaces such as land, floors, boards, walls, and roofs. 369. A square is a flat surface bounded by four equal sides and having four square corners. 370. The unit of square measure is a square, each side of which is bounded by a unit of length ; as, a square inch, a square yard. 370-373] MEASURES 139 A square inch is a square, each side of which is one inch ; a square foot is a square, each side of which is one foot ; a square yard is a square, each side of which is one yard ; a square rod is a square, each side of which is one rod. One Square Inch llnch 371. The area of a figure is the number of square units contained in its surface. TABLE 144 square inches (sq. in.) =1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 80 square yards, or 272 square feet = 1 square rod (sq. rd.). 160 square rods, or 43,560 square feet = 1 acre (A.). 640 acres = 1 square mile (sq. mi.). 8q. ml. A. sq. rd. sq. yd. sq.ft. sq. in. 1 = 640 = 102,400 = 3,097,600 = 27,878,400 = 4,014,489,600. All the units of square measure except the acre are derived from the corre- sponding units of long measure. Thus, 144 (12 x 12) square inches = 1 square foot ; 9 (3 x 3) square feet = 1 square yard ; 30J (5 x 5) square yards = 1 square rod ; 102,400 (320 x 320) square rods, or 640 acres, which are equal to 1 square mile. Surveyors 9 Long Measure 372. Surveyors' long measure is used by surveyors in measuring land, laying out roads, establishing boundaries, etc. 373. The unit of surveyors' long measure is the Gunteifs chain, which is 4 rods, or 66 feet, in length. The chain has 100 links, which may be written as hundredths of a chain. Thus, 4 chains, 27 links = 4.27 chains. TABLE 7.92 inches = 1 link (L). 25 links = 1 rod. 4 tods, or 100 links = 1 chain (ch.). 80 chains = 1 mile (ml.). mi. ch. rd. 1. in. 1 = 80 = 320 = 8000 = 63,360. 140 DENOMINATE NUMBERS L S74-37& Surveyors' Square Measure 374. Surveyors' square measure is used by surveyors in measuring land by acres and sections. It is sometimes called land measure. 375. The unit of land measure is the acre. TABLE 026 square links (sq. 11.) = 1 square rod (sq. rd.). 16 square rods = 1 square chain (sq. ch.). 10 square chains, or 160 square rods = 1 acre (A.). 640 acres = 1 square mile (sq. mi). In some parts of the country 36 square miles, or 6 miles square, is a township. A square mile, or 640 acres, is also called a section in surveying public lands. sq. mi A. sq. ch. sq. rd. sq. 1. 1 = 640 6400 10,2400 64,000,000. 376. United States public lands are surveyed by selecting a north and south line as a principal meridian, and an east and west line intersecting this as a base line. From these, other lines are run at right angles, six miles apart, thus dividing the territory into townships six miles square. 377. The rows of townships running north and south are called ranges. The townships in each range are numbered north and south of the base line, and the ranges axe numbered east and west from the principal meridian. 378. Each township is divided into 36 sqtiares of 1 square mile each. These squares are called sections, and are divided into halves and quarters ; each quarter section is in turn divided into halves and quarters. 379. The numbering of the sections in every township is as in the accompanying diagram. The corners of all sections are perma- nently marked by monuments of stone or wood, and a description of the monument and its location is made in the field notes of the surveyor. ATOWNSHIP A SECTION W- N. i Section (320 A.) 8.W.J (160 A.) W.i of S.E.J (80 A.) ** S.E.J S.E.i 8.1* 380-385] MEASURES 141 Cubic Measure 380. Cubic measure is used in determining the contents or vol- ume of solids. 381. A cube is a regular solid bounded by six equal square sides or faces. Its length, breadth, and thickness are, therefore, equal. 382. The unit of cubic measure is a cube, each side of which is bounded by a unit of length ; as, a cubic inch, a cubic yard. A cubic inch is a cube, each side of which is one inch ; a cubic foot is a cube, each side of which is one foot; a cubic yard is a cube, each side of which is one yard. Cubic Foot. 383. The contents or volume of a cubical body is the number of cubic units it contains. TABLE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu.yd.). cu.yd. cu. ft. cu.in. 1 = 27 = 46,656. The units of cubic measure are derived from the corresponding units of long measure. Thus 1728 (12 x 12 x 12) cubic inches = 1 cubic foot ; 27 (3 x 3 x 3) cubic feet = 1 cubic yard. A cubic foot of water contains nearly 7 gallons and weighs 1000 ounces, or 62| Avoirdupois pounds. Hence, a gallon of water weighs about 8| pounds. WOOD TABLE 16 cubic feet = 1 cord foot (cd. ft.). 8 cord feet, or 128 cubic feet = 1 cord (cd.). A cord of wood is a pile 8 feet long, 4 feet high, and 4 feet thick. A perch of masonry is 16 feet long, 1 feet wide, and 1 foot high, and con- tains 24| cubic feet. A cubic yard of earth is called a load. MEASURES OF CAPACITY 384. Capacity signifies room for things. 385. There are two measures of capacity in general use dry measure and liquid measure. 142 DENOMINATE NUMBERS [ 386-389 Dry Measure 386. Dry measure is used in measuring grain, fruit, vegetables, coal, etc. 387. The unit of dry measure is the Winchester bushel, which is 8 inches deep, 18^ inches in diameter, and contains 2150.42 cubic inches. This is the standard unit in uniform use for measuring shelled grains. The heaped bushel of 2747.71 cubic inches is used for measuring apples, roots, corn in the ear, etc. The British imperial bushel contains 2218.19 cubic inches. TABLE 2 pints (pt.) = 1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). bu. pk. qt. pt. 1=4 = 32 = 64. The dry gallon, or half peck, contains 268.8 cubic inches. Liquid Measure Liquid measure is used in measuring liquids and in estimat- ing the capacity of cisterns, reservoirs, etc. 389. The unit of liquid measure is the gallon, which contains 231 cubic inches. TABLE 4 gills (gi.) = l pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). 81$ gallons = 1 barrel (bbl.). gal. qt. pt. gi. 1 = 4 = 8 = 32. Casks, called hogsheads, pipes, butts, etc., are not fixed measures, their capacity varying for commercial purposes. In the sale of oils and liquors, and in certain other cases, the barrel is also an indefinite quantity. 390-394] MEASURES 143 Apothecaries' Fluid Measure 390. Apothecaries' fluid measure is used in measuring the liquids used in compounding medical prescriptions. TABLE 60 minims (Tt\,) = 1 fluid drachm (f. 3). 8 fluid drachms = 1 fluid ounce (f. 5). 16 fluid ounces = 1 pint (O). COMPARATIVE TABLE OF MEASURES MEASURE Cir. IN. IN ONE GAL. Cu. IN. IN ONE QT. Cu. IN. IN ONE Pr. Cu. IN. IN ONE Gi. Dry Liquid G pk.) 268f 231 67* 57f 33 98} 8* 7& MEASURES OF TIME 391. Time is a measure of duration. Its computations being based upon planetary movements are the same in all lands and among all peoples, 392. The unit of time is the solar day; it includes one revolution of the earth on its axis, and is divided into 24 hours, counting from midnight to midnight again. 393. A solar year is the exact time required by the earth to make one complete rotation around the sun, 365 days, 5 hours, 48 min- utes, 49.7 seconds, or about 365^ days. 394. The solar year is divided in the calendar into 365 days called a common year, except every fourth year, when one day is added to the month of February and the year is called a leap year. Since the fraction that is disregarded when 365 days is counted as a year is less than one fourth of a day, the addition of a day every fourth year is not exactly accurate. The slight error still existing is corrected by excluding from the leap years the centennial years 144 DENOMINATE NUMBERS [394 which are not divisible by 400. Hence, to find whether any year is. a leap year or not, Divide the number of centennial years by 400 and all other years by 4 ; if there is no remainder, the year is a leap year. TABLE 60 seconds (sec.) = 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (da.). 7 days = 1 week (wk.). 365 days = 1 common year (yr.). 866 days = 1 leap year. 100 years = 1 century (C.). yr. mo. da. hr. min. sec. I _ 12 _ / 365 = 8760 = 525,600 = 31,536,000. 1 366 = 8784 = 527,040 = 31,622,400. COMMERCIAL TABLE 30 days = 1 month (mo.). 12 months = 1 year (yr.). The 12 months into which we divide the year are called calendar months. They are of variable length, seven of them containing 31 days, four 30 days, and February 28 days, except in leap years, when it has 29 days. The calendar months, with the number of days they contain, are shown below : 1. January (Jan.) 31 da. 7. July 31 da. 2. February (Feb.) 28-9 da. 8. August (Aug.) 31 da. 3. March (Mar.) 31 da. 9. September (Sept.) 30 da. 4. April (Apr.) 30 da. 10. October (Oct.) 31 da. 5. May 31 da. 11. November (Nov.) 30 da. 6. June 30 da. 12. December (Dec.) 31 da. Standard Time. In 1883 the principal railroads of the United States and Canada adopted what is known as the " Standard Time System." This system divides the United States and Canada into four sections or time-belts, each covering 15 of longitude, 7 of which are east and 7| west of the governing or standard meridian, and the time throughout each belt is the same as the astronomical or local time of the governing meridian of that belt. The govern- ing meridians are the 75th, the 90th, the 105th and the 120th west of Greenwich, and as these meridians are just 15 apart, there is a difference in time of exactly one hour between any one of them and the one next on the east, or the one next on the west ; the standard meridian next on the east being one hour faster, and the one next on the west one hour slower. The time of the 75th meridian is called Eastern Time. The time of the 90th meridian is known as Central Time. 394-399] MEASURES 145 The time of the 105th meridian is known as Mountain Time. Time in the fourth belt, which is governed by the 120th meridian, and extends to the Pacific coast, is called Western or Pacific Time. The changes from one time standard to another are made at the termini of roads, or at well-known points of departure, and where they are attended with the least inconvenience and danger. Thia system has been adopted by most of the principal cities for local use. MEASURES OF VALUE 395. Value is the worth of one thing as compared with another. The general measure of value is money. United States Money 396. United States money has been fully treated on pages 58 to 64 inclusive. Canadian Money 397. Canadian money is the legal currency of the Dominion of Canada; it consists of gold, silver, and bronze coins, and paper 10 mills (m.) = 1 cent (^). 100 cents = 1 dollar ($). The mill is not coined. The gold coins of Canada are the British sovereign and half-sovereign ; the silver coins are the 5, 10, 25, and 50-cent pieces ; the only bronze coin is the cent. The Canadian silver coins are f pure metal and & copper. English Money 398. English or sterling money is the legal currency of Great Britain ; it consists of gold, silver, copper, and bills. 399. The unit of English money is the pound sterling, the value of which in United States money is $4.8665. TABLE 4 farthings (far.) = 1 penny (d.). 12 pence = 1 shilling (s.). 20 shillings 1 pound 8. d. far. 1 = 20 = 240 = 960. f., s., , are the initial letters of the Latin words denarius, solidarius, libra, signifying respectively, penny, shilling, and pound. 146 DENOMINATE NUMBERS [400-403 400 The value in United States money of the different denomi- nations of English money is shown in the following COMPARATIVE TABLE 1 farthing = $ cent. 1 shilling = 24 J cents. 1 penny = 2 s 2 g cents. 1 pound = $4.8665. The farthing is but little used except as a fractional part of the penny. The British gold coins are }} pure gold and ^ alloy ; the silver coins, f pure silver and ? % copper ; the penny and half-penny pieces are pure copper. The gold coins are the sovereign and the half-sovereign ; the silver coins are the crown (equal to 5 shillings), the half-crown, the florin (equal to 2 shillings), the shilling, the six-penny piece, the four-penny piece, and the three-penny piece ; the copper coins are the penny, the half-penny, and the farthing. The guinea (equal to 21 shillings), and the half-guinea are no longer coined. French Money 401. French money is the legal currency of France ; it is a deci- mal currency, and consists of gold, silver, and bronze coins and national bank notes. 402. The unit of French money is the franc, which is equal to 19.3 cents in United States money. The franc is also used in Belgium and Switzerland, and under different names in several other countries. TABLE 10 millimes (m.) = 1 centime (c.). 10 centimes =- 1 decime (dc.). 10 declines = 1 franc (fr.) fr. do. c. m. 1 = 10 = 100 = 1000. 403. The value in United States money of the different denomi- nations of French money is shown in the following COMPARATIVE TABLE 1 centime = $.00193. 1 decime = $.0193. 1 franc = $.193. The millime is not a coin. The gold coins of France are the 5, 10, 50, and 100 franc pieces, which are -fa pure gold and j^ alloy ; the silver coins are the 1, 2, and 6 franc pieces ; also the 25 and 60 centime pieces ; they are ft pure and -jij alloy. The bronze coins are the 1, 2, 5, and 10 centime pieces. French money is read as francs and centimes in the same manner as United States money is read dollars and cents. 404^12] MEASURES 147 German Money 404. German money is the legal currency of the German Empire; it consists of gold, silver, and nickel coins, and paper money. 405. The unit of German money is the mark, which is equal to 23.85 cents in United States money. TABLE 100 pfennige (Pf.) = 1 mark (Rm.). The gold coins of the German Empire are the 5, 10, and 20 mark pieces ; the silver coins are the 1 and 2 mark pieces and the 20 and 50 pfennig pieces ; the nickel coins are the 5 and 10 pfennig pieces ; the copper coins are the 1 and 2 pfennig pieces. The gold and silver coins are T 9 ff pure and T ^ alloy. ANGULAR MEASURE 406. Angular measure is used in surveying, civil engineering, astronomical calculations, and navigation, for measuring angles, determining directions and location of places, latitude, longitude, difference in time, etc. 407. The unit of angular measure is the degree, which, in any circle, is measured by -^-^ of the circumference. 408. A circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the center. 409. The circumference of a circle is the curved line bounding it. 410. The diameter of a circle is a straight line passing through the center and having its end in the circumference. 411. The radius of a circle is a straight line passing from the center to any point in the circumference. 412. Any part of the circumference of a circle is called an arc. 148 DENOMINATE NUMBERS [413 413. Every circle, great or small, is divisible into 4 parts called quarters, which are divisible into 90 equal parts called degrees. Every circle therefore may be divided into 360 degrees. TABLE 60 seconds (") = 1 minute ('). 60 minutes = 1 degree (). 860 degrees = 1 circle (Cir.). . Cir. o 1 = 360 = 21,600 = 1,296,000. Minutes of the earth's circumference are called nautical or geographic miles ; hence, a degree of the earth's surface at the equator contains 60 geographic miles, or 69J statute miles. MISCELLANEOUS MEASURES ENUMERATION TABLE 12 units = 1 dozen (doz.). 12 dozen = 1 gross (gro.). 12 gross = 1 great gross (gt. gro.). gt. gro. gro. doz. units. 1 = 12 = 144 = 1728. Two units are often called a pair , and 20 units a score. STATIONERS' TABLE 24 sheets (sht.) = 1 quire (qr.). 20 quires = 1 ream (rm.). 2 reams = 1 bundle (bdl.). 6 bundles = 1 bale (bl.). bL bdl. rm. qr. sht. 1 = 6 = 10 = 200 = 4800. DRILL EXERCISE 1. Write two like concrete numbers; two unlike concrete numbers. 2. Write two simple denominate numbers; two compound de- nominate numbers. 8. Write two like denominate numbers ; two unlike denominate numbers. 413] MEASURES 149 4. Which is the heavier, a Troy pound or an Apothecaries' pound? a Troy pound or an Avoirdupois pound? Illustrate. 5. Which is the heavier, a Troy ounce or an Avoirdupois ounce ? Illustrate. 6. Write two numbers expressing the weight of a dry medical prescription ; of gold ; of hay ; of vinegar ; of wheat ; of feathers. 7. Write a number expressing volume ; surface ; distance. 8. Write a number expressing Canadian money value ; United States money value ; French money value ; German money value. 9. Write a number expressing the weight of a liquid medical prescription. 10. Write three articles that are sold by the enumeration table. 11. Write a number expressing a quantity of apples in storage in a certain warehouse ; potatoes ; onions ; beef ; fish. 12. Write a number expressing surveyors' square measure ; an- gular measure ; surveyors' long measure ; cubic measure. 13. Write five numbers expressing area; three expressing ca pacity ; two expressing time. 14. Write the next leap year ; the next centennial year. 15. Write the standard unit of weight in the United States in grains. 16. Write the standard unit of English money ; of United States money ; of French money ; of German money. 17. Write the standard unit of dry measure in cubic inches. 18. Write a dry gallon in cubic inches ; a liquid gallon ; a heaped bushel. 19. Write a degree of the earth's surface at the equator in statute miles ; in geographic miles. 20. Write a perch of masonry in cubic inches; a cord of wood in cubic feet ; a township in square miles. 21. Express in United States money the difference between a franc and a quarter of a dollar ; the difference between a pound ster- ling and $ 10. 22. A man has 4 English sovereigns, 3 half-dollars, and 5 francs. Express the total sum in United States money. 150 DENOMINATE NUMBERS [413-416 23. Express 5 reams of paper in sheets ; 5 quires. 24* Express a statute mile in feet ; in inches. 25. Express in Canadian money the cost of 11 gross of lead pen cils at 30 ^ a dozen. DENOMINATE QUANTITIES REDUCTION OP DENOMINATE INTEGERS 414. In reduction the unit or denomination of a number changes, but not the value. When the change is from a higher to a lower denomination the process is called reduction descending) and when from a lower to a higher, reduction ascending. DRILL EXERCISE 1. Find the cost of 90 ft. of cable at $ 1.20 per yard. 2. What will 5 sq. ft. of gold leaf cost at 2^ per square inch ? 8. How many inches in 3 yd. 2 in. ? 4. How many five-cent pieces should be given for an eagle ? 5. How many units in 2 gro. ? 6. Find the cost of 2 bu. of apples at 100 a peck. T In -^ of an acre how many square feet ? 8. in -fa sq. mi. how many square rods ? in -^ sq. mi? 9. In 288 Avoirdupois pounds how many Troy pounds ? How many Apothecaries' pounds ? 10. Express in English money $486.65; in French money $19.30; in German money $23.85. 415. Reduction from a higher denomination to a lower. 416. Example. Reduce 45 5s. Sd. to pence. 45 5s Sd SOLUTION. Since 1 Is equal to 20s., 45 are equal to 45 times 20s., or 900s. 900s. with the 5s. added is ^ equal to 905s. 905s. Since in Is. there are 12&, in 905s. there are 905 12 times 12d., or 10,860d. 10,860A with the Sd. added is 10868c?. equal to 10,868d., or the required result. When possible, add mentally the number of lower denomination to the product as shown in the illustration in the margin. 417-41'.)] DENOMINATE QUANTITIES 151 417. From the foregoing illustration the following rule may be derived : Multiply tTie units of the highest denomination given by the number of the next lower denomination required to make one of this higher, and to the product add the given units, if any, of the lower denomination. Proceed in this manner with each successive result until the required denomination is reached. WRITTEN EXERCISE Reduce to the lowest denomination named : 1. 3 mi. 17 rd. 3 yd. 1 in. 11. 9 T. 5 cwt. 4 Ib. 1 oz. 2. 51 10s. 3d 12. 5 Ib. 8 oz. 13 pwt. 3. 11 bu. 5 pt. 13. 19 rd. 5 in. 4. 5 bu. 1 pk. 7 qt. 14. 14 sq. rd. 5 sq. yd. 3 sq. ft. 5. 43 T 23". 15. 4 A. 31 sq. rd. 5 sq. yd. 3 sq. ft. 6. 17 gal. 2 qt. 1 gi. 16. 5 wk. 367 hr. 5 inin. 31 sec. 7. 5 1. 1. 50 Ib. 2 oz. 17. 7 en. yd. 5 cu. ft. 8. 5 cu. yd. 3 cu. ft. 11 cu. in. 18. 7 Ib. 3 pwt. 4 gr. 9. 175 sq. rd. 15 sq. in. 19. 1 bbl. 2 gal. 1 pt. 10. 1 mi. 15 ch. 43 li. 20. 2 mi. 15 rd. 11 ft. 10 in. 418. Reduction from a lower denomination to a higher. 419. Example. Keduce 473 pt. to bushels. 2 1 473 pt. SOLUTION. Since 2 pt. equal 1 qt., 8 qt. 1 pk., 8 236 qt +1 pt anc * 4 P k * 1 bu- ' tne successive divisors for re- 4- ~ 9Q lr _t- A. f ducing given pints to bushels are 2, 8, and 4, - respectively. 1 P k< Divide 473 pt. by 2 and the result is 236 qt. 473 pt. = 7 bu. 1 pk. w j t h a remainder 1 pt. ; divide 236 qt. by 8 and 4 qt. 1 pt. the result is 29 pk. with a remainder 4 qt. ; divide 29 by 4 and the result is 7 bu. with a remainder 1 pk. Write the last quotient and the several remainders in order and the required result is 7 bu. 1 pk. 4 qt. 1 pt. 152 DENOMINATE NUMBERS 420-42,] 420- Hence the following rule : Divide the given number by the number of the same denomination required to make one of the next higher denomination, and consider the quotient as units of the higher denomination, and the remainder as units of the lower denomination. Proceed in like manner with each successive quotient until the required denomination is reached. The last result and the several remainders written in order will be the answer required. WRITTEN EXERCISE Change to units of higher denominations : 1. 72,920 min. 5. 214,712 in. 9. 9537 sec. & 24,840 gi. 6. 60,720 oz. Avoir. 10. 10,632 sq. rd. 3. 7210 pt. dry meas. 7 52,460 gr. Troy. 11. 8792 cu. in. 4. 40,720 sq. yd. 8. 24,620 da. 12. 34,832 Ib. Avoir. REDUCTION OF DENOMINATE FRACTIONS 421. When the integral unit of a fraction is a denominate num- ber, the fraction is called a denominate fraction. 422. Reduction of denominate fractions from a higher denomination to a lower. 423. Examples. 1. Reduce j^ Troy pounds to the fraction of a pennyweight. SOLUTION. Denominate fractions may be reduced to lower denominations by multiplication in practically the same ' X If x ^- = p^. manner as denominate integers. 1 1 31 Since 12 oz. equal a pound, and 20 pwt. equal 1 oz., the successive multipli- ers for reducing pounds to pennyweights are 12 and 20 respectively. Multiplying r ^ 30 by 12 and 20, by can cellation the result is found to be ff pennyweight 423] DENOMINATE QUANTITIES 153 2. Reduce -f$ of a Troy pound to a compound denominate number. 3 J*_ y Af = _ = 2 * oz SOLUTION. The successive multipliers are 12 ^014 and 20 respectively. Multiplying T 3 g Ib. by 12, 4 the result is 2| oz. Multiplying oz. by 20, the 5 result is 5 pwt. Therefore, ^ of a Troy pound is equal to 2 oz, 6 pwt. S. Eeduce .3165 of a Troy pound to a compound denominate number. SOLUTION. The successive multipliers are 12, 20, and 24 respectively. Multiplying .3165 Ib. by 12, the result is 3.798 oz. Multiplying .798 oz. by 20, the result is 15.96 pwt. Multi- 15.960 pwt. plying .96 pwt. by 24, the result is 23.04 gr. Therefore, 24 .3165 Ib., as a compound denominate number, is equal to a 8 oz. 16 pwt. 23 gr. 192 23.04 gr. ORAL EXERCISE 1. Reduce ^ of a bushel to the fraction of a peck; ^ of a bushel; -f$ of a bushel to a compound denominate number. 2. What part of a shilling is \ of -f^ of a pound sterling ? 3. What decimal of an inch is .08 of a foot ? .016 of a foot ? 4- Reduce ^ of a gallon to the fraction of a pint WRITTEN EXERCISE 1. Reduce -fa of a Troy pound to grains. 2. How many pennyweights in -fa Ib. ? S. Reduce f of a mile to integers of lower denominations. 4. Express fj of an acre as a denominate number. 5. Reduce .1754 of a square mile to lower denominations. 6. Reduce of an acre to lower denominations. 154 DENOMINATE NUMBERS [ 424-427 424. Reduction of a denominate fraction from a lower denomination to a higher. 425. Example. Reduce f of a grain to the fraction of a Troy pound. 5 $1 12 SOLUTION. Denominate fractions ma y De reduced from a lower denomi- nation to a higher by division in prac- 20 12 14400 tically the same manner as denominate integers. Since 24 gr. equal 1 pwt., 20 pwt. equal 1 oz., and 12 oz. equal 1 lb., the successive divisors for reducing grains to pounds are 24, 20, and 12 respectively. ORAL EXERCISE 1. What part of a foot is J of an inch ? 1 of an inch ? 2. What part of a week is ^ of a day ? f of a day ? 8. What part of a gallon is ^ of a pint ? of a pint ? 4- What part of a foot is .5 of an inch ? .025 of an inch ? WRITTEN EXERCISE 1. Reduce % of a penny to tho fraction of a pound sterling. 2. Reduce J of a shilling to the fraction of a pound sterling. 3. Change 2.4 cwt. to the decimal of a ton. Jf. Reduce -| of an inch to the decimal of a yard. 426. Reduction of denominate integers to fractions of higher denominations. 427. Examples. pound sterling. (a) 4 12 20 3 far. 9.5625s. .478125 9s. 6d. 3 far. = 459 far. 1=960 far. 459 -r- 960 = .478125 1. Reduce 9s. 6c2. 3 far. to the decimal of a SOLUTIONS, (a) The successive divisors to reduce farthings to pounds are 4, 12, and 20 respectively. Dividing 3 far. by 4, the result is .75d Putting with this the 6d, the result is 6.75. Dividing 6.76 to find the number f 9 J u t "T A 3 27 - 2T strips, divide the width of the room by 7# strips is practically 8 strips. the width of the carpet yd> divided 22 ft. + 6 in. = 22 J ft. by yd. equals 7f , the number of times 22-J- f t. X 8 = 180 ft. = 60 yd. one strip required. Since fractional widths of carpet cannot be bought, drop the frac- tion and add 1 to the whole number ; 8 strips are required, and ^ of a strip may be cut off or turned under. The length of the room is 22 ft. and there is a waste of 6 in. on each strip for matching ; hence, the length to be bought for each strip is 22$ ft. If there are 22$ ft. in each strip, in the 8 strips there are 8 times 22$ ft. or 180 ft., or 60 yd. WRITTEN EXERCISE 1. How many yards of carpet, 1 yd. wide, laid lengthwise of the room, will be required to cover a floor 10.5 yd. long by 6 yd. wide, if no allowance is to be made for matching ? 2. What will be the cost of the carpet border for a room 16J ft. by 21 ft., if the price be 62^ per yard ? S. How many yards of carpeting f yd. wide will be required to carpet a room 32 ft. long by 25 ft. wide, if the lengths of carpet are laid across the room and 8 in. are lost on each strip in matching the pattern ? How many yards if the strips are laid lengthwise and 6 in. are lost in matching ? If the carpet is laid in the most economical way, what will be the cost at $ 2.55 per yard ? 4. How many yards of Axminster carpeting f of a yard in width, and laid lengthwise of the room, will be required to cover a floor 21| ft. long and 18| ft. wide, making no allowance for waste in matching the design ? 5. Find the cost of a carpet f yd. wide, at $ 2.50 per yard, for a room 22 ft. by 18 ft., if the strips run lengthwise and there is a waste of J of a yard on each strip for matching the pattern. 6. How many yards of ingrain carpeting 1 yd. wide will be required to cover, lengthwise, the floor of a room 26 ft. long by 19 ft. 453-456] PRACTICAL MEASUREMENTS 169 6 in. wide, the waste being 6 in. on each strip ? How many yards of Brussels carpeting 27 in. wide will cover the same room if there is a waste of 18 in. per strip ? 7. What will it cost, at $ 1.15 per yard, to carpet a flight of stairs 11 ft. 4 in. high, the tread of each stair being 10 in. and the risei 8 in.? 8. A hall 8^ ft. wide and 18| ft. long was covered with oilcloth at 75 ^ per square yard. How much did it cost ? PAPERING 454. Wall paper is usually 18 inches wide, and may be bought iiy single rolls 8 yards long or in double rolls 16 yards long. 455. These dimensions are for the wall paper commonly used in America. * Imported papers and some American papers vary in width and length. In practice there is usually considerable waste in cutting and matching the paper, and it is found more economical to buy double rolls. There is no uniform rule respecting the allowance to be made for openings, such as doors and windows. Generally paper hangers estimate the number of full strips that would be necessary for the regular surface of the walls, and divide this number by the whole number of strips that can be cut from a full roll of paper. By this method the ends of the rolls are supposed to be utilized for the surface above doors and above and below windows, and other similar irregu- lar places. 456. It is hardly possible to determine in advance the exact number of rolls of paper required for the walls of any room, but for practical purposes the following method will be found to approxi- mate accuracy : From the perimeter of the room deduct the width of the doors and windows. Find the number of strips necessary for the regular surface of the walls, and divide the result by the whole number of strips that can be cut from a full roll of paper. The quotient will be the number of rolls required. For standard rolls there will be twice as many strips of paper required as there are yards in the length of the regular surface to be covered. Any whole rolls left over after papering may be returned to the seller, but no portion of a roll will ever be taken back. 170 DENOMINATE NUMBERS [ 457-458 457. Example. How many double rolls of paper will be required for the sides and ends of a room 24 ft. long, 18 ft. wide, and 8 ft. high, with 1 door and 3 windows, each 3J ft. wide, making no allow- ance for waste in cutting ? SOLUTION. 24 ft. + 18 ft. x 2 = 84 ft., the perimeter of the room. 3| ft. x 4 = 14 ft., the width of the doors and windows. 84 ft. - 14 ft. = 70 ft., or 23 yd., the length of the regular surface of the walls. A double roll of paper is yd. wide and 48 ft. long. 23 -=- = 46|, or practically 47, the number of strips necessary for the regular surface. 48 ft. -=- 8 f t. = 6, the number of strips in each double roll. 47 -r- 6 = 7 1, or practically 8 double rolls. Hence, 8 is the required number of double rolls of paper. * WRITTEN EXERCISE 1. How many single rolls of paper 8 yd. long and 18 in. wide will it take to cover the ceiling of a room 60 ft. long, 45 ft. wide, if there be no waste in matching ? 2. What is the cost of paper, at $ 1.25 per double roll, for a room 18 ft. long, 12 ft. wide, and 9 ft. high above the baseboard, allowing for 1 door and 2 windows, each 3J ft. wide ? 3. How many rolls of paper 8 yd, long and 18 in. wide will be required for the walls of a room 20 ft. long, 15 ft. wide, and having a height of 8 ft. 9 in., allowing for 1 door 3 ft. by 7 ft., and for 2 windows 3 ft. by 6 ft., and a baseboard 9 in. high ? 4- At $ 1.90 per double roll, what will be the cost of papering a parlor 20 ft. square and 8 ft. high from the baseboard, allowing for 1 door 3 ft. by 7 ft. and 3 windows, each 3 ft. by 6 ft. ? 5. Allowing for 3 windows each 42 in. by 7 ft., and 2 doors each 4 ft. by 11 ft., what will be the cost, at $1.80 per single roll, of papering a room 24 ft. long, 18 ft. wide, and 12 ft. high from the baseboard ? PAINTING AND PLASTERING The unit of painting and plastering is the square yard. Allowances are frequently made for one half or the whole of the area of the openings and the baseboard; but since there is no uniform custom governing such allowances, a written contract definitely referring to this matter should be drawn up ; then complications at the time of settlement will be avoided. 458-461] PRACTICAL MEASUREMENTS 171 WRITTEN EXERCISE 1. At 11 / per square yard, what will be the cost to plaster the sides and ceiling of a room that is 30 ft. long, 24 ft. wide, and 14 ft. high above the baseboard, making full allowance for 4 doors 3 ft. 6 in. by 8 ft. 3 in., and 6 windows 3 ft. 8 in. by 7 ft. ? 2. At 20 ^ per square yard, what will it cost to paint a floor 40 ft. long and 26 ft. wide ? 3. At 22^ per square yard, what will it cost to plaster the sides and ceiling of a room 30 ft. by 24 ft. by 12 ft., if J- of the surface of the sides is allowed for the doors, windows, and baseboard ? 4. What will be the cost at 18^ per square yard for plastering the ceiling and walls of a room 60 ft. long, 30 ft. wide, and 15 ft. high above the baseboard, allowance being made for 6 doors 4 ft. 6 in. wide by 10 ft. 6 in. high above the baseboard, and 12 windows each 3 ft. 6 in. wide by 8 ft. high ? 5. At 21 ^ per square yard, what will it cost to paint both sides of a board partition 90 ft. long and 9 ft. 3 in. high ? 6. Allowing J of the surface of the sides for doors, windows, and baseboard, what will it cost at 12J^ per square yard to plaster the sides and ceiling of a room 30 ft. long, 18 ft. wide, and 15 ft. high ? ROOFING AND FLOORING 459. The unit used in determining the number of square feet in any roof or floor is a square 10 ft. on a side, or 100 sq. ft. 460. Shingles are 16 in. long and on an average 4 in. wide. They are usually laid about 4^- in. to the weather, 1 shingle covering 18 sq. in. of roof. At this rate it requires 8 shingles for each square foot of roof, or 800 for each square of 100 sq. ft. Allowing for the waste that is usual in shingling, about 1000 shingles are estimated for each square of 100 sq. ft. Some shingles run better than this, and from 850 to 900 are regarded as an ample number for a square. 461. A bundle contains 250 shingles ; hence it usually requires about 4 bundles or 1000 shingles for 100 sq. ft. of roof. 172 DENOMINATE NUMBERS [461-464 WRITTEN EXERCISE 1. I wish to floor and ceil a room 25 yd. long, 15 yd. wide. What will be the cost of the material at $ 27 per thousand square feet ? 2. Find the cost at $45 per thousand square feet of the flooring for a room 40 ft. by 30 ft., the waste being i of the area of the floor. 3. Find the cost of laying an oak floor which is 18 ft. by 16- ft., if the labor and other incidentals amount to $25, the price of the lumber is $75 per thousand sq. ft., and an allowance of 42 sq. ft. is made for waste. 4. Counting 1000 shingles for 120 sq. ft., how many will be required to cover the pitched roof of a barn 120 ft. long and 30 ft. wide on each side ? 5. Allowing 800 shingles to a square, how many thousand will be required for the roof of a barn 30 ft. wide on each side and 100 ft. long? What will be the cost at $3.50 per thousand? 6. At $ 10 per square, what will be the cost of the slate for a roof 60 ft. long and 32 ft. wide ? SOLID CONTENTS 462. A rectangular solid is a solid bounded by six rectangular sides or faces. When these sides are squares, the figure is called a cube. 463. The dimensions of solids are length, breadth, and thickness (365). In finding the solid contents of any solid having a uniform length, breadth, and thickness, it is necessary to select a measuring unit. This may be any cube, each side of which is a unit of length (382). The number of cubic units found in the solid is the required solid contents. To illustrate, take the following example. 464. Example. What is the volume of a solid 6 ft. long, 4 ft. high, and 3 ft. wide ? SOLUTION. Select a measuring unit. This may be any cube, each side of which is a unit of length. For convenience, take 1 cu. ft. (a) as a measuring unit. Then lines are drawn as represented in (&), (c) and (d). (&) contains three times as many cubic feet as (a), or 3 cu. ft. (c) contains four times as many cubic feet as (?>), or 12 cu. ft., and the entire figure, or (d), contains six 464-469] PRACTICAL MEASUREMENTS 173 times as many cubic feet as (c), or 72 cu. ft. Therefore, the solid contents of the rectangular solid given is 72 cu. ft. Hence, (d) 1 cu. ft. TJie dimensions of a rectangular solid reduced to units of the same denomination and multiplied together express the solid contents of a solid in cubic units of the same denomination as the units of length. 465. A cylinder is a circular body of uniform diameter, the bases of which are parallel circles. When all the points of one circle are equally distant from all the points of another circle, the circles are said to be parallel circles. 466. The lateral surface of a cylinder is the surface of its curved sides. 467. The lateral surface of a cylinder is equal to the surface of a rectangular body, the length and height of which are equal to the circumference and height of the cylinder. Thus, the lateral surface of the cylinder in the accompanying diagram is the area of the rectangle described by A, B, C, and D back of the cylinder. Hence, 468. To find the area of the lateral surface of a cylinder, Multiply the circumference of the base by the height of the cylinder. 469. To find the solid contents of a cylinder, Multiply the area of the base by the height of the cylinder 174 DENOMINATE NUMBERS [ 469-472 WRITTEN EXERCISE 1. What will be the cost of a sheet-iro^ smokestack 40 ft. high and 2 ft. in diameter, at 15 ^ per square foot ? 2. A bin is 18 ft. long, 4J ft. wide, and 18 ft. high. How many cubic feet in the bin ? 3. At 15^ per cubic yard,, how much will it cost to excavate a cellar 85 ft. long, 43 ft. wide, and 14 ft. deep ? 4. How many cubic feet of stone in a walk 195 ft. long, 4 ft. wide, and 1| ft. thick ? 5. How many cubic feet in a cylinder 10 ft. in diameter and 20ft. long? 6. A rectangular bin contains 259,200 cu. ft. If it is 40 yd. long and 20 yd. wide, how many feet high is it ? 7. Find the cost of digging a round well 25 ft. deep and 8 ft. in diameter, at 35 ^ per cubic yard. BOARD MEASURE 470. In measuring lumber, boards one inch or less in thickness are estimated by the square foot. Thus, a board 18 ft. long, 12 in. wide, and 1 in. thick contains 18 sq. ft. or 18 ft. board measure. 471. In measuring lumber more than one inch in thickness the boards are estimated by the number of square feet of boards, one inch in thickness, to which they are equal Thus, a board 12 ft. long, 12 in. wide, and 2J in. thick contains 2$ times 12 board ft., or 30 board ft. 472. Unless sawed to order, the width of the board is reckoned only to the next smaller half-inch, except in che*rry, black walnut, etc., where the price is 15 ^ per foot and upward. Thus, a board 8J in. in width is reckoned 8 in.; a board 12| in. in width is reckoned 12 in. ; eta J 473-475] PRACTICAL MEASUREMENTS 175 473. When the width of a board tapers uniformly, the average width is found by finding one half the sum of the two ends. Thus, a tapering board 16 ft. long, 12 in. wide at one end, and 6 in. wide at the other, and 1 in. thick, contains f (12 -f 6 -*- 2 -i- 12) of 16 sq. ft., or 12 board ft. 474. Examples. 1. How many board feet in 6 pieces of hemlock, 2 in. thick by 6 in. wide by 18 ft. long ? SOLUTION. Since board feet are 8 X $ X )3 X 18 = 108 board ft. equal to square feet one inch in thick- ness, the length of the board in feet, multiplied by the width of the board in inches, and divided by 12, is equal to the number of board feet in one board, one inch in thickness ; but since the board is 2 in. in thickness, 2 times this result is the number of board feet in each board, and 6 times this result, the number of board feet in the 6 boards. To shorten the work, arrange the factors which are to be multiplied together as shown in the margin, and mentally cancel the 12 in the divisor from any factor or factors in the dividend. 2 x 6 in the dividend is equal to the 12 in the divisor; hence each board contains 18 board ft., and the 6 boards 108 board ft. & Find the number of board feet in 6 pieces of hemlock 4 in. thick by 5 in. wide by 16 ft. long. 2 SOLUTION. Reasoning as in prob- $ lem 1, 6 x 4 x 5 x 16 -* 12 is equal to 0X^4x5x16 = 160 board ft. the number of board feet in the 6 pieces of hemlock. Observe that 6 multiplied by 4 contains 12 twice. Then, 2 x 5 x 16 = 160 board ft. Hence, 475. To find the number of board feet in lumber more than one inch thick, Express the length in feet and the width and thickness in inches. The product of these three dimensions, divided by 12, is equal to the number of feet, board measure. In charging, or billing lumber, the number of pieces are entered first, then the thickness and width in inches, then the feet in length. For example, in recording 6 pieces, 4 in. thick by 6 in. wide and 20 ft. long, the form would be thus : 6 pcs, 4 in. x 6-in.-20 ft., and would be called off by the salesman, "6 four-by-sixes-2Q ft.," four-by-sixes being the name by which he selects and sells stock. Instead of writing "inches" and "feet," lumber billing clerks use (") for inches, and (') for feet; thus, 3 in. by 4 in.-l" ft. long is written, 3" X 4"-17', 176 DENOMINATE NUMBERS [475 ORAL EXERCISE By inspection, determine the number of board feet of lumber in : 1. 5pcs. 3" x 4"-16 f . 11. lOpcs. 2"x 6"-16'. 8. 8pcs. 2" x 6"-20 f . 12. 15 pcs. 3"x 8"-16'. S. 9 pcs. 4" x 6"-20'. 13. 25 pcs. 2" x 6"-20'. 4. 10 pcs. 5" x 7"-12 f . 14. 50 pcs. 3" x 4"-18'. 5. 15 pcs. S" xlO"-20'. Id. 100 pcs. 2" x 6"-18'. 6. 10 pcs. 21" x 8"-18'. 16. 4 pcs. 9" x 10"-16'. 7. 20 pcs. 4" x 6"-16 f 17. 24 pcs. 2" x 4"-20'. 8. 12 pcs. 8' ff xlO"-14'. 18. 6 pcs. 4" x 5"-12'. 9. 17 pcs. 6" x 8 >f -20 f . 19. 10 pcs. 4" x 6"-16 f . 10. 20 pcs. 3" x 4"-14'. m 6 pcs. 2" x 5"-22'. WRITTEN EXERCISE 1. What is the board measure of 7 planks, each 16 ft. long, 15 in. wide, and 3 in. thick ? 2. What will be the cost of plank at $ 18 per M that will cover a floor 24 ft. by 13 ft., if the plank is 2 J in. in thickness ? S. What will be the cost of 10 sticks 2 in. by 4 in., 10 sticks 2 in. by 6 in., 10 sticks 4 in. by 4 in., and 10 sticks 2 in. by 10 in., if they are each 16 ft. long and the cost is $15 per M ? 4. What will be the cost at $ 15 per M of a tapering board 18 ft. long, 1 in. thick, and 1\ in. wide at one end and 16 in. wide at the other ? 5. Find the amount of the following bill of hemlock, price by the M ft. board measure : 26 pcs. 2" x 6"-18' at $ 12 ; 24 pcs. 3" x4"-20' at $ 15 ; 128 pcs. 8" x 4"-14' at $20. NOTE. Always try to shorten the work by mentally eliminating the 12's from the dividend. 6. Find the amount of the following bill of lumber, price by the thousand ft. board measure : 18 pcs. 3" x 4"-20' at $14; 10 pcs. 2 rr x 6"-18 f at $15; 25 pcs. 4" x 6"-16' at $12; 12 pcs. 3" x 5"-20' at $18. 475-478] PRACTICAL MEASUREMENTS 177 7. At $ 32.50 per M, what will be the cost of: 8 scantlings 3" x 4"-18' ; 12 scantlings 4" x 5"-16' ; 8 scantlings 5" x 6"-14 f . 8* At $ 19.50 per M, what will be the total cost of : 9 boards 1" x 2"-14' ; 6 boards 1" X 18"-16 f ; 15 boards 2" x 14"-20'; 8 boards 1J" x 12"-18'. 9. At $ 24 per M, what will be the cost of the lumber required to inclose a field 40 rd. square with a board fence if the boards are 15 ft. long, 5 in. wide, and 1 in. thick, and the fence 5 boards high ? 10. At $21.50 per M, what will be the cost of the lumber in a line fence 160 rd. long if the boards are 11 ft. long, 7 in. wide, and 1 in. thick, and the fence 4 boards high ? WOOD MEASURE 476. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is called a cord. 477. A pile of wood 1 ft. long, 4 ft. wide, and 4 ft. high, or | of a cord, is called a cord foot 478. Example. Find the number of cords of wood in a pile 25 ft. long, 4 ft. wide, and 6 ft. high. 3 SOLUTION. The product of the dimen- 25 X 4 X 75 sions of the pile is equal to the number - = 4 cords - 1 of cubic feet of wood. Since a cord of wood is 8 ft. long, 4 ft. wide, and 4 ft. high, or contains 128 cu. ft., the number of cords in the pile is found by dividing this product by 128. Therefore, the required result is 4^ cords. Hence, 178 DENOMINATE NUMBERS [479-480 479. To find the number of cords of wood in any pile, Divide the product of the length, width, and height, expressed in cubic feet, by 128. WRITTEN EXERCISE 1. How many cords of wood in a pile 108 ft. long, 7 ft. 9 in. high, and 6 ft. wide ? 2. From a pile of wood 71 ft. 6 in. long, 9 ft. 4 in. wide, and 6 ft. 8 in. high, 21f cords were sold. What was the length of the pile remaining ? 3. At $4.75 per cord, what will it cost to fill with wood a shed 34 ft. long, 18 ft. wide, and 10 ft. high ? 4. A pile of wood built 10 ft. high and 25 ft. wide must be how long to contain 125 cords ? 5. A pile of wood 30 ft. long, 4 ft. wide, and 8 ft. high was sold at $ 6 per cord. How much was received for it ? PAVING 480. The unit of paving is the square foot or the square yard. WRITTEN EXERCISE 1. Find the cost of paving a court 150 ft. long and 120 ft. wide at $ 3 per square yard. 2. Which would be the more economical way to pave a street 3 mi. long and 1 rd. wide: with granite blocks at $3.65 per square yard, or with asphalt costing 23^ per square foot, and how much would be saved ? 3. Which would be the more economical, and how much : to pave a walk with stone at 22^ per square foot, or with brick at $ 1.02 per square yard, if the width of the walk is 4 ft. and the length 200 ft. ? 4. How many granite blocks 12 in. by 18 in. will be required to pave a mile of roadway 42 ft. in width ? 5. A street 4975 ft. long and 40 ft. wide was paved with Trinidad asphaltum at $2.G5 per square yard. What was the cost ? 481-491] PRACTICAL MEASUREMENTS 179 GAUGING 481. The process of finding the contents of any regular vessel in gallons, barrels, bushels, etc., is called gauging. 482. In every liquid gallon there are 231 cubic inches. Hence, 483. To find the exact number of gallons in a vessel, Divide the number of cubic inches in the vessel by 231. 484. In every bushel, stricken measure, there are 2150.42 cubic inches. Hence, 485. To find the exact number of stricken bushels in any bin, Divide the number of cubic inches in the bin by 2150.42. 486. In every bushel, heaped measure, there are 2747.71 cubic inches. Hence, 487. To find the exact number of heaped bushels in any bin, Divide the number of cubic inches in the bin by 2747.71. 488. Approximate Rules. For practical purposes the rules given below generally approximate accuracy. 489. Example. What decimal of a stricken bushel is 1 cu. ft. ? O I _ SOLUTION. Since in every bushel there are 2150.42)1728.000 2150.42 cu. in., and in every cubic foot 1728 cu. in., 1720336 a cubic foot k !Bo$ or approximately .8 of a ^bushel. Hence, 490. To find the approximate number of stricken bushels in any number of cubic feet, Multiply by .8 ; and, To find the approximate number of cubic feet in any number of stricken bushels, Divide by .8. 491. Example. What decimal of a cubic foot is a heaped bushel ? 2747.71)1728.0000(.63 SOLUTION. Since a cubic foot contains 1648 626 1728 cu * in> ' an( * a nea P e d bushel 2747.71 cu. in., 70 3740 a cubic foot ma y be reduced to a decimal of a 82 4313 heaped bushel as shown in the margin. The result is .63- cu. ft. Hence, 180 DENOMINATE NUMBERS [ 492-494 492. To find the approximate number of heaped bushels in any number of cubic feet, Multiply by .63; and, To find the approximate number of cubic feet in any number oi heaped bushels, Divide by .63. 493. Example. How many gallons in a cubic foot ? 1728 -T- 231 = 7.48+ SOLUTION. Since a gallon contains 231 cu. in., and a cubic foot 1728 cu. in., the number of gallons in a cubic foot is found by dividing 1728 by 231. The result is 7.48+. Hence, 494. To find the approximate number of gallons in a cistern, Multiply the number of cubic feet by 7~ 2) and from the product sub- tract -^ of the product. ORAL EXERCISE 1. Find the approximate number of bushels of grain contained in a box that is 5 ft. long, 4 ft. wide, and 3 ft. high. 2. How high must a box 10 ft. long and 6 ft. wide be built to hold, approximately, 240 bu. ? 3. A vat 11 in. long, 7 in. high, and 3 in. wide will contain how many gallons of water ? 4. A vat containing 2 gal. of water is 14 in. long and 11 in. wide. How high is it ? WRITTEN EXAMPLES Find the approximate number of bushels of grain required to fill : 1. A bin 18 ft. x 6 ft. x 4 ft. 4. A bin 25 ft. x 12 ft. x 8 ft. 2. A bin 13 ft. x 20 ft. x 5 ft. 5. A bin 12J ft. x 8 ft. x 6 ft. 3. A bin 20 ft. x 8 ft. x 6 ft. 6. A bin.20 ft. x 12 ft. x 8 ft. 7. A cubical cistern is 10 ft. on a side. Find the exact number of barrels of 31^ gal. each that it will contain. 494-000] PRACTICAL MEASUREMENTS 181 8. A farmer exactly filled a bin 9 ft. wide, 12 ft. long, and 7| ft. deep, with wheat grown from a field yielding 32J bushels per acre. How long was the field if its width was 50 rd. ? 9. Find the exact number of gallons of water in a well 6 ft. in diameter, when the water is 9 ft. in depth. 10. A cask is 24 in. at the chime, 30 in. at the bung, and 3 ft. long. Find the exact number of gallons that may be put into it, if the cask is already f full. STONE AND BRICK WORK 495. The unit of stone work is the cubic yard or the perch. 496. A perch of stone or masonry is a rectangular solid 16J ft. long, 1| ft. wide, and 1 ft. high, and contains 24J cu. ft. The number of cubic feet allowed for a perch of stone or masonry varies in different localities. In some places it is considered 16 cu. ft., and in other, places, 25 cu. ft. 497. The number of bricks in walls is usually estimated by the thousand, and 22 common bricks laid in mortar are counted for each cubic foot of wall. A common brick is 8 in. long by 4 in. wide by 2 to. thick, 498. In making estimates for stone and brick work, masons take girt measurements. Whether anything is to be deducted for the area of the windows and other openings is generally fixed by con- tract. In some localities one half the area of such openings is always deducted, while in others nothing whatever is deducted. In estimating material, however, allowance is generally made for the corners and all openings. In taking girt measurements the corners are counted twice, but this is con- sidered offset by the extra work required in building corners ; the work around openings is also more difficult than straight work. 499. To find the number of perches of stone or masonry in a wall, Divide the contents of the wall in cubic feet by 2$. 500. To find the number of bricks for a wall, Multiply the number of cubic feet in the wall by 2$. 132 DENOMINATE NUMBERS [ 600 WRITTEN EXERCISE 1. A cellar is 24 ft. square inside of the wall, which is 9 ft. high and 2 ft. thick. How many perches of 24| cu. ft. each would a mason estimate for the wall? 2. How many cubic yards of masonry in the foundation walls of a house 50 ft. long and 30 ft. wide, outside measurements, if the wall is uniformly 2 J ft. wide and 8 ft. high ? S. How many perches of stone, actual measure, will be required to inclose a field 32 rd. long and 24 rd. wide, with a wall 4J ft. high and 2^- ft. thick, counting 25 cu. ft. to the perch ? 4. What will be the cost, by mason's measure, of building the walls of a block 140 ft. long, 66 ft. wide, and 47 ft. high, outside measurements, at $1.45 per perch of 24| cu. ft., if the walls are 18 in. thick- and no allowance is made for openings ? 5. How many common bricks will be required to erect the walls of a flat-roofed building 120 ft. long, 85 ft. wide, and 22 ft. high, out- side measurements, if the walls are 18 in. in thickness and an allow- ance of 600 cu. ft. is made for openings ? (Solve (1) by mason's meas- ure, making allowance for the openings, and (2) by actual measure.) 6. At $2.25 per perch of 24| ft., how much will it cost, by mason's measure, to build the walls for a building, the length of which is 49.5 ft. and the width 24| ft., the walls to be 14 ft. high from the foundation and 18 in. thick ? 7. How many common bricks will be required in building the four walls for a building 90 ft. long, 50 ft. wide, and 60 ft. high, outside measurements, if the walls are uniformly 1J ft. thick, and 340 cu. ft. is allowed for openings ? (Solve (1) by mason's measure, making allowance for the openings, and (2) by actual measure.) WRITTEN REVIEW /. In estimating the number of posts necessary for a wire fence to inclose a rectangular field 120 rd. long, it is found that to put the posts 12 ft. instead of 16J ft. apart will require 180 more posts. What is the field worth at $90 per acre ? 2. How high must wood be piled in a shed which is 28 ft. long and 16 ft. wide, to contain 28 cords ? 3. What is the cost, at $ 90 per acre, of a rectangular farm having a length twice its width, if the perimeter is 480 rd. ? 500] PRACTICAL MEASUREMENTS 183 4. How much would it cost to plaster the walls and ceiling of a room 25 ft. long, 18 ft. wide, and 12 ft. high, at 27^ per square yard, making an allowance of 396 ft. for doors, windows, etc. ? 5. What will be the cost, at $ 12 per M, of the boards required for a sidewalk 32 rd. long and 4 ft. wide, if the boards are 16 ft. long. 1 in. thick, and 8 in. wide ? 6. Express 15.6 in dollars and cents. 7. At 1|^ per square inch, what will it cost to bronze a cube the depth of which is 2 ft. ? 8. I bought a farm 200 rd. long for $3600. If I paid $ 72 an acre for the farm, how much will it cost to fence it at 25^ per rod ? 9. What will be the cost of excavating a cellar 45 ft. long, 30 ft. wide, and 8 ft. deep, at 35^ per cubic yard ? 10. Estimating that 150 cu. ft. of air should be allowed for each pupil, how many pupils can be accommodated in a schoolroom 45 ft long, 30 ft. wide, and 10 ft. high ? 11. At 9^ per square yard, what will it cost to paint the four sides and bottom of a tank 10 yd. long, 16 ft. wide, and 18 ft. deep. 12. At $ 125 per acre, find the difference in cost between two fields, the first of which contains 80 sq. rd., and the second of which is 80 rd. square. 18. Bought of a produce dealer 900 pounds of wheat at $ 1 per bushel, and gave in payment a pile of wood 16 ft. long, 6 ft. high, and 4 ft. wide. What price per cord did I receive for the wood ? 14. How many cubic feet in 8 pieces of hemlock 24 ft. long, 14 in. wide, and 8 in. thick ? How many board feet ? What part of a cubic foot is a board foot ? 15. A man bought a piece of land 80 rd. square, and after retail- ing 240 sq. rd., sold the remainder at $90 per acre. How much did he receive ? 16. Which would be the more economical, and how much: to pave a sidewalk 1 mi. long and 1 rd. wide with asphalt costing 21 ^ per square foot, or with granite blocks costing $ 2.95 per square yard ? 17. What will it cost at $2.50 per yard to carpet a floor 24 ft. long by 17 ft. wide, if the strips, which are { of a yard wide, are run lengthwise of the room, and there is a waste of 9 in. on each strip for matching the pattern ? PERCENTAGE AND ITS APPLICATIONS PERCENTAGE 501. The arithmetical processes in which the basis of comparison is one hundred are termed percentage. 502. Per cent, usually written " %," is an abbreviation of the Latin words " per centum," and signifies by the hundred. Thus, eight per cent means eight of every one hundred parts, or .08, and is written 8 % ; seven and one half per cent means seven and one half of every one hundred parts, or .07|, and is written 7| %. 503. The essential elements of percentage are the base, the rate, and the percentage. 504. The base is the number upon which the percentage is com- puted. 505. The rate is the number of hundredths of the base to be taken ; it is usually expressed as a decimal. 506. The percentage is the result obtained by taking a certain per cent of the base ; or, It is the product obtained by multiplying the base by the rate. In the expression " 5% of 500 is 25," the base is 600 ; the rate, 6% ; and the percentage, 25. 507. The amount per cent is 100% increased by the rate ; or, 1 plus the rate, expressed as a decimal. 508. The difference per cent is 100% diminished by the rate ; or, 1 minus the rate expressed as a decimal. 509. The amount is the base plus the percentage. 510. The difference is the base minus the percentage. 511. General Principles. The base may either be an abstract or a denominate number; the rate per cent must always be an abstract number; and the percentage, amount, and difference always huvo the same name as the base. 184 512] PERCENTAGE 185 512. Since a per cent is a number of hundredths, it may be expressed either as a decimal or as a common fraction. The prin- ciples of aliquot parts may therefore be used to advantage in many operations in percentage and its applications. TABLE PER CENT DECIMAL VALUE FRACTIONAL VALUE PART OF 100%, OR THE BASE PER CENT 1 >EOIMAL VALUE FRACTIONAL VALUE PART OF 100%, OR THE BASE 1% .01 1 100 Ih 221 % .22| 22J 100 | H% .01| 11 100 'So 28$% .28? 28{ 100 i 11% ou ii i 3U/ 31 1 11 A s% 1 100 ** ? /o I 100 16 21% .021 Jl 100 A 331% .331 100 1 31% .031 Ji 100 * 371% .871 100 1 6}% .06^ 100 A 42?% .42? 100 ? 6|% 06f 100 A 48}% .43f 43| 100 A 81% .081 JL 100 A 60% .60 ao 100 1 %% .09^ 100 A 564% .56} 100 A 10% .10 10 100 A 621% .621 100 ! n*% .111 111 100 i 66|% 66f 66| 100 1 121% .121 'a 100 i 681% .68} 68| 100 H 14?% .14? 100 75% .75 76 100 1 16f% .16| Too i 8H% .81} 100 if 18}% .18| 100 A 831% .831 100 i 20% .20 20 100 r 871% .871 871 100 i 25% .25 26 100 3 83}% .931 93| 100 H: 186 PERCENTAGE AND ITS APPLICATIONS {_ 512-516 DRILL EXERCISE What common fraction in its simplest form is equivalent to : 1. 1% ? 5. 20% ? 9, 100% ? 18. 16f % ? 17. 125% ? 8. 4% ? ft 25% ? m 61% ? ^. 331% ? U. 150% ? 3. 5% ? 7. 50% ? ^. 81% ? ^5. 66|% ? m 21% ? 10% ? A 75% ? 12. 121% ? ig. 871% ? m If % ? Express decimally : 21. 28%. . 101%. 27. 6|%. 50. 182%. 22. 35%. 05. 7f%. $>. 250%. 81. 415%. 9. 50%. ft 9|%. m 137%. ^. 106%. Express as a rate per cent : 88. 86. . 57. . *P. -y. ^. - 513. The operations of percentage are based upon the same general principles as the operations of simple multiplication and division, the base in percentage corresponding to the multiplicand in simple multiplication, the rate to the multiplier, and the percentage to the product: Hence, any two of the elements of percentage being given, the other may be found. 514. The formulse for percentage are derived from the funda- mental principles of multiplication and division, as follows : 1. Multiplicand x multiplier = product ; hence, base x rate per cent = percentage. 2. Product -T- multiplicand = multiplier; hence, percentage -+- base = rate per cent. 3. Product *- multiplier = multiplicand; hence, percentage -r- rate per cent = base. 515. To find the percentage, the base and rate being given. 516. Examples. 1. What is 9% of $500? $500 base. nQ SOLUTION. 9% of a number is .09 of it. There- fore, 9% of $500 is .09 of $500, or $45. $45.00 percentage. 516-617] PERCENTAGE 18? 2. What is 121% of $888.80? SOLUTIONS. () 12|% of a num- ber is .126 of it; therefore, 12i% (a) $888.80 x .125%= $111.10. O f $888.80 is .125 of $888.80, or $111.10. Or, (6) .125% of a number is .12$, or (6) \ Of $888.80 = $111.10. i of it; therefore, \ of $888.80, or $111.10, is the required result. 517. Hence, to find a percentage of a number, Multiply the base by the given rate per cent, expressed decimally. Or, Take such apart of the base as the rate per cent is of 100%. DRILL EXERCISE 1, Formulate a short method for finding 8 \ % of a number. SOLUTION. Since 8% of a number is .08| or ^ of it, to find 8J% of a num- ber, divide by 12. 2. Give a short method for finding the percentage when the base is given and the rate is 121%; 16f%; 25%; 331%; 9 T i-%; 28$%. S. Formulate a short method for finding 75% of a number? 18f % of it; 43f % of it; 62|% o f it; 31^% of it. 4. What aliquot part of a number is 93f % of it? 83% cf it? 66f % of it ? 22f % of it ? 871% of it ? 5. Express as a rate per cent : -fa of a number ; of a number ; | of a number ; -| of a number ; -^ of a number ; -^ of a number ; ^j of a number; -fa of a number; f of a number: ^ of a number. 6. Give a short method for finding 1^% of a number. SOLUTION. 1|% of a number is .01$, or -fa of it. Hence, to find 1|% of a number, point off one place and divide by 8. 7. Give a short method for finding 1|% of a number; 2|%; ; 561%; 6 |%; 20%. ORAL EXERCISE By inspection, find the value of : 1. 37i% of 160. 7. 66|%of930. 13. 18|%of480. 8. 62|%of320. 8. 2|% of 360. 14. 31 J% of 320. 8. 8|% of 720. . 9. 331% of 930. 15. 43f % of 160. 4. 6i%of960. 10. 121% of 880. 16 . 56j%of800. 5. 142% of 210. 11. 6|% of 450. 17. 621% o f 240. 6. 25% of 680. 12. 16f % of 666. 18. 75% of 128. 188 PERCENTAGE AND ITS APPLICATIONS [61b SHORT METHODS 518. Examples. 1. What is 36% of $2500? if and since 48 is 10 % of itself, 120, which is 1 of 480, must be 1 of 100%, J of 100% =25%. Or250/o . or, / c \ (c) Since 480 is 100 % of itself, 1 % of 480 would be T&Q part of it, or 4.80. Since 4.80 is 1 %of 480, 120 = *** would be as many times 1 % as 4.80 is contained times 25 times 1 % = 25 %. in 120 ' which is 25 times; 26 times 1% is 25% * 521. Hence, to find the rate, Divide the percentage by the base and express the number of hun- dredths obtained as a rate per cent. 190 PERCENTAGE AND ITS APPLICATIONS [521 f ORAL EXERCISE What per cent of : 1. $50 are $5? 5. 12 da. are 4 da.? 9. .12 are .24? & 15 hr. are 45 hr.? ft -J- bu. is J bu. ? 10. .24 are .12 ? 3. 24 bu. are 48 bu.? 7. yd. is -^ yd.? ^7. 48 hr. are 36 hr.? 4. 15 A. are 5 A.? 8. 2.4 are 3.6 ? jflR $ 160 are $ 40 ? WRITTEN EXERCISE 1. Of a stock of 800 yd. of prints 240 ygl. were sold at one time and 160 yd. at another. What per cent of the whole stock was still unsold ? 2. Of a regiment of men entering battle 1040 strong only 260 ^ came out unhurt, \ of the remainder having been killed. What per ' cent of the whole regiment was killed ? S. Out of 900 bu. of potatoes put in storage October 15, 45 bu. were found unsound April 1. What per cent of the whole was sound ? 4. A merchant's profits for 1903 were $3800, or $200 in advance ^ of his profits for 1902. Find the per cent of increase in the profits of 1903 over those of 1902. 5. From a cask of lard containing 320 Ib. 70 Ib. were sold at one ~ time and 30% of the remainder at another. What per cent of the whole remained unsold ? ft In a certain school there are 1800 male pupils and 200 female pupils. What per cent more are the male than the female pupils ? 7. A merchant failed in business, having resources amounting* to $ 15,000 and liabilities amounting to $ 75,000. What per cent oi his debts can he pay ? 8. What per cent more is than \ ? 9. What per cent less is | than \ ? 10. A has 25% less money than B. What per cent has B more than A Y 522-624] PERCENTAGE 19l 522. To find the base, the percentage and rate being given. 523. Example. 375 is 12%% of what number ? (a) SOLUTIONS, (a) 12$% of a number Is 3 00Q equal to .125 of it. If .125 of a number is -OPN Q 7 f- rvnrv 876, the whole number may be found by the principles of division of decimals (305; Or ( & ) (6) If 12$% of a number is 376, 1% of 12% = 375. the number is & of 375, or 30, and 100% or 1 % == 30. the whole of the number is 100 times 30, or 100$ = 3000. 800 - Or ' ( c ) (c) 12$% of a number is $ of it. If $ 4 01 ... , orr w of a number is 375, 4, or the whole number. 12|%, or I, of a number =375. . g 8 timeg ^ ^^ f = 8 times 375, or 3000. required result is 3000. 524. Hence, to find the base, Divide the percentage by the rate. ORAL ?. 18 is of what number ? 6. Of what number is 12 5% ? & Of what number is 16 25% ? & 555 is 5% of what number ? S. 3215 is 41^% of what number ? 7. 19 is 16f % of what number? 4. Of what number is 125 62% ? 8. 90 is of what number ? P. A man's yearly expenses are $150, or 12^% of his income. What is his income ? WRITTEN EXERCISE 1. 2i% of 240 is 25% of what number? 2. On a bill of $1280 $64 discount was allowed. What was ^ J the per cent of discount ? 3. Jan. 1 1 paid William Mason & Co. 75% of my indebtedness to them by a New York draft for $5100. Jan. 8 they sent me goods amounting to $ 200. Feb. 1 I sent them a check in full of account. What was the amount of the check ? i 6 10. A certain number plus 17% of itself, plus 6% of 3 times itself, is equal to 270,135. What is the number ? MOORE'S COM. AR. 13 194 PERCENTAGE AM) ITS APPLICATIONS [528-530 528. To find the base, the difference and rate of decrease being given. 529. Example. What number, decreased by 35% of itself, equals 1300? SOLUTION Represent the number by 100%. 35 % = the decrease. 65 % = the number after decrease. 1300 = the number after decrease. Therefore, 65 % = 1300. I%= ft of 1800, or 20. 100% = 100 times 20, or 2000, the required result. 530. Therefore the following rule : Divide, the difference by 1 minus the rate. ORAL EXERCISE What number diminished by : 1. 8% of itself equals 184? 4. 25% of itself equals 33 ? 2. 16|% of itself equals 55? 5. 50% of itself equals 27 J ? \ 3. 12|% of itself equals 77? & f of itself equals 339 ? 7. I sold goods for $ 600 and Ipst 40%. What did they cost ? 8. Brown deposited $850 in a savings bank, which was 15% less than that deposited by his son. How much was deposited by both of them ? 9. An agent earned 15% less in May than he did in June. If ^ he earned $ 370 in the two months, how much did he earn in June ? ' WRITTEN EXERCISE 1. A boat load of wheat was so damaged that it sold for $ 8500, which was 15% less than its original value. What was its value before it was damaged ? 2. After paying 37^% of his debts a man found that the remain- der could be paid with $ 13,025. What was his original indebted- ness? 3. Smith sold two horses for $1500 each, gaining 25% on the first and losing 25% on the second. What did the horses cost him ? $30] PERCENTAGE 195 4. A liveryman paid $456 for a horse and carriage* If the cost of the carriage was 48% less than the cost of the horse, what was the cost of each ? 5. A merchant's sales for Wednesday were 50% greater than his sales for Tuesday, which were 20% less than his sales for Mon- day. If the sales for the three days aggregated $1500, what were the sales for each day ? 6. In selling a suit of clothes for $21.60 a merchant lost 20%. Find the asking price if it was 20% above cost. 7. A man bought a watch, and had $ 125 remaining. He then bought another watch costing twice as much as the first, and still had left 14^% of his money. How much money had he at first ? 8. Divide $1600' between A and B so that B shall have 40% less than A. WRITTEN REVIEW 1. A benevolent lady gave $ 10,500 to three charities. To the first she gave $2500, to the second $4500, and to the third the remainder. What per cent did each receive ? 2. 25% of B's money equals 75% of A's. How much has A if B has $900? ^H 3. A creditor, after collecting 33|% of a claim, lost the remain- der, which was $ 3918,75. What sum was collected ? v \ 4- A has 185% more money than B. How much has each if they together have $ 9625 ? 5. The sum paid for two farms was $19,200. 37^% of the sum .paid for one equals 62 J% of the sum paid for the other.. Find the price of each. 6. If a gain of $4775 was realized on a business at the end of the first year, and a loss of $3586.25 was sustained the second year, what was the per cent of net gain or loss for the two years, the investment having been $63,400 ? 7. After making 7 of the 10 annual payments of the face of a mortgage I find $ 5850 to be still unpaid. How many dollars have been paid ? 8. A has 50% more money than B. What per cent has B less than A ? 196 PERCENTAGE AND ITS APPLICATIONS [ 530 9. The population of a certain city is 238,375. During the last three years the population has increased yearly 25%. What was the population three years ago ? ^ 'V'V ^ C ^\ 10. 80% of a mixture of vinegar and water is vinegar. If there were 10 gallons more of vinegar, the mixture would be 85 % vinegar. How many gallons of water are in the mixture ? \s 11. From an estate the widow received $ 9250, which was \ ; the remainder was divided among three children . aged respectively 15, 12, and 10 years, and they share in proportion to their age. , What per cent of the estate did each of the children receive ? .Z 7 3*7 12. From a farm containing 180 acres 120 square rods, 50% was sold at one time and 50% of the remainder at another time- What per cent of the whole then remained ? 18. After drawing 25% of his deposit from a bank to pay a debt a man finds that he has left in the bank $ 6756.25. What was the amount of his indebtedness, and how much had he in the bank before drawing the check? 14. A. man sold two farms for $8000 each, receiving for one 20% more than it cost and for the other 20% less than it cost. Did he gain or lose by the sale, and how much ? 15. During the first, second, and third years a manufacturer realized gains amounting to 25%, 20%, and 45%, respectively, of his original investment. During the fourth year he lost $9000, which was 25% of his original capital. Find his net gain for the four years. 16. A manufacturer's capital was increased during the first year by profits equal to 25% of his original investment ; the second year by profits equal to 20% of his capital at the beginning of that year; and the third year it was diminished by a loss equal to 25% of the capital at the beginning of that year. If his profits during the three years exceeded his losses by $8000, what was his original investment ? 17. In settling an estate an executor found 1\ % of it to be invested in telegraph stock, 15% in railroad stock, 37|% in United States bonds, $16,750 in real estate, and $7350 cash in bank. Find the total value of the estate. 530] PERCENTAGE 197 18. A horse is worth 25% more than a carriage, and the carriage is worth 300% of the harness. If the horse is worth $37.50 more ' than the carriage, what is the value of each ? 19. A merchant sold 2 horses for $ 140 each. On one he gained 25% and on the other he lost 28-f-%. How much did the horses cost him, and what was the gain or loss ? 20. Express .00025 as a per cent; -^% as a decimal; 36.42 as a per cent. 21. Find 3% of 9 t. 7 cwt. 16 Ib. *i&& I paid for transportation on an invoice of goods $600. I ^/ later sold the goods at a profit of 20% on the full cost, receiving $9696.60. What was the first cost of the goods? What per cent was the value of the goods increased by the transportation charges ? 28. A last will and testament provided that f of an estate dis- tributed should go to the widow and the remainder be so divided among two sons and a daughter that the elder son should receive^. if >> L 10% more than the younger, who should receive 25% more than the?' ^ daughter. W^hat amount was received by each, the estate being valued at $58,000? 24. A father located his son upon a farm, expending for the farm, stock, utensils, and household furniture $19,512.50. The stock cost twice as much as the household furniture, which cost 75% more than the farm utensils, and the cost of the farm was 140% of the cost of the stock. How much was invested in each ? 25. A creditor agrees to receive $ 962.50 for the full amount of a debt. If this settlement is at the rate t)f 25 f on the dollar, what was the original amount of the debt ? 26. A manufacturer failing in business finds that his net resources aggregate $12,600. If he can pay 75 f on the dollar on 20% of his debts, and 60^ on the dollar on the remainder, what is the amount of his indebtedness. 27. Twice ^ of a number is what per cent of 3 times J of it ? 28. A farm is composed of 20% more grazing than grain land, and the timber is 1 of the area. How many acres of each are there if after deducting 12 acres for lawn and garden the area of the farm is 1860 acres ? PERCENTAGE AND ITS APPLICATIONS [ 530-537 29. A man withdrew 25% of his bank deposit and spent 20% of the money to pay for 20% of his indebtedness to Smith & Co. If his indebtedness to Smith & Co. was $1800, what was the original amount of his bank account ? SO. A merchant mixed 100 Ib. of coffee at 25^ per pound with 50 Ib. at 30^ per pound, and sold the mixture for 40/. What per cent of profit does he make ? 531. Applications of percentage. Percentage is applied to two general classes of problems: 1. Those in which time is not an element ; as, Commercial Dis- counts, Gain and Loss, Commission, Insurance, Taxes, and Customs, or Duties. 2. Those in which time enters as an element ; as, Interest, Bank Discount, Present Worth and True Discount, Equation of Accounts, and Exchange. COMMERCIAL DISCOUNTS 532. Discount is an allowance made for the payment of a debt before it becomes due. 533. Commercial discounts are reductions from the fixed or list prices of articles, the amount of a bill of merchandise, or of any other obligation. 534. Commercial discounts embrace trade discounts, time dis- counts, and cash discounts. 535. Trade discounts are reductions from the fixed or list prices of articles. 536. Time discounts are reductions from the amount of a bill of merchandise for payment within a definite time. 537. Cash discounts are reductions made for the immediate pay- ment of a bill of merchandise sold on time. Business houses usually announce their terms upon their billheads; as Terms : 3 months, or 5 % off for cash ; " Terms : 60 days, or 3 % discount in 10 days ; " etc. When bills are paid before maturity, legal interest for the remainder of the time is usually deducted. 537-543] COMMERCIAL DISCOUNTS 199 Trade discounts are deducted from the list price when goods are billed. Time discounts are deducted when a bill is paid. Cash discounts are deducted from the amount of the bill when the sale is made. 538. It is customary for manufacturers, jobbers, and wholesale dealers to have fixed price lists for their goods. Trade discounts are usually made to obviate the necessity of changing these price lists from time to time as the market changes. As the market varies, instead of changing their price lists or issuing new cata- logues, merchants raise or lower their rates of discount. 539. The fluctuations of the market sometimes give rise to two or more discounts known as a discount series. If two or more dis- counts are quoted, the first denotes a discount off the list price, the second off the remainder, and so on. 540. The list price is called the gross price, and the price after the discount has been deducted, the net price. 541. Commercial discounts are usually computed by the rules of percentage, the list price or the amount of the bill or debt corre- sponding to the base, the per cent of discount to the rate, the discount to the percentage, and the net price or net amount of the bill or debt to the difference. 542. To find the net selling price, the list price and discount series being given. 543. Examples. 1. Find the net amount of a bill of $ 450 after a discount of 33^% is made. SOLUTION $450 = the list price. 83i%, or , of |450 = $150, the discount allowed. $450 $150 = $300, the net amount of the bill. The list price of a piano is $ 800. What is the net price if a discount series of 25% and 20% is allowed ? SOLUTION $800 = the list price. 25%, or , of $800 = $200, the first discount. $800 - $200 = $600, the remainder after the first ditooimt 20%, or $, of $600 = $120, the second discount. f 600 - $ 120 = $480, the net price. 200 PERCENTAGE AND ITS APPLICATIONS [544 544. Therefore the following rule : Deduct the first discount from the list price, and each subsequent discount from each, successive remainder. The last remainder is the net selling price. The order in which the discounts of any series are considered is not material, a series of 25%, 15%, and 10% being the same as one of 15%, 10%, and 25%, or 10%, 25%, and 15%. ORAL EXERCISE 1. Find the net cost of a piece of glass listed at $ 3.60 and dis- counted 25%. 2. A merchant sold a bill of goods amounting to $ 4.50 on which he allowed 20% discount. What was the net amount of the bill ? 3. Find the net amount of a bill of $ 18, the discount being 11^%. 4. What is the net amount of a bill of $ 450, the discounts being 33|% and 20% ? 5. A piano listed at $450 is sold less 33^% and 10%. What is the net cost to the purchaser ? 6. Goods listed at $27 are sold less 331% and 16f %. What is the net selling price ? WRITTEN PROBLEMS Find the net amount of the following bills : 1. $ 1550 less 331% and 20%/ : ' 8. $3500 less 20% and 14* %. 2. $840 less 25% and 10%. 9^ 4. $395 less 20% and 20%. i 5. A wholesale dealer offers cloth at $ 2.40 per yard subject to a discount of 25%, 20%, and 5%. How many yards can be bought for $492.48? 'SWA" 6. Find the net price of 2 tons of fence wire listed at 3^ per pound and sold 20% and 25% off. 7. One drummer offers to sell me $1500 worth of iron pipe at a discount of 25%, 10%, and 10% ; another offers to sell me a similar 'quantity of pipe for the same amount less 20%, 20%, and 5%. Which is the better offer, and what is the difference expressed in dollars ? 8. Having bought $1500 worth of merchandise at 20% and 25% off, I sold it for $1500 less 15%, 10%, and 20% off. Did I gain or lose, and how much ? 544-540] COMMERCIAL DISCOUNTS 201 9. Books purchased at 25% and 20% off from the list price were sold at the list price. What was the gain per cent? What was the cost of a shipment which sold for $700 ? 10. A bill of hardware is sold as follows: $25.50 at 20% ; $4.50 at 20% and 25%; $153 at 33^% and 10%; $267.50, net. If a further discount of 2% is allowed for immediate payment, what is the net amount of the bill ? 545. To find the net amount of a bill to render, the terms and dis- count series being given. 546. Ex-mple. Bender a bill for the following transaction: Feb. 18, 1903, E. W, Wells, Medford, Mass., bought of Baker, Tay- lor & Co., Boston, Mass.: 1000 ft. iron pipe at 25^ less 20% and . Boston, Mass.,. Avv 564. To find the cost, the gain or loss and the per cent of gain or loss being given. ORAL EXERCISE Find the cost if : 1. 25 % loss = $ 30. 4. % % loss = $ 100. 7. 15 % gain = $ 150. 2. 20 % gain = $ 1.50. 6. \ % gain = $ 30. 8. 14$% gain = $ 12. 3. 30% loss = $2.10. 6. 125% gain = $3.75. 9. 22% gain = $880. 10-18. Find the selling price in each of the above problems. 19. What must have been the cost of a stock of goods if the owner, by selling at a gain of 12%%, received $450 more than the cost? WRITTEN EXERCISE 1. A dealer sold 35% of a purchase of leather at 14f % gain and the remainder at 5% loss. If his net gain was $87.50, what must have been the cost? ~ 2. A merchant bought goods and paid freight on them equal to 12% of their cost. He then sold them at 6J% profit on the full cost of the goods, receiving 60% of the price in cash, and a note for the remainder. If the amount of the note was $1309, what was the first cost of the goods ? f 1. f&t 3. A dry goods merchant's gain in business for four years aggre- gated 50% of his capital. If his gain was $5000 and he withdrew it and his capital and invested the total in a farm, consisting of 375 A., what was the price paid per acre ?* t- 4. Having bought a house of A at 12%% less than it cost him, I spent $430 for repairs and sold it for $7293, thereby gaining 10% on my investment. How much did the house cost A ? 5. A man sold a horse at 33^% profit. He put with the sum received $50 and bought a piano, which he sold at 20% gain. If his total gain was $ 100, what was the cost of the horse ? MOORE'S COM. AR. 14 210 PERCENTAGE AND ITS APPLICATIONS [565 565. To find the cost, the selling price and the per cent of gain or loss being given. ORAL EXERCISE Find the cost when the selling price at : 1. 5 % gain = $ 105. 4. 125 % gain = $ 225. 7. 140 % gain = $ 480. 2. 20% gain = $240. 5. 12^% loss = $140. 8. 14f% loss = $2400. 3. 16|%gain = $14. 6. 33^% loss = $360. 9. 111% loss = $3200. 10-18. Find the gain or loss in each of the above problems. 19. A fruit dealer, after losing 12^% of his apples by frost, had 150 barrels left. If he bought the apples at $ 2 per barrel and sold at $ 3, what was his gain ? WRITTEN EXERCISE 1. I sold a house to B at 10% profit. B sold it to C, gaining 15%, and C, by selling it to D for $15,939, gained 20% on his pur- chase. How much did the house cost me ? 2. I sold two watches at the same price. On one I gained 25%, and on the other I lost 25%. If my total loss was $10, what was the cost of each ? 3. A sold a stock of silks to B at a gain of 25% ; B sold the same stock to C at a gain of 10%.. If C's cost was $375 more than A's, what did the silks cost A ? WRITTEN REVIEW 1. What amount of money must an attorney collect in order that he may pay over to his client $1700 and retain 15% for his services ? *< 2. In selling an article for $162 an art dealer cleared 12%. At what per cent above cost was it marked if the asking price was $176? 3. A man bought a quantity of apples at $2 per barrel. He sold \ of them at $3 per barrel, -| of the remainder at $3.25 per bar- rel, and the remainder, 750 bbl., at $2.50 per barrel. What was his gain? 4. By selling apples at $2.50 per barrel I gained $200. Had I sold them at $ 2.75 per barrel my rate of gain would have been 37 ', %. How many barrels did I sell ? -^ 565] GAIN AND LOSS 211 5. A dealer bought wheat at 90 ^ per bushel. He sold f of it at 33 \/o gain and the remainder at a loss of $25. If his gain on the whole transaction was 22f %, how many bushels of wheat did he buy? 6. What per cent of gain must be realized on an engine costing $ 1928 in order that it may be sold for $ 2410 ? t 6" % 7. A produce dealer bought 24,000 Ib. of wheat for $360. He sold it at $1.05 per bushel. What was his gain per cent ? J& % 8. A compromised with an insolvent debtor at the rate of 50^ on a dollar. To obtain an immediate payment he allowed a further discount of 5%. What was the amount of his claim, his total loss having been $ 10,505.25 ? \s> 9 Dt 9. An article marked to gain 621% is sold less 25% and 20%. If a collector was afterwards paid 20% for collecting the account, what was the gain or loss per cent ? 10. An article marked 25% above cost is sold at a discount of 16|%. Jf the gain is $25, what is the selling price of the article ? 11. If I made a profit of 16|% by selling a horse at $7,50 above cost, how much should I have received above cost to realize a profit of 25% ? 12. What per cent is gained by buying pork at $17.50 per bar- rel, and retailing it at 12 ^ per pound ? IS. Having bought 75 barrels of apples for $ 187.50, I sold them at a loss of 20%. How much did I receive per barrel ? 14. I lost 25% of a consignment of berries. At what per cent of profit must the remainder be sold in order that I may gain 10 % on the whole ? 15. A Texas farm of 160 acres was bought at $15 per acre; $ 354 were paid for fencing, $ 480 for breaking, $ 626 for a house, and $220 for a barn. At what price per acre must it be sold to realize a net profit of 25% on the investment ? 16. If 25% of the selling price is gain, what is the gain per cent? 17. I sell | of a stock of goods for $27, thereby losing 20%. For what must I sell the remainder to make a profit of 20% on the whole ? 212 PERCENTAGE AND ITS APPLICATIONS [565 18. A banker bought a mortgage at 7\/o less than its face value, and sold it for 3% more than its face value, thereby gaining $981.75. What was the face value of the mortgage ? 19. If I sell J of an acre of land for what |- of it cost, what will be my gain or loss per cent? |U*M J^ 20. B and C each invested an equal amount of money in busi- ness; B gained 121% on his investment, and C lost $5275; C's money was then 42% of B's. How many dollars did each invest ? 21. A manufacturing company's per cent of gain on a self- binder was 25% less than that of the general agent; the general agent's profit was 20%, he thereby gaining $25.30. What did it cost to make the machine? 22. For what must hay be sold per ton to gain 16f %, if, by sell- ing it at $ 18 per ton, there is a gain of 25% ? \ ^ . J V 28. A stock of goods is marked 221% advance on cost, but becoming damaged, is sold at 20% discount on the marked price, whereby a loss of $ 1186.40 is sustained. What was the cost of the goods ? 24. Of a cargo of 8000 bushels of oats, costing 35 ^ per bushel, 25% was destroyed by fire. What per cent will be gained or lost if the remainder of the oats is sold at 45 ^ per bushel ? ' 25. A grocer bought 200 quarts of berries at 11 \$ per quart, and 150 quarts of cherries at 6 \ $ per quart. Having sold the cher- ries at a loss of 30%, for how much per quart must he sell the berries to gain 15% on the whole? 26. Having bought 48 pounds of coffee at the rate of 3J- pounds for 91 ^ and 84 pounds more at the rate of 7 pounds for $ 1.26, I sold the lot at the rate of 9 pounds for $1.53. What was my per cent of gain or loss ? | x^% d^r&O 27. Having paid a retailer $138.60 for a set of furniture, I ascertain that by selling to me he gained 12|%, that the whole- saler of whom he bought gained 10%, that the jobber by selling to the wholesaler gained 16 f %, and that the manufacturer sold to the jobber at 20% above its first cost. How much more than its first cost did I pay ? 565] GAIN AND LOSS 213 28. If I pay $3.20 for 20 gallons of vinegar, how many gallons of water must be added that 40% profit may be realized by selling it at 15^ per gallon? 29. A merchandise account shows that the cost of a stock of goods was $15,000, that the sales to date aggregate $12,000, and that the goods on hand, estimated at cost prices, amount to $ 4500. Find the per cent of gain or loss on the sales. 30. A sold a horse to B and gained 20% ; B sold it to C and gained 25%. If the average gain was $50, what was C's cost ? 81. A grocer buys 10 barrels of apples, each barrel containing 2-J- bushels at $2 per barrel. If the loss by decay amounts to 20%, at what price per peck must he retail them in order to clear 20% ? 32. A grocer mixes 10 pounds of tea costing 36 ^ per pound with 8 pounds costing 45 f per pound. At what price must he sell the mixture to gain 25% upon his outlay ? S3. A stock of- imported silks bought for 200 was sold for $ 1167.96. What was the gain per cent ? 34. What per cent is gained in buying coal at $4.50 for a long ton and retailing it at $ 6 a short ton ? 35. What per cent is gained on quinine costing $2.90 an ounce and sold at 2 $ a grain ? 36. A merchant buys hardware at 25% and 20% off the list prices, and sells at 20% and 10% off the list prices. What per cent of gain does he realize ? 87. A manufacturer sells at 20% and 10% off the list prices. His blundering clerk, in making out the bill for the goods, deducted .30%. If the discount deducted was $450, what should have been the net amount of the bill ? If the mistake passed unnoticed, what was the buyer's gain per cent on the transaction ? 38. A dealer in agricultural implements marked a self-binder at an advance of 25% on the cost. In order to collect an account, he had to pay an attorney 10% of the amount of the debt. If the sale netted him a gain of $25, what was the selling price of the binder ? 214 PERCENTAGE AND ITS APPLICATIONS [ 565-569 89. A retailer buys collars at the rate of $ 1 a dozen, and at the rate of 2 for 25^. What is the per cent of gain ? *C^) 40. A manufacturer sold a retailer a piano and gained 20%. The retailer compromised with his creditors, paying 75^ on the dollar. What was the manufacturer's per cent of loss on the trans- action if he discounted the amount which he could legally collect o% for immediate payment? 41. An insolvent debtor pays his creditors 37^ on the dollar. If his creditors receive $ 3515, what is their joint loss ? 42. A manufacturer sells to a wholesaler at 20% gain; the wholesaler to the retailer at 25% gain; and the retailer to the consumer at 60% gain. Find the cost to the manufacturer of an article for which the consumer pays $ 2.40 more than twice the cost to manufacture. MARKING GOODS 566. In marking goods, it is customary for merchants to use a word, phrase, or an arbitrary arrangement of characters called a key to represent the ten Arabic numerals. In this way the cost and selling price may be written on an article and yet be unintelligible to all except those who know the key. 567. If letters are used, any word or phrase containing ten different letters may be selected. If characters are used, any ten different arbitrary characters may be selected. 568. Merchants generally use two different keys, one to repre- sent the cost and one the selling price. 569. To avoid the repetition of a letter, and to make the key more valuable as a private mark, one or two extra letters called repeaters are used to indicate letters which would otherwise be repeated. To illustrate the method of marking goods, take the following keys : Cost Mark Selling Price Mark SEZIBOHTUA HTIMSKCALB 1234667890 1284667890 Repeaters : G and F. Repeaters : W and O. 569-573] MARKING GOODS 215 It will be observed that the words authorizes and blacksmith, spelled back- wards, are used to represent the cost and selling price respectively ; also that the repeaters in both cases are mere arbitrary letters. The cost and selling price are generally written one above and the other below a line on a tag, or upon a paster or box. Thus, if the above keys are used, * ^ , on a dozen of gloves, would be $ C.oB understood to mean that the cost was $6.00 per dozen, and that the selling price is $7.50 per dozen. WRITTEN EXERCISE Using the keys given in 569, write the cost and selling price of articles costing : 1. $2.50 and selling at 20% gain ; $2.40 and selling at 25% gain. 2. $1.80 and selling at 33% gain ; $4.20 and selling at 16f % gain. 3. 18^ and selling at 16f % gain ; $27 and selling at 30% loss. 4. $4.26 and selling at 16f % gain; $3.60 and selling at 12^% gain. 5. $425 and selling at 20% gain ; $24.90 and selling at 33% gain. 6. $16 and selling at 31^% gain; $2.40 and selling at 37% gain. 570. While the unit of measure varies with the quantities and qualities offered for sale, a large number of manufactured products are sold by the dozen. Jobbers and wholesalers buy a great many articles by the dozen. Retailers buy a great many articles by the dozen, but usually sell them by the piece. 571. To find the cost price of an article, the cost price of a dozen being given. 572. Example. If 1 doz. hats cost $ 25, what will 1 hat cost ? SOLUTION. Since a dozen hats cost $25, 1 hat will cost T ^ of $25, or $2^, which is equal to .$2.08$ or $2.08. It is just as easy to divide a number by 12 as it is by any number of one digit. Hence, in dividing by 12, use the short division method. After dividing the figures in the dividend, consider the remainder as twelfths of a number, and mentally reduce it to an approximate decimal. 573. The following table shows the decimal values of the twelfths which may remain after dividing a number by 12. 216 PERCENTAGE AND ITS APPLICATIONS 573-676 TABLE OF TWELFTHS TWELFTHS SIMPLEST FORM DECIMAL VALUE TWELFTHS SIMPLEST FOBM DECIMAL VALUE A $.08$ A $.58| A i .16| A 1 .66| A i .25 A I .75 A i .33$ u i .88} A 41f H -91| A * .5 H i 1. Familiarity with the above table will give facility in dividing numbers by 12. ORAL EXERCISE By inspection find the cost of one article when billed by the dozen as follows : 1. Shoes at $18.60. 5. Ties at $9. 9. Hose at $3.90. 2. Boots at $42. 6. Coats at $116. 10. Gloves at $ 3.90. 8. Hose at $6.60. 7. Scarf s at $ 1.32. 11. Hose at $5. 4. Hats at $27. 8. Shirts at $11.60. 12. Caps at $ 16.90. 574. To find the selling price of an article, the cost per dozen and the rate per cent of gair being given. 575. Example. Find the selling price of a pair of gloves so as to net a profit of 33^%, the cost per dozen being $7.50. X $7.50 = $2.50. $7.50 + $2.50 = $10. SOLUTION. Since 33 J%, or $, of the cost is profit, the selling price per dozen will be \ more than $ 7.50, or $ 10. $ 10 divided by 12 $ 10 -J- 12 = $ f = 83 equals 83^, or the selling price per pair. 576. Therefore the following rule : Find the selling price of one dozen "by adding to the cost of one dozen such a part of the cost as the rate per cent of gain is a part of 100%. Find the cost per article by dividing the cost per dozen by 12. 576-579] MARKING GOODS 217 WRITTEN EXERCISE Find the selling price per article: Gain to be realized Cost per dozen 1. $15.00 2. $ 9.00 25% 8. $27.50 6J% 4. $28.00 16|% 5. $ 4.00 6. $37.50 Cost per dozen Gain to be realized 20% 7. $7.50 18|% 8. $25.25 20% 9. $17.50 20% 10. $18.90 331% 11. $1.50 16f% 12. $9.60 66} % Cost per Gain to be dozen realized 13. $36.00 25% 14. $116.00 121% 15. $35.50 20% 16. $29.70 17. $28.00 18. $56.00 28^% 577. To find the price at which goods must be marked to insure a given per cent of gain or loss, the cost and discount series being given. 578. Example. A seal sack cost a manufacturer $ 240. At what price must it be marked in order that a discount series of 25% and 20% may be allowed and a gain of 33^% be realized ? SOLUTION Let 100% represent the marked price. A discount series of 25% and 20% is equal to a direct discount of 40%. 100% - 40 = 60%, the amount to be realized. $240 = the cost of the sack. 33|, or | of $240 = $80, the amount to be gained. $240 + $80 = $320, the amount to be realized. Therefore, 60% =$320. l% = $5.33f 100% = $533.33, the marked price. 579. Therefore the following rule : Add the required gain to or subtract the required loss from the cost and divide by 1 minus the rate of discount. WRITTEN EXERCISE 1. What must be the asking price of a watch costing $ 24 in order L - to insure a gain of.33J% and allow the purchaser a discount of 20%? 2. After buying lace at $8 per piece, I so marked it as to allow discounts of 25% and 20% from the marked price, and yet so sell it as to lose but 10% on my purchase. At what price per piece was the lace marked ? 218 PERCENTAGE AND ITS APPLICATIONS (_ 579 8. The cost of manufacturing silk ties being $ 36 per dozen, how much must they be marked that a gain of 16f % ma y be realized by the manufacturer after allowing a discount of 25% and 12i% ? 4. If a carriage be marked 33^% above cost, what per cent of discount can be allowed from the marked price and realize cost ? 5. If the list price of an article is 40% advance on the cost, what other per cent of discount than 14^ % must be allowed to net by the sale? _ WRITTEN REVIEW 1. W. A. Briggs & Co. bought of B. A. Altman & Son invoice of silk hats at $100 per dozen, less 30% and 10%. . What price per hat must be asked in order to gain 33^% ? 2. If goods are retailed at an advance of 25%, what is the sell- ing price per article of goods costing by the dozen as follows : shoes, $48; boots, $38.40; rubbers, $8.16? 8. Briggs, Slote & Co. imported hosiery and knit goods costing per dozen as follows: Hosiery, $2.40, $2.50, $3.84, and $4.80; knit goods, $7.20, $4.08, $16.32, and $10.40. Determine the sell- ing price per article, goods to be sold at a gain of 25%. 4. Using the word handsomely with repeater R for the buying key, and the words black horse with repeater W for the selling key, mark the cost and selling price for the articles in problem 3. 5. What should be the marked price per article of the following goods so as to gain 33^% and allow discounts of 10% and 10% ? Hats, per dozen, $16.20; gloves, per dozen, $8.10; ties, per dozen, $4.05. 6. At what price should the following goods be marked per article so as to allow discounts of 25% and 20% and still net a gain of 331% on the cost ? Hats, $ 9, $ 6.30, $ 15, $ 21 ; gloves, $ 10.80, $12.60, and $8.10, - !.ir 7. What price each must be asked for cocoanuts costing $ 4 per C that an allowance of 16f % for breakage, 20% for decay, and 11|% for bad debts may be made, and still a gain of 33^% be realized ? 8. Having paid 40^ per pound for tea, at what retail price must it be marked that I may allow 12|% for bad debts and gain on the cost ? 579-58C] COMMISSION 219 9. A publisher's prices are 75% above cost. If he allows his agent a commission of 20% and sells at a discount of 10%, what per cent of gain does he make ? 10. What price per pound must be asked for coffee costing 18^ per pound, in order that the seller may deduct 10% from the asking price for bad debts, allow 16f % for loss in roasting, and still gain 20 % on the cost ? COMMISSION 580. Commission is a compensation allowed by one person, called the principal, to another, called the agent, for the transaction of business. It is usually a percentage of the money involved in the transaction. < Thus, in the purchase of goods, it is usually a percentage of the prime cost ; in the sale of goods, a percentage of the gross selling price j in the collection of a debt, a percentage of the amount collected. 581. The person for, whom business is transacted is called the principal, and the person authorized to transact business for another, the agent, broker, commission merchant, or collector, according to the nature of the business transacted. 582. A quantity of goods sent away to be sold on commission is called a shipment ; a quantity of goods received to be sold on com- mission is called a consignment. The person who sends the goods is called the consignor, and the person to whom they are sent, the consignee. 583. Guaranty is a percentage charged by an agent for 'assuming the risk of loss from sales made by him on credit, or for giving pledge of the grade of goods bought. 584. Account sales is- an itemized statement rendered by an agent to his principal, showing in detail the sales of goods and charges thereon, together with the net proceeds remitted or credited. 585. Account purchase is an itemized statement rendered by a purchasing agent to his principal, showing the quantity, grade, and price of goods purchased, and all expenses incurred, together with the gross cost of the transaction. 586. The gross proceeds of a sale or collection is the total amount received by an agent. 220 PERCENTAGE AND ITS APPLICATIONS [ 587-589 587. The net proceeds is the amount remaining after commission and all other charges have been deducted. 588. Computations in commission are performed in accordance with the general rules of percentage, the gross selling price, or the prime cost, corresponding to the base; the per cent of commission, either for buying or selling or for guaranty of quality or credit, to the rate; the commission to the percentage ; the total cost of the goods bought by a purchasing agent to the amount; and the proceeds to the difference. DRILL EXERCISE 1. Given the amount and the rate, how do you find the base ? percentage ? difference ? why, in each case ? 2. Compare the general principles governing commission with those governing abstract percentage. 8. Give the five necessary formulae for performing the operations in commission. 589. To find the commission, the cost or selling price and per cent of commission being given. DRILL EXERCISE 1. Give a short method for finding 10% commission; 2-J% commission. SOLUTION. 10% is -fa of 100%, the whole of a number. Hence, to com- pute commission at 10%, find T V of the base by pointing off one place to the left. 2% is | of 10 %. Hence, to find a commission of 2|%, point off one place to the left and divide by 4. 2. Give a short method for calculating a commission of 3i% ; ; 25%; 331%; 12*96; 6f% ORAL EXERCISE By inspection find the commission in the following problems : Gross Selling Rate of Prime Rate of Price Commission Cost Commission 1. $945.80 10% 7. 9 8480.80 25% 2. $724.80 5% 8. $1200.00 7% 3. $440.40 4% 9. $1500.00 8% 4. $780.20 5% 10. $2500.00 14% 5. $750.60 33J% 11. $2500.00 16% 6. $225.00 3% 18. $2978.95 10% 589-590] COMMISSION 221 WRITTEN EXERCISE 1. A real estate agent sold a farm of 90 acres at $ 125 per acre on a commission of 2%. What was the amount of his commission ? How much did he turn over to his principal ? fyrft}. .2^^ 2. An agent sold 450 barrels of flour at $ 6.25 per barrel on a commission of 3J%. What was his commission? S. A collector succeeded in collecting 80% of a doubtful account of $1500. If he charged 1%% commission, how much did he turn over to his principal ? & / ) ) 4. My Chicago agent buys for me 4500 bushels of wheat at 83J^ per bushel. How much should I remit him to cover the cost of the wheat and his commission of 5/ ? 590. To find the rate of commission, the commission and gross sailing price or prime cost being given. ORAL EXERCISE Find the rate of commission in each of the following examples : Gross Selling Price Commission Prime Cost Commission 1. $750 $7.50 6. $105 $35 5. $216 $6.48 7. $2400 $60 3. $135 $4.04 $125 $25 4. $150 $5.00 9. $920 $23 5. $2500 $50.00 10. $200 $24 11-15. Find the net proceeds in problems 1-5 inclusive. 16-20. Find the gross cost in problems 6-10 inclusive. WRITTEN EXERCISE 1. A lawyer collected a note of $ 2500 and paid to his principal $ 2437.50. What was his rate of commission ? 2. A commission merchant sold a consignment of 1200 barrels of beef at $ 14.50 per barrel. After deducting $ 80 for freight, $ 20 for storage, and his commission, he remits his principal $ 16,952 as the net proceeds of the sale. What was his rate of commission ? f*,r \ v) 222 PERCENTAGE AND ITS APPLICATIONS [591 591. To find the investment or gross sales, the commission and per cent of commission being given. DRILL EXERCISE 1. Formulate a short method for finding the gross sales when the commission is given and the rate is 10% ; 1%%. SOLUTIONS. 10% equals ^ of a number. Hence, if the commission at 10% is given, the gross sales may be found by multiplying the commission by 10, or hy removing the decimal point one place to the right. 7% increased by ^ of itself equals 10%. Hence, to find the gross sales when the commission at 7% is given, remove the decimal point one place to the right and add J. 2. Formulate a short method for finding the gross sales when the commission is given and the rate is 2J% j 3J% ; If % ; 25% ; ; 16f% ; ' ORAL EXERCISE Find the prime cost when : Find the gross sales when : 1. 5% commission =$27.50. <>$* 7. 7%% commission =$90. 2. 2% commission =$22.50. 8. 16f% commission =$150. 3. 3J% commission=$ 14.20. [ 9. &|% commission ==$11. 4. If % commission =$15.50. 10. 25% commission = $140. 5. 16|% commission=$ 75.50. 11. 1J% commission=$110. 6. 71% com mission =$75.75. 12. 16% commission =$ 640. 18-18. Find the gross cost in problems 1-6 inclusive. 19-24- Find the net proceeds in problems 7-12 inclusive. WRITTEN EXERCISE 1. What must an agent's sales for one year aggregate in order that at >% commission his yearly income may be $2700? 2. A Mobile factor earned $99.75 by selling cotton at 2|% com- mission. How many bales, averaging 560 pounds, did he sell, the price being 15 ^ per pound ? 591-593] COMMISSION 223 3. I paid a grain dealer 1^% for buying corn for me at 62^ per bushel. If his commission amounted to $ 83.70, how many bushels did he buy ? 4. An agent charged $433.60 for selling a consignment of canned fruit. If his rate of commission was 2^%, what was the net proceeds ? 592. To find the investment and commission when both are included in the remittance by the principal. 593. Example. I sent my agent $1025 with instructions to deduct his commission of 21% and invest the balance in wheat. How much did he invest, and what was his commission? SOLUTION Represent the actual investment by 100%. 2% = the charges for buying. 100% + 2f % = 102 %, the cost of the investment to the principal. $ 1025 = the cost of the investment to the principal. Therefore, 102 % = $1025. 1% = $10. 100% = $1000, the actual investment in wheat. $1025 $1000 = $25, the commission for buying. ORAL EXERCISE By inspection find the amount to invest and the commission in each of the following problems : Rate of Rate of Rate of Amount Re- Commis- Amount Re- Commis- Amount Re- Commis- mitted sion mitted sion mitted sion L $1030 3% 4. $315 5% 7. $515 3% 2. $105 5% 5. $624 1% 8. $410 2%% 3. $550 10% 6. $205 21% 9. $2075 3% 10. I sent my agent a certain amount with which to buy silks, after deducting his commission of 3%. If his commission was $30, what was the amount of my remittance ? / 3 224 PERCENTAGE AND ITS APPLICATIONS [593 WRITTEN SXERCISE L How many pounds of wool, at 27 ^ per pound, can be bought for $8424, if the agent is allowed 4% for purchasing? #. I remitted $1306.45 to a Boston agent for the purchase of soft hats. If the agent's commission is 4%, and he makes an added charge of 2% for guaranty of quality, how many dozen hats, at $8.50 per dozen, should he send me ? \ (L ^ 8. A city merchant remitted his country agent Jfl> 1093.60 with which to buy butter. If the agent's charges were 3% commission, 5% guaranty, ana $13.60 for inspection, how many pounds, at 25 per pound, did he buy, and what was his commission ? 4- I remitted $300 to an agent for the purchase of hops. If the agent's charges were 5% for purchase and $6 for inspection, how many pounds at 16^ per pound ought he to buy ? ^ j[)j nil ORAL REVIEW 1. If you sell books to the amount of $240 on 33|% commis- sion, what amount do you earn, and what is the net proceeds of the sale? 2. I send you $ 205 with instructions to expend it for wheat at $ 1 per bushel, after retaining your commission of 2|%. How many bushels of wheat will you be able to buy, and what will be your commission ? 8. If I -remit $95 as the proceeds of an account collected by me, how much have I retained, my rate of commission being 5% ? 4- What is the amount of sales when the net proceeds are $ 975 and the commission 21% ? 5. An agent charges 5% commission and receives $250. Find the net proceeds of the consignment. 6. A manufacturer sent his purchasing agent $ 510 with which to buy leather, after deducting his commission. If the agent received $ 10 for his services, what was his rate of commission ? WRITTEN REVIEW 1. Eule a sheet of paper, copy the following account sales, | and make the necessary extensions, etc. : 693] COMMISSION ACCOUNT SALES BOSTON, MASS., Feb. 23, 1904. Sold for the Account of E. W. HARDEN, Worcester, Mass. BY E. A. REED & Co., COMMISSION MERCHANTS. 1904 Jan. 25 100 bbl. S. P. Flour 6.75 28 150bbl. R. P. Floor 6.60 Feb. 18 200 bbl. S. P. Flour 6.80 20 100 bbl. S. P. Flour 6.00 CHARGES Jan. 15 Freight, $135 Cartage, $26 Feb. 20 Storage, $16.50 Insurance, $6.20 23 Guaranty, 1% Commission, 5% Net proceeds, 2. Prepare an account sales under date of Feb. 24 for 5000 bu. of wheat, sold by E. L. Hardy & Co., Boston, Mass., for the account of Welsh Bros. & Co., Springfield, Mass. Sales : Feb. 1, 500 bu. at $1.02; Feb. 15, 1000 bu. at $1.08; Feb. 19, 500 bu. at $1.05; Feb. 22, the remainder at $1. Charges: freight, $95; cartage, $18; storage, $17 50; insurance, \% 5 guaranty, 1% ; commission, 2%. 8. Rule a sheet of paper and copy the following account pur- chase, making the necessary extensions, etc. ACCOUNT PUBCHASE BOSTON, MASS., Feb. 28, 1904. Purchased by F. B. BEBBIMAN & Co., For the Account and Risk of E. L. BROWN, Paterson, N.J. 3 4 5 8 half-ch. G. Tea, 165 Ib. 34^ half-ch. O. Tea, 240 Ib. 41 j* half-ch. J. Tea, 350 Ib. 23^ mats J. Coffee, 600 Ib. 24 ^ CHARGES Cartage $7.90 Commission, 2 % Amount charged to your account, 226 PERCENTAGE AND ITS APPLICATIONS [590 4- In accordance with the foregoing form prepare an account purchase of tea purchased by W. L. Jordan & Co., Feb. 23, for the account and risk of Adams, Kand & Co. Purchases : 10 half-chests J. Tea, 600 lb., at 38^; 5 half-chests 0. Tea, 250 lb., 55^; 5 cases C. Tea, 250 lb., at 55^; 8 half-chests E. B. Tea, 480 lb., 45^. Charges : cartage, $7.50; commission, 2%. s 5. I place a claim of $2580 in the hands of an attorney for collection. If the debtor is a bankrupt having liabilities aggregating $18,000 and resources aggregating $13,500, how much should I receive after my attorney has deducted his commission of 2 % ? x 6. A collector obtained 75% of the amount of an account, and after deducting 12% for fees remitted his principal $495. What was the amount of his commission ? 7. A Hartford fruit dealer sent a Lockport agent $ 1946.70, and /-instructed him to buy apples at $1.40 per barrel. The agent charged / 3% for buying, and shipped the purchase to his principal in six car loads of an equal number of barrels. How many barrels did each car contain ? 8. Find the per cent of commission on a purchase if the gross cost is $2048.51, the commission $87.30, the cartage $20, and other charges $1.21. ' 9. A collector obtained 75% of a doubtful account of $1750. How much was his per cent of commission if, by agreement with the principal, the commission was to be 50% of the net proceeds remitted? v./ : ', 10. A farmer received from his city agent $ 490 as the* net pro- ceeds of a shipment of butter,. If the agent's commission is 3%, delivery charges $ 6.80, and 5 % charge is made for guaranty of quality to purchasers, how many pounds, at 27^ per pound, must have been sold, and how much commission was allowed ? A <><> ^ %* /7 /* * o 11. An agent sold 2000 bu. Alsike clover seed at $7.85 per bushel, on a commission of 5%, and 1200 bu. medium red at $5.20, on a commission of 2|%, taking the purchaser's 3-month s' note for the amount of the sales. If the agent charges 4% for his guaranty of the note, what amount does he earn by the transaction ? 593] COMMISSION 227 12. Find the net proceeds of a sale made by an agent charging if incidental charges and commission charges were each $41.30. 13. Find the gross proceeds of a sale made by an agent charging for commission, 5% for guaranty, $17.65 for cartage, $11.40 for storage, and $3.25 for insurance, if the net proceeds remitted amount to $ 1714.10. 14. I sent $3402.77 to my Atlanta agent for the purchase of sweet potatoes at $ 1.60 per barrel ; his charges were, for commission, 2J%; guaranty, 3%; dray age, 1^ per barrel; and freight, $200. How many barrels did he buy, and how much unexpended money was left in his hands to my credit ? ^^ ^/ // 15. I received from Duluth a cargo of 16,000 bu. of wheat, which I sold at $1.10 per bushel, on a commission of 4%; by the con- signor's instructions I invested the net proceeds in a hardware stock, for which I charged 5% commission. What was the total commis- sion, and how much was invested in hardware ? 16. Having sent a New Orleans agent $1835.46 to be invested in sugar, after allowing 3% on the investment for his commission I received 32,400 pounds of sugar. What price per pound did the sugar cost the agent ? 17. An agent in Providence, R.I., received $828 to invest in prints, after deducting his commission of 3|%. If he paid 7^ per yard for ths prints, how many yards did he buy ? 18. An agent sold, on commission, 1750 barrels of mess pork at $ 16.50 per barrel, and 508 barrels of short ribs at $ 18 per barrel, charging $ 112.50 for cartage and $ 5.55 for advertising. He then remitted to his principal $36,000, the net proceeds. Find the rate of commission, tv 19. Render in full the following account sales, supplying rates per cent for insurance and commission, and showing net proceeds : Feb. 23, Emery Williams & Co., Troy, N.Y., sold for Moody Bros. & Co., Eome, K Y., 12,000 lb.- wool at 35 12,000 yd. woolen goods at 75 f. Charges : freight, $ 450 ; insurance, $ 33 ; commission, $ 264. 20. An agent sold wheat on 5% commission and invested the proceeds in barley at 75^ per bushel on a commission of 5f%. If his total commission was $1200, how many bushels of barley did he buy ? 228 PERCENTAGE AND ITS APPLICATIONS [694-601 INTEREST 594. Interest is that which is paid for the use of money. 595. The essential elements of interest are the principal, the time, the rate, the interest, and the amount. 596. The sum upon which interest is charged is termed the principal ; the period for which the principal bears interest, the time ; the annual rate charged for the use of the principal, the rate of interest ; the product of the rate of interest and the time, the per cent of interest ; the result obtained by taking a per cent of interest of the principal, the interest ; the sum of the principal and interest, the amount. 597. Legal interest is interest computed at the rate established by law to apply when no agreement is made. The legal rate of interest, being established by state statutes, varies in the different states. 598. Usury is any rate of interest in excess of the legal rate. In a number of the states parties may, by special agreement, receive interest at a higher rate than the legal rate. A person taking a usurious rate of interest is liable to certain penalties regulated by state statutes. SIMPLE INTEREST 599. Simple interest is interest allowed for the use of the princi- pal only. 600. The term interest is always understood to mean simple interest. If other forms of interest are meant they are specifically designated ; as, compound interest, periodic interest. 601. For convenience, interest is usually computed on the basis of the commercial year of 12 months of 30 days each, or 360 days. Interest on this basis is called common interest. The practice of taking 360 days as a year, being sufficiently exact for business pur- poses, has the sanction of law in some states and is generally used in all the states. G02-604] INTEREST 602. Simple interest is an application of the principles of abstract percentage with the additional element time introduced. The prin- cipal in interest corresponds to the base in percentage ; the per cent of interest, to the rate ; the interest, to the percentage ; and the sum of the principal and interest, to the amount. The solution of problems in interest is therefore dependent upon the general principles of abstract percentage, 603. There are many methods of computing simple interest, but those given herewith are the most rational and simple. The ordinary- day and the bankers' sixty-day methods are particularly adapted to finding the interest when the time is expressed in days, and the six per cent method to finding the interest when the time is expressed in years and months, or years, months, and days. Ordinary-day Method DRILL EXERCISE 1. What is the interest on $ 1 for 1 year at 2. What part of a commercial year is 60 days ? 6 days ? 8. How many days will it take $1 to yield 1 cent interest? 1 mill interest ? 4. What is the interest on $1 for 60 days at 6% ? for 6 days ? 5. What is the interest on $1 for 1 day at 6% ? 6. What is the interest on $6 for 1 day at 6% ? on $18? on $36? on $300? on $1200? 7. What part of the principal is the interest for 6 days at 6% ? 8. Give a simple way to find the interest on any principal for any number of days at 6%. 604. General Principles. 1. In 6 days at 6% any principal will yield interest equal to .001 of itself ; in 1 day, interest equal to .OOOJ of itself. 2. .001 of any given principal is equal to 6 times the interest for 1 day at 6%. 230 PERCENTAGE AND ITS APPLICATIONS [ 005-606 605. Examples. 1. Find the interest on $750 for 11 da. at 6%. SOLUTION. .001 of the principal, or .750 X 11 =8.250. ^ -7 5 ? i s equal to c t i mes t k e interest 8.250-5-6 = 1.375, or $1.38. for 1 day. Hence, 11 times $.75, or $8.25, is equal to 6 times the interest for 11 days. If 6 times the interest for 11 days is $8.25, the actual interest for 11 days must be of $8.25, or $1.38. 2, Find the interest on $875 for 24 da. at 6%. .875 X 4 = 3.500, or $ 3.50. SOLUTION. .001 of the principal, or $.875, is equal to the interest for 6 times 1 day, or 6 days. Since 24 days are 4 times 6 days, 4 times $ .875 must be the interest for 24 days. 4 times $ .875 = $ 3.50, the required interest. 606. Hence, the following rule may be derived: Point off three integral places in the principal, multiply ~by the number of days, and divide by 6. The result is the required interest at 6%. Or, When it is seen that the time in days and months is a multiple of 6, point off three integral places in the principal and multiply by j of the number of days. The result is the required interest at 6%. WRITTEN EXERCISE s. At 6% per annum find the interest on^ > 1. $ 750 for 73 da. 7. $ 476.87 for 95 da. IS. $ 728.16 for 84 da. 2. $840 for 19 da. 8. $ 925.14 for 72 da. 14. $846.92 for 108 da. 3. $ 920 for 24 da. 9. $ 724.18 for 75 da. 15. $ 1246.45 for 24 da. 4. $ 780 for 36 da. 10. $ 420.10 for 11 da. 16. $ 1432.1 8 for 36 da. 5. $ 920 for 42 da. 11. $ 500.60 for 7 da. 17. $ 1945.62 for 18 da. 6. $ 924 for 17 da. 12. $ 702.45 for 17 da. 18. $ 7514.95 for 12 da. Using the exact number of days, find the interest at 6% on : 19. $ 170 from July 15 to Sept. 1 ; from Apr. 6 to Oct. 9. 20. $ 1750 from Jan. 1 to Feb. 8 ; from May 15 to July 9. 21. $ 2470 from Apr. 7 to July 1 ; from Apr. 2 to Aug. 1. 22. $7562 from July 2 to Sept. 5 ; from Mar. 2 to Apr. 30. 28. $2172 from Jan. 2 to July 9; from Sept. 2 to Dec. 1. 24. $2400 from Oct. 1 to Dec. 1 ; from May 8 to Aug. 1. 25. $ 2(575 from Oct. 5 to Nov. 1 ; from Sept. 9. to Dec. 1. 607-611] INTEREST 231 607. To find the interest for any number of days at any rate per cent per annum. 608. General Principle. .001 of any given principal is the interest for 1 da. at 36%. 609. Example. What is the interest on $750 for 16 da. (a) at 6% ? (6) at 4% ? (c) at 9% ? (d) at SOLUTIONS, (a) .001 of $750 = $.75, or (a) .750 X 16 = 12. the interest for 6 days. $.75 x 16 = $ 12, or 12 -f- 6 = 2, or $ 2. the interest for 6 times 16 days. $ 12 -=- 6 = $ 2, or the required interest. (6) .001 of $750 = $.75, the in- : L *- terest for 6 days at 6%, or for 1 day 12 -h 9 = 1.33, or $1.33. at 36%. $.75 x 16 = $12, or the in- terest for 16 days at 36%. 4% is of 36 %. Hence, $ of $ 12, or $ 1.33, is the interest at 4 %. (c] 750 X 16 = 12 () Tne interest on tne principal for 16 days at 36 % is $ 12. 9 % is \ of 36 %. Hence, = 3, O *3. ^ of 3 12) or $ 3, is the interest at 9 %. (d) The interest on the principal (d) .750 X 16 = for 16 days at 360/o is $12 . 4jo/ fl i s | 12 -- 8 = 1.50, or 9 1.50. of 36%. Hence, \ of $12, or $1.50, is the interest at 4^%. 610. Hence the following rule may be derived : Point off three integral places in the principal, multiply by the number of days, and take such a part of the product as the given rate per cent is of 36%. 611. Deducing a rule for each of the ordinary rates of interest, we have the following : Point off three integral places in the principal, multiply by the number of days, and to find the interest at 6%, divide by 6; at 3%, divide by 12; at 4%, divide by 9; at 41%, di- vide by 8 ; at 5%, divide by 7.2 (8 and .9) ; at 7%, divide by 6 and add \ of the quotient; at 7J%, divide by 6 and add \ of the quotient; at 8%, dirhlc by 4.5 (9 and .5) ; at 9%, divide by 4'> ati 10%, divide by 3.6 (6 and .6) 232 PERCENTAGE AND ITS APPLICATIONS [611 WRITTEN EXERCISE Find the amount. Principal Time Bate 1. $9000 91 da. 4% 8. $1700 73 da. 4% S. $2750 81 da. % 4. $2400 15 da. 8% 5. $1750 21 da. 8% 6. $4200 84 da. 10% 7. $2972.50 31 da. 9% 8. $1750.90 87 da. 9% Principal Tim* Bate 9. $2431.75 35 da. 9% 10. $1862.15 34 da. 9% /f. $2417.50 41 da. 18. $7500.75 16 da. 18. $2400 19 da. / $1840.75 71 da. 15. $2417.92 76 da. 16. $1695.14 93 da. 3% Bankers' Sixty-day Method DRILL EXERCISE 1. How many months will it take $ 1 to yield 1 cent interest at 6% ? How many days ? 2. What is the interest on $75 for 60 days at 6% ? on $250 ? on $920? on $780? on $240.75? on $21729.75? 8. What part of a number is the interest at 6% for 60 days ? ^. What is the simplest way to find the interest on any princi- pal for 60 days ? 5. What part of 60 days are 30 days ? 20 days ? 15 days ? 10 days ? 6. What is the interest on $84 for 30 days ? for 20 days ? for 15 days ? for 10 days ? 7. What is the simplest way to find the interest on any princi- pal for 30 days at 6%? for 20 days ? for 15 days ? for 10 days ? 8. How long will it take $1 at 6% to yield 1 mill interest ? 9. What part of the principal is the interest at 6% for 6 days ? 10. What is the interest on $175 for 6 days at 6% ? on $215? on $240? on $378? on $7560? on $8925.75? on $4928.79? 11. What is the simplest way to find the interest on any princi- pal for 6 days at 6%? 611-612] INTEREST 233 12. What is the interest on $ 240 for 3 days ? 2 days ? 1 day ? 13. How many months at 6% will it take $ 1 to yield 10 cents interest ? How many days ? 14. What part of the principal is the interest at 6% for 600 days? 15. What is the interest on $ 800 at 6% for 600 days ? on $ 9500 ? on $2465? on $5619? on $4500? on $217.40? on $924.68? on $275.19? 16. What is the simplest way to find the interest on any princi- pal at 6% for 600 days ? 300 days ? 200 days ? 150 days ? 75 days ? 120 days ? 100 days ? 50 days ? 17. In how many days will the interest at 6% equal the prin- cipal ? 18. What is the interest on $1 for 6000 days at 6% ? on $24 ? on $97? on $55? on $372.50? on $920.75? on $860.44? 19. Pointing off 3 integral places in the principal gives the inter- est for how many days at 6% ? 2 places ? 1 place ? taking the prin- cipal for the interest ? 20. What is the interest on $5695 for 6 days at 6%? for 60 days ? for 600 days ? for 6000 days ? 612. General Principles. 1. Pointing off 3 integral places from the right in the principal gives the interest at 6% for 6 days. 2. Pointing off 2 integral places from the right in the principal gives the interest at 6% for 60 days. 3. Pointing off 1 integral place from the right in the principal gives the interest at 6% for 600 days. 4. Writing the principal for the interest gives the interest at 6% for 6000 days. Thus, the interest on $7621 for 6 days at 6% is $7.52 ; for 60 days, $75.21 ; for 600 days, $762.10 ; for 6000 days, $7621. ORAL EXERCISE Find the interest at 6% on: 1. $360 for 6 days ; for 3 days ; for 2 days ; for 1 day. 2. $900 for 60 days; for 30 days; for 20 days; for 15 days; for 12 days; for 10 days. 234 PERCENTAGE AND ITS APPLICATIONS [ 612-615 8. $100 for 18 days; for 24 days; for 36 days; for 42 days; for 48 days ; for 54 days ; for 66 days. 4. $200 for 180 days; for 240 days; for 420 days; for 480 days ; for 546 days ; for 660 days. 6. $240 for 7 days. SOLUTION. 7 days are \\ times 6 days. Hence, to find the interest for 7 days, point off 3 places from the right in the principal- and add g. $ .24 -f J of itself = $.28, the required interest. 6. $360 for 7 days ; for 8 days ; for 9 days. 7. $990 for 4 days. SOLUTION. 4 days are less than 6 days. Hence, to find the Interest for 4 days, point off 3 places from the right in the principal and subtract |. $.99 \ of itself $.66, the required interest. 8. $240 for 4 days; for 5 days; for 40 days; for 50 days; for 40 days ; for 80 days ; for 90 days ; for 70 days ; for 50 days. 613. To find the interest at 6% for aliquot parts of 6 or 60 days, or aliquot parts more or less than 6 or 60 days. 614. Examples. 1. What is the interest on $ 1240 for 30 days at6%? $ 12 40 SOLUTION. .01 of the principal, or $12.40 is the interest for 'oA 6 da y s ' 30 da y s is $ of 60 days. Since $12.40 is the interest e interest for 3Q days is O f $ 12.40, or $6.20. & What is the interest on $2400.60 for 80 days at 6 % ? $ 24 0060 SOLUTION. .01 of the principal, or $ 24.006, is 8 0020 ^ e * nterest * or ^ days. 80 days are $ more than 60 days. Hence, 4 more than $24.006, or $32.0080, or $32.01. $3201j is the required in \ erest 8. What is the interest on $360 for 5 days at 6% ? $ .360 SOLUTION. .001 of the principal, or $ .36 is the interest for 6 .060 days. Since 5 days are J less than 6 days, the interest for 5 days b $ 1100.59 for 48 da. f 710.18 for 420 da. $ 2100 for 42 da. $ 317.42 fgrjjffl da. ' $ 111.49 for 18 da. S. Find the total amount of interest at 6% on: $ 519 for 24 da. $ 1900 for 36 da. $ 1100 for 18 da. $ 1600.53 for 54 da. $ 170.50 for 240 da. $ 1700 for 120 da. f 11 for 540 da. $ 214.18 for 18 da. $ 210.40 for 66 da $ 210.90 for 180 da. $ 167.90 for 540 da. $ 1100 for 72 da 618-621] INTEREST 237 4. Find the total amount of interest at 6% on : $ 121 for 18 da. $ 760 for 240 da. $ 900 for 30 da. $ 745 for 600 da. $ 500 for 42 da. $ 800 for 60 da. $ 600 for 120 da. $ 360 for 72 da. $ 788 for 66 da. $ 20 for 36 da. $ 350 for 180 da. $ 89 for 54 da. 619. To find the interest at 6% for any number of days. 620. Examples. 1. What is the interest on $660 for 11 days at6%? SOLUTION. Sometimes aliquot parts may be subdivided so as $6.60 to make two or more aliquot parts. Subdividing 11 we have 10 and 1, or \ of 60 and \ of 6. The interest for 60 days is $6.60, and for 10 days $ 1.10. The interest for 6 days is $.66, and for $ 1 21 1 day $ -11- Adding the interest for 10 days and the interest for 1 day, the required interest is found to be $ 1.21. 2. What is the interest on $240 for 53 days at 6% ? SOLUTION. 53 days are 1 day less than 9 times 6 days. The $.240 interest for 6 days is $ .24, and for 54 days $ 2.16. If the interest 2.160 for 6 days is $ .24, the interest for 1 day is $.04. If the interest .04 for 54 days is $2.16, and the interest for 1 day is $.04, the in- $2 12 terest for 53 days is the difference between $2.16 and $.04, or $2.12. 5. What is the interest on $240 for 127 days at 6% ? SOLUTION. 127 = 60 x 2 + 6 + 1. The interest for 60 days is $ 2.40. Hence the interest for 120 days is twice $ 2.40, or $4.80. The interest for 6 days is $.24, and the interest for 1 day is $ of $ .24, or $ .04. Adding the interest for 120 days, 6 days, and 1 day we have $ 5.08, or the interest for 127 days. 621. Therefore the following rule : Find the interest on the principal for 6 days by pointing off three integral places from the right in the principal, and for 60 days by pointing off two integral places. For any number of days take such a part of the interest for 6 days, or for 60 days, as the given number of days is a part more or less than 6 days, or 60 days; or as many times the interest for 6 days, or 60 days, as the required number of days will contain 6 or 60 any multiple of 6 or 60 days. 238 PERCENTAGE AND ITS APPLICATIONS [621-624 WRITTEN EXERCISE Principal Time 13. $325.50 29 da. 14. $211.10 57 da. 16. $440 25 da. 16. $ 309.09 83 da. 17. $1200 14 da. 18. $100 53 da, 622. To find the interest at any rate per annum. 623. Examples. 1. What is the interest on $840 for 54 days Find the interest at 69 1 o on: 1. Principal $900 Time 53 da. 7. Principal $ 775.10 Time 17 da. 2. $ 287.10 47 da. 8. $211 43 da. 3, $ 1890 69 da, 9. $500 67 -da. 4- $14.50 81 da. 10. $450 58 da. 5. $2JL 91 da- 11. $ 700.80 126 da. 6. $59109 31 da. 12. $600 47 da. SOLUTION. The interest at 6 % is found to be $ 7.56. Since 7.560 8 % is more than the assumed rate 6 %, the interest at 8 % is I 2.52 more than the interest at 6%. Adding | of $7.56 to itself, the $ 10.08 result is found to be $ 10.08, or the interest at 8%. 2. What is the interest on $ 2100 for 180 days at 5% ? $21.00 63.00 10.50 $52.50 SOLUTION. The interest at 6 % is found to be $ 63. Since 5 % is \ less than the assumed rate 6 %, the interest at 5 % is \ less than the interest at 6 %. Taking \ of $ 63 from itself, the result is found to be $52.50, or the interest at 5%. 624. Hence the following rule : Add or subtract from the interest at itself as the given rate is greater or less than such a part of DRILL EXERCISE 1. Given the interest at 6%, how may the interest at 7% be found? SOLUTION. 7 % is \ more than 6 %. Hence, the interest at 6 % increased by \ of itself is equal to the interest at 7 %. 2. Formulate a short method for changing 6% interest to 8% interest; to 5% interest; to 4% interest; to 9% interest; to 10% interest; to 1\% interest. 624-625] INTEREST 239 3. If the interest at 6% is $60, what is the interest at 7%? at 5%? at8%? at4J%? at 7*%? 4. If the interest at 6% is $ 240, what is the interest at 9% ? at 10% ? at 3% ? at 41% ? at 625. General Principles. 1. 6% interest increased by ^ of itself equals 7% interest; by ^ of itself, 7-i-% interest; by \ of itself, 8% interest ; by \ of itself, 9 % interest. 2. 6% interest diminished by of itself equals 5% interest; by J of itself, 41% interest; by \ of itself, 4% interest. 6% interest may be changed to 10% interest by dividing by 6 and moving the decimal point 1 place to the right ; to 12% interest by multiplying by 2 ; to 3% interest by dividing by 2 ; to any other rate of interest by dividing by 6 and multiplying by the required rate. WRITTEN EXERCISE Find the interest on : 1. $ 1750 for 15 da. at 6%.^ " 17 f $ 3741.85 for 6 da. at 7%/ 18. $ 5178 for 9 da. at 9%. 19. $ 732 for 11 da. at 6%. 20. $ 1174.51 for 42 da. at 8%. 21. $340 for 70 da. at 10%. ' 22. $ 1478 for 80 da. at 6%. 28. $ 2150 for 96 da. at 4J%. 24. $ 1200 for 53 da. at 6%. 25. $ 1500 for 87 da. at 7%. 26. $420 for 41 da. at 5%. 27. $360 for 81 da. at 6%. 28. $ 2347.50 for 18 da. at 7%. 29. >$ 1112.49 for 25 da. at 8%. 80. $ 1300 for 13 da. at 6%. 81. $ 17,000 for 3 da. at 5J%. 32. $ 195.50 for 33 da. at 10%. 2. $ 1125 for 24 da. at 8. $ 742.50 for 30 da. at 6%. 4. . $ 900 for 93 da. at 5. $660 for 63 da. at 8%. r 6. $ 136.42 for 33 da. at 9%.j / 7. $ 1000 for 21 da. at 10%. 8. $ 2000 for 12 da. at 5%. 9. $ 351.23 for 40 da. at 4J %. 10. $ 1368 for 50 da. at 3%. 11. $93.40 for 150 da. at 0%. 12. $ 550 for 75 da. at 7%. 18. $ 842.50 for 45 da. at 6%. 14. $ 800 for 27 da. at 5%. 15. $ 1725 for 57 da. at 9%. 16. $ 125 for 55 da. at 6%. 240 PERCENTAGE AND ITS APPLICATIONS [ 625-626 33. $ 1050 for 43 da. at 1%. 87. 9 60 for 50 da. at 5%. 34. 9 1560 for 44 da. at 1%%. ^ * 38. $ 930 for 83 da. at 6% . 85. 9 180 for 47 da. at 6%. 89. $ 750 for 84 da. at 6%. 36. $120 for 49 da. at 9%. 40. $550 for 72 da. at 7%. 41. Find the total amount of interest on : 550 for 18 da. at 6%. 9 250 for 50 da. at 6%. 9 810 for 40 da. at 7%. 9 593.25 for 80 da. at 7%. $1000 for 41 da. at 1\%. $1966 for 75 da. at 5%. 9 342.50 for 42 da. at 5%. $ 450 for 83 da. at 8%. $ 1362.50 for 45 da. at 6%. $ 990 for 63 da. at 6%. 42. Find the total amount of interest on : $ 720 for 9 da. at 10%. $ 1124 for 15 da. at 3%. 9 7500 for 3 da. at 1%. $ 550 for 45 da. at 1\% $ 216 for 93 da. at 8%. $ 160 for 27 da. at 6%. $504 for 54 da. at 6%. $240 for 31 da. at 8%. $ 600 for 4 da. at 44$. $ 540 for 41 da. at 9%. 43. Find the total amount of interest on : $ 1452 for 8 da. at 3%. $ 1400 for 26 da. at 6%. 9 1728 for 10 da. at 6%: $ 1700 for 29 da. at 8%. $ 2150.42 for 17 da. at 7%. 9 1900 for 37 da. at 7%. 9 519 for 24 da. at 8%. $ 2100 for 43 da. at 6%. $ 1600 for 23 da. at 1\%. $ 3100 for 53 da. at 3%. SHORT METHODS 626. Interest is a product of which the rate and time are factors. Since the rate, being a constant factor, may be ignored, it will be observed that it will make no difference if, for convenience, the prin- cipal in dollars and the time in days be interchanged. Thus, the interest on $600 for 93 days is the same as the interest on $93 for 600 days. Since the interest for 600 days is ^ of the principal, ^ of $93, or $9.30, is the required interest on $600 for 93 days. The interest on $150 for 88 days is the equivalent of the interest on $88 for 150 days. Since 150 is \ of 600, the required result may be found by taking ^ of 88 and dividing the result by 4, obtaining $2.20 as the required interest. 626-628] INTEREST 241 By inspection, find 1. $600 for 93 da. 2. $300 for 42 da. 3. $200 for 66 da, 4. $150 for 44 da. 5. $120 for 55 da. 6. $60 for 89 da. 7. $30 for 56 da. 8. $20 for 84 da. 9. $15 for 124 da, 10. $10 for 66 da. 11. $6000 for 139 da. 12. $ 3000 for 142 da.. ORAL EXERCISE the interest at 6% on: 18. $2000 for 186 da. 14. $ 1500 for 64 da. 15. $1000 for 126 da. 16. $750 for 88 da. 17. $1200 for 155 da. 18. $2400 for 11 da. 19. $1800 for 31 da. 20. $3600 for 51 da. 21. $4200 for 11 da. 22. $5400 for 7 da. 23. $240 for 21 da. 24. $360 for 17 da. 25. $420 for 13 da. 26. $4200 for 103 da. 27. $ 3600 for 108 da. 28. $ 1200 for 39 da. 29. $3000 for 145 da. SO. $ 1000 for 246 da. 81. $ 6000 for 159 da. 32. $ 1800 for 39 da. 88. $2400 for 51 da. 84. $ 7200 for 19 da. 35. $ 4800 for 17 da. 86. $480 for 11 da. 627. When the rate is not six per cent, many times it is desirable to increase or diminish the principal or time, instead of the interest, by the proper fraction. 628. Examples. 1. Find the interest on $ 1500 for 84 da. at 8%. SOLUTION. Since 8% is $ more than 6%, if the principal is increased by $ of itself, and the interest computed for the given time at 6%, the result will be equal to the interest at 8%. Increasing $1500 by \ of itself, the result is $2000. Interchanging the dollars and days, the problem in its simplest form is equivalent to $84 for 2000 days at 6%. Since a principal will double itself in 6000 days, it will yield an interest equal to of itself in 2000 days. of 84 equals 28, making the required interest $28. 2. Find the interest on $799.59 for 45 da. at 8%. SOLUTION. Since 8% interest is more than 6% $ 7.9959 = $ 8. interest, if we increase the time by of itself and com- pute the interest on the principal for this time at 6 %. the result will be the interest at 8 %. 45, increased by $ of itself, equals 60. .01 of any number is the interest for 60 days. Hence, $8 is the required interest. 242 PERCENTAGE AND ITS APPLICATIONS [ 628-629 3. Find the interest on $844.20 for 80 da. at SOLUTION. 4^% interest is \ less than 6% in. $ 8.4420 = $ 8.44. terest. Hence, if we decrease the time by of itsell and compute the interest on the principal for the remainder at 6%, the result will be the interest at 4* %. 80 days decreased by \ of itself equals 60 days. .01 of the principal is the interest for 60 days at 6%. Hence, the required interest is $8.44. WRITTEN EXERCISE Find the interest on : Principal Time 1. $1200.00 79 da. 2. $783.60 45 da. 3. $425.80 45 da. 4. $1600.00 35 da. 6. $3200.00 78 da. 6. $2700.00 48 da. Rate Principal 7. $799.59 8. $111.10 9. $2400.00 10. $2400.00 11. $3800.00 12. $1200.00 Time 48 da. 48 da. 59 da. 38 da. 73 da. 66 da. Rate Six Per Cent Method DRILL EXERCISE 1. What is the interest on $ 1 for 1 yr. at 6% ? 4 yr. ? 5 yr. ? 8yr.? 2. What is the interest on $10 for 2 yr. at 6% ? on $30? on $25 ? on $80 for 5 yr. at 6% ? 3. What is the interest on $1 for 1 mo. at 6% ? for 2 mo.? for 4 mo. ? for 6 mo. ? for 8 mo. ? for 3 mo. ? for 7 mo. ? 4. What is the interest on $1 for 6 da. at 6% ? for 1 da. ? 5. How many mills will $1 yield in 12 da. at 6% ? in 24 da. ? in 9 da. ? in 15 da. ? 6. What is the interest on $1 for 1 yr. at 6% ? 1 mo. ? 1 da. ? 7. At 6%, what is the interest on $1 for 1 yr. 2 mo. 6 da.? on $20? on $2000? on $3000? on $1500? on $7500? 8. Give a simple way to find the interest on any principal for any given number of years, months, and days. 629. General Principles. $1 in 1 year at 6% will yield $.06 interest ; in 1 month, $ .005 interest ; in 1 day, $ .000 \ interest. 630-632] INTEREST 243 630. To find the interest on any principal for any time and rate by the six per cent method. 631. Example. What is the interest on $650 for 2 yr. 4 mo. 12 da. at 6% ? OP y 2 12 6^)0 SOLUTION. The interest on $ 1 for 1 .005 x 4 .02 .142 .0001 x 12 .002 1 300 .142 yr. at 6% is $ .06, and for 1 mo., or ^ yr., it is ^ of $.06, or $.005, and for 1 da. ~- of a month ^ - 242 PERCENTAGE AND ITS APPLICATIONS [ 628-629 S. Find the interest on $844.20 for 80 da. at $8.4420 = $8.44. Time Rate Principal Time 79 da. 1\% 7. $799.59 48 da. 45 da. 8% 8. $111.10 48 da. 45 da. 4$, 9. $2400.00 59 da. 35 da. 1\% 10. $2400.00 38 da. 78 da. 1\% 11. $3800.00 73 da. 48 da. 8% 12. $1200.00 66 da. SOLUTION. 4|% interest is J less than 6% in. terest. Hence, if we decrease the time by of itsell and compute the interest on the principal for the remainder at 6%, the result will be the interest at 4* %. 80 days decreased by | of itself equals 60 days. .01 of the principal is the interest for 60 days at 6 %. Hence, the required interest is $8.44. WRITTEN EXERCISE Find the interest on : Principal Time Rate Principal Time Rate 1. $1200.00 2. $783.60 8. $425.80 4. $1600.00 35 da. 1\% 10. $2400.00 38 da, 5% 5. $3200.00 Six Per Cent Method DRILL EXERCISE 1. What is the interest on $ 1 for 1 yr. at 6% ? 4 yr. ? 5 yr. ? 8yr.? 2. What is the interest on $10 for 2 yr. at 6% ? on $30? on $25 ? on $80 for 5 yr. at 6% ? 8. What is the interest on $1 for 1 mo. at 6% ? for 2 mo.? for 4 mo. ? for 6 mo. ? for 8 mo. ? for 3 mo. ? for 7 mo. ? 4. What is the interest on $1 for 6 da. at 6% ? for 1 da. ? 5. How many mills will $1 yield in 12 da. at 6% ? in 24 da. ? in 9 da. ? in 15 da. ? 6. What is the interest on $ 1 for 1 yr. at 6% ? 1 mo. ? 1 da. ? 7. At 6%, what is the interest- on $1 for 1 yr. 2 mo. 6 da.? on $20? on $2000? on $3000? on $1500? on $7500? 8. Give a simple way to find the interest on any principal for any given number of years, months, and days. 629. General Principles. $1 in 1 year at 6% will yield $.06 interest ; in 1 month, $ .005 interest ; in 1 day, $ .OOOJ interest. 630-632] INTEREST 243 630. To find the interest on any principal for any time and rate by the six per cent method. 631. Example. What is the interest on $650 for 2 yr. 4 mo. 12 da. at 6% ? 06 X 2 12 650 SOLUTION. The interest on $1 for 1 yr. at 6% is $.06, and for 1 mo., or ^ yr., it is T L of $.06, or $.005, and for 1 da., or .0001 x 12 .002 1 300 ^ of a mon th, it is ^ of $ .005, or $ .OOOJ. .142 9100 If the interest on $1 for 1 yr. is $.06, for 92.300 2 yr. it is twice $.06, or $.12. If the in- terest for 1 mo. is $.005, for 4 mo. it is 4 times $.005, or $.02. If the interest for 1 da. is $.000, the interest for 12 da. is 12 times $.000, or $.002. If the interest on $1 for 2 yr. is $.12, for 4 mo. $.02, and for 12 da $.002, the interest on $1 for 2 yr. 4 mo. 12 da. is $.142. If the interest on $1 is $.142, the interest on $650 is 650 times $.142, or $92.30. 632. Hence the following rule may be derived : Multiply the interest on $ 1 for the given time at 6% by the number of dollars in the principal, and the result is the interest at 6%. Change 6% to any other rate of interest by 625. WRITTEN EXERCISE Find the interest by the 6% method. Kate Principal 1. $750.50 Time 4 yr. 11 mo. Kate 6% Principal 4. $1116 Time 3 yr. 11 mo. & $3560.00 9 yr. 10 mo. 8% 1 5. $17,500 2 yr. 1 mo. 3. $610.15 7 yr. 11 da. 1% 6. $2400 7 yr. 1 mo. 7. On the 16th of September, 1904, I borrowed $3500 at 8%, interest. How much will settle the loan Jan. 1, 1910 ? 8. My note for $875.25, given 2 yr. 9 mo. 27 da. ago, bearing 4% interest, is due to-day. What is the amount of interest due ? 9. July 16, 1903, I borrowed $2750 at 5% interest, and on the same day loaned it at 7-J% interest. If full settlement is made Jan. 4, 1905, how much will be gained ? 244 PERCENTAGE AND ITS APPLICATIONS [ 632-635 10. Find the amount of interest at 6% by the six per cent method onr^fc $ 680, for 2 yr. 6 mo. 10 da. $500, for 3 yr. 1 mo. 27 da. $ 1895, for 1 yr. 7 mo. 7 da 9 895, for 5 yr. 11 mo. 11 da. $468, for 5 yr. 5 mo. 1 da. $1650, for 1 yr. 10 mo. 23 da $ 1000, for 11 yr. 1 mo. 20 da $ 1463, for 9 yr. 1 mo. 9 da. $ 645, for 4 yr. 4 mo. 5 da. $ 365, for 4 y r. 1 mo. 25 da. EXACT INTEREST 633. Exact interest is interest computed for the exact time in days on the basis of 365 days to a common year and 366 days to the leap year It is used by the United States government and by a few merchants and bankers. Aside from the uses in government calculations, exact interest is rarely com- puted. While it is enforcible, being strictly legal, the greater convenience of the 360-day rules so commend them to public favor as to lead to their common use. 634. To change common interest to exact interest 635. On a basis of 12 periods of 30 days each, a year's interest is taken for too short a period, since a year, exclusive of a leap year, contains 365 days. The time is, therefore, five days or -g-f-g-, equal to ^ too short, and the interest taken on that basis is proportion- ately too great. To correct this error and obtain the exact interest, Subtract -part from any interest computed on the 360-day basis. WRITTEN EXERCISE Find the exact interest of: 1 $954 for 63 days at 7%. 6. $681.80 for 90 days at 10%. 8. $630 for 50 days at 6%. 7. $500 for 48 days at $6%. 8. $800 for 33 days at 5%. 8. $ 1200 for 31 days at 5%. 4. $137.50 for 93 days at 8%. 9. $1500 for 55 days at 1\% 6. $210.54 for 100 days at 9%. 10. $4500 for 75 days aft &%.A b 11. $920 from Apr. 15 to July 25, at 6%. IS. $ 1756.90 from May 5 to Aug. 2, at 6%. IS. $ 2500.75 from June 25 to Dec. 8, at 6%. 14. $3200 from Oct. 15 to Nov. 25, at 6%. 15. $ 2500 from Apr. 16 to June 7, at 6%. ' 030-639] INTEREST 245 PROBLEMS IN INTEREST 636. The four distinct elements considered in ilrorest are the principal, rate per cent, time, and interest or amount. Simce the fourth element is practically the product of the first three, if any three of the elements are given, the other may be found in accordance with the general principles of percentage. 637. To find the rate per cent, the principal, interest, and time being given. 638. Example. At what rate per cent must $2100 be loaned for 2 yr. 5 mo. 6 da, to gain $459.90 ? SOLUTIOW Let 1 % equal the rate. $ 51.10 = interest on $ 2100 for the given time at 1 %. $459.90-*- $51.10 = 9. The interest at 1 % is contained in the given interest 9 times. Therefore the required rate is 9 times 1 %, or 9%. 639. From the above solution the following rule may be derived : Divide the given interest by the interest an the given principal for the given time at 1%. ORAL EXERCISE Find the rate of interest : Principal Interest Time Principal Interest Time 1. $600 $72 2yr. 4. $200 $24 4 yr. 2. $500 $60 Syr. 6. $400 $16 6 mo. & $300 $60 5yr. 6. $100 $24 Syr. WRITTEN EXERCISE 1. At what rate will $ 1260 yield $ 13.44 interest in 96 days ? 2. The interest for $2400 for 1 yr. 8 mo. 6 da. is $262.60. Find the rate of interest. 3. If I pay $518.75 interest on $1250 for 5 yr. 6 mo. 12 da., what is the rate per cent ? *| V^. 4. A lady deposited in a savings bank $3750, on which she received $93.75 interest semiannually. WKat per cent of interest did she receive on her monev ? ^T ?, 246 PERCENTAGE AND ITS APPLICATIONS [640-644 640. To find the time, the principal, interest, and rate of interest being given. 641. Example. In what time at S% will $2000 gain $400 interest ? SOLUTION Let 1 year represent the time. 8 % of $ 2000 = $ 160, the interest on the given principal for 1 year. 400 -r- 160 = 2.5. Since the interest for 1 year is contained in the given interest 2.6 times, the required interest must be 2.5 times the assumed time. 1 year x 2.6 = 2.5 year, or 2 years 6 months, the required time. 642. From the above solution the following rule may be derived : Divide the given interest by the interest on the principal for 1 year at the given rate per cent. ORAL EXERCISE Find the time in each of the following problems : Principal Interest Rate Principal Interest Bate L $700 $84 6% 4. $750 $ 7.50 6% 2. $250 $90 4% d. $900 $67.50 5% 8. $400 $44% 6. $600 $ 6.00 6% WRITTEN EXERCISE 1. How long will it take $360 to gain $53.64 at 5%? 2. How long should I keep $466.25 at 8% to have it amount to $610.48? Vlo'1*/ 8. A debt of $1650 was paid with 5|% interest on Aug. 30, 1888, by delivering a check for $2316.85. At what date was the debt contracted ? 643. To find the principal, the Interest, rate of interest, and time being given. 644. Example. What principal will yield $400 interest in 2 yr 6 mo. at S%? 644-647] INTEREST 247 SOLUTION Let $ 1 represent the principal. $ .20 = the interest on $ 1 for 2 yr. 6 mo. at 8%. 400 -*- .20 = 2000. The interest on the required principal is 2000 times the interest on the assumed principal. Therefore the required principal is 2000 times $1, or $2000. ORAL EXERCISE Find the principal in each of the following problems : Interest Kate Time Interest Rate Time 1. $40 6% 6yr. 8 mo. 4. $24 9% 8 mo. & $42 6% Syr. 6 ma . 6. $32 6% 6 mo. 12 da. S. $50 7i% 240 da. 6. $50 1\% 24 da. WRITTEN EXERCISE 1. What principal at 1% will gain $ 1080 in 3 yr. 6 mo. ? ^ 2. What principal at 4% will yield $455 in 3 yr. 6 mo. 18 da. ? S. A dealer who clears 121% annually on his investment is forced by ill health to give up his business. He lends his money at 7%, by which his income is reduced $1512.50. How much had he invested in his business ? ^'J ~&~d 645. To find the principal, the amount, rate per cent, and time being given. 646. Example. What principal will amount to $508 in 4 yr. 6 mo. at 6%? SOLUTION Let $ 1 represent the principal. $ 1.27 = the amount of $ 1 for 4 yr. 6 mo. $ 508 = the amount of a certain principal for 4 yr. 6 ma $508 -4- $1.27 = 400. Since the given amount is 400 times the assumed amount, the required prin- cipal must be 400 times the assumed principal. 400 times $ 1 = $ 400, the required principal. 647. From the above solution the following rule may be derived : Divide the given amount by the amount of $ 1 for the given time and rate. 248 PERCENTAGE AND ITS APPLICATIONS [647-650 ORAL EXERCISE Find the principal in each of the following problems : Amount Time Kate Amount Time Kate 1. $1120 2yr. 6% 5. $1025 3 mo. 10% 8. $2080 6 mo. 8% 6. $1212 60 da. 6% S. $4090 90 da. 9% 7. $218 2 yr. 4^% 4. $3120 6 mo. 8% 8. $367.50 3 yr. 1\% WRITTEN EXERCISE 1. Find the principal that will amount to $3360 in 3 yr. at 4%. - 2. What sum of money put to-day at 6% interest will amount in 7 mo. 12 da. to $4148? 8. Owed a debt of $ 5310 due in 1 yr. 6 mo. 18 da. I deposited in a bank that allowed me 4% interest a sum sufficient to cancel my debt when due. Find the sum deposited. 4. I borrowed a certain sum for 2 yr. 6 mo. with the understand- inff that I was to pay interest at the rate of 8%. If at maturity I gave my check for $2400, what was the sum loaned me ? PERIODIC INTEREST 648. Periodic interest is simple interest on the principal and on any interest remaining unpaid. 649. When interest is payable annually, it is called annual .interest; when payable semiannually, semiannual interest; when pay- able quarterly, quarterly interest; etc. 650. In some states annual and other periodic interest is sanc- tioned by law, but in many states it cannot be legally enforced. To secure periodic interest in any state, it must be specified by contract. Periodic interest is sometimes secured by a note or a series of notes ; in such cases the principal only is secured by one of the series (if not by mortgage or otherwise), while each of the other notes is drawn for one interest payment, and matures on the date at which such payment is due. By such arrangement, periodic interest can be enforced in states where it would otherwise be regarded as illegal. 651-655] INTEREST 249 651. To find periodic interest. 652. Example. Find the interest on $400 for 2 yr. at 6%, pay- able semiannually. SOLUTION $48 = the simple interest for the whole time. $12 = the semiannual interest. 1 yr. 6 mo. = the period for which 1st interest remained unpaid. 1 yr. = the period for which 2d interest remained unpaid. 6 mo. = the period for which 3d interest remained unpaid. 3 yr. = the period for which one semiannual interest draws interest. $2.16 = the simple interest on $12 for 3 yr. $48 + $2.16 = $50.16, the semiannual interest due. 653. From the above analysis the following rule may be derived : To the simple interest on the principal for the full time add the interest on one period's interest for the aggregate time for which the payments of interest were deferred. WRITTEN EXERCISE 1. Find the quarterly interest on $1600 for 2 yr. at 6%. 2. What is the difference between the simple and the annual interest of $ 2000 for 3.yr. at 6% ? 8. Find the amount of interest due at the end of 4 yr. 9 mo. on a note for $1155 at 6%, interest payable annually, but remaining unpaid. 4. On a note of $ 1750, dated Aug. 1, 1898, given with interest payable annually at 10%, the first three payments were made when due. How much remained unpaid, debt and interest, Jan. 1, 1905 ? COMPOUND INTEREST 654. Compound interest is the interest on the principal and on the principal increased by the interest at the expiration of regular intervals. 655. Interest may be added to the principal annually, semi- annually,, or quarterly, according to agreement. Compound interest is not recoverable by law, but a creditor may receive it if tendered without incurring the penalty of usury ; and a new obligation may 250 PERCENTAGE AND ITS APPLICATIONS [655-658 also be taken at the maturity of a compound interest claim for the amount so shown to be due, and such new obligation will be valid and binding. Most savings banks allow compound interest on balances remaining on deposit for a full interest term. 656. To find compound interest 657. Exampla Find the compound interest on $600 for 3 yr. 6 ino. at 6%. SOLUTION $600 x $ .06 = $ 86.00, the interest for the first year. $ 600 + $ 36 = $ 636. 00, the amount for the first year. $636 x $.06 = $ 38.16, the interest for the second year. $ 636 + $ 38.16 = $ 674.16, the amount for the second year. $674.16 x $ .06 = $ 40.45, the interest for the third year. $674.16 + $40.45 = $714.61, the amount for the third year. $ 714.61 x $ .03 = $ 21.44, the interest for 6 mo. $714.61 + $21.44 = $736.05, the amount for the full time. $ 736.05 $ 600 = $ 136.05, the compound interest for the full time. 658. Hence the following rule inay be derived : Find the amount of the principal and interest for the first period and make that the principal for the second period, and so proceed to the time of settlement. Subtract the principal from the last amount f and the remainder will be the compound interest. If the time contains fractional parts of a period, find the amount due for the full periods, and to this add its interest for the fractional period. WRITTEN EXERCISE 1. What is the compound interest on $1200 for 4 years at 7% if the interest is compounded annually ? Vt ^ ' ^ 2. What is the compound interest on $600 for 3 years at 5% if the interest is compounded quarterly ? 8. Find the compound interest on $ 400 for 4 years at 4%, interest payable semiannually. 4. Find the amount at compound interest on $ 500 for 3 yr. 4 mo. at 5%, interest payable annually. 5. Find the compound interest at 6% on $2000 for 1 yr. 5 mo., interest payable quarterly. 658] INTEREST 251 COMPOUND INTEREST TABLE Showing the amount of $1 at compound interest at various rates per cent for any number of years, from 1 year to 50 years, inclusive. Yrs. 1 per ct. Hperct 2 per ct. 2^ per ct. 3 per ct. 3 per ct. 4 per ct. 1 1.0100000 1.0150000 1.02000000 1.02500000 1.03000000 1.03500000 1.04000000 2 1.0201 000 11.0302 250 1.0404 0000 1.0506 2500 i.ot;o90000 1.07122500 1.08160000 3 1.0303010 L0456784 1.0612 0800 1.0768 9062 1.0927 2700 1.1087 1787 1.12486400 4 1.0406 040 1.0613 636 1.08243216 .1038 1289 1.1255 0881 1.14752300 1.16985856 5 1.0510 101 1.0772 840 1.1040 8080 .1314 0821 1.1592 7407 1.1876 8631 1.21665290 6 1.0615202 1.0934433 1.1261 6242 .1596 9342 1.19405230 1.22925533 1.2653 1902 7 1.0721 354 1.1098450 1.1486 8567 .1886 8575 1.22987387 1.2722 7926 1.31593178 8 1.0828567 1.1264926 1.1716 5938 .2184 0290 1.26677008 1.31680904 1.3685 6905 9 1.0936 853 1.1433900 1.19509257 .2488 6297 1.30477318 1.3628 9735 1.42331181 10 1.1046 221 1.1605408 1.2189 9442 .2800 8454 1.3439 1638 1.4105 9876 1.48024428 11 1.1156683 1.1779489 1.24337431 .3120 8666 1.38423387 1.45996972 1.53945406 12 1.1268250 1.1956 182 1.2682 4179 .3448 8882 1.42576089 1.51106866 1.60103222 13 1.1380933 1.213.-) 524 1.2936 0663 .3785 1104 1.46853371 1 .5639 5606 1.66507351 14 1,1494742 1.2317 557 1.3194 7876 .4129 7382 1.5125 8972 1.6186 9452 1.73167(545 15 1.1609 690 1.2502 321 1.34586834 .4482 9817 1.5579 6742 1.67534883 1.80094351 16 1.1725 786 1.2689 855 1.3727 8570 .4845 0562 1.6047 0644 1.73398601 1.87298125 17 1.1843044 1.2880203 1.4002 4142 .5216 1826 1.65284763 1.79467555 1.94790050 18 .1961 475 1.3073406 1.4282 4625 .5596 5872 1.70243306 1.857485)20 2.02581(552 19 .2081 090 1.3269507 1.4568 1117 .5086 5019 1.75350605 1.92250132 2.10(i84918 20 .2201 900 1.3468 550 1.48594740 1.63&6 1644 1,8061 1123 1.9897 8886 2.1911 2314 21 1.2323919 1.3670578 1.51566634 .6795 8185 1.8C029457 2.05943147 2.2787 6807 22 .2447 159 1.3875637 1.5459 7967 .7215 7140 1.91610341 2.1315 1158 2.3699 1879 23 .2571 630 1.4083772 1.57689926 .7646 1068 1.97358651 2.2061 1448 2.4647 1555 24 1.2697346 1.4295 028 1.6084 3725 .8087 2595 2.0327 9411 2.2833 2849 2.5633 0417 25 1.2824 320 1.4509454 1.64060599 .8539 4410 2.0937 7793 2.3632 4498 2.6658 3633 26 1.2952563 1.4727095 1.6734 1811 .9002 9270 2.1565 9127 2.4459 5856 2.7724 6979 27 1.3082089 1.4948 002 1.7068 8648 .9478 0002 2.2212 8901 2.5315 6711 2.8833 6858 28 1.3212 910 1.5172222 1.7410 2421 .9964 9502 2.2879 2768 2.6201 7196 2.9987 0332 29 1.3345039 1 .5399 805 1.77584469 2.0464 0739 2.3565 6551 2.7118 7798 3.11865145 30 1.3478490 1.5630802 1.81136158 2.0975 6758 2.4272 6247 2.8067 9370 3.24339751 31 1.3613274 1.5865 264 1.84758882 2.1500 0677 2.5000 8035 2.9050 3148 3.3731 3341 32 1.3749407 1.6103243 1.88454059 2.2037 5694 2.5750 8276 3.0067 0759 3.5080 5875 33 1.3886901 1.6344792 1.92223140 2.2588 5086 2.6523 3524 3.11194235 3.64838110 34 1.4025 770 1.6589964 1.9606 7603 2.3153 2213 2.73190530 3.2208 6033 3.7943 1634 35 1.4166028 1.6838 813 1.99988955 2.3732 0519 2.81386245 3.3335 9045 3.9460 8899 36 1.4307688 1.7091 395 2.0398 8734 2.4325 3532 2.8982 7833 3.4502 6611 4.10393255 37 1.4450765 1.7347766 2.0806 8509 2.49334870 2.9852 2668 3.5710 2543 4.2680 8986 38 1.4595272 1 .7607 983 2.1222 9879 2.5556 8242 3.0747 8348 3.6960 1132 4.4388 1345 39 1.4741225 1.7872 103 2.1647 4477 2.6195 7448 3.1670 2698 3.8253 7171 4.6163 6599 40 1.4888637 1.8140184 2.2080 3966 2.6850.6384 3.2620 3779 3.9592 5972 4.8010 2063 41 1.5037524 1.8412 287 2.25220046 2.7521 9043 3.3598 9893 4.0978 3351 4.9930 6145 42 1.5187899 1.8688471 2.2972 4447 2.8209 9520 3.4606 9589 4.2412 5799 5.1927 8391 43 1.5339778 1.8968 798 2.3431 8936 2.8915 2008 3.5645 1677 4.38970202 5.40049527 44 1.5493 176 1.9253330 2.39005314 2.9638 0808 3.6714 5227 4.5433 41(50 5.6165 1508 45 1.5648107 1.9542 130 2.4378 5421 3.03790328 3.7815 9584 4.7023 5855 5.8411 7568 46 1.5804589 1.9835262 2.4866 1129 3.1138 5086 3.89504372 4.86694110 6.0748 2271 47 1.5962634 2.0132 791 2.5:5634351 3.19169713 4.01 18 9503 5.0372 8404 6.3178 1562 48 1.6122261 2.0434 7S3 2.5870 7039 3.2714 8956 4.13225188 5.21358898 6.5705 2824 49 1.6283483 2.0741 305 2.63881179 3.35327680 4.25621944 2. 30(50 6459 6.8333 4937 50 1.6446318 2.1052424 2.6915 8803 3.43710872 4.38390602 5.58492686 7.10668335 252 PERCENTAGE AND ITS APPLICATIONS [658 COMPOUND INTEREST TABLE ' Showing the amount of $1 at compound interest, at various rates per cent, for any number of years, from 1 year to 50 years, inclusive. Yrs. 4 per ct. 5 per ct. 6 per ct. 7 per ct. 8 per ct. 9 per ct. 10 per ct. 1 1.04500000 1.0500 000 1.0600000 1.0700000 1.0800000 1.0900000 1.1000000 2 1.09202500 1.1025000 1.1236000 1.1449000 1.1664000 1.1881 000 1.2100000 3 1.14116612 1.1576 250 1.1910 160 1.2250430 1.2597 120 1.2950290 1.3310000 4 1.19251860 1.2155063 1.2624770 1.3107960 1.3604 890 1.4115816 1.4641 000 5 1.2461 8194 1.2762 816 1.3382256 1.4025517 1.4(593281 1.5386 240 1.6105 100 6 1.3022 6012 1.3400956 1.4185 191 1.5007 304 1.5668743 1.6771 001 1.7715610 7 1.36086183- 1.4071 004 1.5036303 1.6057 815 1.7138 243 1.8280391 1.9487 171 8 1.4221 0061 1.4774554 1.5938481 1.7181 862 1.8509302 1.9925626 2.1435 888 9 1.4860 9514 1.5513282 1.6894 790 1.8384 592 1.9990046 2.1718 933 2.3579 477 10 1.55296942 1.6288946 1.7908477 1.9671 514 2.1589250 2.3673637 2.5937 425 11 1.62285305 1.7103394 1.8982986 2.1048 520 2.3316 390 2.5804 264 2.8531 167 12 1.69588143 1.7958563 2.0121 965 2.2521 916 2.5181 701 2.8126 648 3.1384284 13 1.77219610 1.8856491 2.1329283 2.4098 450 2.7196 237 3.0658046 3.4522 712 14 1 1.8519 4492 1.9799316 2.2609 040 2.5785 342 2.9371 936 3.3417 270 3.7974 983 15 1.93528244 2.0789 282 2.3965582 2.7590315 3.1721691 3.6424 825 4.1772482 16 12.02237015 2.1828 746 2.5403 517 2.9521 638 3.4259426 3.9703059 4.5949 730 17 2.11337681 2.2920 183 2.6927 728 3.1588 152 3.7000 181 4.3276334 5.0544 703 18 j 2.2084 7877 2.4066 192 2.8543 392 3.3799323 3.9960 195 4.7171 204 5.5599 173 19 2.30786031 2.5269 502 3.0255 995 3^165 275 4.3157011 5.1416613 6.1159390 20 2.4117 1402 2.6532 977 3.2071 355 3^8696 845 4.6(509571 5.6044 108 6.7275000 21 2.5202 4116 2.7859626 3.3995636" '4.1405 624 5.0338 337 6.1088077 7.4002499 22 2.6336 5201 2.9252607 3.6035 374 4.4304017 5.4365404 6.6586004 8.1402749 23 2.7521 6635 3.0715 238 3.8197 497 4.7405299 5.8714 637 7.2578 745 8.9543024 24 2.87601383 3.2250 999 4.0489 346 5.0723 670 6.3411 807 7.9110 832 9.8497 327 25 3.00543446 3.3863549 4.2918 707 5.4274 326 6.8484 752 8.6230807 10.8347 059 26 3.1406 7901 3.5556 727 4.5493 830 5.8073 529 7.3963532 9.3991 579 11.9181765 27 3.28200956 3.7334 563 4.8223459 6.2138 676 7.9880615 10.2450 821 13.1099942 28 3.42969999, 3.9201291 5.1116 867 6.6488 384 8.6271064 11.1671 395 14.4209 936 29 3.58403649 4.1161356 5.4183 879 7.1142571 9.3172 749 12.1721 821 15.8630 930 30 3.7453 1813 4.3219424 5.7434912 7.6122550 10.0626 569 13.2676785 17.4494023 31 3.9138 5745 4.5380 395 6.0881 006 8.1451 129 10.8676694 14.4617 695 19.1943425 32 4.08998104 4.7649 415 6.4533867 8.7152 708 11.7370 830 15.7633 288 21.1137768 33 4.2740 3018 5.0031 885 6.8405899 9:3253 398 12.6760 496 17.1820284 23.2251 544 34 4.46636154 5.2533480 7.2510 253 9.9781 135 13.6901 336 18.7284 109 25.5476 699 35 4.66734781 5.5160 154 7.6860 868 10.6765 815 14.7853443 20.4139 679 28.1024369 36 4.8773 7846 5.7918 161 8.1472520 11.4239422 15.9681 718 22.2512250 30.9126805 37 5.0968 6049 6.0814069 8.6360 871 12.2236 181 17.2456 256 24.2538 353 34.0039486 38 5.3262 1921 6.3854773 9.1542 524 13.0792 714 18.6252 756 26.4366 805 37.4043434 39 5.5658 9908 6.7047 512 9.7035 075 13.9948 204 20.1152977 28.8159817 41.1447778 40 5.81636454 7.0399887 10.2857 179 14.9744 578 21.7245 215 31.4094 200 45.2592 556 41 6.0781 0094 7.3919 882 10.9028610 16.0226699 23.4624832 34.2362679 49.7851 811 42 6.3516 X548 7.7615 876 11.5570327 17.1442568 25.3394 819 37.3175 320 54.7636 992 43 6.6374 3818 8.1496669 12.2504546 18.3443548 27.3666.404 40.6761 098 60.2400 692 44 6.9361 2290 8.5571 503 12.9854 819 19.6284596 29.5559 717 44.3369 597 66.2640 761 45 7.24824843 8.9850078 13.7646 108 21.0024518 31.9204494 48.3272 861 72.8904837 46 7.5744 1961 9.4342 582 14.5904 875 22.4726 234 34.4740 853 52.6767 419 80.1795 321 47 7.9152 6849 9.9059711 15.4659 167 24.0457 070 37.2320 122 57.4176486 88.1974853 48 8.2714 5557 10.4012 697 16.3938 717 25.7289065 40.2105 731 (52.5852 370 97.0172 338 49 8.6436 7107 10.9213 331 17.3775040 27.5299300 43.4274 190 68.2179083 106.7189572 50 9.0326 3627 11.4(573998 18.4201 543 29.4570 251 46.9016 125 74.:;r>7520i 117.3908529 059-660] INTEREST 253 Application of Compound Interest Table 659. To find the amount of any given principal for any given num- ber of years : Multiply the given principal by the amount of $1 at the given rate, as shown by the table. For periods beyond the scope of the table, multiply together the amounts shown for periods, the sum of which will equal the time required. For example, to find the compound amount of $ 1 for 100 years : Multiply the amount for 50 years by itself; to find the amount for 75 years, multiply the amount for 50 years by the amount for 25 years; etc. If interest is to be compounded semiannually, take one half the rate for twice the time. If interest is to be computed quarterly, take one quarter of the rate for 4 times the time / etc. 660. To find the compound interest on principals of $ 100 or less, use four of the decimal places shown in the table; on principals of $1000 or less, use five of the decimal places ; and so on. WRITTEN EXERCISE By the use of the compound interest table solve the following problems : 1. Find the amount of $ 1750 compounded annually for 10 yr. 6 mo. at 5%. 2. Find the compound interest on $800 for 16 yr. 4 mo. 18 da. at 8%, interest compounded annually. 8. Find the compound interest at 6% on $500 for 1 yr. 3 mo., interest payable quarterly. 4. Find the compound interest at x 8% on $1200 for 2 yr. 5 mo., interest payable quarterly. 5. What principal will, in 8 yr. at 5%, amount to $4107.26, if interest is compounded semiannually ? ORAL REVIEW 1. What principal in 2 yr. at 6% will yield $24 interest? $72 interest ? $ 84 interest ? 2- What principal in 8 yr. at 5% will amount to $280? 254 PERCENTAGE AND ITS APPLICATIONS [660 8. What sum of money invested at 6% will in 2 yr. 3 mo. yield $ 135 interest ? $ 405 interest ? 4- At what rate would $ 300 in 3 y r. yield $ 45 interest ? & A lady deposited in a savings bank $750, on which she received $15 interest semiannually. What per cent of interest did she receive on her money ? 6. To satisfy a debt of $400 which had been standing 2 yr., I gave my check for $440. What was the rate of interest charged? 7. How long will it take $100 to gain $12 at 6% ? 8. How long must $ 550 be on interest at 7 % to amount to $ 570 ? 9. In what time will money bearing 8% interest double itself? SOLUTION. In order to double itself the interest accumulated must be equal to 100% of the principal. Since the principal increases 8% per annum, it will require as many times one year to increase 100%, or to double itself, as 8%i* contained times in 100%, or 12 times, equal to 12 yr. 6 mo. 10. In what time will any sum bearing interest at 4% double itself? at 5% ? at 10% ? at 9% ? 11. How many days will $6000 require to yield $71 interest? 12. In how many days will the interest at 6% be one half of the principal ? double the principal ? WRITTEN REVIEW 1. A note of $ 1260 is 151 days past due. What amount will settle the note and interest, money being worth 6% per annum ? 2. The interest on a certain sum in 12 \ years is \ of that sum. What is the rate of interest? Lf- H 3. A certain principal placed at simple interest for 64 days amounts to $606.40. If the same principal would amount to $624.90 in 249 days, what is the rate of interest? What is the principal ? 4- What monthly rent should be charged for a house costing $10,240 in order that 6% interest on the investment may be realized ? \ . 3^0 5. At the end of five years the accrued interest on a certain principal is found to be of the sum drawing interest. What is the rate of interest ? H JO 660] INTEREST 255 6. A merchant charges interest at the rate of 6 % per annum on overdue accounts. He received a check for $ 1205.40 in settlement for an overdue account of $1200. How long overdue was the account? ^.1 ibk . 7. A man borrowed $12,000 at 5% and with it immediately bought a house which he rented for $1800 per year. What was his yearly per cent of net gain or loss ? In how many years will his net gains aggregate the sum borrowed ? 8. May 16 I bought 300 barrels of flour, at $7 per barrel; July 28 I sold 50 barrels, at $8 per barrel ; Oct. 30, 100 barrels, at $6.75 per barrel ; and Feb. 13 following, the remainder, at $7.80 per barrel. Allowing interest at 6%, what was my gain ? 9. Find the simple interest on 40 8s. 5d. for 2 years at 6%. Give the answer in United States money. 10. Find the compound interest, by the table, on $2000 for-^x 4 years at 6%, interest payable quarterly. 5^S\- ^\\ 11. A man invested $ 16,000 in business, and at the end of 3 yr. 3 mo. withdrew $ 22,240, which sum included investment and gains. What yearly per cent of interest did his investment pay ? \ 0^ I f\ 12. Find the interest of that sum for 11 yr. 8 da. at 10i$f, which will, at the given rate and time, amount to $ 1715.08. 18. Sold an invoice of crockery on a credit of 2 months; the bill was paid 3 mo. 18 da. after the date of purchase, with interest at 8%, by a check for $ 1963.45. How much was the interest ? 14. A man having $ 21,000 invested it in real estate, from which he received a semiannual income of $ 787.50. He sold this property at cost and invested the proceeds in a business which yielded him $ 472.50 quarterly. How much greater rate per cent per annum did he receive from the second investment than from the first ? 15. In order to engage in business, I borrowed $3750 at 6%, and kept it until it amounted to $4571.25. How long did I keep the money ? \-~ ^ 16. In what time will interest at 8% equal f of the principal ? 17. A building which cost $ 10,500 rents for $ 87.50 per month. What annual rate of interest on his investment does the owner receive if he pays yearly taxes amounting to $102.50; insurance, $21.25; repairs, $136.80; and janitor's services, $ 56.95 ? 256 PERCENTAGE AND ITS APPLICATIONS [ 660-666 18. A merchant sold a stock of glassware on one month's credit; the bill was not paid until 3 mo. 21 da. after it became due, at which time the seller received a draft for $4716.21 for the bill and interest thereon at the rate of 5%. Find the selling price of the goods. 19. Oct. 12, 1904, I purchased 2700 bushels of wheat at $1.05 per bushel, and afterwards sold it at a profit of 6%. On what date was the wheat sold if my gain was equivalent to 10% interest on my investment ? 20. I am offered a house that will rent for $ 27 per month, at such a price that, after paying $ 67.20 taxes and other yearly expenses amounting to $ 24.85, my net income will be 8J% on my investment. What is the price asked for the house ? PRESENT WORTH AND TRUE DISCOUNT 661. The present worth of a debt payable at a future time with- out interest is such a sum as being put at simple interest at a legal rate will amount to the given debt when it becomes due. 662. The true discount is the difference between the face of the debt due at a future time and its present worth. To illustrate, suppose A owes B $212 to be paid for one year after date. Should A care to cancel the indebtedness at once, the sum which he ought to pay should be such that, if put out at legal interest by B, it would at the end of the year amount to $ 212. Suppose that B can receive 6 % on his money. At this rate $ 1 put at simple interest would, at the end of one year, amount to $ 1.06. If $ 1 in one year amounts to $ 1.06, it will take as many times $ 1 to amount to $ 212 as $ 1.06 is contained times in $ 212, or 200 times. Hence, $200 must be the sum which A ought to pay now to cancel a debt which at the end of one year amounts to $212. The $ 200 to be paid is called the present worth, and the difference between $212 and $200, or $ 12, the true discount. 663. Computations in present worth and true discount come under the case of interest problems, in which the amount, the rate per cent, and the time are given, to fin*> 16. $2400.00. BT. PAUL, MINN., Aug. 31 ,-1903. Six months after date I promise to pay to the order of John W. Bell, Twenty-four Hundred Dollars, with interest at 8%, after one month. Value received. OLIVER JONES. Discounted Sept. 5, 1903, at 8%. 17. $ 800.00. CLEVELAND, O., Jan. 31, 1903. One month after date we promise to pay to the order of Hale & Bly, Eight Hundred Dollars, with interest at 5%. Value received. HART & COLE. Discounted Feb. 10, 1903, at 6%. \ 699] BANK DISCOUNT 269 18. $1200.00. SPRINGFIELD, MASS., Mar. 5, 1903. Four months after date we jointly and severally promise to pay to the order of Shaw Bros. & Co., Twelve Hundred Dollars, with interest from date. Value received. RAYMOND D. DANN, Discounted May 1 at ^\%. CHARLES L. KINSLEY. / 2- 19. $2576.25. CHICAGO, ILL., May 20, 1903. At ninety days' sight pay to the order of Ourselves, Twenty-five Hundred Seventy-six and -ffa Dollars, value received and charge to the account of To SPEAR BROS. & Co., F. E. KoGERS & Co. San Francisco, Cal. Accepted May 25, 1903. Discounted May 30, 1903, at 6%. "4-f/^i* It&s 270 PERCENTAGE AND ITS APPLICATIONS [699 22. $ 795j%. ROCHESTER, N. Y., May 15, 1904. ' Sixty days after date pay to the order of Ourselves, Seven Hundred Ninety-five and -ffo Dollars, value received, and charge to the account of BOWEN, MERRILL & Co. To GRAY & SALISBURY, Buffalo, N. Y. Accepted May 20, 1904. Discounted May 21, 1904, at 6%. Col- lection charges 28. A note for $1200, dated Boston, Mass., Mar. 4, 1903, and payable 3 months after date, was discounted Apr. 5 at 5%. Find the bank discount and proceeds. $4. What is the proceeds of a note for $ 3500, dated Feb. 2, 1903, and due in 4 months, without interest, if the note is discounted Apr. 20, at 4% ? 25. Find the date of maturity, 'the bank discount, and proceeds of a note for $ 1800, dated Feb. 18, 1903, payable in 90 days, and discounted Apr. 25, 1903 at 6%. 26. Paul Harmon's bank account is overdrawn $3596.11. He now discounts at 6% : a 90-day note for $450, a 60-day note for $1754.81, a 30-day note for $851.95, a 20-day note for $345.25, a 10-day note for $100; proceeds of all to his credit at the bank. What is the condition of his bank account after he receives these credits? \ - (XQ * vv 27. C. H. Good & Co.'s bank account is overdrawn $7,564.19. They discount at 6% : a 90-day note for $ 3975.21, a 60-day note for $1546.19, and a 20-day note for $2546.85; proceeds of all to their credit at the bank. What is the condition of their bank account after they receive credit as above ? v , , 28. Assuming that the model note, page 259, was discounted March 4, 1903 at 6%, find the bank discount and net proceeds. 29. Assuming that the model draft, page 261, was discounted March 9, 1903 at 5%, find the bank discount and net proceeds; collection charges, %. 700-702] BANK DISCOUNT 271 700. To find the face of a note, the proceeds, time, and rate of dis- count being given. 701. Example. What must be the face of a note, payable in 30 days, in order that when discounted at 6% the proceeds will be $ 895.50? SOLUTION Let $ 1 represent the face of the note. $ .005 = the bank discount on $ 1 for 30 da. $1 - $.005 = $.995, the proceeds of $1 due in 30 days. $895.50-7- $.995 = 900. The given proceeds is 900 times the proceeds of $ 1. Therefore the required face is 900 times $1, or $900. 702. Hence the following rule : Divide the proceeds of the note by the proceeds of $ 1 at the given rate for the given time. * WRITTEN EXERCISE 1. I wrsh to borrow $ 650.08 of a bank. For what sum must I issue a 90-day note to obtain the amount, discount being at 6%? 2. A man wishes to borrow $ 594 cash. For how much must he draw a note, so that when discounted at 6% on 60 days' time, with- out interest, the proceeds will be the sum wanted ? 8. A 30-day 6% interest-bearing note was discounted 10 days after it was drawn up. If the rate of discount was 6% and the bank discount $ 13.40, what was the face of the note ? 4. A merchant bought goods to the amount of $ 2376. For how % much must he draw his 60-day note, without interest, that when dis- counted at 6% he may pay for his purchase with the proceeds ? 5. A note dated Mar. 15, 1903, payable in 3 months with inter- est at 7i%, was discounted Apr. 16, 1903, at 10%. If the proceeds were $ 2404,25, what must the face" have been ? 6. You have $650.80 to your credit at a bank; you give your check for $1872.40, after which you discount a 30-day note for $850.80, proceeds to your credit at the bank. You then discount a 90-day note, made by H. C. Davis, proceeds to your credit, when you find yourself indebted to the bank $ 24.74. If discount be at 6%, what must have been the face of the note made by Davis ? 272 PERCENTAGE AND ITS APPLICATIONS [ 703-706 PARTIAL PAYMENTS 703. A partial payment is a payment of a part of the amount due on a note, mortgage, or other interest-bearing obligation. Partial payments should be acknowledged by indorsement on the back of a note, as follows : 1 ! 6 u ^g T- 3 ! s 1 H 02 Ci O **H ^ *. ^> ^ - I 3 M 02 i s | w w s^ ~ ,g ,g \ M 1 ^ 1 | Sometimes special receipts are given for partial payments on notes and other similar obligations. A debtor, or his authorized agent, may make a payment, either in part or in full, of any obligation, and such payment may be received by the creditor, or his authorized agent. 704. Various rules are in use for finding the balance due on claims on which partial payments have been made, but only the United States Rule and the Merchants' Rule have more than local application. UNITED STATES RULE 705. The United States Rule is very generally used when partial payments are made on interest-bearing obligations that run for more than one year. It is the rule that has been adopted by the Supreme Court of the United States, and by most of the separate states. 706. General Principles. The United States Eule recognizes the following general principles : 1. Accrued interest must be paid before the principal may be diminished. 2. Interest must not be charged upon interest. 707-708] PARTIAL PAYMENTS 273 707. To find the balance due by the United States Rule for partial payments. 708. Example. A note, the face of which was $2500, bearing interest at 6%, was given Nov. 1, 1899, and settled Aug. 5, 1904. Find the balance due, the following payments having been made : Dec. 5, 1900, $ 600 ; Jan. 5, 1902, $ 500 ; May 1, 1903, $ 100 ; July 5, 1904, $800. SOLUTION Face of note $2500.00 Interest from Nov. 1, 1899, to Dec. 5, 1900 (1 yr. 1 mo. 4 da.) 164.17 Amount due Dec. 5, 1900, time of first payment . . 2664.17 Payment of Dec. 5, 1900 600.00 New principal, or amount to draw interest from Dec. 5, 1900, 2064. 17 Interest from Dec. 5, 1900, to Jan. 5, 1902 (1 yr. 1 mo.) . ' 134.17 Amount due Jan. 5, 1902, time of second payment . . 2198.34 Payment Jan. 5, 1902 500.00 New principal, or amount to draw interest from Jan. 5, 1902 1698.34 Interest from Jan. 5, 1902, to May 1, 1903 (1 yr. 3 mo. 26 da.) $134.73 The interest exceeds the payment, and a new principal is not formed. Interest from May 1, 1903, to July 5, 1904 (1 yr. 2 mo. 4 da.) 120.02 Total interest due July 5, 1904 254.75 Amount due July 5, 1904, the time of the fourth payment 1953.09 Sum of payments May 1, 1903, and July 5, 1904 . 9QO.OO New principal, or amount to draw interest from July 5, 1904 1053.09 Interest from July 5, 1904, to Aug. 5, 1904 (1 mo.) . . 5.27 Amount due Aug. 5, 1904, the final date of settlement . 1058.36 CONDENSED FORM FOR WRITTEN WORK DATES INTEREST PERIODS PRINCIPALS INTERESTS AMOUNTS PAYMENTS Yr. Mo. Da 1899 11 1 1900 12 5 1 yr. 1 mo. 4 da. $2500.00 $164.17 $2664-17 $600 1902 7 5 1 1 2064.17 134.17 2198.34 500 1903 5 1 1 3 26 1698.34 134.73 100 1904 7 5 1 2 4 1698.34 120.02 1953.09 800 1904 8 5 010 1053.09 5.27 1058.36 Ans. MOGUL'S COM. AR. 18 \ 274 PERCENTAGE AND ITS APPLICATIONS [709 709. RULE. Find the amount of tlw principal to the time when the payment or the sum of the payments shall exceed the interest then due. From this amount deduct tlw payment or the sum of the payments made. Consider the remainder as a new principal, and proceed as before to the time of settlement. 7-u-M WRITTEN EXERCISE /? ^-.7. On a mortgage for $650, made Aug. 10, 1894, and bearing i* interest at 6%, payments were indorsed as follows: Feb. 2, 1896, 7 * $100; June 20, 1898, $50; Nov. 1, 1900, $250. How much was T* | due Mar. 31, 1903 ? .|ft On a claim for $3000, dated Sept. 15, 1900, bearing interest at 8%, payments were made as follows : Jan. 1, 1901, $300 ; July 20, 1901, $500; Feb. 2, 1902, $ 125; Apr. 20, 1903, $1800. How much was due at final settlement Jan. 1, 1904? lr "" 3. On the note below indorsements were made as follows : Apr. 1901, $300; Nov. 20, 1902, $1000; Mar. 18, 1903, $600; Mar. '"1904, $ 1100. What was the balance due Jan. 1, 1905 ? $4000.00 Chicago, 111., Mar. 15, 1901. v On demand we promise to pay to the order of Williston, Burgess & Hart, Four Thousand Dollars, at Union National Bank, with interest at 6%.. Value received. HOUGHTON, DUTTON & Co*: Ji&7.si \ 4. A note of $ 1500, dated June 20, 1902, bearing interest at had payments indorsed upon it as vfollows : Dec. 5, 1902, Apr. 2, 1903, $30; July 20, 1903, $500; Dec. 31, 1903, $400. the amount due Apr. 1, 1904. 5. On a mortgage for $4500, dated May 1, 1899, and bearing in- at 7%, the following payments were made: Feb. 2, 1900, Aug. 5, 1900, $75-, Aug. 5, 1901, $2000; Dec. 20, 190L', $300. How much was due at final settlement Apr. 1, 1903? -J4 *-'*" .JM - " 6. On the note below payments were indorsed as follows : Oct. 1, . Co 709-713] PARTIAL PAYMENTS 275 /^>* 1901, $750; Apr. 1, 1902, $150; Oct. 1, 1902, $365.90; Mar. 25, 1903, $150; Nov. 1, 1903, $200. How much was due Apr. 6, 1904? $2500.00 r 7 *-_ Boston, Mass., Apr. 6, 1901. f^, ^ Two years after date for value received, I promise to pay toj^ :' the order of C. W. Frey & Co., Twenty-five Hundred Dollars, interest at 6%. F. M. EJ.LEBY. MERCHANTS' RULE 710. When partial payments are made on interest-bearing that run for one year or less, the amount due at final settlement is usually found by the Merchants' Rule. 711. General Principles. The Merchants' Eule recognizes the following general principles : 1. The face of the note draws interest to the time of settlement. 2. Interest is allowed on each payment from the time -such pay- ment is made to the date of settlement. The Merchants' Rule is varied in its use by different creditors, and hence is rather more an agreement, founded upon custom or otherwise between debtor and creditor as to mode of settlement, than a strict rule of law. 712. To find the balance due by the Merchants' Rule for partial payments. 713. Example. A note for $ 900, dated May 5, 1903, payable on demand, shows that the following payments have been made : June 20, $200; Aug. 15, $300; DeC. 1, $200. What is due Dec. 31, 1903, money being worth 6% ? SOLUTION Face of note ......... $900.00 Interest from May 5, 1903, to Dec. 31, 1903(7 ino. 26 da.) . 35.40 Amount of note at date of final settlement, Dec. 31, 1903 . 935.40 First payment ......... $200.00 Interest on payment from June 20 to Dec. 31 (6 mo. 11 da.) 6.37 Second payment ......... 300.00 Interest on payment from Aug. 15 to Dec. 31 (4 mo. 16 da.) 6.80 Third payment ......... 200.00 Interest on payment from Dec. 1 to Dec. 31 (30 da.) . . 1.00 Value of payments at the date of final settlement . . 714.17 Balance due Dec. 31, the date of final settlement . . $221.23 276 PERCENTAGE AND ITS APPLICATIONS CONDENSED FORM FOR WRITTEN WORK 713-714 DATES INTEREST PERIODS PRINCI- PAL PAY- MENTS INTERESTS AMOUNT OF PRINCIPAL AMOUNTS OF PAYMENTS BALANCE Mo. Da. Mo. Da. 6 5 7 26 $900 $35.40 $ 935.40 $935.40 6 20 6 11 $200 6.37 $206.37 8 15 4 16 300 6.80 306.80 12 1 80 200 1.00 201.00 714.17 12 31 $221.23 Ans. 714. EULE. Find the amount of the principal to the date of settlement regardless of any payments made. Find the amount of each payment from the time it was made to the time of settlement- Subtract the sum of the payment amounts from the amount . of the principal and the result will be the balance due. WRITTEN EXERCISE 1. A note for $2100 dated Apr. 15, 1903, payable on demand, with interest, bears the following indorsements: June 20, $300; Sept 1, $200; Nov. 25, $750; Dec. 18, $300, What is due Jan. 31, 1904, money being worth 7% ? 2. What is the balance due Apr. 15, 1904, on a note for $525 dated Jan. 1, 1903, bearing 6% interest if the following indorse- ments were made thereon: Mar. 2, 1903, $75; July 15, 1903, $200; Sept. 20, 1903, $100; Jan. 3, 1904, $50? 8. A note for $ 1600 dated Mar. 2, 1903, payable in 6 months .with interest at 6% has the following indorsements: Apr.~3, $450; June 15, $320; July 19, $179.85; Aug. 3, $400. What is due at the maturity of the note ? 4. A note for $950.75 dated Mar. 8, 1903, bears the following indorsements: Apr. 16, $250; June 8, $150; Aug. 2, $200; Sept. 30, $100; Nov. 2, $90. ri What was due Dec. 8, 1903, at 6% ? At 5% ? At8%? i-V C \ 6. A note for $ 1900 dated Jan. 25, 1903, was indorsed as follows ; Apr. 2, $900; May 3, $750; July 3, $100. What remained due Sept. 25, 1903, money being worth G % ? 716-721] EQUATION OF ACCOUNTS 277 EQUATION OF ACCOUNTS 715. Equation of accounts is the process of finding the date when the balance of an account can be paid without loss or gain to either party. 716. Accounts having items on but one side, either debit or credit, are called simple accounts, and the equating of such accounts is called simple equation, or equation of bills. 717. Accounts having both debit and credit items are called compound accounts, and the equating of such accounts is called com- pound equation, or equation of accounts. 718. The term of credit is the time that must elapse before a debt becomes due. If the term of credit is given in days, the exact number of days must be added to the date of the purchase or sale ; if given in months, the number of months, regardless of the number of days included, must be added to the date of the purchase or sale. 719. Book accounts bear legal interest after they become due ; and notes, even if not containing an interest clause, bear interest after maturity. 720. The average term of credit is the time that must elapse before several debits due at different times may be equitably dis- charged in one sum. 721. The average date of payment, the equated date, or due date is the date on which payment or settlement may be equitably made. To illustrate, suppose A stands charged as follows : Jan. 10, $200. Jan. 20, $200. Jan. 30, -$200. If the first charge were not paid on Jan. 10, it would be subject to interest from Jan. 10 to the date of settlement ; if the third charge were paid before Jan. 30, discount should be allowed for the number of days between the date of settlement and Jan. 30 ; if the second charge were paid Jan. 20 no interest would be charged or discount allowed. It will "be seen that the interest on the first charge from Jan. 10 to Jan. 20 is equal to the discount on the third charge from Jan.. 20 to Jan. 30. Hence, the whole account can be equitably paid on Jan. 20, the average date of payment. 278 PERCENTAGE AND ITS APPLICATIONS [ 722-725 722. The focal date is any assumed date of settlement with which the dates of several accounts are compared for the purpose of finding the average term of credit, or due date. Any date may be used as a focal date, and the result will be the same. This is true because all items are equally affected when a different focal date is used. Any rate per cent may be used and the result will be the same. As a matter of convenience always use 6%, and base all computations on the commercial year of 360 days. 723. Only personal accounts are equated. The occasion for equating personal accounts arises from two causes : 1. If any item of any account is not paid when due, the holder of the account suffers a loss. 2. If an item is paid before it is due, the holder of the account realizes a gain. 724. Accounts are equated to ascertain a date when the settle- ment may be made without loss or gain to either the holder of the account or the maker of it. 725. The face value of an item is always to be used in equating accounts. An item not subject to a term of credit is worth its face value the day it is dated. This is always true of an interest-bearing note. Items subject to a term of credit and non-interest-bearing notes are worth their face value at maturity. DRILL EXERCISE 1. How long may $ 2 be kept to balance the use of $ 4 for 10 days ? $ 1 for 20 days ? $ 6 for 15 days ? 2. The use of f 40 for 1 month is equivalent to the use of what sum for 2 months ? 3. If I use 1 6 of B's money for 30 days, how much of my money should he use for 10 days in return for the accommodation ? 4. If I pay one half of an account 10 days before the whole account is due, how long after the whole account is due may I have in which to pay the balance ? 6. If I pay $10 of an account 30 days before it is due, how long may I keep $ 5, the balance account, after maturity ? $ 20 ? 725] EQUATION OF ACCOUNTS 279 6. Jan. 10 Mason & Brown sold F. E. Rogers on account 30 days merchandise amounting to $400. (a) When is the account due? (6) If on Jan. 25 F. E. Rogers paid $ 200, on what date is the balance due? SOLUTIONS, (a) The account is due 30 days after Jan. 10, or Feb. 9. (6) If a payment of $200 was made on Jan. 25, Mason & Brown have had the use of $200 for 15 days ; hence, they should extend the time on the remain- ing $200 15 days beyond the original maturity. Feb. 9 plus 15 days equals Feb. 24, the date on which the remaining $ 200 of the account should be paid. 7. May 5 F. C. Clark & Co. sold Charles H. Jones & Son on account 30 days merchandise amounting to $ 400. May 20 Charles H. Jones & Son made a payment of $ 100 on account. On what date should the balance be paid without loss or gain ? SOLUTION. The original charge matures 30 days after May 5, or June 4. The amount paid, $100, is | of the amount remaining unpaid, $300. Since $100 is paid 15 days before it is due, the $300 may be kept of 15 days, or 5 days, after it is due. June 4 plus 5 days equals June 9, the date on which the remaining $ 300 should be paid without loss or gain. 8. Nov. 1 George B. Thayer & Co. sold E. M. Williams on account 30 days merchandise amounting to $ 600. (a) When is the account due ? (6) If a payment of $ 300 is made on Nov. 1, on what date is the balance due ? 9. Nov. 1 W. D. Lyman sold F. C. Hill on account 30 days merchandise amounting to $ 600. If a payment of $ 200 is made on Nov. 11, on what date is the balance due ? A payment of $ 100 ? A payment of $400? 10. Mar. 20 W. K. Frey purchased a bill of goods of you amount- ing to $300. If no term of credit is given, how much was legally due Mar. 30 ? 11. Smith & Brown bought goods of you as follows : Mar. 20, $300. Mar. 30, $300. (a) How much is legally due on the above account Mar. 30 ? (6) On what day can the amount of the account, $600, be paid without interest ? SOLUTIONS, (a) If the account is settled Mar. 30, on the charge of Mar. 20 there is 10 days' interest due. The interest on $ 300 for 10 days is $.50, which, added to the amount of the account, $ 600, makes the amount legally due $ 600.50. 280 PERCENTAGE AND ITS APPLICATIONS [ 725-727 (6) 10 days' interest on $300 is equivalent to 5 days' interest on $600; therefore if the whole account had been paid 5 days before Mar. 30, or Mar. 25, only the face of the account, $ 600, would have been due. PROOF. The interest on the first charge from Mar. 20 to Mar. 25 is $.25, and the discount on the second charge from Mar. 25 to Mar. 30 is $.25. The interest and discount being equal, the face of the account, $600, can be paid Mar. 25 without loss or gain to either party. 12. If I owe $200 due May 1, and $400 due May 31, at what time can both debts be equitably paid ? SIMPLE EQUATIONS 726. To find the equated time of an account when the items are all on one side and are subject to no terms of credit. 727. Example. Robert S. Campbell is charged on the books of Spencer, Mead & Co. as follows : Nov. 1, To mdse., $ 60. Nov. 7, To mdse., 120. Nov. 13, To mdse., 180. Nov. 19, To mdse., 240. On what date may the amount of the account, $600, be paid without interest ? SOLUTION. For convenience 1 -rev ^^^ ^ 2 r ^^^ 3.00 ; -T 300 ^ J 00 SOLUTION Take Apr. II, 1903, as the focal date. Dates Mar. 1 Mar. 16 Dates Mar. 26 Apr. 11 Items $600 300 $900 Interest Periods 41 da. 27 da. Credits Interest Periods 17 da. 00 da. Interests $4.10 1.35 $5.45 Interests $.85 .00 .85 $ 900 - $ 600 = $ 300, the balance of the account. $ 5.45 - $ .85 = $ 4.60, the interest due the holder of the account Apr. 11, 1904. The interest on $ 300 for 1 day = $ .05. $4.60-7- $.05 = 92. Therefore $300 will yield $4.60 interest in 92 days. Apr. 11, 1904, Jaines B. Halsey not only owes the balance of the account, $ 300, but 92 days' interest on this amount. Therefore the face of the account was due 92 days before Apr. 11, 1904. Apr. 11, 1904, 92 days = Jan. 10, the equated date of payment. PROOF. To prove the correctness of the above work it is necessary to show that the payment of $ 300 on Jan. 10, 1904, will result in neither a loss nor gain to either debtor or creditor. This may be done by equating the account again with Jan. 10 as the focal date. SOLUTION Debits Dates Mar. 1 Mar. 16 Items $600 800 Interest Periods Interests 51 da. $6.10 65 da. 3.25 Total debit interest, $8.35 286 PEKCENTAGE AND ITS APPLICATIONS [733 Credits Bates Mar. 25 Apr. 11 Items $300 300 Interest Periods 75 da. 92 da. Interests $3.75 $4.60 $8.35 Total credit interest, From the above statement we see that if the holder of the account had, on Jan. 10, received cash for each item charged in the account, he would have had the use of $ GOO for 51 days and $ 300 for 65 days, and would have gained $ 8.35 interest ; but the first payment of $ 300 was made Mar. 25, and the second Apr. 11. By not receiving payment on Jan. 10 he would not have the use of $300 for 75 days and 92 days; hence he would have lost $8.35 interest. Since this loss is can- celed by a gain of $8.35, it is shown that the balance of the account may be equitably settled Jan. 10. 2. Find the equated date for paying the balance of the following account : z *Z~6,j~4., 600 >C? /SLsC^L^j!^ .100 /) 1*0 ^ tj .300 = SOLUTION Take May 1, 1904, as the focal date. Debits Dates Feb. 1 Feb. 10 Terms of Credit 90 da. 60 da. Due Dates May 1 Apr. 10 Items $600 300 $900 Interest Periods 00 da. 21 da. Interests $ .00 1.05 $1.05 Credits Dates Feb. 20 Mar. 30 Items $300 300 $600 Interest Periods 71 da. 32 da. Interests $3.55 1.60 $5.15 $ 900 - $ 600 = $ 300, the balance of the account. $5.15 - $ 1.05 = $4.10, the interest due Ira B. Perkins May 1. The interest on $ 300 for 1 day = $ .05. $4.10 -=-$.05 = 82. 733-735] EQUATION OF ACCOUNTS 287 Therefore, $ 300 will yield $ 4.10 interest in 82 days. ' On May 1 there is 82 days' interest due Ira B. Perkins. Hence, the face of the account is not due until 82 days after May 1. May 1 plus 82 days equals July 22, the equated date of payment. PROOF Debits Due Dates Items Interest Periods Interests May 1 $600 82 da. $8.20 Apr. 10 800 103 da. 5.15 $13.35 Credits Due Dates Items Interest Periods Feb. 20 $300 153 da. Mar. 30 300 114 da. The above statement shows that on July 22, 1904, the loss suffered by the holder of the account is canceled by the gain realized. Therefore, the account is proved to be equitably due July 22, 1904. In proving the equation of accounts, the equitable settlement of which is found to come at a date within the account or between its extreme dates, the difference between the interest and discount of the debit items from their respective dates to the due dates must be offset or balanced by the difference between the interest and discount on the credit items from their respective dates to the due date, within one half cent of the interest or discount on the balance for one day. 734. General Principles. 1. If the larger interest is on the larger side of the account, it shows that the holder of the account has suffered a loss because items were not paid when due; if on the smaller side, it shows that the holder of the account has gained because items were paid before they were due. 2. A loss is offset by dating back; a gain is offset by dating forward. 735. Hence the following rule : Find the balance of the account and also the excess of inter- est from the latest date as the focal date. If the balance of account and excess of interest are on the same side f date back ; if on opposite sides t date forward. 288 PERCENTAGE AND ITS APPLICATIONS [ 735 ORAL EXERCISE 1. Oct. 1, Henry Ball & Co. sold F. E. Gorham a bill of merchan- dise amounting to $800. Terms: 30 days. Oct. 11, F. E. Gorham made a payment of $400 on account. When is the balance, $400, equitably due ? SOLUTION. By the terms of the contract the account would mature Oct. 31. Since a payment of $ 400 is made 20 days before maturity, Henry Ball & Co. have gained the use of $400 for 20 days. To offset this gain they should allow F. E. Gorham 20 days beyond the original maturity of the account in which to pay the balance. Oct. 31 plus 20 days is equal to Nov. 20, the date on which the balance of the account may be equitably paid. Find the time for equitably paying the balance of the following accounts. Terms: cash. Dr. Or, 2. Oct. 1, $800; Oct. 11, $400. 8. Apr. 12, $300; Apr. 17, $150. 4. July 5, $900; July 20, $300. Find the time for equitably paying the balance of the following accounts. Terms : 30 da. Dr. Or. 5. Oct. 1, $600; Oct. 6, $200. 6. May 10, $400; May 6, $ 200. 7. June 15, $800; June 25, $600, WRITTEN EXERCISE Find the equated date of payment of each of the following accounts. Prove all work. 1. E. M. ELDRED & Co. 1903 1903 Jan. 20 To mdse. 600 Feb. 8 By cash 300 Feb. 25 To mdse. 300 Mar. 20 By cash 300 8 735] EQUATION OF ACCOUNTS VICTOR H. BROWN & Co. 289 1903 1903 Jan. 15 To mdse. 600 Jan. 25 By cash 1000 30 To mdse. 300 Feb. 15 By cash 200 Feb. 8. To mdse. 600 20 To mdse. 300 B. N. SHERWOOD & SON 1903 1903 Apr. 8 To mdse. 420 Apr. 18 By cash 240 20 To mdse. 180 20 By cash 60 May 15 TO mdse. 540 June 2 By cash 300 June 2 To mdse. 60 W. I. PARKER 1903 1903 Aug. 5 To mdse. 200 Sept. 8 By cash 240 20 To mdse., 2 mo. 360 Oct. 5 By 60-da. note 240 Sept. 15 To mdse., 30 da. 360 (no interest) REED & HAMLIN 1903 1903 June 20 To mdse., 30 da. 300 July 1 By cash 100 30 To mdse., 60 da. 180 Aug. 1 By cash 100 Aug. 1 To mdse., 30 da. 480 Sept. 1 By cash 100 Sept. 20 To mdse., 30 da. 120 Oct. 1 By cash 100 1904 Jan. 1 By cash 100 NOTE. Interest may be computed on one of the five similar credit items for the aggregate number of days. MOORE'S COM. AR. 19 290 PERCENTAGE AND ITS APPLICATIONS 736-738 EQUATION OF ACCOUNTS SALES 736. An account sales is equated in practically the same man- ner as an ordinary ledger account. The agent's charges constitute the debits of the account, and the gross sales the credits. The agent's charges include freight, cartage, storage, commission, insurance, advertising, guaranty, etc. 737. When equating accounts sales, agents generally consider such charges as freight, cartage, storage, and insurance, as not due until they have been paid. 738. When goods are sold promptly, agents usually consider commission and guaranty as due on the date of the last sale. When, the sales are large and there are long intervals between them, the commission or guaranty is considered due on the average due date of the sales. When goods are sold for cash, or on short time, the account sales is seldom averaged. WRITTEN EXERCISE 1. Find when the net proceeds of the following account sales are due by equation. Consider the commission as due on the date of the BOSTON, MASS., Oct. 8, 1903. PARKER, MONTGOMERY & Co. Sold for the account of 'W. D. SPRAGUE, Buffalo, KY. 1903 Sales Sept. 23 95 bbl. 6.<, cash Oct. 1 200 bbl. 6., 1 mo. 18 65 bbl. 5.80, 60 da. Nov. 3 110 bbl. 6.80, 30 da. 25 130 bbl. 6.75, cash Charges Sept. 24 Freight 62 60 20 Cartage 30 Oct. 28 Cash advanced 2000 Nov. 15 Cooperage 5 25 Commission, 4% 738-742] CASH BALANCE 291 2. Using the foregoing form for a model, arrange the following narrative in the form of an account sales and find when the net pro- ceeds are due by equation. Consider the commission as due on the average due date of the sales. R. J. Briggs & Co., Boston, Mass., sold for the account and risk of B. Sornmers & Co., Chicago, 111., 1000 bbl. potatoes as follows : Nov. 2, 1903, 400 bbl. peach blows at $ 3, cash ; Dec. 1, 1903, 300 bbl. pink eyes at $3.50, 30 da.; Jan. 1, 1904, 100 bbl. peach blows at $ 3.60, cash ; Jan. 25, 1904, 200 bbl. pink eyes at $ 3.50, 30 da. The charges were as follows: Nov. 1, 1903, freight, $ 350; Nov. 1, 1903, cartage, $50; Nov. 1, 1903, insurance and advertising, $100; commission and guaranty, 3%. CASH BALANCE 739. Cash Balance treats of showing the balance or amount due on an account at any given date. 740. The cash balance of an account on which interest is not charged is the difference between the two sides of the account in the ledger. The cash balance of an account on which interest is charged is the difference between the two sides of the account after interest has been added to the items past due, or deducted from the items not due at the date of settlement 741. Each item of an account equitably draws interest from the time it becomes due to the date of settlement, and each item paid before maturity is equitably entitled to discount for the time from the date of payment to the date it is due. Whether interest is charged on the items of a running account or not is usually regulated by the custom of the business, or an agreement between the parties thereto. As a rule, retailers do not charge interest on the items of run- ning accounts, but frequently the balance of a closed account is considered in- terest-bearing from the date the balance is brought down. Wholesale dealers usually charge interest on the items of an account at the expiration of the time specified in the terms of credit. 742. Example. When money is worth 6% per annum, what is the cash balance due on the following account June 23, 1903 ? 292 PERCENTAGE AND ITS APPLICATIONS [ 742-743 30/9 300 SOLUTION Dates Jan. 1 Jan. 31 Terms of Credit Due Dates Items Interest Periods 1 mo. Feb. 1 $ 600 142 da. 10 da. Feb. 10 1800 133 da. $2400 Credits Dates Items Interest Periods Interests i^b. 19 $300 124 da. $6.20 ?eb. 28 300 115 da. 5.75 tor. 6 300 109 da. 5.45 Interests $ 14.20 39.90 $54.10 $900 $ 17.40 $2400 + $54.10 = $2454.10, the amount due on account June 23, 1903, had no payments been made. $900 + $ 17.40 = $917.40, the value of the payments on June 23, 1903. $2454.10 - $917.40 = S 1536.70, the cash balance of the account June 23, 1903. 743. From the foregoing explanation the following rule may be derived : Find the maturity of each item of the account. Compute the interest on each item from the date it becomes due to the date of settlement. To the sum of the debit items add the sum of the debit inter- ests ; also to the sum of the credit items add the sum of the credit interests. Subtract these totals and the result is the cash balance. 743] CASH BALANCE 293 WRITTEN EXERCISE /. If money be worth 7% per annum, what is the cash balance due on the following account, July 1, 1903 ? HENRY HARRISON & Co. 1903 1903 Jan. 31 To mdse. 450 Jan. 2 By mdse. 600 Mar. 30 To mdse. 450 Feb. 13 By cash 300 Mar. 29 By mdse. 300 2. What is the cash balance of the following account, Apr. 1, 1903, if the money be worth 8 % per annum ? BENJAMIN TRACY & SON 1902 1902 Aug. 4 To mdse., 1 mo. 200 Oct. 1 By cash 150 Sept. 1 To mdse., 2 mo. 400 Nov. 1 By cash 150 Oct. 31 To mdse., 4 mo. 600 Dec. 1 By cash 150 Dec. 3 To mdse. 300 1903 Jan. 1 By cash 150 Feb. 1 By cash 150 Mar. 1 By cash 150 3. Equate the following account and find the cash balance due Apr. 1, 1903, if money be worth 7% per annum. Prove the work. BROWN, SHIPLEY & Co. 1902 1903 Sept. 9 To mdse. 600 Jan. 2 By cash 500 Oct. 1 To mdse., 2 mo. 300 Mar. 16 By 2-mo. note Dec. 13 To mdse., 1 mo. 150 (on interest) 100 1903 Apr. 30 By 3-mo. note Jan. 31 To mdse. , 1 mo. 450 (no interest) 300 May 1 By cash 200 NOTE. To find the cash balance of an equated account. Find the difference between the equated date of payment and the date of settle- ment, and compute the interest on the balance of the account for this time. The sum of the interest thus found and the balance of the account is the cash balance due at the date of settlement. 29-4 PERCENTAGE AND ITS APPLICATIONS [ 744-746 BANKERS' CASH BALANCE 744. Many bankers balance their accounts with their corre- spondents at regular intervals, monthly, quarterly, semiannually, or yearly, allow interest on all sums that have been credited, charge interest on all sums that have been debited, and bring the cash bal- ance down to a new account to subsequently draw interest the same as the regular items in the account. 745. Some bankers and trust companies balance their accounts with depositors at regular intervals and allow interest on the bal- ances credited. 746. Example. Find the balance due on the following account Apr. 1, 1904, settlements being made quarterly with interest at 6%. ffff 2. 200 BA: T K ACCOUNT CURRENT DATES DEBITS CREDITS CREDIT BALANCES DAYS CREDIT INTERESTS 1904 Jan. 1 800 800 3 40 4 100 700 4 47 8 1000 1700 3 85 11 600 1200 30 6 Feb. 10 800 2000 10 3 33 20 150 1850 14 4 32 Mar. 5 200 2050 25 8 54 30 50 2000 2 67 800 2800 24 58 800 2000 + 24 .68 = 2024 .58 746-747] CASH BALANCE 295 SOLUTION. By arranging the debit and credit items in the order of their dates the balance of the account at each of the dates may easily be determined. The account shows a credit balance of $ 800 from Jan. 1 to Jan. 4, or for 3 days, when the amount is reduced to $ 700 by the charge of $ 100. The interest on $ 800 for 3 days is $ .40. The account shows a credit balance of $ 700 from Jan. 4 to Jan. 8, or for 4 days, when the amount is increased to $1700 by the credit of $1000. The interest on $700 for 4 days is $.47. Continuing in this manner to Apr. 1, it is found that on that date the account shows a credit balance of $2000, and that on the daily balances there has accumulated $24.58 interest. The cash balance of the account is then found to be $2000 plus $24.58, or $2024.58. NOTE. Had there been a debit balance on any of the above dates, two extra columns would have been required in the operation, one for the debit balances and one for the debit interests. The difference between the debit and credit interests would then be the balance of accrued interest. 747. Hence the following rule may be derived : Arrange the debits and credits in the order of their dates and find the 'balance of the account at each date. Find the interest on each balance for the period that it remains unchanged. If the balance of interest and the balance of the account are on tJie same side, take their sum; if on the opposite sides, take their difference. The result obtained is the cash balance due. WRITTEN EXERCISE 1. Find the balance due on the following bank account July 1, 1903, at 4%. Dr. CENTRAL NATIONAL BANK, Springfield, Mass. Cr. 1903 1903 Apr. 15 To cash 200 Apr. 1 By cash 1200 20 To cash 200 May 4 By cash 900 May 20 To cash 300 June 1 By cash 500 June 10 To cash 300 20 By cash 420 #. Find the balance due on the following bank account, Apr. 1, 1903, at 3%. 296 PERCENTAGE AND ITS APPLICATIONS 747-753 Dr. MERCHANTS NATIONAL BANK, Kochester, N.Y. Cr. 1903 1903 Jan. 2 To cash 600 Jan. 1 By cash 1500 Feb. 8 To cash 480 31 By cash 1200 21 To cash 240 Feb. 15 By cash 120 Mar. 20 To cash 180 Mar. 31 By cash 400 8. The Security Trust Company, Rochester, N.Y., allows inter- est to its depositors on daily balances at 4% per annum, payable quarterly. Find the cash balance of the following account with George W. Snyder, Apr. 1, 1903: Jan. 1, 1903, deposited $900; Jan. 8, drew out $200; Jan. 12, deposited $750; Jan. 15, drew out $475; Feb. 9, deposited $721.90; Feb. 24, drew out $121.90; Mar. 15, deposited $795.98 ; Mar. 30, drew out $400. SAVINGS-BANK ACCOUNTS 748. A savings bank, as its name implies, is an institution organ- ized for the purpose of encouraging economy and thrift and caring for the savings of the people. 749. The deposits in savings banks are practically payable on demand. Savings banks generally reserve the right to require depositors to notify them from 30 to 60 days before making a withdrawal. 750. The interest term is the time between dates at which divi- dends of interest are declared. Dividends of interest are usually declared semiannually. 751. If interest is not withdrawn, it is placed to the credit of the depositor on the books of the bank, and draws interest the same as any regular deposit. In this way savings banks pay their depositors compound interest. No interest is allowed on fractional parts of a dollar. 752. The interest days are the days on which interest is allowed to commence. 753. Savings banks are not uniform in their practice of allow- ing interest on deposits made after the beginning of the interest term. In some savings banks deposits begin to draw from the first 753-757] SAVINGS-BANK ACCOUNTS 297 of each quarter ; in others, from the first of each month. The latter method is preferable for persons having a small income. Monthly interest days usually begin on the first day of each month ; quar- terly interest days on Jan. 1, Apr. 1, July 1, and Oct. 1 ; semiannual interest days on Jan. 1 and July 1. 754. Nearly all savings banks allow interest on only those sums that have been on interest for the full time between the interest days. Thus, if the interest begins quarterly, only those sums that have been on deposit for the full quarter draw interest ; if monthly, only those sums that have been on deposit for the full month draw interest. 755. Savings banks furnish each depositor with a small book called a pass book, in which are entered all amounts deposited and all amounts withdrawn, together with the interest credited to the depositor at the expiration of the interest term. 756. To find the balance due a depositor when there are no with- drawals. 757. Example. The interest term of Wildey Savings Bank is 6 months. A deposited in this bank Dec. 20, 1903, $200; Feb. 10, 1904, $100; Apr. 1, 1904, $50; June 8, 1904, $50. No with- drawals having been made, what was due July 1, 1904, if interest at 4% per annum be reckoned on the deposits (a) from the first of each quarter ? (6) from the first of each month ? (a) DATES DEPOSITS DAILY BALANCES INTEREST DAYS SMALLEST QUARTERLY BALANCES QUARTERLY INTERESTS 1903 Dec. 20 200 200 Jan. 1 1904 Feb. 10 100 300 Apr. 1 50 350 Apr. 1 200 2.00 June 8 60 400 July 1 400 Julyl 360 3.50 6.50 400. 405.50 SOLUTION. If interest begins on the first of each quarter, only the smallest balance for any quarter will draw interest. The deposit of Feb. 10, being made 298 PERCENTAGE AND ITS APPLICATIONS [757 after the beginning of the first quarter, will not begin to draw interest until the beginning of the second quarter. Hence, the only sum that draws interest for the first quarter is the deposit of Dec. 20, $200. The interest on $200 for one quarter is $2. The deposits of Feb. 10 and Apr. 1, together with the smallest balance for the first quarter, will draw interest for the second quarter. The deposit of June 8, being made after the beginning of the second quarter, will not draw interest until the beginning of the third quarter. Hence, the sum to draw interest for the second quarter is $350 ($200 + $ 100 + $50). The interest on $350 for one quarter is $3.50. $2 -f $3.50 = $5.50, the interest to be added to the account July 1. $400 + $5.50 = $405.50, the balance due on the account July 1. (W DATES DEPOSITS INTEREST DATS SMALLEST MONTHLY BALANCES MONTHLY INTERESTS 1903 Dec. 20 200 Jan. 1 200 1904 Feb. 10 100 Feb. 1 200 1.00 Apr. 1 June 8 50 50 Mar. 1 Apr. 1 May 1 June 1 July 1 300 350 350 350 400 1.50 1.75 1.75 1.75 3)7.75 2.58 5.17 400. 405.17 SOLUTION. Only the smallest balance on deposit each month will draw interest. The amount deposited Dec. 20 will not begin to draw interest until Jan. 1. The smallest balance for each month is as shown above. The aggregate interest on the smallest monthly balance is found to be $7.75 at 6%, or $5.17 at 4%. $400, the balance on deposit July 1, + $5.17 = $405.17, the balance due the depositor July 1, 1904. WRITTEN EXERCISE 1. A made the following deposits in a savings bank : July 1, 1903, $50; July 30, $50; Aug. 20, $100; Oct. 5, $200; Nov. 8, $ 150 ; Dec. 15, $ 200. If the interest term is 6 months, what is the balance due A Jan. 1, 1904, interest being allowed on balances from the first day of each quarter at 4% per annum ? 757-759] SAVINGS-BANK ACCOUNTS 299 2. J. M. Carroll made the following deposits in the Security Savings Bank: Dec. 18, 1903, $400; Jan. 5, 1904, $200; Mar. 8, 1904, $ 100; May 20, 1904, $ 30; July 1, 1904, $40. If the interest term is 3 months, what is the balance due J. M. Carroll July 1, 1904, interest at 4% being computed from the first day of each quarter ? 758. To find the amount due depositors when there are withdrawals. 759. Example. Find the balance due July 1, 1904, on the follow- ing account. Deposits: Dec. 10, 1903, $600; Apr. 10, 1904, $200; May 20, $150. Withdrawals: Mar. 10, $300; May 1, $50. The interest term is 6 months and interest at the rate of 4% per annum is allowed from the first day of each quarter. DATES DEPOSITS WITHDRAWALS DAILY BALANCES INTEREST DATS SMALLEST QUARTERLY BALANCES QUARTERLY INTERESTS 1903 Dec. 10 600 600 Jan. 1 1904 Mar. 10 300 300 Apr. 1 300 Apr.l 300 3.00 Apr. 10 200 500 Mayl 50 450 May 20 150 600 July 1 300 3.00 6.00 600. 606.00 SOLUTION. Interest begins quarterly and the interest days are Jan. 1, Apr. 1, and July 1. The smallest balance for the first quarter is $300, and the smallest balance for the second quarter is $ 300. The quarterly interest on these -two balances aggregates $6.00. $600, the amount on deposit July 1, plus $6.00, the interest due on that date, equals $ 606.00, the balance of the account July 1, 1904. WRITTEN EXERCISE 1. W. E. Small deposits in a savings bank as follows : Jan. 1, $ 400 ; Feb. 2, $ 200 ; Mar. 10, $ 150 ; Apr. 2, $ 60 ; May 18, $ 200 ; during the same time he withdrew as follows: Jan. 10, $50; Feb. 4, $ 50 ; Apr. 5, $ 50 ; June 30, $ 80. The interest term is 6 months. What interest at 4% per annum, to commence from the first of each quarter, should be added to the account July 1 ? 300 PERCENTAGE AND ITS APPLICATIONS [ 759-764 2. In the Home Savings Bank the interest term is 6 months, and the interest days are Jan. 1, Apr. 1, July 1, and Oct. 1. Find the balance due on the following account J uly 1, 1904, at 4 % per annum. 00 JJ2. STOCKS 760. A joint stock company is a partnership in which the affairs of the business are conducted by officers chosen by the stockholders. 761. A corporation is a fictitious person. It consists of several natural persons who, in the name .of the corporation, are authorized by law to transact business. The instrument which defines the rights and duties of the corporation is called a charter. It is issued by government under seal. 762. Stocks is a general term applied to shares in the capital stock of banks, insurance, railroad, and other incorporated or joint stock companies. 763. A stock certificate is a written or printed instrument of a joint stock company or corporation issued to the stockholder, certi- fying the value of each share and the number of shares such cer- tificate represents. 764. A share represents simply a certain component part of the capital stock. It is commonly $25, $50, or $100. The stock certificate represents the number of shares specified thereon. 765-774] STOCKS 301 765. The capital stock of a company is the sum of all the shares issued at their par value. 766. The common stock of a corporation is the stock which is ordinarily issued to the incorporators. 767. Preferred stock is stock on which dividends are paid before any allowance is made for dividends on the common stock. Preferred stock is sometimes issued to take up the floating indebtedness of a corporation. Agreed dividends are declared on it at certain intervals out of the net earnings, and before any dividend can be declared on the common stock. Such stock is frequently issued upon the reorganization of railroads or the consolidation of joint stock companies. 768. The par value of stocks is their face value ; their market value is the sum at which they are quoted in the market. 769. Stocks are above par or at a premium when they are worth more than their face value ; below par or at a discount when they are worth less than their face value. 770. Stock quotations are published prices or rates per share that stocks sell for. Thus, when stock is 3% above par it is quoted at 103 ; when it is 2 % below par it is quoted at 98. 771. A dividend is a pro rata division of profits among the stock- holders of a company or corporation. The income from stocks is in the nature of dividends, and is dependent upon the prosperity of the company or corporation. Dividends are declared at a certain per cent on the par value of the capital stock of the company, either quarterly, semiannually, or annually. The dividend on preferred stock is often at a different rate from that on common stock. 772. An assessment is a sum levied pro rata upon the stock- holders of a corporation to cover losses, etc. 773. Stock brokers are persons who act for others in buying and selling stocks at a stock exchange. For this service they charge a certain rate per cent commission, called brokerage, on the par value of the stocks dealt in. 774. Brokerage is usually \% of the par value of the stock dealt in. Occasionally il is as high as \ % or | %, or as low as y 1 ^ %. 302 PERCENTAGE AND ITS APPLICATIONS [775-781 775. Stocks are generally bought and sold eitker " regular way," or "cash," or "buyer three," or "seller three." Stock soM "regular way " is to be paid for and delivered the next day ; stock sold " cash " is deliverable the day sold. When stock is bought " seller three," the seller of the stock may deliver it on any one of the three days following the transaction, at his option, but cannot be required to deliver it till the third day. When stock is sold " buyer three," the buyer may demand delivery at any time within three days, but is obliged to take and pay for it by the third day. If a stock pays a dividend while a transaction is being executed, the dividend belongs to the purchaser of the stock. 776. A margin is a deposit made with a broker by a person who wishes to speculate in stocks, such deposit being used by the broker to protect himself against loss. The margin is usually 10% of the par value of the stock dealt in. A wishes to speculate, and deposits with B, his broker, $1000 as a margin, directing B to buy 100 shares of a stock quoted at 90. B would pay for the stock $9000, $1000 of which is the margin furnished by A ; B furnishes $8000, and charges the usual rate of interest on that sum for " carrying " the stock. In case the quoted value of the stock drops below 90, the margin must be made good by A's depositing an additional amount. If A fails to make good his margin, B may sell the stock to protect himself from losing any of the money he has fur- nished. 777. Collateral consists of stocks, notes, etc., given in pledge as security when money is borrowed. 778. If any one has sold stock he does not own, in the hope of realizing a profit by buying it in at a lower price, he is said to be "short." 779. If stock has been sold " short," and the seller buys it in to realize a profit or to protect himself against loss, he is said to " cover his short sales." 780. Stock sold Ex. Div. means that a recently declared divi- dend is received by the seller. 781. When a corporation increases the quantity of its stock with- out increasing the value of its property, which the stock is supposed to represent, the stock of such a corporation is said to be watered to the extent of the increase. 782] STOCKS 303 782. The following list, showing the highest, lowest, and closing quotations of certain stocks, and net changes from closing prices of the previous day, is reproduced from the Wall Street edition of the New York Sun under date of Jan 17, 1907. High- Low- THE STOCK MARKET Clos- Net High- Low- Clos- Net est est ing CVge est est ing (. W brokerage ^ % ? 310 PERCENTAGE ANDT ITS APPLICATIONS [796-800 8. On Jan. 17 a speculator deposited $4000 as a margin, and by his orders the broker bought 200 shares Atchison, Topeka & Santa Fe at 108J, 100 shares Atlantic Coast Line at 133|, and 100 shares Chicago & Alton at 27J. On Jan. 25 (the margin being nearly ex- hausted) the broker " sold him out " at the following quotations : Atchison, Topeka & Santa Fe at 98f, Atlantic Coast Line at 119|, and Chicago & Alton at 24J. How much did the speculator lose ; interest 6 %, brokerage BONDS 797. A bond is a written or printed obligation under seal issued by a company or corporation, municipal or state government, or by the federal government. It is conditioned to pay a certain sum of money at a specified time and at a fixed rate of interest, payable at regular intervals. Bonds of business corporations are usually secured by mortgages on their real estate. Municipal bonds are issued by vote of the people or their repre- sentatives, and for their payment a sinking fund is accumulated by a yearly rate per cent levied on all the real property within the limits of the municipality. 798. Government bonds are bonds issued by the federal govern- ment. Their names are usually derived from the interest they bear and the time when due. Thus " U.S. 4's, 1912," is understood to mean "United States bonds bearing 4% interest, and due in 1912"; and "U.S. 3's, 1925," is understood to mean " United States 3% bonds due in 1925." 799. A coupon bond is a bond that has coupons or certificates of interest attached. When the interest becomes due, these coupons are detached and surrendered upon receipt of the interest repre- sented by them. The interest coupons on government coupon bonds are payable to the bearer, and will be cashed by any bank or banker in the United States. Coupon bonds may be converted into registered bonds of the same issue. 800. A registered bond is one which is payable to the owner as registered in the books of the corporation or government issuing it. Registered bonds can be transfered only by assignment and registry on the books. The interest on registered bonds is paid by checks, payable to the order of the registered owner, and sent to him. The checks for interest on government bonds are readily cashed by banks and bankers. 801-8C3] BONDS 311 801. As with stocks, the par value of bonds is their face value ; the market value is the amount at which they are quoted in the market. Bonds are above par or at a premium when they are worth more than their face value ; below par or at a discount when they are worth less than their face value. 802. Bond quotations are the market prices or rates that the bonds sell for. The income from bonds, unlike that from stocks, is fixed; that is, it is in no way affected by the general conditions of the corporation, so long as the corporation is solvent. Bonds are usually quoted flat; that is, the quoted price is for the bond as it is at the time of the quotation, including accrued interest, except that after the closing of the books registered bonds are quoted less the interest. The interest then due belongs to the holder of the bonds at the time the books are closed. 803. The list herewith shows part of one day's bond sales (so many dollars par value of each) on the floor of the New York Stock Exchange ; it is taken from the New York Sun, Wall Street edition, under date of Jan. 17, 1907 : Adams Exp 4s 500 102^ Chi & E 111 s f 6's 111 Central 4s 1953 2000 103| Penna cv 3s 20000 96| 3000 102* 1000 lOOf 111 Cent L div 3|s 1000 96| Am Ice deb 6s Chi Mil & St P 4s 1000 89f Penna 3s 1915 1000 .... 89 1000 106 3000 87 J 20000 92 Am Tobacco 6s Chi R I & Pac RR Mo K & T s f 4|s 22000 93 3000 110J gold 5s 19000 . 87 3000 92| 39000 110 1000 90 Mo Pacific 5s 1920 24000 93 2000 110 3000 . 90f 1000 . . . 105 20000 92| Am Tobacco 4s 2000 90 N Y Central 3^s 2000 93 3000 77f 5000 90 4000 93 \ 7000 93| 6000 . 78 Erie c v 4s ser A 58000 93 \ Union Pacific 4s registered 10000 100 North Pacific 3s 2000 101^ 500 75| 5000 1004 4000 73| 4000s 15. . .10 If Bait & Oh gold 4s 16000 101 15000 100 37000 99 5000 73 5000 73 U S 3s cpn 3500 103 Ches & Ohio 6s 2000 . . . 99| 2000 73 U S Steel s f 5s 1000 115 10000 100 Or S L fdg 4s 6000 98 Ches & Ohio 4|s 69000 .... 991 4000 94 17000 98 \ 1000 104 Erie prior lien 4s - 5000 94} 4000 98 Chi & Alton 3s 4000 97 1 , Penna 4s 1921 11000 98 l 4000 76 Green Bay & West 1000 107 11000 98 Chi Bur & Q 4s deb ser B 3000 106| 1000 98 1 2000.. . 96i 2000.. . m 3000.. ..1061 19000.. . 984 312 PERCENTAGE AND ITS APPLICATIONS [804 NOTE. Brokers usually charge ^% brokerage for transactions in bonds. That rate is to be understood if none is specified. 804. Example. What rate per cent per annum interest will Chicago & Alton 3j's yield on the investment, if bought through a broker at the price quoted in the list, p. 311 ? SOLUTION $76 -f .125 = cost of $ 100 par value of the bonds. $3.50 = the income on $100 worth of the bonds. $8.60 -*- $76.125 (cost of bonds) = 4.5977+ %, the rate of income. WRITTEN EXERCISE 1. If a broker invested on his own account in Chesapeake 6c Ohio 5's as quoted in the list, p. 311, what per cent of income would he receive ? 2. Find the proceeds of the United States 3's coupon sold through a broker. S. How much must be invested in Chesapeake & Ohio 4-'s to produce a semiannual income of $1350; regular brokerage? 4. What per cent income will be produced by $ 358,750 invested in Adams Express Company's 4's at the market quotation, allowing the regular brokerage? 5. How much must you invest through a broker in Pennsyl- vania 4i-'s, 1921, at the last quotation, so that you may have an income of $2700 per year? This income is what per cent of the investment ? 6. A sells through a broker one $ 5000 Pennsylvania 3, 1915, at the last quotation, and loans the proceeds at 5%. How much will his yearly income thereby increase? 7. How many $ 1000 Illinois Central 4's, 1953, bought at 98J and sold at the list quotation, will yield $2375 gain, usual brokerage both ways ? 8. A has an annual income of $ 880 on an investment in Balti- more & Ohio $500 gold 4's. How many does he own? If they were bought at the quotation in the list, through a broker, what rate per cent per annum does he receive on his investment ? 804] BONDS 318 9. Which would be the better investment, Erie convertible 4's, series A, at the last quotation in the list, or Missouri Pacific 5's, 1920, if both were purchased through a broker? How much better? WRITTEN REVIEW 1. What will be the cost, including \ / brokerage, of 250 shares Denver & Rio Grande, 300 shares Atchison, Topeka & Santa Fe, 50 shares New York Central, 40 shares Ontario & Western, 125 shares Louisville & Nashville, 150 shares American Tobacco pre- ferred, all at the closing quotations on the list, p. 303? 2. What annual income is derived from investing $27,500, exclu- sive of brokerage, in Consolidated Gas at 137^, if it averages 5 % annual dividends ? 3. How much must be invested, exclusive of brokerage, in Amal- gamated Copper at 115J, so that an annual income of $ 2500 may be realized if a 4 % yearly dividend is declared ? 4. A bought through his broker, at |-%, 500 shares Pacific Coast, for which he paid the broker $ 60,812.50. With what market quo- tation does the price he paid agree ? How much was the brokerage? 5. In example 1 if the average annual dividend was 41 %, what was the rate per cent of interest on the investment? 6. In example 2 what is the rate per cent interest on the invest- ment ? 7. On Jan. 2 I deposited with my broker $6000 as a margin, and he bought for me 250 shares Erie at 40 f, 200 shares Missouri Pacific at 88^, and 150 shares Brooklyn Rapid Transit at 82. On Jan. 17 the stocks were quoted late in the day at the closing figures in the list, p. 303. How much must I deposit to make my margin good ? If the broker had " sold, me out" because I could not make my margin good, how much would I have lost ? ' 8. A sold 500 shares Louisville & Nashville stock at 135J, through a broker, and bought with the proceeds of the sale Chicago, Milwaukee & St. Paul 4 % bonds at 106, through a broker. How many $500 bonds did he get, and how much unexpended balance was there due him ? 314 PERCENTAGE AND ITS APPLICATIONS [804-806 9. Sold three $ 1000 American Tobacco 6's at 110J, and with the proceeds bought Northern Pacitic at 158. Later in the day I sold the stock at 160J. How much did I gain, allowing the usual brokerage on all the transactions ? How much did the broker have belonging to me ? 10. American Ice Securities pays an annual dividend of 7 % ; Delaware & Hudson, 9%; Baltimore & Ohio, 6%; Erie, 2d pre- ferred, 4%. If bought at the closing quotations in the list, p. 303, with no brokerage, what is the rate per cent of income on each ? 11. The American Smelting Kefining Co. had reported net earn- ings; during the fiscal year 1906, of $ 10,161,358. Its common stock was $50,000,000. If 5% of the net earnings is set aside as surplus fund, a 7 % dividend is declared, and the balance carried to undi- vided profits, what sums go respectively to surplus fund, to divi- dend, to undivided profits ? 12. What sum invested in Chicago & Eastern Illinois sinking fund 6's at 100 1 will produce a yearly income of $3000, no brokerage ? 18. X owns 200 shares Reading, 1st preferred, which cost him $ 18,000. He realizes annually 5 % on his investment. What rate of dividend was declared ? H. The net earnings of the Canadian Pacific for the fiscal year 1906 were $22,973,312, and the capital stock was $105,307,100. If 50 /o of the net earnings is carried to surplus fund, what even per cent of dividend may be declared, and how much will be left as undivided profits ? 15. Y owned 300 shares of the Canadian Pacific stock. Z owned 250 shares. If Y bought his stock at 188f, and Z bought his stock at 190, what is the ra,te per cent of income on each man's invest- ment, making no allowance for brokerage ? What does each receive as dividend ? INSURANCE 805. Insurance treats of those computations arising from con- tracts guaranteeing security against loss or damage. 806. The parties to insurance are the insured or assured and the insurer or underwriter. 807-817] INSURANCE 815 807. The insured or assured is the person protected, or insured, against loss or damage. 808. The insurer or underwriter is the party that guarantees security against loss or damage. Insurers or underwriters are usually incorporated companies. 809. A policy is a written contract between the insured and the insurer. It sets forth the conditions under which the risk is taken, the liability of the insurance company, the time the insurance is to continue, the premium. 810. A valued or closed policy is one in which a fixed value is given to the thing insured. A valued or closed policy is the ordinary form used in general fire insurance. 811. An open policy is one in which no fixed value is given to the thing insured. In an open policy additional insurance may be entered at any time at rates and under conditions agreed upon. 812. The premium is the sum paid for insurance. 813. The term of insurance is the period of time for which the risk is taken or the property insured. 814. Premium rates are sometimes given as a specified number of cents per $ 100, and sometimes as a certain per cent of the sum insured. They depend upon the nature of the risk and the length of time for which the policy is issued. Insurance is usually effected for a year or a term of years. 815. Short rates are certain rates of premium charged by insur- ance companies for terms less than one year. Short rates are pro- portionately higher than yearly rates. 816. An insurance agent is one who acts for an insurance com- pany in obtaining insurance, collecting premiums, adjusting losses, reinsuring, etc. 817. An insurance broker is a person who negotiates insurance for others, for which he receives a brokerage from the company taking the risk ; he is considered, however, an agent of the insured, not of the company. 316 PERCENTAGE AND ITS APPLICATIONS [ 818-825 818. Insurance companies are distinguished by the way in which they are organized; as stock companies, mutual companies, and mixed companies. 819. A stock insurance company is one whose capital has been contributed and is owned by the stockholders, who share the gains and are liable for the losses. 820. A mutual insurance company is one in which the gains and losses are shared by the insured parties. 821. A mixed insurance company is one which combines the features of both stock and mutual companies. In mixed companies all gains above a limited dividend to the stockholders are divided among the policy holders. PROPERTY INSURANCE 822. Property insurance is the insurance of property against any specified casualty. 823. Property insurance includes : 1. Fire insurance, or indemnity for loss of, or damage to, property by fire. 2. Marine insurance, or indemnity for loss of, or damage to, a ship or its cargo by any specified casualty at sea or on inland waters. 3. Live stock insurance, or indemnity for loss of, or damage to, horses, cattle, etc., and from lightning or other casualty. 4- Transit insurance, or indemnity for loss of, or damage to, goods transported from one place to another by land or by both land and water. 824. Insurance policies are sometimes classified as ordinary poli- cies and average clause policies. 825. Under an ordinary policy the company will pay the full amount of any loss or damage that does not exceed the sum covered by the policy. Thus, if a house worth $ 12,000 is insured for $ 9000, and a fire occurs by which a loss of $ 7000 is sustained, the company is bound to pay the full loss, or 825-832] INSURANCE 317 $ 7000 ; but if the loss should be $ 10,000, or any sum in excess of $ 9000, the company will pay only the $ 9000 specified in the policy. 826. Under an average clause policy the company will pay only such a proportion of the loss as the policy is of the entire value of the thing insured. Thus, if a vessel valued at $ 12,000 is insured for $ 8000, and a fire occurs by which a loss of $6000 is sustained, the company will pay two thirds (i^o(j) of $ 0000, or $ 4000 ; but if the loss is total, the company will pay the full $ 8000, which is two thirds of the entire valuation, $ 12,000. 827. Marine insurance policies usually contain the average clause. 828. Almost all insurance companies will not issue a policy above a certain fixed sum ; and they will issue only one policy covering the same property. Therefore, if a person owns a valuable building, he must ordinarily have it insured in several different companies, in order to protect his interests. 829. If property that is insured in several companies is damaged by fire to the extent of the total amount of the insurance, each com- pany must pay the full amount of its policy. If the loss is less than the total amount of the insurance, each company must pay such a portion of the loss as its policy is a part of the entire insurance. 830. To cancel a policy is to annul the contract between the insurer and insured. In case a policy is terminated at the request of the insured, he is charged the short rate premium. If, however, it be terminated at the option of the company, the lower long rate will be charged, and the company will refund the premium for the unexpired time of the policy. 831. Salvage is an allowance made to those rendering voluntary aid in saving vessels or cargoes from marine casualties. Insurance companies usually reserve the privilege of rebuilding, replacing, or repairing damaged property. 832. Computations in property insurance are made in accord- ance with the general principles of abstract percentage, the amount insured corresponding to the base ; the rate of the premium to the rate; and the premium to the percentage. 318 PERCENTAGE AND ITS APPLICATIONS [ 832-836 DRILL EXERCISE 1. Find the cost of insuring a barn and contents for $4000 at 2. At 2%, what amount of insurance can I procure for $ 74? 3. If $25 is paid for insuring property worth $1000, what is the rate ? 4. State a formula for finding the premium when the amount insured and the rate of premium are given. 5. Given the premium and rate of premium, how may the amount of insurance be found? 6. Given the premium and the amount insured, how may the rate of premium be found ? 7. A dealer paid $ 125 for insuring a cargo of grain at 1\% on | of its value. Find the value of the grain. 833. To find the cost of insurance. 834. Example. How much will it cost to insure a store and contents for $42,000 at l\% ? SOLUTION $42,000 = the amount insured. $42,000 = $630, the premium charged. WRITTEN EXERCISE 1. A store is valued at $12,000 and the contents at $18,000. Find the cost of insuring f of the value of the store at f %, and f of the value of the contents at f %. 2. An insurance company, having insured a block of buildings for $ 200,000 at 75^ per $ 100, reinsured $ 60,000 with another com- pany at |%, and $80,000 with another at f %. What amount of premium did it receive more than it paid ? 835. To find the rate of insurance. 836. Example. I paid $ 30 for insuring a house worth $ 6400 at | valuation. What was the rate ? SOLUTION f of $6400 = $4800, the face of the policy. $ 30 -r- 4800 = .00625, or f , the rate of insurance. 836-838] INSUKANCE 319 WRITTEN EXERCISE 1. The cost of insuring f of a cargo of wheat worth $ 24,000 was $ 240. What was the rate of insurance ? 2. I insured f of a stock of goods worth $ 4500, and paid $ 18 premium. What was the rate of insurance ? 837. To find the amount insured. 838. Example. A man paid $280 to insure a stock of goods for 3 months. If the rate of premium was J%, for what amount was the policy issued? SOLUTION Let 100 % represent the amount of the policy. $280 = the premium paid. | % = the rate of premium. . Therefore|%= $280. t%=*40; |%orl%=$320. 100% = $32,000, the amount of the policy. WRITTEN EXERCISE ft 1. A gentleman paid $ 35.60 per annum for insuring his house at 2% on -f of its value. What was the value of the house? f/-J 2. A ranchman paid a premium of $ 76.00 for insuring f of his herd of cattle at 60^ per $100. If the cattle were valued at $40 per head, how many had he ? /^ - 3. The contents of a factory were insured for a certain sum at IY%. Later the goods were damaged by fire and losses paid by the company to the amount of $ 18,750, which was f of the amount in- sured. If the amount insured was | of the value inventoried, what was the total value of the goods ? /,* ^^ WRITTEN REVIEW 1. Find the cost of insuring a cargo of wheat valued at $ 24,000 at li%. 2. How much insurance, at 1%, can be procured for $ 90 ? 8. If it cost $ 663 to insure a certain block for $ 44,200, what will be the cost, at the same rate, to insure a block valued at $ 105,000 if $ 1.50 extra be charged for the policy in the latter case ? 320 PERCENTAGE AND ITS APPLICATIONS [838 4. How much will it cost to insure a factory for $ 42,000 at f %, and its machinery for $16,500 at 1J%, the charge for policy and survey being $ 2.50 ? 5. The premium on a cargo of 3000 tons of coal valued at $ 3.50 per ton, and insured at f of its valuation, is $ 47.25. Find the rate of insurance. ^L^ 6. If a store and its contents are valued at $ 27,000, for how much must it be insured at 1^-%? to cover loss and premium in case of total destruction ? /\ u\\ \ v \ 7. A cargo of teas, valued at $ 33,000, was insured for $ 18,000 in a policy containing an "average clause." In case of damage to the amount of $ 21,000, how much should the company pay ? */ 8. The steamer Norseman, valued at $ 90,000, is insured for $75,000, at 21%. What will be the actual loss to the insurance company in case the steamer is damaged to the amount of $ 20,000 ? 9. A speculator bought 2000 barrels of flour, and had it insured for 80% of its cost, at 3J%, paying a premium of $429. At what price per barrel must he sell the flour, to make a net profit of i^X 10. I insured my grocery store, valued at $ 13,500, and its con- tents, valued at $ 33,000, and paid $ 350 for premium and policy. If the policy cost $ 1.25, what was the rate per cent of premium ? AQ 11. A canal boat load of 8400 bushels of wheat, worth 90^ per bushel, is insured for f of its value, at If % premium. In case of the total destruction of the wheat, how much will the owner lose ? 12. A stock of goods valued at $ 30,000 was insured for 18 months, at 1% ; at the end of 12 months the owner surrendered the policy. If the "short rate" for 6 months was 65^ per $100, what should be the return premium ? 18. The German Insurance Company insured the Field block for $ 105,000, at 60^ per $ 100; but thinking the risk too great, it rein- sured $40,000 in the Home, at f %, and $45,000 more in the Mutual, at J%. How much premium did each company receive ? What was the gain or loss of the German ? What per cent of premium did it receive for the part of the risk not reinsured ? 14. A block of stores and contents was insured for $ 220,000 and became damaged by fire and water to the amount of $150,000. Of the risk, $ 40,000 was taken by the Hartford Co., $ 65,000 by the 838-844] INSURANCE 321 Manhattan, $ 35,000 by the ^Etna, and the remainder was divided equally between the Phoenix and the Provident. What was the net loss of each company, if the premium paid was 1J % ? 15. A factory worth $45,000 is insured, with its contents, for $ 62,500 ; $ 30,000 of the insurance is on the building, $ 12,500 on machinery worth $20,000, and $20,000 on stock worth $35,000. A fire occurs by which the building and the machinery are both damaged, each to the amount of $ 15,000, and the stock is entirely destroyed. How much is the claim against the company, if the risk is covered by an " ordinary " policy ? How much if the policy contains the " average clause " ? PERSONAL INSURANCE 839. Personal insurance is the insurance of person. It includes : 1. Life insurance, or indemnity for loss of life. 2. Accident insurance, or indemnity for loss from disability occasioned by accident. 3. Health insurance, or indemnity for loss occasioned by sickness. 840. Life insurance policies are usually either life policies or endowment policies. 841. A life policy agrees to pay to the beneficiary named in it a fixed sum of money on the death of the insured. The beneficiary is the one to whom the insurance is guaranteed to be paid. 842. An endowment policy guarantees the payment of a fixed sum of money at a specified time, or at death, if the death occurs before the specified time. 843. Life insurance companies are known as stock, mutual, mixed, and cooperative. Losses sustained by stock and mixed companies are paid either from reserve funds, or by assessment on the stock- holders ; those sustained by mutual and cooperative companies are paid by pro rata or fixed contributions of the policy holders. 844. Life insurance may be made payable to any one named by the insured. If made payable to himself, at his death it becomes a part of his estate and is liable for his debts ; if payable to another, MOORE'S COM. AR. 21 822 PERCENTAGE AND ITS APPLICATIONS [844-847 that other cannot be deprived of the benefit of the insurance, either by the will of the person taking out the insurance, or by his creditors. 845. Any one having an insurable interest in the life of another may take out, hold, and be benefited by, a policy of insurance upon the life of that person ; and any one may take out a policy in his own name and then assign it to any creditor or to any one having an insurable interest. 846. The following table shows the rates of one of the leading life insurance companies : ANNUAL PREMIUMS FOR AN INSURANCE OF $1000 LIFE POLICIES, PAYABLE AT DEATH ONLY ENDOWMENT POLICIES, PAYABLE AS INDI- CATED OR AT DEATH, IF PRIOR & < Continuous Premiums 10 Premiums 15 Premiums 20 Premiums C. J 8 o 10 cj "3 > O 8 8 {H g <( WSJ o 1 o 8-flfi O CT O (M t- O ^ Ml a > 5 1 5 (X) OOO C$*5 r-l Q IH 25 I I a a g ^ M 1 1 1 1 tf 1 -S I H 2 I CJD 5 O -^ fH C ^3 O w o?s ^ 5 39 M 5 h s o; a x S 'i S J 03 i O 869-870] CUSTOMHOUSE BUSINESS 333 869. In the assessment book, p. 332, the full city rate of $ 1.50 per $ 100 applies to the improved real estate when not otherwise specified. The suburban rate (see table, p. 331) is levied on " unimproved prop- erty " chiefly areas relatively small fronting on unimproved streets. The farm rate is levied on tracts of relatively large area used for farming and usually lying in the rear of lands fronting upon some improved street. The state tax of 4 mills per dollar on vehicles to hire and money at interest is collected by the city and sent to the state j but the state then returns f of it to the city. WRITTEN EXERCISE 1. From the table of rates and the assessment book, pp. 331, 332, how much tax is levied on the property 1400-1408 Chestnut St. ? 2. What is Thomas Dolan's total tax? If he had failed to make return for the value of his property, how much would his tax have been ? (See last paragraph on p. 331.) 3. What is the total tax of the persons listed on p. 332 ? 4. How much of this total tax would the state finally keep ? 5. If the trustees of the William Blanchard estate did not pay the real estate tax till Dec. 5, how much would the penalty be ? 6. What is the penalty if Anna Blanchard does not pay her tax till Aug. 5 ? 7. If the Land Title Trust Co. paid their real estate tax in March, what was the discount ? How much did they have to pay ? 8. If the Corporation of Haverford College paid their city tax March 20, their suburban tax March 30, and their farm tax July 5, what was the total discount ? NOTE. The teacher can easily vary the number of examples as desired, "by replacing example 3 with a few specific examples, and by giving additional examples based on the regulations for discounts, penalties, etc. CUSTOMHOUSE BUSINESS 870. A customhouse is an office established by the national government for the transaction of business relating to duties, or customs, and for the entry and clearance of vessels. 334 PERCENTAGE AND ITS APPLICATIONS [ 871-876 871. A port of entry is a port at which a customhouse is estab- lished for the legal entrance of vessels and merchandise. The waters and shores of the United States are divided into collection dis- tricts in each of which there is a port of entry which is also a port of delivery ; other ports than those of entry may be specified as ports of delivery. Duties are paid (or secured to be paid), and clearances made, at ports of entry only, but after vessels have been properly entered, their cargoes may be discharged at any port of -delivery. 872. Duties, or customs, are taxes levied by the national govern- ment upon imported goods. They are of two kinds, ad valorem and specific. 873. An ad valorem duty is a certain per cent levied on the appraised value of the goods, which is the market value in the country from which they are imported. Ad valorem duties are not computed on fractions of a dollar; if the cents in an invoice are less than 50, they are rejected ; if 50 or more, they are counted as another dollar. On pages 338-340 it is assumed (unless otherwise stated) that the appraised value corresponds to the invoiced cost. 874. A specific duty is a tax levied upon the number, weight, or measure of goods, regardless of their value ; as, a fixed sum per bale, ton, barrel, etc. Upon some goods both specific and ad valorem duties are levied. Before specific duties are finally determined, allowances are made for tare, leakage, etc. 875. An invoice, or manifest, is a written account of the particular goods sent to the purchaser or consignee, showing the quantity and the actual cost or value of the goods. All invoices must be made out in the weights and measures of the place or country from which the goods are imported, and in the currency of that country or in the currency actually paid for them. When the value of foreign currency is fixed by law, such value must be taken in estimating the duties. 876. The value in United States money of the foreign currency of the different nations of the world is proclaimed by the Secretary of the Treasury every three months. The following values of for- eign coins were proclaimed Apr. 1, 1907 : 87C-880] CUSTOMHOUSE BUSINESS VALUES OF FOREIGN COINS 335 COUNTRY STANDARD MONETARY UNIT VALUE IN U. S. GOLD Argentina .... Austria-Hungary Gold Gold Gold Peso Crown Peso $.965 .203 .9733 Brazil ..... Chile Denmark, Norway, Sweden . Egypt France, Belgium, Switzerland . German Empire Great Britain, India Japan ..... Gold Gold Gold Gold Gold Gold Gold Gold Gold Milreis Peso Crown Pound Franc Mark Pound sterling Yen Peso .546 .365 .268 4.943 .193 .238 4.8665 .498 .498 Netherlands .... Newfoundland .... Philippine Islands . Gold Gold Gold Gold Florin Dollar Peso Milreis .402 1.014 .50 1.08 Russia Gold Ruble .515 Turkey Gold Gold Piaster Peso .044 1 034 The drachma of Greece, the lira of Italy, the peseta of Spain, the bolivar of Venezuela are of the same value as the franc. The dollar, of the same value as our own, is the standard of the British possessions of North America (except Newfoundland), of Liberia, and of Colombia. The libra of Peru has the same value as the British pound sterling. The gourde of Haiti has the same value as the peso of Argentina. 877. A tariff is a schedule of goods, and the legal rates of import duties imposed by law on the same. 878. A free list is a list of such articles as are exempt from duty. 879. Tonnage is a tax levied upon a vessel, independent of its cargo, for the privilege of coming into a port of entry. 880. Duties are collected at the port of entry by a customs officer appointed by the United States government, and known as the Collector of the Port. Under him are deputy collectors, surveyors, and appraisers, and many inspectors, weighers, gangers, etc. 336 PERCENTAGE AND ITS APPLICATIONS [ 881-888 881. A naval officer, appointed only at the more important ports, receives copies of all manifests, countersigns all documents issued by the collector, and certifies his estimates and accounts. 882. The surveyor is the outdoor executive officer of the port. He supervises the inspectors, controls the unlading of foreign mer- chandise, etc. 883. The appraiser examines imported merchandise and de- termines its dutiable value; that is, the foreign market value at the time of exportation. 884. The public store is a place provided for the examination of imported merchandise. One package of every invoice of merchandise, and at least one package out of every ten similar packages, must be sent to the public store for examination. Bulky and heavy articles are examined at the wharf where they are unloaded. Weighable and gaugeable goods paying only specific duties are seldom sent to the public store for examination. 885. Warehousing is the depositing of imported goods in a gov- ernment or bonded warehouse. 886. A bonded warehouse is a place provided by law for the storage of dutiable merchandise. Goods may be withdrawn from a bonded warehouse for export without the payment of the duties. If goods on which the duty, amounting to $50 or more, has been paid are exported, the amount of duty, less 1 %, is refunded ; the sum so refunded is called a drawback. 887. Smuggling is the act of bringing foreign goods into a coun- try illegally without paying the required duty. Smuggling is a crime, for the prosecution and punishment of which stringent laws are enacted. 888. A customs broker is a person familiar with customs law and practice, who makes entries and transacts similar business for importers. He frequently acts as agent or attorney for his principal. 889-893] CUSTOMHOUSE BUSINESS 337 889. Tare is an allowance made for the box, bag, crate, or other covering of the goods. Leakage is the allowance made for waste of liquids imported in barrels or casks. 890. The gross weight is the weight before any allowances for tare, etc., are made. 891. Net weight is the weight after all allowances have been made. The ton by law consists of 2240 avoirdupois pounds in all cases where it is used in the customs. 892. To find a specific duty. 893. Example. Find the specific duty on 160 gallons of wine at $ 2 per gallon ; leakage, 10 % SOLUTION 10% of 160 gallons = 16 gallons leakage. 160 gallons 16 gallons = 144 gallons, the net quantity. $2 x 144 = $288, the specific duty. WRITTEN EXERCISE 1. Find the total specific duty on 1250 bushels barley at 30 $ per bushel; 400 bushels onions at 40^ per bushel; 1260 pounds cheese at 6^ per pound ; 2500 bushels wheat at 25^ per bushel. 2. Find the total duty on the following : 900 pounds unground cayenne pepper at 2^ per pound; 1200 bushels malt at 45 / per bushel ; 800 pounds butter at 6 $ per pound. 8. If the duty on plate glass is 8^ per square foot, how much will be the charge on an importation of 175 boxes, each containing 25 plates 16 by 24 inches in size ? 4- Find the specific duty on 1656 pounds macaroni, at 1 J t per pound; 900 pounds hops at 12^ per pound; 3150 pounds filler tobacco, unstemmed, at 35^ per pound; and 165 pounds hemp cordage at 2 f per pound. 5. After being allowed 10 % for leakage, a wine merchant paid $864 duty at f 2 per gallon, on 12 casks of wine. How many gal- lons did each cask originally contain ? MOORE'S COM. AR. 22 338 PERCENTAGE AND ITS APPLICATIONS [ 894-895 894. To find an ad valorem duty. 895. Example. What is the duty on an invoice of leather goods from Vienna, the dutiable value being 15,240 crowns and the rate of duty 35% ad valorem? SOLUTION $.203 = the value on 1 crown in United States money. $.203 x 15,240 = $3093.72, the dutiable value in United States money. 35 % of $ 3094 = $ 1082.90, the ad valorem duty. WRITTEN EXERCISE 1. Find an ad valorem duty of 35% on an importation invoiced at 17,450 francs. 2. What is the duty at 50% ad valorem, on a consignment of 650 dozen cotton gloves invoiced at 90 francs per dozen ? 3. Find the duty at 60% ad valorem on 3 cases of silk goods from Berlin, invoiced at 4692 marks each. 4. I imported from England 20 cases woolen goods, weighing 390 pounds each; tare 10% ; invoiced at 410 per case. What was the total duty at 44^ per pound and 60% ad valorem ? 5. I received by steamer Raglan from Liverpool the following invoice of goods: 768 yards velvet, invoiced at 1, 12s. per yard; 2150 yards lace, invoiced at 3s. 4d. per yard ; 1200 yards broadcloth, invoiced at 15s. per yard ; 3520 yards carpet, invoiced at 11s. 6d. per yard. If the duty on the velvet was 60%, on the lace and broadcloth 35%, and on the carpet 50%, how much was the total duty to be paid ? WRITTEN REVIEW 1. What is the duty on 1000 yards Brussels carpet 27 inches wide, invoiced at 6s. 9d per yard ; duty 28 ^ per square yard spe- cific and 40% ad valorem? 2. If the duty on flannel is 22^ per pound specific and 30% ad valorem, how much must be paid on an invoice of 2150 yards, weighing 420 pounds, and valued in Canada, whence it was imported, at 75 ^ per yard ? 3. Find the duty at 40% ad valorem on 3 dozen clocks, invoiced at $ 21.50 each, and 6 dozen watches, invoiced at $ 35 each. 895] CUSTOMHOUSE BUSINESS 339 4- Find 35% ad valorem duty on 250 cases German toys in- voiced at 175 marks per case. 5. I imported from Belgium 300 meters Brussels carpet, f of a yard wide, at 5 francs per meter. I paid a specific duty of 28 ^ per square yard and an ad valorem duty of 30%. What was the total duty? NOTE. A meter is equivalent to 1.0936 yards. 6. What is the amount of duty chargeable on 4000 pounds worsted yarn invoiced at 490, when the rate of duty is 38^^ per pound and 40% ad valorem ? 7. A merchant imported 10 gross table knives costing 15s. per dozen in Sheffield, England. What was the duty at $2.40 per dozen specific and 40% ad valorem ? Find the dutiable value and compute the duty on the following entries of merchandise : 8 Manifest No.. INWARD FOREIGN ENTRY OF MERCHANDISE Imported by ; tf / rT^^^ .Master, : In the steamer Arrived _^ Number* Packages and Contents Quantity Duty Total t **-** **,** 340 PERCENTAGE AND ITS APPLICATIONS [ 895-896 9. Manifest No.. Invoiced at yW INWARD FOREIGN ENTRY OF MERCHANDISE -^ / ^/-JP *~X T ^ y ' Imported by /J^f^Sz^Zstf^f yl^-^r^f?^ In . Fr Arrived Packages and Contents Quantity Duty Total 10. A merchant imports 1200 yards Brussels carpet, | of a yard wide, invoiced at 200. Compute a duty of 28 ^ per square yard and an ad valorem duty of 40%. If freight charges and losses aggre- gated $ 185.50, at what price per yard must the carpet be sold to gain 20%? 11. A Boston merchant imported mandolins invoiced in Germany at 40 marks each. If he paid an ad valorem duty of 45%, what price must he sell them for to gain 20% on the cost? EXCHANGE 896. Exchange treats of methods of making payments at distant places without the transmission of money. Settlements are effected by means of written orders called bills of exchange, express money orders, telegraphic money orders, letters of credit, etc. and the risk and expense of sending the money itself is avoided. 897-901] EXCHANGE 341 897. An exchange center is some recognized money center The principal exchange centers of the United States are New York, Boston, Philadelphia, Chicago, St. Louis, Baltimore, Cincinnati, and San Francisco ; of Europe, London, Paris, Antwerp, Geneva, Amsterdam, Hamburg, Frankfort, Berlin, and Vienna. . Exchange is of two kinds : domestic, or inland, and foreign. DOMESTIC EXCHANGE 899. Domestic exchange is exchange payable in the country in which it is drawn. Domestic bills of exchange are commonly called drafts. The business of making payments by means of drafts and bills of exchange is usually conducted through the medium of banks and bankers. 900. Funds may be remitted from one place to another place in the same country in six different ways without the transmission of money, as follows : 1. By a postal money order. 4. By a bank draft. 2. By an express money order. 5. By a check. 3. By a telegraphic money order. 6. By a sight draft of a cred- itor on a debtor. 901. A postal money order is an order drawn by the postmaster, or his clerk, at one office, directing the postmaster of another office to pay to the person named in his private letter of advice the sum specified in the order. Applications for postal money orders must be in writing, and must state the amount of each order wanted, the name and address of the person to whom the order is to be paid, and the name and address of the remitter. At the present time the maximum amount for which a single postal money order may be issued is $ 100, and the rates charged are as follows : $2.50 or less .... 3^. $30.00 to $40.00 . . . 15?. $2.50 to $5.00 . . . 5?. $40. 00 to $50. 00 . . . 18?. $ 5.00 to $10. 00 . . . 8?. $50.00 to $60. 00 . . .20?. $10. 00 to $20. 00 . . 10?. $60. 00 to $75.00 . . . 25?. $20.00 to $30.00 . . 12?. $75. 00 to $100. 00 . . . 30?. The payee who desires a money order to be paid to another person must fill out and sign the form of transfer which appears on the face of the order. More than one transfer is prohibited by law. 342 PERCENTAGE AND ITS APPLICATIONS [ 901-904 If a money order is lost, a certificate should be obtained from both the paying and issuing postmasters stating that it has not been paid and will not be paid. The Post Office Department at Washington will then issue another order upon application. 902. An express money order is an order drawn by the agent of the express company at any given office directing another agent of the company at some designated place to pay to the person named therein a certain sum of money. Express money orders may be obtained for any number of dollars, and the rates at the present time are the same as for postal money orders. Express money orders are transferable by indorsement, the same as notes, checks, etc. 903. A telegraphic money order is an order drawn by a telegraph agent at any given office instructing the agent at some designated office to pay to the person named in the telegraphic message the sum specified, upon his personal application and proper identification. At the present time telegraphic transfer rates are as follows : $60 or less . . . 60^. $200 to $300 . . $1.50. $50 to $100 . . . 1%. $300 to $400 . . $1.75. $100 to $200. . . $1.25. $400 to $500 . . $2.00. Over $ 500, special rates. The rates in the above table are entirely apart from the cost of telegraphic service, which is based upon distance and the number of words contained in the 904. A bank draft is an order written by one bank directing another bank to pay a specified sum of money to a third party, or to his order. 905-909] EXCHANGE 343 905. Nearly all banks keep money deposited with some other bank, called a correspondent, at one or more commercial centers against which they draw drafts to sell to their customers for remit- tance purposes. These drafts pass as cash in the section tributary to the commercial centers upon which they are drawn. Banks usually make a charge called exchange for the trouble ol keeping funds on deposit at commercial centers and drawing drafts against these funds. These charges range from r L% to %. On many small drafts a definite charge ranging from 10 $ to 50 ? is. frequently made. Some banks make no charge for drafts sold to regular depositors. 906. Instead of making remittances by bank drafts merchants frequently send their personal checks in payment of bills. 907. A check is an order on a bank by a depositor for the pay- ment of money ; except that it is drawn by a person, it is very much like a bank draft. Rochester, N. V-, ^f^>^i/> ^ Vt*.t. No.. ALLIANCE NATIONAL BANK Pay to the order of. Dollars. 908. Commercial drafts play a prominent part in facilitating the payment of bills at distant places. Commercial drafts, which include sight and time drafts, were discussed on pages 260 and 261. Exercises in discounting time drafts were given in connection with bank discount, pages 269 and 270. 909. Formerly domestic exchange was at a premium or discount in the city where purchased according as the balance of trade between that city and the one on which the draft was drawn was in favor of or against the former city. If the drawer city owed the drawee city, exchange on the latter would be at a premium at the former place; if the balance of trade was in favor of the drawer city, the co .dition 344 PERCENTAGE AND ITS APPLICATIONS [909-912 of exchange would be reversed in the two places. For a number of years past, however, domestic exchange has been practically at par. 910. Bankers usually make a charge called collection for collect- ing out-of-town drafts deposited with them. 911. Sometimes unaccepted time drafts are left with a bank for collection, and sometimes they are offered for discount. 912. Banks are usually willing to accept for discount the unac- cepted drafts of responsible parties when they are properly indorsed. WRITTEN EXERCISE 1. W. J. Boone & Co., of San Francisco, Cal., have bills to pay as follows: T. W. Brooke, Dayton, O., $650; E. L. Grey son & Sun, Cedar Eapids, la., $ 46.53; Barnes & Snyder, Bolton, Mo., $48.50; and their traveling salesman, W. H. Post, is wanting $100 for ex- penses at Denver, Col. They pay the amounts by remitting as fol- lows : T. W. Brooke and E. L. Grey son & Son, express money orders; Barnes & Snyder, postal money order; and W. H. Post, by tele- graphic transfer in a ten-word message. If the cost of the telegram was 50^, what was the total amount required? 2. Barnum & Co., of St. Paul, drew a sight draft of $1400 on Martin & Cole, 415 High St., Boston, on account of an invoice of hides shipped to them valued at $3000, as per bill of lading attached to the draft. They sold the draft at the bank at |% discount. What were the proceeds ? 8. A commission merchant of Charleston, S.C., bought a ninety- day commercial draft at 1% discount for $800 drawn on a Boston firm. If money be worth 6%, what did the draft cost him ? SOLUTION $ .015 = the bank discount on $ 1 for 90 da. $.005 = the commercial discount on $ 1. $ .015 + $ .005 = $ .02, the total discount on $ 1. $ 1 - $ .02 = $ .98, the proceeds of $ 1. $ .98 x 800 = $ 784, the cost of the draft. 912-913] EXCHANGE 345 4. A commission merchant holds, subject to the order of his principal, 5005. His principal directs him to remit the amount by New York draft after deducting the cost of the draft. If the bank charges exchange at the rate of ^%, what will be the face of the draft ? 5. Gates & Son, of Memphis, drew a sight draft on Perrin & Boon, Portland, Me., for $8750.85, which they sold at the Cotton Exchange Bank at f % discount. How much were the proceeds ? 6. I drew a 60-day draft on one of my customers and sold it to a broker at f % discount, receiving $ 1354.18 as proceeds. What was the face of the draft, money being worth 6% ? 7. Jno. W. Williams, of Boston, Mass., remitted Janis Bros. & Co., of Milwaukee, $1750 by draft on New York, exchange 15^ per each $100; Martin & Co., of Allentown, Pa., by American Express money order, $ 89.75 ; and Theodore Emens, $ 28.50, by post office money order. Find the total cost of exchange. 8. Thomas, Bailey & Co., of St. Louis, drew a sight draft for $ 1900 on Slocum, Wilde & Co., 291 Milk St., Boston, Mass., on ac- count of an invoice of molasses shipped them, valued at $3506, as per bill of lading attached to draft. They sold the draft at a bank at \/o discount. What were the proceeds ? 9. A wholesale grocer owed for an invoice of $5425.40, pur- chased in New York, subject to a discount of 6% if paid within 10 days. Within the required time he discounted the bill and remitted for balance as follows : A sight draft which he bought of E. M. Brooks on Gunn & Baker for $4000, at \/ discount, and a bank draft for the remainder, the exchange being 10^ for each $100. How much was required to settle the bill, and how much was gained by discounting it ? 10. Hedman & Son drew a 60-day draft on Johnson Manufactur- ing Co. for $2500, and had it discounted at a bank at 6%. If the rate of collection was J %, what were the proceeds of the draft ? FOREIGN EXCHANGE 913. Foreign exchange is exchange payable in another country than that in which it is drawn. It is by means of the system of foreign exchange that the people of the various nations pay their debts to one another. 346 PERCENTAGE AND ITS APPLICATIONS [ 914-917 914. The business of foreign exchange was brought about by the fact that goods are exported and imported by the nations of the earth, and the fact that investors put money in the enterprises and securities of nations foreign to their own. During the year 1906 the goods imported by the United States amounted to $1,226,563,843, and the goods exported to $1,717,953,382; hence that year foreign countries owed us $491,389,539 for goods we exported in excess of what we owed them for goods imported. The imports and exports were paid for chiefly through the medium of foreign exchange. This same means is used in paying the sums invested by Americans in foreign countries, the sums invested by foreigners in this country, the amounts spent by Americans abroad and by foreigners traveling in this country, and the sums of money sent by foreigners in this country to their families in Europe. The grand total of our foreign exchange is thus extremely large. 915. The following are some of the more common forms of foreign exchange: a draft, check, bill of exchange, money order, circular letter of credit, traveler's cheque, an order (either written or cabled) to pay certain persons money in some foreign country. The. transportation of gold or specie from one country to another is also an important part of the system of foreign exchange. Exchange of any kind whatever may be made payable in the money of the country in which payment is to be received, or in the money of a country in which a great financial city is located. London is the greatest financial center of the world, and many drafts on Germany, France, Norway, Sweden, Russia, China, India, and other countries are payable in sterling exchange. New York is the financial center of all America, and many drafts or bills of exchange drawn on Canada, Mexico, and the countries of South America are payable in New York funds. 916. A bill of exchange is a draft drawn payable in a foreign country. If it is drawn to cover the value of goods exported, a bill of lading and an insurance certificate usually accompany it ; such a bill is known as a documentary bill of exchange. If the bill of lading and the insurance certificate are not attached, the bill of exchange is known as a clean bill of exchange. 917. Frequently bills of exchange are drawn in sets of two or sometimes three, called first, second, and third of exchange. When bills are drawn in sets of two they are sent by different mails, so that if one is lost the other may be presented. If three bills are drawn, the third one is kept by the purchaser as a memorandum. 918-020] EXCHANGE 34? A SET OF EXCHANGE & ^u^el. /SOtTis^ ' f///f/ Jrt'/f />//////// s7/// /*y/ //ss .y//////' // sfs'f/'////f// r iii 1 far ^^^ms^i^^ YV/Sf//S/{'~7>-is/.j/ /////////// J/ssf// /<" ///c X////V /'// 918. Bills of exchange are many times used as a means of col- lecting debts due in foreign countries. The method employed is similar to that used in collecting debts by means of commercial drafts. 919. The par of exchange is the established value of the stand- ard unit of money of one country expressed in that of another. It is of two kinds : intrinsic and commercial. 920. The intrinsic par of exchange is the real or intrinsic value of the coins of one country as compared with those of another. Thus, the pound sterling of Great Britain contains 113 grains of pure gold, and the dollar of the United States contains 23.22 grains of pure gold. Since 113 grains are 4.8665 times 23.22 grains, the pound sterling is worth $4.86^. 348 PERCENTAGE AND ITS APPLICATIONS [921-927 921. The commercial par of exchange, commonly called the course of exchange, is the market value of the standard unit of money of one country in the currency of another. 922. The course of exchange is usually governed by the relative state of indebtedness of merchants of different countries and' the sup- ply of gold and silver ; hence it may be at a premium or at a discount. If the merchants of England owe the merchants of the United States more than is usual, exchange in London on New York quickly advances to a high rate, while exchange in New York on London declines. 923. Normally foreign exchange rates fluctuate between gold- exporting and gold-importing points. If it costs less to export gold than to buy a bill of exchange, the gold itself is shipped. However, gold is seldom actually shipped in quantities of less than $ 20,000. If the rate paid for demand sterling in New York drops below $4.84| per pound, it is cheaper to import gold in large quantities than to have remittances made from abroad by means of draft. If exchange in New York goes above $4.88 per pound sterling, then gold can be exported at less cost than to buy a draft on London. 924. In the exportation and importation of gold there must be considered the risk of loss by shipwreck, the loss because of abrasion, the cost of transportation, and the charge for insurance. The cost of sending gold from New York to London is usually as follows : Insurance and freight, each %; abrasion on $5 gold pieces, %; on $10 gold pieces, %; on $20 gold pieces, &% to $ %. 925. Quotations of foreign exchange are given by means of market value equivalents, no reference being made to the intrinsic par value. 926. Exchange on Great Britain is quoted by giving the exchange value of 1 in dollars and cents. Thus, when exchange on London is quoted at 4.86|, a bill for 1 will cost 4.86|. 927. Exchange on France, Belgium, and Switzerland is usually quoted by giving the exchange value of $ 1 in francs. Thus, when exchange on Paris is quoted at 5.18, $1 will buy 5.18 francs. Notice that such quotations differ from all others in character ; a change from to 5.19 is Si fall in the rate. 928-933] EXCHANGE 349 928. Exchange on the Netherlands is quoted by giving the ex- change value of 1 guilder, or florin, in cents. When exchange on Amsterdam is quoted at 41, 1 guilder is equal to 41 cents. 929. Exchange on Germany is quoted by giving the exchange value of 4 reichsmarks in cents. When exchange on Berlin is quoted at 96, 4 reichsmarks are equal to 96^ cents. 930. Dealers in foreign exchange refer to changes in the quota- tions as being of so many points. A point is one hundredth of a cent, or one unit in the fourth decimal place. If the quotation for London exchange is 4.8255, the addition of 75 points would make it 4.833. 931. Quotations are sometimes made in the form 93^- + ^ 01 93J 3*2-, or 5.17 y 1 ^, or the like. The fraction means the fraction of 1 % of the rate. A -}- fraction always makes the rate higher ; a fraction, lower. Thus 93| - & means $ .935 - ^ % of $ .935, or $ .93471 ; but 5.17 - ^ means Fr. 5.17 + ^% of Fr. 5.17, or Fr. 5.17323. See 927. 932. The law of England requires that revenue stamps be affixed to all drafts drawn for more than five days' sight. Revenue stamps are required also on drafts for more than three days' sight on Holland, and on drafts at one or more days' sight on France or Germany; but the three days' sight "letters of delegation" on Germany are exempt. The required amount of stamps is -^ % of the face of the draft ; their cost may therefore be included by adding to the quoted rate. 983. A cable transfer is a telegraphic order to pay a certain per- son in a foreign country a certain sum of money. Such transfers are made by cipher codes. The rate charged for a cable transfer is the quoted or market rate of exchange, plus a commission of about i%, plus the telegraph and cable charges. By cable transfer, a merchant who desires to ship wheat to London can com- plete the transaction in a few hours. He can ship the wheat, telegraph the fact to the consignee atLondon, obtain particulars concerning the condition of the market, and, if he thinks best, have the wheat sold at once, " to arrive," and the proceeds remitted through a London banker. A bill does not appear in the transaction. A very large amount of business in foreign exchange is done by means of the cable. Operators in this line conduct their 'business upon a very close mar- 350 PERCENTAGE AND ITS APPLICATIONS [933-936 gin, and calculate the outcome of their transactions to a nicety. This is possible because there is little delay during which the rates might materially change so as to cause a loss. During a part of each day the cable enables persons in New York, Philadelphia, Chicago, London, Berlin, and other money centers, to con- clude transactions without material delay. 934. Arbitrage of exchange is the calculation of the relative values of exchange at the same time at two or more places with the purpose of taking advantage of the difference in price. It is con- ducted (largely and most profitably by cable) by buying simulta- neously in the cheaper and selling in the dearer market. Suppose that a merchant owes a debt of 1500 in London, and that direct exchange on London is 4.87, and direct exchange on Paris is Fr. 5.24| to the dollar, and that Paris exchange on London is Fr. 25 to the pound. Then Fr. 37,500 in Paris will purchase 1500. Fr. 37,500 will cost, in U.S. money at 5.24$, $7149.67. Direct exchange on London at $4.87| for 1500 = $7312.50; 7312.50 $7149.67 = $ 162.83, amount gained by arbitrage or indirect exchange. If the German mark could, because of some condition, purchase more ster- ling in proportion in Berlin than could the franc in Paris, then the merchant would gain still more by buying marks at the lowest rates. But if the mark could be had for less by remitting via Paris than by direct exchange on Berlin, the payment of the 1500 would be made in London through both Paris and Berlin. 935. A letter of credit is a circular letter issued by a banking house to a person who desires to travel abroad. The letter is usu- ally addressed to the foreign correspondents of the bank issuing it, requesting them to furnish the traveler such funds as he may require up to the aggregate amount named in the letter. When the traveler desires funds, he goes to any correspondent mentioned in the letter of credit, and draws a draft on that correspondent for the amount desired. The draft is signed in the presence of the correspondent, who care- fully compares the signature with the one on the letter, and, if they are found to agree, the draft is cashed and the amount inscribed on the back of the letter. The last draft drawn is attached to the letter itself. The difference between a bill of exchange and a letter of credit is that the former is payable at a certain designated place, at a specified time, and in one amount, while the latter is payable at several places, at different times, and in variable amounts. 936. Travelers' cheques are a substitute for letters of credit and bills of exchange. They are similar in form to bank bills. They are issued for fixed printed amounts, with the equivalent of each 936-940] EXCHANGE 851 denomination in the money of the principal European countries, and are payable to order, after being signed and countersigned by the purchaser or holder. They are cashed without discount or com- mission by an extended list of banks and bankers, and are received in settlement of hotel bills by the principal hotels in Europe. 000000 Travelers' Cheque 937. To find the cost of a foreign bill of exchange. 938. Examples. 1. Find the cost of a draft on London for 380 10s. 6d. sterling, exchange being quoted at $ 4.86f . SOLUTION 380 10s. 6d. = 380.525. 4.86| = the market quotation of 1. -#4.86| X 380.525 = $ 1851.73, the cost of the draft. 2. Find the cost of a sight draft on Paris for 3108 francs, exchange being quoted at 5.18. SOLUTION $1 = 5.18 francs. 3108 francs -=-5.18 francs = 600. Therefore a bill for 3108 francs will cost $600. 939. To find the face of a foreign bill of exchange. 940. Example. The cost of a bill of exchange on London was $3654.47. When exchange was quoted at $4.86J, what was the face of the bill ? 352 PERCENTAGE AND ITS APPLICATIONS 940 SOLUTION $4.86| = the market value of 1. $3654.47^$ 4.86 } =751.175. Therefore $3654.47 = 751.175. .175 = 3s. 6d. Hence the face of the bill was 751 3s. 6d. WRITTEN EXERCISE 1. Find the cost of a sight draft on London for 400 when exchange is quoted at 4.865. 2. I bought a bill of exchange on Paris and paid $ 2156. What was the face of the bill, exchange being quoted at 5.17f ? 3. A New York importer who owed a Dresden manufacturer 21,320 reichsmarks, bought a bill of exchange on Berlin at 95J, and paid for the same by check. What was the face of the check ? 4- What is the face of a bill of exchange on London which can be bought for 15807.25 if quoted at $4.85, brokerage \% ? 5. An exporter sold through a broker a bill of exchange on Hamburg at 95f , and received $ 5953.49 as net proceeds. What was the face of the bill, brokerage \%1 6. Hibbard & Co., of Brooklyn, purchased a bill of exchange on London at 3 days' sight for 342 12s. 6d. at 4.86 J. How much did the bill cost ? s 7. An importer purchased a sixty-day bill of exchange on Bre- men at 95| for $446.20. What was the face of the bill ? 8. A New York diamond merchant purchased a bill of exchange on Amsterdam at 3 days' sight for 63,892 guilders at 40. What did the bill cost? 9. I purchased a bill of exchange on Paris for 33,250 francs and paid $6412.72. What was the course of exchange ? 10. An importer purchased a bill of exchange on Amsterdam for 3575 guilders and paid $1443.41 for it. What was the course of exchange ? 940-944] EXCHANGE 353 11. A Manchester, England, manufacturer drew a bill of exchange at 3 days' sight for 450 10s. Scl on a Rochester, N.Y., merchant. The draft was presented to the drawee by a local bank, and paid by check. What was the face of the check, exchange being 4.85 J? 12. Langdon & Perry, of New York, owed on foreign invoices as follows: T. C. Shepherd Sons, London, 1800 8s. ; J. L. Von Buesche, Berlin, 1600 marks ; Perrie, Buzzell & Co., Paris, 4016 francs ; F. Gonzalez, Mexico, 816 dollars. They bought at their bank : exchange on London at 4.86; on Berlin, 96^; on Paris, 5.19 *-; on Mexico, 79, and issued one check to cover the total purchase. What was the amount of the check ? 941. To find the cost of exporting gold. 942. Example. Strawbridge & Clothier, Philadelphia, are quoted 4.89.L on London for a sight bill of 6540. They export the gold instead of buying the exchange, paying 1 % insurance, 1 % freight, T 3 g / abrasion. How much did they save by shipping the gold ? SOLUTION $4.89| x 654 = $32,013.30, the cost of the draft. 4.8665 = intrinsic par of exchange. 4. 8665 =.02129. 4.8665 + .02129 =4.88779, the rate for exporting gold. $4.88779 x 6540 = $31,966.16, the cost of gold shipment. $32,013.30 - $31,966.15 = $47.15, saved by gold shipment. 943. To find the proceeds of a bill of exchange. 944. Examples. 1. John J. Rose & Co. sell to Brown Bros. & Co. a sight draft for Mk. 1852^%. What did they receive at 93 T \ ? SOLUTION $ .93^ = $.933125 = the rate for 4 marks. $.933125 x 1852^ -=- 4 = $432.22, the proceeds. 2. J. Griffith & Co. sold to Brown Bros. & Co. at 5.22} -^ a sight draft for Fr. 14,645^. What are the proceeds ? SOLUTION $1 = Fr. 5.22 J- - & = Fr. 5.22451. See 931. Fr. 14,645^ - Fr. 5.22451 = 2803.23. Therefore the proceeds are $2803.23 NOTE. The following examples show how a banker determines what rate to quote to a customer. MOORE'S COM. AR. 23 354 PERCENTAGE AND ITS APPLICATIONS [944 S. On Jan. 3, 1907, there is drawn by the Baldwin Locomo- tive Works, Philadelphia, on the London & Brazilian Bank, London, a commercial bill of exchange at 90 days after sight for 617 8s. 3d. At 4.865 market quotation what should a banker pay for it, allowing for interest at 3%, revenue stamps, and a commission of NOTE. Interest on London exchange is reckoned on a basis of $4.85 per pound 365 days to the year. English law allows 3 days grace. SOLUTION Market rate on London = $4.865 Disct. rate 93 ds. (3 ds. grace) = 4.85 x .03 x -fa = .03707 Com . % per pound = .02433 Cost per pound revenue stamps ^ % of rate = .00243 .06383 The banker's quotation $ 4. 80 1 17 617 8s. 3d = 617.4125. $4.80117 x 617.4125 = $2964.30, the sum the banker would pay. 4. A bill of exchange on Paris at 30 days after sight is drawn in favor of Lazard Freres for Fr. 16,764^^ with documents. Interest rate, 3%; market quotation, Fr. 5.165; commission, \% of rate. What are the proceeds ? SOLUTION Market quotation on Paris Fr. 5.165 Discount, 30 ds. at 3 % = \ % of rate as . 01291 Com. |% of rate = .00646 Revenue stamps ^%= .00258 The banker's quotation $ 1 = Fr. 5.18695 Fr. 16,764.70 -$- Fr. 5. 18695 = 3232.09. Hence the proceeds are $ 3232. 09. 5. A documentary bill of exchange is drawn on Amsterdam at 60 days after sight. What rate per guilder can Brown Bros. & Co. offer for it, allowing for discount rate, 3J%; market quotation, ; commission, \%', revenue stamps, -fa% ? SOLUTION Market quotation (40| + A) = $.401376 Revenue stamps ^% of rate = .0002 Discount, 60 ds. at 3^% of rate = .00234 Commission \ % of rate .0005 _ .00304 The banker's quotation " $.398336 944] EXCHANGE 355 WRITTEN EXERCISE 2. The Baldwin Locomotive Works of Philadelphia sell a sight draft for 416 9s. 4d. on the London & Brazilian Bank to Brown Bros. & Co. at 4.81-|-. What amount of money does it bring ? 2. J. S. Griffith & Co. sell to Brown Bros. & Co. at 5.20f ^ a draft on Credit Lyonnais, Havre, for Fr. 19,745^. What amount do they receive for it ? 8. The Baldwin Locomotive Works draw a draft on the London & Brazilian Bank at 90 days sight for 314 3s. 4d. How much will Brown Bros. & Co. pay for it ; market quotation, 4.826 ; commission, | % ; interest rate, 3 % ; revenue stamps, -$ % ? 4. John J. Eose & Co. sell to Brown Bros. & Co. the draft shown on page 347. Market rate, 91^; interest rate, 3^-%; commission, \/o', revenue stamps, V%- What amount of money do they receive ? NOTE. Interest on German exchange is computed at 95 cents per 4 marks, 360 ds. to a year. 5. John Wanamaker sells in New York a documentary bill of exchange for 640 12s. 6d, at 90 days sight, on a London bank. What should he receive, allowing for 3 % interest, revenue stamps, and a commission ^%, if the quotation is $4.835? 6. The Rogers Locomotive Works of Paterson, N.J., sell to Brown Bros. & Co. a documentary bill of exchange on a Paris bank for Fr. 18,765^%, drawn 90 days after sight. What are the pro- ceeds, if market rate is 5.17J + -fa ; interest rate, 3 % ; commission, \ % ; revenue stamp, -fa%? 7. The Wm. H. Hostman Co. bought from John Heckemann, Bremen, a bill of goods amounting to $4695.45 on 60 days time, 2%%, 10 days. They send him a draft to-day at sight, so as to take advantage of the discount. What is the face of the draft, in marks, if demand exchange is 94J, which includes broker's commission ? 8. James McCreery & Co. sell a documentary bill of exchange on a Geneva banker for Fr 14,764^ at 90 days after sight, to Lazard Freres. What are the proceeds, the rate of exchange being 5.16^-; rate of interest, 3% ; commission, \/ ; revenue stamps, ; and cost of collecting and sending funds to Paris, 356 PERCENTAGE AND ITS APPLICATIONS [ 944 9. Tiffany & Co. import an invoice of statuettes and bronzes from Florence, Italy, amounting to 14,725^- lire. They buy a three days after sight draft of Brown Bros. & Co. on a Flor- ence banker to pay the bill. What will they pay Brown Bros. & Co. for it if the market rate is 5.17|-; commission, \%\ revenue stamps, -fa % ; discount rate, 10. B. Altman & Co., New York, are quoted 4.89f, which in- cludes commission, for a sight bill on London for 7560. They buy the gold and ship it, instead of buying the draft. How much is saved if they pay | % insurance, \ % freight, and ^ % abrasion ? 11. You are a clerk in the office of Brown Bros. & Co., and a merchant hands you a bill on London for 312 3s. 4d at 60 days after sight; a bill at 90 days after sight on Havre, France, for Fr. 15,612^; a bill on Hamburg at 30 days after sight, for Mk. 3412^; a sight bill on Amsterdam for 1345^ guilders. The posted rates are 4.831 on London ; 5.16 -^ on France ; 92 \ on Ham- burg; 40^ + -^ on Amsterdam. The foreign rate of interest is 3 %. You are to allow for a commission of \%> and for revenue stamps on time paper, fa%. How much should you pay the merchant? 12. Altman & Co., New York, are quoted a rate of 4.89J, which includes commission, on London, for a sight bill amounting to 846010s. Instead of buying the bill they export the gold. If they pay \% insurance, \% freight, -fa% for abrasion, how much do they gain or lose by shipping the gold instead of buying the exchange ? 18. New York pates on Amsterdam are 41-J; on Paris are 5.15J. A merchant owes a debt of Fr. 13,450 in Paris. He pays it by re- mitting via Amsterdam at Fr. 2.14 to the guilder. What is saved by the indirect exchange (934) ? 14- Mr. W. of Philadelphia goes to London for a visit and directs his Philadelphia broker to remit him $ 10,000. Brown Bros. & Co. quote London exchange at 4.89 and Berlin exchange at 951, both rates including the commission. In Berlin, exchange on London is quoted at Mk. 20.3 to the pound. How much more or less in English money does Mr. W. receive by indirect exchange than by direct ex- change, if there is a charge of \% for remitting from Berlin to London ? SHARING PROPORTIONAL PARTS 945. Sharing is the process of dividing a number into shares proportional to other given numbers. DRILL EXERCISE 1. A and B agree to perform a certain piece of work for $ 160. If B can earn $ 10 while A earns $ 6, how much should each receive as his share of the $160 ? SOLUTION $160 is to be divided iiito shares proportional to 6 and 10. The numbers 6 and 10 may be regarded as shares. The whole will then be represented by 6 + 10, or 16 shares. 16 shares = $ 160. 1 share = % 10. 6 shares = $60, or A's part. 10 shares = $ 100, or B's part. 2. Divide 240 into shares proportional to 8 and 16. 8. Divide $ 600 into shares proportional to 1, 3, and 6. 4. Divide a profit of $ 1200 among three partners in a business in which A invests $ 2 as often as B invests $4, and C$6. 6. Divide $ 150 into shares proportional to ^ and . SOLUTION Fractions must be similar to be compared. \ and \ = ft and ft, respectively. ft and ft stand in the same relation to each other as 8 and 2, respectively. Hence the whole may be represented by 6 shares. 6 shares = $150. 1 share = $30. 3 shares = $90, or the first part. 2 shares = $60, or the second part. 6. Divide $ 780 among three persons, whose shares are to be in proportion to ^, , and . 857 358 SHARING [ 945 7. Three men engage in business. A puts in $ 2000 ; B, $3000 ; and (J, $ 4000. They gain $ 900. What should be each man's share of the gain ? 8. A bankrupt owes $500 to A, $2000 to B, and $1500 to C. If his net resources are $2000, what will each of his creditors receive ? 9. Divide $2100 among A, B, and C so that A's part will be twice B's part and one half of C's part. 10. Three boys bought a watermelon for 24^, of which price Charles paid 9^, John 8^, and Walter If. Ralph offered 24^ for one fourth of the melon, which offer was accepted and the melon divided. How should the 24^ received from Ralph be divided among the other three boys? WRITTEN EXERCISE 1. A will provided that an estate be divided among three per- sons in proportion to their ages, which are 15, 18, and 20 years, respectively. If the amount received by the youngest person was $3000, what was the value of the estate ? / o ^ 2. A, B, and C engage in trade for one year. A puts in $ 6000, B $3000, and C $2000. If their gain for the year is $4400, what is each man's share ? ji^ i - - 8. A, B, and C rent a pasture for $ 740. A put in 3 cows for 4 months ; B, 5 for 6 months ; and C, 8 for 4 months. How much should each pay? ! ^ 00 ^3.0 4- An estate was so divided between A and B that A's part was to B's part as \ is to \. If B received $ 2400 more than A, what was the value of the estate ? ) 2_ o 5. A man bequeathed his property to his wife and two daugh- ters. The wife received $5 for every $3 received by the elder daughter and for every $2 received by the younger daughter. What was the value of the estate, the younger daughter having received $1500 less than her sister? 6. Divide the simple interest on $ 2600 for 6 years 3 months at 4% among A, B, and C in proportion to ^, ^, and ^, respectively. 945-951] PARTNERSHIP 359 7. A man left his property to be divided among his three sons and two daughters in proportion to their ages. The sons are aged 20, 16, and 10 years, respectively ; the daughters 18 and 8 years, respectively. If the share of the youngest child was $ 7200, what was the value of the property ? 8. So divide $30,000 among A, B, C, and D, that their portions shall be to each other as 1^, 3, 4^, and 6. 9. Coe, Hall, Tell, and Lee have a contract to dig a ditch which Coe can dig in 35 days, Hall in 45 days, Tell in 50 days, and Lee in 60 days. How long will it take all together to do the work ? If f 100 be paid for the work and all join till it is completed, how much should each get ? PARTNERSHIP 946. Partnership is an association resulting from an agreement between two or more persons to place their capital or services, or both, in some enterprise or business, and to share the gains and bear the losses in certain proportions. 947. Partnerships may be formed by: 1. Oral agreement. 2. Written agreement, (a) under seal, or (b) not under seal. 3. Im- plication. All important partnership agreements should be in writing, and all of the conditions relating to the partnership should be definitely stated. 948. The partners are the persons associated in any business. Collectively they are called &jirm, a house, or a company. 949. Partners are of four classes : 1. Real or ostensible. 2. Dor- mant, silent^ or concealed. 3. Limited. 4. Nominal. 950. A real or ostensible partner is one who appears to the public to be, and who actually is, a partner. 951. A dormant or concealed partner is practically a real partner whose connection with the partnership is concealed from the public. A concealed partner is responsible for the debts of a firm if his connection with the partnership becomes known. 360 SHARING [ 952-902 952. A limited partner is one whose responsibility is limited instead of general. In case of failure of the firm, the general partner is individually responsible for all the debts of a firm, while the limited partner is responsible only for the amount named in the partnership agreement. Limited partnerships are forbidden by the laws of some states. In the states where they are permitted, it is generally provided that at least one member of a firm must be a general partner. 953. A nominal partner is one whose name appears to the public as a partner, but who has no investment and receives no share of the gains. A nominal partner is responsible to third parties for the debts of a firm. 954. The capital or stock is the money or other equivalent property invested in the business. Capital is frequently real estate, personal property, time, skill, etc. 955. Resources or assets are the entire property of a firm, includ- ing accounts receivable, bills receivable, etc. 956. Liabilities are the entire debts of a firm. 957. The net capital is the excess of the resources over the liabilities. 958. The net insolvency is the excess of the liabilities over the resources. When the resources of a firm exceed the liabilities, the business is said to be solvent ; when the liabilities exceed the resources, the business is said to be insolvent, or bankrupt. 959. The net investment of a person is his investment minus all withdrawals for personal use. 960. A business statement contains an itemized list of all resources and liabilities, of losses and gains, the present worth of the business, and the net gain or net loss for any given period. 961. The net gain is the excess of total gains over the total losses for any given period. 962. The net loss is the excess of total losses over the total gains for any given period. 962-964] PARTNERSHIP 361 DRILL EXERCISE 1. A commenced business with a cash investment of $6000. At the end of the year his net capital is $3500. What is his net gain or loss, no withdrawals having been made? 2. B began business with an investment of $7500. At the end of one year his net capital is found to be $ 9500. Find the net gain or loss. 8. C began business Jan. 1 with a cash investment of $2500. At the close of the year his net insolvency is $1700. Required the net gain or loss. 4. At the beginning of a year D's net insolvency was $600; at the close of the year his net capital was $150. Required his net gain or loss for the year. 5. E's insolvency at the beginning of the year was $ 4000, and at the close of the year is $ 3500. What has been his net gain or loss for the year ? 6. F's loss for one year is $ 700. His insolvency at the end of the year is $ 300. Required the net capital at the beginning. 7. G's net gain for one year is $ 2000 ; his net capital at the end of the year is $ 1500. What was the net capital or net insolvency at the beginning of the year ? 8. Jan. 1, 1903, H's resources were $3000, and his liabilities $2000; one year later his resources were $2000 and his liabilities $ 3000. What was the net gain or loss for the year ? 9. Fs insolvency at the end of one year is $ 3000. If he gained $ 1200 during the year, what was his net capital or net insolvency at the beginning of the year ? 10. J's capital at the end of one year was $5200. If his gain for the year was $6900, what was his net capital or net insolvency at the beginning of the year ? 963. To divide the gain or loss when each partner's investment has been employed for the same period of time. 964. Example. A and B enter into partnership to carry on a commission business for one year, A investing $7000 and B $ 4000. 362 SHARING [ 964 During the year they gain $ 3300. What should be each man's share of the gain ? SOLUTION The total investment = $ 7000 + $ 4000, or $ 11,000. A's investment = rVV'oV or A o * tlie total investment. B's investment = fWo^ or T 4 i of tne total investment. Each partner receives such a part of the gain as his investment is a part of the total investment. Therefore, A's share of the gain = ^ T of $ 3300, or $2100. B's share of the gain = of $ 3300, or $ 1200. ORAL EXERCISE Find each man's gain or loss in each of the following problems, the gains being shared or losses borne in proportion to investments. 1. A invested $300 and B $200; they gained $150. 2. C invested $ 800 and D $ 300 ; they gained $330. S. E invested $1000 and F $ 800; they lost $ 360. 4. G invested $ 1200 and H $ 900 ; they gained $ 560. 5. I invested $ 900 and J $ 700; they lost $ 320. 6. K invested $1500 and L $ 1200; they gained $540. 7. M invested $2000 and N $ 800; they lost $560. 8. invested $1000 and P $500; they gained $ 186. WRITTEN EXERCISE 1. A and B unite in the purchase of a house costing $ 4200 ; A pays $1800 and B $2400. The property rents for $294 per annum. What share of the rent ought each to receive ? 2. A, B, and C enter into partnership for the purpose of carry- -^r) ing on a manufacturing business. A joint capital of $65,000 is formed of which A furnishes f, B | of the remainder, and C what still remains. Their net gain for one year is equivalent to 25 % of the net capital invested. Required each man's share of the net gain. 964] PARTNERSHIP 3G3 S. Two men bought a mine for $ 20,000, of which sum A paid 112,500 and B the remainder. They sold the mine for $42,000. How much of the gain should each man receive ? What part of the selling price should each receive ? 4. A, B, and C engage in business, A investing $ 1250 ; B, $ 750 ; and C, $1000. They gain $ 957.30. How much of the gain should each receive ? How much is each man worth after receiving his share of the gain ? 5. B, C, and D unite their capital amounting to $12,000 in a business venture and realize a gain of $1500, of which B received '- $750; C, $500; and D, $250. What was the investment of ^ each? 6. In a partnership A invested $5000 and received f of the gain ; B invested $ 2400 and received -^ of the gain* The gains and losses were as follows: merchandise, gain, $840.30; expense, loss, $ 310.40 ; real estate, gain, $ 265.61. What was the net gain ? What was the gain of each partner ? After receiving his share of the gain, what was each partner worth ? 7. In a partnership C invested $4000 and D $2000. It is agreed that if gains are realized they are to be shared according to the investment, each partner receiving such part of the gain as his investment is a part of the total investment; that if losses occur they are to be borne equally. At the end of one year the gains and losses were as follows: merchandise, gain, $534.20; expense, loss, $325.60; real estate, loss, $675; interest, gain, $34.25; dis- count, loss, $ 56.35. What was the net gain or loss ? What was each partner worth after his share of the net results of the business was carried to his account ? 8. In a partnership E invested $7654, F $8000, and G $7000; the gains and losses were to be shared equally. During a month the gains and losses were as follows : merchandise, gain, $ 2318 ; stocks and bonds, gain, $735; expense, loss, $1140; interest, gain, $342. What was the net gain? What was each partner's interest at closing ? 9. H's net loss for one year was $17,290. His insolvency at the close of the year was $10,000. Find the net capital at the beginning. 364 SHARING [ 965-967 965. To divide the gain or loss according to the amount invested and the time the investment is employed. 966. The best method of solving partnership problems is to treat them from the accountant's standpoint. In connection with the so- lution of each problem a ledger page may be used, an account opened with each partner, the net gain or loss properly divided, and the ledger accounts with the partners closed. 967. Examples. 1. In a partnership, A invested $2000 for 8 months, B, $3000 for 6 months, and C, $4000 for 5 months. A gain of $ 1350 was realized. Find the gain of each partner. SOLUTION Dr. A Cr. Present worth 2400 2000 Net Gain 400 = 2400 2400 Dr. B Cr. Present worth 3450 Net Gain 3000 450 3450 3450 Dr. Cr. _ Present worth 4500 Net Gain 4000 500 4500 4500 A's investment, $ 2000 for 8 mo. = $ 16,000 for 1 mo. B's investment, $ 3000 for 6 mo. = 18,000 for 1 mo. C's investment, $4000 for 5 mo. = 20,000 for 1 mo. Firm's investment = $ 54,000 for 1 mo. Each partner should receive such apart of the gain as his investment for 1 mo. is a part of the total investment for 1 mo. Therefore, A's gain = JV OflHfR) of 1350, or $ 400. B's gain = & or J G^) of $ 1350, or $450. C's gain = tf ($${{$) of $ 1350, or $ 500. 967] PARTNERSHIP 365 2. A and B were partners in a manufacturing business. Their investments and withdrawals for one year were as follows : Jan. 1, A invested $2000; Apr. 1 he withdrew $500; July 1 he invested $1000; Oct. 1 he withdrew $700. Jan. 1, B invested $3000; May 1 he withdrew $1000; Sept. 1 he invested $500. During the year the firm gained $1780. What was each man's share of the gam? SOLUTION Dr. A Cr. Apr. 1 500 Jan. 1 2000 Oct. 1 700 July 1 1000 Jan. 1 Present worth 580 Jan. 1 Net Gain 780 3780 3780 Dr. Cr. May 1 1000 Jan. 1 3000 Jan. 1 Present worth 3500 Sept. 1 500 Jan. 1 Net Gain 1000 4500 4500 A's Investment $ 2,000 for 3 mo. = $ 6,000 for 1 ino. 1,500 for 3 mo. = 4,500 for 1 mo. 2,500 for 3 mo. = 7,500 for 1 mo. 1,800 for 3 mo. = 5,400 for 1 mo. A's total investment = $ 23,400 for 1 mo. B's Investment $3,000 for 4 mo. = $ 12,000 for 1 mo. 2,000 for 4 mo. = 8,000 for 1 mo. 2,500 for 4 mo. = 10,000 for 1 mo. B's total investment = $ 30,000 for 1 mo. $23,400 + $30,000 = $53,400, the firm's investment for 1 mo. A's gain = f f of $ 1780, or $ 780. B's gain = f$ of $ 1780, or $ 1000. WRITTEN EXERCISE 1. Three persons traded together and gained $900. A in- vested in the business $ 1000 for 6 months ; B invested $ 750 for 10 months ; and C invested $ 1200 for 5 months. How should the ^ gain be divided ? 2 7 366 SHAKING [ 967-969 2. A, B, and C were partners. A had $ 800 in the business for ^1 year, B had $1000 in for 9 months, and C had $2000 in for 8 ^ ^ ^months. How should a gain of $2150 be divided ? f 7 V /? A A, B, and C hired a pasture for 6 months for $ 95.10; A put .- in 75 sheep, and 2 months later took out 40 ; B put in 60 sheep, and at the end of 3 months put in 45 more ; C put in 200 sheep, and after 4 months took them out. What part of the rent should each pay ? 4. A commenced digging a ditch, and after wording 6 days was joined by B, after which the two worked together 9 days, when they ^ 2 were joined by C. The three then worked 12 days, and at the end of that time A left the job and D worked with the other two 3 days, . r when the work was completed.- If $ 92 was paid for the work, how much should each receive ? / v5. Martin and Eaton were partners one year, Martin investing at first $ 5000 and Eaton $ 3000 ; after 6 months Martin drew out $ 3000 and Eaton invested $ 1500 ; they gained $ 3600. What was the gain of each and the present worth of each, at the time of the dissolution of the partnership ? '- 6. A and B engaged in a grocery business for 3 years from March 1, 1901, On that date each invested $ 1600 ; June 1, of the same year, A increased his investment $400, and B withdrew $300) Jan. 1, 1902, each withdrew $1000; Jan. 1, 1903, each invested $ 1500. How should a gain of $ 7500 be divided at the expiration of the partnership contract ? PARTNERSHIP SETTLEMENTS 968- A partnership settlement is an adjustment of the net value of a business among the partners when a partnership is dissolved, either by mutual consent or by limitation of contract. 969. In most partnership affairs a business statement would bo required in connection with the finding of the condition of the busi- ness at the close of any given period. WRITTEN EXERCISE 1. Copy the following statement of resources and liabilities, fill- ing out the missing terms : 969] PARTNERSHIP 367 000 .^/00 2-000 t To be written in red ink. 368 SHARING [ 069 Using the foregoing statement form as a model, make statements to solve the questions embodied in each of the following problems : 2. At the time of closing business, the resources of a firm were : cash, $931.50; merchandise, per inventory, $13,196.25; notes and accounts due it, $8154; interest on same, $211.50; real estate, $11,150. The firm owed, on its notes, acceptances, and bills out- standing, $7142, and interest on the same, $348.50; and there was an unpaid mortgage on the real estate of $ 2500, with interest accrued thereon of $88.50. If the invested capital was $ 22,500, what was the net solvency or insolvency of the firm at closing, and how much has been the net gain or net loss ? 3. Burke, Brace, and Baldwin became partners, each investing $15,000, and each to have one third of the gains or sustain one third of the losses. Burke withdrew $2100 during the time of the partnership, Brace $ 1800, and Baldwin $ 2000. At the close of busi- ness their resources were : cash, $ 3540 ; merchandise, $ 14,785 ; notes, acceptances, and accounts receivable, $ 16,250 ; real estate, $ 28,500. They owed on their outstanding notes $8125," and on sundry personal accounts $1950. Find the present worth of each partner at closing. ( 4. Parsons and Briggs became partners Apr. 1, 1901, under an agreement that each should be allowed 6% simple interest on all investments, and that, on final settlement, Briggs should be allowed 10% of the net gains, before other division, for superintending the business, but that otherwise the gains and losses be divided in pro- portion to average investment. Apr. 1, 1901, Parsons invested $18,000, and Briggs $4000; Jan. 1, 1902, Parsons withdrew $5000, and Briggs invested $3000; Aug. 1, 1902, Briggs withdrew $1500; Dec. 1, 1902, the partners agreed upon a dissolution of the partner- ship, having resources and liabilities as follows : Resources Liabilities Cash on hand and in bank, $ 1,101.05 Notes and acceptances, $6,520.00 Accounts receivable, 16,405.50 Outstanding accounts, 21,246.50 Bills receivable, 2,550.00 Rent due, 1,200.00 Interest accumulated on same, 287.41 Mdse. per inventory, 9,716.55 If only 80% of the accounts receivable prove collectible, what has been the net gain or net loss ? What has been the net gain or net loss of each partner ? What is the firm's net insolvency at dissolu- tion ? What is the net insolvency of each partner ? 909] PARTNERSHIP 369 Liabilities Mortgage on real estate, $ 7,000.00 Interest accrued on same, 210.00 Notes outstanding, 26,950.00 Interest accrued on same, 811.75 Due Barnes, Clay & Co., 33,560.00 5. A and B became partners for one year, A investing f of the capital, and B f , the agreement being that the gains or losses shall be apportioned according to average net investment, and that each partner be allowed 6% interest per annum on all investments, and be charged interest at the rate of 6% on all sums withdrawn. At the end of the year the firm had resources and liabilities as follows : Resources Mdse., per inventory, $ 21 ,460.00 Real estate, 15,000.00 Cash, 1,950.00 Bills receivable, 13,146.50 Interest accrued on same, 519.25 Accounts due the business, 11,218.50 Furniture, 1,320.00 Delivery wagons and horses, 2,100.00 It is found that 33 \fo of the accounts due the firm are uncol- lectible. If the firm's losses during the year have been $12,000, how much was invested by each partner ? What is the present worth or net insolvency of the firm, and of each partner, at closing ? 6. Mason and Kivers were joint partners, each investing an equal part of $ 9000. At the end of the year the resources and liabilities were as follows: cash on hand, $2212.45; merchandise on hand, $7278.54; bills receivable, $943.50; interest accrued on bills receiv- able, $22.70; office safe, $160; accounts receivable, $2956.20; 5% of the accounts receivable were estimated as not collectible ; accounts payable, $1147; unpaid freight bill, $64.50; bills payable $560; interest accrued on bills payable, $17.25. What was the net gain ? What was each partner's present worth at the close of the year ? 7. D and E are partners, each investing $9000. The losses and gains, respectively, are to be borne or shared equally. At the end of one year the following is a list of their resources and liabili- ties : merchandise on hand, $ 6235.42 ; personal accounts due the firm, $4785.15; cash on deposit, $4756.20; cash in safe, $543.82; bills receivable on hand, $2658.90; N. Y. C. & H. E. E. E. stock, $ 3600 ; First National Bank stock, $ 2000 ; bills payable outstand- ing, $4298.75; due sundry persons on account, $3215.60. During the year D withdrew $1800 and E withdrew $ 2000. What is the net gain and the present worth of each partner ? w f^i. v- ^ SHARING - [90 llr^ < 'A and B are equal partners in a business, the losses and gains of which are to be borne and shared equally. The following is the condition of the business at the close of one year.. Resources : cash, $ 4275 ; merchandise, per inventory, $ 5476..20 ; accounts receivable, $ 2356 ..75. Liabilities : bills payable, $ 2240 j Davis & Weller, $ 140Q. The net gain for the year is $4267^5. What was the investment of each partner at the beginning of business ? 9. The following is the trial balance of the firm of Austin & Leland, at the close of one year's business : Dr. Charles Austin, Proprietor, $350.00 William Leland, Proprietor, 327.00 Cash, 5,647.27 Merchandise, 3,187.56 Bills receivable, 9,000.00 Bills payable, Expense, 475.00 Interest, 76.46 Discount, Accts. receivable, 3,427.50 Accts. payable, Or. $0,885.24 6,385.24 5,909.00 64.65 3,756.65 Inventories : Mdse. (goods on hand) , Mdse. (unpaid freight bill), Expense (coal on hand), Expense (unpaid gas bill), $8,764.50 82.26 $28.46 10.50 It is estimated that 10% of the accounts receivable cannot be collected. Make a business statement. What is the present worth of the business? What is the net gain? What is the present worth of each proprietor? rf? 10 x 12 x 8 WRITTEN EXERCISE 1. If 15 men earn $ 607.50 in 18 days, how much should 21 men earn in 12 days ? 2. If 6 men, working for 12 days, dig a ditch 80 rods long, how many rods of such ditch should 15 men dig in 21 days ? 3. If 5 men, working 6 days of 12 hours per day, can cut 24 acres of corn, how many acres of corn should 8 men cut in 5 days, if they work 10 hours per day ? 4. If $145.35 interest accrue on $510, at 6%, in 4 yr. 9 mo., how much interest will accrue at the same rate and time on $1350? 5. If $760, put at interest at 10%, accrue $9.50 interest in 45 days, in how many days will $1140 accrue $17.67 interest at 6% ? STORAGE 1002. Storage is a charge made for storing movable property in a warehouse. 1003. The rates of storage may be fixed by an agreement between the parties to a contract, but they are often regulated by boards of trade, chambers of commerce, associations of warehousemen, and by legislative enactment. 1004. The term of storage is the period of time for which storage charges are made. 1005. Storage may be charged at a fixed rate per package, etc., or at a fixed sum per term ; but it is usually charged by the day, week, or month, and a fractional term is counted as a full term. CASH STORAGE 1006. Cash storage is the term applied to cases in which the storage is paid or estimated at the time of the withdrawal of goods from the warehouse. 1007. In cash storage all goods delivered are deducted from the oldest receipt on hand. In private bonded warehouses of the United States goods may be taken out at any time in quantities not less than an entire package, or if in bulk, of not less than one ton, by the payment of duties, storage, and labor charges. The storage charges are computed for periods of one month each, a fractional part of a month being counted the same as a full month. 1008. Example. At 5 ^ per barrel per month, or fraction thereof, how much should be paid for the following storage of apples, storage being charged at each delivery ? Receipts Deliveries Sept. 1, 2000 bbl. Oct. 20, 1000 bbl. Nov. 20, 300 bbl. Nov. 2, 300 bbl. Dec. 20, 1000 bbl. 381 382 STORAGE [ 1008 SOLUTION Date Receipts and Deliveries Bate Storage Sept. 1 received 2000 bbl. Oct. 20 delivered 1000 bbl., which were in storage 49 da. 10 <* $ 100 1000 bbl., balance in storage Nov. 2 delivered 300 bbl., which were in storage 62 da. 16 J* 45 700 bbl., balance in storage Nov. 20 received 300 bbl. 1000 bbl., balance in storage Dec. 20 delivered 1000 bbl., 700 of which were in storage 110 da. 20 ^ 140 300 of which were in storage 30 da. 6 ^ 15 Total storage $300 WRITTEN EXERCISE 1. Find the storage of the following at 2 ^ per month, or fraction thereof, storage being calculated at each delivery. Receipts Deliveries Sept. 5, 200 cases Oct. 1, 100 cases Sept. 30, 400 cases Nov. 2, 200 cases Nov. 1, 100 cases Dec. 1, 200 cases Nov. 8, 200 cases Dec. 10, 400 cases 2. At a warehouse there was received and delivered flour as follows : Receipts Deliveries Jan. 3, 150 bbl. Jan. 23, 250 bbl. Jan. 20, 200 bbl. Mar. 1, 400 bbl. Feb. 1, 300 bbl. The storage charge on the above was 5^ per barrel for the first 10 days, or fraction thereof, and 3^ per barrel for each subsequent period of 10 days, or fraction thereof. What sum must be paid in settlement ? 3. The receipts and deliveries of goods at a storage warehouse were as follows : Receipts Deliveries Sept. 2, 100 bbl. Sept. 20, 100 bbl. 25, 200 bbl. 30, 100 bbl. Oct. 19, 350 bbl. Oct. 10, 100 bbl. 31, 150 bbl. 20, 100 bbl. Nov. 7, 200 bbl. 30, 100 bbl. Nov. 20, 500 bbl. 1008-1010] CREDIT OR AVERAGE STORAGE 383 The contract required the payment of 6^ per barrel for the present term of 30 days, or fraction thereof, and 3j per barrel for each subsequent term of 30 days, or fraction thereof. Find the storage bill. CREDIT OR AVERAGE STORAGE 1009. Credit or average storage is the term applied to cases in which the storage is not paid or estimated until the last withdrawal is made. 1010. Example. The storage charges being 2^ per barrel per month of 30 days, what will be the bill in the following transaction ? Receipts Deliveries July 19, 200 bbl. Aug. 15, 100 bbl. 31, 300 bbl. Sept. 17, 400 bbL SOLUTION From July 19 to July 31 = 12 da., 200 bbl. stored for 12 da. = 1 bbl. stored for 2,400 da. July 31 300 bbl. received. From July 31 to Aug. 15 = 15 da., 600 bbl. stored for 15 da. = 1 bbl. stored for 7,500 da. Aug. 15 100 bbl. delivered. From Aug. 15 to Sept. 17 = 33 da., 400 bbl. stored for 33 da. = 1 bbl. stored for 13,200 da. Sept. 17 400 bbl. delivered. Total, 1 bbl. stored for 23,100 da. 28,100 -v- 30 = 770 terms of 30 da. each. $.02 x 770 = $15.40, the total storage bilL WRITTEN EXERCISE 1. There was received at a warehouse : May 30, 4000 bu. wheat ; June 5, 2600 bu. oats ; June 24, 3500 bu. barley ; July 18, 5000 bu. corn. If all of this was shipped July 20, what was the storage bill, the charge being 1-|^ per bushel per term of 30 days 7 average storage ? 2. What will be the storage charge, at 4-J ^ per barrel, for a term of 30 days' average, in the following transaction ? 384 STORAGE [ 1010 Receipts Deliveries Feb. 8, 180 bbl. flour Mar. 1, 100 bbl. apples 27, 100 bbl. apples 28, 190 bbl. flour Mar. 8, 60 bbl. potatoes Apr. 15, 60 bbl. potatoes 13, 300 bbl. flour 15, 60 bbl. flour 29, 230 bbl. flour 8. A farmer received for pasture: Apr. 30, 12 head of cattle-, May 15, 14 head of cattle; May 23, 27 head of cattle; June 9, 5 head of cattle; June 30, 8 head of cattle; July 16, 40 head of cattle. All were delivered July 25, and the charges were 75^ per head for each week of 7 days' average pasturage. How much was his bill ? 4. A drover hired a pasture of a farmer, agreeing to pay $4.20 per head of stock pastured for each average term of 30 days. What was the amount of the bill, the receipts and deliveries being as follows ? Receipts Deliveries June 15, 21 head of cattle July 1, 30 head of cattle 27, 20 head of cattle 20, 15 head of cattle July 5, 15 head of cattle 30, 15 head of cattle 29, 40 head of cattle Aug. 21, the remainder 31, 40 head of cattle 5. Find the storage charges, at 3^ per barrel, for a term of 30 days' average, on the following: Receipts Deliveries Sept. 2, 1620 bbl. Sept. 13, 520 bbl. 16, 2920 bbl. 26, 966 bbl. Oct. 25, 1470 bbl. Dec. 2, 4524 bbl. APPENDIX METRIC SYSTEM OF MEASURES 1011. The metric system is a decimal system of denominate numbers. It is in use in nearly all the European states, in South America, Mexico, and Egypt. It is also used in parts of Asia, is authorized by law in the United States, and is almost universally used in scientific treatises. 1012. The fundamental unit of the metric system is the meter, a measure of length, which is equal to about one ten-millionth of the distance from the equator to the pole. It is defined by law as being the length of the bar of platinum which is carefully preserved at Paris. Accurate copies of the meter have been procured by the governments of all civilized nations. 1013. Among the advantages claimed for the metric system are: 1. It employs only five unit words and seven prefixes. 2. Every word used suggests its measure. 3. It is consistent, uniform, simple, and complete, and would do away with the present inconsistent system of compound numbers. 4. Being a decimal system it makes arithmetical operations relat- ing to measure much more simple. 5. It gives to the nations a uniform system of measures and thus materially facilitates trade and exchange. 1014. The principal units of the metric system are : 1. The meter for lengths. 2. The square meter for small surfaces such as floors, ceilings, etc. 3. The are, of 100 square meters, for large surfaces such as land measurements. 4. The cubic meter for solids. 5. The liter for capacities. 6. The gram for weights. 386. APPENDIX [ 1015-1018 1015. Each of the metric units is divided and multiplied deci- mally. The higher orders are indicated by four Greek prefixes, as follows : deka, meaning 10 ; hecto, meaning 100 ; kilo, meaning 1000 ; myria, meaning 10,000. The lower orders are indicated by three Latin prefixes, as follows : deti, meaning .1 j centi, meaning .01 j milli, meaning .001. Metric Long Measure 1016. The unit of long measure is the meter. TABLE 10 millimeters (mm.) 10 centimeters 10 decimeters 10 meters 10 dekameters 10 hektometers 10 kilometers = 1 1 1 1 1 1 1 centimeter (cm.) = decimeter (dm.) = meter (m.) = dekameter (Dm.) = hektometer (Hm.) = kilometer (Km.) = myriameter (Mm.) = .01 .1 1. 10. 100. 1000. 10,000. meter, meter, meter, meters, meters, meters, meters. In the above and each of the following tables the units in common use are indicated by the blackfaced type. In the metric system all abbreviations which indicate a fractional part of a standard unit begin with a small letter, while all those which indicate a multiple of a standard unit begin with a capital letter. Metric Square Measure 1017. The units of square measure are the square meter for small surfaces and the are for land measurements. 1018. The units of square measure are the square of the units of long measure ; hence 100 units of any given denomination are required for 1 of the next higher. 100 sq. millimeters 100 sq. centimeters 100 sq. decimeters 100 sq. meters TABLE = 1 sq. centimeter (sq. cm.) = = 1 sq. decimeter (sq. dm.) = = 1 sq. meter (sq. m.) = 1 sq. dekameter (sq. Dm.) = 100 sq. dekameters = 1 sq. hektometer (sq. Hm.) = 10.000. 100 sq. hektometers = 1 sq. kilometer (sq. Km.) = 1,000,000. .0001 sq. meter. .01 1. meter = 1 centare. 100. meters = 1 are. " = 1 hectare. 1010-1024] METRIC SYSTEM OF MEASURES 887 1019. The centare, the are, and the hectare, are used only in measuring land. Metric Cubic Measure 1020. The units of cubic measure are the cubic meter, for ordinary solids, and the stere, for wood measurements. 1021. The units of cubic measure are the cube of the units of long measure; hence 1000 units of any given denomination are required for 1 of the next higher. TABLE 1000 cu. millimeters (cu. mm.) 1 cu. centimeter (en. cm.) .000001 cu. meter. 1000 cu. centimeters 1 cu. decimeter (cu. dm.) = .001 '* " 1000 cu. decimeters = 1 cu. meter (cu. m.) 1. " Metric Measures of Capacity 1022. The unit of capacity, for both liquid and dry measures, is the liter. 1023. The liter is a cube whose side is 1 decimeter; hence a liter is a cubic decimeter. TABLE = .01 liter. = .1 " = 1. " = 10. liters = 100. " = 1,000. " Measures of Weight 1024. The unit of weight is the gram, which is equal to 1 cubic centimeter of pure water at its greatest density ; that is, at a tem- perature just above freezing. TABLE 10 milligrams (mg.) = 1 centigram (eg.) = .01 gram. 10 centigrams = 1 decigram (dg.) = .1 *' 10 decigrams = 1 gram (g.) = 1. " 10 grams = 1 dekagram (Dg.) = 10. grams. 10 dekagrams = 1 hektogram (Hg.) = 100. 10 hektograms = 1 kilogram (Kg.) = 1,000. 10 kilograms = 1 myriagram (Mg.) = 10,000. 10 myriasrams = 1 quintal (Q. ) = 100,000. 10 quintals = 1 tonneau (T.) = 100,0000. " 10 milliliters (ml.) 10 centiliters 10 deciliters 10 liters 10 dekaliters 10 hektoliters = 1 centiliter (cl.) = 1 deciliter (dl.) = 1 liter (1.) = 1 dekaliter (Dl.) = 1 hektoliter (HI.) = 1 kiloliter (Kl.) . 338 APPENDIX [ 1025 1025. An act of Congress requires all reductions from the metric to the common system, or the reverse, to be made according to the following TABLE OF EQUIVALENTS Long Measure 1 inch = 2.54 centimeters. 1 centimeter = .3937 of an inch. 1 foot = .3048 of a meter. 1 decimeter = .328 of a foot. 1 yard = .9144 of a meter. 1 meter = 1.0936 yards. 1 rod = 5.029 meters. 1 dekameter = 1.9884 rods. 1 mile = 1.6093 kilometers. 1 kilometer = .62137 of a mile. Square Measure 1 sq. inch = 6.452 sq. centimeters. 1 sq. centimeter = .155 of a sq. inch. 1 sq. foot = .0929 of a sq. meter. 1 sq. decimeter = .1076 of a sq. foot. 1 sq. yard = .8361 of a sq. meter. 1 sq. meter = 1.196 sq. yards. 1 sq. rod = 25.293 sq. meters. 1 are = 3.954 sq. rods. 1 acre =40.47 ares. 1 hektare = 2.471 acres. 1 sq. mile = 259 hectares. 1 sq. kilometer = .3861 of a sq. mile. Cubic Measure I cu. inch = 16.387 cu. centimeters. 1 cu. centimeter = .061 of a cu. inch. 1 cu. foot = 28.317 cu. decimeters. 1 cu. decimeter = .0353 of a cu. foot 1 cu. yard = .7645 of a cu. meter. 1 cu. meter = 1.308 cu. yards. 1 cord = 3.624 steres. 1 stere = .2759 of a cord. Measures of Capacity 1 liquid quart = .9463 of a liter. 1 liter = 1.0567 liquid quarts. 1 dry quart = 1.101 liters. 1 liter = .908 of a dry quart. 1 liquid gallon = .3785 of a dekaliter. 1 dekaliter = 2.6417 liquid gallons. 1 peck = .881 of a dekaliter. 1 dekaliter = 1.135 pecks. 1 bushel = .3524 of a hektoliter. 1 hektoliter = 2.8375 bushels. Measures of Weight 1 grain, Troy = .0648 of a gram. 1 gram = .03527 of an ounce, Avoir. 1 ounce, Avoir. = 28.35 grams. 1 gram = .03215 of an ounce, Troy. 1 ounce, Troy = 31.104 grams. 1 gram =15.432 grains, Troy. 1 pound, Avoir. = .4536 of a kilogram. 1 kilogram = 2.2046 pounds, Avoir. 1 pound, Troy = .3732 of a kilogram. 1 kilogram = 2.679 pounds, Troy. I ton (short) = .9072 of a tonneau. 1 tonneau = 1.1023 tons (short). 1025-1028] POWERS AND ROOTS 389 WRITTEN EXERCISE 1. What will be the cost in. Paris of a cargo of 38,500 bu. United States wheat at 10 francs 60 centimes per hektoliter ? 2. How many avoirdupois pounds in 10 myriagrams 4 kilo- grams ? 3. Reduce 250 hectares to common units. 4. A pile of wood 56 meters long, 18J meters wide, and 3f meters high was sold at $6 per cord. How 'much was received for it? 5. At 21^ per liter, what will 150 quarts of olive oil cost ? 6. If the cost of 50 liters of wine was 800 francs, what was the price per gallon in United States money ? 7. A merchant bought silk at $ 1.20 per meter and sold it by the yard at a gain of 20%. What was the selling price per yard ? 8. A man bought 50 kilograms of sugar for $ 5.51 and sold it at a gain of 20%. What did he receive per pound ? 9. A pile of wood 8 meters long and 2 meters wide contains 56 steres. Find the height of the pile in meters ; in feet and inches. 10. Reduce 954 miles to kilometers. 11. How many meters of carpet 70 centimeters wide will be required for a room 7 meters long and 5 meters wide, if the strips run crosswise and 4 meters are lost in matching the pattern ? how many yards ? 12. How many hectares in a field 150 meters on a side ? how- many acres ? POWERS AND ROOTS 1026. The power of a number is the product arising from multi- plying the number by itself one or more times. 1027. A perfect power is a number that can be exactly produced by the involution of some number as a root. Thus, 25 and 8 are perfect powers, since 5 x 5 = 25, and 2x2x2 = 8. 1028 The square of a number is its second power. 390 APPENDIX 1029. The cube of a number is its third power. [ 1029-1033 1030. An exponent is a small figure written at the right of a number to indicate how many times the number is to be used as a factor. Thus, 2 2 is equivalent to 2 x 2 and is read the second power of 2 ; and 5 3 is equivalent to 5 x 5 x 5 and is read, 'the third power of 5. 1031. The root of a number is one of the equal factors which multiplied together will produce the given number. SQUARE ROOT 1032. The square root of a number is one of the two equal factors which multiplied together will produce the given number. The accompanying diagram is a square 14 ft. on a side. Its area is, by inspection, found to be made up of : 1. The tens of 14, or 10 2 , equal to 100 sq. ft., as shown by the square within the angles, a, >, c, d. 2. Twice the product of the tens by the units of the same number or 2 x (10 x 4), equal to 80 sq. ft., as shown by the surface within the angles e, /, gr, h and z, j, k, I. 3. The square of the" units, 4 ft, equal to 16 sq. ft., as shown by the square within the angles to, x, y, z. a d c h b c f (J i I w z J k X y 14 ft. = 10 ft. and 4ft. Hence a square 14 ft. on each side will contain : 10 2 = 100 sq. ft. 2 x (10 x 4) = 80 sq. ft. 4 2 = 16 sq. ft. 142 = 196 sq. ft. 1033. Therefore the following general principle may be stated : The square of any number, composed of two or more figures, is equal to the square of the tens plus twice the product of the tens multiplied by tlw units plus the square of the units. 1034-1036] POWERS AND ROOTS 391 1034. In extracting the square root of a number, the first impor- tant step is to separate the figures of which the number is composed into groups. The squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. From the above it is evident : 1. That the square of any number will contain at least one place or one order of units. 2. That the square of no number represented by a single figure will contain more than two places. 3. That if the number of which the square root is sought be separated into periods of two figures each, beginning at the units, the number of periods and partial periods so made will represent the number of unit orders in the root. 4. That the square of any number will contain twice as many places or one less than twice as many places as its root. 5. That where the product of the left-hand figure multiplied by itself is not greater than 9, then the square will contain one less than twice as many places as the root. 1035. Example. Find the square root of 625. 6 OK /OK SOLUTION. The number consists of one full and one partial period ; hence, its root will contain two places. The given number, 625, is the second power of the root to 45)2 25 be extracted ; therefore the first figure of the root, which will 2 25 be the highest order of units in that root, must be obtained from the first left-hand period. The first, or tens' figure, of the root will be the square root of the greatest perfect square in 6. Hence, 2 is the tens' figure of the root. Subtracting the tens, the remainder, 225, must be equal to twice the tens multiplied by the units plus the square of the units. Twice the 2 tens is equal to 4 tens. 4 tens is contained in the 22 tens of the remainder 5 times ; hence, 5 is the units' figure of the root. Twice the tens mul- tiplied by the units plus the square of the units is equivalent to twice the tens plus the units multiplied by the units. Therefore, 5 units are annexed to the 4 tens and the result, 45, is multiplied by 5. Therefore, the square root of 625 is 25. 1036. From the foregoing explanations the following rule may be derived : Beginning at the right, separate the given number into periods of two places each. Take the square root of the greatest perfect square con- tained in the left-hand period for the first root figure; sub- 392 APPENDIX [ 1036-1037 tract its square from, the left-hand period, and to the remainder bring down the next period. Divide the number thus obtained, exclusive of its units, by twice the root figure already found for a second quotient or root figure. Place this figure at the right of the root figure before found, and also at the right of the divisor. Multiply the divisor thus formed by the new root figure. Subtract the result from the dividend, to the remainder bring down the next period, and so proceed until the last period has been brought down, considering the entire root already found as so many tens in determining subsequent root figures. Whenever the divisor is greater than the dividend, place a cipher in the root, and also at the right of the divisor ; bring down another period, and proceed as before. When the root of a mixed decimal is required, form periods from the decimal point right and left, and if necessary supply a decimal cipher to make the decimal periods of two places each. Any root of a common fraction may be obtained by extracting the root of the numerator for a numerator of the root, and the root of the denominator for the denominator of the root. To find a root, decimally expressed, of any common fraction, reduce such fraction to a decimal, and extract the root to any number of places. WRITTEN EXERCISE Find the square root of : 1. 196. 5. 5625. 9. 125.44. 2. 225. 6. 42436. 10. 50.2681. 3. 576. 7. 15625. 11. 4. 1225. 8. 1048576. 12. APPLICATIONS OF SQUARE ROOT 1037. It has been shown that the area of a square is the product of its two equal sides. Hence, The side of any square is the square root of its area. 1038-1039] POWERS AND ROOTS 393 1038. The hypothenuse of a right-angled triangle is the side opposite the right angle. 1039. The square formed on the hypothenuse of a right-angled triangle is equal to the sum of the squares formed on the base and perpen- dicular. Hence, The hypothenuse of a right-angled tri- angle is the square root of the sum of the squares of the other two sides; and The base or perpendicular of a right- angled triangle is the square root of the dif- ference between the square of the hypothe- nuse and that of the given side. WRITTEN EXERCISE 1. The base of a figure is 60 ft. and the perpendicular 80 ft. What is the hypothenuse ? 2. A farm of 80 acres is in the form of a rectangle, the length of which is twice its width. How many rods of fence will inclose it ? 3. How many rods of fence will inclose a triangular field whose base is equal to its perpendicular and whose area is 20 acres ? 4. If a farm is 1 mile square, how far is it diagonally across from corner to corner? Express the result in rods, feet, and inches. 5. What is the width of a street in which a ladder 60 ft. long can so be placed that -it will reach the eaves of a building 40 ft. high on one side of the street, and of another building 50 ft. high on the opposite side of the street ? 6. How many feet of fence will inclose a square field containing 16 acres ? 7. How far apart are the opposite corners of a rectangular field having a width equal to f of its length and containing 30 acres ? 8. What is the length of one side of a square field, the area of which is one acre ? 394 APPENDIX [ 1040-1043 CUBE ROOT 1040. The cube root of a number is one of the three equal factors which multiplied together will produce the given number. Thus, a cubic foot equals 12 x 12 x 12, or 1728 cubic inches, the product of its length, breadth, and thickness ; and since 12 is one of the equal factors of 1728, it must be its cube root. 1041. The first point to be settled in extracting any root is the relative number of unit orders or places in the number and its root. The cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 From the above it is evident : 1. That the cube of any number expressed by a single figure cannot have less than one nor more than three places or unit orders. 2. That each place added to the number will add three places to its cube. 3. That if a number be separated into periods of three figures each, begin- ning at the units, the number of places in the root will equal the number of periods and partial periods, if there are any. 4. That the cube of any number will contain three times as many places or one or two less than three times as many places as its roots. 1042. Since 57 equal 50 -f 7, the cube of 57 may be determined in the following mannqr : 50+7 50+7 (50 x 7) +7* 60=126,000 cu. ft. 50 2 +(50x7) 3 x (50 2 x 7) = 52,500 cu. ft. 502+2 x (50 x 7) +7* 3 x (50 X 7 2 ) = 7,350 cu. ft. 50+7 73= 343 cu . ft. (50*x7)+2x(50x7 2 )+7 8 67 3 =185,193 cu. ft. 50 3 +2x(50 2 x7) + (50x7 2 ) 503+3 x (50 2 x 7) +3 x (50 x I 2 ) + 7* =185,193 1043. Hence the following general principle may be stated : The cube of a number is equal to the cube of the tens, plus three times the square of the tens, multiplied by the units, plus three times the tens multiplied by the square of the units, plus the cube of the units. 1044] POWERS AND ROOTS 395 1044. Example. Extract the cube of 15625. SOLUTION. The given num- ber consists of two periods of three figures each, therefore its 3t 2 u-f3tu 2 4-u 3 = 7625= remainder. cube root wil1 contain two t s +3t 2 u+3tu 2 +u 3 =15.625(2 5 t 3 = 8 or 8000 t 2 =400 3t 2 =1200 3t= 60 3 1 2 +3 1=1260 trial divisor. 3t 2 u=6000 3tu 2 =1500 u 8 = 125 3t 2 u+3tu 2 +u 3 =7625. places. Since the given number is a product of the root taken three times as a factor, the first fig- ure, or highest order of the root, must be obtained from the first left-hand period, or highest or- der of the power. The greatest cube in 15 is 8 and the cube root of 8 is 2 ; hence, 2 is the tens' figure of the root. Subtracting the cube of the root figure thus found and bringing down the next figure, the entire remainder is found to be 7825. Referring to the general principle stated above, we find that having sub- tracted from the given number the cube of its tens, the remainder, 7625 must contain three times the product of the square of the tens by the units plus three times the product of the tens by the square of the units plus the cube of the units. If a cube (^4), 20 inches in length on each side, is formed, its solid contents will equal 8000 cubic inches, and it will be shown that the remaining 7625 cubic inches are to be so added to cube (^4) that it will retain its cubical form. In order to do this, equal additions must be made to the three adjacent sides; and these three sides being each 20 inches in length and 20 inches in width, the addi- tion to each of them in surface, or area, is 20 2 , and to the three sides 3 x(20 2 ), as shown in the squares (U). It will also be observed that the three oblong blocks, as shown in ((7), will be required to fill out the vacancies in the edges, and also the small cube (Z>), to fill out the. corner. Since each of the oblong blocks has a length of 2 tens, or 20 inches, the three will have a length of 3 x 20 inches. Observe now that the surface to be added to the cube (^4), in order to include in its contents the 7625 remaining cubic inches, has been nearly, but not exactly, obtained ; and since cubic contents 396 APPENDIX [ 1044-1045 divided by surface measurements must give units of length, the thickness of the three squares (5), and of the three oblong pieces (O), will be determined by dividing 7625 by the surface of the three ^ squares plus the surface of the three oblong blocks. This division may give a quotient too large, owing to the omission in the divi- sor of the small square in the corner ; hence, such surface measure taken as a divisor may with propriety be called a trial divisor. So using it, 5 is obtained as the second, or unit figure of the root. Assuming this 5 to be the thickness of the three square blocks (J5), and both the height and thickness of the three oblong blocks ((7), gives for the solid contents of the three square blocks (.B), 6000, and for the solid contents of the three oblong blocks ((7), 1500 ; these two added together equal 7500. Again referring to the general principle stated above, we find that the only element required to complete the cube is the cube of the units. Now, by reference to the illustrative blocks, observe that by placing the small cube (Z>) in its place in the corner, the cube is complete. And since (D) has been found to contain 5 x 5 x 5, or 125 cubic inches, add this sum to the 7500 obtained above, and the result is 7625. Subtracting 7625 from the re- mainder in the problem, nothing remains ; hence, it has been shown that the cube root of 15,625 is 25. By the operation is also proved the correctness of the general principle as stated. 1045. From the foregoing explanations the following rule may be derived : Beginning at the right, separate the given number into periods of three figures each. Take for the first root figure the cube root of the greatest perfect cube in the left-hand period ; subtract its cube from this left-hand period and to the remainder bring down the next period. Divide this remainder, using as a trial divisor three times the- square of the root figure already found, considered as tens, so obtaining the second or units' figure of the root ; next sub- tract from the remainder three times the square of the tens multiplied by the units, plus three times the tens multiplied by the square of the units, plus the cube of the units. In examples of more than two periods proceed as above, and after two root figures are found, treat both as tens for finding the third root figure. For finding subsequent root figures treat all those found as so many tens. 1046-1046] POWERS AND ROOTS 397 In case the remainder, at any time after bringing down the next period, be less than the trial divisor, place a cipher in the root and proceed as before. Should the cube root of a mixed decimal be required, form periods from the decimal point right and left. If the decimal be pure, point off from the decimal point to the right, and if need be annex decimal ciphers to make full periods. The fourth root may be obtained by extracting the square root of the square root. The sixth root is obtained by taking the cube root of the square root or the square root of the cube root. WRITTEN EXERCISE Extract the cube root of : 1. 1728. 4. 65939264 7. ^il}?- 2. 15625. 5. T 6 2V 8. 1264.295441. 8. 110592. ft 9. APPLICATIONS OF CUBE ROOT 1046. It has been shown that the solid contents of a cube is the product of its three equal sides. Hence, Tlie side of any cube is the cube root of its solid contents. WRITTEN EXERCISE 1. How many square inches in the six faces of a cubical block whose solid contents are 6400 cubic inches ? 2. A cubical cistern contains 3375 cubic feet. What is its depth ? 8. What must be the height of a cubical bin that will hold 1000 bushels of wheat ? 4. A square cistern the capacity of which is 420 barrels, has a depth of only \ its width. Find its dimensions. 5. A cubical cistern contains 630 barrels. How deep is it ? APPENDIX COMPOUND INTEREST TABLE FOR ANNUAL PAYMENTS Showing how much $1.00 per annum will amount to, compounded annually, in any number of years from 1 to 50 years. (Compare with pages 251, 252.) Tears 3 per ct. 3s per ct. 4 per ct. 4* per ct. 5 per ct. 6 per ct. Years 1 1.030 1.035 1.040 1.045 1.050 1.060 1 2 2.091 2.106 2.122 2.137 2.153 2.184 2 3 3.184 3.215 3.247 3.278 3.310 3.375 3 4 4.309 4.363 4.416 4.471 4.526 4.637 4 5 5.468 5.550 6.633 6.717 5.802 5.975 5 6 6.663 6.779 6.898 7.019 7.142 7.394 6 7 7.892 8.052 8.214 8.380 8.549 8.898 7 8 9.159 9.369 9.583 9.802 10.027 10.491 8 9 10.464 10.731 11.006 11.288 11.578 12.181 9 10 11.808 12.142 12.486 12.841 13.207 13.972 10 11 13.192 13.602 14.026 14.464 14.917 15.870 11 12 14.618 15.113 15.627 16.160 16.713 17.882 12 13 16.086 16.677 17.292 17.932 18.599 20.015 13 14 17.599 18.296 19.024 19.784 20.579 22.276 14 15 19.157 19.971 20.825 21.719 22.658 24.673 15 16 20.762 21.705 22.698 23.742 24.840 27.213 16 17 22.414 23.500 24.645 25.855 27.132 29.906 17 18 24.117 25.357 26.671 28.064 29.539 32.760 18 19 25.870 27.280 28.778 30.371 32.066 35.786 19 20 27.677 29.270 30.969 32.783 34.719 38.993 20 21 29.537 31.329 33.248 35.303 37.505 42.392 21 22 31.453 33.460 35.618 37.937 40.431 45.996 22 23 33.427 35.667 38.083 40.689 43.502 49.816 23 24 35.459 37.950 40.646 43.565 46.727 63.865 24 25 37.553 40.313 43.312 46.571 60.114 68.156 25 26 39.710 42.759 46.084 49.711 63.669 62.706 26 27 41.931 45.291 48.968 62.993 67.403 67.528 27 28 44.219 47.911 61.966 56.423 61.323 72.640 28 29 46.575 50.623 65.085 60.007 65.439 78.058, 29 30 49.003 53.430 68.328 63.752 69.761 83.802 30 31 61.503 56.335 61.702 67.666 74.299 89.890 31 32 54.078 59.341 65.210 71.756 79.06 96.343 32 33 56.730 62.453 68.858 76.030 84.067 103.184 33 34 69.462 65.674 72.652 80.497 89.320 110.435 34 35 62.272 69.008 76.598 85.164 94.836 118.121 35 36 65.174 72.458 80.702 90.041 100.628 126.268 36 37 68.159 76.029 84.970 95.138 106.710 134.904 37 38 71.234 79.725 89.409 100.464 113.095 144.059 38 39 74.401 83.550 94.026 106.030 119.800 153.762 39 40 77.663 87.510 98.827 111.847 126.840 164.048 40 41 81.023 91.607 103.820 117.925 134.232 174.951 41 42 84.484 95.849 109.012 124.276 141.993 186.508 42 43 88.048 100.238 114.413 130.914 150.143 198.758 43 44 91.720 104.782 120.029 137.850 158.700 211.744 44 45 95.502 109.484 125.871 145.098 167.685 225.508 45 46 99.397 114.351 131.945 152.673 177.119 240.099 46 47 103.408 119.388 138.263 160.588 187.025 255.565 47 48 107.541 124.602 144.834 168.859 197.427 271.958 48 49 111.797 129.998 151.667 177.503 208.348 289.336 49 50 116.181 1&5.5H3 158.774 186.536 219.815 307.756 50 ANSWERS Page 15. 1. 43,400. 2. 46,987. 3. 48,099. 4. 29,273. 5. 21,771. 6. 23,564. 7. 25,606. Page 19. 1. 447,136,427. 2. 387,213,476. 3. 484,730,888. 4. 503,980,350. 5. 458,853,797. 6. 475,095,610. 7. 78,841. 8. 84,551. 9. 69,394. 10. 79,473. 11. 76,930. 12. 53,143. Page 20. 13. Vertical totals : 34,134; 22,798; 26,057; 47,779; 34,788; 51,426. Horizontal totals : 23,764; 21,751; 15,089; 26,428; 31,480 ; 29,679 ; 30,061 ; 16,318 ; 22,412. Total, 216,982. 14. Vertical totals: Clothing, $4652.21; dry goods, $5500.32; furnishings, $849.08; millinery, $2357.92; household utensils, $ 4011.16. Horizontal totals : Monday, $2659.05 ; Tuesday, $2883.10 ; Wednesday, $2847.60; Thursday, $2613.38; Friday, $2937.30; Saturday, $ 3430.26. Total, $ 17,370.69. Page 23. 1. Vertical totals : Armories, $ 180,280.28 ; metropoli- tan sewer, $492,597.24; abolition of grade crossings, $393,663.73; metropolitan water, $1,946,951.37; highways, $1,157.89. Hori- zontal totals: 1895-1896, $392,774.63; 1896-1897, $404,963.27; 1897-1898, $ 386,627.68 ; 1898-1899, $ 425,974.50 ; 1899-1900, $647,621.81; 1900-1901, $756,688.62. Total, $3,014,650.51. 2. Vertical totals: Shoes, $2253.49; gloves, $1527.23; hats, $1409.44; dress goods, $3002.61; clothing, $3211.52. Horizontal totals : A to D Ledger, $ 1183.12 ; E to H Ledger, $ 1224.73 ; I to L Ledger, $1881.51; M to P Ledger, $1386.58; Q to T Ledger, $3078; U to Z Ledger, $2650.35. Total, $11,404.29. 3. Vertical totals: Domestics, $5760.89; notions, $3791.25; woolens, $6408.90; dress goods, $4961.63. Horizontal totals: Monday, $2056.71; Tuesday, $2481.41; Wednesday, $4749.31; Thursday, $ 2661.46 ; Friday, $ 4490.56 ; Saturday, $ 4483.22. Total, $20,922.67. Page 24. Vertical totals : Eegistered letters, 11,420 ; ordinary letters, 94,667 ; postal cards, 9338 ; book packets, 3397 ; parcels, 400 ANSWERS 1516; newspapers, 138,689. Horizontal totals: Monday, 45,717; Tuesday, 37,584 Wednesday, 42,788; Thursday, 47,162; Friday, 51,665 ; Saturday, 34,111. Total, 259,027. 5.1,154,276,889. 6.662,377,884. 7.788,754,622. 8.837,865,199. Page 29. 1. 3380 Ib. 2. 1650 Ib. 3. 214 Ib. 4. 2520 Ib. Page 30. 5. $1428.73. 6. $7428.96. 7. $654.97. Page 31. 8. $1722.32. 9. $409.40. 10. $463.73. 11. $368.15. Page 33. 1. E. W. Allen, $1416.84; C. W. Briggs, $814.95; L. M. Comer, $1030.73; 0. D. Day, $1477.43; A. L. Emery, $453.81; B. C. Foley, $907.83; J. I. Good, $1087.51; L. O. Hall, $1539.44; Chas. E. Irwin, $1929.54; Chas. H. Jones, $827.95. Total new balances, $11,486.03. Total old balances, $9054.87. Total checks, $3391.08. Total deposits, $5822.24. 2. D. T. Ames, $ 10,514.15 ; M. T. Ballou, $ 7602.30 ; W. T. Collins, $2663.89; Dorman & Co., $2985.63; Evans & Son, $1972.10; Farley Bros., $ 2162.01 ; Grant & Snow Co., $ 7574.90 ; Hall & Smith, $7578.60; J. T. Irwin, $1780.92; M. I. Jamison, $5338.33. Total new balances, $50,172.83. Total old balances, $36,732.21. Total checks, $8232.94. Total deposits, $21,673.56. Page 35. 1. $2000. 2. $4225. 3. $755. 4. Gained $1638. 5. Lost $96. 6. $1056. 7. 150 A. Page 40. 1. 462. 2. 1584. 3. 264. 4. 14,190. 5. 14,025. 6. 1386. 7. 4686. 8. 7062. 9. 11,000. 10. 26,708. 11. 3256. 12. 2200. 13. 8030. 14. 4752. 15. 6644. 16. 8250. Page 41. 1. 1,480,016 ems. 2. 6,856,080 men. 3. 5,412,969 links. 4. 106,272 pairs. 5. $2,329,992. 6. $53,816. 7. 4,557,168 books. 8. 11,751,810 Ib. 9. 125,244 Ib. 10. $154,716,975. Page 43. 1. 768. 2. 1435. 3. 67,680. 4. 2921. 5. 2226. 6. 77,184. 7. 47,082. 8. 5220. 9. 1222. 10. 1776. 11. 30,524. 12. 10,679. 13. 4250. 14. 2088. 15. 69,255. 16. 136,000. 17. 5248. 18. 4644. 19. 1734. 20. 51,114. 21. 8676. 22. 3976. 23. 4095. 24. 135,030. Page 44. 1. 24,442. 2. 46,748. 3. 45,904. 4. 66,825. 5. 124,440. 6. 133,179. 7. 184,518. 8. 291,450. 9. 124,313. 10. 112,420. 11. 207,739. 12. 379,080. 13. 50,061. 14. 197,290. 15. 577,521. 16. 145,754. Page 45. 1. 106,889. 2. 23,150. 3. 16,535. Page 46. 1. $3933. 2. 549,120ft. 3. $498.96. 4. 26,708 Ib. 5. 44,660 Ib. 6. Gained $1678.20. ANSWERS 401 Page 48. 1. 621 A. 2. 54. Page 49. 3. 75,629. 4. 13. 5. 16 da. 6. 715 A. 7. 1105^ cd. 8. 16,107. 9. 66. 10. Gained $2500. Page 52. l. 3, 3, 2, 2, 2, 2. 2. 2, 2, 31. 3. 11, 7, 3, 2, 2. 4. 17, 17. 5. 5, 3, 3, 3. 6. 5, 3, 3, 3, 191. 7. 7, 5, 5, 3, 3. 8. 3, 3, 7, 2, 2. 9. 3 and 317. 10. 3 and 509- Page 53. i. 11. 2. 12. 3. 7. 4. 16. Page 54. 5. 12 ft. 6. 120 boards. Page 55. 1. 480. 2. 450. 3. 624. 4. Jan. 1, 1904. 5. 252 A. Page 56. 1. 92f 2. 25. Page 57. 3. 15f . 4. 37f 5. 45 bu. 6. 5 bbl. 7. 720 yd. 8. 5J pc. 9. 8800 bu. 10. 3| mi. 11. 5 sections. 12. 360 bbl. 13. 500yd. 14. $120. 15. 100 bbl. Page 62. 1. $11,693,823.17. 2. $1,417,548.52. 3. $1,193,532.63. 4. $1,384,147.19. 5. $24,320.65. 6. $1088.60. 7. $4073.76. 8. $384.51. 9. $2094.15. Page 63. 1. $209.11. 2. $523.89. 3. $326.33. 4. $1256.81. 5. $5763.09. Page 64. 1. Wheat, 350 bu.; oats, 425 bu.; corn, 175 bu. 2. $91.10. Page 68. 1. 71,116,542. 2. 1,436,942,736. 3. 13,644,817,552. 4. $4265.91. 5. $299.64. 1. 195,448. 2. $2081.48. Page 69. 3. $1387. 4. 10 yr. 5. 6,075,486. 6. 20 bbl. 7. $7500; $900. 8. $ 4500 and $ 4745. 9. $540.19. 10. 240 bbl. 11. $4960. 12. 1,450,950,624. 13. 2,046,757,518. Page 73. 1. 4f . 2. IJA. 3. Ajyyu.. 4. *f|i. 5. *. 6. AJA. 7. L3Q.. a A^L. 9. 6|9. 10 . JJJLL. 11. AfA. 12. 6ioi 1. 142 T V 2. 17^. 3. 27f|. 4. 13f|. 5. 38^. 6. 8^. 7. 24fV. 8. 53||. 9. 23Jf 10. 16ff Page 74. l. f. 2. ^ * if- * A* 5 - rVA- 6 7 - f 8. }. 9 . |. 10 . A . Page 76. 1. T V%, A 2 ir, AV 2. IAA, f||, fff . 3. & ffi, ||5. * ** li ** 5 - * *> ^ 6 - Wt itti At 7 - tfc ff *f 8- b *fc Pi- page 77. 1. Iff 2 . lA|f 3. If 4. 1^ 5. If. 6. Iff * m- 8, 2^. 9. IJAf 10. Iff. 11. HO.. 12 . !40. Page 79.. 1. 34,120. 2. 22,578f. 3. 12,934ff 4. 12,355|. 402 ANSWERS Page 81. l. 151&. 2. T ^, 3. 7ff. 4. lO^fe. 5. 20. 6. llfi 7. 969f 8. 29J, 9. 25ff. 10. 47fV acres. Page 84. 1. 25. 2. 7^. 3. 44. 4. 30. 5. 248. 6. 26f 7. 462f. 8. 941. 9. 1^. 10. 11 11. $ 84|. 12. $.63}. 13. $43f. 14. $11. 15. $200. 16. 2 7 4- Page 86. l. 26811 2. 505|. 3. 339f. 4. 679}. 5. 756}. 6. 316|. 7. 5421. 8 . 6351 9. 90}. IQ. 1533f. 11. 1831f. 12. 939}. Page 89. 1. $72. 2. 5 shares. 3. 40 families. 4. 28 1 bu. 5. 5 da. 6. $41 7.1 A. 8. 782| sacks. 9. 18 da. 10. 13 fields. I. 2f. 2. 7if 3. $14,000. 4. 3bu. 5. 2 Ib. 6. $135,000. Page 90. 7. $52,500. 8. $70. 9. $300. 10. $2. 11. $5. 12. 336 trees. 13. A, $ 6, B, $ 15, and 0, $ 16. 14. $35, $40. 15. 1050 horses. 16. 6} bu. 17. 221 da. ia C. $38f, J. $47f 19. 114fft. 20. Colt, $94; cow, $30. Page 91. 21. 25|ft. 22. 405 sheep. 23. Carriage, $324; horse, $ 216. 24. 6| da. 25. 5 T \ 5 T da. 26. 42f da. 27. A, $ 1260 ; B, $420; C, $840. 28. Gained $ 10,106.56. 29. A, $2800; B, $3500. 30. 2161. 31. 1200. 32. 240 bu. 33. 20 da. Page 92. 34. $2501 T V 35. 28 da. 36. 15 hr. 37. A, $19.75; B, $15.80. 38. L6se52^. 39. 60 oranges. 40. 28 bu. and 80 bu. 41. $900. 42. 255} bu. 43. $11.91 gain. 44. Gained $38f 45. $438f Page 93. 46. Gained $ 98.32. 47. $395.54; $9.17. 48. Gained $3.14. 49. $3277.40. 50. $25,250. Page 96. 1. .26. 2. .27. 3. .0006. 4. .04. 5. 5.7. 6. 500.05. 7. .00022. 8. 5000.005. 9. 1,000,000.000001. 10. .500 or .5. II. .00005. Page 97. 12. 7.7. 13. 2.002. 14. 2000.002. 15. 11.000107. 16. 83.0504. 17. 710.00243. 18. 54,054,054.0054054054. 19. .37, .0004, 1.097, 3.0893, 9.17. Page 98. 1. ff 2. ^ 3. ^. 4. {fo 5. Jffa 6. 7- iW&- 8. IMfa 9. . 10. flHHfr 11. jfffa. 12. 13. ^. 14. ytV 15. ^. 16. 5^. 17. 13^- 18- 11} 19. 31^. 20. 16^. 21. 81& 22. G&ftftp 23. 35^. 24. 15^. 25. 28J. 1. .0625. 2. .15. 3. .275. 4. .09375. 5. .1375. 6. .5652173 + . 7. .0525. 8. .46875. 9. .024. 10. .9375. 11. .015625. 12. .96875. 13. .028. 14. .95. 15. .94. ANSWERS 403 Page 99. i. 848.1816. 2. 1652.461772. 3. 12,638.517852. 4. 1,000,608.012354001. 5. 57,697.358230005. 6. 385.8225yd. 7. 41.885 cd. 8. 136.33 thousand feet. 9. 84.423 T. 10. Num- ber of thousand feet, 101.184; total cost, $1403.75. 11. 376. 12. 926 2 or926f Page 100! l. .52977. 2. 1.27848. 3. 5.5264. 4. 1.546548. 5. .81. 6. .198. 7. 754.6005. 8. .3148. 9. 385,994.01246. 10. 1000.0099. 11. 2.99985. 12. 102.93702. Page 101. l. 0. 2. 117.843385. 3. .2375. 4. .6. 5. .0009. 6. 7231.98325125. 7. .0018044. 8. $554.63. 9. $336.33. 10. $14,856.56. 11. $3510.71. 12. $496.05. Page 102. 13. $ 1068.57. 14. $ 3537.58. Page 105. 1. 10,011,112.1010001. 2. 6,330,303.3333. 3. 40,448,404.48. 4. 1,056,000.605. 5. 5,655,500.1005. 6. 30,003.36303603. Page 106. 1. 2.92125. 2. 751.383957246471. 3. $127.12. Page 107. 4. 6.875 da. 5. 24.93-f thousand feet. 6. 7 yr. 7. Gained $5513.80. 8. Gained $ 400.75. 9. $1124.34. 10. 48 da. 11. Entire school, 1000 pupils ; bookkeeping department, 500 ; short- hand and typewriting department, 375; English department, 125. 12. $2238.75. Page 108. 13. Total gain, $ 5950 ; net gain, $ 5000. 14. $7500. Page 112. l. 371 Ib. 2. 81.5 yd. Page 113. 3. 115.2 A. 4. 689yd. 5. 123 Ib. 6. $40.96. Page 115. 1. $ 2166. 2. $ 5163.97. 3. $ 1142.33. Page 117. 1. $2643.52. 2. $6315.38. 3. $2462.43. 4. $8789.88. 5. $8256.38. Page 118. 1. $20; $19.13. 2. $28.35; $28. 3. $30.17; $8. 4. $28.13; $60. 5. $121.50; $16. 6. $630.63; $180. 7. $31.04; $62.50. 8. $11.17; $12. 9. $148.28; $247.50. 10. $61.40; $125. Page 119. 1. $8.25. 2. $14.66. 3. $158.76. 4. $123.18. > $53. 6. $38.75. 7. $19.80. 8. $100.32. 9. $140.81. 10. $399. 1. $2.68. 2. $4.71. 3. $78.70. 4. $1.42. 5. $1985.26. 6. $1609.78. 7. $903.96. 8. $21.06. 9. $12.65. 10. $19.51. Page 120. 1. $32. 2. $29.71. 3. $52.52. 4. $39.29. 5. $24.66. 6. $22.63. 7. $55.67. 8. $58.33. 9. $174.44 10. $132.36. 404 ANSWERS 9. Page 128. Page 129. Page 130. Page 131. Page 132. $ 97.44. Page 133. l. 2. 3. 1. 5. 10. 9 11,607.65. $ 322.67. $ 452.57. 4. $458.33. 2. $429.16. 6. (a) $170.29 $540.26. $524.03. 3. $168.68. $943.54. 7. $987.11. ; (6) $136.23. 11. 4. .8. (a) $ 1040.65. $440.48. $381.70; (6) $305.37. 12. (a) 10. 20's, 1; 10's, 9; 5>s, 7; 2>s, 9; 1's, 4; halves, 4; quar- ters, 3 ; dimes, 3 ; nickels, 2 ; pennies, 14. (b) 10. 10's, 9 ; 5's, 4 ; 2's, 9; 1's, 4; halves, 4; quarters, 6; dimes, 5; nickels, 4; pennies, 3. (a) 11. 20's, 11 ; 10's, 7 ; 5's, 11 ; 2's, 13 ; 1's, 5; halves, 6; quarters, 6 ; dimes, 7; nickels, 7; pennies, 15. (b) 11. 20's, 8; 10's, 7; 5's, 8; 2's, 11 ; 1's, 5 ; halves, 9 ; quarters, 9 ; dimes, 11 ; nickels, 7 ; pennies, 17. Page 151. 1. 193,555m. 2. 12,363d. 3. 709 pt. 4. 175 qt. 5. 155,243". 6. 561 gi. 7. 180,002 oz. 8. 238,475 cu. in. 9. 6,860,715 sq. in. 10. 9543 1. 11. 296,065 oz. 12. 1373 pwt. 13. 3767 in. 14. 3859J sq. ft. 15. 182,727.75 sq. ft. 16. 4,345,531 sec. 17. 194 cu. ft. la 40,396 gr. 19. 269 pt. 20. 129,832m. Page 152. 1. 7 wk. 1 da. 15 hr. 20 min ; or 1 mo. 20 da. 15 hr. 20min. 2. 24 bbl. 20 gal. 1 qt. 3. 112 bu. 2 pk. 5 qt. 4. 8 A. 66 sq. rd. 3 sq. yd. 4 sq. ft. 72 sq. in. 5. 3 mi. 124 rd. 2 yd. 8 in. 6. 1 T. 17 cwt. 95 Ib. 7. 9 Ib. 1 oz. 5 pwt. 20 gr. 8. 73 yr. 3 mo. 1 wk. 1 da. 9. 2 hr. 38 min. 57 sec. 10. 66 A. 72 sq. rd. 11. 5 cu. ft. 152 cu. in. 12. 17 T. 8 cwt. 32 Ib. Page 153. 1. 432 gr. 2. 10 pwt. 3. 213 rd. 1 yd. 2 ft. 6 in. 4. 110 sq. rd. 5. 112 A. 40 sq. rd. 29 sq. yd. 51.84 sq. in. 6. 43 sq. rd. 19 sq. yd. 2 sq. ft. 36 sq. in. Page 154. 1. ^. 2. A^fa. 3. .12 T. 4. .017361+ yd. Page 155. 1. .327 T. 2. .0625 A. 3. fff Ib. 4. .29791+ Ib. 5. im T. Page 156. 1. 90 3s. 2. 17 mi. 46 rd. 1 yd. 2 ft. 3. 33 A. 2 sq. rd. 17 sq. ft. 7 sq. in. 4. 672 Ib. 1 oz. 12 pwt. 8 gr. 5. 211 Ib. 11 oz. 18 pwt. 21 gr. Page 157. 1. 14 gal. 2 gi. 2. 396 A. 78 sq. rd. 3. 3 yd. 2 ft. IJin. 4. 6 Ib. 9 oz. 7 pwt. 17 gr. 5. $44.31. 6. 6 T. 5 cwt. 7. Gained 507 5s. ANSWERS 405 Page 159. 1. 96 da. 2. 236 da. 3. 323 da. 4. 87 da. 5. 223 da. 6. 233 da. 7. 49 da. 8. 241 da. 9. 81 da. 10. 402 da. 11. 523 da. 12. 883 da. 13. 5 yr. 7 mo. 3 da. 14. 17 yr. 15. 4 yr. 4 mo. 5 da. 16. 2 yr. 10 mo. 25 da. 17. 3 yr. 1 mo. 3 da. 18. 7 yr. 4 ino. 23 da. 19. 3 yr. 1 mo. 1 da. 20. 18 yr. 8 da. 1. $1924.39. 2. $1958.25. Page 160. 3. 54 T. 16 cwt. 47 Ib. ; $699. 4. $70.20. 5. $577.78; $650; $433.33. 6. $20,589.84. 7. $97.50 gain. 8. $73.88. 9. $ 122.80 by first method ; $122.79 by second method. Page 161. l. 2bu. 3pk. 2. 1 T. 57 Ib. 12 oz. 3. 4 15s. 3f far. 4. 3 yr. 46 da. 7 hr. 30 min. 5. 1 Ib. 3 oz. 5 pwt. 11 gr. 6. $13,050. 7. $19.97. 8. $5. 9. 1061 3s. 9d. 1 far. 10. 38 3d 2.4 far.; 513 14s. 3d. 3.36 far. 1. 12. 2. 770 11s. 5d. 3 far. 3. 4 A. 83 sq. rd. 6 sq. yd. 644 sq. in. 4. 101 sq. rd. 2 sq. yd. 21.6 sq. in. 5. 123 mi. 162 rd. 3 yd. 1 ft. 4 in. Page 162. 6. $6.20 gain. 7. Gained 3^. 8. $ 72.58 gain. 9. $19.16. 10. $54. 11. $72.81. 12. Gained $360.67. 13. $1209.60 gain. 14. $9.77. 15. $88.89. . Page 164. l. 75 ft. 4.7808 in. 2. $ 335. 3. $ 17.33. 4. 440.6094ft. 5. 1250ft. 6. 168 ft. 4 in. Page 166. 1. 4 A. 116 sq. rd. 2. 19 A. 122 sq. rd. 3. 34 A. 4. 10 A. 56 sq. rd. 5. 15 A. 6. 12 A. 12 sq. rd. 7. 446 A. 5 sq. ch., or 446 A. 80 sq. rd. 8. 249 A. 9 sq. ch., or 249 A. 144 sq. rd. 9. 390 A. 10. $3277.97. 11. 357 rd. 5 ft. 6 in. 12. 414 ft. lOf in. 13. Ark., 177 A. 80 sq. rd. ; N. and S. Dak., 124 A. 40 sq. rd. ; all other states, 133 A. 20 sq. rd. 14. $ 10.56. 15. $ 121,500. 16. 57 sq. rd. 21 sq. yd. 108 sq. in. 17. $100. 18. $262.35. Page 168. 1. 63 yd. 2. $ 15.63. 3. 128 yd. ; 130 yd. ; $327.25. 4. 65|yd. 5. $150. 6. 61| yd. ; 82J yd. Page 169. 7. $9.46. 8. $13.10. Page 170. 1. 75 rolls. 2. $8.75. 3. 14 rolls. 4. $15.20. 5. $39.60. Page 171. 1. $23.99. 2. $23.11. 3. $45.32. 4. $77.61. 5. $38.85. 6. $24.17. Page 172. 1. $182.25. 2. $63. 3. $49.75. 4. 60,000 shingles. 5. 48 M. ; $ 168. 6. $ 192. Page 174. 1. $37.70. 2. 1458 cu. ft. 3. $284.28. 4. cu.ft. 5. 1570|cu. ft. 6.36ft. 7. $16.29. 406 ANSWERS Page 176. l. 420 board ft. 2. $14.04. 3. $11.20. 4. 27 5. $108.39. 6. $23.42. 7. $24.18. 8. $21.78. 9. $132. 10. $132.44. Page 178. 1. 39i| cd. 2. 26 ft. 9& in. 3. $227.11. 4. 64ft. 5. $45. 1. $6000. 2. With asphalt ; save $45,883.20. 3. With brick; save $85.33. 4. 147,840 blocks. 5. $58,594.44. Page 180. 1. 345.6 bu. 2. 1040 bu. 3. 768 bu. 4. 1920 bu. 5. 480 bu. 6. 1536 bu. 7. 237.476+ bbl. Page 181. 8. 64 rd. 9.6+ ft. 9. 1903.5648 gal. 10. 22.3074 gal. Page 182. 1. 81^- or 81.45+ perches. 2. 118 cu. yd. 14 cu. ft., or 118^4 cu. yd. 3. 836.1 perches. 4. $1701.68. 5. 284,460 bricks ; 280,104 bricks. 6. $283.50. 7. 546,920 bricks ; 535,040 bricks. 1. $8100. 2. 8ft. 3. $7200. 4. $32.58. Page 183. 5. $25.34. 6. $75.92. 7. $57.60. 8. $120. 9. $140. 10. 90 pupils. 11. $21.36. 12. $4937.50. 13. $5. 14. 149J cu. ft. ; 1792 board ft. ; -jfccu.ft. 15. $3465. 16. Cheaper to pave with asphalt; $10,260.80. 17. $165. Page 188. 1. $60,000. 2. $2700. 3. Wheat, 8320 bu. ; oats, 3120 bu. ; barley, 13,520 bu. Page 189. 4. $17,820. 5. $14.04. 6. $4.40. 7. $576.60; 658 bu. 8. $40,500. Page 190. l. 50%. 2. 25%. 3. 95%. 4. 5f%. 5. 54ii%. 6. 800%. 7. 20%. 8. 100%. 9. 50%. 10. 33^%. Page 191. 1. 24. 2. 5%. 3. $1900. Page 192. 4. 2040 qt. 5. $40,300. 6. $3750. 7. 200 trees. 8. $10.56. 9. $54. 10. 370f bu. Page 193. 1. $600. 2. $400. 3. $20,000. 4. $4.50. 5. $28. 6. 500 pupils. 7. 36,080. 8. $ 8000. 9. 560 bu. 10. 200,100. Page 194. 1. $10,000. 2. $20,840. 3. $3200. Page 195. 4. Horse, $ 300 ; carriage, $ 156. 5. Monday, $500; Tuesday, $ 400 ; Wednesday, $600. 6. $32.40. 7. $175. 8. A, $1000; B, $600. 1. 23f%; 42%; 33|%. 2. $300. 3. $1959.38. 4. A, $7125; B, $2500. 5. $7200; $12,000. 6. 1% gain. 7. $13,650. 8. 331%. Page 196. 9. 122,048. 10. 40 gal. 11. 27 V%; 21|^%; 18 T f T %. 12. 25%. 13. Amount of indebtedness, $ 2252.08 ; in bank, $9008.33. 14. Lost, $666.67. 15. $23,400. 16. $64,000 17. $60,250. ANSWERS 407 Page 197. 18. Carriage, $150; horse, $187.50; harness, $50. 19. Cost, $308; lost, $28. 20. ^%; .000125; 3642%. 21. 5 cwt. 61.48 lb., or 5 cwt. 61 Ib. 7.68 oz. 22. First cost, $7480.50; 8 r98 4 7%- 23 - Wife > $21,760* daughter, $10,000; younger son, $12,500; elder son, $13,750. 24. $1750; $3062.50; $6125; and $8575. 25. $3850. 26. $20,000. 27. 133%%. 28. Grazing, 504 A. ; grain, 420 A. ; timber, 936 A. Page 198. 29. $7200. 30. 50%. Page 200. 1. $826.67. 2. $567. 3. $2400. 4. $252.80. 5. 360yd. 6. $72. 7. First; $|. 8. Gained $18. Page 201. 9. 66f%; $420. 10. $374.75. Page 202. 1. $680.24. 2. $1246.80. 3. $322.87. 4. Amount of bill to render, $4863.75; amount to be remitted, $4620.56. 5. Amount of bill to render, $2199; amount to be remitted, $2133.03. Page 203. 6. $222.30; trade discount, $91; cash discount, $11.70. 7. $234.38. 8. $270. Page 205. l. 12%. 2. $1000. 3. $165.75. 4. $920. 5. $1000. 6. 2|%. Page 206. 1. Lead pipe, 35^; iron pipe, 13^; bath tubs, $8.64, $7.20, and $5.76. 2. $1317.50. 3. $126. 4. Second is $ 9 cheaper. 5. $189.47. Page 207. 1. Gain, $ 24.75 ; selling price, $ 173.25. 2. $ 6406.25. 3. Gained, $74. 4. $105 gain. 5. $37.50 gain. Page 208. 1. 2000 bu. 2. 29^%. 3. 49J%. 4. 6J%. 5. 331%. 6. 50%. 7. 77J%. Page 209. 8. 29^%. 1. $5000. 2. $2750. 3. $40. 4. $7085.71. 5. $150. Page 210. 1. $10,500. 2. $ 60 and $ 100. 3. $1000. 1. $2000. 2. 22f%. 3. $1218.75. 4. 400 bbl. Page 211. 5. 1000 bu. 6. 25%. 7. 16|%. 8. $20,010. 9. 22% loss. 10. $625. 11. $11.25. 12. 37|%. 13. $2. 14. 46f%. 15. $31f. 16. 331%. 17. $74.25. Page 212. 18. $9350. 19. 16|% loss. 20. $10,000. 21. $110. 22. $16.80. 23. $59,320. 24. 3f% loss. 25. 15^. 26. 18J|% loss. 27. $58.60. Page 213. 28. 9|f gal. 29. 14f % gain. 30. $300. 31. 30 32. 50^. 33. 20%. 34. 491%. 35. 23l2V%- 36 - 37. $1080; 2f%. 38. $250. 408 ANSWERS Page 214. 39 Page 215. 1. $.ST. $EH. , 50%. 40. d}> Tj 1 Ti A < jp _CJ._LW\_ < 141. 4 $E.IA l. $5985 2. Z.OA , . 42. $6. $I.EA fl AG BW' 1 l i ;i. PI.BW EO $ $T.MB' . $IER. $ M.LB $ ELUA $.TH' $HA. LB *'$ M .LC' $ M.BS *' $ SHB' $ IO.TB ' $.TH ' ' $ B.OE ' $ **. J.J.O.UO, J-.-l-L^ lllCO, ^o ^ Page 276. 1. $635.92. 2. $122.29. 3. $279.64. 4. At 6%, $183.49; at 5%, $179.70; at 8%, $191.08. 5. $180.93. Page 282. i. Oct. 18, 1903. 2. Nov. 15, 1903. 3. June 23, 1903. 4. Dec. 23, 1903. Page 284. 1. Apr. 11, 1903. 2. June 5, 1903. 3. July 6, 1903. 4. Jan. 24, 1904. 5. May 17, 1903. 6. Nov. 18, 1903. 7. July 22, 1903. Page 288. l. Dec. 9, 1902. Page 289. 2. Feb. 6, 1903. 3. Apr. 18, 1903. 4. Sept. 10, 1903. 5. Aug. 8, 1903. Page 290. 1. Dec. 7, 1903. Page 291. 2. Dec. 31, 1903. Page 293. 1. $313.18. 2. $613.78. 3. Equated date, Jan. 4, 1902 ; cash balance, $435.15. Page 295. 1. $2034.71. 2. $1730.26. Page 296. 3. $1985.?3. Page 298. 1. $752.50. Page 299. 2. $781.04. 1. $10. Page 300. 2. $1264.74. Page 304. l. $3637.50. 2. $7275. 3. $17,381.25. 4. $8820. 5. $26,400. 6. $25,967.50. Page 305. 7. $ 92,868.75. 8. $ 2917.50. 9. $ 80,500, par value; $ 83,263.75, market value. 10. 110. l. 400 shares; $ 40,000. 2. 500 shares. 3. 740 shares. 4. 300 shares ; $ 1200. 412 ANSWERS Page 306. 1. $64,837.50. 2. $29,718.75. 3. $180. 4. $51,700. Page 307. 1. 5.268+ %. 2. N.Y. Air Brake, better by 1.481-1-%. 3. 4.593+%. 1. $1440. 2. $75,475.50. Page 308. 3. Total dividend, $9,892,460; C's dividend, $110. 4. Total dividend, $251,250; F's dividend, $93.75. 5. Surplus fund, $47,085.79; dividend, $800,000; undivided profits, $498,- 222.46. 1. 9%. 2. 6 % dividend ;$ 11,135.60, undivided profits. Page 309. 3. 13%; $ 567,274.83 undivided profits. 1. $1352.06. 2. $415.38 loss. Page 310. 3. $3782.55. Page 312. 1. 4.347+%. 2. $3600.62. 3. $62,625. 4. 3.897+%. 5. $ 63,825 invested ; 4.23 + %, rate of income. 6. $57.50. 7. 50 bonds. 8. 44 bonds; 3.936+ %, rate of income. Page 313. 9. Mo. Pacific, better by .741 +%. l. $81,940.63. 2. $1000. 3. $72,031.25. 4. Quotation, 121 J ; brokerage, $ 62.50. 5. 5.024+%. 6. 3.636+%. 7. $ 1004.05 to make margin good ; $ 1079.05 loss if sold out. 8. 127 bonds ; unexpended balance, $485.62. Page 314. 9. $ 36.25 gain; $3343.75 on hand. 10. Am. Ice Sees., 8.139+ % ; Del. & Hud., 4.176+ % ; Bait. & Ohio, 5.068+ % ; Erie, 2d pf., 6.324+ %. 11. Surplus fund, $508,067.90; dividend, $ 3,500,000 ; undivided profits, $ 6,153,290.10. 12. $ 50,437.50. 13. 41%. 14. Dividend, 10 %; undivided profits, $ 955,946. 15. Y's rate of income, 5.298+ %; Z's 5.263+ %; Y's dividend, $ 3000 ; Z's, $ 2500. Page 318. 1. $117. 2. $840. Page 319. 1. 1J % 2 - I %- l- $4450. 2.475. 3. $37,500. 1. $300. 2. $8000. 3. $1576.50. Page 320. 4. $ 523.75. 5. f %. 6. $27,411.17. 7. $ 11,454.55. 8. $18,125. 9. $8.66. 10. j %. 11. $1967.96. 12. $60. 13. G., $ 630; H., $ 150; and M., $ 337.50 ; $ 142.50 gain; .71 %. 14. H., $26,522.73; M., $43,099.43; A., $23,207.39; Phoenix, $ 26,522.73 ; Provident, $ 26,522.73. Paee321. 15. $ 47,500 and $ 39,375. Page 325. 1. $158.10. 2. $280.40. 3. $333.40. 4. $6386.91 ANSWERS 413 due from bank; $1386.94 more from bank than from insurance company. 5. $4610.64 more than premiums; $3947.08 more than the premiums and interest. 6. $ 680.05 loan ; $ 1555.70 due from bank. 7. $ 6948 on ordinary policy ; $ 24,885.60 on endowment policy. 8. $4265.92 more than premiums; $3985.33 loss on policy. 9. $29.60 less in premiums, per $1000; $1276.08 in principal and interest, per $1000. Page 326. 10. $28,026.25. 11. $4011.05 more than premiums; $3891.27 less than amount received by A. 12. $440.55, loan value; 9 years^47 days, extended insurance; $925, amount of paid up policy. 13. $ 2596.60, loan value ; 13 years, extended insurance ; $ 3710, paid up policy. Page 328. l. $18.75. 2. $1134; B. $13.44. Page 329. l. Rate, \\%\ $120. 2. 50^ per $100; $11.25. 1. Bate, %; $62.50. 2. $3550. 3. 4.5 mills ; $70. 4. $35.35. 5. Kate, .012898 ; $ 82.39. Page 330. 6. 21 mills; 2 mills. 7. 4.5 mills; $110.63. 8. Kate, 5| mills; $54.05. 9. 1.25669 10. $24,519.60; $612.99. Page 333. 1. $24,300. 2. $7273.56 total tax; $10,910.34 if he failed to make return. 3. $76,647, total city tax; $790, total suburban tax; $2182.50, total farm tax; $44.25, tax horses and cattle ; $ 26,743.10, total tax money at interest; $ 5.60 tax ve- hicles to hire ; grand total, $106,412.45. 4. $6687.18. 5. $66. 6. $94.48. 7. $243, discount; $24,057, total paid for real estate tax. 8. $4.74. Page 337. 1. $1235.60. 2. $610.50. 3. $933.33. 4. $1238.64. 5. 40 gal. Page 338. l. $1178.80. 2. $5645.50. 3. $2010. 4. $27,031.80. 5. $10,656.40. 1. $866.80. 2. $576.30. 3. $1317.60. Page 339. 4. $3644.55. 5. $155.90. 6. $2494. 7. $463.20. 8. $1168.80. Page 340. 9. $210.08. 10. $1.80. 11. $16.82. Page 344. l. $848.87. 2. $1393. 3. $784. Page 345. 4. $5000. 5. $8696.16. 6. $1378.30. 7. $18/1.30, 8. $1890.50. 9. $5080.98; $344.42 gain. 10. $2471.88. 414 ANSWERS Page 352. 1. $1946. 2. 11,160.89 francs, or 11,160 francs 89 centimes. 3. $5103.48. 4. 1195 17s. 6d. 5. 25,000 marks. 6. $1666.87. 7. 1864 marks 8. $25,876.26. 9. 5.185 francs. 10. $.40f. Page 353. 11. $2186.21. 12. $10,558.96. Page 355. 1. $ 2003.72. 2. $3790.30. , 3. $1501.87. 4. $3506.93. 5. $3056.63. 6. $3594.11. 7. Mk. 19,429.43. 8. $2827.12. Page 356. 9. $2851.18. 10. $119.37. 11. $5824.17. 12. $82.15 gained. 13. $8.16. 14. 15 14s. 6Jd. more. Page 358. 1. $10,600. 2. A, $2400; B, $1200; C, $800. 3. A, $120; B, $300; C, $320. 4. $12,000. 5. $15,000. 6. A, $150; B, $200; C, $300. Page 359. 7. $64,800. 8. A, $3000; B, $6000; C, $9000; D, $12,000. 9. llfff da.; C, ^$ 32.67; H, $25.41; T, $22.87; L, $ 19.05. Page 362. 1. A, $126; B, $168. 2. A, $6500; B, $6500; C, $3250. Page 363. 3. A's share of gain, $ 13,750 ; B's share of gain, $8250; A's share of selling price, $26,250; B's share of selling price, $15,750. 4. A's gain, $398.88; C's gain, $319.10; B's gain, $239.32; A's present worth, $ 1648.88 ; C's present worth, $1319.10; B's present worth, $989.32. 5. B, $6000; C, $4000; D, $2000. 6. Net gain, $795.51 ; A's share of net gain, $530.34; B's share of net gain, $265.17; A's present worth, $5530.34; B's present worth, $2665.17. 7. Net loss, $488.50; C's present worth, $3755.75; D's present worth, $1755.75. 8. Net gain, $2255; E's present worth, $8405.66; Fs present worth, $8751.67; G's present worth, $7751.67. 9. $7290. Page 365. 1. A, $276.92; B, $346.15; C, $276.93. Page 366. 2. A, $596.53; B, $559.25; C, 994.22. 3. A, $17.40; B, $29.70; C, $48. 4. A, $36; B, $32; C, $20; D, $4. 5. M's gain, $1737.93; E's gain, $1862.07; M's present worth, $3737.93; E's present worth, $6362.07. 6. A, $4548.39; B, $2951.61. 1. A's present worth, $12,025.17; B's present worth, $9525.18; firm's present worth, $21,550.35; firm's net gain, $4050 35. ANSWERS 415 Page 368. 2. Net resources, $23,564.25; net solvency, $23,564. 25 1 net gain, $1064.25. 3. Burke, $17;533.33; Brace, $ 17,83-3.34; Baldwin, $ 17,633.33. 4. Loss, $22,747.09; Brigg's loss, $5907.62; Parson's loss, $16,839.47; net insolvency, $2187.09; Brigg's present worth, $127.38; Parson's insolvency, $2314.47. Page 369. 5. A's in vestment, $3865.80; B's in vestment, $2577.20; firm's insolvency, $5557; A's insolvency, $3334.20; B's insolvency, $ 2222.80. 6. Net gain, $ 2636.83 ; Mason's present worth, $ 5818.42; Eiver's present worth, $5818.41. 7. Net gain, $2865.14; D's present worth, $8632.57; E's present worth, $8432.57. Page 370. 8. $2100. 9. Present worth of business, $ 16,766.57 ; net gain, $4673.09 ; A's present worth, $8371.78 ; L's present worth, $8394.79. 10. Gain of each, $1825; E. H. Hill's present worth, $7425; N. P. Pond's present worth, $7825. Page 373. 1. $3200.25. 2. $1910.40. 3. $1565. Page 374. 1. 7th meeting, dues, $550; interest, $16.50; loan, $550; profit per share, $.03; book value of each share, $7.105. 2. 13th meeting, dues, $400; interest, $26.40; loan, $400; profit per share, $ .066 ; book value of each share, $ 13.429. Page 375. 3. 17th meeting, dues, $800; interest, $ 96 ; loan, $1600; profit per share, $.12; book value of each share, $18.02. 25th meeting, dues, $ 800 ; interest, $156; loan, $800; profit per share, $.195; book value of each share, $27.3175. 4. $1 profit for 24th month, or $12.16 total profit for 24 months; $212.16, book value after paying 25th month's dues. 5. $ 18.40. 6. $ 22. Page 376. 1. L's profit, $475.20; M's, $206.25; N's, $158.40; O's, $110; P's, $ 16.50. Lowest terms, respectively, 36, 25, 16, 4, and 1. Profit per share, respectively, $3.96, $2.75, $1.76, $.44, and $.11. 2. S's profit, $83.60; T's, $37.62; Q's, $104.50; U's, $12.54; V's, $9.50; E's, $14.25. Lowest terms, for S, 16; for T, 9 ; for Q, U, and V, 4 ; for R, 1. Profit per share, respect- ively, $ 1.52, $ .855, $ .38, $ .095. 3. $ 274.70. Page 379. 1. 733J ft. 2. 478 bu. 3. $2812.50. 4. 2 yr. 9 mo, 23 da. (221 da.). 5. 140 A. Page 380. 1. $567. 2. 350 rd. 3. 26f acres. 4. $384.75. 5. 93 da. 416 ANSWERS Page 382. l. $40. 2. $64. 3. $64.50. Page 383. i. $211. 2. 36.09. Page 384. 3. $483.96. 4. $368.48. 5. $321.80. Page 389. l. 143,814.44 francs. 2. 229.2784 Ib. 120 sq. rd. 4. $6431.23 5. $29.81. 6. 11.69. 3. 617 A. 7. $1.32. 8. 6. 9. 3i meters; 11 ft. 5.79 in. 10. 1535.2722 kilometers. 11. 54 meters; 59.0544yd. 12. 2J hectares; 5.55975 A. Page 392. i. 14. 2. 15. 3. 24. 4. 35. 5. 75. 6. 206. 7. 125. 8. 1024. 9. 11.2. 10. 7.09. 11. f|. 12. ff Page 393. i. 100 ft. 2. 480 rd. 3. 273.137 + rd. 4. 452 rd. 9 ft. .56 -f in. 5. 77.88 + ft. 6. 3339.364 ft. 7. 100 rd. 8. 208.710 ft. Page 397. l. 12. 2. 25. 3. 48. 4. 404. 5. f 6. H- 7. 1. 8. 10.813 + . 9. ff l. 2067.954 + sq. in. 2. 15ft. 3. 10ft. 9 -h in. 4. 7 f t. 7 + in. deep j 15 ft. 2 + in. square. 5. 13 ft. 10 + in. - , / tf'S" 'obe "~^ 3 ^ -W*"f v"^ * 031 t // - & 5^1870 UNIVERSITY OF CALIFORNIA LIBRARY '