LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIFT OF Class MODEBN Commercial Arithmetic BY F. J. SCHNECK CHICAGO POWERS & LYONS NEW YORK SAN FRANCISCO APH 4 1911 GIFT COPYRIGHT, 1902 BY POWERS & LYONS " or -HE UNIVERSITY OF PREFACE A course in business arithmetic should train the pupil to figure correctly, easily, and rapidly, and should fit him to solve the problems that arise in the ordinary course of business. To this end Modern Commercial Arithmetic gives a brief review of the fundamental operations, fractions and decimals^ intro- ducing short practical methods. The mechanical part of arith- metic is illustrated and explained by diagrams, examples, operations, and notes. The intellectual part is developed in the pupil's mind by mental problems, questions, and state- ments. A student should solve a problem from his knowledge of the facts or conditions of the problem and the principles involved; therefore, rules and cases are superseded by develop- ment exercises which will make him thoughtful and independent. In the business office problems do not come tabbed with article and rule, but the business man must first discover the principle that is involved and then by a process of reasoning determine the result. It has been the object of this work to present the problems as nearly as possible as they are presented in the business office. When the pupil changes from the school to the office he will find the change in the method of thought involved as slight as possible. The student in school has no time to waste and therefore this work contains no puzzles or catch problems. Subjects that do not arise in ordinary business transactions are omitted. The author's aim has been to present a work that would give the pupil such instruction as he needs, and to set it forth in the manner he will meet it in the business office. It is hoped and believed that an inspection and trial of the work will show that he has succeeded. 210143 CONTENTS PAGE NOTATION AND NUMERATION 7 Arabic Notation 7 Roman Notation 8 ADDITION 10 How to Make Groups 10 Cipher Method 19 Civil Service Method 21 SUBTRACTION 23 MULTIPLICATION 26 Cross Multiplication 29 DIVISION 32 THE EQUATION 34 CANCELLATION , 37 FRACTIONS 40 Decimal Divisions and Decimal Fractions 42 How to Write Decimals 43 Addition of Decimals 45 Subtraction of Decimals 46 Division of Decimals 47 Reduction of Fractions 49 Addition of Common Fractions 54 Subtraction of Fractions 55 Multiplication of Fractions 57 Division of Fractions 59 The Three Problems of Fractions. 61 UNITED STATES MONEY 65 OPERATIONS WITH ALIQUOT PARTS.... 68 PRICE, COST, AND QUANTITY 71 DENOMINATE NUMBERS 76 Linear Measure 77 Square Measure 77 4 CONTENTS 5 DENOMINATE NUMBERS PAGB Cubic Measure 78 Surveyors' Measures 78 Measures of Capacity 79 Measures of Weight 79 Troy Weight 80 Apothecaries' Weight 80 Measures of Time 80 Measures of Angles 82 Measures of Values 82 Reduction of Denominate Numbers...., 83 Reduction of English Money 87 Fundamental Operations 87 Subtraction of Dates , 90 Comparison of Weights and Measures 91 Papers and Books 92 Price, Cost, and Mixed Quantities 95 PRACTICAL MEASUREMENTS 102 Land Measurements 104 Papering 115 Carpeting 116 Measurement of Solid Figures 118 Brick and Stone Work 120 Wood 121 Lumber 122 Practical Rules for Dealers in Farm Produce 125 Square Root 128 PERCENTAGE .' 139 PROFIT AND LOSS 150 COMMISSION AND BROKERAGE 155 TRADE DISCOUNT 160 MARKING GOODS 164 STORAGE 166 INSURANCE 169 Property Insurance 169 Personal Insurance 172 Table of Rates 175 INTEREST 177 Cancellation Method 179 1000-Day Method 180 Banker's 60-Day Six Per Cent Method 182 6 CONTENTS INTEREST PAGE Ordinary Six Per Cent Method , 185 Periodic Interest 190 Compound Interest 191 Compound Interest Table 193 Partial Payments 201 TRUE DISCOUNT 207 BANKING BUSINESS 211 Bank Discount 211 Bank Deposits and Checks 214 Collateral Notes 218 Domestic Exchange 219 The Clearing House 222 Foreign Exchange 227 ACCOUNTS AND BILLS 229 Bills Trade Discount 233 Equation of Bills 235 Equation of Accounts 241 Accounts Current 246 Account Sales 249 PARTNERSHIP 251 STOCKS AND BONDS 257 TAXES 266 MISCELLANEOUS REVIEW PROBLEMS... .. 272 MODERN COMMERCIAL ARITHMETIC NOTATION AND NUMERATION 1. The writing of numbers is called Notation. 2. The reading of numbers is called Numeration. 3. Numbers may be written in three ways: 5, five,.V. The first method was used by the Arabs, and it is called the Arabic system of notation. THE ARABIC SYSTEM 4. There are ten figures used in this system. Each of the following figures is a digit, and represents a number: 1, 2, 3, 4, 5, 6, 7, 8, 9. is a figure, but it does not represent a num- ber. It is used with other figures to represent numbers. In counting, units are grouped into units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, etc. 10 of one group make 1 of the next larger group. Thus, 10 units make 1 ten, 10 tens make 1 hundred, 10 hundreds make 1 thousand, 10 thousands make 1 ten-thousand, 10 ten- thousands make 1 hundred-thousand, 10 hundred-thousands make 1 million, etc. The number 1234567 contains 7 units, 6 tens, 5 hundreds, 4 thousands, 3 tens of thousands, 2 hundreds of thousands, and 1 million. It may be described as 7 units of the first order, 6 of the second, 5 of the third, 4 of the fourth, 3 of the fifth, 2 of the sixth, and 1 of the seventh. 5. Principles. 1. Orders of units increase from right to left in a tenfold ratio. 2. Orders of units decrease from left to right in a tenfold ratio. 7 8 MODERN COMMERCIAL ARITHMETIC 6. In reading numbers, three figures are grouped into a period and read as follows : Millions, Thousands, Units 203, 203, 203 The number is read: 203 million, 203 thousand, 203. EXERCISES 7. Eead: 3002, 42,005, 40,305, 400200, 40^204, 400Q20, 402,030, 4003001, 4010,050, 50003002, 50030020, 50300200, 300020070, 301201701, 110001010. Remark. Numbers above hundreds of millions are seldom used in business. When used, such numbers are read as millions. Thus, 2467,845^27 is read 2467 million, 845 thousand, 427. Write: Twenty thousand, eighty. One hundred thousand, forty-six. Three million, thirty thousand, fifteen. Two hun- dred four million, eighteen thousand, one hundred fifty. Two thousand eighty-five million, seventy thousand, three. Twenty million, fourteen thousand, forty. One hundred three million, one hundred three. One thousand four million, one thousand, four. One thousand five million, five hundred five thousand, five hundred five. ROMAN NOTATION 8. This system was used by the Komans. Letters used: I V X L C D M Values: 1 5 10 50 100 500 1000 9. Principles. 1. Eepeating I, X, C, or M repeats its value. Thus, III = 3, XX = 20, CCC = 300. These letters are not repeated more than three times, although we find IIII on clocks, and 400 is sometimes written CCCC. 2. When I is "before V or X, X before L or C, C before D or M, the values of the letters are subtracted. Thus, IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900. 3. When one letter is placed after another of greater value, their values are added. Thus, VI = 6, XI = 11, LX = 60, CI = 101. DOTATION AND NUMERATION 9 4. A dash placed over a letter multiplies its value by 1000. Thus V = 5000, XI = 11000, LX = 60000. 10. How to Write Numbers. 1. Use the expressions IV, IX, XL, XO, CD, CM as one letter. 2. Write the letter, or expression, whose value is nearest that of the required number, but less than the required num- ber, and add letters to this expression until the required number is obtained. To write 19, first write 10 (X), then add 9 (IX). To write 28, first write 10 (X), add 10 (X), add 5 (V), add 3 (III). To write 49, write XL, and add IX. To write 99, write 50 (L), add 40 (XL), add 9 (IX). 11. Write: 29, 34, 49, 51, 69, 89, 91, 219, 289, 391, 1047, 1863, 1899, 1900, 20678, 4685, 3569, 56435, 4567, 1365, 3709. Eead: XXXIX, XCV, XOIX, CIX, CXIX, DOVI, DCCXXIX, DCCCXL, CMXIV, MDCCXXVII, MDCCCLXI, MCMXOIX. ADDITION 12. Addition is the most difficult process in arithmetic, for it is not done until it is correctly done. When the columns of numbers added are long, mistakes are likely to occur. Care and practice are necessary to enable one to add correctly and with a reasonable degree of rapidity. Probably the first test or trial an employer will give an applicant for a position as book- keeper will be to add, and the test may be important. Those who add well generally perform the other operations well. It is worth while to learn to add correctly and rapidly. 13. The mental process in adding consists in grouping digits of the same order. No matter how many the numbers added may be, the whole work is to group and combine digits. In adding by any method, results only should be mentioned or thought of. One should not name, even mentally, the digits combined. In adding the digits 7, 2, 4, 6, one by one, think "9, 13, 19," not "7 and 2 are 9, 9 and 4 are 13, 13 and 6 are 19." Instead of adding digits one by one, one may group two or more digits and combine the groups. Thus, in adding 4, 5, 6, 2, 3, 7, 4, 6, the digits may be combined into groups of two each. Then "17, 27, 37," thinking of results only. Of course, one may think of the groups and the digits in the groups, but they are minor subjects of thought. The attention should be on the results. HOW TO MAKE GROUPS 14. Making groups and combining them is the whole of the group method. In combining the groups it is important that they be made quickly and to advantage. It is easy to com- bine 10's and 20's, and it is therefore advantageous to make groups of 10's and 20's. 10 ADDITION 11 Groups of two digits that produce 10 98765 12345 These groups should be recognized at sight. Thus, when 7 the pupil sees , he should think "10." These groups should 3 be so well known that the pupil may give his whole attention to combining the groups. Groups of three digits that produce 10 : 11112223 12342343 87656 5 4 4 Groups of three digits that produce 20 : 99998887 98768767 23454566 All possible groups of two digits each. There are only 45 such groups : 111111111222222 987654321234567 223333333444444, 893456789456789 555556666777889 567896789789899 The pupil may know the sum of each of these groups, but he should accustom himself to think of the sum of each group instead of the two numbers in each group. He should think results only, and should be thoroughly drilled on these groups. Simplest Group Method 15. The ordinary pupil has learned to add by combining digits one by one. Thus, in adding 2, 5, 7, 3, 4, 2, 6, 3, 7, 2, he thinks: 7, 14, 17, 21, 23, 29, 32, 39, 41. The simplest group method is to combine two or more 12 MODERN COMMERCIAL ARITHMETIC digits whose sum is 10 or less than 10, and add as above*. Thus, tl^e pupil in adding the above numbers would group and think as follows, the numbers in parentheses show the groups made: (2, 5) 7, (7, 3) 17, (4, 2) 23, (6, 3) 32, (7, 2) 41. EXAMPLE. Add the following column of numbers, making groups of 10 or less: 435. 274 7~; IQfr EXPLANATION. The'parentheses show the groups made. 921 Begin at the bottom and add upward, naming results 7g 6 only. (2, 5) (6, 1) 14, 21, (4, 5,) 30. Write 0, carry 3. Add 205 (3, 7) 10, (8, 2) 20, 26, (7, 3) 36. Write 6, carry 3. Add (4, 2) 9, 16, (9, 1), 26, (4, 2) 32. Write 32. 3260 EXAMPLE. Add the following: 6708 4352 1637 EXPLANATION. Add as in the preceding example. 9452 (2, 6) (3, 4) 15, (2, 7) 24, (2, 8) 34. Write 4, carry 3. Add 3614 3, (5, 3) 11, (7, 1) 19, (5, 3) 27, 32. Write 2, carry 3. 3, 2373 8, 16, (3, 6) 25, (4, 6), 35, (3, 7) 45. Write 5, carry 4. 4, 4836 (2, 4) 10, (2, 3) 15, (9, 1) 25, (4, 6) 35. Write 35. 2552 35524 Straight Grouping and Irregular Grouping 16. Making groups of digits in the order in which they are written may be called straight grouping. Selecting digits out of their regular order to make groups may be called irregular grouping. To make groups equal to 10 or 20, it may be neces- sary to skip about in the column. The advantage in combining groups of 10 and 20 justifies such irregular grouping. If the digits in a column run 3, 0, 6, 7, 4, we may group 3 and 7, and 6 and 4. If the digits run 8, 8, 7, 4, we may group 8, 8, and 4, and leave 7 to be made into another group or to be added separately, PROBLEMS 17. Add the following columns of numbers, making groups of 10 or less: ADDITION 13 1. &. 3. 4. 5. 6. 7. 5843 25364 465371 386754 369625 36825691 46387625 2671 72459 245827 538432 537167 35718356 18253751 9326 72459 274336 276345 362841 35624352 26715436 1473 57531 524386 534785 275146 35625436 63726354 5845 14328 295437 369328 264375 25634253 63527152 3264 46534 624852 574183 536271 53462534 26372453 5728 73291 735968 263715 296345 25635417 35715487 4532 52618 243143 497462 527164 45278164 53716482 2674 45423 538427 513728 356278 37452735 37184527 3426 28936 253673 354372 357281 53672845 25735427 8. 9. 10. 11. 12. 26748353 25749574 39615385 25647154 27845398 17639052 30816748 25674926 37185378 26894536 73620874 20981632 15674892 37882567 25037524 10835627 35718253 71565735 61782035 26735526 16738625 35618536 25637526 35629816 28019635 19035276 51673871 26018635 52673815 27618763 52176351 56271463 63728165 73625437 18674536 93827362 26173815 37227384 93026375 27133926 29637256 34685647 24674468 34256709 45136784 25135487 45789643 23125376 35867657 98787656 15463787 23543476 56984765 25769804 56875674 25416748 35984627 35682536 35987164 37618835 13. u* 15. 16. 17. 15367283 15427685 24537657 43519687 35239675 27742625 76865432 76583251 65763452 76972541 27531874 76574532 76457361 45347648 34135268 77615524 74563424 76593425 76583242 76854325 37106352 76484323 35246745 76953420 74693542 35628163 98063514 63203516 17035263 87063521 46245163 90584615 76845361 25761536 37825163 27462738 27093546 16725830 35627916 25673541 36547684 70561825 26174382 25364534 46874536 25543423 63572514 27615437 27645362 25639653 38762453 18674523 28764358 26517094 25763542 29861542 17625376 35245142 37617284 25160939 25167253 17620935 37825391 25635427 20198352 35426155 15269354 26175243 47654372 25617542 36812563 36782546 37652453 37905615 36873526 26714526 25634516 53672534 90635243 63752635 36254177 24536426 35745246 35649182 63725416 73612803 52719352 26748234 25436145 26415536 14 MODERN COMMERCIAL ARITHMETIC 18. 19. 21. 43523543 45315436 165247 253762 375263 67865434 26794572 167354 352435 635426 32145673 72326436 156345 635645 435261 87694351 76945332 896547 457654 365426 37094532 27840354 265346 256354 486745 7.6456324 25653543 254155 244362 256341 96352743 37615224 175243 524152 263435 25637184 26185643 864532 367453 365473 37689452 37614523 163725 367281 354263 27610934 27615437 908564 367352 532718 28165437 15426435 256371 453626 376281 28919735 26514326 187645 291853 109673 36537462 25637745 902876 267352 536745 38964532 45673542 873452 145352 536476 25763415 25346732 253645 547635 348273 53672543 35426634 273654 351674 536018 25735462 25639064 534614 425134 234162 82453142 35261743 256173 356472 291864 24316423 35241364 156274 852637 254163 24719342 16354276 526376 256173 253452 84. 25. 27. 367251 256372 356271 253645 109354 376534 254364 256453 256453 755463 534265 547654 345263 264273 453432 276354 243524 756473 356473 253647 345243 254365 345253 764534 251652 371852 634524 271563 254362 253624 735261 278435 253462 156372 357453 356453 482617 251736 251637 253746 352617 356453 926504 251647 352671 356576 453664 345634 354625 356427 946732 534167 352463 245362 834925 261832 256173 352671 764836 874693 768546 675846 764836 536281 648376 534253 356453 457154 376283 473865 381965 409756 381794 361748 356738 389164 361748 657436 358467 876452 571684 356271 254615 345261 357152 356046 647835 280193 355627 861744 357849 609654 351709 356173 254363 524534 645352 764352 267453 352634 ADDITION 15 All Possible Groups of Three Digits Each 18. There are 165 such groups: 1111111111111111111 1111111112222222233 1234567892345678934 1111111111111111111 3333344444455555666 5678945678956789,6. 7- 8- 1111111222222222222 6777889222222223333 9789899234567893456 2222222222222222222 3334444445555566667 7894567895678967897 2222233333333333333 7788933333334444445 8989934567894567895 3333333333333344444 5555666677788944444 6789678978989945678 4444444444444444555 4555556666777889555 9567896789789899567 5555555555556666666 5566667778896666777 8967897898996789789 6667777778889 8897778898899 8997898998999 NOTE. If the pupil will learn to think the sum of each of these groups at sight, instead of adding one by one, or two by two, he will be able to add three digits at a time. He may omit groups whose sum exceeds 20. Combining Groups between 10 and 20 19. It is easy to combine groups of 10's and 20's. It requires practice to combine rapidly and accurately groups like 15, 17, 19. In combining such groups, particular atten- 16 MODERN COMMERCIAL ARITHMETIC tion should be given to the unit figures, for mistakes are gener- ally made in combining units, not in combining tens. All the combinations (45) that can be made with numbers between 10 and 20 are as follows : 11 11 11 11 11 11 11 11 11 12 12 12 12 12 11 12 13 14 15 16 17 18 19 12 13 14 15 16 12 12 12 13 13 13 13 13 13 13 14 14 14 14 17 18 19 13 14 15 16 17 18 19 14 15 16 17 14 14 15 15 15 15 15 16 16 16 16 17 17 17 18 19 15 16 17 18 19 16 17 18 19 17 18 19 18 18 18 19 19 19 The unit figures in these groups are the same as those in the 45 combinations of the nine digits. The sum of the tens is 2 in each group. If the sum of the units in any group is 10 or more, the tens in that group will be increased by 1. Think what the unit figure will be. Then the tens figure will be either 2 or 3. If the combinations of the digits have been learned, the combinations of these numbers will soon be mas- tered. Combining 16 and 19 is the same as combining 6 and 9, and adding 2 tens to the sum. Combining any two numbers between 10 and 20 is the same as combining their unit figures, and increasing the sum by 2 tens. Device for Drill Work 2O. Let the teacher draw on the board a diagram like the following : In the space between the two rings are all the numbers between 10 and 20. In the center may be placed, successively, each of the num- bers between 10 and 20. The pupil should be required to combine the num- ber in the center with each number between the rings, naming only the sum of the numbers. The pupil should begin at o.ie point and go around the ring. Thus he may begin at 12, and say: 25, 27, 30, etc. This device will give thorough 'and rapid ADDITION 17 drill, and the pupil may profitably use it outside of the reci tation. All Combinations of Numbers between 10 and 20 with all Numbers between 10 and 100 21. If the pupil can make these combinations readily, he will be in possession of the "lightning method." If the pupil has learned the combinations under Art. 19, he can readily learn the combinations under this article. The wheel for drill work under Art. 20 may be used to develop all the possible combinations here. With the numbers between 10 and 20 in the wheel, and the number 13 in the center, as shown in the diagram, the pupil may make 81 combinations. Thus, 13 beginning with , which he can see on the wheel, he may con- tinue mentally, 13 13 13 13 13 13 13 13 22 32 42 52 62 72 82 92 13 Then he may take , the next combination on the wheel, and combine 13 13 13 13 24 34 44 54 etc. Next he may combine 13 13 13 13 17 27 37 47 and so on. He may use each number between 10 and 20 in the center. There are no more combinations of units under this article than there are under Art. 19. The only difference is in the tens. PROBLEMS 22. Solve the problems under Art. 17, making groups of numbers from 10 to 20 inclusive. Also add the following columns. NOTE. Some teachers may prefer to omit this exercise now, and teach the pupil to add by the cipher method or by the method of re- jecting tens. 18 MODERN COMMERCIAL ARITHMETIC 1. g. 3. 4. 5. 6. 7. 746092 356189 849756 357925 672109 261752 251673 615427 256352 256381 987536 267154 256371 377163 635867 263451 253645 386728 904625 109725 376281 251673 376184 456098 378194 476386 837625 156253 906498 378294 783902 389173 371839 274681 378294 274891 356745 278357 287467 173823 490879 387164 547993 371846 275637 190896 389523 779174 265784 537683 453789 809251 437892 709261 345672 456728 245681 907896 475829 356281 167493 267356 908957 534tfi7 264357 785536 671526 390895 378456 267453 908576 378164 467584 467582 367153 178098 190758 256173 381745 ] 189047 356174 637456 190467 378594 352635 467189 356453 578264 390183 354671 267184 345671 278590 675987 461736 254367 356174 256738 354672 567438 546781 689053 563748 253672 356478 534672 356271 386478 467590 409873 536728 356278 156253 273884 534721 254671 356713 356183 356472 256173 356174 276453 371809 309814 456173 356172 309735 253781 367184 276153 908153 456153 460981 376154 124563 908945 356278 267493 251674 748367 8. 9. 10. 11. 12. 13. 14. 554367 356904 378904 356076 365243 377815 251436 635736 356253 346891 380965 356253 368256 356189 536715 253671 356872 268467 901645 276489 256378 356187 256184 356209 378915 256109 350987 536173 356187 356274 356278 256173 256374 366453 356274 904687 378987 367187 789762 356274 378190 378167 859045 356287 153617 354613 674657 356184 678098 352464 456374 465745 341745 389178 467589 671523 526354 657890 351647 456374 467563 152463 351672 256153 356456 789153 467819 378918 678193 609871 261873 678163 254167 409871 564073 378109 367194 360192 108391 357184 356273 371835 674839 789463 908758 478567 675876 678354 617356 256173 371835 785094 367184 367184 153723 478194 367855 361738 901746 352617 361846 453152 768970 345241 768970 152435 678706 132453 670078 451324 670789 453314 352415 676097 678796 345241 253425 678709 679687 352415 674533 463718 894675 678969 352453 678698 256104 453253 609687 908968 675446 467364 354672 674352 352673 356278 162543 908718 674098 361014 ADDITION 19 IB. 16. 17. 18. 19. 367183 356280 467587 467584 567483 467384 374837 389106 671908 367184 361893 267134 235243 676987 253617 453671 360985 473645 342515 678678 567483 901235 367183 253671 356183 268493 256109 356173 906879 467354 368193 367281 467384 453674 281947 960687 453627 352415 657869 906879 567483 352633 109273 456378 907685 758495 356475 617382 354609 470194 367495 467382 638485 133749 351848 546473 905736 351635 451736 251674 467583 101886 461735 905681 253471 456183 901857 567685 567685 678509 891025 675847 467584 467584 567485 391046 908679 906870 567006 564732 152453 674536 514235 671884 155367 251453 617352 184761 352785 906875 647382 905768 574857 567485 678968 906879 678596 132453 102947 647283 467384 362718 253647 895047 850495 102847 564738 291837 647859 905748 356173 785686 605674 172959 637152 675448 152673 718938 188965 536718 354678 996574 471836 768006 675869 567281 387596 162839 452637 152435 261745 352674 786960 678957 456374 568676 787967 857465 860978 564735 The Cipher Method 23. By this method the student makes groups of 10's and 20's, and combines them with the other digits in groups or one by one. By practice in irregular grouping, the student can put many of the digits into groups of 10 or 20. The following columns show how such groups may be made: 53 20 MODERN COMMERCIAL ARITHMETIC PROBLEMS 24. Look through the columns of figures under Articles 17 and 22, and make groups of 10's and 20's. Skip about if necessary, but do not try to put all the digits into such groups. NOTE. The teacher should assign some of the problems under Articles 17 and 22, to be added by the cipher method. Method of Rejecting 10's 25. By this method the student rejects the 10's (does not hold them in memory), but retains them on a piece of paper or on the fingers. The mind is thus relieved from keep- ing account of the 10's, and addition becomes an easy operation. The student adds, one by one or by groups, until he has from 10 to 20, rejects 10, begins again with what he has left and adds until he has from 10 to 20, rejects 10, and so on. EXAMPLE. Add the following column of numbers, rejecting 10's: EXPLANATION. Begin at the bottom. Add 6, 2 and 8 ; having more than 10, drop 10 and begin again with the 6 left (for every 10 dropped, makes a mark on a slip of paper). ""^ Add 6 and 7; having more than 10, drop 10 and begin ~* again; add 3 and 9, drop 10 and begin over; add 2, 2, 4, and 6, drop 10 and begin over; add 4 and 6, drop 10; add 4, 2, and 4, drop 10. There are no units left. Write in ' ~* units' place. 5 tens have been dropped. Add the second column, including the 5 tens dropped from the first. Add 5, 3, and 5, drop 10; add 3, 6, and 3, drop 10; add 2, 4, 8, drop 10; add 4, 3, 7, drop 10; add 4, 3, 8, drop 10; wr ite ^the 5 left, and carry the 5 dropped. Add the third column, including the 5 dropped from the second. Add 5, 7250 8) dr P 10; add 3 7> dr P 10; add 4 > 5 > 6 > dr P 10; add 5) 7 ' drop 10; add 2, 9, drop 10; add 1, 8, 7, drop 10; add 6, 6, drop 10 ; write the 2 left, also write the 7 dropped. NOTE. Hold the units clearly in mind, but merely notice the tens. The tens are to be dropped from memory, and are to be recorded by some device. When adding 7 and 9, think 6 and record 1 ten as being dropped. If the next number to be added is 8, think 4 and record 1 ten. Give close attention to the units, give just enough at- tention to the tens to record them. ADDITION 21 Device for Recording 10's 26. The following cut shows how the 10's may be recorded. Number the joints of the thumb, beginning at the end, 5, 10, 15; the joints of the first finger, 1, 6, 11; the second finger, 2, 7, 12; the third finger, 3, 8, 13; and the fourth finger, 4, 9, 14. To record 1 ten, place the thumb on the joint of the little fingermarked 1. In like manner, 2 tens, 3 tens, and 4 tens may be recorded on the other fingers. To record 5 tens, straighten the thumb. To record 6 tens, place the thumb on the second joint of the little finger. In like manner, 7 tens, 8 tens, and 9 tens may be recorded on the other fingers. To record 10 tens, straighten the thumb. And so on. By a little practice, the pupil will be able to record the tens on his hand and know when he is through with a column just how many tens have been dropped. If the count ends when his thumb is straightened, he must know, of course, whether it has been straightened once, twice, or three* times. If any column con- tains more than 15 tens, the student may repeat the count on his hand. Civil Service Method of Recording Partial Sums 27. If the sum of each column added be written in full, instead of writing but one figure, the accountant can leave his work after adding part of the number of columns and resume it without re-adding any of the columns. The addition of any column may be verified without re-adding the other columns to find the figure carried. The following illustrates the method : 4628 7394 25 Tne sums f the columns are 25, 30, 28, 26, 5462 30 an( * ^ ne y are written under one another, each 6856 28 succeeding number one place to the left. The 4785 26 partial sums are then added and the result placed under the numbers added. 29125 MODERN COMMERCIAL ARITHMETIC EXERCISES 28. Look through the columns under Articles 17 and 22, and combine the digits, dropping the tens and thinking of the units only. Do not record the tens, but drop them. Add the columns under Articles 17 and 22 by repeating the above exercise, keeping on the hand a record of the tens dropped and writing the proper figure under each column. Horizontal Addition 29. Sometimes numbers that are to be added are found written in horizontal lines. They can be added without re- writing them in vertical columns. In adding horizontally, from right to left or from left to right, care must be taken to com- bine only like orders of units. PROBLEMS SO. Add the following : 1. 45, 34, 67, 35, 98, 56, 74, 37, 25, 74, 34, 64. 2. 123, 435, 609, 524, 274, 315, 376, 903, 457. 3. 260, 789, 635, 256, 235, 170, 585, 370, 245. 4. 1245, 3465, 9075, 1260, 4561, 2351, 6754, 4532. 5. 2745, 706, 486, 3450, 235, 5687, 250, 1274, 378. 6. 652, 3476, 6789, 376, 4523, 8097, 560, 4563, 276. 7. 476, 6453, 254, 8609, 560, 3124, 3654, 450, 231. 8. 3500, 3540, 350, 3456, 873, 2578, 2154, 687, 3542. 9. 5476, 9050, 576, 3542, 1456, 684, 9085, 541, 355. 10. 35672, 5609, 90504, 526, 500, 8050, 45763, 375. 11. 3540, 6750, 4535, 548, 46876, 78045, 362, 4752. 12. 45, 687, 50943, 475, 9967, 4500, 450, 35460. 13. 4536, 567, 9075, 32165, 45, 768, 75, 35025, 268, 4575. 14. 550, 6735, 98, 675, 558, 75, 35450, 265, 4235. 15. 3750, 790, 89, 563, 4576, 256, 950, 5425, 635. 16. 4576, 9056, 75, 356, 87, 2457, 675, 9050, 350. 17. 5490, 5467, 580, 25, 69, 599, 4563, 2875, 654. 18. 4678, 6780, 47, 387, 5674, 3265, 745, 87, 4765. 19. 367, 90, 9835, 453, 69, 2543, 657, 45, 76255. 20. 54376, 9004, 5476, 899, 654, 37459, 362, 7623. SUBTRACTION Subtraction by Addition 31. EXAMPLES. 786 minuend 243 subtrahend 543 remainder 243 + 543 = 786 subtrahend -f- remainder = minuend 243 + ? = 786 To find the remainder, find a number which added to the subtrahend will produce the minuend. The best way to find that number is to add to the subtrahend, digit by digit, until the minuend is produced. Solving the example at the top of this page, one would say, beginning at the right hand, 3 (sub- trahend) and 3 (write it in the remainder) are 6 (minuend). 4 (sub.) and 4 (write in rem.) are 8 (min.) 2 (sub.) and 5 (write in rem.) are 7 (min.) EXAMPLE. From 968 take 425. OPERATION 968 EXPLANATION. 5 (sub.) and 3 (rem.) are 8 (min.) 2 425 (sub.) and 4 (rem.) are 6 (min.). 4 (sub.) and 5 (rem.) are 9 (min.). 543 EXAMPLE. From 7023 take 4326. EXPLANATION. 6 (sub.) and 7 (rem.) are 13 (which 7023 g ives tne figure in the minuend). Carry 1. 2 (sub.) and 1 4.39R (carried) and 9 (rem.) are 12 (min.). 4 (sub.) and 1 (car- ried) and 2 (rem.) are 7(min.) This process is called sub- 2697 traction by addition. It is easy, and is productive of the highest degree of accuracy and rapidity. 23 24 MODERN COMMERCIAL ARITHMETIC PROBLEMS 1. 4670-2463 = ? 9. 507433-285362 = ? 2. 8609-3726 = ? 10. 429728-136475 = ? 3. 2432-1716 = ? 11. 16078435- 1356428 = ? 4. 2586-1654 = ? 12. 24736842-21687584 = ? 5. 130765-124587 = ? IS. 42687903-26875486 = ? 6. 728062-257465 = ? 14. 87065435-84736247 = ? 7. 709354 - 467152 = ? 15. 80035463 - 78200456 = ? 8. 370816-281029 = ? 16. 56190364-49015246 = ? 32. Two or more subtrahends : EXAMPLE. From 86798 take 21342, 26584, and 22765. EXPLANATION. The operation is similar to the process f " Subtraction *>y Addition." Instead of adding the remainder to one subtrahend we add it to three subtra- 2134, hends. Thus, 5, 8, 2 (sub.) and 3 (rem.) are 18 (min.). 6, 4, 4 (sub.) and 1 (carried) and 4 (rem.) are 19 (min.). %"<65 7, 5, 3 (sub.) and 1 (carried) and 1 (rem.) are 17 (min.). 2, 16143 6> * (sub.) and 1 (carried) and 6 (rem.) are 16 (min.). 2, 2, 2 (sub.) and 1 (carried) and 1 (rem.) are 8 (min.). 33. This method is often convenient in finding the balance of an account. The above problem may be written and solved as follows : ( 21342 subtrahends \ 26548 ( 22765 remainder 16143 minuend 86798 Prom the following statement find the net gain : GAINS LOSSES $4260 $324 1273 1673 1647 246 365 113 426 Net gain $7971 $7971 NOTE. Add up the gains. Place the sum under the losses. Add losses, and write for the net gain such digits as added to the losses will produce $7971, and balance the account. SUBTRACTION 25 34. PROBLEMS 1. From 9467 take 2135, 1682, 1478, and 2692. 2. From 12784 take 6253, 3786, and 2057. 3. From 28650 take 10652, 8549, 3267, and 1026. 4. From 156709 take 24680, 13793, 16745, and 24863. 5. From 84907 take 16243, 14786, 9352, and 27054. 6. From 246532 take 73855, 12736, and 96854. 7. From 42584 take 12653, 11964, and 13867. 8. From 89756 take 21684, 14793, 16248, and 13728. 9. From 126087 take 43568, 13752, 28537, and 21864. 10. From 254375 take 62873, 94852, 14695, and 17586. 11. From 48692 take 12071, 9825, 4687 and 14650. 12. From &4508.75 take $1328.14, $924.68, $156.90, $437.15 and $260. 54. 13. From $39575.40 take $4681.25, $11060.50, $16245.30 and $716.90. 14. From $12825.70 take $625.25, $513.42, $765.34 and $1486.76. 15. From $28746.80 take $9458.28, $7134.35, $863.47, $6479.58, $143.92. 16. From $137248. 75 take $41980.40, $3714.24, $9652.18, $4?3.62 and $31967.86. 17. From $125876.85 take $1463.25, $4563.54, $3287.43 and $458.35. 18. From $981.44 take $135.72, $368.27, $94.25, $126.36, $72.83, $43.28. 19. From $3574.50 take $1247.31, $719.85, $64.27, $435.63 and $246. 33. 20. From $42867.25 take $19763.45, $1564.22, $867.14, $938.53 and $6537.48. MULTIPLICATION 35. The product is composed of the same kind of units as is the multiplicand, and the multiplier is an abstract number. But as the product of two numbers is the same whichever factor is used as a multiplier, either factor may be regarded as the multiplier. EXAMPLE. 246 multiplicand 3 multiplier 738 product 36. Know the multiplication table up to 10 x 10. Multiplication Table 123456789 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 26 MULTIPLICATION 27 Numbers between 10 and 20 37. EXAMPLE. Multiply 12 by 13. EXPLANATION. 2 x 3 = 6. The product of the OPERATION units is 6, which write. The product of tens by 12x13 = 156 units is 10 X 2 + 10 X 3 = 50, or 5 tens, which write. The product of the tens is 1 (hundred), which write. NOTE. The product of any two numbers between 10 and 20 is composed of three parts : the product of the units, the product of the units by the tens, and the product of the tens. The product of the units by the tens, in tens, is always the sum of the units. The product of the tens is always 1 (hundred). Therefore, take the product of the units for the units, the sum of the units for the tens, and 1 for the hundreds. If the product or the sum of the units is more than 9, carry as usual. EXAMPLE. Multiply 14 by 15. OPERATION EXPLANATION. 5 X 4 = 20. Write 0, and carry 2. 14x15 = 210 5 + 4 = 9. 9 + 2 (carried) = 11. Write 1, carry 1. 1 (hundred) + 1 (carried) = 2, which write. PROBLEMS Find the product of : 1. 13 x 14. 7. 14 x 17. 18. 19 x 19. 2. 14 x 15. 8. 15 x 18. 14. 14 x 18. 8. 13 x 15. 9. 16 x 19. 15. 17 x 15. 4. 15x16. 10. 17x18. 16: 17x14. 5. 13 x 17. 11. 15 x 19. 17. 12 x 18. 6. 13 x 19. 12. 16 x 18. 18. 18 x 19. NOTE. The pupil should learn to perform these operations men- tally, and the teacher may assign other problems of the same kind. Multiplying by 10, 100, 1000, Etc. 38. If one cipher (0) be annexed to a. number., by what is the number multiplied? If two ciphers (00) be annexed to a number, by what is the number multiplied? If three ciphers (000) be annexed? 28 MODERN COMMERCIAL ARITHMETIC MENTAL PROBLEMS Find the product of : 1. 87 x 100. 8. 872 x 10000. 5. 368 x 10000. 2. 605 x 1000. 4. 9651 x 100. 6. 7524 x 1000. Multiplying Numbers with Ciphers at the Eight Hand 39. EXAMPLE. Multiply 860 by 2400. OPERATION EXPLANATION. Multiply the digits, 86 x 24 = 2064 then to the product annex as many 2064 x 10 x 100 = 2064000. ciphers as are found at the right of both factors. PROBLEMS Find the product of: 1. 27 x 1300. 8. 5200 x 137000. 5. 1240 x 1400. 2. 680 x 12400. 4. 2600 x 18000. 6. 3500 x 16000. Multiplying by a Number Near 100, 1000, etc. 40. EXAMPLE. Multiply 246 by 99. EXPLANATION. Multiply 246 by 100 by OPERATION annexing two ciphers. 99 is 1 less than 100. 246 x 100 = 24600 Therefore, multiply 246 by 99 by subtract- 24600 - 246 = 24354 ing once 246 from 24600. Or, annex two ciphers and subtract 246. EXAMPLE. Multiply 425 by 997. OPERATION 425000 EXPLANATION. Annex three ciphers and sub- 425 x 3 = 1275 tract 3 times 425. 423725 EXAMPLE. Multiply 425 by 1003. OPERATION 425000 EXPLANATION. Annex three ciphers and add 3 425 x 3 = 1275 tim es 425. 426275 MULTIPLICATION 29 PROBLEMS NOTE. Do not write the multiplier, and write the multiplicand but once. Find the product of : 1. 875 x 99. 5. 3075 x 995. 8. 2073 x 105. 0. 6032 x 998. 6. 732 x 102. 9. 358 x 1003. 8. 587 x 97. 7. 864 x 1004. 10. 2561 x 106. 4. 6802x996. CROSS MULTIPLICATION When Each Factor Contains Two Figures 41. EXAMPLE. Multiply 21 by 23. Steps in the operation: 1. Units by Units 21 \ 3x1 = 3 Units. 23' OPERATION 21 x 23 = 483 2 - Tens b y Units 21 2x3 + 2x1 = 8 Tens. /^ 23 3. Tens by Tens /21 2x2 = 4 Hundreds. V23 EXPLANATION. In multiplying units and tens by units and tens, there are three steps: finding the product of units by units, the product of tens by units and the product of tens by tens. The product of the units is 3, which write for the units of the complete product. The product of the tens by the units is 8 tens (3X2 + 2X1), which write for the tens of the complete product. The product of the tens by the tens is 4 hundreds, which write for the hundreds of the complete product. If any partial product be more than 9, carry as usual. EXAMPLE. Multiply 48 by 35. EXPLANATION. First step, 5X8 = 40, units. OPERATION Write 0, carry 4. Second step, 5x4 + 3x8 + 4 (car- 48 x 35 = 1680 ried) = 48 tens. Write 8, carry 4. Third step, 3x4 + 4 (carried) = 16 hundreds, which write. 30 MODERN COMMERCIAL ARITHMETIC PROBLEMS NOTE. Write the factors in a horizontal line and perform the operations mentally. Find the product of: 1. 42x34. 6. 74x53. 11. 37x46. 16. 37x89. 2. 53 x 27. 7. 67 x 58. 12. 48 x 53. 17. 46 x 74. 3. 62 x 43. 8. 95 x 36. 13. 75 x 26. 18. 38 x 92. 4. 73 x 28. 9. 46 x 58. 14. 83 x 47. 19. 79 x 62. 5. 68 x 94. 10. 29 x 35. 15. 64 x 93. 20. 88 x 69. When the Multiplicand Contains More Than Two Figures 42. EXAMPLE. Multiply 235 by 24. EXPLANATION. There is one more step in the OPERATION operation than there are figures in the multiplicand. 235 x 24 = 5640 Omitting the carrying figures, the steps are as fol- lows: 235\ 235 1. ) 5 X 4 = 20 units. S. X 2x4 + 3x2 = 14 hundreds. 24' 24 2. X 3X4 + 5X2 = 22 tens. 4- ( ' ' 2x2 = 4 thousands. Write the proper figures and carry as usual. EXAMPLE. Multiply 4376 by 57. NOTE. The steps in the operation may be illustrated thus: 4376\ 4376 4736 4376 /4376 57/ 57 57 57 \57 EXAMPLE. Multiply 43765 by 57. NOTE. The steps in the operation may be illustrated thus: 43765\ 43765 43765 43765 43765 /4376S 57/ 57 57 5^ 57 \57 PROBLEMS NOTE. Write the factors in a horizontal line and perform the operations mentally. This method is convenient in extending bills. 1. 234 x 23. 3. 567 x 38. 5. 4306 x 42. 7. 1236 x 63. 2. 426 x 34. 4. 1214 x 35. 6. 7308 x 54. 8. 4253 x 76. MULTIPLICATION 31 9. 1684x57. 10, 2173x65. 11. 4287 x 29. 12. 64261x36.^ 18. 23784 x 75. 14. 40265 x 84. 15. 25738 x 73. 16. 6381 x 76. mt~ 17. 9017x39. s# #v IS. 65182x82. <-- \fjfa4 19. 52781 x 87. * / ft I *' ; >f 0. 6910x94. Multiplying by Two Figures One of Which is 1 43. This is usually given as a special method, but in its simplest form it is cross multiplication. PROBLEMS Find the product of : 1. 236x13. 5. 8316x31. 9. 237x19. 18. 1378x12. 2. 472x14. 6. 7026x18. 10. 1384x91. 14. 7813x16. 8. 1238x17. 7. 1682x41. 11. 2765x71. 15. 836x61. 4. 2036x21. 8. 2361x51. 12. 1683x15. 16. 7825x81. To Multiply by 11 44. Although usually given as a special method this is simply cross multiplication. PROBLEMS Find the product of : 1. 18x11. 5. 315x11. 9. 356x11. 18. 1362x11. 2. 26x11. 6. 246x11. 10. 943x11. 14. 2480x11. 8. 135x11. 7. 719x11. 11. 257x11. 15. 7156x11. -f 76x11. 5.263x11. 10.645x11. 1(5.3062x11. DIVISION 45. Illustration of Terms and Proof of Division 40-5 8 dividend - divisor = quotient 5x8 =40 divisor x quotient = dividend 40 + 8 5 dividend - quotient = divisor 128 - 5 = 25 and 3 remainder, written f , 3 to be divided by 5. divisor x quotient -f remainder (if any) = dividend Hence to prove division, multiply together the divisor and quotient and to the product add the remainder. If the result equals the dividend the work is correct. 46. Principles. 24-4 = 6 48-4 = 12 12-4 = 3 1. Multiplying the dividend multiplies the quotient; divid- ing the dividend divides the quotient. 24-4 = 6 24-8 = 3 24-2 = 12 2. Multiplying the divisor divides the quotient; dividing the divisor multiplies the quotient. 24-4 = 6 48+8 = 6 12-2 = 6 3. Multiplying or dividing both dividend and divisor by the same number does not change the value of the quotient. Short Long Division 47. This is the ordinary method of long division shortened by introducing the "subtraction by addition" method and by multiplying and subtracting at the same time. DIVISION 33 EXAMPLE. Divide 721098 by 291. EXPLANATION. The first figure of tho quo- tient is 2. Multiply the divisor by 2 and subtract the product from 721 as follows: 2X1 = 2 and 9 (which write for the remainder) are 11. 2x9 = 18-and 1 (carried) are 19 and 3 (write in remainder) are 22. 2x2 = 4 and 2 (carried) are 6 and 1 (write in remainder) are 7. Bring down the next figure of the dividend, making the remainder 1390. Multiply 291 by 4 and subtract. 4x1 = 4 and 6 are 10. 4 X 9 = 36 and 1 are 37 and 2 are 39. 4X2 = 8 and 3 are 11 and 2 are 13. Bring down the next figure of the dividend, making the remainder 2269. Multiply 291 by 7 and subtract. 7X1 = 7 and 2 are 9. 7 X 9 = 63 and 3 are 66. 7 X 2 = 14 and 6 are 20 and 2 are 22. Bring down the 8. Multiply 291 by 8 and subtract. OPERATION 291)721098(2478 1390 2269 2328 0000 PROBLEMS Find the quotient of: 1. 5. 6. 7. 8. 9. 10. 11445 + 10664- 30128 + 11312644- 1691823- 313194106 - 1243414986 - 58327. 302418 - 954. 205683- 629. 2151090- 4185. 35. 24. 56. 7856. 357. 7153. 11. IS. 14. 15. 16. 17. 18. 19. 20. 1085. 317. 654. 423. 2136. 4874. 2576. - 16843. 1091510 - 135676- 6220848 * 4687238- 7098643 -' 9364725 - 537654-*- 1068432- 2764379- 4268. 876543 - 769. 48. Dividing by 10, 100, 1000, etc. Principle. Cutting off one figure from the right of the divi- dend divides it by 10, cutting off two figures divides it by 100, cutting off three figures divides it by 1000, etc. The part of the dividend cut off is the remainder. PROBLEMS Find the quotient of: 1. 46800- 100. '.; 6. 420070- 1000. 2. 73000- 1000. 7. 53781936- 100000. 3. 2860- 100. 8. 800167829- 10000. 4. 796800-10000. 9. 670197356-1000000. 5. 63875- 1000. 10. 5179036527 + 1000000. THE EQUATION 49. The sign of equality (=) between two equal numbers or expressions forms an Equation. 7 + 2=9. 4x5 + 3 = 6x3 + 5. Cost of 25 cows = cost of 100 sheep. minuend - subtrahend = remainder dividend -*- divisor = quotient Axioms 50. Truths so simple that they do not admit of proof are called Axioms. 1. Things equal to the same thing are equal to each other. If $600 = cost of 25 cows, and $600 = cost of 100 sheep, then cost of 25 cows = cost of 100 sheep. 2. If equals are added to equals the sums are equal. If cost of 25 cows = cost of 100 sheep, then cost of 25 cows +$100 = cost of 100 sheep + $100. If A = B, then A + C = B + C. 3. If equals are subtracted from equals the remainders are equal. If cost of 25 cows = cost of 100 sheep, then cost of 25 cows -$100 = cost of 100 sheep -$100. If A=B, then A-0 = B-C. 4. If equals are multiplied by equals the products are equal. If cost of 25 cows = cost of 100 sheep, then cost of 100 cows = cost of 400 sheep. If A = B, then A x C = B x C. 34 THE EQUATION 35 5. If equals are divided by equals the quotients are equal. If cost of 25 cows = cost of 100 sheep, then cost of 5 cows = cost of 20 sheep. If A = B, then A + C = B + C. 51. The process of changing a number from one side of an equation to the other is called transposition. (1) A-10 = B Add 10 to both sides (members) of the equation (Axiom 2,) and (2) A-10 + 10 = B + 10, or A = B + 10 In the first equation, 10 is in the first member ; in the sec- ond equation, 10 is in the second member. The sign of 10 in the first equation is , in the second +. 10 has been changed from one member of the equation to the other, and its sign has been changed from to +. (3) A + 10 = B Subtract 10 from both members of the equation, and (4) A = B-10 10 has been changed from one member of the equation to the other, and its sign has been changed from + to . 52. Principle. A number may be transposed from one member of an equation to the other by changing its sign from + to - , or from to + . Solution of the Equation 53. If in any equation there is a term whose value is not given, as A = 3 + 7, finding the value of that term is called solv- ing the equation. NOTE. It is a common error to take the signs of addition, subtrac- tion, multiplication and division in the order in which they come. The signs of multiplication and division have the preference over the signs of addition and subtraction, and the operations indicated by the former are to be performed before those indicated by the signs of the latter, thus: 2 + 9 X 6 = 56, not 66 36 MODERN COMMERCIAL ARITHMETIC In every equation the plus or the minus sign must be understood to affect the result of the whole operation indicated between it and the next plus or minus, or between it and the close of the expression. If a problem is stated in the form of an equation, solving the equation is solving the problem. PROBLEMS 54. Solve the following equations: 1. Cost of horse = $100 4- $6 x 10. (Find the cost of the horse.) 2. A = 65 4-32 -7. 3. Cost of 2 cows = $18 + $3 x 10: (Find cost of one cow.) 4. 2 A = 18 + 3 x 10. (Find value of A.) 5. Cost of stove + $10 = $40. (Find cost of stove.) 6. Cost of horse - $15 = $60. 7. Value of 3 horses = $210. (Find value of a horse.) 8. Value of 3 horses - $15 = i 9. Value of 3 horses 4- $10 = 1 10. Value of 8 cows = $1920 + 6. 11. Value of 5 sheep 4- $12 = $42. 12. A 4-6x8 = 504- 23. 18. Dividend = 12 x 360 4- 40. 14. Subtrahend - 165 = 586. 15. How many cows at $25 per head = $600? CANCELLATION 55. Cancel means to mark out. Cancellation is division by marking out or crossing out factors. The dividend consists of two factors : the divisor and quotient. If one of these factors be marked out, the other factor will remain. The divisor itself maybe separated into factors. When both divisor and dividend contain like factors, such factors may be cancelled without changing the value of the quotient. Cancellation is the process of rejecting common factors from both dividend and divisor. (See Principle 3, Art. 46.) 56. EXAMPLE. Divide 6 x 8 x 12 x 15 x 20 by 2 x 3 x 4 x 5 x 30. Instead of dividing the product of the first set of factors by the product of the second set, we may cancel the common factors and then divide if a divisor remains uncancelled. EXPLANATION. Write OPERATION the factors of the dividend $ , 12 10 ^0 above a horizontal line and = 48 the factors of the divisor * 49 30^ below. Then reject equal factors from both dividend and divisor. The factors 2 and 3 in the divisor cancel 6 (2X3) in the dividend. 4 and 5 in the divisor cancel 20 (4 X 5) in the divi- dend. 15 in the dividend cancels 15 (one of the factors of 30) in the divisor, leaving the factor 2 instead of 30 in the divisor. This 2 can- cels 2 (one of the factors of 8) in the dividend, leaving 4 instead of 8 in the dividend. In the dividend there remain the factors 4 and 12, which produce 48, the required quotient. Exact Divisors 57. 1. 2 is an exact divisor of any even number. 2. 3 is an exact divisor of any number the sum of whose digits is divisible by 3. 37 38 MODERN COMMERCIAL ARITHMETIC 3. 5 is an exact divisor of any number ending with 5 or 0. 4. 9 is an exact divisor of any number the sum of whose digits is divisible by 9. 5. No even number is an exact divisor of an odd number. PROBLEMS 58. Cancel when you can, multiply when you must. Divide : 1. 45 x 16 x 60 x 27 by 15 x 8 x 12 x 9 x 4. 2. 96 x 128 x 72 x 64 by 16 x 48 x 27. 8. 33 x 57 x 72 x 216 by 19 x 11 x 8 x 27 x 16. 4. 872 x 365 x 496 by 654 x 175 x 428. 5. 384 x 495 x 350 by 352 x 330 x 210. 6. 345 x 432 x 120 by 210 x 360 x 216. 7. 882 x 225 x 168 by 556 x 315 x 112. 8. 616 x 512 x 420 by 560 x 448 x 315. 9. A man exchanged 24 loads of wheat, each weighing 2480 pounds, worth 80 cents per bushel of 60 pounds, for 59 barrels of sugar, each weighing 420 pounds. What was the price of the sugar per pound? 10. A farmer sold 18 loads of hay, each weighing 2860 pounds, at $21 per ton of 2000 pounds. He received in pay- ment 8 loads of phosphate, each load containing 12 bags, and each bag 240 pounds. What was the price of the phosphate per 100 pounds? 11. How many city lots, each containing 21 square rods, and valued at $8 per square rod, are equal in value to 15 fields, each containing 1120 square rods, valued at $32 per acre of 160 square rods? 12. How many pieces of cloth, each containing 126 yards, valued at 16 cents per yard, are equal in value to 35 pieces of cloth, each containing 288 yards, valued at 14 cents per yard? 18. If 18 barrels of beef, each containing 200 pounds, are worth $288, what will 75 pounds cost at the same rate? 14. If 52 men can dig a ditch in 42 days, working 9 hours a day, how many days will be required by 24 men to do the same work, if they work 7 hours per day? CANCELLATION 39 15. How many boxes of tobacco, each weighing 42 pounds, valued at 90 cents per pound, are equivalent to 80 boxes of tobacco, each weighing 149 pounds, valued at 81 cents per pound? 16. Divide 120 x 540 x 695 by 380 x 175. 17. 250 x 25 x 84 x 21 = 365 x 80 x 32 x ? 18. The factors of the dividend are 940, 760, 145, and 724. The factors of the divisor are 190, 724, 235, and 180. Find the quotient. 19. A bicyclist rode 8 miles an hour for 18 days of 10 hours each and walked back at the rate of 3 miles per hour. How many days did it take Jrim to get back, if he walked 12 hours a day? 20. How many days' work, of 10 hours each, at 15 cents per hour, will be required to pay for a pile of wood 48 feet long, 4 feet wide, and 9 feet high, at $4.50 per cord? 21. How many boards 16 feet long and 12 inches wide, at $24 per thousand, must be given in exchange for 160 scantling 2 inches by 4 inches and 18 feet long, at $14 per thousand? 22. How many bricks 2 inches by 4 inches by 8 inches will be required to lay a wall 14 feet long, 6 feet high, and 1^- feet thick? 23. How many village lots 6 rods by 8 rods, worth $6 per square rod, are equal in value to a farm 80 rods by 90 rods, at $96 per acre? 24. At $26 per thousand, how many sticks 4 inches by 6 inches by 14 feet are equal in value to 130 boards 18 feet long and 12 inches wide, at $45 per thousand? 25. How many days, of 9 hours each, must a man work, at 18 cents per hour, to pay for a fot 120 feet by 165 feet, at $8 per square rod? FRACTIONS 59. If a whole is divided into two or more equal parts, the parts are called fractions of the whole. One of the parts is a fraction. Several of the parts are a fraction. Division is the process of separating a unit or a number into equal parts. One-seventh of 1, one-ninth of 1, one-tenth of 1, are frac- tions, for they are each equal parts of a unit. One-seventh of 4, one-ninth of 4, one-tenth of 4, are frac- tions, for they are each equal parts of a number. A fraction is one or more of the equal parts of a unit, or one of the equal parts of a number. f shows 2 of the 3 equal parts into which a unit is divided, or it shows that 2 is divided into 3 equal parts. A fraction is division indicated by the sign /. 1-^-2 and \ are the same in value, f \ and 27 -f- 36 are the same in value. 60. In a fraction, the dividend (number above the line) is called the Numerator. It enumerates, or tells, how many of the equal parts of a unit are included in the fraction, or it shows what number has been divided into equal parts. The fraction f shows 3 of the 5 equal parts into which the unit is divided, or it shows that 3 is divided into 5 equal parts. 61. The divisor (number below the line) is called the Denominator. It names the parts into which a unit or a num- ber is divided. If the denominator is 4, it shows that a unit or a number is divided into fourths. 62. A fraction indicates division. The value of a fraction is therefore the quotient of the division indicated. A fraction is a quotient. The numerator and denominator are called the terms of a fraction. The dividend and divisor are called tb terms of a division. 40 FRACTIONS 41 Comparing fractions with division, the numerator is the dividend, the denominator is the divisor, and the fraction is the quotient. Principles of Division 63. 1. Multiplying the dividend multiplies the quotient, dividing the dividend divides the quotient. 2. Multiplying the divisor divides the quotient, dividing the divisor multiplies the quotient. 3. Multiplying or dividing both dividend and divisor by the same number does not change the value of the quotient. Principles of Fractions 64. 1. Multiplying the numerator multiplies the fraction, dividing the numerator divides the fraction. 2. Multiplying the denominator divides the fraction, divid- ing the denominator multiplies the fraction. 3. Multiplying or dividing both numerator and denominator by the same number does not change the value of the fraction. Terms 65. A Proper Fraction is one whose numerator is less than its denominator; as, f, , |. An Improper Fraction is one whose numerator equals or exceeds its denominator; as, f, f, y. A Mixed Number is one expressed by an integer and a fraction ; as, 4f , read four and three-fifths. A Complex Fraction is one which has a fraction in one or s 5 9 2 both of its terms ; as, -|> ^ ? - fv *t C A Compound Fraction consists of two or more single frac- tions joined together by the word of; as, f of f of f . PROBLEMS Write the following in the form of fractions : 1. One-half over six-fifths. 2. Four over eight-seventeenths. 42 MODERN COMMERCIAL ARITHMETIC 3. Mne-tenths of eight and two-thirds. 4. Seven-ninths of four-sevenths. 5. Six and seven-eighths over nine and three-fourths. DECIMAL DIVISIONS AND DECIMAL FRACTIONS 66. If 1000 is divided into 10 equal parts, what is one of the parts called? Perform the operation by pointing off one figure. If 100 is divided into 10 equal parts, what is one of the parts called? Perform the operation by pointing off one figure. If 10 is divided into 10 equal parts, what is one of the parts called? Perform the operation by pointing off one figure. If 1 is divided into 10 equal parts, what is one of the parts called? Perform the operation by pointing off one figure. If .1 is divided into 10 equal parts, what is one of the parts called? Point off one figure as before. Put a cipher between the period (decimal point) and the one to show that one v place more has been pointed off. If .01 be divided into 10 equal parts, what is one of the parts called? Point off as before. Insert another cipher. If .001 is divided into 10 equal parts, what is one of the parts called? Point off as before, and insert another cipher. The divisions of a number into tenths, hundredths, thou- sandths, etc., are Decimal Divisions, or Decimal Fractions. PROBLEMS Divide by pointing off (insert ciphers when necessary) and read the quotients : 1. 1+ 10. 10. 10-1000000. 2. 1 + 100. 11. 25 + 100. ft I-*- 1000. 12. 25 + 1000. 4. 3 + 10. IS. 25-*- 10000. 5. 3- 100. 14. 25+ 100000. .3 + 1000. 15. 25 + 1000000. 7. 1+ 10000. 16. 136+ 1000. 8. 1+ 100000. 17. 136+ 10000. 9. 1 + 1000000. 18. 1378 + 1000000. DECIMAL DIVISIONS AND DECIMAL FRACTIONS 43 67. By what must 1 be divided to produce .01? Read .01. By what must 1 be divided to produce .001? Read .001. By what must 1 be divided to produce .0001? Read .0001. Read .0007, .00001, .00005, .000001, .000006, .000021, .00055. DECIMAL SCALE OF ARABIC NOTATION 1 1 1 I 1 1 . i 1 I 1 1 1 I 1 I J i : 1 1 I v I ! ! I I 7654321234567 68. Repeat the scale from millions to millionths, from mil- lionths to millions. The decimal point is always before tenths. Repeat the scale from the decimal point each way. The name, or denomination, of a decimal is that of its right- hand order of units. Thus, .0001 is one ten 'thousandth. How to Write Decimals 69. According to our system of writing we begin at the left and write toward the right. This is the case with script and also with figures. For the sake of economy of time and to be consistent, pupils should learn to write decimals in the same manner and according to the following rules: 1st. Fix the decimal point. 2d. Think of the number of places required to make a frac- tion of the given denominator. 3d. Think of the number of places given in the numerator. 4th. Beginning at the right of the decimal point, write as many ciphers as are required to make the number of places given -equal to the number required, and follow these by the numerator, or the figures given. 44 MODERN COMMERCIAL ARITHMETIC 7O. Write decimally: 1. 3 thousandths. 1st. Fix the point. 3d. It requires three places to make thousandths. 3d. One place is given. 4th. At the right of the point write two ciphers, then the 3. 2. 135 millionths. 3. 3 millionths ; 345 hundred-thousandths ; 45 millionths. 4. 23456 millionths; 356 hundred-millionths. 5. Eighteen ten-thousandths. Twenty-seven hundred-thou- sandths. One hundred sixty-five millionths. Thirty-four ten- thousandths. One hundred eight hundred-thousandths. Eighty millionths. 6. Twenty-five and twenty-six thousandths. NOTE. The word and is used to connect a whole number and a decimal. This expression is written 25.026. 7. One hundred and seventy-eight ten-thousandths. One hundred seventy -eight ten-thousandths. 8. Four hundred sixty-three and four hundred sixty-three ten-millionths. Four hundred and forty-seven ten-thousandths. Four hundred forty-seven ten-thousandths. 9. Six hundred-thousandths. Six hundred thousandths. Three hundred ten thousandths. Three hundred ten-thou- sandths. 10. Three hundred eighty thousand and thirteen thousand four hundred ninety- six hundred-thousandths. EXERCISES IN NUMERATION 71. The numerator of a decimal is the number expressed by the figures of the decimal. It is the number divided into tenths, hundredths, thousandths, etc. The denominator of a decimal is indicated by the decimal point and by the number of figures following the decimal point. The decimal point stands for 1 ; each figure in the decimal stands for a cipher following that 1. The denominator, if expressed, would be 1 with as many ciphers annexed as there are figures at the right of the decimal point. FRACTIONS 45 EXERCISES Read the following: 1. .00125. NOTE. There are two parts to the operation: reading the numer- ator and reading the denominator. Read the numerator as if it were a whole number. To read the denominator, begin at the decimal point and numerate toward the right, thus : tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths. The decimal is one hundred twenty-five hundred-thousandths. 2. .000367, .1709, .00231, .100236, .00002, .000367. 3. .07, .0028, .6400, .056843, .0086003, .00008. 4. 18.34. NOTE. This is read 18 and 34 hundredths. In reading mixed numbers, it is necessary to connect the integral and fractional parts by and. 5. 127.0034, 13.2006, 1780.0073, 146000.146. 6. 400.04, 10000.010, 10.010, 200.003, .203. 7. .00800654, 3800.0076, 56.7003804, 4800.7063. 8. 3674583.000437, 2687.015008, 3268.007583. 9. 1200.0012, 3984.05307, 100300.00301, 37.0000045. 10. 256784370.4600736, 4300.56032, 140000.00007554. 72. Compare .3, .30, .300, .3000 as to value. Compare .3, .03, .003, .0003 as to value. Principles: 1. Annexing ciphers to a decimal does not change its value. 2. Inserting a cipher between the decimal point and the decimal figures divides the decimal by 10. 3. Decimal orders of units increase and decrease in value the same as do orders of units in whole numbers. Addition of Decimals 73. Principle. Only like orders of units can be added. EXAMPLE. Add .265, 13.7, and 1.3787. OPERATION EXPLANATION. Write the numbers so that the .265 decimal points fall in a vertical line, and units of 13.7 the same order will stand in the same column. Add 1.2787 as in integers, and put the decimal point in the sum 1*> 34-37 directly under the points in the numbers added. 46 MODERN COMMERCIAL ARITHMETIC PROBLEMS Find the sum of : 1. .14, 1.268, 3.72, 4.682, 15.79. 2. 31.06, 128.374, 175.009, 38.063, 53.034. 3. 504.017, 3.86, 7.128, 130.065, 14.586. 4. .0038, 100.16, 9.054, .69786, 1687.98745. 5. 432.867, 576.09, 78.659, 9.85, 15.462. 6. .3286, 14.567, 284.007, 1275.49, 36.5976. 7. 12, 14.825, .7364, 129, 16.004, 157.3697. 8. 47.25, 6.0078, 174.6, 9.678, 23.00159. Subtraction of Decimals 74. Principle. Only like orders of units can be subtracted. EXAMPLE. From 48.73 take 25.6274. OPERATION EXPLANATION. Write the numbers as in addi- 48.73 tion. Consider ciphers as annexed to the minuend, 25.6274 and subtract. Put the decimal point in the remainder 23. 1026 directly under the points in the numbers subtracted. PROBLEMS Find the remainder of: 1. .832 -.126. 5. 179.86-138.0583. 2. 15.06-7.2584. 6. 4085.75-927.6485. S. 460.85-53.265. 7. 928.6482-25.7595. 4. 246.7385-39.74. 8. 16843.7652-483.28. Multiplication of Decimals 75. Tens x tens = hundreds. 10 x 10 = 100. Tens x hundreds = thousands. 10 x 100 = 1000. Hundreds x hundreds = ten-thousands. 100 x 100 = 10000. Hundreds x thousands = hundred-thousands. 100 x 1000 = 100000. Tenths x tenths = hundredths. .1 x .1 = .01. Tenths x hundredths = thousandths. .1 x .01 = .001. Hundredths x hundredths = ten -thousandths. .01 x .01 = .0001. Hundredths x thousandths = hundred-thousandths. .01 x .001 = .00001. FRACTIONS 47 Principle. The product contains as many decimal places as there are decimal places in the factors. EXAMPLE. Multiply 1.256 by .32. OPERATION -^ 25g EXPLANATION. Multiply as in whole numbers. Point 32 off as many decimal places in the product as there are decimal places in both factors. 2512 NOTE. If there are not enough figures in the product 3768 to point off, prefix ciphers to the product. Thus, .2 X .04 ."40192 = - 08 - PROBLEMS Find the product of: 1. .83x1.4. 11. 1000x10. 2. 1.75x.23. 12. .001x10. 8. . 272 x. 081. 18. 1.001 x .0001. 4. .064x1.5. 14. . 0001 x. 001 5. . 0053 x. 029. 15. 10.001 x .00001. 6. 2.0038 x .00016. 16. 1000 x .001. 7. 4.062 x. 0037. 17. 10001.0001xl.001. 8. .0155x1.8. 18. 10000.0001x10000. 9. 27.08 x. 125. 19. 5005.005x5000. 10. 523 x. 00017. 20. 500.0005x5000.000005. Division of Decimals 76. .05 x .005 = .00025. Hence, .00025 + .05 = .005, and .00025 + .005 = .05. Principle. The quotient contains as many decimal places as the number of decimal places in the dividend exceed those in the divisor. EXAMPLE 1. Divide .00036 by .004. EXPLANATION. Divide as in whole numbers. Point off in the quotient as many decimal places as the number of decimal places in the dividend .004).00036(.09 exceed those in the divisor. 5 3 = 2. Prefix one cipher to the quotient, and point off two places. SUGGESTIONS. 1. If the quotient does not contain enough figures to point off, prefix ciphers. 48 MODERN COMMERCIAL ARITHMETIC 2. Before dividing, make the number of decimal places in the divi- dend at least equal to the number of places in the divisor, by annexing ciphers to the dividend. 3. When all the figures of the dividend have been used and there is a remainder, annex ciphers to the dividend and continue the division. 4. Ordinarily it is not necessary to extend the division to more than four decimal figures in the quotient. PROBLEMS Find the quotient of: 1. 68.125-25. 6. .0357-51000. 2. 16.025 -* .045. 7. 625-*- .0025. 5. 52.848-09. 8. 20-75. 4. 75 -.00125. 9. 723.68-143. 5. .0065-125. 10. 2.652-17. EXAMPLE 2. Divide 638.25 by 300. OPERATION 300)6.3825 EXPLANATION. Cut off the two ciphers in the divisor and point off two places in the dividend, begin- 2.1275 ning at tne decimal point. Then divide 6.3825 by 3. 11. 675-50000. 15. 865-124000000. 12. 428.6-200. 16. 97.281-900. IS. .373-4000. ^17. 57800-8000000. U. 1728-1200. 18. .6307-700. Ten problems are given in each of the following groups. The object is to drill the pupil in pointing off. State the quotient in the form of a decimal fraction in each of the following problems : 19. 20. 21. 2-2. 1-2. 4-25. 2 -.02. 1-200. 400 -.25. 20-2. .1-.2. .4-2500. 2 -.2. .01-20. .004 -.0025. 20 -.002. .0001 -.2. .4 -.000025. .2-2. 100 -.002. 4000 -.000025. .2 -.002. .0001-200. .0004-250000. .2 -.2. 10 -.0002. 40-25000. .2 - .200. .001 - .00002. 4 - .000025. 20 + 2000. 1000 - .0002. .00004 - .000025. FRACTIONS 49 22. 23. 24. 33 + 11. 250-12500. .08 + 16. .0033 + 1100. 2.5 * .00125. 80 + 1600000. 33000 -# .0011. 2500 + .0000125. 8000 -*- .000016. 3300 + 110000. .00025 + 125000. .0008 + .00016. .33 + .000011. 2.5 + 125. .00008 * 16000. .00033 + 110000. .025 + .000125. 8 + 1600. .00033 + .011. .0025 + 1250000. 80000 + 160. .0033 * .000011. 25000 + .0000125. 800 * .000016. 330 + .00011. .25 + .000125. .0008 * 160000. .000033 + 1100000. 250 + 12500000. .000008 + .0016. REDUCTION OF FRACTIONS 77. Fractions may be written as decimals or as common fractions. Principles. 1. A fraction is an indicated division. 2. The denominator of a decimal when expressed is 1 with as many ciphers annexed as there are orders of units at the right of the decimal point. PROBLEMS 1. Write .64 as a common fraction. 2. Write T Vo as a decimal. 3. Write .032 as a common fraction. 4. Write 4 + 5 as a decimal. Perform the operation in dicated. 5. Write as a decimal. Perform the operation indicated. 6. Write J as a decimal. Write the following as common fractions : 7. .268, 10. .0205. 18. .0036. 8. .3758. 11. .09. 14. .25738. 9. .046. ./#. .049. 15. .0013. Write as mixed numbers : 16. 2.65. 18. 10.15. 00. 101.101. ,?7. 17.364. 19. 610.0018. 21. 13.085. Write as decimals : 22. f. 24. |. 05. f. 28. |. 05. |. 05. f. 07. I. 00. -|. 50 MODERN COMMERCIAL ARITHMETIC NOTE. When the division is not exact, the remainder may be expressed as a common fraction, or the sign + may be placed after the decimal to show that the division is not complete. Thus, J = .333J, or .3333-f. Common fractions in their lowest terms cannot be reduced to pure decimals if their denominators contain any prime factors other than 2 or 5. 30. I. 34. A. 38. V 42. I 31. f. 35. ft. 39. \%. 43. 5 V ^2. f. *0. *V 40. . 44- ih- S3. f. 37. . 41. A- 4S. U. NOTE. .251 = .25125. .23? = .2375. 46. .17J. 48. .27^. 50. .24f. 52. .96?. 47. .24f. 49. .16f. 57. NOTE. 3 J = 3. 3333 +. 4. 0| = 4. 05. 54. 7i 50. 14f. 5^. 25.0| . 60. 55. 17.0f. 57. 20.00f. 59. 9.0 T V- ^^. 78. Whole numbers may be written as fractions thus: 5x2 5x14 5x20 5 = --= -- = tt --= Changing a whole number to a fraction by giving a denom- inator to the whole number is simply dividing the number by that denominator. If a number is changed to a fraction with a given denom- inator, the number must be multiplied by that denominator. PROBLEMS 9 V 9 1. 3= '-. *15= t . 5. 11,-. V V V a. 12 = -. 4- o = . 6 '" 18= 2^ 7. Write 7 as 13ths. 10. Write 5 as 36ths. 8. Write 16 as 30ths. 11. Write 8 as 41sts. 9. Write 4 as 27ths. 12. Write 12 as 120ths. FRACTIONS 51 79. Mixed numbers may be written as common fractions thus: H-l + iorf. 7|-Y+|orY- NOTE. Multiply the whole number by the denominator of the fraction, add the numerator, and write the sum over the denominator. PROBLEMS ^ Write as common fractions : 1. 7f 4- 15f 7. 120 T 6 f . 10. 163 V 0. 12|. 5. 20|. 8. 25}f. .n. 16ft. 5. 6f. 6. 14ft. & 62 T V ^. 24 r 8 T . 80. Common fractions may be written as whole or mixed numbers thus : 1 = 64-3 = 2. I = 7 * 3 = 2f *ft 6 = 245 -*- 12 = 20ft. NOTE. Perform the operations indicated. PROBLEMS Write as whole or mixed numbers : i. i. 4. *i*. 7. YJ. ^ W- - 81. Common fractions may be written in their lowest terms. 12*6 2 f 1 - ^g ^ g - tf ** Tl *" T- Principle. Dividing both numerator and denominator by the same number does not change the value of the fraction. NOTE. If both terms be divided by their greatest common divisor, or by as many successive divisors as possible, the fraction will then be expressed in its lowest terms PROBLEMS Write the lowest terms of the following: * H- 4. if r. m- 10. m 2. H. 5. rfft. 8. T Y*V 11. H*. 3. iVV 6. jffr. 9. |41|. 12. 52 MODERN COMMERCIAL ARITHMETIC 82. Common fractions may be written in higher terms, 3x3 5 x 6 ' Principle. Multiplying both terms of a fraction by the same number does not change the value of the fraction. Thus, change $ to 27ths. 27 9 x 3 " * T * NOTE. Multiply both terms of the fraction by a number that will change the given denominator to the required denominator. To find such a number, divide the required denominator by the given denom- inator. PROBLEMS Change : 1. T V to 48ths. 5. f to 147ths. 9. || to 72ds. 2. T 5 T to SOths. 6. T V to 225ths. 10. f f to 319ths. 8. T 9 T to 55ths. 7. T t, I, A- ^ A, A, * 3 8, A- ^- i> I > T\> A' ^- i A A* A- 5- A, A, A> H- 6 - A, T 6 *, A, A- 89. To change fractions to fractions having their L.C.D. /. Find the L.C.D. 77. Change each fraction to a fraction having the L.C.D. as its denominator. PROBLEMS Change to fractions having their L.C.D. i- H, A, A, 4- 7. T ' 8 , &, A, ii, W- 4, A, A, li- 5 - A. A, A, A- * A, A. ii If- 9 - H if' H ^. A, A, *l, H- ^. H, |f, if, if, if. 5. A, it, A, A ^- +, U, A, !* e. A, H, A, *+ ^- H, If, Ai A- MENTAL PROBLEMS Reduce to their simplest forms: J. H- -4- i"A. 7. T y 8 - 10. TA. 15. |f *. AV 5. if- 5 - ti 11 54 MODERX COMMERCIAL ARITHMETIC Reduce : M. 4 to 9ths. 17. J to 18ths. 0. J to 72ds. 27. I to 64ths. *- 05. | to 120ths. / T to 126ths 18. | to 40ths. 19. ft to 96ths. 00. 4 to 49ths. 05. T V to 55ths. . 07. } to 70ths. A- to 200ths. Reduce to integers or mixed numbers: 28. -H. 31. y. 29. i|A. 50. ff Q / 160 QV 3 Jf.. 32 . #> W- ^. w- 42. *%&. Reduce to the fractional form : 44- n -47- 2i- 50. 12f 55. 124. 55. 15J. 56. 10|. 57. 144. 58. 5P. 60. Find the L. C. D. of: i and T V- ^. i J and ft- 63 - t and "A- ft and ^V 04- A and ^. ft and A- 05. TV and ft. 69. |, i, and i, and T ^, and ADDITION OF FEACTIONS 9O. Principles. 1. Only similar fractions can be added. 2. Dissimilar fractions must be reduced to similar fractions before they can be added. PROBLEMS EXAMPLE 1. Find the sum of |, | and ft. - TV A,*, H- ti 4, A, A- *, i A, A- VS V, H, U- fi> H, W> H- Find the sum of : 1. i, M, TV ^. t, I, i, H- ^. i A, A, H- 4- I, *, T 6 ., A- 5- ft, A, ^ A- FRACTIONS 55 EXAMPLE 2. Find the sum of 3, 7|, 6|, and 5f . = 21. 21 + Iff = 224f . NOTE. Add fractions and integers separately, then add the results. 11. 124, 6if, 15H- # 82^, 1*A, 13H- 12. 123f, 168&, 64^, 65 T V 16. 3^ T , 9H, 4&, ^. IS. 27M, 19H, 45*V, 90/r- ^ 6 A> H 28 A, 63H, SUBTRACTION OF FRACTIONS 91. Principles. 1. Only similar fractions can. be sub- tracted. 2. Dissimilar fractions must be reduced to similar fractions before they can be subtracted, PROBLEMS EXAMPLE 1. Find the value of f - T V |-iV = M-H=*V Find the value of: i- I-T'T- *. A -A. *. !- T \. 5. t-H. 5. n-M- i-i- 5. l-f 7. i-A- ^ - ^. i-f. A-f ' i-i ^. 4 -! ' f-f i*. I -f MULTIPLICATION OF FRACTIONS 96. Principles. 1. Multiplying the numerator multiplies the fraction. 2. Dividing the denominator multiplies the fraction. PROBLEMS Find the product of: 1. \ x 2, or 2 x f . f x 2 = ^y? = f (Principle 1. ) 2. f x 4, or 4 x f. 5. T 9 ^ x 7, or 7 x -fy. 3. T \ x 6, or 6 x T V 6. T 8 T x 4, or 4 x ^ T . 4. T \ x 8, or 8 x T \. 7. |f x 6, or 6 x f |. x 3, or 3 x x 3 = = *' ( Princi P le 2 - ) 9. Jf x 9, or 9 x fj. ff x 12, or 12 x ff. if x 17, or 17 x if. ^. i ^ i 5? T x 3, or 3 x ^ T . j x 21, or 21 x Jf VfXll, or 11 x^ 75. fxf AX 58 MODERN COMMERCIAL ARITHMETIC Mixed numbers may be reduced to improper fractions. M. If x 2| x 3f 24. & x 2ft x 3ft. 23. 25. EXAMPLE 1. Find the product of 12f x 14 or 14 x 12f . OPERATION 14 168 Bf 176| EXPLANATION. Multiply the integral and the fractional parts of the mixed number separately by the whole number, and add the products. 12 X 14 = 168. f X 14 = 8. 168+ 8| = 176. 26. 14fxl8. 27. 13 T 6 T xl4. 28. 15xl2f. 29. 42x7 T \. SO. 62 T 6 T x21. 31. 135ft x 48. 32. 206ft x 56. 33. 89x17^- 34. 156x4 ft. 35. 73x62|. 86. 28 T 6 T x45. 37. 35 T %x39. 38. 64y\x75. 39. 56fxl4. 40. 48fx25. 41. 31xl4|. 42. 44xl6 T \. 43. 39xl2 T V 44- 17x23if. 45. SlxlSff- EXAMPLE 2. Find the product of 12 x 15J. OPERATION 2 t H NOTE. The connecting lines in the diagram 180 =12x15 show the steps in the operation. 7i= |x!5 P\/h \l l ll UX^ 191f = 12i x 46. 16 T Vxl4|. 47. 128 T 3 T x 42|. 48. 169ft x 28ft. 49. 54|x72|. 50. 24|x36|. 51. 180|^xl4f. 52. 1684| x 132| . 53. 426ft x 96 1 . 54. 28ftx26-|. 55. 33/ T xllJ. 56. 7fxl46|. 57. 18|xl20J. FRACTIONS 59 MENTAL PROBLEMS Find the product of : 1. fx7. U. 45 x$. 27. ix-ft- 39. & x & & | x 18. 15. T . ^- X y-g . ^tt>. X ^ X ^. 5. |x64, 21. fxf. 54. AX A- -*. ixfxf. 9. SSx^j. ^. |x|. 35. A* A- ^ 7 - T 3 irx|xf 10. 56 x|. . . fxff 56. Ax|f. 4*. *xfxf. 11. 32 x A- ^- *xA- 57 - 4 X V- ^- |x>xf ^. 20x4. 25. Ixi. 55. T \ x 1. 50. JLxA-xj. O o lies lSJlo ^. 30x3. ^0. AX A. o 1 A o DIVISION OF FEACTIONS 97. Principles. 1. Dividing the numerator divides the fraction. 2. Multiplying the denominator divides the fraction. EXAMPLE 1. Find the quotient of T f -f- 6. OPERATION 12 -j- 6 EXPLANATION. To divide }J by 6 If * 6 = 17 = T S T- Or, is to take i of j j t| x J = ft. But ^ 12 is f inverted, and f is the divisor writ- is. _._ g _ . _ ^2^ = ^. ten as a fraction. Hence if we write 1 ' x 6 the (ji v i sor as a fraction, invert the divisor, and multiply the dividend by the inverted divisor, we divide by the divisor. Then, to divide by a fraction, invert the divisor and multiply. NOTE. Divide when you can, multiply when you must. PROBLEMS Find the quotient of : 1. ||-5. 6. ff-6. 11. Jf* 18 - 2. 3-9. 7. ff -4-4. 12. ^yt + 26. 3. if -16. 8. T \-4. 13. -fi-27. 4. tt + 8. 9. |f + 9. 1. H + 8. 5. ^-6. /#. T^-3. 75. H-3. 60 MODERN COMMERCIAL ARITHMETIC Mixed numbers may be changed to improper fractions. 16. lf-f-3. 18. 2f-3. 20. 4f-f-5. 17. 3H + 15. 19. 5 T V + 17. 21. If + 6. EXAMPLE 2. Find the quotient of 19J -*- 6. OPERATION EXPLANATION. 6 is contained in 19 three times, with a remainder of 1. Change 1 to \ and add it to |, 3^\ making \. \ divided by 6 is / ? . NOTE. Instead of reducing the mixed number to an improper fraction, it is sometimes more convenient to divide as above. 22. 129| + 8. 26. 17| + 7. 30. 57-ft + S. 23. 1384^-5 27. 31 T 6 F + 5. 31. 268 T V + H 24. 21f + 6. 28. 43 T 4 T . 32. 78561 + 7. 25. 3804 + 10. 29. 15f + 9. 33. 12032/ T -15. NOTE. Invert the divisor and multiply. S5. 4 + 14- 39. lA-f. 43. 123 T 8 T -65J. 0/y 164 7/2.4 / ^T Q 4 2 O i . ir -f- Tf. JfJ.. -y -I VV . JfU . O-g- ~^~ 3^ Divide: ^. f V of T 8 T by | of 44. 50. f of | by f of f of if. 47. A of if by || of r V 8 . 51. if of if by \\ of if. 4^. 3| of 2 T V by |f of if 50. 4| of 6 T 8 T by 4| of 5|. 40. 11-fr of 14 T 3 T by 2J of f |. 55. || of H ty 4t of H- MENTAL PROBLEMS Find the quotient of : 1. _? T ^3. ^. 8 + 4. ^. J + f ^. 12 + f. ^. ^f-4-6. ^. 9 + -^. ^. f + f. 50. 7 + |. 3. fi+7. ^. 12 + A- ^- * + A- ^- ** 4. -^ + 5. 14. 15 + f 04. i + f. 54. | + T v 5. 6^. 6> 75. 25 + 2f ^5. f+|. 55. f+f|. ^. T 6 T -15. ^. 18 + |. 0<5. 16 + |. ^- 4 -i 7. T \-14. 17. J+15. 07. | + 9. 57. T \-i- ^. Y^4. -/5. 24 + |. 28. ^ + 3. &?. 4 + 9. 9. if-lO. 70. 30 + f. ^- t + f ^- 15 + 4. 70. 6 + f 00. f + 8. ' 50. f + f . 40. 24 + f . OF THE UNIVERSITY OF FRACTIONS 61 THE THREE PROBLEMS OF FRACTIONS 98. 1. To find a part of a number: What is - of 48? 2. To find what part one number is of another: 6 is what part of 15? 3. To find a number when a part of it is given : f of a num- ber is 12; what is the number? 12 is f of what number? Solution by the Equation Each of the above problems may be stated as an equation. Is means =, of means x. Representing the number to be found by ?, the above problems may be stated thus : I. ? = x48. 2. 6 = ?xl5. 3. f x? = 12; 12 = | x ?. In equations 2 and 3, the product of two numbers and one of the numbers is given to find the other number. PROBLEMS Write each of the following problems as an equation and then solve the equation : 1. Findf of 270. (? = fx270.) 2. 21 is what part of 36? (21 = ?x36.) 8. Of 48, 27 is what part? (27 = ? x 48.) 4. 25 is f of what number? 5. f of a number is 35 ; what is the number? 6. What is T \ of 128? 7. 342 is | of what number? 8. T 6 5- is f of what number? 9. Of T 4 s, -fa is what part? 10. What is |f of 180? II. What part of 86 is 14? 12. What part of 32f is 12? 18. Of 64 days, f is what part? 14. 18| is $ of what number? 15. 24 pounds is what part of 65 pounds? 16. A cow cost 27f dollars, and a horse 78^- dollars. The cost of the cow was what part of the cost of the horse? 62 MODERN COMMERCIAL ARITHMETIC 17. A desk cost 18f dollars, which was T 9 i of the cost of a table. What was the cost of the table? 18. The value of 26 cords of wood at 3f dollars per cord is what part of the value of 35 tons of coal at 4| dollars per ton? 19. A gain of / r of a stock of goods is a gain of what part of of the goods? 20. Different kinds of coffee are mixed in the following parts: 14y 5 F pounds, 18 T \ pounds, 21 T 4 F pounds. Each part is what part of the whole mixture? 21. If 31 gallons of cider make of gallons of jelly, the number of gallons of jelly is what part of the number of gallons of cider? MENTAL REVIEW 99. Solve mentally : 1. Add \ and ^ . 2. From take f 8. Add J, i, t and f 4. Find the product of , -^, , | and |. 5. Change T \ to 84ths. #. Reduce \$% to lowest terms. 7. What will 12 pounds of tea cost at f dollars per pound? 8. What will -& of a ton of hay cost at $18 per ton? 9. Find the cost of 9 ounces of butter at $.20 per pound. 10. Find the cost of 15 eggs at $.14 per dozen. 11. If 1 pound 14 ounces of cheese cost $.23, what is the price per pound? 12. If a horse eats f bushel of oats in a day, how long will 30 bushels last him? IS. Divide 14 by 4? 14. A crock of butter weighs 8 pounds, and the crock alone weighs 1^ pounds. What is the value of the butter at cents per pound? 15. What will 9| cords of wood cost at 4| dollars per cord? 16. At 36 bushels to the acre, what is the yield of 1$ acres? 17. | is what part of f ? FRACTIONS 63 18. What is T \ of 32? 19. 18 is what part of 40? 20. From | take -J. ^^. Find the cost of f of a piece of cloth of 36 yards at 1^ dollars per yard. 22. At If dollars per day of 10 hours, what will a man earn in 7 hours? 23. If 10 bushels of apples will make 32 gallons of cider, how much cider will 7 bushels make? 24. Find the cost of 750 feet of lumber at 9J- dollars per thousand. 25. | is what part of -y~? 26. A lady bought 5f yards of silk for 8| dollars, what was the price per yard? 27. At If dollars per yard, how much cloth can be purchased for $14? 28. 32 is of what number? 29. What part of 64 is 42? 80. | is what part of f ? 31. A grocer mixed 7 ounces of coffee at $. 30 per pound with 9 ounces at $.40 per pound. What is the pound of mixed coffee worth? 82. Divide T \ by 8. 88. Add f and f . 34. A couch is worth $9. For how much must it be sold to gain T y 35. At $4.50 per week, what will five days' board cost? 86. Sold a book for $2.25 and gained f . What did the book cost me? 37. Find the cost of f of a yard of silk at $.60 per yard. 88. 14 is what part of 7? 89. Multiply f by 6. 40. 15 is .75 of what number? 41. Find the cost of .8 pounds of tea at $.45 per pound. J$. Find the cost of f of a ton of coal at $6.50 per ton 43. Change f to a decimal. 44- Change . 625 to a common fraction. 64 MODERN COMMERCIAL ARITHMETIC 45. Bought a wagon for $40 and sold it at a gain of .20 of the cost. Find the selling price. 46. If I wish to gain $.25 on a dollar, how must I mark an article that cost me $4.80? 47. Sold a table for $6 and made a gain of .25 of the pur- chase price. Find the purchase price. 48. Find the cost of 425 pickles at $.30 per hundred. 49. Change -fc to a decimal. 50. | is what part of .40? 51. Change .2 to a common fraction. 52. A man borrows $350, and agrees to pay .06 of the sum for its use. How much should he pay the lender? 53. .08 is what part of .16? 54. At $. 12 apiece, how many brushes can be bought for 4f dollars? 55. In selling combs at $.20, I lost .20 of the cost. What did they cost me? 56. What is .04 of f ? 57. | is what part of .80? 58. Find the cost of 10 ounces of meat at $.12 per pound? 59. A rod 10 feet long is lengthened by .03 of itself. What is its length then? WEITTEN REVIEW 1OO. Solve the following: 1. If a furnace consumes a ton of coal in 9 days, in how many weeks will it consume 9 tons? 2. If I pay $36.40 with wheat worth $f per bushel, how many bushels must I give? 3. If a train goes 146.54 miles in 3 hours, what is the rate of speed per minute? 4- A contributed $5800 to the capital of a company; B, $7800; C, $9600; and D, $5500. What part of the whole did each put in? 5. From a tract of 49-f- acres of land, how many lots of f of an acre each can be laid out? FRACTIONS 65 6. A agreed to keep B's horse 14 weeks for $18. If A keeps the horse 11 days, how much ought B to pay? 7. A field of 29 acres produced 3450 bushels of potatoes. What was the average yield per acre? 8. At $.87 per bushel of 60 pounds, what will 4780 pounds of wheat cost? 9. If apples lose .70 of their weight in drying, how many pounds of apples must be used to make 300 pounds of dried apples? 10. If a bushel of wheat of 60 pounds will make 44 pounds of flour, and 16 pounds of feed, and the miller takes .10 of the grist for grinding, how many bushels of wheat must a cus- tomer take to the mill to get 10 barrels of flour of 195 pounds each? How much feed will he get? UNITED STATES MONEY 101. Money is a measure of value and a medium of exchange. A watch is worth 10 dollars. The dollar is the unit of measure. A piece of cloth is 10 yards long. The yard is the unit of measure. A butcher wants a hat worth 3 dollars, a ring worth 5 dol- lars, and a book worth 3 dollars. Can he take 3 dollars' worth of meat to the hatter, 5 dollars' worth to the jeweler, and 3 dollars' worth to the bookseller and exchange for the things he wants? Why? What can he do that he may practically ex- change his meat for these things? Why does the butcher exchange his meat for money if they both have the same value? Money is the medium by which the butcher makes the exchange. 102. The unit of United States money is the dollar. The first Congress of the United States made the dollar the unit of value. It determined the value of the dollar by pro- viding for the coinage of silver dollars to contain 371.25 grains of pure silver (with certain alloy) and of gold pieces to contain 24.75 grains of pure gold (with certain alloy) to the dollar. 66 MODERN COMMERCIAL ARITHMETIC The value of the coins was determined by the amount of metal they contained, and by the value trade and custom gave them. The coin determined the value of the dollar, the dollar did not determine the value of the coin. 103. Ratio. By the first coinage law the weight of a silver dollar was 15 times the weight of a gold dollar. The ratio of weight then was 15 to 1. In 1836, Congress passed a bill mak- ing the coinage ratio 16 to 1, so that since then a silver dollar weighs 16 times as much as a gold dollar. 104. The denominations and scale of United States money are shown by the following TABLE 10 mills = 1 cent (0, c., or ct.). 10 cents = 1 dime. 10 dimes = 1 dollar ($). 10 dollars = 1 eagle. United States money is based on the decimal scale. It is expressed as dollars, cents, and mills. The terms dime and eagle are not commonly used. Dollars are written as integers, cents as hundredths, and mills as thousandths. The sign ($) is prefixed to expressions of United States money; as, $7, $.07, $.007. Cents and mills are sometimes written as common fractions ; as, $12.25, $12 r V3^, 2, 25, 250. ;>. 56 by 2^, 25, 250, li, m, 125. 6\ 486 by If, 16*, 166*, 8*, 83*. 7. 126 by li, 12*, 125, 2*, 25, 250. 8. 156 by '-8*, 83*, -2*, 25, 250, 33*. 9. 256 by 6*, 62, 8*, 83*, 25, 333*. ALIQUOT PARTS 69 10. $26.40 by 50, 33, 25, 16|, 11. $1.84 by 33, 250, 16f, 125, 83, 6. 72. $3.36 by 5, 3, 2, 33J, 25, If, m, !, 16f, 125, 8J, 62^, 83, 6J, 50, 333, 250, 166|. IS. $7.68 by If, 250, 16f, 25, 166f, 2|, 1^, 333, 12|, 33$, 125, 3i, 8i, 6f, 83^, 6. 74. $10.56 by 62|, 6J, 83^, 125, 8J, 12|, IGf . 15. $50, $.33^, $.02^, $.16f, $62.50, $1.25 by 576. 112. Multiplication Table 2 o 6f 101 _L /v"5" 16f 25 33* 3 H 10 18f 25 37i 50 4 10 13* 25 33* 50 66f 5 12f 16f 31i 41* /<) 1 U/v'2' saj 6 15 20 37| 50 75 100 7 17* 4 23* 43f 58i 87* 116f 8 20 2tff 50 66| 100 133* 9 22| 30 ^56i 75 112^ 150 10 25 33^ 62| 83i 125 166f NOTE 1. This table can bo easily learned and will prove conve- nient in mental operations. NOTE 2. Black figures indicate multipliers and multiplicands. Intersecting points of horizontal and vertical columns give the prod- ucts. MENTAL DRILL 1. Multiply each multiplicand by its corresponding multi- plier. 2. Multiply each multiplicand by each of all the multipliers. 3. Multiply each of all the multiplicands by each multiplier. 70 MODERN COMMERCIAL ARITHMETIC MULTIPLICANDS MULTIPLIERS MULTIPLICANDS MULTIPLIERS $ 6.72 .05 $ .05 48 7.20 .02| .03J 96 7.68 .03 .33 144 8.16 .83J .02| 192 8.64 2.50 .25 240 9.12 .16f 2.50 288 9.60 .25 .16$ 336 10.08 .62 1.66| 384 10.56 1.25 .12 432 11.04 .33J 1.25 528 12.96 1.66f .83^ 576 13.44 .12^ .62| 524 NOTE. The product will be the same whichever factor is used as the multiplier. DIVISION WITH ALiaUOT PARTS 113. EXAMPLE. Divide 245 by 3, 33, and 333 respect, ively. OPERATION EXPLANATION. 10 is 3 times 3J. 245 -*- 3 245 -f- 3 = 73. 5 equals as many times 3 as there are 10's in 245, 245 -^ 33 = 7.35 or 24. 5. 24 5 X 3 == 73. 5. In like manner, 100 245 - 333 = . 735 is 3 times 33J, and 1000 is 3 times 333^. NOTE. To divide by an aliquot part of 10, 100, or 1000, multiply the dividend by the number that shows what aliquot part the divisor is of 10, 100, or 1000, and point off 1, 2, or 3 places, as the divisor is part of 10, 100, or 1000. MENTAL PROBLEMS 1. Divide all the dividends by each divisor. 2. Divide each dividend by all the divisors. DIVIDEND DIVISOR DIVIDEND DIVISOR DIVIDEND DIVISOR .76 125 1.82 8 4.25 83J 7.63 6i 2.57 7.82 5 87.64 .83 5.47 50 32.45 1.25 .82 3* 17.63 .62-J- 1.26 33 18.24 1.66| 3.14 Q 1 "2" 5.67 2.50 ALIQUOT PARTS 71 DIVIDEND DIVISOR DIVIDEND DIVISOR DIVIDEND DIVISOR .91 25 4.39 i Q i -*- & 2" 14.75 .33* .85 250 35.87 .06^ 7.53 .50 2.13 16} 42.64 .25 12.15 .02* 4.16 166} 3.65 .08* 8.24 .03J .53 1 O_L J./V* 16.52 .16| 124.16 .05 PEJCE, COST, AND QUANTITY 114. Business computations deal with price, cost, and quantity. Solution by the equation: Let P = price, C = cost, Q = quantity. Then, P x Q = C. .Hence, C + Q = P, and C -*- P = Q. MENTAL PROBLEMS 1. If the price is 8 cents, and the quantity 12, what is the cost? 2. If 16 articles cost 48 cents, what is the price? 3. If the price is 3 cents and the cost 36 cents, what is the quantity? Find the term not given: PRICE QUANTITY COST PRICE QUANTITY COST 4. $.08 15 ? 10. $.12* 16 ? 5. .11 ? $1.32 11. ? 21 $1.05 6. .12 10 ? 12. .25 12 ? 7. .06 ? 2.40 IS. ? 14 .98 8. ? 15 6.00 14. .11 ? 1.65 9. ? 8 1.60 15. .08 ? 4.00 EXERCISES Tell how to find the term not given . 1. P (25#), Q. 7. C, P. 2. P (62*#), C. 8. C, Q (33*). 3. C, Q (12*). 9. C, P (83*$. 4. C, P ($16}). 10. Q, P ($2.25). 5. Q(112*), P. 11. P, Q (166}). 6. P ($1.16}), Q. 12. C, Q(62*). 72 MODERN COMMERCIAL ARITHMETIC 18. P (250), C. 82. Q, P (500). 14. C, P ($6.25). 83. P ($83 J), 0. 15. Q, P ($2.125). &. Q (6), C. 16. Q (83*), P. 85. Q, P ($2.50). 17. Q (25), C. 86. Q (75), P ($2.33J). 18. C, P (66f0). 57. Q, P (62i#). ^. Q (33 *), C. *. Q (83*), C. 20. P (750), Q. 89. C ($1.66f), P (30). &/. Q (75), P. 40. C, Q (25). 0. C, P (750). ,41. C, P (8J0). S. Q (75), C. 42. P, Q (2.62|). 24. Q, P ($1.83J0). ?. Q (8J), 0. 5. Q (50), P. . Q (12|), P ($25). 26. C, Q (6J). #T. P (2^0), Q. 27. C, P (.1250). 4^. C, P (*1.08i). 28. C ($65.13), P. 47. P, Q (116f). 29. Q (75), P (250). 48. Q (66f), 0. SO. Q, (62|), P ($1.86). 49. Q, P (6#). W. P (460), C ($16.89). BO. Q (8*), P. NOTE. To multiply by 2.33J, multiply by 2 and by .33J separately and add the products. Treat 1.16, 1.12J, 2.83J, 2.62J, etc., in a sin ilar manner. Articles Bought by 100 (C) or 1000 (M) 115. Q -5- 100 = Q in hundreds. (Point off two places.) Q -*- 1000 = Q in thousands. (Point off three places.) P per 100 x Q in hundreds = Cost. P per 1000 x Q in thousands = Cost. EXAMPLE 1. Find the cost of 384 laths at $.33 per 0. FORMULA SOLUTION PperlOOxQ 384x$.33j ___ =C ost. - I55 - fcl.28. EXAMPLE 2. Find the cost of 2415 laths at $3.25 per M. FORMULA SOLUTION P per 1000 x Q n 2415 x $3.25 ,. Q _ - - PRICE, COST, AND QUANTITY 73 PROBLEMS Find the cost of: 1. 781 brick at 85 cenbs per C. 2. 2107 feet pine at $18.50 per M, 3. 6385 feet hemlock at $14.60 per M. 4. 1343 posts at $12.25 per C. 5. 1560 pineapples at $8J per C. 6. 2752 pounds coal at 25^ per C. 7. 687 feet oak at $32 per M. 8. 3250 shingles at $3.33 per M. 9. 964 pounds beef at $6.25^ per C. 10. 4738 feet timber at $23.50 per M. Articles Bought by the Ton 116. Price per ton -*- 2 = price per 1000 pounds. Price per 1000 pounds x Q + 1000 = Cost. EXAMPLE. Find the cost of 2685 pounds of hay at $12 per ton. FORMULA SOLUTION P per ton x Q $12x2685 2 x 1000 C 8t - 2x1000 PROBLEMS Find the cost of: 1. 6842 pounds of coal at $5.20 per ton. 2. 4975 pounds steel at $33.33 per ton. 3. 2360 pounds sugar at $83 per ton. 4- 15837 pounds old iron at $6.25 per ton. 5. 6974 pounds salt at $5.75 per ton. 6. 3798 pounds; price per ton, $6.90. 7. 8790 pounds; price per ton, $12.50. 8. 350 pounds; price per ton, $9.60. Articles Bought by the Bushel 117. EXAMPLE. F*nd the cost of 2100 pounds of wheat at 70 cents per bushel of 60 pounds. FORMULA pounds x P per bushel n SOLUTION pound per bushel S ' 2100 + 60 x 70^ = $24. 50. NOTE. Use cancellation. 74 MODERN COMMERCIAL ARITHMETIC TABLE OF BUSHEL WEIGHTS POUNDS POUNDS POUNDS Apples 56 Corn (shelled). . . 56 Potatoes 60 Barley 48 Corn (ear) 70 Eye 56 Beans 60 Flaxseed 56 Timothy seed, . .45 Buckwheat 48 Oats 32 Turnips 56 Clover seed 60 Onions 57 Wheat 60 NOTE. These weights are used in most of the States. PROBLEMS Find the cost of a load of : 1. Oats, weighing 2146 pounds, at 350 per bushel. 2. Potatoes, weighing 3257 pounds, at 480 per bushel. 3. Apples, weighing 2980 pounds, at 220 per bushel. 4. Turnips, weighing 3425 pounds, at 480 per bushel. 5. Barley, weighing 4160 pounds, at 360 per bushel. 6. Beans, weighing 3290 pounds, at $1.85 per bushel. 7. Buckwheat, weighing 1846 pounds, at 580 per bushel. . 8. Corn, weighing 2163 pounds, at 650 per bushel. 9. Flaxseed, weighing 3375 pounds, at 400 per bushel. 10. Onions, weighing 1956 pounds, at 400 per bushel. 11. Rye, weighing 2742 pounds, at $2.50 per bushel. 12. Wheat, weighing 3094 pounds, at 750 per bushel. IS. Find the total value of the following produce: 3 loads of wheat weighing 3122, 2659, and 3380 pounds respectively, at 850 per bushel; 1 load of barley, weighing 2755 pounds, at 720 per bushel; 4 loads of potatoes, weighing 3062, 2587, 3420, and 2970 pounds respectively, at 420 per bushel ; 2 loads of beans, weighing 3160 pounds each, at $2.12-^ per bushel; and 1 load of apples, weighing 2875 pounds, at 480 per hundred pounds. NOTE. Coal is sold by the ton, and by the bushel of 80 Ibs. 14. Find the cost of a load of coal, weighing 2260 pounds, at 22 cents per bushel of 80 pounds. What is the equivalent price per ton? FRACTIONS 75 15. Find the cost of 2493 pounds of coal at $5.75 per ton. What is the equivalent price per bushel? 16. A wagon loaded with potatoes weighs 4750 pounds, and the wagon alone weighs 1426 pounds. What are the potatoes m worth at 33$ per bushel? 17. A cart loaded with coal weighs 3492 pounds, and the cart weighs 1280 pounds. Find the cost of the coal at 25$ per bushel. Find the equivalent price per ton. 18. A wagon loaded with 28 bags of wheat weighs 4960 pounds, the wagon weighs 1420 pounds, and each bag weighs 2 pounds. Find the value of the wheat at 72$ per bushel. 19. Find the cost of 20 bags of beans, weighing 118 pounds each, at $1.85 per bushel. 20. What is the cost of three loads of turnips, weighing 2240, 2875, and 2680 pounds respectively, at 28$ per bushel? 21. How many pounds of shelled corn, at 48$ per bushel, can be bought for $22.50? 22. Find the cost of a load of drying apples, weighing 3120 pounds, at 18$ per bushel. 23. A wagon loaded with onions weighs 5570 pounds, the wagon and crates weigh 1808 pounds. Find the cost of the onions at 45$ per bushel. 24. A man bought a load of oats for $21.35. If the oats weighed 1952 pounds, what price did he pay per bushel? 25. A wagon loaded with 30 bags of beans weighs 5870 pounds, the wagon weighs 1842 pounds, and each bag weighs 2 pounds. Find the cost of the beans at $1.25 per bushel. DENOMINATE NUMBERS DEFINITIONS 118. Some things are counted; as, dollars, eggs, tickets. Some things are measured; as, time, area, volume, weight. 119. When the name of the objects counted or measured is used with the expressed number (6 apples, 3 feet), the num- ber is a Concrete dumber. 120. A Unit of Measure is any standard by which we determine the number or the amount of anything. A dozen, a bushel, a pound are units of measure. A shovelful, a boxful, a dipperful are also units of measure. The dozen, bushel, and pound are definite and established units of measure. The shovelful is not a definite or established unit of measure. Some units of measure were established by law (pound, yard, gallon). Some were established by custom (hour, dozen, degree). 121. A number whose unit of measure is established by law or custom is a Denominate Number. 122. A number that expresses units of measure of the same kind is a Simple Denominate Number (7 pounds) . 123. A number that expresses units of measure of similar kind but of different values is a Compound Denominate Num- ber (7 pounds, 6 ounces). MEASURES OF EXTENSION 124. Extension means length, length and breadth, or length, breadth and thickness. This leaf is a volume, or solid; it has length, breadth, and thickness. This page is a surface; it has length and breadth only. The edge of the page is a line ; it has length only. 76 DENOMINATE NUMBERS 77 125. Linear Measure TABLE 12 inches (in.) = 1 foot. ft. 3 feet = 1 yard. yd. 5 yards or 16 feet = 1 rod. rd. 320 rods = 1 mile. mi. mi. rd. yd. ft. in. 1 = 320 = 1760 = 5280 = 63360. Scale. 320, 5|, 3, 12. In measuring cloth, the yard is divided into quarters. Yards and quarters are sometimes written thus: 12 2 (12f), 15 3 Square Measure 26. TABLE 144 square inches (sq. in.) = 1 square foot. sq. ft. 9 square feet = 1 square yard. sq. yd. 3()i square yards = 1 square rod. sq. rd. 160 square rods = 1 acre. A. 640 acres = 1 square mile. sq. mi. Scale. 640, 160, 30|, 9, 144. A square 1 in. on a side is an inch square; a square inch. A square 1 ft. on a side is a foot square; a square foot. A square 1 yd. on a side is a yard square; a square yard. A square inch is equivalent to an inch square. A square foot is equivalent to a foot square. A square yard is equivalent to a yard square. NOTE. The units of square measure need not be squares. A square foot may be either round or oblong. It is called a square foot because it was derived from, and is equivalent to, a foot square. A square foot measures as much as a foot square. The number of units of square measure in a surface is its Area. 78 MODERN COMMERCIAL ARITHMETIC 127. Cubic Measure TABLE 1.728 cubic inches (cu. in.) = 1 cubic foot. cu. ft. 27 cubic feet = 1 cubic yard. cu. yd. 128 cubic feet = 1 cord. C. Scale. -27, 1728. The units of cubic measure are the cubic inch, cubic foot, and cubic yard. These units need not be cubes. A cubic inch is equivalent to a cube an inch on each edge. A cubic foot is equivalent to a cube a foot on each edge. A cubic yard is equivalent to a cube a yard on each edge. The number of units of cubic measure a solid contains is its Solid Contents or Volume. 3 FT 128. Surveyors' Linear Measure TABLE 7.92 inches (in.) 25 links 4 rods or 100 links 80 chains Scale. SO, 4, 25, 7.92. Surveyors' Square Measure 129. United States government land when surveyed is divided into townships tracts of land 6 miles square. A town- ship is divided into 36 equal squares, square miles. Each square mile is called a section. A section is divided into half- sections, quarter-sections and quarter quarter-sections* DENOMINATE NUMBEBS 79 625 square links (sq. 16 square rods 10 square chains 640 acres TABLE 1.) = 1 square rod. sq. rd. = 1 square chain, sq. ch. = 1 acre. A. = 1 square mile. sq. mi. Scale. GIO, 10, 16, 625. MEASURES OF CAPACITY 13O. Liquid Measure TABLE 4 gills (gi.) = 1 pint. pt. 2 pints 4 quarts gal. 1 = 131. = 1 quart. qt. = 1 gallon. gal. qt. pt. gi. 4 = 8 = 32 Scale. 4, 2, 4. Dry Measure 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 Scale. 4, 8, 2. MEASURES OF WEIGHT Avoirdupois Weight Avoirdupois weight is used for ordinary purposes. TABLE 16 ounces (oz.) =1 pound. Ib. 100 pounds =1 hundredweight, cwt. 20 hundred-weight=l ton. T. T. cwt. Ib. oz. 1 = 20 = 2000 = 32000 Scale. 20, 100, 16. 80 MODERN COMMERCIAL ARITHMETIC Troy Weight 133* Troy weight is used by jewelers. Ib. 1 TABLE 24 grains (gr.) = 1 pennyweight, pwt. 20 pennyweights = 1 ounce. oz. 12 ounces = 1 pound. Ib. oz. - 12 = pwt. 240 = 5760 Scale. 12, 20, 24. Apothecaries' Weight 134. Apothecaries' weight is used by druggists. TABLE 20 grains (gr.) = 1 scruple, sc., or 9 3 scruples = 1 dram. dr., or 3 8 drams = 1 ounce. oz., or 5 12 ounces = 1 pound. Ib., or Ib Ib. oz. dr. sc. gr. 1 = 12 = 96 = 288 = 5760 Scale. l^ 8, 3, 20. MEASURES OF TIME 135. 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 year. yr. 366 days = 1 leap year. 1. yr. 100 years = 1 century. cen. pr. mo. da. hr. min. 1 = 12 - 365 = 8760 = 525600 = Soak. 366, 24, 60, 60. sec. 31536000 DENOMINATE NUMBERS 81 136. The earth rotates from west to east. The Day meas- ures the time of one complete rotation of the earth on its axis. 137. A straight north and south line passing through both poles and through any point on the earth's surface is the meridian of that point. 138. It is noon at a place when the meridian of the place is under the direct rays of the sun. 139. A.M. (ante-meridian) denotes the half -day before noon. P.M. (post-meridian) denotes the half-day after noon. 140. In astronomy, the day begins at noon; in business, it begins at midnight. 141. The earth revolves around the sun in equal periods of time. The Year measures the time of one revolution of the earth around the sun. 142. A year consists of 365 da. 5 hr. 48 min. 49.7 sec. The Common Year is 365 da. The Leap Year is 366 days. 143. If, to make up for the 5 hr. 48 min. 49.7 sec. dropped from each common year, we add one day to each fourth year, we would add 44 min. 41.2 sec. too much. In 100 years we would add 18 hr. 37 min. 10 sec. too much. If we omit adding a day every 100 years, we would lose 5 hr. 22 min. 50 sec. In 400 years we would lose 21 hr. 31 min. 20 sec. If, then, we add one day for each 400 years, we will gain 2 hr. 28 min. 40 sec. ; and in 4000 years we would gain 24 hr. 46 min. 40 sec. So we omit adding a day once in 4000 years. Rule for Leap Year. Century years divisible by 400, and other years divisible by four, are leap years, except the year 4000. 144. The day added to leap year becomes the 29th of February. 145. In business, 30 days are usually considered a month, and 12 months a year. 146. The common year contains 52 weeks and 1 day; the leap year 52 weeks and 2 days. Each year begins one day later . in the week than the preceding year, except the year following leap year, which begins two days later in the week. 82 MODEKN COMMERCIAL ARITHMETIC 147. Days in the Months. February has 28 days, except in leap year, when it has 29 days. Thirty days hath September, April, June, and November; All the rest have thirty -one, Excepting February alone, Which has four and twenty-four, Till leap year gives it one day more. MEASURES OF ANGLES AND AECS 148. A Circumference is the bounding line of a circle. An Arc is any part of a circumference. 149. sl^ of any circumference is a Degree of the circum- ference. If the space about a point be divided into 360 equal parts or angles, by straight lines meet- ing at the point, each angle is an jfg^ angle of 1 degree. TABLE 60 seconds (")= 1 minute. (') 60 minutes = 1 degree. () 360 degrees = 1 circumference, cir. Scale. 360, 60, 60. The length of a degree of longitude at the equator is nearly 70 miles. MEASURES OF VALUE Canada Money 150. The table of the currency of Canada is the same as that of the United States (see p. 66), although English money also is used. English or Sterling Money 151. The unit of English money is the Pound or Sovereign. TABLE 4 farthings (far. ) = 1 penny (d. ) = 12 pence = 1 shilling (s.) = shillings = 1 pound () = $4. 8665 Scale. -20, 12, 4. DENOMINATE NUMBERS 83 French Money TABLE 10 centimes (ct.) = 1 decime (dc.) = fO declines = 1 franc (fr.) = 153. German Money TABLE 100 pfennigs = 1 mark 154. COUNTING TABLE 12 things = 1 dozen. doz. 12 dozen = 1 gross. gr. 12 gross = 1 great gross. Gr. gr. REDUCTION OF DENOMINATE NUMBERS 155* The process of changing a number expressed in one denomination to an equivalent expressed in another denomina- tion is called Keduction. Change 8 dimes to cents; 3 dollars to dimes; 3 dollars and 8 dimes to cents. 156. Changing a number from a higher to a lower denom- ination is Eeduction Descending. 1. Change 2 ft. to inches; 2 yd. to feet; 2 yd and 2 ft. to inches. 2. Change 2 qt. to pints; 2 pk. to quarts; 2 pk. 2 qt. to pints; 2 bu. to pecks; 2 bu. 2 pk. 2 qt. to pints. 3. Change 4 bu. 3 pk. to quarts; 1 pk. to pints. 4. Change 2 hr. to seconds; 2 da. to minutes. NOTE.- Reduction descending is performed by multiplication. 157. Changing a denominate number from a lower to a higher denomination is Eeduction Ascending. 1. Change 60 cents to dimes; 1200 cents to dollars; 50 dimes to dollars; 287 cents to dimes and cents; 365 cents to dollars, dimes and cents. 84 MODERN COMMERCIAL ARITHMETIC 2. Change 36 in. to feet; 67 in. to feet and inches; 98 in. to yards, feet, and inches. 3. Change 64 qt. to pecks, then to bushels; 42 qt. to bushels, pecks, and quarts; 89 pt. to higher denominations. NOTE. Reduction ascending is performed by division. 158. Under several of the denominate tables is a line of equivalents. Tell how one equivalent is found from another, the highest denomination from the lowest, the lowest from the highest. 159. Principles. 1. To perform reduction descending, multiply by the numbers in the scale from the given to the required denomination. 2. To perform reduction ascending, divide by the numbers in the scale from the given to the required denomination. Model Solutions 160. EXAMPLE 1. Reduce 12 yd. 2 ft. 9 in. to inches. OPERATION 12yd. 3 ft. (multiply) 3 t NOTE. The product is of the same denom- 2 ft. (add) ination as the multiplicand. The multiplier 3g ft must be an abstract number, so 12 in the first 12 in* (multiply) case an(i ^ * n the second are considered ab- stract numbers and the multipliers. 456 in. 9 in. (add) 465 in. EXAMPLE 2. Eeduce 892 in. to higher denominations. OPERATION Divide 12 in. | 892 in. Divide 3 ft. 74 ft. 4 in. remainder Divide 5| yd. j 24 yd. 2 ft. remainder 4 rd. 2 yd. remainder .-. 892 in. = 4 rd. 2 yd. 2 ft. 4 in. NOTE. The divisors are considered abstract numbers. DENOMINATE NUMBERS 86 EXAMPLE 3. Eeduce f ft. to the fraction of a rod. OPERATION EXPLANATION. Divide | by 3 and the f ft. x J x T 2 T = -fa ft. quotient by o. EXAMPLE 4. Eeduce .8 pt. to the decimal of a bushel. OPERATION P ' I * E_J EXPLANATION. Divide .8 pt. by 2, the quo- " *$ \ -^ V- tient by 8 and this quotient by 4, as abstract 4 pk. | .05 pk. numbers. .0125 bu. EXAMPLE 5. Eeduce rd. to integers of lower denom- inations. OPERATION i (rd.) x 5i = 2 yd. f (yd.) x 3 = 2i ft. | (ft.) x 12 = 3 in. .-. ird. =2 yd. 2ft. Sin. EXAMPLE 6. Eeduce .7 bu. to integers of lower denomina- tions. OPERATION .7 bu. (multiplier) NOTE. Since the decimal is to be _ P*F; reduced to integers, multiply only by the 2.8 pk. (multiplier) decimal part of the multiplier. The 8 qt. product is of the same denomination as 6.4 qt. (multiplier) the multiplicand, but when the product 2 pt. is used again as a multiplier it is consid- ~^ pj. ered as an abstract number. .7 bu. =2pk. 6 qt. .8 pt. PROBLEMS Eeduce to higher denominations: 1. 4256 in. 8. 12863 sq. in. 15. 6952 sq. in. 2. 86579 in. 9. 6871 sc. 16. 68754 min. 3. 684 pt. dry. 10. 9478 1. 17. 61453 gr. Troy. I*. 1272 gi. -11. 42735 in. 18. 567389 cu. in. 5. 1298 gr. Troy. 12. 6853 gi. 19. 1593 pt. dry. 6. 15652 gr. apoth -IS. 735 sc. 20. 11268 pwt. 7. 489754 sec. 14. 627841. 86 MODERN COMMERCIAL ARITHMETIC Reduce to the lowest denomination given : 21. 8 rd. 2 yd. 1 ft. 4 in. 22. 3 mi. 80 rd. 4 yd. 2 ft. 28. 2 Ib. 10 oz. 16 gr. Troy. 24. 4 hr. 17 min. 40 sec. 25. 12 Ib. 6 dr. 2 sc. 26. 16 rd. 12 ft. 7 in. 27. 17 Ib. 2 pwt. 28. 2 da. 26 min. 29. 4 Ib. 7 oz. 1 sc. 80. 16 20". 31. 20 15' 20". #. 5 gal. 3 qt. 1 pt. 3 gi. 88. 14. bu. 3 pk. 5 qt. 1 pt. 34. 3 cu. yd. 15 cu. ft. 806 cu. in. 35. 20 gal. 2 qt. 1 pt. 86. 16 gal. 1 pt. 2 gi. 87. 2 bu. 2 pk. 1 pt. 88. 5 cu. yd. 12 cu. ft. 39. 14 gal. 2 qt. 2 gi. 40. 7 Ib. 3 oz. 2 dr. Reduce to lower denominations : 41. frd. 45. |da. 42. J- mi. 40. .76 Ib. apoth. 43. .86 rd. -47. T \ cu. yd. 44- t Ib. Troy. 48. | bu. -40. 50. .94 gal. ^ sq. yd. } gal. .55 sq. rd. Reduce to a fraction of the highest denomination : 53. I pt. dry. 54. .62 gi. 55. I yd. 56. Jdr. 57. | pwt. 58. .56 cu. ft. 59. J 1. 00. .85 gr. Troy. 61. Jgi. 00. i rd. 05. .96 pwt. 0-4. .35 dr. EXAMPLE 7. Reduce 2 pk. 6 qt. 1 pt. to the decimal of a bushel. OPERATION 2 | 1 pt. 8 | 6.5 qt. 4| 2.8125pk. .703125 bu. Or, 2 pk. 6 qt. 1 pt. = 45 pt. 1 bu. = 64 pt. 45 p t. + 64 = .703125 pt. EXPLANATION. Reduce 1 pt. to quarts and annex the result to 6 qt. Reduce quarts to pecks and annex the result to 2 pk. Reduce pecks to bushels. Reduce to decimals of the highest denomination : 65. 4 oz. 12 pwt. 16 gr. 68. 12s. lOd. 66. 12 hr. 40 min. 30 sec. 69.. 2 pk. 6 qt. 1 pt. 67. 2 qt. 1 pt. 2 gi. 70 14s. 6d. 2 far. DENOMINATE NUMBERS 8? REDUCTION OF ENGLISH MONEY 161. EXAMPLE. Reduce 12 sov. 12s. 6d. to United States money. EXPLANATION. Call each OPERATION ..... 12 SOY. 12s. 6d. = 12.624 sov. f llhn ' 5 ' a <***- * & A Q- duce pence to farthings and 1 SOV. = 4.ODDD ,, ,, ,, ., -,,, 10 CM &A Qt*fK EXAMPLE 5. Multiply 2 dollars 5 dimes 8 cents by 7. OPERATION 258 $25 dimes 8 cents 7 7 18 6 $18 dimes 6 cents EXAMPLE 6. Multiply 4 gal. 3 qt. 1 pt. 2 gi. by 6. OPERATION 4 gal. 3 qt. 1 pt. 2 gi. 6 29 gal. 2 qt. 1 pt. gi. EXAMPLE 7. Divide 8 dollars 5 dimes 5 cents by 3. OPERATION 3)855 3) $8 5 dimes 5 cents 285 $2 8 dimes 5 cents DENOMINATE KUMBERS EXAMPLE 8.- Divide 5 bu. 1 pk. 7 qt. by 4. OPERATION 4)5 bu. 1 pk. 7 qt. 1 bu. 1 pk. 3 qt. H pt. PROBLEMS 1. Add 2 Ib. 6 oz. 16 pwt. 18 gr., 8 Ib. 11 oz. 17 pwt. 21 gr., 14 Ib. 9 oz. 12 pwt. 20 gr. 2. From 1 mi. 60 rd. 4 yd. 1 ft. 4 in. take 130 rd. 2 yd. 2 ft. 9 in. 8. Multiply 12 gal. 3 qt. 1 pt. 3 gi. by 8. 4. Add 2 cu. yd. 16 cu. ft. 987 cu. in., 8 cu. yd. 20 cu. ft. 1265 cu. in. 5. Divide 7 Ib. 10 oz. 12 pwt. 16 gr. by 4. 6. Divide 20 bu. 4 qt. by 2 bu. 3 pk. 4 qt. NOTE. Reduce both expressions to quarts, then divide. 7. Divide 48 gal. 1 qt. 3 gi. by 3 gal. 2 qt. 1 pt. 3 gi. 8. Multiply 6 bu. 3 pk. 5 qt. by 14. 9. From 12 da. 16 hr. 30 miri. 14 sec. take 9 da. 20 hr. 50 sec. 10. Add 1 mi. 165 rd. 2 yd. 8 in., 3 mi. 120 rd. 4 yd. 2 ft. 11. Divide 12 bu. 3 pk. 7 qt. into 8 equal parts. 12. How many jugs holding 1 gal. 1 qt. 1 pt. each can be filled from a barrel containing 54 gal. 2 qt. 1 pt. 18. Add 3 Ib. 6 oz. 7 dr. 2 sc. 16 gr., 4 dr. 2 sc. 12 gr., 1 Ib. 9 oz. 1 sc. 8 gr., 11 oz. 2 sc. 18 gr., 1 Ib. 8 oz. 5 dr. 14. Multiply 3 da. 12 hr. 30 min. 14 sec. by 12. 15. Subtract 1 mi. 190 rd. 4 yd. 8 in. from 2 mi. 60 rd. 2 yd. 2 ft. 9 in. 16. How many times is 2 dr. 2 sc. 12 gr. contained in 1 Ib. 2 oz. 5 dr. 1 sc. 12 gr. 17. Add 6 gal. 2 qt. 1 pt. 3 gi., 12 gal. 3 qt. 2 gi., 17 gal. 1 qt. 1 pt. 3 gi. 18. Multiply 3 mi. 80 rd. 5 yd. 2 ft. 8 in. by 16. 19. Subtract 1 yr. 6 mo. 15 da. 9 hr. 45 min. 28 sec. from 2 yr. 3 mo. 12 da. 4 hr. 20 min. 90 MODERH COMMERCIAL ARITHMETIC 20. Multiply 6 Ib. 9 oz. 12 pwt. 16 gr. by 13. 21. Add 31 mi. 65 rd. 3 yd. 1 ft., 196 rd. 2 yd. 2 ft. 9 in., 3 mi. 145 rd. 5 yd. 1 ft. 8 in., 7 mi. 98 rd. 4 yd. 2 ft. 10 in. 22. Add 60 A. 90 sq. rd. 20 sq. yd. 5 sq. ft., 12 A. 120 sq. rd. 20 sq. yd. 8 sq. ft., 16 A. 80 sq. rd. 16 sq. yd. 5 sq. ft. 28. Subtract 3 gal. 3 qt. 1 pt. 2 gi. from 8 gal. 1 qt. 3 gi. 24. Multiply 7 bu. 3 pk. 6 qt. 1 pt. by 21. 25. Add 1 yr. 6 mo. 13 da. 12 hr. 40 min., 2 yr. 7 mo. 15 da. 13 hr. 30 min., 9 mo. 25 da. 17 hr. 48 min., 8 mo. 24 da. 14 hr. SUBTRACTION OF DATES 163. EXAMPLE. Find the difference in time between April 6, 1900, and Nov. 11, 1899. OPERATION EXPLANATION. Write the time as years, months yr. mo. da. and days. April is the fourth month and is written 1900 4 6 as 4 mo. November is the eleventh month and is 1899 11 11 written as 11 mo. One month is also called 30 days. 4 25 As the number of days in a month varies, this method, called compound subtraction, may not give the exact number of days' difference. Thus, counting 30 days to a month, the difference between the above dates as shown in the opera- tion is 145 days, but the true difference is 146 days. To find the exact number of days, count the number of days by months from the first date to the second. Thus, 19 (days in November after November 11), 31 (December), 31 (January), 28 (February), 31 (March), and 6 (days counted in April) are 146. PROBLEMS Find the difference in time between the following dates, by compound subtraction: 1. March 4, 1899, and Dec. 11, 1901. 2. June 21, 1899, and April 2, 1903. 8. Aug. 30, 1897, and May 15, 1899. 4. Sept. 26, 1895, and Nov. 12, 1897. 5. Oct. 20, 1900, and Feb. 6, 1904. DENOMINATE NUMBERS 91 Find the actual difference in days between : 6. Jan. 7, 1900, and July 12, 1900. 7. Oct. 16, 1900, and Jan. 17, 1901. 8. Dec. 6, 1901, and March 14, 1902. 9. May 4, 1902, and Oct. 30, 1902. 10. Sept. 23, 1900, and Feb. 26, 1901. COMPARISON OF WEIGHTS AND MEASURES 164. TABLE TROY APOTHECARIES' AVOIRDUPOIS 1 lb. = 5760 gr. = 5760 gr. = 7000 gr. 1 oz. = 480 gr. = 480 gr. = 437.5 gr. 1 bu. (32 qt.) = 2150.42 cu. in. 1 gal. = 231 cu. in. 1 qt. (dry) = 67 cu. in. 1 qt. (liquid) = 57| cu. in. 1 cu. ft. water = 62^- lb. avoir. 1 gal. water = 8$ lb. avoir. Which is heavier, a pound Troy or a pound avoirdupois? an ounce Troy or an ounce avoirdupois? Which is larger, a quart dry measure or a quart liquid measure? NOTE. Large fruits, vegetables, coal, etc., are measured by the heaped bushel, or the bushel of 40 quarts. PROBLEMS 1. How many liquid quarts in a bushel? 2. Change 1 lb. Troy to the fraction of a pound avoirdupois. 8. Change 8 lb. 12 oz. avoirdupois to apothecaries' weight. 4. Change 1 lb. avoirdupois to Troy integers. 5. How many prescriptions of 1 dr. 2 sc. 12 gr. can be filled from 14 oz. avoirdupois? 6. A 60 qt. liquid measure is equivalent to what in dry measure? 92 MODERK COMMERCIAL ARITHMETIC 7. A bin that holds 620 bu. of wheat will hold how many bushels of potatoes? 8. A tank that holds 1280 gal. will hold how many bushels of wheat? 9. Since 1728 cu. in. make 1 cu. ft., and 1 gal. contains 231 cu. in., how many gallons in 1 cu. ft.? 10. A cistern that contains 384 cu. ft will hold how many gallons? PAPERS AND BOOKS 165. TABLE 24 sheets = 1 quire 20 quires or 500 sheets = 1 ream 2 reams = 1 bundle 5 bundles = 1 bale NOTE. The 480-sheet ream is now used rarely in this country except for stationery, and odd and fancy papers. 166. Book Paper. The paper out of which books, circu- lars, and pamphlets are usually made is called Book Paper. It is sold in large unfolded sheets. 167. Flat, Linen, and Ledger Papers. The paper out of which billheads, letterheads, blank books, writing books, etc., are made is called Flat Paper. Flat paper is more expensive than book paper. Both book and flat papers come in sheets of various sizes. 168. SIZES OF BOOK PAPER NOTE. 22 in. by 32 in. may be written 22" X 32", as in the fol- lowing table : 22" x 32" 25" x 38" 28" x 42" 36" x 48" 24" x 36" 25" x 40" 32" x 44" 38" x 50" 169. SIZES OF FLAT PAPER 14" x 17" 17" x 22" 17" x 28" 18" x 46" 15" x 19" 16" x 26" 20" x 28" 22" x 31" 16" x 21" 19" x 24" 21" x 32" 23" x 36" 18" x 23" DENOMINATE NUMBERS 93 Paper Folding 17O. This table shows the number of leaves into which book paper is folded in making books : NAME OF FOLD LEAVES PAGES Folio 2 4 Quarto (4to) 4 8 Octavo (8vo) 8 16 Duodecimo (12mo) 12 24 16mo 16 32 18mo 18 36 24mo 24 48 32mo . 32 64 EXERCISE IN FOLDING 1. Make a folio ; a quarto. 2. An octavo may be % the width and ^ the length, or ^ the width and |- the length of the sheet. Make both kinds. 3. A duodecimo may be ^ the width and % the length, or ^ the width and -J- the length of the sheet. Make both kinds. 4. Make a 16mo ^ the width and ^ the length of the sheet. 5. A 24mo may be \ the width and \ the length, or \ the width and \ the length, or J the width and the length, or the length and -J- the width of the sheet. Make a 24mo of each kind. PROBLEMS 1. Give two possible sizes of the pages of quarto books made from paper (a) 24" x 36", (b) 25" x 38", (c) 28" x 42", (d) 32" x 44". 2. Give the two sizes of the pages of a 12mo book made from paper 24" x 36" SOLUTION 24" -3 = 8" 36" + 4 = 9" Page, 8" x 9" Or, 24" -^4 = 6" 36" ^ 3 = 12" Page, 6" x 12" 3. Give the possible sizes of the pages of a 24mo book made from paper 24" x 36". 94 MODERN COMMERCIAL ARITHMETIC SOLUTION 3x8 = 24 4x6 = 24 2x12 = 24 24" +3=8" 36" + 8 = 4f Page, 4" x 8" Or, 24" +8=3" 36" + 3 = 12 " Page, 3 " x 12" Or, 24" +4=6" 36" +6=6" Page, 6 " x 6" Or, 24" +6=4" 36" +4=9" Page, 4 " x 9" Or, 24" + 2 = 12" 36" + 12 = 3 " Page, 3 " x 12" Or, 24" + 12 = 2" 36" + 2 = 18 " Page, 2 " x 18" NOTE. Pages 3" X 12" and 2" X 18" would be very rare. 4. Find the possible sizes of the pages of an 18mo book made from paper 28" x 42" 5. What size of page can be made by folding a sheet 22" x 28" into an 8vo? 6. What size of page can be made by folding a sheet 22" x 32" into a 16mo book? 7. What size of page can be made by folding a sheet 32" x 44" into a 24mo book? 8. What size of page can be made by folding a sheet 32" x 44" into a 32mo book? 9. What size of pages can be printed from a sheet of paper 28" x 42", using the 18mo form? 10. What size of pages can be printed from all the sheets in the table of book paper, using the 24mo form? 11. What size of billheads can be made from flat paper 24" x 38", using the 24mo form? 12. Using the 24mo form, what size of flat paper should be purchased to make letterheads o-J-" x 8J" with the least waste? IS. A publisher wishes to print a 16mo book 4^" x 6-J-". What size of paper should he buy? 14. What size of paper should be purchased to make a 24mo book 5" x 7i"? 15. What size of book paper should be bought to make a 12mo blank book 8" x 11"? DENOMINATE NUMBERS 95 PRICE, COST, AND MIXED QUANTITIES 171. Merchants and manufacturers often have occasion to form a compound or mixed substance by combining different ingredients or similar ingredients of different qualities. Thus, a grocer may mix Rio coffee with Java coffee, a confectioner may mix two or more kinds of candy, and a manufacturer of paint mixes different oils, colors, and leads. In such cases it is necessary to find the price per pound, quart, etc., of the mix- ture. If the dealer wishes to make a mixture, to be sold at a certain price, it is necessary for him to know what quantities of each ingredient to put into the compound to make the resulting mixture of the required value. To Find the Price of a Mixed Quantity 172. EXAMPLE. A grocer mixed in equal quantities coffees worth 120, 150, and 180 per pound respectively. At what price per pound should he sell the mixture? FORMULA EXPLANATION. The quantity is 3 Cost (of mixture) lb ' the cost is 45 ^ ( 12 + 15 + 18 )> and ^ L = frice. the price is therefore, 45^ -r- 3, or 15^ Quantity ^ MENTAL PROBLEMS 1. At what price should the mixed candy be sold, if in making it, candies selling for 80, 100, and 150 respectively are mixed in equal quantities? 2. Coffee worth 150 per pound is mixed in equal quantities with coffee worth 220 per pound. What is the value per pound of the mixture? 3. 100 lb. of sugar worth 6-J-0 per pound are mixed with 50 lb. worth 50 per pound. What is the value of a pound of the mixture? NOTE. If the quantities mixed have a common divisor, the quo- tients of the common divisor may be taken instead of the quantities themselves. Thus, instead of 100 lb. and 50 lb., 2 lb. and 1 lb. may be used. 96 MODERN" COMMERCIAL ARITHMETIC 4. 20 qt. of wine worth 200 per quart are mixed with 10 qt. of. cider worth 50 per quart. What is the price of the mix ture per gallon? 5. 40 gal. of rum at $2.25 per gallon are mixed with 5 gal. of water. "What is the mixture worth per quart? 6. Find the price of mixed nuts if the lot is made up of equal quantities worth 90, 120, and 180 respectively. 7. 100 Ib. of tea worth 250 per pound are mixed with 75 Ib. worth 180 per pound. What is the value of a pound of the mixture? 8. 10 Ib. of pepper worth 400 per ounce are mixed with 15 Ib. worth 300 per ounce. What is the value of the mixture per ounce? 9. Beans worth $2.15 per bushel are mixed in equal quanti- ties with beans worth $2.05 per bushel. Find the price per quart of the mixed beans. 10. Syrup worth 450 per gallon is mixed in equal quantities with syrup worth 750 per gallon. Find the price of the mix- ture per quart. PROBLEMS 1. A paint dealer mixed 300 gal. of oil worth 650 per gallon with 250 gal. worth 87|0 per gallon, and 140 gal. worth 700 per gallon. Find the price of 1 gallon of the mixed oil. 2. A druggist made a composition drug using ingredients of the following weights and values: 14 Ib. 9 oz. at 300 per ounce, 9 Ib. 6 oz. 4 dr. at 280 per ounce, 21 Ib. 6 oz. at 250 per ounce, and 7 dr. 2 sc. at 100 per dram. Find the price per dram of the mixture. 3. A manufacturer mixed 1 T. 14 cwt. of wool at 680 per pound with 1 T. 15 cwt. 75 Ib. at 550 per pound, and 960 Ib. at 500 per pound. What was the value per pound of the mixed wool? 4. A dealer mixed 10 bbl. of wine worth 900 per gallon with 8 bbl. worth 750 per gallon, 7 bbl. worth 500 per gallon, and 40 gal. of water. What was the liquor worth per gallon? 5. A patent-medicine manufacturer mixed ingredients of the following volumes and values: 18 gal. at $1.85, 2 gal. DENOMINATE NUMBERS 97 water, 1 gal. at $3.75, 1 pt. at $1.30, 2 gr. at 200, and a drug worth $2.40 which added nothing to the bulk of the liquid. If he sold the mixture for twice what it cost him, what was the price per pint? To Find What Quantities Must Be Used to Produce a Mixture of a Given Price 173. EXAMPLE. A grocer wishes to sell mixed tea for 250 per pound, and desires to mix teas worth 450, 350, 200, and 100 per pound. What proportional quantities of each may he use? OPERATION 1. 2. 3. 4. 5. 6. 7. 25 45 35 20 10 15 5 3 1 3 1 20 10 5 15 20 10 4 2 2 4 EXPLANATION. For convenience, write the required price at the left of a vertical line, and the given prices at the right. Write the difference between the given and required prices in the next vertical column at the right. Thus, 45 25 = 20, 35 25 = 10, 25 20 = 5, 25 10 = 15. Draw a horizontal line separating the prices that are greater than the required price from those that are less. The numbers in column 2 show the number of cents gained or lost by putting 1 Ib. of tea at a given price into the mixture, and selling it at 25^ per pound. Thus, if 1 Ib. of 45^ tea is sold for 25^, 20^ is lost. If 1 Ib. of 10^ tea is sold for 25^, 15^ is gained, etc. The horizontal line separates the gains from the losses. The gains and losses must be equal. If 1 Ib. of 45^ tea and 1 Ib. of W tea be put into the mixture, 20^ will be lost on one, and 15^ gained on the other. But if 15 Ib. of the tea OQ which 20^ is lost per pound is mixed with 20 Ib. of the tea on which 15j^ is gained per pound, the gains and losses will be equal (20 X 15 = 15 X 20). And if 5 Ib. of the tea on which 10^ is lost per pound are mixed with 10 Ib. of the tea on which 5^ is gained per pound, the gains and losses will be equal. By comparing prices, one above and one below the horizontal line, a balance of gains and losses is kept, and the figures in columns 3 and 4 show how many pounds of each 98 MODERN" COMMERCIAL ARITHMETIC kind of tea may be mixed. The numbers in any vertical column may be multiplied or divided by any number, as that will not affect the comparative quantities of the two substances mixed. Reducing col- umns 3 and 4 to their simplest form, columns 5 and 6 are obtained. These columns show that 3 Ib. of the 45^ tea and 4 Ib. of the W, 1 Ib. of the 35^, and 2 Ib. of the W may be mixed and sold at 25^ with- out gain or loss. This result is shown in column 7. Suppose the prices given were 45^, 40^, 35^, 30^, 20^, 10^, and the required price 25^. The solution would be: 25 1. 2. i 3. 4. 5. 6. 7. 8. 9. 10. 11. 45 40 35 30 20 15 10 5 15 15 5 5 3 1 1 1 3 1 1 1 20 10 5 35 20 15 10 5 4 1 2 1 3 5 Column 2 shows the difference between the given prices and the required price. By taking these numbers in pairs and reversing them^ columns 3, 4, 5, and 6 are obtained, which show the comparative quan- tities that may be mixed. These columns are reduced, by dividing the numbers in each by the greatest number that will divide them, to columns 7, 8, 9, and 10. And the number of pounds of each kind that may be used are shown in column 11. As the numbers in any one of columns 7, 8, 9, or 10 may be multiplied by any number, an indefinite number of changes may be made in column 11. PROBLEMS 1. A merchant has lots of pepper worth 20^, 28^, 35^, and 40^ per pound, and wishes to form a mixture worth 30^ per pound. How many pounds of each may he use? Give five answers. If he uses 50 Ib. of the first, how many pounds of each of the others must he use? 2. Syrup worth 40^, 50^, 65^, and 70^ per gallon is mixed and sold at 60^ per gallon. What comparative quantities of each are mixed? 3. A dealer mixed wines worth $1.25, $1.10, 90^, and water, so that the mixture was worth $1 per gallon. What quantities of each were used? Give three answers. DENOMINATE NUMBERS 99 4. How many pounds of each kind of candy worth 60, 80, 100, and 150 may be mixed to form a compound worth 1 20 per pound? 5. Drugs worth 450 per ounce, 600 per ounce, and 100 per dram were mixed and sold at 500 per ounce. If 8 Ib. of the second were used, what quantities of the others were used? 6. What proportional quantities, worth 850, 750, 650, 550, and 400, may be mixed with 80 Ib. worth 500 so that the mixture shall be worth 700 per pound? 7. Oils worth 650, 87-^-0, and 700 were used to form a mixture worth 750. What proportional quantities of each were used? 8. Coffees worth 12^-0, 150, 180, and 250 per pound were mixed and sold for 200 per pound. What comparative quanti- ties of each were used? 9. A druggist mixed chemicals worth 200 per ounce, $3.30 per ounce, 60 per ounce, and 80 per dram. If the compound was worth 450 per oijnce, what quantities of each did he use? BUSINESS PROBLEMS 1. A dealer sold 24800 bu. wheat for 4135 sov. 12s. What was the price per bushel in United States money? 2. A manufacturer made 20 gross of silver spoons, each weighing 10 pwt. 16 gr. How much did the silver in the spoons cost him at 900 per ounce avoirdupois? 3. A gold dollar weighs 25.8 gr. and is -fa pure gold. A manufacturer made, from United States gold coin, 750 gold chains 18 k. fine. If each chain weighed 3 oz. 16 pwt. 18 gr., what was the value of the coin melted to make the chains? NOTE. Pure gold is said to be 24 karats fine. Gold that contains fa alloy has |f of pure gold and is 18 karats (18 k.) fine. 4. How many vases, each containing 8 oz. 16 pwt. of pure silver can be made from 95 Ib. 4 oz. (Troy) of silver? 5. Find the cost, at $1.10 per cwt, of three loads of meal weighing as follows: 1 T. 7 cwt. 60 Ib., 2 T. 80 Ib., 1 T. 12 cwt. 75 Ib. 100 MODERN COMMERCIAL ARITHMETIC 6. What will be the freight on 75 bbl. of oil, each weighing 450 lb., at $2 per ton? 7. What will be the freight on 8765 bu. of wheat (60 lb. = 1 bu.) at 750 per ton? 8. A druggist bought 150 lb. of drugs, by avoirdupois weight, at $7.50 per pound, and sold them at 80 per scruple. What did he gain? 9. A dealer bought nuts at $2 per bushel (32 qt.) and sold them at 80 per qt. liquid measure. Find his gain per bushel. 10. A grocer bought peas for $2.40 per bushel, and sold them at 100 per quart. What did he gain per bushel? What part of his purchase price did he gain? 11. What will be the cost, at 300 per bushel of 60 lb. , of 5 loads of potatoes, weighing 2472 lb., 3185 lb., 2817 lb., 3025 lb., 2960 lb.? 12. Find the cost of 18 gal. 3 qt. 1 pt. of wine, at 220 per quart. IS. A cask of brandy containing 46 gal. 2 qt. 1 pt. was bought for $108 and sold at 200 per gill. What was gained? H. A grocer bought 8 bu. of beans by dry measure and sold them by the liquid quart. How many liquid quarts did he gain? 15. How many feet of fence will be required to inclose a field 25 ch. 41 1. long and 21 ch. 40 1. wide? 16. If fence wire weighs 12 oz. to the rod, how many pounds of wire will be required to build a fence around a field 19 ch. 20 1. long and 14 ch. 30 1. wide, if the fence be made 8 wires high? 17. Change $425 to equivalents in French money; to equivalents in German money. 18. Change 1486 francs to equivalents in United States money. 19. Change 2538 marks to equivalents in United States money. 20. If pine is estimated to weigh 3000 lb. per M, and maple 4000 lb. per M, how much freight must be paid, at $1.20 per ton, on 4560 ft. of pine and 5872 ft. of maple? DENOMINATE NUMBERS 101 21. Find the cost of 1865 ft. of lumber at $17.60 per M. 22. If it requires 31 Ib. of coal per day to heat a house, how much will it cost per week to heat it, the price of coal being $5.50;per ton? 23. A farmer raised 230 bu. 2 pk. 6 qt. from 9 bu. 2 pk. 4 qt. of seed. What was the yield from one bushel of seed? 24. Change 83 sov. 12s. 8d. to United States money. 25. Find the total cost of: 12 Ib. 9 oz. lard at 11^ per pound. 9 Ib. 6 oz. steak at 14^ per pound. 7 Ib. 4 oz. mutton at 9^ per Ib. 8 Ib. 12 oz. pork at 10^ per pound. 2 gal. 2 qt. 1 pt. molasses at 60^ per gallon. 7 Ib. 5 oz. cheese at 12-J^ per pound. 20 ft. cord at 8^ per yard. 100 pickles at 10

Altitude 130 rd., base 115 rd. 5. Base 86 ft., altitude 74 ft. 6. Altitude 126 yd., base 245 yd. 7. How many square yards in a triangular park whose alti- tude is 362 ft. and base 581 ft.? 8. The area of a triangle is 65 sq. yd. and its altitude is 18 ft. Find its base. 9. Find the area of the four sides of a square pyramid. Each side is 40 yards at the base, and the distance from the mid- dle of the edge of the base to the vertex is 80 ft. 10. How many feet of lumber will be required to cover a triangular floor whose base is 45 ft. and altitude 37 ft.? 11. Find the area of a triangular field if one of its sides is 124 rd. and the distance from this side to the vertex opposite is 56 rd. 12. I wish to lay out a triangular park that will contain 3 A. If I make the base 36 rd., how far opposite must the apex be placed? 13. The base of a triangle is 17 ft. and the area is 75 sq. ft. What is its altitude? 14- The three sides of a triangle are 6, 8, and 10 rd. If the area is 24 sq. rd. , what is the distance of each side from the apex opposite? 15. Find the area of a triangle if the altitude is 14 rd. and the base is 29 rd. To Find the Area of a Trapezoid 193. A quadrilateral having two sides parallel is a Trape- zoid. A diagonal of a trapezoid divides it into two triangles having the same altitude. PRACTICAL MEASUREMENTS 109 EXPLANATION. Area of triangle A B C = J of B C X alt. of the trapezoid. Area of triangle A C D = J of / A D X alt. of the trapezoid. Area of trapezoid A B C D = \ of (AD + BC) X alt. of the A trapezoid. Principle. The area of a trapezoid is equal to one-half of the product of the sum of the parallel sides by the altitude. PROBLEMS Draw and find the area of trapezoids of the following dimensions: 1. Parallel sides 22 ft. and 16 ft., altitude 14 ft. 2. Parallel sides 36 yd. and 24 yd., altitude 25 yd. 3. Parallel sides 128 rd. and 75 rd., altitude 37 rd. 4. Altitude 246 ft., parallel sides 426 ft. and 538 ft. 5. Altitude 42 rd., parallel sides 85 rd. and 96 rd. 6. The area of a trapezoid is 4 acres. The sum of the parallel sides is 64 rd. What is the altitude? 7. How many acres in a field in the form of a trapezoid whose altitude is 38 rd. and whose parallel sides are 56 rd. and 62 rd.? 8. Find the arsa of a park in the form of a trapezoid, 60 rd. long, 8 rd. wide at one end and 12 rd. at the other. 9. One side of a lot is 16 rd. long, the side parallel to it is 12 rd. long, and the perpendicular distance between them is 8 rd. How much is the lot worth at $2 per square yard? 10. The area of a trapezoid is 4 A. If the altitude is 18 rd., what is the sum of the parallel sides? 11. The parallel sides of a field are 46 rd. and 28 rd., and the distance between them is 26 rd. What is the area? 12. How many square feet in a board 18 ft. long, 24" wide at one end, 14" at the other? 18. How many square feet in the two gable ends of a house 32 ft. wide, if the peak of the roof is 12 ft. above the plate? 110 MODERN COMMERCIAL ARITHMETIC 14- How many square inches in a sheet of metal having two parallel sides 20" and 32" long, if the distance between them is 19"? To Find the Area of a Regular Polygon 194. A plane figure whose sides and angles are respectively equal each to each is a Kegular Polygon. HEXAGON OCTAGON (AC, apothem; AB, radius.) 195. The Perimeter of a polygon is the sum of all its sides. 196. The Radius of a regular polygon is the distance from the center to any vertex. 197. The Apothem of a polygon is the perpendicular dis- tance from the center to any side. 198. The radii of a regular polygon divide it into equal triangles. The apothem of the polygon is then the altitude of each of these triangles. The perimeter of the polygon is the sum of the bases of the triangles. Therefore, the area of a regular polygon is equal to one-half the product of its perim- eter and apothem. Principle. The area of a regular polygon is equal to one- half the product of its perimeter and apothem. PROBLEMS 1. Find the area of a regular hexagon whose sides are 5 ft. and whose apothem is 4 ft. 4 in. 2. The perimeter of a regular octagon is 48 ft., and the perpendicular distance from the center to one side is 7 ft. 3 in. What is the area of the octagon? PKACTICAL MEASUREMENTS 111 3. The side of a regular pentagon is 10 yd. and the apothem is 7 yd. 1 ft. 5 in. What is the area? 4- The apothem of a regular octagon is 12 ft. and each side is 9.94 ft. What is the area? 5. Find the area of a regular hexagon, if the perimeter is 36 ft. and the apothem is 5| ft. 6. Find the area of a regular hexagon each of whose sides is 8" and whose apothem is 6.93". 7. If the area of a regular octagon is 174 sq. ft. and the length of one side is 6 ft., what is its apothem? 8. If the area of a regular hexagon is 374.4 sq. ft. and the length of one side is 12 ft., what is the apothem? NOTE. To find the area of any regular polygon, multiply the square of its side by its number in the following table: Triangle 433013 Nonagon 6.181824 Pentagon 1.720477 Decagon 7694209 Hexagon 2.578076 Dodecagon 11.196152 Octagon 4828427 9. Find the area of a regular pentagon whose perimeter is 40ft. 10. Find the area of a regular decagon one of whose sides is 15 yd. 11. If a triangle is 7 ft. on a side, what is its area? 12. If the side of a regular dodecagon is 15 ft., what is its area? 13. Find the area of a regular nonagon whose sides are each 22 ft. H. The side of a regular pentagon is 44 yd. What is its area? 15. What is the area of the floor of an octagonal room 16 ft. on a side? To Find the Circumference and Diameter of a Circle 199. A plane figure bounded by a uniformly curved line is a Circle. 3OO. The line that bounds a circle is its Circumference. 112 MODERN COMMERCIAL ARITHMETIC 201. Every point in the circumference is equally distant from the center. 202. A straight line from the center to the circumference is the Kadius. 203. A straight line from one side of a circle through the center to the opposite side is the Diameter. Principles. 1. Circumference = diameter x 3.1416. 2. Diameter = circumference + 3.1416. For ordinary purposes, circumference = diameter x 3^. PROBLEMS 1. What is the circumference of a circle whose diameter is 28 ft.? 2. Find the diameter if the circumference is 49 yd. 8. Find the diameter if the circumference is 56 yd. 4. The radius is 7 ft. ; find the circumference. 5. The circumference is 68 yd. ; find the radius. 6. The diameter is 42 ft. ; find the circumference. 7. The circumference is 95 yd. ; find the radius. 8. The radius is 7 rd. ; find the circumference. 9. The circumference is 123 rd. ; find the diameter. 10. The radius is 11 ft. 7"; find the circumference. To Find the Area of a Circle 2O4. If the number of sides of a regular polygon be suffi- ciently increased, the polygon will become a circle, the perimeter will become the circumference, and the apothem will become the radius of the circle. Principle. The area of a circle is equal to one-half the product of its circumference and radius. circumference x radius Area = 2 PRACTICAL MEASUREMENTS 113 205. Inscribe a circle within a square. How does the 'diameter of the circle compare in length with a side of the square? How does the circumference of the circle compare in length with 4 times the length of the side of the square? It is just how many times the side of the square or the diameter of the circle? How does the circle compare in area with the area of the square around the circle? If the side of the square is 10 ft., what is the area of the square? What is the diameter of the inscribed circle? What is ^ the circum- ference? What is the radius? What is the area? PROBLEMS Find the area of circles of the following dimensions : 1. Circumference 50 ft., diameter 15.915 ft. 2. Circumference 1 mi., diameter 101.856 rd. 3. Circumference 60 ft. 8. Diameter 85 ft. 4- Radius 28 in. 9. Diameter 18 rd. ,5. Radius 4 ft. 10. Diameter 24 ft. '6. Circumference 26 yd. 11. Radius 12 rd. 7. Circumference 34 yd. 12. Radius 16 yd. 13. How many acres in a circular park 60 rd. in diameter? H. How many acres in a race-course 1 mi. in circumference? 15. How many square feet in the bottom of a cistern 6J ft. In diameter? 16. Find the area of the side and bottom of a cistern, if it is 6 ft. deep and 20 ft. in circumference. 17. Find the entire surface of a cylinder 8" in diameter and 18" long. 18. How many square feet in a rug 35 ft. in circumference? Measurements by the Square Yard 206. The cost of plastering, ceiling, painting, and paving is usually computed by the square yard. It is customary to allow for one-half of the area of open- ings, as for doors and windows. 114 MODERN COMMERCIAL ARITHMETIC PROBLEMS 1. Find the cost of plastering the four walls of a room 16 ft. square and 9 ft. high, at 200 per square yard. 2. How many square yards in the ceiling of a room 22 ft. by 18 ft.? 3. The walls of a room 20 ft. by 18 ft. and 10 ft. high have 3 ft. of wainscoting and 7 ft. of plaster. Find the number of square yards in the floor, ceiling, and wainscoting, and the number of square yards of plaster on the walls. 4. A room 18 ft. square and 10 ft. high has 3 doors each 3 ft. by 7J ft., and 4 windows each 3 ft. by 6 ft. How many square yards in the walls and ceiling, making one-half allowance for doors and windows? 5. Find the cost of ceiling, at 200 per square yard, 5 rooms of the following dimensions: 12 ft. by 14 ft., 18 ft. by 21 ft., 13 ft. by 16 ft., 14 ft. square, and 15 ft. by 18 ft. 6. Find the cost of paying a walk 4-^ ft. wide, 8 rd. long, at 600 per square yard. 7. How many square yards of paying in a street 85 ft. wide and 60 rd. long? 8. How many brick 8 in. by 4 in. will be required, if laid flat, to pave a walk 4 ft. 8 in. wide and 25 ft. 4 in. long? 9. A block of buildings measures 320 ft. by 420 ft. How many square yards in an 8 ft. walk surrounding the block? 10. How many square yards of paving in a courtyard in the form of a trapezoid, the parallel sides being 98 ft. and 76 ft. respectively, and the distance between these sides being 48 ft.? 11. Find the cost of plastering a circular cistern 8 ft. in diameter and 6 ft. deep, at 300 per square yard. 12. How many square yards of plastering on a cistern 7 ft. deep, 10 ft. long, and 8 ft. wide? 13. A room 20 ft. square and 9 ft. high has 8 windows 4 ft. by 8 ft., and 7 doors 3 ft. by 8 ft. How many square yards in the walls, ceiling, and floor, allowing % space for doors and windows? PEACTICAL MEASUKEMENTS 115 14. Find the cost, at 28^ per square yard, of ceiling the sides and overhead of 6 rooms of the following dimensions : 16 ft. by 18 ft., 12 ft. by 13 ft., 9 ft. by 12 ft., 8 ft. by 14 ft., 13 ft. by 14 ft., and 11 ft. square, each being 9 ft. high. 15. Find the cost of paving, at 75^ per square yard, a hexagonal court 150 ft. on a side. 16. How many square feet of cement in a circular court 62 ft. in diameter? 17. How many square yards of painting on the outside of a house 50 ft. long, 32 ft. wide, and 21 ft. high? The gable ends are 14 ft. high, the cornice is estimated at 40 sq. yds. and no allowance is made for windows. 18. How many squares of painting, 100 ft. to the square, on the outside of a brick chimney the slant height of which is 60 ft., 12 ft. square at the bottom, and 6 ft. at the top ? 19. How many square yards of plastering on the inside of the above chimney if the walls are 1| ft. thick? 20. How many square feet of painting on 12 stairs, each haying 20 steps 3 ft. wide with a tread of 9" and a rise of 8"? Papering 207. A roll of wall paper is usually 8 yd. long and 18 in. wide. Double rolls are counted as two rolls. 208. To find the number of rolls of paper required to paper a room : 1. Find the number of strips of paper required. 2. Find the number of strips that can be cut from a roll. 3. Divide the number of strips required by the number that can be cut from a roll. NOTE. Sometimes waste occurs in matching, so that it may not always be possible to estimate the exact number of rolls required. PROBLEMS 1. How many rolls of paper, 8 yd. long and 18 in. wide, will be required to cover the walls of a room 20 ft. square and 9 ft. high, making allowances for 6 windows 3 ft. wide, and 4 doors 3 ft. wide? 116 MODERN" COMMERCIAL ARITHMETIC 2. How many rolls of paper will be necessary for a room 22 ft. long, 15 ft. wide, and 8 ft. high, allowing only for 4 doors 3|- ft. wide? 3. How many rolls of paper must be used to cover the ceil- ing of a room 16 ft. by 21 ft., if the paper runs crosswise with the room? 4. Find the number of rolls of paper required to cover the walls and ceiling of a room 18 ft. square and 10 ft. high. A roll will make two strips for the wall. In the middle of each side is a door 3J ft. wide, and on each side of each door and 2 ft. from it, is a window 3 ft. wide. 5. How much paper will be required for the walls and ceil- ing of a room 12 ft. by 13 ft. and 8 ft. high? Allow for 2 doors and 4 windows each 3^- ft. wide and no waste in matching. 6. How many yards of plain paper 30" wide must be used to cover the walls of a room 32 ft. by 28 ft., and 14 ft. high, allowing 1 ft. for border, 3 ft. for wainscoting, and for 4 doors each 4 ft. wide, and 8 windows each 3-J- ft. wide? 7. How many yards of plain paper 3 ft. wide will be neces- sary to cover the walls and ceiling of a hall 120 ft. long, 12 ft. wide, and 13 ft. high? 8. How many rolls of paper will be used for the walls and ceiling of a room 17 ft. by 23 ft. and 10 ft. high, allowing 1 ft. for border, 2 ft. of paper on each strip for matching, and for 5 doors and 6 windows, each 4 ft. wide? Carpeting 2O9. Carpets are usually either 1 yd. or f yd. in width. Allowance must be made for waste in matching the patterns in carpets. As carpets are sold in strips and matched by strips, the number of strips required to cover the floor must be found. Sometimes it is necessary to cut off or turn under part of a strip, and the part cut off or turned under must be included in the estimate. Sometimes there is less waste when the strips run one way in the room than when they run the other way. PRACTICAL MEASUREMENTS 117 In finding the length of border to be put around a carpet the entire distance around the room must be taken, as there is a waste at each corner in making the border. PROBLEMS 1. A floor 21 ft. by 24 ft. is covered with carpet 1 yd. wide, without waste in matching. Find the cost of the carpet at $1.10 per running yard. 2. If the above room is covered with carpet f yd. wide, what will it cost at 90^ per yard? 3. Find the cost of carpeting a room 20 ft. by 27 ft., with carpet f yd. wide, at 80^ per yard, if the strips run length- wise ; if the strips run crosswise. 4. How many yards of carpet 1 yd. wide will be required for a room 24 ft. by 16 ft., if the strips run lengthwise and J of a yard in each strip is wasted in matching? 5. Find the least number of yards of carpet required to cover floors of the following dimensions, and tell which way the strips should run, allowing no waste for matching: ROOM WIDTH OF CARPET 40 ft. by 42 ft. 1 yd. 22 ft. by 15 ft. f yd. 27 ft. by 34 ft. f yd. 13 ft. by 15 ft. 1 yd 14 ft. by 18 ft. f yd. 6. Find how much more carpet would be required in each case if it were laid in the opposite direction. 7. Find the least number of yards of carpet necessary to cover floors of the following dimensions, allowing no waste for matching : ROOM WIDTH OF CARPET 14 ft. by 22 ft. 1 yd. 21 ft. by 24 ft. f yd. 13 ft. by 17 ft. 1 yd. 12ft. by 16 ft. f yd. 22 ft. by 25 ft. f yd. 118 MODERN COMMERCIAL ARITHMETIC MEASUREMENT OF SOLID FIGURES To Find the Volume of a Prism 210. A solid has length, breadth, and thickness. 211. A solid that has two parallel equal bases, and three or more sides that are parallelograms, is a Prism. TRIANGULAR PRISM RECTANGULAR PRISM 212. The area of a rectangle is equal to the product of its length and breadth. 213. A rectangular prism that is one unit high has a vol- ume equal to the product of its length and breadth. Its solid contents equals the area of its base. If the prism is 3 units in height, its volume is equal to 3 times the area of its base, or the product of its length, breadth, and thickness. Principles. 1. The volume of a rectangular prism is equal to the product of its length, breadth, and thickness. 2. The volume of any prism is equal to the product of the area of its base by its altitude. PROBLEMS 1. Find the contents in cubic feet of prisms having the fol- lowing dimensions: (a) 12 ft. square and 16 ft. high. (b) Base 8 ft. by 14 ft., altitude 7 ft. (c) Length 20 ft., breadth 16 ft., height 8 ft. (d) Area of base 137 sq. ft., altitude 9 ft. PRACTICAL MEASUREMENTS 119 #. The triangle that forms the base of the prism has a base of 8 ft. and an altitude of 5 ft. The altitude of the prism is 10 ft. Find the contents. 8. How many cubic feet in a rectangular solid 12 ft. by 8 ft. by 14 ft? 4. Find the solid contents of a hexagonal prism 12 ft. on a side and 16 ft. high. 5. The bottom of a bin is a trapezoid, the parallel sides being 14 ft. and 16 ft. and 10 ft. apart. How many cubic feet does it contain if it is 8 ft. deep? 6. How many cubic feet of water can be put into a tank in the form of an octagonal prism, if it is 9 ft. high and each side of the tank is 8 ft.? 7. A gas reservoir has a base in the form of a dodecagon 16 ft. on a side. If it is 40 ft. high, how many cubic feet of gas will it hold? To Find the Volume of a Cylinder 214. A solid having two equal parallel circles for its bases and a uniformly curved surface for its side is a Cylinder. Principle. The volume of a cylinder is equal to the product of the area of its base by its altitude. PROBLEMS 1. Find the contents of cylinders with the following dimen- sions : (a) Altitude 7 ft., base 6 ft. in diameter. (b) Circumference of base 25 ft., altitude 6 ft. (c) Altitude 26 ft., diameter of base 4 ft. (d) Circumference of base 42 ft. , altitude 7 ft. 2. How many cubic feet in a pillar 8 ft. in diameter and 50 ft. high? 3. A cistern 7 ft. deep and 28 ft. in circumference contains how many cubic feet? 120 MODERN COMMERCIAL ARITHMETIC 4. The diameter of a pipe is 2-J- ft. How much water will flow through it in an hour, if it flows with a velocity of 90 ft, per minute? 5. How many cubic feet of iron in a pipe 80 ft. long and 3 ft. in diameter, if the iron is " thick? 6. I wish to make a cylindrical cistern that will hold 1000 cubic feet of water. If I make it 12 ft. in diameter, how deep must it be ? Brick and Stone Work 215. Brick work is commonly estimated by the 1000 brick. Masonry is commonly estimated by the cubic foot and by the perch. NOTE. A perch is 16J cu. ft. Sometimes 24| cu. ft. are called a perch. Usually a deduction is made for one-half of the openings. 216. In estimating the amount of work done in laying stone and brick, the length of the wall is found by measuring around the wall on the outside, or by measuring on the inside and adding 8 times the thickness of the wall. The corners are thus counted twice. 217. In estimating the amount of material used, find the exact length of the wall by measuring on the inside and adding 4 times the thickness of the wall for the corners, or by measur- ing on the outside and subtracting 4 times the thickness of the wall for the corners. Principle. To find the contents of a wall, find the entire length of the wall ; find the product of the length, height, and thickness. Make deductions for openings. PROBLEMS 1. A house is 28 ft. by 22 ft. How many cubic feet in the cellar wall, if the wall is 18 in. thick and the cellar is 7 ft. deep? 2. What will be the cost, at S0

-f- 12 = 10. Likewise, as parts of 12, 3 = J, 4 = J, 6= J, 8 = |, 9 = 5, 15 = li, 16 = 1J, 18 = 1J. PROBLEMS Find the number of feet of lumber in boards, not more than 1 in. thick, of the following dimensions: 1. Length 15 ft., width 16 in. 5. Length 24 ft., width 16 in. 2. Length 14 ft., width 9 in. 6. Length 22 ft., width 6 in. S. Length 18 ft., width 10 in. 7. Length 8 ft., width 9 in. 4. Length 20 ft., width 15 in. 8. Length 14 ft., width 13 in. Find the number of board feet in timbers and plank of the following dimensions: 9. Width 8 in., length 16 ft., thickness 2| in. 10. Width 12 in., length 14 ft., thickness 3 in. 11. Width 14 in., thickness 4 in., length 10 ft. 12. Length 18 ft., width 12 in., thickness 9 in. 13. Length 20 ft., 8 in. square. 14. Length 16 ft., 2 in. by 6 in: 15. Length 18 ft., 2 in. by 4 in. 16. Length 14 ft., 2 in. by 8 in. 17. Length 15 ft., 2 in. by 4 in. 18. Find the cost, at $18.50 per M, of 32 scantling, each 14 ft. by 2 in. by 4 in. 19. Find the cost, at $36 per M, of 12 timbers, each 18 ft. long and 10 in. square. 20. Find the cost, at $14 per M, of a load of hemlock tim- bers of the following dimensions : 24 scantling 16' x 2" x 4" 12 scantling 14' x 2" x 6" 8 timbers 12' x 6" x 8" 10 timbers 16' x 4" x 4" 20 plank 18' x 2" x 10" 6 timbers . 14' x 4" x 6" 124 MODERN COMMERCIAL ARITHMETIC 21. Find the cost, at $28 per M, of the following pieces of pine: 22 plank 16' x 2 " x 12" 18 plank 14' x 3 " x 10" 25 scantling 18' x 2 " x 4" 6 sticks 14' x 4 " x 4" 40 boards 12' x 1|" x 8" 36 rafters 18' x 2 " x 6" 48 sleepers 16' x 2 " x 8" 8 sticks 24' x 6 "x 8" 22. Find the cost of the following chestnut lumber at $45 per M: 9 pieces 12' x 1" x 10" 12 pieces 14' x 1 "x 12" 10 pieces 16' x 1" x 14" 6 pieces 12' x 1 " x 14" 8 pieces 8' x 2 "x 6" 9 pieces 6' x !" x 8" 23. What is the cost of the following boards at $18 per M: 16 boards 12' x f" x 12" 12 boards 16' x $" x 10" 20 boards 18' x f " x 12" 10 boards 14' x 1 " x 9" 24 boards 12' x i" x 8" 15 boards 9' x 1" x 10" Capacity of Bins and Cisterns 223. TABLE' 2150.4 cu. in. = 1 bu. 231 cu. in. = 1 gal. 311 gal. = 1 bbl. FORMULAS Length x breadth x thickness x 1728 = bushels 2150.4 Length x breadth x thickness x 1728 2 = gallons 231 NOTE. All dimensions should be in feet. PRACTICAL MEASUREMENTS 125 PROBLEMS 1. Find the contents (dry measure) of the following: (a) A bin 8 ft. square and 6 ft. deep ; (b) a box 9 ft. by 3 ft. 4 in. by 10 in. ; (c) a box 6 ft. long, 4 ft. high, 4^- ft. wide. 2. How many bushels of wheat will a bin hold that is 8 ft. long, 5 ft. wide, and 5 ft. high? 8. How many bushels of potatoes will a bin hold that is 9 ft. square and 6^ ft. deep? 4. How many bushels of apples will a wagon box hold that is 14 ft. by 3 ft. 4 in. by 2 ft. 3 in.? 5. Find the contents, liquid measure, of a tank 9 ft. long, 4 ft. wide, and 3 ft. deep. 6. How many barrels will a cistern hold that is 7 ft. by 8 ft. by 6 ft.? 7. A vat 8^ ft. by 6 ft. by 4 ft. will hold how many gallons of water? 8. A cylindrical cistern 7 ft. in diameter and 7 ft. high will hold how many barrels of vinegar? 9. If 3 measures of grapes will make 2 measures of wine, how many gallons of wine can be made from 3 bu. of grapes? 10. How many gallons of alcohol will be required to fill a pipe 18 ft. long and 1-J- in. in diameter? 11. How many gallons of oil in a cylindrical can 2 ft. high and 14 in. in diameter? 12. If 1 cu. ft. of water weighs 62 lb., how many gallons in 1 T. of water? 13. A tank is 22 ft. in circumference and 8 ft. high. How many barrels of beer (28 gal.) will it contain? PRACTICAL RULES FOR DEALERS IN FARM PRODUCE 224:. The following rules are approximate: 1. To find the contents, in bushels, of a bin or box, multiply the number of cubic feet by .8. 2. To find the volume of a bin required to hold a given number of bushels, divide the number of bushels by .8, or multiply by f . 126 MODERN COMMERCIAL ARITHMETIC 3. To find the contents, in 40-qt. bushels, of a bin, multiply the number of cubic feet by . NOTE. The 40-qt. or "heaped" bushel is used for apples, potatoes, etc. 4> To find the volume of a bin required to hold a given number of 40-qt. bushels, divide the number of bushels by . 5. To find the contents, in gallons, of a tank or cistern, multiply the number of cubic feet by 7-|. 6. To find the volume, in cubic feet, of a tank required to hold a given number of gallons, divide the number of gallons by 7V 7. To find the number of shelled bushels in a crib of unshelled corn, multiply the number of cubic feet by .45. NOTE. Pupils should be required to give the reasons for the above rules. NOTE A bushel is to a cubic foot as 56 is to 45. Hay 225. The weight of hay, per cubic foot, in a load, shed, mow or stack, varies with the kind of hay, the height of the pile, pressure or treading in packing, and time of settling. The higher the pile the more compact the hay will be. Principles. 1. To find the weight of hay in a load, or low pile, allow 540 cu. ft. for a ton. 2. To find the weight of hay in an ordinary mow, or low stack, allow 400 cu. ft. for a ton. 3. To find the weight of hay in mow bottoms and in bot- toms of high stacks, allow 325 cu. ft. per ton. PROBLEMS 1. How many bushels of wheat in a bin 6 ft. square and 4 ft. deep? 2. How many bushels of apples in a wagon box 14 ft. long, 3 ft. wide, and 30 in. high. 3. How many bushels of oats in a bin 8 ft. long, 6 ft. wide, and 3 ft. deep? 4. At 35^ per bushel, find the value of a bin of potatoes 10 ft. long, 7 ft. wide, and 5 ft. high. PRACTICAL MEASUREMENTS 127 5. How many cubic feet of space must be provided for 500 bushels of apples? If the bin is 4 ft. deep and 8 ft. wide, how long must it be? 6. I wish to build a bin that will hold 120 bushels of wheat. If I make it 6 ft. square, how high must it be? 7. How many cubic feet in a cylindrical cistern 5 ft. in diameter and 6 ft. deep? How many gallons would it hold? 8. How deep must a cistern be to hold 60 bbl. of water if it is 7 ft. in diameter? 9. How many bushels of shelled corn in a crib of unshelled corn 18ft. by 6ft. by 7ft.? 10. If a bin will hold 400 bushels of wheat, how many bushels of apples will it hold? 11. A wagon box 14 ft. long, 3 ft. wide and 2-J- ft. high is full of corn in the ear. How many bushels of shelled corn will the load make? 12. How many bushels of turnips in a bin 7 ft. long, 5 ft. wide, and 4 ft. deep? 13. I wish to make a wagon box to hold 40 bu. of apples. If I make it 14 ft. long and 3 ft. wide, how high must it be? 14- How many hundredweight of hay in a load 20 ft. long, 12 ft. wide, and 7 ft. high? 15. The top of a high hay stack has been removed. The bottom is 20 ft. square and 12 ft. high. How many tons does it contain? 16. A pile of hay 40 ft. long, 15 ft. wide, and 9 ft. high contains how many tons? 17. If a stack of hay is 24 ft. in diameter, and 12 ft. high, what is it worth at $12 per ton? 18. I wish to build in a barn a bay that will hold 100 tons of packed hay. If I make it 40 ft. wide and 12 ft. deep, how long must it be? 19. How many cubic feet of space will be required to hold 400 bu. of potatoes? If the bin is made 12 ft. square, how deep must it be? 20. A bin that will hold 250 bu. of wheat will hold how many bushels of potatoes? 128 MODERN COMMERCIAL ARITHMETIC SQUARE ROOT AND ITS APPLICATIONS 226. To square a number is to multiply it by itself. The square of 5 is 25. 36 is the square of 6. What is the square of 8, 9, 11, 12, 15, 20, 25? 227. One of the two equal factors of a number is the square root of the number. Thus, 10 and 10 are the two equal factors of 100, and 10 is the square root of 100. 12 is the square root of 144. What is the square root of 81, 121, 225, 625? 228. To indicate the square of a number we write a small figure 2 at the upper right hand side of the number. Thus, 12 2 means 12 x 12, or 12 squared. 15 2 = 225. Give the value of 8 2 ; 10 8 ; II 2 ; 13 2 ; 21 2 ; 19 2 . 229. To indicate the square root of a number, we write the character ^/ at the left of the number. \/100 means the square root of 100, or 10. \/144 = 12. Give the value of \/225; v/625; \/900; v/169; \/400. 230. Principles. 1. If the side of a square represents a number, the square itself will represent the square of the number. 2. If a square represents a number, a side of the square will represent the square root of the number. MENTAL PROBLEMS 1. I wish to make a square table with a surface of 16 sq. ft. ; what musfc be the length of a side? 2. What is the length of a side of a square board that con- tains 225 sq. in.? 3. What is the length in rods of a square field that contains 10 acres? 4- A square floor contains 400 sq. ft. of space ; what is the length of one side? PRACTICAL MEASUREMENTS 129 The diagram compares a square with the square of a number and the side of the square with the square root of the same number. The large square marked T 2 equals the square of the tens of the num- ber. The area of the square is 100 and the square of 10 is 100. Each of the rectangles marked T X U equals the product of the tens by the units. The small square marked U 2 equals the square of the units. The complete square equals the square of the tens plus twice the product of the tens and units, plus the square of the units, or T 2 + 2TU + U 2 . 13 = 10 + 3, or IT -+ 3U. 13 2 = 10 2 + 2 (10 X 3) + 3 2 , or T 2 + 2TU + U 2 . Any number may be considered as composed of tens and units. 125 is composed of 12 tens and 5 units. 125 2 = T 2 + 2TU + U 2 , or 120 2 + 2 (120 X 5) + 5 2 . 2654 = 265 tens + 4 units. 2654 2 == 2650 2 + 2 (2650 X 4) +- 4 2 . The following example also shows the relation of a number to its square. EXAMPLE. Find the square of 10 plus 3. T X U = 30 U 2 = 9 CO T 2 = 100 TXU o 10 3 OPERATION 10 + 3 10 + 3 10 2 (10 x 3) + 3* (10 x 3) 10 2 + 2 (10 x 3) + 3 2 = 169 = T 2 + 2TU + IP Principle. The square of a number composed of tens and units is equal to the square of the tens + 2 x the tens x the units + the square of the units. PROBLEMS Write out the squares of the following : 1. 5T + 4U. 8. 60 + 8. 2. 40 + 3. 4- ^0 + 5. 5. 80 + 4. 130 MODERN COMMERCIAL ARITHMETIC Write by inspection the square root of : 6. 100 + 60 + 9. 8. 100+ 80 + 16. 10. 100 + 140 + 49. 7. 100 + 100 + 25. 9. 100 + 120 + 36. 11. 400 + 200 + 25. To Find the Square Root of a Number 231. Finding the square root of a number is an operation in " fitting and trying" like long division. We try and then find whether we are right. We can tell something about the square root of a number by inspection. Study the following table : . 1 2 = 1 10 2 = 100 100 2 = 10000 9 2 = 81 99 2 = 9801 999 2 = 998001 It will be seen that the square of a number of 1 figure cannot con- tain more than two figures ; that the square of a number of 2 figures cannot contain less than three nor more than 4 figures ; that the square of a number of 3 figures cannot contain less than 5, nor more than 6 figures. If any number be separated into periods of 2 figures each, beginning at the right, there will be just as many periods as there are figures in the square root of the number. Thus, in the square root of 1 ' 46 ' 75 ' 83 there are 4 figures ; in the square root of 21 ' 38 ' 00 there are three figures. The first figure of the square root of a number will always be the highest figure whose square can be formed in the first (left hand) period. If the first period is 1, the first figure of the root will be 1. If the first period is 6, the first figure of the root will be 2. If the first period is 28, the first figure of the root will be 5. What will be the first figure of the root if the first period is 36; 40; 49; 55; 60; 65; 75; 85; 98? Find the number of figures in the root, and the first figure which is always sure to be right, then proceed to find the next figure. This is found somewhat as a quotient figure in division is found. After two figures of the root are found, call them both tens consider them as one and find the next figure, and so on. EXAMPLE 1. Find the square root of 169. OPERATION EXPLANATION. There are two periods and 1'69 I 13 there will be two figures in the root. The first figure of the root must be 1. Take the square of 1 (1 ten) from the whole square (169) 20 23 69 69 and 69 is left. 69 is 2 X the tens -f- the square of v the units, or 2TU X U, or U(2T + U). Now find the next figure, or U, the number wfeich multiplied by twice the tens plus itself will produce 69. At the PRACTICAL MEASUREMENTS 131 left of 69 write 2 X the tens, or 20, and by inspection the next figure is found to be 3. Add 3 to twice the tens already found and get 23, (2T + U). Multiply this by 3 and it is found that the estimate was correct. EXAMPLE 2. Find the square root of 6770404. OPERATION EXPLANATION. There will be four 6'77'04'04 I 2602 figures in the root. The first figure of . the root is 2, for the greatest square in the first period is 4. Taking the square 40 46 277 276 of 2 from the first period, 2 is left. Bringf down the next period and from 5200 10404 the result, 277, find the next figure of 5202 10404 the root. 277 is twice the first figure X the second + the square of the second + a possible remainder. To find the second figure divide 277 by twice 2 tens, the first root figure, or 40, and get 6, which is probably the second figure of the root. Add 6 to 40, and then multiply 46 by 6 instead of multiplying 40 by 6 and adding 6x6. Subtract 276, (46 X 6), from 277 and obtain 1. Bring down the next period and the result is 104. Divide this by twice the first figure of the root, which is now 26 tens, to find the second figure. Twice 26 tens are 520 (26 X 2 with one cipher annexed). But 520 is not contained in 104, so the next figure of the root is 0. Call 260 tens the first figure of the root, and proceed to find the last figure. Bring down the next period and 10404 is the dividend. Divide it by twice the found part of the root (260 tens), which is 5200 (260 X 2 with one cipher annexed). The quo- tient is 2. Add 2 to the "trial divisor," 5200, and the "complete divisor" is 5202. Then divide 10404 by 5202, and the quotient is exactly 2. Steps. 1. Beginning at the left point off into periods of 2 figures each. 2. Find the first figure of the root. 3. Take the square of the first figure from the first period and bring down the next period. 4- At the left write twice the part of the root already found and annex one cipher. 5. Estimate the next figure of the root. 6. Add this new figure to the trial divisor of the fourth step. 132 MODERN COMMERCIAL ARITHMETIC 7. Divide and bring down the next period. 8. Kepeat 4, 5, 6, 7, and so on. NOTES. 1. If at any time the product of the divisor and last fig- ure of the root is greater than the dividend, the quotient figure is too great and must be made less the same as in long division. 2. If there is a remainder after finding the last integral figure of the root ciphers may be annexed, and the root continued as a decimal. PROBLEMS Find the square root of : 1. 576. 4. 1522756. 7. 15876. 2. 5625. 5. 7 to 3 decimal places. 8. 2645731. 3. 42436. 6. 2. 9. ff. NOTE In extracting the square root of a fraction the root of each term may be found separately, or the fraction may first be reduced to a decimal. 10. Extract the square root of -Jf. 11. What is the length of the side of a square lot that con- tains 15 A.? 12. A man wishes to set out 15625 trees in a square so that there shall be the same number of trees in rows each way. How many trees should be planted in a row? IS. How many hills of potatoes, each 3 ft. from the others in rows 3 ft. apart, can be planted in an 8-A. square lot? 14. At $1.40 per rod, what will it cost to fence a field of 28 A. twice as long as it is wide? 15. A man wishes to erect a building having 2900 feet of floor. If he wishes its length to be four times its width, what must be its dimensions? To Find the Area of a Triangle by Square Root 232. EXAMPLE. Find the area of a triangle whose sides are 12 ft., 8 ft., and 16 ft., respectively. Steps in the Operation. 1. Add the three sides and divide the sum by 2. 2. From this half-sum subtract each side separately. PRACTICAL MEASUREMENTS 133 3. Find the product of the three remainders and the half- sum. 4. Extract the square root of the product. This will give the area. OPERATION 8 + 12 + 16 = 36 36+ 2 = 18 18- 8 = 10 18-12= 6 18-16= 2 18 x 10 x 6 x 2 = 2160 N/2160 = 46.475, area in square feet PROBLEMS 1. The sides of a triangle measure 21 ft., 40 ft., and 45 ft. Find the area of the triangle. 2. How many acres in a triangular field whose sides measure 48 rd., 62 rd., and 78 rd.? 3. How many square feet in the gable end of a building that is 80 ft. wide, if the length 01 each side of the roof is 50 ft.? 4. How many acres in a triangular field which measures 36 rd., 24 rd., and 42 rd.? 5. What is the diameter of a circular field that contains 1 acre? NOTE. Diameter 2 X .7854 = area. 6. Each side of a triangle is 62 ft. What is its area? 7. If the area of a circular field is 10 A., what is its diameter? 8. How many rods of netting must be purchased to enclose a circular park of % acre? 9. What is the diameter of a pipe if the area of a cross- section is 8 sq. ft.? 10. What is the area of a triangle whose sides are 23 rd. , 28 rd., and 31 rd.? 11. A fence 100 yd. long will enclose how much land in the form of a circle? How much in the form of a square? 134 MODERN COMMERCIAL ARITHMETIC 12. Find the diameter of a circle equal in area to a square 12 ft. on a side. IS. Find the side of a square equal in area to a circle whose diameter is 100 ft. 14. If a cistern is to be 6 ft. deep, what must be its diam- eter in order that it may contain 50 bbl.? 15. The three sides of a field are 75 rd., 60 rd., and 53 rd. What is its area? 16. I wish to build a bin that will hold 1000 bushels of wheat. If I make it 4 ft. deep, how large square must it be? 17. Find the area of a triangle whose sides are 19 yd., 21 yd., and 24yd. 18. Find the contents in bushels of a bin 5 ft. deep, whose bottom is an equilateral triangle 8 ft. on a side. 19. Find the number of square feet of roofing on a house 40 ft. square. The roof consists of 4 parts, each slanting from a side of the house to the peak in the center. The distance from the peak to one corner of the roof is 80 ft. 20. What is the area of a triangle whose sides measure 20ft., 24ft., and 30 ft.? The Eight Triangle 233. The Eight Triangle, or a Eight-Angle Triangle, is a triangle having one right angle. 234. The side opposite the right angle is the Hypotenuse. If the base is 4 and the altitude 3, the hypotenuse is 5. NOTE. As shown in the diagram, if a square is drawn on each side of a right triangle the square on the side of the hypotenuse is equal to the sum of the squares on the other two sides. This is true of any right triangle. Principle. In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. PRACTICAL MEASUREMENTS 135 Hypotenuse 2 = base 2 -f altitude 2 B 2 = H 2 - A 2 A 2 = H 2 -B 2 EXAMPLE 1. The hypotenuse of a right angle is 10 and the altitude is 6. What is the base? OPERATION B 2 = 10 2 - 6 2 B 2 = 100- 36, or 64 B = \/64, or 8 EXAMPLE 2. The base of a triangle is 12 ft., the altitude is 16 ft. Find the hypotenuse. OPERATION H 2 = 16 2 + 12 2 H 2 = 2564- 144, or 400 H = v/400, or 20 PROBLEMS 1. The hypotenuse is 15 yd., the base is 12 yd. What is the altitude? 2. If the altitude is 15 ft., the base 20 ft., what is the hypotenuse? 3. How many acres in a field in the form of a right triangle if the hypotenuse is 70 rd. and the base is 56 rd.? 4. Find the diagonal of a square 22 ft. on a side. 5. Find the distance from the lower southeast corner of a room to the upper northwest corner, if the room is 18 ft. by 16 ft., and 10ft. high. 6. I wish to put a steel brace against a brick wall. If the brace is to meet the wall 18 ft. above the base, and is to stand 7 ft. from the foot of the wall, how long must the brace be? 7. How long must a wire be to reach from the top of a stack 60 ft. high to a stake 45 ft. from the base of the stack? 8. Find the area of the surface of the sides of a squaro pyramid 125 ft. on a side, if the apex is 90 ft. above the base. 136 MODERN COMMERCIAL ARITHMETIC 9. There are two poles 90 ft. apart, each 110 ft. high. How far is it from the top of one to the middle of the other? 10. What must be the length of the rafter of a house 28 ft. wide, if the peak of the roof is to be 10 ft. above the plate and the roof is to project 1|- ft.? 11. What is the size of the largest square that can be cut out of a circular sheet of metal 2 ft. in diameter ? 12. Find the area of a right triangle whose base is 64 ft. and hypotenuse 80 ft. 13. What is the distance from a point to the top of a pole 120 ft. high, if the pole is 90 ft. from the place of meas- urement? 14* What is the length of the diagonal of a room 18 ft. by 28 ft? 15. Find the area of the six faces of a hexagonal pyramid 8 ft. on a side, the distance from the apex to any corner at the base being 20 ft. REVIEW PROBLEMS IN MENSURATION 1. Find the area of a circle 15 ft. in diameter. 2. How many square feet in the floor of a hexagonal room 10 ft. on a side? 3. How many feet of lumber in a board 18 ft. by 11 in. and 1| in. thick? 4- What is the contents in bushels of a bin 9 ft. square and 6 ft. deep? 5. How many cords of wood in a pile 38 ft. long, 7 ft. high and 4 ft. wide? 6. How much liquid will be required to fill a pipe f in. in diameter and 25 ft. long? 7. How many square feet of sheet iron will be required to build a smokestack 45 ft. high and 14 in. in diameter? 8. What is the value, at $68 per acre, of a triangular field which measures 36 rd., 42 rd., and 54 rd.? 9. What will it cost, at 20^ per square foot, to pave the intersecting diagonal walks of a park 40 yd. square, if the walks are 6 ft. wide.? PRACTICAL MEASUREMENTS 137 10. How many cubic feet of marble in a cylindrical monu- ment 9 ft. in diameter and 32 ft. high? 11. How many bricks in a chimney 6 ft. square and 48 ft. high, if the chimney wall is 1 ft. thick? 12. Find the number of yards of carpet, f yd. wide, neces- sary to cover the floor of a room 21 ft. by 25 ft. 13. If water weighs 62 Ib. per cubic foot, what is the weight of the water in a tank which measures 4 ft. by 2 ft. by 14ft.? 14. How many feet of lumber in a timber 8" x 9" and 28 ft. long? 15. How many feet of fence will enclose a square field of 8 A.? 16. Find the number of cubic feet of masonry in the walls of a cellar 16 ft. square on the inside and 9 ft. deep, if the wall is 20 in. thick. 17. How many loads of earth must be removed in digging a cellar 20 ft. wide, 46 ft. long, and 7 ft. deep? 18. Find the capacity in barrels of a tank 4 ft. wide, 10 ft. long, and 5 ft. deep. 19. What will be the cost, at $28 per M, of a stick of timber 26 ft. long, 12 in. wide, and 9 in. thick? 20. How many square yards of plastering are there on the walls and ceiling of a room 21 ft. wide, 27 ft. long, and 11 ft. high, allowing for 7 doors 4 ft. wide and 8 ft. high, and for 8 windows 4 ft. wide and 7 ft. high? 21. How many cords of stone will be used in building a foundation wall for a factory 80 ft. by 110 ft., if the wall is to be 10 ft. high and 2 ft. thick? 22. How many brick will be used in erecting a brick house 24 ft. wide, 36 ft. long, and 18 ft. high, if the wall of the house is made 1^ ft. thick, and, allowing for 3 doors 3^ ft. by ?i ft., and for 24 windows each 3 ft. by 6 ft.? NOTE. 22 brick will make a cubic foot of wall. 28. How many feet of lumber in a stick 14 ft. long, 9 in. wide and 6 in. thick? 138 MODERN COMMERCIAL ARITHMETIC 24. How much lumber will be required to build a fence 6 boards high around a square lot of 4 acres, if the boards are 6 in. wide and are to be nailed to posts 4 in. square and 8 ft. long, set 8 ft. apart? How many posts? 25. A bin 6 ft. by 9 ft. by 5 ft. will contain how many bushels of potatoes? 26. A builder has planned a house 27 ft. by 45 ft. How many feet of lumber will be required : (a) For the sills, which are to be 8 in. by 3 in.? (b) For the studs, which are to be 2 in. by 4 in. by 18 ft., and are to be placed at intervals of 18 in.? (c) For the sleepers, or joists, of 2 stories, which are to be 2 in. by 10 in., and are to be laid 18 in. apart? (d) For the plates, which are to be 4 in. by 4 in.? (e) For the rafters, which are to be 3 in. by 5 in. and are to be 1-J- ft. apart? The roof is to project 2 ft. and the peak is to be 10 ft. above the plates. (/) For the 2 floors, if the flooring is to be 1J in. thick? (g) For the siding for the outside? (h) How many square feet of roofing will there be? \i) How many square yards of plastering will there be on the walls and ceiling of both stories, the lower rooms being 9 ft. high, and the upper ones 7 ft. high? (/) If the cornice is to be 2 ft. wide, what will it cost at $80 per M? 27. A cellar 22 ft. by 34 ft. and 9 ft. deep (inside measure- ments) is to be built. (a). If the wall is to be 20 in. thick, how many cubic feet of masonry will it contain? (b) How many cords of stone will be required? (c) How many loads of earth must be removed in digging the cellar? (d) How many square yards of plastering will there be on the walls of the cellar? (e) How many square feet of cement will there be on the cellar bottom? PERCENTAGE 235. OPERATIONS WITH HUNDREDTHS PROBLEMS 1. Find .08 of 250; .25 of 730; .60 of 840. 2. Find 3 hundredths of $600 ; 12 hundredths of $180. 3. A merchant gained .05 of $1260. How much did he gain? 4. A dealer lost .06 of $790. How much did he lose? 5. 18 is how many hundredths of 24? 6. 9 is how many hundredths of 36? 7. .05 of a number is 60. Find the number. 8. A man began business with $1200 and gained .20 on his investment. How much had he then? 9. .15 of a number is 75. What is the number? 10. 12 is how many hundredths of 48? 11. 17 is how many hundredths of 136? 12. Find .22 of 184. 13. 10 is .02 of a number. What is .01 of the number? What is the number? H. 25 is .04 of a number. What is the number? 15. 18 is .35 of what number? 16. What is .45 of 560? 17. A merchant gained $42 on $350. How many hun- dredths of his investment did he gain? 18. A dealer lost .08 of his iuvescment. How many hundredths of his investment had he left? 19. A dealer gained .12 of his capital. How many hun- dredths of his original capital had he then? 20. $7.50 is how many hundredths of $25? 21. $46.30 is how many hundredths of $578.75? 22. What is .55 of 1125? 23. A man lost .14 of his money. How many hundredths had he left? 139 140 MODERN COMMERCIAL ARITHMETIC ^4. A man borrowed $850 and paid .06 of that amount each year for the use of the money. How much did he pay each year? 25. Find 43 hundredths of 700. APPLICATIONS OF PERCENTAGE 236. Per cent means hundredths. 237. In many business operations we say per cent instead of hundredths. Read the examples on the preceding page, and read per cent instead of hundredths. The meaning will be the same. 238. Percentage means a process involving hundredths, or per cent. The examples on the preceding page are examples in percentage. 239. Operations in percentage involve certain business terms, a certain knowledge of the way in which business is done, and decimals. How Per Cent Is Expressed 240. Since per cent means hundredths, it is commonly expressed as a decimal. It may also be expressed as a common fraction. Thus, .05 = 1 | Tr = -fa. 241. The sign of per cent is %. 242. Expression of hundredths : .01= l%= T *o- -05= 5%= T fo = *V .50 = 50% = T 5 oo=i. .10= 10% = T Vo = T V .75 = 75%= T VTr = f. -20= 20%=fVo = t- .25 = 25% = T Vo = i. 1.25 = 125% = U = H- .04 = 4% = T fo = fa. 1.50 = 150% = f| = H. 243. Hundredths as aliquot parts: 12|% = |. 33^% = J. 66f % = |. 16*% = I 37i% = f . 75 % = f . 20 % = f 50 % = i. 87^% = |. 25 % = i. 62|% = f . 125 % = 1. NOTE. When more convenient, these fractional parts should be used instead of the decimal equivalents. PERCENTAGE 141 244. Some expressions : *% is read one-third of one per cent, or one-third per cent. It is written decimally .00*, not .*. 8% = .08; .8% = .008; .08% = .0008; .008% = .00008. The sign % indicates two decimal places. Adding the sign % is equivalent to pointing off two decimal places, or dividing by 100. EXERCISES Express decimally: 1. 18%. 8. 6*%. 5. 66*%. 7. f%. 9. 7*%. 2. 93%. 4. 7%. 6. 25f%. 8. f%. 10. .024%. Express as common fractions : 1. 33*%. 5. 87*%. 5. 16|%. 7. *%. P. f%. 0. 83*%. 4. 125%. 6. |%. *. i%. ^. f%. Express as a per cent : *. * 3. *. 5. *. 7. f. P. A- 0. . 4. J. ^. i A f. *?. T V- 245. An example in percentage involves: (a) A decimal that indicates a part of a number. (b) The part of the number indicated by the decimal. (c) The number of which a part is indicated. The decimal = rate per cent, or rate The part = percentage The number = base 246. The decimal that shows how many hundredths of a number are taken is the Eate Per Cent. 247. The number of which the hundredths are taken is the Base. 248. The number that is a given number of hundredths of the base is the Percentage. 249. When the percentage is added to the base, the sum is called the Amount. 250. When the percentage is subtracted from the base, the remainder is called the Difference. 142 MODERN COMMERCIAL ARITHMETIC 251. To Find the Percentage MENTAL PROBLEMS 1. Find i of -36; 25% of 36. 2. What is y 1 ^ of 750? 10% of 750? 3. What is .02 of 800? 2% of 800? 4. What is 33% of 72? of 48? of 60? 5. Find 16f % of 96. 6. Find 25% of 28; of 44; of 64. 7. Find 62% of 72; of. 144. 8. Find 66f% of 15; of 24. 9. Find 40% of 250. 10. Find 12| % f 40 yd. FORMULA Base x rate = percentage PROBLEMS Find the percentage, the amount, and the difference: 1. 24% of 360. 11. 12|% of 840 cows. 2. 42% of $700. 12. 37|% of 944 horses. 3. 65i% of $840. 13. 16f % of $2563.92. 4. 83% of $762.50. 14. 84% of 4672 Ib. 5. 29|% of $538.20. 15. 112% of $354.62 6. 25% of $684.40. 16. 108^% of 6457. 7. 75% of $932.60. 17. 125% of 6820. 8. 33% of $638.60. 18. 35f% of $921.30. 9. 87i% of $9734. 19. 53% of $749.80. 10. 66f% of $1248. 20. 28% of 4648. 21. A drover bought 780 sheep, and sold 30% of them. How many had he left? 22. A dealer bought 34% of a stack of hay containing 18 T. 14 cwt. How much hay did he buy? 28. 23-|% of a barrel of oil leaked out. If the barrel held 48 gal., how much was lost? 24. How much lead will be obtained from 580 tons of ore, if the ore yields 15% of metal? PERCENTAGE 143 25. If cloth will shrink 4-^%, what will be the shrinkage on 128 yd.? 26. If .27% of coloring is to he mixed with white paint, how much coloring must be used with 3 T. 7 cwt. 80 Ib. of paint? 27. In making a certain medicine, .04% of arsenic is used. How much arsenic will be used in making 420 Ib. of the medicine? 252. To Find the Bate MENTAL PROBLEMS 1. 8 is what part of 100? 20 is what part of 100? 2. Of 100, 15 is what part? 35? 17? 82? 8. 4 is what part of 5? 7 of 9? 8 of 15? 4. 4 is how many hundredths of 5? 3 of 4? 4 of 16? 5. 4 is what per cent of 5? of 8? of 16? of 32? 6. 8 is what per cent of 16? of 24? of 12? 7. 6 is what per cent of 12? of 24? of 18? 8. 9 is what per cent of 18? of 27? of 36? 9. 12 is what per cent of 24? of 18? of 36? of 60? 10. What per cent of 44 is 11? 22? 33? 11. What per cent' of 60 is 15? 12? 20? 30? 45? 12. What per cent of 4 is f ? |? f ? 4? FORMULA P (percentage) = E (rate) x B (base) PROBLEMS 1. What per cent of 270 is 90? 2. What per cent of 38 is 27? 3. 45 is what per cent of 60? 4. Of 165, 15 is what per cent? 5. What per cent of $860 is $520? 6. $750 is what per cent of $500? 7. 140 sheep are what per cent of 560 sheep? 8. What per cent of $900 is $1200? 144 MODERN COMMERCIAL ARITHMETIC 9. 3465 is what per cent of 4587? 10. What per cent of 860000 is 580? 11. What per cent of 720 is 24000? 12. 480000 is what per cent of 12000000? 13. T 5 T is what per cent of T \? 14. What per cent of .86 is .32? 15. What per cent of 6.25 is 98.75? 16. The United States silver dollar weighs 26.729 gr,, and the Japanese dollar weighs 26.9564 gr. The weight of the American dollar is what per cent of the weight of the Japanese dollar? 17. A hank has $257326 cash on hand, and its deposits are $10658730. The bank report will state that the cash on hand is what per cent of the deposits? 18. A bank has received $14873580, and can repay only $3672500. What will a man receive who has deposited $1260? 19. The following table shows the population and number of deaths for a period of time in three cities : POPULATION NUMBER OF DEATHS 2768000 138 1875000 116 2162000 154 Find the death rate (per cent) for each city. 20. Three substances are mixed in the proportion of 5, 21, and 38. Each substance is what per cent of the mixture? How many pounds of the first substance in 85 Ib. of the mixture? 21. In a school of 55 pupils, there were 16 absences in 10 days. What was the rate per cent of attendance? 22. If out of 76000000 passengers 420 are killed, what per cent of the passengers are killed? 23. If a bushel of apples weighs 50 Ib., and 10 bu. of apples will make 32 gal. of cider weighing 8 Ib. per gallon, what per cent of the apples becomes cider? 24. From a substance weighing 8 Ib. 4 oz. (Troy), 7 dr. of a mineral were obtained. What per cent of mineral did the substance yield? PERCENTAGE 145 353. To Find the Base MENTAL PROBLEMS 1. 25 is of what number? 2. 25 is 20% of what number? 3. 30 is 33% of what number? 4. 16 is 8% of what number? 5. 12 is 40% of what number? 6. Of what number is 12 60%? 7. 15 is 75% of what number? 8. Of what number is 14 7%? 9. Of what number is 20 40%? 10. Of what number is 25 37|%? 11. Find the number of which 20 is 20%. 12. Find the number of which 40 is 5%. 13. Find the number of which 60 is 12%. 14. Of what number is 18 6%? 15. Of what number is 27 33%? 16. 100 is 12|% of what number? 17. 32 is 62i% of what number? 18. 72 is 9% of what number? FORMULA PROBLEMS 1. Find the number of which 10608 is 17%. 2. $43.40 is 32% of what sum? 3. 472 is 33% of what number? 4. f is 40% of what number? 5. Of what number is 675 16f %? 6. Of what number is 924 .056%? 7. Of what number is 87 %? 8. Find the number of which 3.045 is 24%. 9. .0000315 is .08% of what number? 10. Find the number of which 530100 is .57%. 11. A business pays a net gain of $2865 per annum. A buyer offers for the business a sum of which the net gain shall be 8%. What is the value of his offer? 146 MODERN COMMERCIAL ARITHMETIC 12. A company with a capital of $685750 pays annual divi- dends to the amount of 444573.75. The next year the dividends are $76242.30, and the company wish to report a capitalized value of their business that shall yield the same per cent of dividends as did the capital the year before. What will be the reported capital of the company the second year? 13. A mine pays $623750 annually. If invested money pays on an average 4% of itself as profits, what is the cash value of the mine? H. A certain business pays 5% dividends. How much must be invested in the business to produce an annual income of $1375? 15. If wine contains 11 J% of alcohol, how much wine must be purchased by a distiller who wishes to make 586 gal. of alcohol from the wine? 16. If a certain paint contains .7% of zinc, how much paint can be made from 200 Ib. of zinc? 17. A solution contains .003% of arsenic. How many grains of the solution must be taken to get 12 gr. of arsenic? 18. If an agent receives 3% commission for selling goods, how many goods must he sell to earn $275? 19. A manufacturer wishes to make a composition metal to consist of 3S% of gold, 5% of silver, 4% of tin, and the remainder copper. If he has 120 Ib. (Troy) of gold, how much of each of the other metals must he add to it? 20. A custom miller takes 8% of the grist for grinding. How many pounds of oats must be taken to the mill to get 300 Ib. of feed? 254:. To Find the Amount and Base MENTAL PROBLEMS 1. 10 is what per cent of 10? 2. Any number is what per cent of itself? 3. The base is always what per cent of itself? 4. The percentage is what of the base? 5. What is the base plus the percentage called? 6. What is 100% plus the rate per cent called? PEKCENTAGE 147 7. If the rate is 8%, what is the amount per cent? 8. If the base is 100 and the rate 8%, what is the amount? ?. The amount in the last question is what per cent of the base? 10. If the amount per cent is 110%, what per cent is the base? 11. If the amount is 110 and the amount per cent is 110%. what is the base? 12. If the amount per cent is 115% and the base is 300, what is the amount? 13. If the amount per cent is 115% and the amount is 345, what is the base? 14. If the amount per cent is 107% and the base is 400, what is the amount? 15. If the amount per cent is 107% and the amount is 428, what is the base? 16. If the rate per cent is 5 % and the base is 600, what is the amount? 17. If the rate per cent is 5% and the amount is 630, what is the base? FORMULA B -f percentage = amount 100% + rate per cent = amount per cent Base x amount per cent = amount Amount -f- amount per cent = base PROBLEMS Find the base : 1. When the amount per cent is 112^ and the amount is 1000. 2. When the amount is 1680 and the amount per cent is 105. 3. When the amount is $29.38 and the amount per cent is 113. 4- When the amount per cent is 103 and the amount is 460. 5. When the amount per cent is 102^ and the amount is 65.83. 148 MODERN COMMERCIAL ARITHMETIC 6. When the amount is $1268 and the rate per cent is 4. 7. When the rate is 12^-% and the amount is $1475.65. 8. When the amount is 763 and the amount per cent is 100.7. 9. When the amount is $528.25 and the rate is 12|%. 10. W T hen the rate is .06% and the amount is $1194.50. 11. What number increased by 25% of itself is 855? 12. If it costs 3% of the price of iron to buy it, how much iron can be bought for $1648? IS. If a jeweler adds to gold an alloy equal to 21% of the weight of the gold, how much pure gold must be used in mak- ing 31 Ib. (Troy) of the alloyed gold? 14. If metal bars will expand -J % when heated, how long should they be made if, after expanding, they should be 18 ft. long? 15. If a business pays 8% per annum, how much must a man invest in the business so that at the end of the year his interest in the business shall be worth $10000? 255. To Find the Difference and Base MENTAL PROBLEMS 1. What is the base less the percentage called? NOTE. 100% less the rate per cent is called the Difference Per Cent. 2. If the rate is 8%, what is the difference per cent? 3. If the base is 100 and the rate is 8%, what is the differ- ence? 4. In the above example, the difference is what per cent of the base? 5. If the difference per cent is 90% and the base is 100, what is the difference? 6. If the difference is 90, and the difference per cent is 90%, what is the base? 7. If the difference per cent is 85% and the base is 300, what is the difference? PEKCEKTAGE 149 8. If the difference "per cent is 85% and the difference is 255, what is the base? 9. If the difference per cent is 95% and the base is 400, what is the difference? 10. If the difference per cent is 95% and the difference is 380, what is the base? 11. If the rate is 6% and the base is 600, what is the difference? 12. If the rate is 6% and the difference is 564, what is the base? FORMULAE 100% - rate per cent = difference per cent Base percentage = difference Base x difference per cent = difference Difference -*- difference per cent = base PROBLEMS Find the base : 1. When the difference per cent is 92 and the difference is 1656. 2. When the difference is $245 and the difference per cent is 86. 3. When the difference is 750 and the rate is 15%. 4. When the rate is 12| per cent and the difference is $435. 5. When the difference per cent is 79 and the difference is $692.40. 6. When the difference is 804 and the difference per cent is 67. 7. When the rate is 6% and the difference is $379.25. 8. When the difference is $119.36 and the difference per cent is 97.25. 9. When the rate is .0036 and the difference is $274.86. 10. What number diminished by 13% of itself equals 1827? 11. If the waste in melting a metal and making it into articles is 2%, how much metal must be used to make 1865 Ib. of the articles? 150 MODERN COMMERCIAL ARITHMETIC 12. If cloth, shrinks 3% in length "and a man wishes to use a piece 136 yd. long after shrinking, how much cloth must he buy? 18. How many pounds of alloyed gold must be taken to obtain 86 Ib. of pure gold if it contains 22 %% of alloy? 14- A customer ordered 380 of a certain article, and remitted money enough to pay for them, with instructions to send as many articles as the money would buy in case there should be any discount on the goods. If a discount of 5% is allowed, how many articles should be sent him? 15. If it costs 3% of the price of goods to sell them, how many goods must be sold to net the dealer $1000? PROFIT AND LOSS QUESTIONS 256. 1. Are goods ever sold at a loss to the dealer? 2. Show how each of the following may cause a dealer to sell below cost: fire, flood, competition, out-of-style goods, lack of storage, desire to advertise, supply of perishable goods. NOTE. The rate of gain or loss is usually computed as a per cent. S. Comparing the terms of percentage with those of profit and loss, to what does the cost (to the dealer) compare? The whole gain or loss corresponds to what term in percentage? The per cent of profit or loss? The selling price if at a gain? The selling price if at a loss? MENTAL PROBLEMS 1. Sold a hat that cost $2 at a gain of 20%. What was the gain? What was the selling price? 2. Sold a cow that cost $24 at a gain of 12|%. What was the gain? The selling price? 8. Sold an article that cost $30 at a loss of 33%. What was the loss? The selling price? 4. Sold an article that cost $50 at a gain of 15%. What was the gain? The selling price? PERCENTAGE 151 5. Bought a book for $2 and sold it for $3. How much was gained? What per cent of the purchase price was gained? 6. Bought a watch for $12 and sold it for $16. How much was gained? What part of the purchase price was gained? What was the gain per cent? 7. Gain or loss is reckoned at a certain per cent of what sum? 8. Bought a horse for $50 and sold it for $40. How much was lost? What part of the cost was lost? What was the loss per cent? 9. If an article costs $2.40, for how much must it be sold to gain 25%? 10. A man bought a wagon for $45 and paid $15 for repairs. If he then sold the wagon for $70, what was his gain per cent? 11. If you buy a book for $2. 60 and know that the dealer made a profit of 30%, how much did the book cost the dealer? 12. At what price must an article be bought so that it may be sold for $2.80 and make a profit of 40%? IS. A man wishes to sell shoes for $2.50 and make a profit of 25%. What is the most he can afford to pay for the shoes? 14. A shoe dealer who sells at a profit of 20% made $1200. What did he pay for the goods sold? What did he sell them for? 15. If you know that a dealer makes $60 on the sale of a piano, and that he sells at a profit of 15%, you know that the piano cost him how much? 16. If an article is marked $4.80 and you know that the dealer makes a profit of 20%, you know that the article cost him how much? 17. If an article cost $1.50, how must it be marked to insure a gain of 30%? 18. A merchant buys eggs at 12^ per dozen and sells them at 14^ per dozen. What per cent does he gain? 19. Bought an article for its list price, and sold it for f its list price. What per cent was gained? 20. Bought an article for $24 and sold it for $20. What was the per cent of loss? 152 MODERN COMMERCIAL ARITHMETIC 21. If an article is bought for $8 and sold at a loss of what is the selling price? 22. If an article is bought 20% below cost and sold 20% above cost, what is the gain per cent on the investment? TABLE FOR MENTAL DRILL GAIN % GAIN Loss % Loss SELLING PRICE P URCHASE PRICE 1 10 $20 9 9 2 ? $15 $75 ? 3 12^- 9 9 $200 4 5 5 $25 9 $18 9 $ 20 6 9 9 $20 $ 18 7 8 9 9 $ 1.20 8 10 9 $90 9 9 ? $12 $92 ? 10 14 ? $ 5.70 ? 11 15 $ 4.50 9 9 12 20 9 9 $ 6.50 13 7 9 ? $ 14 14 33i $25 9 ? 15 6 9 ? $ 1.20 16 ? $24 $96 9 17 12 ? 9 $ 15 18 9 $12 9 $ 42 19 10 $22 ? 9 20 ? $ 6 $48 ? 21 16 9 $23.20 9 22 15 ? $17 9 23 9 $ 3 9 $ 18 n 37| $40 9 9 PROBLEMS 1. For how much must goods that cost $875.38 be sold so as to gain 21|%? 2. A merchant bought dishes for $1.15 per dozen. At what price must he sell them apiece to gain 20%? 3. A dealer bought wheat at 98^ per bushel and sold it at $1.10 per bushel. What per cent did he gain? PERCENTAGE 153 4. If a man buy 2400 brick for $8, for how much per M must he sell them to gain 15%? 5. On an investment of $2875 a man gained $213.50. What was his gain per cent? 6. Jan. 1, 1897, a man invested $3800. Jan. 1, 1900, his investment was worth $4425. What was his gain per cent per annum? 7. A furniture dealer sold furniture that cost him $1125, at 12% below cost. What was his loss? 8. If a horse is bought for $85 and sold for $75, what is the loss per cent? 9. A man invested $1680 and afterward sold his investment for $1475. What was his loss per cent? 10. A grocer bought a barrel of sugar containing 420 Ib. for $16. What will be his gain per cent if he sells it at 50 per pound? 11. A grocer bought 64 Ib. of ginger for $12.80. For how much per pound must he sell it to gain 25%? 12. A produce dealer bought 250 bu. of wheat at 850 per bushel, 360 bu. of oats at 320 per bushel, 415 bu. of corn at 280 per bushel, and 16 bu. of beans at $1.80 per bushel. If he sold the wheat at 900 per bushel, the oats at 300 per bushel, the corn at 300 per bushel, and the beans at $1.95 per bushel, what was his entire gain per cent? 13. A real estate dealer bought a lot for $2600 and paid $85 for a fence, $30 for digging a drain, $160 for street repair, and $3250 for a house. For how much must he sell the house and lot to gain 15% on his investment? H. A druggist bought alcohol at $2 per gallon and sold it at 350 per pint. What was his gain per cent? 15. If collars are bought for $1.10 per dozen and sold at 150 apiece, what is the gain per cent? 16. If plates must be sold at 200 apiece, for how much must they be purchased to insure a gain of 20%? 17. A fruit dealer bought 200 pineapples at 7f apiece. 13 of them spoiled, and he sold the remainder at 80 apiece. What was his gain per cent? 154 MODERN COMMERCIAL ARITHMETIC 18. A dealer lost of a load of fruit. At what per cent of profit must the rest be sold that he may gain 20% on the whole load? 19. Bought a farm of 260 acres for $28 per acre, and paid $320 for fencing, $850 for repairs, $175 for draining, $82 for taxes. At what price per acre must it be sold to gain 20% on the investment? 20. If shoes are bought for $3 per pair, how must they be marked that the dealer may make a reduction of 10% from the marked price and gain 20% on the cost? 21. What per cent is gained on goods marked 30% above cost and sold at 20% off from the marked price? 22. What is the gain or loss per cent on goods marked 25% above cost and sold at 25% off from the marked price? 28. It costs a publishing house $1.64 to produce a book. If the house wishes to make a profit of 25% on the first cost of the book, and allows the agents 40% of the sales of the book, at what price must the house have the agents sell the book? 24. At what price per dozen must caps be bought that the dealer may sell them at 30^ apiece and gain 20%? MENTAL EXERCISES Tell how: 1. To find the gain when the cost and per cent of gain are given. 2. To find the gain per cent when the cost and selling price are given. 3. To find the gain per cent when the gain and selling price are given. 4. To find the gain per cent when the gain and cost are given. 5. To find the loss when the cost and per cent of loss are given. 6. To find the loss per cent when the cost and selling price are given. 7. To find the loss per cent when the loss and cost are given. PERCENTAGE 155 8. To find the loss per cent when the loss and selling price are given. 9. To find the selling price when the gain and per cent of gain are given. 10. To find the selling price, the loss and loss per cent being given. 11. To find the cost, the gain and gain per cent being given. 12. To find the cost, the loss and loss per cent being given. 18. To find the cost, the selling price and gain per cent being given. 14- To find the cost, the selling price and loss per cent being given. COMMISSION AND BROKERAGE 257. A person who does business for another is an Agent. The book agent sells books for the publisher. The clerk in a store sells merchandise for his employer. The traveling salesman takes orders for goods to be filled by his house. A commission merchant resides in some city and receives goods to be sold by him. A man in New York may receive farm prod- uce from a farmer, sell the stuif at the market price, take out his charges, and remit the balance to the farmer. A broker buys and sells bonds, stock certificates, etc., for others. Some agents receive a salary. Some receive a percentage on the things they buy or sell. 258* The pay that an agent receives is called Commission. 259* The pay a broker receives is called Brokerage. 260. The agent's commission is computed on what he buys or sells. 261. The person for whom the agent transacts business is the Principal. A company, firm, or corporation is considered in business as a person. 262. Goods sent by a principal to an agent is a Consign- ment. 263. The principal is the Consignor, the agent is the Consignee. 156 MODEKN" COMMERCIAL ARITHMETIC 264. For taking the risk of loss from sales on credit or for pledging the quality of the goods bought, the agent makes a charge called Guaranty. 265. The charges that an agent may make to his principal are commission, guaranty, freight, storage, insurance, inspec- tion, etc. 266. The whole amount received by an agent from a sale or a collection is the Gross Proceeds. 267. The sum that remains after the agent has deducted all his charges is the Net Proceeds. 268. Sometimes a principal sends to his buying agent a sum of money (remittance) that includes the agent's commis- sion and' the sum to be invested. The agent is to invest as much as he can and still have enough left for his commission. 269. The sum actually invested in goods is the Prime Cost, or Net Cost. The net cost -lus all charges of the agent is the Gross Cost. 270. Comparing the terms of commission and brokerage with the terms of percentage : The gross proceeds = ? The net cost = ? The rate of commission = ? The commission = ? Purchase price + commission, or remittance to agent for investment = ? Selling price commission = ? 271. To Compute Commission and Brokerage MENTAL PROBLEMS 1. An agent receives 40% commission on his sales. If he sells and delivers $200 worth of books, how much does he keep for his commission? 2. If an agent sold $2500 worth of goods on 10% commis- sion, how much commission did he receive? How much did he send to his principal? PERCENTAGE 157 3. An agent receives 12|% commission on all the goods he sells. How many dollars' worth of goods must he sell to earn $100? 4. If a buying agent receives 5% commission for buying cotton, how much money must his principal send him to pay for $1 worth of cotton and the commission? If the principal sends $5.25 to pay for cotton and the agent's commission, how much cotton will the agent buy? If the agent receives a remit- tance of $1050, how many dollars' worth of cotton will he send his principal? 5. If an agent receives 3% commission for buying wool, what does $1 worth of wool cost the principal? How much must the principal send the agent to pay in full for $100 worth of wool? If the principal remits $206, how much wool will he receive? 6. If an agent's commission is 2% on all goods bought, how much will $2 worth of goods cost the principal? This cost is what per cent of the purchase price of the goods? 7. A farmer sent a commission merchant 500 bu. of wheat to be sold on a commission of 2%. If the wheat is sold at 80^ per bushel, how much should the farmer receive? 8. If an agent charges 2% commission for buying silk, how much money must the principal send to pay in full for $500 worth of silk? 9. If a collector receives 1% for collecting, how much must he collect to earn $200 per month? The principal receives what per cent of the sum collected? 10. If an agent's commission for buying flax is 4%, what per cent of the sum to be invested in flax must the principal send the agent? 11. How much does an agent earn who sells $3000 worth of goods on a commission of 7%? What sum does the principal receive? What per cent of the sales does he receive? 12. What will $1100 worth of goods cost a principal, if he pays his agent 6% commission? What per cent of $1100 will the goods cost the principal? 13. How many dollars' worth of goods must an agent sell 158 MODERN COMMERCIAL ARITHMETIC to earn $160, if his commission is 4%? If the principal wishes to receive $192 as a result of the sale, how many dollars' worth of goods must he send the agent? 14. If the agent's commission is 2% for buying, the prin- cipal must send the agent what per cent of the value of the goods the principal wishes to receive? If the principal wishes to receive $100 worth of goods, how much money must he send the agent? If the principal sends the agent $500, how many dollars' worth of goods will he receive? 15. If the agent's commission for buying is 3%, how many dollars' worth of goods will $412 secure the principal? PROBLEMS Find the commission, guaranty, and the gross cost: 1. Purchase price $875.60, rate of commission 2^%. 2. Prime cost $954, rate of commission 3%. 3. 360 bu. wheat at 900, commission 2%, guaranty %. 4. 684 bbl. apples at $1.85, commission 3%, guaranty 1%. 5. Prime cost $1287.50, commission 5|%, guaranty l-J-%, insurance 1%, freight $8.25. 6. 17685 ft. oak at $28.50 per M, commission 7%, guar- anty 1%. 7. 345 bbl. flour at $5.80, commission 250 per barrel, .guaranty 2%. 8. 72860 Ib. hay at $12 per ton, commission 3%, guaranty 2%, charges 37-J-^ per ton. 9. 127 yd. carpet at $2.40, commission 3%. Find the commission, guaranty, and net proceeds: 10. Gross proceeds $867.25, commission 3^%. 11. Sold 215 bu. corn at 450, commission 2-J% , guaranty % . 12. 11680 Ib. hay at $11 per ton, commission 2%, guar- anty %. 13. 75600 ft. cherry at $41 per M, commission 4%. 14. Gross proceeds $1825.30,commission 5%, guaranty 1%%. 15. 1625 books at $3.75, commission 25%, charges 50 per volume. PERCENTAGE 159 16. Gross proceeds $2368, commission 22%, guaranty 3%. 17. Gross proceeds $736.80, commission 18%, guaranty , freight $45, insurance $6.50. 18. 65 T. coal at $5.25, commission 5%, guaranty 1%. Find the prime or net cost: 19. Gross cost $2838.71, commission 7%. 20. Commission $365.25, rate of commission 2^%. 21. Gross cost $723, commission 6%, guaranty 1%, freight $15. 22. Gross cost $1235.40, commission 8%, guaranty 2%. 23. Commission $117.30, rate of commission 4%. 24. Guaranty $213.75, rate of guaranty 3%. 25. Gross cost $6872, commission 12|%, guaranty 2%%. 26. Total charges $157.25, commission 5%, guaranty 1%, freight $12.15. 27. Gross cost $1673.20, commission and guaranty 14%. 28. Commission $275, rate of commission 2^%. 29. Guaranty $218, rate of guaranty 3%. SO. Total charges $753, commission 4%, freight $27. 31. Commission $513, rate of commission 9%. Find the rate of commission : 32. Commission $412, gross proceeds $13800. 33. Commission $357, net proceeds $6783. 84. Guaranty $47, gross proceeds $1342.86. 35. Guaranty $195, net proceeds $2705. 36. An agent sold $4682 worth of goods on 8% commis- sion. He charged 1% for guaranty and $27 for expenses. Find the commission, guaranty, and net proceeds. 37. An agent bought $5387 worth of goods on 13% com- mission. He charged 2% for guaranty and $125 for expenses. Find the gross cost. 38. A principal sent his agent $2965 to invest in lace, tak- ing out commission and expenses. If the commission was 6% and the expenses $17, how much lace did the principal receive? 39. A real estate agent sold a farm for $12825 on a com- mission of 3%, and the stock on the farm for $2360 on a com- mission of 4%. Find the commission. 160 MODERN COMMERCIAL ARITHMETIC 40. A principal sent his agent $1475, with instructions to invest it in oil. If his commission is 4%, how much oil will he purchase? 41. How many dollars' worth of merchandise can be pur- chased for $3691, if the agent's commission is 5%? J$. A commission merchant received a consignment of 831 bbl. of flour to sell on a commission of 4%. He sells the flour at $7. 30 per barrel and pays $14.50 for cartage. How much should he remit his principal? 4B. An agent received $2935 to invest in wool, taking out his commission of 3^%. How much commission did he receive? 44- An agent bought 28735 bu. of corn at 45^ per bushel, on a commission of 1|%. If he paid $793.28 for freight, $112 for storage, and $41 for cartage, how much should he receive from his principal? 45. One agent charged $72 for selling a lot for $2400, and another agent charged 3^% commission for selling a lot valued at $2700. Which agent charged the higher rate of commission? TEADE DISCOUNT 272. Manufacturers and dealers issue catalogues and cir- culars containing a description of their goods and a price list. The List Price is the price given in the catalogue or circular. Dealers often sell for less than the list price. What the dealer "throws off" is the Trade Discount. The list price may be put high enough so that the dealer can allow a discount and yet make a desired profit. A purchaser is pleased to buy goods at a discount. Sometimes, after getting out a price list, the market price of goods falls, and instead of changing the price list the dealer may retain the old price list and offer a discount. If the price varies, the dealer may vary the discount. If the price falls three times in succession, the dealer may offer three different discounts. PERCENTAGE 161 When there are more trade discounts than one, they are called a discount series. 273. Trade discount is reckoned as a per cent of the list price. The list price less the discount is the Net Price. In a discount series, the first discount is reckoned on the list price, and each subsequent discount is reckoned on the net price the price after deducting the preceding discount. It makes no difference in what order the discounts of a series are considered. A series of 20%, 15%, and 10% is the same as 10%, 20%, and 15%. But 20%, 15%, and 15% is not the same as 20% and 30%. 274. Discounts are usually aliquot parts of 100, and opera- tions should be performed with aliquot parts when practicable. MENTAL PROBLEMS 1. Find the net price if the list price is $12 and the trade discount is $25%. 2. Find the net cost of 10 boxes of soap at $4 per box, less 12^-% discount. 3. If on a sale of $60 worth of goods a discount of 15% is allowed, what is the amount of discount? What is the net cost? 4. A dealer sold goods listed at $80, less 25%, 20%, 12%. What was the net price after the first discount? After the second? After the third? 5. Find the net selling price of an article marked $6, less a discount of 15%. 6. What is the cost of 25 yd. of cloth at $2, less discounts of 20% and 10%? 7. What is the cost of goods listed at $75 and sold at dis- counts of 20%, 25%, 33 J%, and 50%? 8. What will a $5 book cost if discounted at 15%? 9. Find the cost of goods listed at $20, less 30% and 10%. 10. What is the cost of 20 cases of glass at $27, less dis- counts of 33% and 12|%? 162 MODERN COMMERCIAL ARITHMETIC 11. Find the single discount equivalent to the discount series of 20%, 5%, and 25%. SOLUTION. The net price after the first discount is 80% of the list price. After the second discount the net price is 76 % of the list price. After the third discount the net price is 57% of the list price. If the net price is 57% of the list price, the discounts must be equivalent to 100% 57%, or 43%. Find the single discount equivalent to the following series: 12. 10%, 10%. 22. 50%, 10 %, 20 %. 13. 10%, 20%. 23. 25%, 20 %, 12%. 14. 15%, 20%. 24. 20%, 5 %, 25 %. 15. 20%, 5%. 25. 12%, 50 %. 16. 5%, 10%, 20 %. 26. 10%, 33%, 20 %. X 17. 10%, 10%, 10 %. 27. 10%, 16|%. 18. 20%, 20%, 20 %. 28. 24%, 25 %. IP. 20%, 10%, 121%. 29. 35%, 20 %. V 20. 6%, 10%. 50. 13%, 33*%. 21. 25%, 10%. 31. 12%, 25 %, 331%. 32. What must I ask for an article so that I may allow a discount of 20% and still receive $40 for it? NOTE. The net price is what per cent of the asked price? 83. Cloth costing $4 per yard must be marked at what price to gain 25% and allow a discount of 20%? 34* At what price must I mark goods costing $18, that I may allow a discount of 25% and sell them at a gain of 331%? PROBLEMS 1. What is the net cost of a bill of goods amounting to discounted at 30% and 18%? NOTE. Each discount may be taken out separately, or one equiv- alent discount may be used. 2. Find the net cost of a bill of goods invoiced at $825 and discounted at 25%, 20%, 10%. PERCENTAGE 163 3. Three houses list similar goods at $1260. One offers dis- counts of 25% and 25% ; another offers 20%, 20%, and 10% ; and the third offers 35% and 15%. What is the net price of each house? V' 4. I bought goods at 30% and 20% from the list price of $1475, and sold them at 20%, 10%, and 20% from the list price. How much did I gain? 5. A dealer offers bicycles at $100, subject to discounts of 40%, 20%, and 15%. How much will 1 doz. bicycles cost? 6. A book costs $5. What must be the marked price to gain 20%, and allow discounts of 20% and 25%? REMARK. The book must be sold for $5 plus 20 % of $5, or $6. A discount of 20% and 25% is equivalent to a discount of 40%. If 40% discount is allowed, then $6 is 60% of the marked price. 7. Carpet costing $4 per yard must be marked at what price /\to gain 25% and allow discounts of 15% and 20%? 8. Goods listed at $265, and bought at discounts of 28% and 12^%, must be marked at what price to gain 33-J-% and allow discounts of 24% and 16|%? 9. A dealer sells goods at a discount of 25% from the marked price and makes a profit of 25%. At what per cent above cost did he mark the goods? 10. Find the cost: LIST PRICE DISCOUNT (a) $ 126.50 15 %, 12% (V) $ 872.60 33%, 25 % (c) $ 150.25 30 %, 20 %, 15 % (d) $1285 40 %, 33 %, 12|% () $1673.80 15 %, 20 %, 16|% (/) $4896.20 35 %, 22 %, 8 % (g) $5325.75 24 %, 16 %, 12 % (4) $ 436.18 16f%, 12|% 0) $1120 30 %, 25 %, 15 % 0') $1736 45 %, 164 MODERN" COMMERCIAL ARITHMETIC 11. At what per cent above cost must goods be marked to allow a discount of: (a) 25 % and make 20 % on the cost? (b) 10 % and make 25 % on the cost? (c) 20 % and make 15 % on the cost? (d) 10 % and make 8 % on the cost? (e) 33% and make 12|% on the cost? (/) 40 % and make 25 % on the cost? (g) 10 % and make 50 % on the cost? (h) .25 % and lose 15% on the cost? (i) 50 % and neither gain nor lose? (/) 15 % and lose 10% on the cost? 12. Which is the better discount offer, 25%, 20%, and 15%, or 50%. MABKING GOODS 275. Merchants often mark the cost and the price of their goods with symbols instead of figures. Sometimes the selling price is in figures and the original cost in symbols. 276. In marking by symbols, ten letters of the alphabet are used to correspond to the ten figures of the Arabic notation. A word or phrase containing ten letters, no two being alike, is taken as a "key." Some words and phrases used as keys are: blacksmith, handsomely, what prices, cash profit, black horse. The value of each letter of the key is determined by its position in the word or phrase. Thus : blacksmith 1234567890 For $1.25 write Wc\ for $2.68 write lsi\ for $4.77 write cmm. Instead of repeating a letter, as in cmm, a letter not found in the key may be used as a repeater. Thus, $4.77 might be written cmr 9 using r to show that m is repeated. The last two letters in the expression represent cents. If a period is added, it shows that there are no cents. Thus, Ilk. means $125. PERCENTAGE 165 When the selling price is given in figures, and the cost mark is used, the selling price is written above the cost mark. Thus, ' means that the selling price is $1.50 and the cost is $1.25. ollc When both cost and selling price are given, two keys may be used, one for the cost and the other for the selling price. Thus, using "what prices" for the selling price and "black- filj^") Q smith" for the cost mark, -=~-^ means that the selling price is ollc $1.50 and the cost is $1.25. PROBLEMS 1. Write the following prices, using "handsomely" as the key: $1.25, $2.25, $4.17, $1.87, $2,65, 75^, 36^, $1.95, $3.33, $4.35. 2. Interpret with the same key: liss, smy, II, ahe, nos, das, eyy, may, hams. 3. Mark all of the above prices 25% advance, using the same key. 4. With the same key for the cost price and "what prices" for the selling price, mark the following : COST GAIN COST GAIN (a) $1.50 25 % (/) $2.00 15 % (b) .80 20 % (3) !- 60 25 % (c) 1.75 33J% (Ji) .90 16|% (d) 2.40 12*% (0 1.20 75 L (e) 1.25 33J% 0') 1-40 30 % MENTAL PROBLEMS Use the key "blacksmith" and y as a repeater. Give the cost mark for the following: 1. $1.45. 4. $ .85. 7. $7.50. 10. $3.20. 2. 2.25. 5. 1.60. 8. 8.25. 11. 6.85. 3. 5.20. 6. .90. 9. 2.15. 12. 1.75. 166 MODERN COMMERCIAL ARITHMETIC STORAGE 277. Storage is the price or amount charged for storing goods in a warehouse. Storage is usually computed by the weight, measure, or quantity of the goods stored. Sometimes it is computed as a per cent of the value of the goods. 278. The Term of Storage is the time for which the charge is made. Storage may be charged by the day, week, or month, and a fractional part of a term is counted as a full term. 279. Average Storage. When the quantity of goods in storage varies on account of additions to or withdrawals from the stock, charge is made on each quantity for the actual num- ber of days it is stored, and an average term of storage and an average quantity stored is ascertained. EXAMPLE. A man deposits 400 bu. of wheat, and after 20 days withdraws 100 bu. ; the rest he withdraws in 15 days. What is the storage charge at 1^ per bushel per month? SOLUTION. The storage of 400 bu. for 20 days is equivalent to the storage of 1 bu. for 8000 ^days, the storage of 300 bu. for 15 days is equivalent to the storage of 1 bu. for 4500 days, and the whole storage is equivalent to the storage of 1 bu. for 12500 days. The average time reduced to months is (12500 -f- 30 = 416f) 41? months. The charge for 1 bu. for 417 months is $4. 17 ; therefore the charge for 400 bu. for 20 days and 300 bu. for 15 days is 14.17. 280. Sometimes storage is paid on each withdrawal from the warehouse. This is Cash Storage. Sometimes storage is not paid until the last withdrawal is made. This is Credit Storage. PROBLEMS 1. (Simple Storage.) Find the storage on 4500 bu. of wheat for 3 mo. 20 da., at 1$ per bushel per month; 1735 bu. of corn for 4 mo., at f^ per bushel; 2863 bu. of oats for 4 mo. 15 da., at per bushel; 1160 bbl. of flour for 3 mo. 25 da., at 6^ per barrel; 845 bu. of potatoes for 1 mo. 18 da.., at 10. NOTE. Fractional terms count as whole terms. PERCENTAGE 167 2. (Average Cash Storage.) A dealer deposits in a ware- house 460 bbl. apples Oct. 3 ; 632 bbl. pears Oct. 6 ; 535 bbl. potatoes Oct. 15 ; and 782 bbl. apples Oct. 25. He withdraws the whole on Feb. 1. What are the storage charges at 4^ per barrel per month? OPERATION 460 bbl. for 121 da. = 1 bbl. for 55660 da. 632 bbl. for ? da. = 1 bbl. for ? da. 535 bbl. for ? da. = 1 bbl. for ? da. 782 bbl. for ? da. = 1 bbl. for ? da. Total storage = 1 bbl. for ? da. or ? mo. 8. Find the storage charges, at 1-J-^ per bushel per month, on the following deposits : July 21, 8140 bu. ; Aug. 1, 1670 bu. ; Aug. 15, 1985 bu.; Sept. 1, 2430 bu. All was withdrawn Dec. 20. 4. (Average Credit Storage.) Find the storage charges, at 3^ per bbl. per month, on the following : DEPOSITS WITHDRAWALS Sept. 1, 810 bbl. Sept. 12, 260 bbl. Sept. 15, 1460 bbl. Sept. 25, 483 bbl. Oct. 24, 735 bbl. Dec, 1, the remainder OPERATION 810 bbl. stored for 11 da. = 1 bbl. for ? da. 260 bbl. withdrawn 550 bbl. stored for 3 da. = 1 bbl. for ? da. 1460 bbl. deposited 2010 bbl. stored for 10 da. = 1 bbl. for ? da. 483 bbl. withdrawn 1527 bbl. stored for 29 da. = 1 bbl. for ? da. 735 bbl. deposited 2262 bbl. stored for 38 da. = 1 bbl. for ? da. Total storage = 1 bbl. for ? da. 168 MODERN COMMERCIAL ARITHMETIC 5. The deposits and withdrawals of goods at a warehouse were as follows : DEPOSITS WITHDRAWALS June 1, 240 bales June 18, 160 bales June 28, 673 bales July 3, 540 bales July 10, 490 bales Aug. 15, the remainder Find the storage bill at 6^ per bale per month. 6. Find the storage bill of the following, at 3

may amount to $697.67 in 1 yr. 4 mo.? 190 MODERN- COMMERCIAL ARITHMETIC PERIODIC INTEREST 329. Simple interest is simply interest on the principal. 330. Periodic Interest is interest on the principal and interest on the simple interest due at certain interest periods. 331. Interest maybe due annually, semi-annually, quar- terly, etc. 332. Annual Interest is interest on the principal payable annually, and simple interest on the interest that remains unpaid. 333. Semi- Annual Interest is interest on the principal pay- able semi-annually, and simple interest on the interest that remains unpaid. 334. Quarterly Interest is interest on the principal payable quarterly, and simple interest on the interest that remains unpaid. 335. In some States periodic interest cannot be legally collected. To secure periodic interest, a contract should specify it. EXAMPLE 1. Find the annual interest on $500 for 4 yr. at 6%, if no interest is paid until the principal is due. SOLUTION Int. on $500 for 1 yr. = $ 30 Int. on $500 for 4 yr. = 120 Int. on $30 (unpaid int.) for 3 yr., 2 yr., and 1 yr., or for 6 yr. = 10.80 Annual int. = $130.80 EXAMPLE 2. What is the semi-annual interest on $400 for 3 yr. 4 mo., at 6%? SOLUTION Int. on $400 for 6 mo. = $12 Int. on $12 for 2 yr. 10 mo., 2 yr. 4 mo., 1 yr. 10 mo., 1 yr. 4 mo., 10 mo., and 4 mo., or for 9 yr. 6 mo. = 6.84 Int. on $400 for 3 yr. 4 mo. = 80 Semi-annual int. = $86.84 INTEREST 191 PROBLEMS 1. What is the annual interest on $475, for 4 yr. 8 mo. , at 6 % ? 2. Find the semi-annual interest on $263, for 2 yr. 7 mo., at 6%. 3. Find the amount of interest due at the end of 5 yr. 3 mo., on a note for $218, at 6% annual interest. 4. What will $125 amount to in 2 yr. 8 mo., with interest at 8%, payable quarterly? 5. Find the annual interest on $436, for 5 yr,. 8 mo., at 5%. 6. Find the semi-annual interest on $1080, for 2 yr. 7 mo. 15 da., at 5%. 7. What will $125 amount to in 1 yr. 9 mo. 12 da., with quarterly interest, at 6%? 8. Find the amount of $528 for 2 yr. 9 mo. 18 da., with interest at 4%, payable semi -annually. 9. What will be the annual interest on $1750 for 3 yr. 11 mo. 8 da., at 4%? 10. Find the amount of $318 for 2 yr. 5 mo. 21 da., at 6% interest, payable semi- annually. 11. What will $162 amount to in 1 yr. 10 mo. 25 da., at 5% quarterly interest? 12. What will be the amount of $435 for 4 yr. 8 mo., at 4J% annual interest? 18. What sum will amount to $500 in 1 yr. 8 mo., at 6% semi-annual interest? COMPOUND INTEREST 336* Compound Interest is interest upon the principal and on the interest combined with the principal at regular intervals. For the purpose of finding the compound interest, the sim- ple interest is added to the principal at regular intervals, and the amount becomes the new principal on which the interest is computed. Interest may be compounded annually, semi-annually, or quarterly. 337. Compound interest cannot be enforced by law. Savings banks usually allow compound interest. 192 MODERN COMMERCIAL ARITHMETIC EXAMPLE. Find the interest on $250 for 2 yr. 6 mo. 15 da., at 6%, compounded annually. SOLUTION $250 = prin. 15 int. for first year $265 = amt. for first year 15.90 = int. for second year $280.90 = amt. for second year 9.13 = int. for 6 mo. 15 da. $290.03 = amt. for 2 yr. 6 mo. 15 da. 250 = original prin. $ 40.03 = compound int. NOTES. 1. As in the above example, the last interest period may be but part of a full period. 2. If interest is compounded semi-annually or quarterly, take one- half or one-fourth of the annual rate per cent. PROBLEMS 1. What is the interest on $640 for 2 yr. 8 mo., at 7%, com- pounded semi-annually? 2. What is the interest on $1230 for 1 yr. 3 mo. 15 da., at 8%, compounded quarterly? 8. Find the interest on $390 for 3 yr. 7 mo., at 5%, com- pounded annually. 4. Find the amount of $750 for 2 yr. 9 mo. 18 da., with 6% interest, compounded annually. 5. Find the interest on $375 for 6 yr. 8 mo., at 5%, com- pounded annually. 6. Find the amount of $520 for 3 yr. 9 mo., at 8% inter- est, compounded semi-annually < 7. What is the interest on $640 for 2 yr. 3 mo., at 10% interest, compounded quarterly? 8. What is the amount of $328 for 7 yr. 4 mo., at 5% inter- est, compounded annually? INTEREST 193 Compound Interest Table 338. Persons who have to make many computations in compound interest usually use a printed table like the following : TABLE SHOWING THE AMOUNT OF $1 AT COMPOUND INTEREST 1TEAR 2?o S 4^ 5^ 6# 7J6 1. 1.020000 1.030000 1040000 1.050000 1.060000 1.070000 2. 1.040400 1.060900 1.081600 1.102500 1.123600 1.144900 3. 1.061208 1.092727 1.124864 1.157625 1.191016 1.225043 4. 1.082432 1.125508 1.169858 1.215506 1.262477 1.310796 5. 1.104080 1.159274 1.216652 1.276281 1.338225 1.402551 6. 1.126162 1.194052 1.265319 1.340095 1.418519 1.500730 7. 1.148685 1.229873 1.315931 1.407100 1.503630 1.605781 8. 1.171659 1.266770 1.368569 1.477455 1.593848 1.718186 9. 1.195092 1.304773 1.423311 1.551328 1.689479 1.838459 10. 1.218994 1.343916 1.480244 1.628894 1.790847 1.967151 11. 1.243374 1.384233 1.539454 1.710339 1.898298 2.104852 12. 1.268241 1.425760 1.601032 1.795856 2.012196 2.252191 13. 1.293606 1.468533 1.665073 1.885649 2.132928 2.409845 14. 1.319478 1.512589 1.731676 1.979931 2.260904 2.578534 15. 1.345868 1.557967 1.800943 2.078928 2.396558 2.759031 16. 1.372785 1.604706 1.872981 2.182874 2.540351 2.952163 17. 1.400241 1.652847 1.947900 2.292018 2.692772 3.158815 18. 1.428246 1.702433 2.025816 2.406619 2.854339 3.379932 19. 1.456811 1.753506 2.106849 2.526950 3.025599 3.616527 20. 1.485947 1.806111 2.191123 2.653297 3.207135 3.869684 NOTES. 1. Any principal multiplied by the amount of $1 for any given time, at any given rate, is the amount of the principal, for the given time and rate. 2. The amount of $1 for any given number of years is equal to the product of the amounts of $1, for such periods of years whose sum is equal to the given number of years. To find the amount of $1 for 40 yr., multiply together the amounts for 20 and 20 yr., or for 15, 15, and 10 yr., etc. 3. For semi-annual interest, take | the rate for twice the time. 4. For quarterly interest, take \ the rate for 4 times the time. PROBLEMS Solve the following problems by the use of the table: 1. Find the compound interest on $632, for 15 yr., at 5%. 2. Find the interest on $1285, for v 9 yr. 9 mo., at 6%, compounded semi-ammally. 3. Find the amount of $750, for 3 yr. 8 mo., at 8%, com- pounded quarterly. 194 MODERN COMMERCIAL ARITHMETIC 4. What is the interest on $196, for 16 yr. 5 mo., at 6%, compounded annually? 5. What is the amount of $224, for 6 yr. 8 mo., at 8% interest, compounded semi- annually? 6. Find tho amount of $3567, for 3 yr. 6 mo., at 4%, com- pounded semi-annually. 7. What is the interest on $2687, for 7 yr. 9 mo. 15 da., at 5%, compounded annually? Find the compound interest on: PRINCIPAL RATE 8. $1428 8% 9. $ 732 7% 10. $ 523 6% 11. $ 176 12% 12. $ 391 5% 13. $ 746 3% 14* $ 412 4% 15. $ 834 4% TIME 9 yr. 7 mo. 14 yr. 10 mo. 24 yr. 5 yr. 7 yr. 35 yr. 16 yr. 9 mo. 14 yr. 8 mo. 5 mo. PAYABLE quarterly annually annually quarterly semi-annually annually annually semi-annually Compound Interest Amount Table TABLE SHOWING THE AMOUNT OF $1 INVESTED AT THE BEGINNING OF EACH YEAR FOR A SERIES OF YEARS, AT COMPOUND INTEREST YEAR %% 2*0 80 8W 4 W 1. 1.020000 1.025000 1.030000 1.035000 1.040000 1.045000 2. 2.060400 2.075625 2.090900 2.106225 2.121600 2.137025 3. 3.121608 3.152515 3.183627 3.214942 3.246464 3.278191 4. 4.204040 4.256328 4.309135 4.362465 4.416322 4.470709 5. 5.308120 5.387736 5.468409 5.550152 5.632975 5.716891 6. 6.434283 6.547430 6.662462 6.779407 6.8982S4 7.019151 7. 7.582969 7.736115 7.892336 8.051686 8.214226 8.380013 8. 8.754628 8.954518 9.159106 9.368495 9.582795 9.802114 9. 9.949721 10.203381 10.463879 10.731393 11.006107 11.288209 10. 11.168715 11.483466 11.807795 12.141991 12.486351 12.841178 11. 12.412089 12.795552 13.192029 13.601961 14.025805 14.464031 12. 13.680331 14.140441 14.617790 15.113030 15.626837 16.159913 13. 14.973938 15.518952 16.086324 16.676986 17.291911 17.932109 14. 16.293416 16.931926 17.598913 18.295680 19.023587 19.784054 15. 17.639285 18.380224 19.156881 19.971029 20.824531 21.719336 16. 19.012070 19.864730 20.761587 21.705015 22.697512 23.741706 17. 20.412312 21.386348 22.414435 23.499691 24.645412 25.855083 18. 21.840558 22.946007 24.116868 25.357180 26.671229 28.063562 19. 23.297369 24.544657 25.870374 27.279681 28.778078 30.371422 20. 24.783317 26.183273 27.676485 29.269470 30.969201 32.783136 INTEREST COMPOUND INTEREST AMOUNT TABLE. Continued. 195 YEAR Z% 6 1% 8 % 10 1.... 1.050000 1.060000 1.070000 1.080000 1.090000 1.100000 2.... 2.152500 2.183600 2.214900 2.246400 2.278100 2.310000 3.... 3.310125 3.374616 3.439943 3.506112 3.573129 3.641000 4.... 4.525631 4.637093 4.750739 4.866601 4.984710 5.105100 5.... 5.801912 5.975318 6.153290 6.335929 6.523334 6.715610 6.... 7.142008 7.393837 7.654021 7.902803 8.200434 8.487171 7.... 8.549108 8.897468 9.259802 9.616627 10.028473 10.435888 8.... 10.026564 10.491316 10.977988 11.467557 12.021036 12.579476 9.... 11.577892 12.180795 12.816448 13.466562 14.192929 14.937424 10.... 13.206787 13.971642 14.783599 15.625487 16.560293 17.531167 11.... 14.917126 15.869941 16.888451 17.957126 19.140719 20.384283 12.... 16.712982 17.882137 19.140963 20.475296 21.953384 23.522712 13.... 18.598631 20.015066 21.550488 23.194920 25.019189 26.974983 14.... 20.578563 22.275970 24.129022 26.132113 28.360916 30.772481 15.... 22.657491 24.672528 26.888053 29.304283 32.003398 34.949729 16.... 24.840366 27.212880 29.840217 32.730225 35.973704 39.544702 17.... 27.132384 29.905652 32.999032 36.430243 40.301338 44.599173 18.... 29.539003 32.759992 36.378965 40.426263 45.018458 50.159090 19.... 32.065954 35.785591 39.995492 44.741964 50.160119 56.275029 20.... 34.719251 38.992727 43.865177 49.402921 55.764530 63.002529 NOTES 339. A Note, or a Promissory Note, is a written promise to pay a sum certain, at a time certain. The sum may be paid in money, or in other valuable things, as mentioned in the note. The time of payment may be a fixed date (as June 1, 1903) ; it may be the date of the occurrence of an event that is sure to hap- pen (as the death of a person) ; it may be any date on which the person entitled to payment may ask for it (payment on demand). 340. Form of Notes 1450.00. Buffalo, N. Y., June 1, 1902. Six months after date, for value received, I promise to pay T. B. Smith, or order, four hundred fifty dol- lars ($450.00). D. A. WEST. $500.80. Detroit, Mich., June 1, 1902. Four months after date, for value received, I prom- ise to pay R. L. Gordon, or bearer, five hundred j 8 dollars ($500.80), with 6 per cent interest. WALTER JOHNSON. 196 MODERN COMMERCIAL- ARITHMETIC 341. The Maker or Drawer is the person who signs the note. 342. The Payee is the person to whom the note is made payable. 343. The Face of the note is the sum promised to be paid. 344. A note should contain: 1. Time when, and place where made. 2. Time when payable. 3. The sum to be paid. 4. The expression, "for value received." To prevent forgery the sum to be paid should be written in words. If the words "for value received" are omitted, the maker cannot be compelled to pay unless the owner of the note can show that the maker received a valuable consideration for mak- ing the note. If no place of payment is mentioned, the note is payable at the maker's place of business. A note may be made payable "on demand," and is then payable whenever its owner calls for its payment. 345. A note may contain: 1. The words "or order," or "or bearer." 2. The words "with interest," or "with use." If the words "with interest," or "with use," are omitted, the note will not draw interest, but if it is not paid at matu- rity, it will draw interest at the legal rate from the time it becomes due. If a note contains the words "with interest," but does not mention the rate, it will draw interest at the legal rate where the note was made. If a note contains the words "or bearer," it is payable to whoever presents it for payment. If a note contains the words "or order," it is payable to the person mentioned as payee, or to whomever he orders it to be INTEREST 197 paid. A payee may order a note paid to another by indorse- ment. 346. An Indorsement is a writing on the back of a docu- ment. 347. A note may be indorsed: 1. To make it payable to another person. 2. To make sure that the note will be paid. 3. To show that the note has been paid. 4. To show that a partial payment has been made. Men frequently buy and sell notes. If a note is made pay- able to J. Smith, or bearer, Smith may sell it to another per- son, who may also sell it. The maker will pay whoever presents the note for payment. But, if T. Jones, who buys the note of Smith, thinks that the maker is not "good" for the amount of the note, he may require Smith to indorse the note by writing his name on the back of it. That would legally bind Smith to pay the note in case the maker should refuse to pay it if proper demand were made for its payment. If a note is made payable to "J. Smith, or order," and Smith sells the note to Jones, Smith may write on the back of the note, "Pay to Jones. J. Smith." The note is said to be transferred to Jones, and is payable to him. Smith, by his indorsement, is responsible for the payment of the note if the maker refuses to pay Jones. When a note is paid, it is returned to the maker, who may destroy it. But the destruction of a note is no proof that it has been paid. If a note is lost, the payee may still require the maker of the note to pay the debt. When a note is paid, unless it is made payable to the bearer, the maker often requires the holder, or owner, to write his name on the back, which makes the note payable to the maker. The maker then has the indorsed note as a receipt to show that it has been paid. A treasurer who pays out money on the order of some other person should always require the payee of an order to indorse it. It then becomes a receipt for the treasurer. 198 MODERN COMMERCIAL ARITHMETIC If a part payment is made on a note, the payee writes on the back of the note, above his name, a statement of the amount received in payment. 348. The person who writes his name on the back of a note is an Indorser. 349. A note may be indorsed in one of three ways : 1. The payee may write only his name on the back of the note. That makes the note payable to the bearer, and also makes the indorser liable for its payment if the maker refuses to pay it. This is called indorsement "in blank." 2. The payee may write, "Pay to James Wise," and sign his name. That makes the note payable to James Wise, and also makes the indorser responsible for its payment. This is called a "full indorsement." The payee may indorse a note thus: "Pay to James Wise, or order. J. Smith." James Wise may indorse the note in a similar manner in favor of Wilson Niles, and so on. A note may have several indorsers, each 01 whom becomes individually responsible for its payment. An indorsement to transfer a note makes the indorser responsible for payment unless the indorsement is made "with- out recourse." 3. If the payee wishes to transfer a note, but does not wish to become responsible for its payment except to a limited degree, he may indorse the note thus: "Pay to K. A. Wall, or order, without recourse to me. Anson Brown." This is called indorsement "without recourse." 350. A note may be indorsed for transfer, for security, or for transfer and security. Write and indorse notes that will illustrate each of these. 351. If the maker of a note refuses to pay it, he is said to dishonor the note. In order to make an indorser legally responsible for the payment of a note that has been dishonored by its maker, the INTEREST 199 holder of the note must demand payment of its maker at maturity, and give the indorser, within a reasonable time, notice of its dishonor by its maker. If there are several indorsers, each should be notified. Notice may be given to the indorser by letter or verbally. If the parties to the note are of different States, the owner of the note should mail a protest to the indorser. A protest is a written statement, made by an officer who takes oaths, giving notice to the indorser of the note that it has not been paid. 352. A note that may be transferred from one party to another by indorsement and give the holder the right to sue for its payment in his own name is a Negotiable Note. Such a note must contain the words "or order," "bearer," or "or bearer." 353. A note that cannot be transferred by indorsement is a Non-Negotiable Note. A note made payable to James Smith is not negotiable. 354. In some States three days are allowed by law for the payment of a note in addition to the time mentioned in the note. These three days are called Days of Grace. 355. A note given for a number of months is due on the expiration of that number of calendar months. Thus, a note given on February 1, for three months, is due May 1. 356. A note given for a number of days is due on the expiration of that number of days. Thus, a note given on February 1, for ninety days, is due on May 2 in an ordinary year, and on May 1 in a leap year. 357. A note maturing on a legal holiday should be paid on the day previous. If that day is a legal holiday also, the note should be paid on the day before. If Monday is a legal holi- day, notes maturing on that day should be paid on Saturday. In a State where days of Grace are not allowed, if the day of maturity of a contract falls on Sunday or a holiday, it is due the day following. 358. In all States a note or a closed account will out- law become void in a certain number of years after it becomes due, if nothing be paid on it. The time required for a note or an account to outlaw varies in the different States from two years to twenty years. 200 MODERN COMMERCIAL ARITHMETIC 359. What is the value of a note at its maturity? If it does not draw interest, its value is its face. If it draws interest, its value is its face plus the interest. All notes draw interest after they become due. PROBLEMS 1. Find the value of this note at maturity : $263.80. Rochester, N. Y., May 1, 1902. Four months after date, for value received, I prom- ise to pay Thomas Byron, or order, two hundred sixty- three r %o dollars, with interest at 6 per cent. L. C. JOHNSON. 2. Find the amount due on this note Sept. 1, 1903: $480.00. Cleveland, Ohio, Jan. 17, 1902. Six months after date, I promise to pay A. C. Berry, or order, four hundred eighty dollars, for value re- ceived with interest. M. F. SWAN. NOTE. When interest is mentioned but no rate is given in these problems, Q% is to be understood. Below are given the data of several notes. Find the due date arid the amount due at time of settlement, assuming that interest is at 6%, and that no interest is paid till the time of settlement : SETTLEMENT Oct. 6, 1902 when due Sept. 30, 1901 Feb. 14, 1900 5 mo. after date June 16, 1901 Aug. 30, 1902 Dec. 16, 1900 April 1, 1901 Nov. 25, 1900 FACE DATE TIME 3. $ 625 Jan. 3, 1900 3 mo. 4. $ 590. 50 Feb. 12, 1898 90 da. 5. $ 268. 25 Dec. 7, 1899 6 mo. 6. $1120 Oct. 3, 1897 1 F- 7. $ 375. 60 May 5, 1898 60 da.* 8. $ 214. 75 Nov. 28, 1899 30 da. 9. $3620 July 29, 1899 4 mo.* 10. $ 493 March 30, 1898 9 mo.* 11. $ 318 Sept. 1, 1899 3 mo. 12. $ 422 April 3, 1900 30 da. *Interest is not mentioned in the note. PARTIAL PAYMENTS 201 PARTIAL PAYMENTS 36O. Payment of a portion of the amount due on a note is often made. Such a payment is called a Partial Payment. Several partial payments are often made on a note. The amount and date of each payment should be indorsed on the back of the note. Partial payments may be made on mortgages and accounts, and may be made before or after the obligation becomes due. 361. Mercantile Rule MENTAL PROBLEMS 1. What will a debt of $500 amount to in 6 mo., interest at 6%? If $200 be paid 4 mo. before the debt is finally paid, to what will the partial payment amount at the time of settlement? If such a payment be made, what will be the net amount due on the note at the end of 6 mo.? 2. A debt of $1000 was settled 1 yr. after it became due, but 3 mo. after it became due $200 was paid on it, and 3 mo. later $400 was paid on it. What was the amount of the original debt at date of settlement, interest at 6%? What was the amount of each payment at date of settlement? What was the amount left to be paid at the time of settlement? Principles. 1. The amount of the debt equals the face of the debt plus the interest on the same till the time of settle- ment. , 2. The amount of each payment equals the face of the pay- ment plus the interest on the same till the time of settle- ment. 3. The amount due at the time of settlement equals the amount of the debt less the sum of the amounts of the pay- ments. EXAMPLE. The following payments were made on a note dated Jan. 1, 1900, for $500, with 6% interest: April 3, 1900, 202 MODERN COMMERCIAL ARITHMETIC ); May 12, 1900, $80; June 25, 1901, $90. What remained due July 15, 1902? EXPLANATION. Write the dates in consecutive order, putting the last first. Begin at the bottom and subtract each date from the one immediately above it, and place the remainders in consecutive order. Write the amounts of the payments and the debt opposite the dates. Figure the interest and find the balance due as shown in the operation. OPERATION Subtraction of Dates Cr. Dr. Yr. Mo. Da. P'm'ts Int. Debt Int. 1898 7 15 time of settlement 1897 6 25 time of third payment $ 90.00 1896 1896 1896 5 4 1 12 3 1 time of second payment time of first payment time of note 80.00 240.00 $500.00 2 6 14 time from note to settlement $240.00 $32.88 $500.00 $76.17 2 2 3 2 12 3 time from first p'm't to settlem't time from second p'm't to settlem't 80.00 90.00 10.44 5.70 1 20 time from third p'm't to settlem't $410.00 $49.02 $500.00 $76.17 49.02 76.17 $459.02 $576.17 459.02 Amount due $1.7.15 PROBLEMS 1. What was the balance due Sept. 14, 1900, on a note for 3, dated March 6, 1896, interest at 6%, if the following payments were made: Dec. 3, 1896, $115; Feb. 12, 1897, $120; Oct. 18, 1898, $208? & $975.50. New York City, Jan. 14, 1897. Six months after date I promise to pay L. C. Stone, or order, nine hundred seventy -five ffo dollars, for value received, with interest at 6 per cent. C. E. BABCOCK. Find the amount due on the above note Jan. 1, 1900, the following payments having been made: Aug. 3, 1897, $86; Jan. 12, 1898, $175; Dec. 23, 1898, $215; July 5, 1899, $328. PARTIAL PAYMENTS 203 8. $1285.00. Philadelphia, Pa., May 2, 1898. Four months after date, for value received, 1 prom- ise to pay to the order of G. W. Beam twelve hundred eighty-five dollars. A. T. BRONSON. The note was not paid when due, but the following pay- ments were made: Oct. 4, 1898, $164; Jan. 13, 1899, $245; July 6, 1899, $338. What remained due Sept. 23, 1899, money being worth 6%? 4. Find the balance due Oct. 28, 1902, on a mortgage given Nov. 1, 1897, for $2600, interest at 6%, the following pay- ments having been made: July 5, 1898, $685; Oct. 14, 1899, $1260; March 7, 1900, $240; May 1, 1901, The United States Rule 362. The mercantile rule is often used by common consent in settling short-time notes and accounts. The rule sanctioned by the Supreme Court of the United States is called the United States Kule. The principle of the United States Rule is that each payment should be applied to pay the interest due at that time, and the balance of the pay- ment should be used to diminish the principal. If at any time the payment is less than the amount of interest due, the pay- ment is not considered made at that time, but it is added to the next payment or payments, when the sum of the payments shall at least equal the interest due at the date of last payment. Most States have adopted the United States Rule. MENTAL PROBLEM On Jan. 1, I owe $1000. How much will I owe May 1, interest at 6%? If on May 1, I pay $120, how much will I still owe on that date? How much will I owe Nov. 1? If I pay $177 on Oct. 1, how much will I still owe? How much will I owe on the first of the next March? EXAMPLE. A mortgage was given Jan. 12, 1896, for $6820, with interest at 6%. The following payments were made: 204 MODERN COMMERCIAL ARITHMETIC Oct. 15, 1896, $380; Feb. 17, 1897, $650; July 13, 1897, $760; Jan. 15, 1898, $1290. How much remained due Nov. 1, 1898? OPERATION Subtraction of Dates Year Month Day 1898 11 1 time of settlement 1898 1 15 time of fourth payment 1897 7 13 time of third payment 1897 2 17 time of second payment 1896 10 15 time of first payment Debt 1896 7 12 time mortgage was given $6820.00 Payment i time from mortgage to first payment $380.00 4 2 time from first to second payment 650.00 4 26 time from second to third payment 760.00 6 2 time from third to fourth payment 1290.00 9 16 time from fourth payment to settlement Face of debt $6820.00 Int. to first payment 105.71 Amt. of debt at first payment $6925.71 First payment 380.00 Amt. of debt after first payment $6545.71 Int. to second payment 133.09 Amt. due at second payment $6678.80 Second payment 650.00 Amt. due after second payment $6028.80 Int. to third payment 146. 70 Amt. due at third payment $6175.50 Third payment 760.00 Amt. due after third payment $5415.50 Int. to fourth payment 164.27 Amt. due at fourth payment $5579.77 Fourth payment 1290.00 Amt. due after fourth payment $4289.77 Int. to time of settlement 204.48 Amt. due at settlement $4494.25 PROBLEMS 1. On a debt of $3245, beginning Sept. 3, 1897, and bearing 5% interest, the following payments were made: Jan. 14, 1898, $630; Aug. 9, 1898, $560; Feb. 14, 1899, $780. How much was due April 2, 1900? PARTIAL PAYMENTS 205 Find the amounts due : & Date of Debt Face Int. Payments Settled Aug. 15, 1896 $1245 4# Dec. 1, 1896, $ 208 April 1, 1897, 315 Nov. 15, 1897, 120 July 6, 1898, 160 Aug. 4, 1899 3. May 6, 1898 1790 6# Aug. 8, 1898, 218 Jan. 19, 1899, 350 Dec. 14, 1899, 190 July 12, 1900 4. Oct. 4, 1897 645 6# Feb. 2, 1898, 175 Oct. 8, 1898, 225 Sept. 29, 1899, 130 Oct. 16, 1900 5. Jan. 6, 1896 3255 5# Oct. 12, 1896, 250 May 4 1897, 380 Feb. 26, 1898, 1650 Dec. 7, 1898 363. When a payment is less .than the interest due on the debt, the payment is not deducted from the amount due, but it is added to the next payment, and the sum is considered as one payment bearing the date of the latter payment. EXAMPLE. A mortgage for $2500 was given Jan. 4, 1896. The following payments were made: July 7, 1896, $350; Feb. 15, 1897, $50; Oct. 4, 1897, $540; May 4, 1898, $60; Dec. 22, 1898, $425. How much was due April 1, 1899? OPERATION Face of debt $2500.00 Int. to first payment 41.25 Amt. due at first payment $2541.25 First payment 35000 Amt. due after first payment $2191.25 Int. to second payment* 159.96 Amt. due at second payment $2351.21 Second payment 590.00 Amt. due after second payment $1761.21 Int. to third payment* 131.21 Amt. due at third payment $1892.42 Third payment 485.00 Amt. due after third payment $1407.42 Int. to settlement 42.93 Amt. due at settlement $1450. 35 *This is the sum of two payments with the date of the latter, for the former was less than the interest due. The pupil can generally tell by inspection whether the interest due is greater than the payment Subtraction of Dates Year Month Day 1899 4 1 1898 12 22 1898 5 4 1897 10 4 1897 2 15 1896 7 7 1896 1 4 6 3 7 8 7 19 7 7 18 3 9 206 MODERN COMMERCIAL ARITHMETIC PROBLEMS Find the amounts due : 1. Date Face Int. Payments Settled March 20, 1897 $1540 6# Sept. 14, 1897, $ 85 June 6, 1898, 50 Dec. 7, 1898, 320 Feb. 3, 1899, 245 Nov. 1, 1899 2. Dec. 21, 1898 2435 4# May 3, 1899, 140 Jan. 17, 1900, 60 June 1, 1900, 75 Sept. 4, 1900, 420 Nov. 3, 1900 3. Aug. 1, 1896 630 6# Oct. 8, 1896, 115 Feb. 16, 1897, 50 Jan. 1, 1898, 25 Dec. 11, 1899, 325 April 16, 1900 I Nov. 22, 1897 1182 6# May 7, 1898, 25 Sept. 3, 1898, 85 March 21, 1899, 256 Aug. 4, 1899, 120 July 3, 1900 5. Feb. 4, 1895 2090 Sfo Oct. 26, 1895, 235 July 13, 1896, 90 Jan. 12, 1897, 460 May 4, 1897, 150 Jan. 29, 1898 6. Sept. 7, 1896 875 *% April 26, 1897, 145 Oct. 6, 1897, 130 May 18, 1898, 225 Dec. 30, 1898, 160 March 1, 1899 7. May 3, 1898 1470 % Nov. 29, 1898, 110 Sept. 28, 1899, 160 Jan. 8, 1900, 440 Aug. 8, 1900 8. July 2, 1896 2100 *% Dec. 1, 1896, 220 Aug. 9, 1897, 165 Feb. 4, 1898, 235 Nov. 28, 1898, 50 April 15, 1899 9. June 19, 1897 1360 % Nov. 10, 1897, 135 March 11, 1898, 290 Feb. 16, 1899, 325 April 3, 1900, 165 Sept. 21, 1900 10. Jan. 4, 1898 2900 *% June 13, 1899, 180 Aug. 16, 1899, 70 Oct. 24, 1899, 390 Sept. 6, 1900, 625 Dec. 21, 1900, 750 Jan. 10, 1901 OF THE UNIVERSITY OF INTEREST 207 364. TRUE DISCOUNT MENTAL PROBLEMS 1. What will $1 amount to in 1 yr. at 6%? In 2 yr.? In 3yr.? 2. What will $100 amount to in 1 yr. at 6%? In 2 yr.? 8. What sum now will amount to $106 in 1 yr. at 6%? To $112 in 2 yr.? To $118 in 3 yr.? 4. What is the present value of $106 due in 1 yr., if money is worth 6%? What is the present value of $112 due in 2 yr.? Of $118 due in 3 yr.? 5. What is the present value of $212 due in 1 yr., if money is worth 6%? Of $208 due in 1 yr., if money is worth 4%? 6. If you owe $106 on a credit of 1 yr., how much ought you to pay if you pay the debt now, money being worth 6%? 7. If you owe $115 on 3 yr. credit, how much should you pay now to cancel the debt, money being worth 5%? 8. If you owe $112 on 2 yr. credit, how much should your creditor allow if you pay the debt now, money being worth 6 % ? 9. If you owe $210 on 1 yr. credit, how much discount should you be allowed if you pay the debt now, money being worth 5%? 10. If you owe now $210, which is to be paid in 2 yr., with interest at 5 % , how much should you pay to cancel the debt now? 365. $100 in cash is worth more than $100 to be paid in 1 yr., without interest. It is worth as much as $100 to be paid in 1 yr., with interest. If a man has a sum of money due him at a future time without interest, he can afford to allow a dis- count from the face of the debt if it is paid now. An equitable deduction made from the face of a debt, due at a future time, without interest, is called Time Discount, or True Discount. 366. What a dbt, due in the future, without interest, is worth now, is the Present Worth of the debt. The present 208 MODERN COMMERCIAL ARITHMETIC worth of a debt is that sum which put at interest now will amount to the face of the debt when it falls due. Face of debt present worth = discount Face of debt discount = present worth EXAMPLE. I owe $2935.50 to be paid in 6 mo., without interest. What is the present worth and the true discount, if money is worth 6 % ? OPERATION $1 now = $1.03 due in 6 mo. $2935.50 -f- 1.03 = $2850, present worth $2935.50 - $2850 = $85.50, true discount EXPLANATION. $1 now = $1.03 due in 6 mo. Each $1.03 found in $2935.50 is equivalent to $1 due now. Therefore $2935.50 -f- 1.03 = the number of dollars due now. NOTES. 1. The present worth of an interest bearing debt is the face of the debt, if discounted at the interest bearing rate. 2. When an interest bearing debt is discounted, the amount of the debt when it becomes due should be taken as the face of the debt to be discounted. PROBLEMS 1. $840.00. St. Louis, Mo., July 6, 1900. Eight months after date I promise to pay G. O. Black, or order, eight hundred forty dollars, for value received. WILLIAM JOHNSON. How much was the above note worth July 6, if money was worth 6%? #. $1250.00. Charleston, S. C., Aug. 1, 1900. One year after date I promise to pay H. A. Wells, or order, twelve hundred fifty dollars, for value received. A. D. NOYSE. What was the above note worth Oct. 18, 1900, money being worth 6%? Find the present worth and the true discount of the fol- lowing : 3. $1217 on 90 da. credit, money worth 4%. 4. $625.80 due in 1 yr. 4 mo. 32 da., money worth 6%. INTEREST 209 5. $3084 payable in 7 mo. 29 da., money worth 8%. 6. $215.25 payable in 1 yr. 6 mo. 18 da., money worth 6%. 7. $1028 payable in 60 da., money worth 7%. 8. $2135 payable in 2 yr. 8 mo., money worth 4%. 9. $716 payable in 3 yr. 3 mo., money worth 6%. 10. $937 payable in 1 yr. 4 mo., money worth 6%. 11. What sum will amount to $1438 in 2 yr. 7 mo., at 6%? 12. I gave my note for $3625, for 1 yr., with 4% interest. If the note is discounted at 6 % , what should I pay now? 13. On Jan. 1, 1900, I agreed to pay $1275 in 1 yr. 4 mo., with interest at 4%. Four months later the debt was dis- counted at 6%. What did I pay? H. Which is better and how much, to buy sheep at $8.50 on 7 mo. time, or to pay $8. 25 cash, money being worth 6 % ? 15. A man was offered a house for $18500 cash, or $19400 due in 10 mo. If money was worth 6%, which was the better offer? 16. When money is worth 4%, what cash offer is equivalent to an offer of $2550 on 6 mo. credit? 17. An agent bought a house for $1200. He kept it 14 mo., paid $185 for repairs, and sold it for $1500, on 9 mo. credit. What was his gain or loss, if money was worth 4%? 18. A merchant sold $1475 worth of goods on 8 mo, credit. If he sold the goods at a gain of 20%, what was his actual gain per cent, money being worth 6%? 19. Which is the better bargain for the purchaser, and how much better, $1000 worth of goods bought on 8 mo. time, or 5 % off for cash, money being worth 6 % ? 20. Find the present worth of a debt of $6580, $2000 of which is due in 8 mo., $1500 in 14 mo., and the remainder in 1 yr. 8 mo., money being worth 6%. NOTE. Find the present worth of each payment. '21. George is 17 yr. old. How much must be invested for him, at 5% simple interest, that he may have $10000 of prin- cipal and interest when he becomes of age? 22. I bought a stock of goods on 8 mo. time. After hold- ing them 6 mo., I sold them at an advance of 25%, giving a 210 MODERN COMMERCIAL ARITHMETIC credit of 10 mo. What was my gain per cent, money being worth 6%? 28. A man agreed to pay a debt of $1200 in 6 equal semi- annual payments, with simple interest at 4% per annum. Two months later he paid the present worth of the debt, discounted at 6% per annum. What was the amount of the discount? 24. On a bill of goods for $7850, a trade discount of 20%, 15%, and 10% and a credit of 6 mo. is allowed. What should be the total discount for cash payment, money being worth 5%? BANKING BUSINESS BANK DISCOUNT 367. A business man frequently takes notes for one, two, or three months, or longer, without interest or with interest. If he wishes to procure the money on a note before it becomes due, he may present it to a bank, which will purchase it from him. The amount that a bank deducts from the face of the note for advancing the money is called Bank Discount. 368. For discount banks take the legal interest on the amount due at maturity, for the time between the date of dis- counting and the date when the note becomes due. 369. The time for which a note is discounted is the Term of Discount. 370. The amount due on a note at maturity less the bank discount is the Proceeds of the note. The proceeds of a note are the sum received by its owner when he has it discounted at a bank, 371. In discounting a note, its value, for discount pur- poses, is its future worth (what it will be worth when it falls due). If a note does not draw interest, its future worth is its face. If a note draws interest, its future worth is its princi- pal plus the interest to maturity. 372. Difference betiveen True and Bank Discount. True discount is interest on the present worth of a debt. Bank dis- count is interest on the future worth of a debt. MENTAL PROBLEMS 1. What is the bank discount on a note for $500, if the term of discount is 2 mo. and the interest rate for discount is 6%? What are the proceeds? 211 212 MODERN COMMERCIAL ARITHMETIC 2. What is the bank discount on a non-interest bearing note for $200, due in 2 mo., money being worth 6%? 3. What is the bank discount on an interest bearing note for $200, due in 2 mo., money being worth 6%? 4. A debt of $300, without interest, is due in 2 mo., and money is worth 6%. What is the future worth of the debt? What is the present worth by true discount? What is the interest on the present worth for 2 mo.? What is the sum of the interest on the present worth and the present worth? What is the bank discount? What are the bank proceeds? What is the interest on the proceeds for 2 mo.? On the interest on the interest on the proceeds for 2 mo.? What is the sum of the proceeds, interest on the proceeds, and interest on the interest on the proceeds? Proceeds + int. on proceeds + int. on int. on proceeds = future worth. Present worth + int. on present worth = future worth. 373. If a merchant intends to have a note discounted at a bank, he has the maker make the note payable at the bank with which the merchant does business. Then when the merchant wishes to have the note discounted, he indorses it and presents it to the bank. The note becomes payable to the bank, and the bank pays the merchant the proceeds of the note, which is the future worth less the interest on the future worth for the term of discount. 374* A note made payable at a bank is called a Bank Note, and is usually in this form : ^350.00. Brooklyn, N. Y., Jan. 1, 1900. Two months after date I promise to pay to the order of Richard Roe three hundred fifty dollars, at the First National Bank, Brooklyn, N. Y. Value received JOHN DOE. Due March 1. BANKING BUSINESS 213 EXAMPLE 1. The above note was indorsed in blank by Richard Roe and presented to the First National Bank of Brooklyn, Feb. 1, 1900. What were the term of discount, the bank discount, and the proceeds, money being worth 6%? OPERATION Term of disc. = 28 da., time from Feb. 1 to March 1 Bank disc. = $1.63, int. on $350 for 28 da. Proceeds = $350 - $1.63, or $348.37 EXAMPLE 2. $200.00. Syracuse, N. Y., June 1, 1900. Ninety days after date I promise to pay to the order of J. L. Johnson two hundred dollars, for value received, with interest at 6 per cent, at the State Bank, Syracuse, N. Y. LESTER BROWN. Due Aug-. 29. Tliis note was discounted by the bank July 12. Find the bank discount and the proceeds. OPERATION Term of disc. = 48 da., time from July 12 to Aug. 29 Future worth = $203, face plus interest for 90 da. Bank disc. = $1.62, int. on $203 for 48 da. Proceeds = $203 - $1.62 = $201.38. PROBLEMS Find the discount and the proceeds of the following notes: Rate of Face Date Time Interest Discounted Discount 1. $185 Jan. 2, 1900 4 mo. 4# Jan. 18, 1900 6# 2. 225 Feb. 21, 1900 90 da. None April 2, 1900 5% S. 436 Jan. 10,1900 3 mo. 5% Feb. 1,1900 6# 4. 250 April 2, 1900 6 mo. 8# May 3, 1900 6% 5. 712 July 6, 1900 30 da. None July 6, 1900 4% 6. 456 March 15, 1900 2 mo. None March 28, 1900 6^ 7. 575 June 12, 1900 60 da. 1% July 2, 1900 1% 8. 326 Feb. 9, 1900 90 da. % Feb. 21, 1900 6^ 9. Find the discount and the proceeds of the following: $485.00. Omaha, Neb., Feb. 6, 1900. Three months after date, for value received, I promise to pay Thomas Wentworth, or order, four hundred eighty-five dollars, with interest at 6 per cent, at the First National Bank. H. R. BECK. Discounted Feb. 15, at 6%. MODERX COMMERCIAL ARITHMETIC 10. 1318.00. Philadelphia, Pa., Jan. 2, 1900. Ninety days after date, for value received, 1 prom- ise to pay to the order of C. D. Eaton three hundred eighteen dollars, at the Girard Bank. HAROLD SPENCER. Find the proceeds if discounted Jan. 17, at 6%. 11. 640.00. Lexington, Ky., April 4, 1900. Four months after date", for value received, I promise to pay to the order of James Nelson six hun- dred forty dollars, with interest at 5 per cent, at the Commercial Bank. L. R. WATSON. Find the proceeds if discounted April 30, at 6%. 12. $525.00. Charleston, S. C., May 4, 1900. Six months after date, for value received, I promise to pay L. C. Westcott, or order, five hundred twenty- five dollars, with interest at 6 per cent, at the Mer- chants' Bank. R A. VANCE. Find the proceeds if discounted June 12, at 6%. Find the discount and the proceeds of the following notes : Rate of Discounted Discount Feb. 1, 1900 6# April 30, 1900 % June 25, 1900 5$ Aug. 28, 1900 8# Aug. 3, 1900 % April 28, 1900 6^ July 25, 1900 6^ June 18, 1900 % Jan. 12, 1900 6# Feb. 24, 1900 6# Face Date Time Interest IS. $1160 Jan. 4, 1900 4 mo. None U. 475 April 17, 1900 90 da. 6# 15. 329 June 22, 1900 30 da. None 16. 1275 Aug. 4, 1900 3 mo. 8# 17. 738 July 20, 1900 90 da. None 18. 426 April 6, 1900 60 da. None 19. 375- June 7, 1900 4 mo. 4* 20. 526 March 5, 1900 6 mo. 6# 21. 840 Jan. 12, 1900 30 da. None 22. 237 Feb. 20, 1900 60 da. None BANK DEPOSITS AND CHECKS 375* Savings banks receive money from individuals and pay interest on such deposits. The depositor receives a bank book in which the sums deposited are credited to him. When he draws out money, the amount is debited to him. BANKING BUSINESS 215 Commercial banks also receive deposits. Some commercial banks pay interest on deposits, and some do not. Deposits are often made for safe-keeping. 376. When a depositor in a commercial bank wishes to draw on his deposit for his own use, he writes a check, of which the following is a form : New York, Jan 2, 1900. FIRST NATIONAL BANK OF NEW YORK. Pay to Self- One Hundred Eighty Dollars. $180.00. AMOS WENTWORTH. If Amos Wentworth wishes to pay $180 from his deposit, to J. Higginson, he writes a check as follows: New York, Jan. 2, 1900. FIRST NATIONAL BANK OF NEW YORK Pay to the order of J. Higginson One Hundred Eighty Dollars. $180.00. AMOS WENTWORTH. In this case J. Higginson must indorse the check. 377. The one who signs a check is called the Drawer. 378. The one to whom the check is made payable is the Payee. 379. The one to whom the check is addressed is the Drawee. 380. The drawer must be a depositor. It sometimes hap- pens that a depositor of good financial standing draws out more than he has deposited. The payee may be the drawer, bearer, or any person named in the check. 381. When a person deposits money in a bank, the bank holds the money subject to his order in the form of a check. Assume that you have $500 deposited in a bank, and wish to draw out $100 for yourself and wish to pay Wm. Springer $200. Write two checks that will accomplish the result. Write a check so that only James Wells can draw the money. 216 MODERN COMMERCIAL ARITHMETIC 383. Many banks issue Certificates of Deposit, of which the following is a form: No. 165. CERTIFICATE OF DEPOSIT. FIRST NATIONAL BANK OF CLEVELAND. Cleveland, Ohio, Jan. 2, 1900. John Doe has deposited in this Bank three hundred dollars, payable to his order on the return of this Certificate properly indorsed, with interest at 3 per cent. H. WILSON, Teller. RICHARD ROE, Cashier. BANK LOANS 383. A depositor may borrow money from a bank and give his note for the sum borrowed. Usually the bank requires the borrower to deposit some security, as stocks, bonds, etc., or to have some responsible party indorse the borrower's note. Such notes are called Bank Notes. 384. If the First National Bank of Detroit is willing to loan James Wilson $100 on his note, the form might be as follows : $100.00. Detroit, Mich., Jan 2, 1900. Two months after date I promise to pay the First National Bank of Detroit, Mich. , one hundred dollars. Value received. JAMES WILSON. Due March 2. 385. Bank notes are usually for one, two, or three months, or for 30, 60, or 90 days. They do not draw interest, but interest is paid in advance in tbe form of discount. Thus, on the above note James Wilson would receive $100 - $1 (discount for 2 mo.), or $99. PROBLEMS Find the net amount received from the bank as proceeds of each of the following notes : Face Time Rate of Discount Face Rate of Time Discount 1. $ 340 60 da. 5% 7. $400 30 da. 8% 2. $ 250 3 mo. 6% 8. $360 2 mo. 4% 8. $ 475 1 mo. 8% 9. $520 90 da. 6% 4. $ 600 90 da. 4% 10. $175 1 mo. 7% 5. $ 850 30 da. 6% 11. $230 ' 60 da. 6% 6. $1200 2 mo. 6% 12. $150 90 da. 8% BANKING BUSINESS 217 386. If the First National Bank of Detroit is willing to loan James Wilson $100 on a note indorsed by Wm. Sully, the form of the note might be as follows: $100.00. Detroit, Mich., Jan. 2, 1900. Two months after date I promise to pay Wm. Sully, or order, one hundred dollars, at the First National Bank of Detroit, Mich. Value received, Due March 2. JAMES WILSON. Wm. Sully indorses the note, thus making it payable to the bank, and also making himself responsible for its payment in case James Wilson fails to pay it. Wilson can secure $99 ($100 - $1 discount) on the note. Sully indorses the note to accommodate Wilson, and he is called an accommodation indorser. 387. Such notes, if not paid when due, draw interest from the day of maturity to the day of payment. 388. If an indorsed note is not paid at maturity, the bank immediately sends the indorser a notice of protest. PROBLEMS 1. You wish to borrow $200 from a bank that will accept your note if indorsed by Lewis Ross. Write the note for three months. If discounted at 6%, what would be the net amount received? 2. On May 1, 1900, E. H. Westcott borrowed $350 from the Third National Bank of Cleveland, Ohio, for 3 mo., on his note indorsed by J. C. Stone. Write and indorse the note. Find the proceeds if discounted at 5%. 3. Wm. Walton borrowed $350 from the Citizens' Bank of Boston, Mass., on a 2 mo. note, dated June 4, 1900, and indorsed by E. H. Jones. Write and indorse the note. Find the amount received by Walton if the discount is 6%. If he immediately put this sum at interest at 6%, how much would he lack of having enough to pay the note at its maturity? If Walton did not pay the note, and the indorser paid it Sept. 12, 1900, what sum did he pay, interest at 6%? 218 MODERN COMMERCIAL ARITHMETIC COLLATERAL NOTES 389. A Collateral Note is one whose payment is secured by making a deposit of personal property. FORM OF NOTE $25.00. Buffalo, N. Y., July 8, 1902. Two months after date, for value received, I prom- ise to pay H. B. Swan, or order, twenty-five dollars, with interest at 6 per cent. As security for the pay- ment of the same I hare deposited herewith a Smith- Premier typewriter, No. 75863, with permission to sell the same, if the note and interest thereon are not paid at maturity. WALTER C. CLARK. RECEIPT FOR COLLATERAL Buffalo, N. Y., JulyS, 1902. Received from Walter C. Clark a Smith-Premier typewriter, No. 75863, to secure the payment of a note for twenty -five dollars, given this day by said Walter C. Clark to me. H. B. SWAN. 390. Life insurance policies, stock certificates, notes, mort- gages, etc., may also be assigned, or transferred to secure the payment of notes. PROBLEMS 1. $120.00. Chicago, 111., July 2, 1902. Four months from date, for value received, I prom- ise to pay F. E. Welsh, or order, one hundred twenty dollars, with interest at 6 per cent. To secure the payment of this note I deposit herewith a diamond ring marked H. K. and bought of Wilson Bros., this city. HAROLD KNOX. What was due on the note Oct. 1, 1902? 2. On July 22, 1902, A. L. Ross gave an interest bearing note for $265 for 2 mo. to J. A. Briggs, and deposited as security for its payment a box of jewelry with permission to sell the same if the note was not paid at maturity. The note was not paid when due, .and the jewelry was sold Oct. 15 for $340. How much money should Briggs pay Eoss? BANKING BUSINESS 219 8. A gave B a 9-mo. note for $375, interest at 6%, and assigned a life insurance policy as security for payment. What was the total amount due B 4 mo. after the note became due? 4. C assigned a stock certificate to D bo secure the payment of a loan of $225 made May 1, 1902. What amount did C pay on Aug. 20, 1902, to redeem his stock certificate? 5. On June 3, 1902, K. H. Empie assigned a chattel mort- gage to C. D. Klein to secure the payment of a loan of $130. What amount was due Klein on Aug. 27, 1902? DOMESTIC EXCHANGE 391. A keeps a meat market and owes B for cattle. C works for B by the month. B has no money to pay C until the end of 8 mo. C wishes to trade at the market, and asks B for money. B writes the following order : Avon, N. Y., April 4, 1900. Mr. A: Please pay to C forty dollars in meats, and charge to my account. B. If A accepts the order C can trade at the market, and A will also be paying $40 on his debt to C, but there will be no money used. 392. A owes B $100. B owes C $100. C owes A $100. How do these men stand financially after the following order is delivered? "Mr. C : Please pay to B $100. A." Or after this order? "Mr. A: Please pay to C $100. B." Or after this order? "Mr. B: Please pay to A $100. C." 393. If A in Boston owes B in St. Louis $100, he may pay the debt in one of five ways without sending money. (a) A may go to the postoffice in Boston and buy a Postal Money Order for $100. He would fill out an application blank for a money order, in which he would state the sum to be paid, to whom to be paid, and where to be paid. The postmaster would then give A an order, which would direct the postmaster at St. Louis to pay $100 to the person to be named by the postmaster of Boston in a letter of advice. A would send the order to B. The Boston postmaster would tell the St. Louis postmaster in a "letter of advice" to pay the order to B. B T 220 MODERN COMMERCIAL ARITHMETIC upon receiving the order, would take it to the postoffice and get $100. The postmaster at Boston would charge A $100.30 for the order. At the present time money orders are issued, for any amount up to $100, at the following rates : $ 2.50 or less 3^ 30.00 to $ 40.00 15? 2.50 to $ 5.00 5? 40.00 to 50.00 IS? 5.00 to 10.00 8? 50. 00 to 60.00 20? 10. 00 to 20.00 W 60.00 to 75.00 25^ 20.00 to 30.00 12? 75.00 to 100.00 30J* (b) A may buy an Express Money Order from an express company in Boston. Express money orders are similar to pos- tal money orders. The agents of the company at different offices of the company issue orders on one another as do the postmasters. The rates for express money orders are the same as for postal money orders. (c) A may go to a telegraph office in Boston and buy a Tele- graphic Money Order. The agent of the telegraph company at Boston would telegraph to the company's agent at St. Louis to pay B $100. The rates for telegraphic money orders are higher than the rates for the other orders mentioned, but the exchange is made much more quickly. (d) A may buy from a Boston bank a Draft made payable to B. A would pay the Boston bank $100 plus the charge for "exchange." The Boston bank would write an order for some bank with which it has money deposited, to pay B $100. This order would be a Check of the Boston bank, but when a bank draws its check the paper is called a draft. The Boston bank would give the order, or draft, to A, who would send it to B. B could take it to a bank and have it cashed. The form of the draft might be : FIRST NATIONAL BANK OF BOSTON. Boston, Mass., Jan. 2, 1900. Pay to the order of B One Hundred Dollars. $100.00. T. A. WILSON, Cashier. To the Mercantile National Bank, New York City. BANKING BUSINESS The draft might be made payable to A, who would then indorse it in favor of B. Drafts and checks are much alike in form, but a check is drawn by a party, not a bank, on a bank, while a draft is drawn by a bank or banker on another bank. (e) If A has money deposited in a Boston bank, he may send B a check like the following : Boston, Mass., Jan. 2, 1900. First National Bank of Boston, Mass. : Pay to the order of B One Hundred Dollars. $100.00. (Signed) A. If B has a bank account with a St. Louis bank, he may indorse the check and deposit it with the bank, which will probably collect it for him without charge. When B indorses the check he becomes responsible for its payment, and since he has a deposit in the bank, the bank is safe in accepting the check. If B has no bank account, he may indorse the check and present it to a bank. The bank will pay the check after ascertaining that it is good. If B has no bank account, he may indorse the check and send it direct to the Boston bank, which will send him a draft for $100 less a charge for exchange. This draft B may have cashed by a bank. Cost of Drafts 394. Banks sell drafts, or, as it is termed, sell "exchange." The cost of a draft is the face of the draft plus the charge for exchange. 395. New York or Chicago exchange, drafts on New York or Chicago banks, usually sell at a premium of about .1%. A draft for $500 would probably cost $500.50. On small drafts a definite charge, as 10$ or 15^, may be made. 222 MODERN COMMERCIAL ARITHMETIC PROBLEMS 1. Find the cost of a draft for $850 bought in Buffalo and drawn on a New York bank, if the Buffalo bank charges a premium of |% as exchange. 2. What will be the cost of a draft on Chicago for $1280, when exchange is at a premium of J%? 3. A merchant of Omaha owes a wholesale house in Mil- waukee $3800. He buys of his banker a NQW York draft, at a cost of 80 per $100. How much does the draft cost? 4. Find the cost of a draft for $1570 on a Chicago bank, the rate of exchange being .1% premium? 5. Find the cost of a Boston draft for $9700, at the rate of 100 per $100. 6. An agent in New Orleans, wishing to pay his principal in New York $3265, bought a New York draft at-J-% discount. What did the draft cost him? 7. When exchange is at \J premium, what will a draft for $1 cost? What will be the face of a draft that $5.01 will buy? What will be the face of a draft that $851.70 will buy? 8. An agent has $2500 of his employer's money which he is to remit at the expense of the employer. What will be the face of the draft, if exchange is % discount? 9. What will be the cost of a draft for $2635, at 120 per $1000? 10. A principal directs his agent to send him the money he has on hand, deducting charges for the draft. If the agent has $4500 of his employer's money, and exchange is at % premium, what will be the face of the draft? The Clearing House 396. Each large city is a money center. The banks in the villages and smaller cities deposit money with, remit drafts pay- able to, and sell drafts drawn upon the banks in the money centers. In New York State money is usually sent from one village to another by a draft drawn on a New York bank. BAJSTKIKG BUSINESS 223 A man in Livonia wishes to send $100 to a man in Koches- ter. He pays the Bank of Livonia $100.50 for a draft for $100 on the Chemical Bank of New York. The draft is sent to Rochester and is cashed by a bank, which sends the draft to the Manhattan Bank of New York and is credited with a deposit of $100. A man in Dundee wishes to send $100 to a man in Livonia. He pays the Dundee National Bank $100.50 for a $100 draft on the Manhattan Bank of New York, and sends the draft to Livonia. The Liviona man takes the draft to the Bank of Livonia, which cashes the draft and then sends it to the Chem- ical Bank of New York to pay for the draft which the Bank of Livonia first drew on the Chemical Bank. The Chemical Bank now has a draft for $100 on the Man- hattan Bank, and the Manhattan Bank has a draft for $100 on the Chemical Bank. These drafts are sent to the Clearing House, where they cancel each other. The New York banks send agents to the clearing house to exchange the drafts held by each on the others. If one bank owes all the others $50000, after the drafts are cancelled, it pays that sum to the clearing house. If several banks owe one bank $50000, the clearing house pays the bank that sum. The clearing house settles the accounts between the various banks. It receives from the debtor banks and pays to the creditor banks. The amount of money exchanged is small in proportion to the value of the drafts exchanged. The "clearings" (cancella- tions) at the New York clearing house for 1899 were $57,368,- 230,771, and the balances paid in money were $3,085,971,370. Each of the great American cities has a clearing house, but the one at New York does more business than all the others combined. Fluctuation of Exchange 397. On small sums the rate of exchange is usually at a uniform premium. This premium is to pay the banks for their trouble. Banks usually buy New York and Chicago drafts at par, that is, the banks make no charge for collecting the drafts. 224 MODERN COMMERCIAL ARITHMETIC 398. The rate of exchange on large sums varies in the differ- ent cities". If Denver banks owe New York banks large sums, they would be obliged to send the money to New York, which would cause delay and expense. If then a man in Denver wishes to buy a draft on New York, he would have to pay more than the usual rate of exchange, for the Denver banks would then have to send more money to New York. But if a man in New York wishes to buy a draft on Denver, he might get it at a discount, for then the New York banks would procure a portion of their money from Denver immediately. Thus on large drafts between distant cities the rate of exchange varies, or fluctuates. PROBLEMS Find the cost of each of the following drafts : Face Rate of Exchange Face Rate of Exchange 1. $ 2300 |% premium 6. $1684 -j-% premium 2. $18250 -J% discount 7. $3790 \% discount 8. $ 6580 At par 8. $1260 At par 4. $ 750.90 % premium 9. $ 538 .1% discount 5. $ 1585 % premium 10. $2520 100 per $100 Commercial Drafts 399. The check of one bank on another to effect exchange is called a Bank Draft. The order or draft of one individual or firm on another, asking payment of a debt through the agency of a bank, is called a Commercial Draft. A commer- cial draft is made payable at a certain bank, and the bank becomes the means by which one party collects from another. 400. A draft may be made payable at sight, or when pre- sented for payment. Such a draft is called a Sight Draft. SIGHT COMMERCIAL DRAFT Detroit, Mich. , Jan. 2, 1900. At sight, pay to the order of the First National Bank of Detroit Eighty-Five Dollars. $85.00. To A. B. Wade, E. E. LAMSON. Lansing, Mich. BANKING BUSINESS 225 Wade owed Lamson $85, and this is the means that Lamson took to procure payment. Wade paid the bank $85, and the bank paid Lamson $85 less % discount for collecting. 4O1. A draft may be made payable a certain number of days after sight, or after date. Such a draft is called a Time Draft. TIME COMMEKCIAL DRAFT St. Louis, Mo., Jan. 2, 1900. Thirty days after date pay to the order of the Third National Bank, St. Louis, One Hundred Fifty Dollars. $150.00. To M. D. Clark, R. A. WALL. Kansas City, Mo. Clark owed Wall $150, due in 30 da., and Wall asked pay- ment by means of this draft. He presented the draft to the bank and asked to have it discounted. The bank sent the draft to Clark, who "accepted" it agreed to pay it on matur- ity. Clark wrote across the face of the draft: "Accepted, Jan. 3, 1900. M. D. Clark," and returned the draft to the bank. The bank paid Wall .$150 less the bank discount on $150 for 30 days, and an additional discount of \% for collecting. If the drawer of a commercial draft indorses it and is a responsible party, the bank will pay him the proceeds of the draft (the face less the discount for time and for collecting) without waiting for the drawee to pay or accept the draft. If the drawee should refuse to accept and pay the draft, the drawer by his indorsement becomes responsible for payment. A draft that is not accepted by the drawee when duly pre- sented is said to be "dishonored." In States that allow grace, grace is allowed on time drafts. PROBLEMS Find the proceeds of the following drafts : 1. Albany, N. Y., July 6, 1900. At sight pay to the order of the First National Bank, Albany, two hundred dollars. To H. W. Reed. JAMES KIMM. The bank discounted the draft at 226 MODERN COMMERCIAL ARITHMETIC & New York City, Aug. 14, 1900. Sixty days after sight pay to the order of the Man- hattan National Bank of New York one hundred eighty dollars. To J. A. Weed, . L M ' RCWE - New York. The bank immediately cashed the draft, discounting it at 6% for the time and % for collecting. $. Boston, Mass., Sept. 17, 1900. Thirty days after date pay to the order of the Chemical National Bank, Boston, one hundred seventy- five dollars, and charge to my account. To James Elliot, H. W. LONGWOOD. Lynn, Mass. The bank advanced the money, taking out time discount, money being worth 6%. 4. A Chicago broker bought a 60-da. commercial draft for $1800 on a New York company, at -J-% discount. Find the cost of the draft, money being worth 6%. 5. Find the face of the 90-da. commercial draft that can be bought for $825, premium -J-%, interest 6%. EXPLANATION. A draft for $1 will cost II + $.0025 premium $.015 interest, or 1.9875. If 1.9875 will buy a draft for $1, $825 will buy a draft whose face is as many dollars as $.9875 is contained times in $825 or $833.44. 6. Barr and Creelman drew a 60-da. draft on the Wilson Mfg. Co., for $2360, and sold the draft to the Merchants' Bank, at % discount, and interest at 6%. What were the proceeds? 7. A $580 sight draft was sold at a premium of 1%. Find the proceeds. 8. A Denver broker drew a 60-da. draft on a Chicago firm for $2400. Twenty days after date he sold the draft at a dis- count of \ % , interest at 6 % . What did he receive for the draft? BANKING BUSINESS 227 FOREIGN EXCHANGE 402. Foreign Exchange is that in which drafts drawn in one country are payable in another. It differs from domestic exchange in the currency used and in the manner of quoting the rate of exchange. 403. In foreign exchange drafts are called Bills of Exchange. 404. The rate of exchange on Great Britain is quoted by giving the value of 1 sov. in dollars and cents. Thus, when exchange on London is quoted 4.87, a draft for 1 sov. will cost $4.87. 405. Exchange on France is usually quoted by giving the value of $1 in francs. The quotation 5.15 means that $1 is worth 5.15 francs. Sometimes exchange is quoted by giving the value of 1 franc in cents. The quotation 19.6 means that 1 franc is worth 19.6^. 406. In Germany, the exchange quotation 96 means that 4 marks are worth 96^. The quotation 24 means that 1 mark is worth 24^. 407. The monetary unit of Mexico is the dollar. The gold dollar, which contains nearly the same amount of pure gold as onr dollar, is worth $.983 in TJ. S. money; and the silver dollar, which contains nearly the same amount of pure silver as our silver dollar, is worth from 44^ to 49^ in TJ. S. money, according to the fluctuation in the price of silver as compared with gold. 408. The gold in a sovereign is worth $4.8665 in Ameri- can coined gold. The quotation 4.8665 is, therefore, at par. It is the par of exchange. The par of exchange on France is 5.18J, or 19.3, whichever way the quotation is made. The par of exchange on Germany is 95.2, or 23.8. 409. The Commercial Rate of Exchange may be at par, above par, or below par. It is the market value in one country of drafts on another. 410. Bills of exchange were formerly drawn in triplicate, that is, three bills were drawn of the same tenor and date, one 228 MODERN COMMERCIAL ARITHMETIC of which being paid the others became void. At present many bills are drawn in duplicate, and in many cases only a single bill is drawn. The object of sending two or three bills by different routes or at different times is to make sure that one copy will reach its destination. PROBLEMS 1. What will be the cost in New York of a draft on London for 420 sov. 14s., exchange at 4.86? 2. When exchange on Liverpool, England, is quoted at 4.86|, what is the face of a draft that $1285 will buy? 3. What will be the cost of a bill of exchange on Berlin for 2540 marks, exchange at 95^-? 4. Find the cost of a bill of exchange drawn on Paris for 35800 francs, exchange at 5.16. 5. What will be the cost in Paris of a bill of exchange drawn on New York for $875, exchange at 5.17? 6. A man owes a London merchant $4560. What is the face of the draft he should send, exchange at 4.86J? 7. A man in New York owes $3480 in U. S. money to a dealer in Mexico. If exchange on Mexico is %% premium, what will be the cost of a draft to pay the debt? 8. A speculator in Chicago wishes to buy $10000 worth of Mexican stock, payable in silver. What must be the face of the draft if a Mexican dollar is worth 49^ in U. S. money? What will the draft cost if exchange is at f % premium? 9. I wish to invest $10000 in II. S. money in Mexican silver bonds. How many dollars' worth of the bonds can I buy if exchange is % premium and the par of exchange is 49? 10. A merchant in Mexico owes a dealer in New Orleans $18950 in U. S. money. How many Mexican silver dollars will be required to pay the debt, exchange at % discount and the par of exchange at 49? ACCOUNTS AND BILLS 411. A owes B $100. To A this item is a debt, to B it is a credit. 4d2. That which one owes another is a Debt. It is a debt to the one owing it and a credit to the one to whom it is owed. 413. That which one has paid to another is a Credit. It is a credit to the one who has paid it, and a debt to the one who has received it. What is bought or received is a debt, or debit. What is sold or paid is a credit. A person is debited to what he receives, and credited ly what he parts with. 414. A record of one or more business transactions by two individuals or firms showing the proper debits and credits is an Account. Thus, a merchant keeps an account with each person deal- ing with him. He heads a page of his ledger with the name of a customer. In a column headed "Dr." (debtor) he puts the values of what the customer receives. In a column headed "Cr." (creditor) he puts the sums that the creditor pays. 415. The difference between the amount of the debits and credits of an account is called the Balance of the Account. If the balance is on the debtor side of the account, it shows that the customer has received more than he has paid for. It shows how much more he should still pay. It shows how much more must be added to the credit side of the account to bal- ance the account. If the balance is on the credit side of the account, it shows that the customer has paid for more than he has received. It shows how much must be added to the debit side of the account to balance the account. 416. The following account is taken from the account book of John Doe: 229 230 MODERN COMMERCIAL ARITHMETIC DR. (has received). WM. SMITH. CR. (has paid). 1900 1900 Jan. 2 To lumber, $14.00 Jan. 3 By cash, 12.50 4 To flour, 7.80 5 By labor, 3.75 5 To feed, 6.20 6 By balance, 11.75 $28.00 12800 It is found that the Dr. side is $11.75 greater than the Or. side, and that $11.75 must be added to the Cr. side to balance the account. This shows that Wm. Smith should pay John Doe $11.75. BILLS 417. A detailed statement of goods sold or of services ren- dered is called a Bill. A bill of goods is also called an Invoice. 418. The following abbreviations and terms are frequently used: account agent amount balance bought charged Company commission collect on delivery consignment acct. agt. ami. ML bot. cJigd. Co. com. G. 0. D. con. dft. disc. 'exch. frt. guar. i. e. Pp., or pp. pages mem. Messrs. N. B. net No. P., or p. draft discount exchange freight guaranteed that is merchandise memorandum Gentlemen take notice without discount number page pay't pd. per plcg. P. 0. payment paid by, or by the package postoffice pr. pair pc. rec'd piece received retft R. R. receipt railroad sliip't sunds. shipment sundries inst. prox. ult. present month next month last month @ at ft number a/c c/o n/c o/c account care of new account old account ACCOUNTS AND BILLS 231 419. When goods are sold it is customary for the creditor to render a statement to the debtor. On January 2, 1900, James Wilson sold Allen Jones 4 Ib. coffee at 22^, 15 Ib. sugar at 6^; Jan. 9, 2 bu. potatoes at 45^; Jan. 11,5 gal. oil at 80. The following statement was made out: Rochester, N. Y., Jan. 11, 1900. ALLEN JONES, In Account with JAMES WILSON. (Or, To James Wilson, Dr.) Jan. 2 4 Ib. coffee, @ 22?, $ 88 2 15 Ib. sugar, @ 6^, 90 9 2 bu. potatoes, @ 45^, 90 11 5 gal. oil, @ 8?, 40 Total, 13 08 A. S. Ward bought of H. D. West, Geneva, N. Y., May 7, 1900, 18 yd. calico at 9^, 20 yd. flannel at 60^; May 8, 12 yd. gingham at 10^; May 14, 15 yd. factory at 7^, 10 yd. cambric at 6^. May 16 Ward paid $10. The following bill was rendered : Geneva, N. Y., May 16, 1900. A. S. WARD, To H. D. WEST, Dr. May May 7 7 8 14 14 16 18 yd. calico, @ 9^, 20 yd. flannel, @ 600, 12 yd. gingham, @ 100, 15 yd. factory, @ 70, 10 yd. cambric, @ 60, Total, Cr. By Cash, Balance due, 1 -1 12 1 62 00 05 60 $13 1 1 62 20 65 $16 10 47 00 74 $ 6 Receipting Bills 4 2O. When a bill is paid it is receipted by the creditor or his agent. When paid in cash, the following may be written at the end of the bill : Received payment [or, Paid], JOHN DOE. Or, Received payment, JOHN DOE, Per James Fox. [Fox is agent]. 232 MODERN COMMERCIAL ARITHMETIC When paid by a note the receipt may be : Received payment [or, Paid] by note due Sept. 2, JOHN DOE, Per James Fox. When a bill is not paid, after "Total," "Balance," or "Bal- ance due," the words "Please remit," or "Kec'd payment," without the signature, may be written. PROBLEMS Eender and receipt bills for the following transactions : 1. R. B. Hillman bought of Young & Taylor, July 5, 1900, 14 Ib. steak at 120, 18 Ib. lard at 100, 6 Ib. pork sausage at 110; July 7, 20 Ib. roast at 130, 15 Ib. tallow at 70; July 9, 4 Ib. steak at 12^0, 12 Ib. pork chops at 120. July 8, Hillman sold Young & Taylor a pig for $4.50, and paid the balance due July 10. 2. Stevens & Bacon, Cleveland, Ohio, sold R. S. Wall, Aug. 7, 1900, 4 ranges at $65, 5 wood stoves at $18; Aug. 10, 7 plows at $6.80, 12 shovels at 900; Aug. 13, 6 forks at 400. Aug. 15, Wall paid the bill by a note due Dec. 8, 1900. 8. A. B. Clark, agent for the Building Company, Detroit, Mich., sold Charles Hawes, Sept. 3, 1900, 25600 ft. pine floor- ing at $22.50 per M, 16700 ft. hemlock roofing at $14 per M; Sept. 8, 6900 ft. oak joists at $24.75 per M, 2150 lath at 400 per C. Sept. 6, Hawes sold the Building Company 8500 ft. pine logs at $9 per M, and 1600 ft. hemlock logs at $8 per M, and paid the balance due Sept. 12. 4. On August 27, 1902, L. Mitchell & Co., Chicago, sold to F. E. Baker, St. Louis, Mo., terms cash, the follow- ing items: 1640 ft. A flooring @ $24 per M, 920 ft. C flooring @ $18 per M. 2467 ft. fencing @ $16 per M, 5428 ft. scantling @ $13 per M, 1432 ft. timber @ $9.37 per M, and 860 ft. timber @ $8.62 per M. ACCOUNTS AND BILLS 233 BILLS TEADE DISCOUNT New York, Sept. 3, 1900. MANNING & JONES, Albany, N. Y., Bought of NEW YORK SUPPLY COMPANY, Terms: 30 days; 10% for cash. Sept. 1 1 200 yd. Brussels carpet, @ $2.40, 80 yd. wool cloth, @ 90^, > $480 72 00 00 $552 00 3 160 yd. velvet carpet, @ $3, 480 00 Less 25%, 120 00 $360 00 Less 20%, 72 00 288 00 Total, $840 ^0 Rec'd paym't Oct. 1, 1900, NEW YORK SUPPLY COMPANY. EXPLANATION. The full cost of each item is given. Then the dis- count is taken out, and the net cost written in the proper column. By the "Terms" a credit of 30 da. is given. If the bill is paid at date of last item a discount of 10% is allowed. The bill was paid and receipted Oct. 1. If it had not been paid by Oct. 3 (30 da. after date), it would have drawn interest from that date. PROBLEMS Render the following bills properly discounted and re- ceipted : 1. H. A. Wood bought of Sawyer & Jones, Oct. 8, 1900, on account 30 da. : 24 tables at $12.50, less 25% and 15%; 16 lounges at $7.60, less 30% and 10% ; and 12 chairs at $2.50, less 20% and 12%. The bill was paid when due. 2. H. C. Kimball sold to Booth & Jessop, for cash, Oct. 9, 1900: 12 pianos at $275, less 33% ; 15 organs, at $110, less 25% ; 25 violins at $8.50, less 20%. 3. C. D. Beam sold Austin Neff, Oct. 11, 1900, on 3 mo. credit: 200 bu. barley at 52^; 180 bu. wheat at 74^; 520 bu. corn at 45^; 260 bu. oats at 40^. A trade discount of 20% was allowed. Neff paid the bill Nov. 11, and received the proper time discount, money being worth 6%. 234 MODERN COMMERCIAL ARITHMETIC 4. C. S. Fanchild bought of Wheeler & Wilson, Buffalo, N. Y., July 2, 1902, 2 doz. tables at $2.60 each, on 2 mo. ; 20 sewing machines at $18, 12^% off; 4 doz. chairs at 750 apiece, on 30 da.; 8 desks at $45, 20% and 12|% off; 6 stands, at $28, 15% off. Find the amount due Aug. 1, 1902, money being worth 6 % . 5. A. C. Taylor, Eochester, N. Y., sold to Wright & Young, July 1, 1902: 12400 ft. pine at $24 per M; 4650 ft. chestnut at $45 per M; 5900 ft. oak at $38 per M. On July 15, 1902, 24300 ft. oak at $35 per M; 36500 lath at 900 per C; 12420 ft. hemlock at $16 per M. A credit of 30 da. was allowed and the bill was paid Sept. 1, 1902, interest at 6%. 6. Eender the bill and find the amount due August 21, 1902, interest and discount at 6%: Geo. C. Thirp, Chicago, HI., sold to W. A. Parsons, May 1, 1902, 6 pc. cotton, 60, 65, 70, 58, 45, 75 yd., at 110; 8 pc. gingham, 52, 60, 46/64, 68, 42, 70, 55 yd., at 120; June 10, 1902, 4 pc. shirting, 40, 46, 52, 58 yd., at 90; July 23, 1902, 4 pc. sateen, 51, 60, 73, 58 yd., at 70. A credit of 2 mo. was allowed on each item. 7. Wetmore & Wales, Cleveland, Ohio, sold to J. C. John- son, on 2 mo. time, 10% off for cash, May 3, 1902, 15 diction- aries at $5.40, less 12|% ; 25 atlases, at $4.50, less 10% ; 4 doz. geographies at 800, less 15%; 5 doz. arithmetics at 500, less 16f %. Make out the bill for cash payment. 8. The St. Louis, Mo., Lumber Co. sold M. T. Evans, April 2, 1902, 14650 ft. pine at $26 per M, 3860 ft. clapboards at $25 per M; May 7, 1902, 960 posts at $8 per C, 11900 ft. flooring at $32 per M, 7250 shingles at $3.20 per M. A credit of 1 mo. was allowed, and discounts of 10% and 12|-%. Make out the bill for payment Sept. 1, 1902. 9. The Metallic Tubing Co., New York City, sold 1ST. M. Butler, April 18, 1902, 28 ft. |--in. copper tubing at $1.20, less 15% ; 120 ft. 1-in. copper tubing, at $1.60, less 16f % ; 80 ft. l|-in. steel tubing, at 600, less 15%. A credit of two months was allowed. Make out the bill for settlement Sept. 1, 1902, interest at 6 % . 10. Werner & Jones, Detroit, Mich., sold H. A. Greenwood, ACCOUNTS AND BILLS 235 May 1, 1902, 148 yd. silk at $1.30, less 16f % ; 45 yd. lace at 800, less 20% ; 100 yd. cashmere at 250, less 12% ; 320 yd. flannel at 330, less 20% ; 115 yd. lining at 110, less 8% ; 200 yd. crash at 60, less 10%. A credit of 30 da. was allowed. Make out the bill for payment Sept. 1, 1902. EQUATION OF BILLS 421. Debts draw interest after they become due. 422. Merchants frequently sell goods on credit, that is, the bill for payment of the goods does not become due, does not draw interest, till after a certain period of time; as, 1 mo., 2 mo., 60 da., etc. 423. The time that must elapse before a debt becomes due is called the Term of Credit. 424. A debt ought to be paid when due. If it is not paid when due, it should draw interest from the time it is due till the time it is paid. If it is paid before it is due, it should be discounted for the time between the date of payment and the date when it becomes due. 425. If a bill contains several items due at different times, each item need not be paid as it falls due, but the total of the bill may be paid on a certain date, so that the interest on the debts falling due after that date will equal the discount on the debts falling due before that date. Payment on such a date would be just to both debtor and creditor. That date is called the due date, average time, or equated time. 426. The Average Time, Equated Time, or Due Date is the date on which several debts, or items of a bill, may be can- celled by one payment. 427. Finding the equated time is called Equating or Averaging Bills. 428. The process of finding the equated or average time is called the Equation, or Average of Bills. 429. The time between the equated time and the due date of the earliest payment is called the Average Term of Credit. 236 MODERN COMMERCIAL ARITHMETIC 43O. The time between the equated time and the due date of the latest payment is called the Average Term of Discount. A owes B $100 to be paid on Jan. 1, $100 to be paid Jan. 11, and $100 to be paid Jan. 21. When may he pay $300 and avoid paying interest? How does the interest on $100 from Jan. 1 to Jan. 11 compare with the discount on $100 from Jan. 11 to Jan. 21? EXAMPLE 1. On Jan. 1, 1900, N. E. Spenser bought of James Hayes mdse. as follows : Date When Due Amount Jan. 16, 1900 $100 Jan. 25, 1900 75 Feb. 9, 1900 200 March 1, 1900 150 Total.. On what date can Spenser pay $525 and avoid the payment of interest? EXPLANATION. If he pays it Jan. 16, the date when the first item becomes due, he should be allowed a discount on the other three items for payment in advance. If he waits until March 1, the date when the last item becomes due, he should pay interest on the first three items. He wishes to find a date, between Jan. 16 and March 1, on which he may pay 525, so that the discount on the first items shall equal the interest on the latter items. Suppose he does not pay until March 1, the due date of the last item. That would be the natural time of settlement. Then the amount he would owe is shown as follows (int. at OPERATION Due Date Items Term of Interest Interest Jan. 16 Jan. 25 Feb. 9 March 1 $100 75 200 150 44 da. 35 da. 20 da. da. .73 .44 . .67 .00 3525 $1.84 $525 + $1.84 = 526.84, due March 1 How many days before March 1 should he pay the 525, so that he will not have to pay any of the 1.84 interest? As many days as it will take for the interest on 525 to amount to 31.84. In how many ACCOUNTS AKD BILLS 237 days will $525 produce $1.84 interest, at 6%? $1.84 -j- $.875 (int. on $525 for 1 da.) = 21 da. Then, to avoid paying any interest he should pay $525 21 da. before March 1, 1900. 21 da. before March 1 is Feb. 8, which is the average time, or due date, for paying the whole amount. 21 da. is the term of discount. PROOF. To prove that Feb. 8 is the equated time, it is only neces- sary to show that the interest on the money due before that time is equal to the discount on the money due after that time : Due Items Term of Interest to Feb. 8 Interest Jan. 16 Jan. 25 $100 75 23 da. 14 da. $.38 .18 $.56 Term of Discount from Feb. 8 Discount Feb. 9 March 1 200 150 1 da. 21 da. $.03 .53 $.56 EXAMPLE 2. Wood & Wilson sold goods to J. E. Almj as follows : Jan. 4, 1900 $225 Jan. 22 340, on 2 mo. credit Feb. 6, 160 Feb. 27, 180, on 60 da. credit Eind the average date, or the date from which the whole sum due should draw interest. OPERATION Due Date Items Term of Interest Interest Jan. 4 March 22 Feb. 6 Apiil 28 $225 340 160 180 114 da. 37 da. 81 da. Oda. $25.65* 12.58 12.96 000 $905 $51.19 *For convenience find the interest by the 1000-day method, interest at 36%. That is, multiply the dollars by the days and point off three places. Interest on $905 for 1 da. is $.905. $51.19 -*- $.905 = 56.5 da., or 57 da., the term of discount. 57 da. before April 28 is March 2, the average time. 238 MODERN COMMERCIAL ARITHMETIC EXPLANATION. For convenience arrange the work in columns as above. Add the proper term of credit (calendar months when it reads mouths, and the actual number of days when it reads days) to each credit item, and that will give the true due date of each item. As in the preceding example, assume that the account was settled when the last item became due. The terms for which the various items would then draw interest are 114, 37, 81, and days, respectively. Find the inter- est on the items, .by the 1000-day method, at 36% . The total interest is $51.19. It will take as many days for $905 to produce $51.19 interest as the interest on $905 for 1 da. is contained times in $51.19. The interest on $905 for 1 da. is found by pointing off three places. $ 905 is contained 56.5 times in $51.19. Therefore, $905 will produce $51.19 interest in 56.5 da., or 57 da. 57 da. is the average term of discount, and 57 da. before April 28, or March 2, is the average time of pay- ment. Steps in the Operatiom. 1. Find the due date of each item by adding the proper term of credit. When it reads days, add the number of days; when it reads months, add the number of months. 2. Assume a settlement on the latest due date, which is called the Focal Date. Find the term of interest for each item the number of days from each due date to the focal date. 3. Find the interest on each item for its term of interest, and the interest on the sum of the items for 1 da. 4. Divide the total interest due on the items by the interest on the sum of the items for 1 da. NOTE. Steps 4 and 5 may be briefly stated : Multiply each item by the term of interest in days. Divide the sum of these products by the sum of the items. 5. Find the average time by counting back from the focal date the number of days in the average term of discount. NOTES. 1. Any date may be taken as a focal date. Some take the earliest due date, and discount the items due in the future for pay- ment in advance. But it is more in line with business practice to take the latest due date and add interest to the items not paid when due. An account ought to be settled after it is made. It cannot be settled before it is made. Therefore it is better to take the latest instead of the earliest due date as a focal date. 2. In finding the average term of credit, a fraction of a day of one-half or more is counted as a full day. ACCOUNTS AND BILLS -239 PROOF Due Items Term of Interest Interest Jan. 4 Feb. 6 $225 160 57 da. 24 da. $12.83 3.84 $16.67 Term of Discount Discount March 22 April 28 340 180 20 da. 57 da. $ 6.80 10.26 $17.06 The difference between the interest and discount is $.39. The interest or discount on the sum due, for 1 da., is $.905, therefore the operation is correct. It should be remembered that the division in the operation of finding the equated time in the example was not exact, and that a fraction of a day was called a whole day. That frac- tion of a day in the average term makes the difference between the interest and the discount in the proof. The difference between the interest and the discount in the proof should always be less than the interest on the debt for one-half of a day. The Product Method 431. The interest on the items for the terms of credit by the 1000-day 36% method is found by taking the products of the number of dollars by the number of days in the terms of credit and pointing off 3 places. The product method is similar to this interest method. EXAMPLE. When is the following statement due by equation? W. A. MILLARD, To C. E. BIGELOW, Dr. June 2 To Mdse. , on 30 da., $150 June 19 ToMdse., 220 July 3 To Mdse., 1 mo., 180 July 22 ToMdse., 140 OPERATION Due Date Items Term of Interest Products July 2 July 19 July 22 Aug. 3 $150 X 220 X 140 X 180 X 32 da. = 15 da. = 12 da. = Oda. = $4800 3300 1680 $690 $9780 $9780 - due date. = 14+. 14 da. before Aug. 3 is July 20, average 240 MODERN COMMERCIAL ARITHMETIC EXPLANATION. The arrangement of the work is similar to that in the interest method. The latest due date is Aug. 3. If the first item is not paid till Aug. 3, the debtor will have the use of 150 for 32 da., which is equivalent to the use of 150 X 32 da., or 4800 for 1 da. If the items are not paid till Aug. 3, the debtor has had the equivalent of the use of 9780 for 1 da. He should pay the 690 long enough before Aug. 3 that the creditor will have the equivalent of the use of 9780 for 1 da. 9780 -r- 690 = 14+. If the debtor pays 690 14 da. before Aug. 3, or on July 20th, no interest will be due either party. * PROBLEMS When are the following bills due by equation? Prove each operation : 1. E. N. BAKER, To WM. GRAY, Dr. 1900 Jan. 2 ToMdse., $140 16 To Mdse., 1 mo. credit, 125 Feb. 7 ToMdse., 30 da. credit, 160 26 To Mdse., 60 da. credit, 115 THOMAS GOODE, To STEVENS & BACON, Dr. 1900 Jan. 4 ToMdse., 2 mo., $85 29 To Mdse., 145 March 5 ToMdse., 1 mo., 175 26 To Mdse., 130 3. J. H. ROWE, To JAMES MURDOCK, Dr. 1900 April 2 ToMdse., 30 da., $120 16 ToMdse., 2 mo., 240 May 1 To Mdse., 30 da., 90 24 To Mdse., 118 30 To Mdse., 70 4- C. A. WOOD, To POTTER & BOWEN, Dr. 1900 May 3 ToMdse., 70 15 ToMdse., 1 mo., 110 June 1 To Mdse., 60 da., 265 11 ToMdse., 30 da., 210 30 ToMdse., 180 ACCOUNTS AND BILLS 5. GEO. CLARK, To SAMUEL TAYLOR, Dr. 241 1900 June 3 To Mdse. 3 mo., 8275 19 To Mdse. 30 da. 120 28 To Mdse. 60 da., 90 July 5 To Mdse. 310 20 To Mdse. 1 mo., 400 6. J. O. MORGAN To 0, D. WRIGHT, Dr. 1900 July 3 To Mdse. 30 da., $230 9 To Mdse. 2 mo. 170 Aug. 1 To Mdse. 30 da., 260 17 To Mdse. 30 da., 220 Sept. 1 To Mdse. 1 mo., 180 7. H. H. SHORT, To R. P. REED, Dr. 1900 Sept. 3 To Mdse. $300 10 To Mdse. 1 mo., 150 26 To Mdse. 2 mo., 140 Oct. 1 To Mdse. 1 mo., 190 18 To Mdse. 30 da., 250 8. BENTON COY, To RAYMOND HOWE, Dr. 1900 Sept. 1 To Mdse. $430 13 To Mdse. 30 da., 160 29 To Mdse. 1 mo., 90 Oct. 22 To Mdse. 130 Nov. 2 To Mdse. 220 EOTATION OF ACCOUNTS 432. Accounts are equated in the same manner as are bills, but in accounts there is usually a debit and a credit side. Instead of finding the amount of the debit side as in a bill, we find the balance of the two sides. The equated time of an account is the time on which the balance of the account is due, or the time from which the balance should draw interest. The focal date should be the latest due date in the account. 242 MODERN COMMERCIAL ARITHMETIC EXAMPLE. What should be the face and the date of a note given to settle the following account? DR. J. C. BARR. CR. 1900 1900 May 1 To Mdse., $100 May 14 By Cash, $ 90 May 16 To Mdse., 120 May 21 By Note, 100 June 1 To Mdse. 300 June 8 By Cash, 200 OPERATION Due Items Term Int. Paid Items Term Int. May 1 May 16 June 1 $100 120 300 38 da. 23 da. 7 da. $3.80 2.76 2.10 May 14 May 21 June 8 $ 90 100 200 25 da. 18 da. da. $2.25 1.80 .00 $520 390 $8.66 4.05 $390 $4.05 Balance, $130 $4.61 Int. on $130 for 1 da. is $.13. $461 -5- $.13 = 35 r %, or 35 da. 35 da. before June 8 is May 4, the average date. The face of the note should be $130, and the da*te May 4, 1900. PROOF Dr. Items Term Int. Cr. Items Term Int. May 1 May 16 June 1 $100' 120 300 3 da. 12 da. 28 da. $ .30 (disc.) 1.44 8.40 May 14 May 21 June 8 $ 90 100 200 10 da. 17 da. 35 da. $ .90 1.70 7.00 $9.54 $9.60 9.54 Balance of int., $ .06 EXPLANATION. Find the difference between the discount and the interest on each side, of the account, then the difference of discount or interest between the two sides should be less than one-half of the interest or discount on the balance of the account for one day. PROBLEMS When should interest begin on the following accounts? Prove each operation. 1. DR. ROBERT A. WALKER. CR. 1900 1900 Jan. 2 To Mdse., $400 Jan. 4 By Cash, 1300 Jan. 27 To Mdse., 300 Feb. 6 By Cash, 250 ACCOUNTS AND BILLS 243 DR. J. C. MILLER. CR. 1900 1900 May 1 To Mdse., $200 May 25 By Note, on int., $180 May 12 To Mdse., 30 da., 300 June 12 By Cash, 400 June 1 To Mdse., 350 July 6 By Cash, 300 June 18 To Mdse., 2 mo., 200 3. DR. A. B. CLAYTON. CR. 1900 1900 June 4 To Mdse., 30 da., 1310 June 7 By Cash, $150 June 27 To Mdse., 2 mo., 160 July 3 By Note, 100 July 6 To Mdse., 150 July 16 By Cash, 100 Aug. 8 To Mdse. , 240 Aug. 15 By Cash, 120 Aug. 13 To Mdse. , 400 Aug. 20 By Cash, 375 4. DR. L. C. BRADLEY. CR. 1900 1900 July 6 To Mdse. , $250 July 23 By Cash, $400 July 17 To Mdse. , 1 mo. , 260 Aug. 17 By Cash, 100 Aug. 4 To Mdse., 2 mo., 300 Aug. 24 By Cash, 230 Aug. 28 To Mdse., 400 Sept. 4 By Cash, 300 OPERATION Due Items Term Interest Paid Items Term Interest July 6 Aug. 17 Aug. 28 Oct. 4 $ 250 260 400 300 90 da. 48 da. 37 da. Oda. $22.50 12.48 14.80 .00 July 23 Aug. 17 Aug. 24 Sept. 4 $ 400 100 230 300 73 da. 48 da. 41 da. 30 da. $29.20 4.80 9.43 9.00 $12iO 1030 $49.78 $1030 $52.43 49.78 Balance, $ 180 Balance, $ 2.65 Int. on $180 for 1 da. is $.18. $2.65 -J- $.18 = 14, or 15 da., the average term. 15 da. after Oct. 4 is Oct. 19, the average due date. EXPLANATION. This problem differs from the preceding only in that the balances of the items and the interest are on opposite sides, instead of on the same side of the account. The balance of the items shows that there is still due $180. The balance of the interest shows that $2.65 more interest has been paid than is due. Therefore, $180 should not be paid until as many days after Oct. 4 as it will take $180 to produce $2.65 interest, which is found to be 15 da. 244 MODERN COMMERCIAL ARITHMETIC Principle. When the "balance of the account and the bal- ance of interest are on the same side of the account, date back from the focal date; when the balance of the account and the balance of interest are on opposite sides of the account, date forward. 5. DR. H. A. GIBBS. CR. 1900 1900 Oct. 1 ToMdse. 1 mo., $200 Oct. o By 'Cash, $175 Oct. 23 To Mdse. 360 Oct. 10 By Mdse., 300 Oct. 30 To Mdse. 1 mo., 400 Nov. 1 By Cash, 400 Nov. 6 To Mdse. 320 Nov. 13 By Cash, 350 Nov. 17 To Mdse. 480 Nov. 19 By Cash, 200 6. DR. O. S. CONNOR. CR. 1900 1900 Oct. 19 To Cash, $350 Oct. 4 By Mdse. 2 mo., $700 Oct. 30 To Note, on int. , 460 Oct. 26 By Mdse. 1 mo., 100 Nov. 9 To Cash, 400 Nov. 5 By Mdse. 1 mo., 500 Nov. 21 To Mdse., 520 Nov. 24 By Mdse. 1 mo., 680 Nov. 30 To Cash, 250 Nov. 29 By Mdse. 230 Dec. 18 To Mdse., 420 Dec. 20 By Mdse. 800 7. DR. A. S. WISE. CR. 1900 1900 Nov. 1 To Mdse., 3 mo., $620 Nov. 9 By Cash, $200 Nov. 13 To Mdse., 1 mo., 175 Nov. 23 By Note, 2 mo., 550 Dec. 6 To Mdse., 2 mo., 340 no interest, Dec. 10 To Mdse., 1 mo., 450 Dec. 27 By Mdse., 975 Dec. 21 To Mdse.. 520 Dec. 29 By Mdse., 140 Dec. 31 To Mdse., 380 Dec. 31 By Cash, 250 8. DR. J. A. NEWTON. CR. 1900 1900 Jan. 4 To Mdse. $310 Jan. 10 By Note, on int., $250 Jan. 25 To Mdse. 530 Feb. 6 By Cash., 625 Feb. 9 To Mdse. 160 Feb. 13 By Mdse. , 100 Feb. 26 To Mdse. 315 Feb. 28 By Cash, 240 March 5 To Mdse. 650 March 7 By Cash, 325 ACCOUNTS AND BILLS 245 9. DR. WILLARD DOWN. CR. 1900 1900 May 7 To Mdse., 1 mo., $425 May 10 By Note, on int. , $350 May 26 To Mdse. 2 mo., 375 May 29 By Cash, 400 June 2 To Mdse 1 mo., 540 June 12 By Note, 2 mo. , 550 June 21 To Mdse. 250 no int., June 30 To Mdse. 190 July 3 By Cash, 270 July 6 To Mdse. 460 July 25 By Cash, 500 10. DR. D. M. SUTTON. CR. 1900 1900 June 2 To Mdse. , 1 mo. , $160 June 19 By Mdse., 1 mo., $350 June 12 To Mdse., 325 June 30 By Cash, 100 July 6 To Mdse., 250 July 21 By Cash, 475 July 20 To Mdse., 340 July 28 By Mdse., 120 Aug. 3 To Mdse. , 520 Aug. 8 By Cash, 450 Aug. 25- To Mdse., 425 Aug. 31 By Cash, 200 11. DR. R. C. PERRY. CR. 1900 1900 Sept. 1 To Note, on int. , $250 Aug. 24 By Mdse. 60 da. $275 Sept. 14 To Mdse., 1 mo.. 325 Sept. 8 By Mdse. 1 mo. 380 Oct. 1 To Cash, 240 Sept. 29 By Mdse. 30 da. 350 Oct. 19 To Cash, 360 Oct. 9 By Mdse. 1 mo 220 Oct. 30 To Cash, 150 Oct. 27 By Mdse. 1 mo. 460 Nov. 20 To Cash, 280 Nov. 13 By Mdse. 150 12. Find the balance of the following account, when the balance was due by equation, and what the balance amounted to if nob paid until June 1, 1900, money being worth 6% : DR. D. A. BROOKS. CR. 1900 1900 Jan. 4 To Mdse., $300 Jan. 9 By Cash, $200 Jan. 18 To Mdse., 160 Jan. 25 By Cash, 240 Feb. 1 To Mdse., 420 Feb. 10 By Cash, 360 Feb. 16 To Mdse. , 350 Feb. 28 By Cash, 210 IS. Find when the balance of the following account was due, and what was paid to settle the account April 1, 1900, interest at 6 % : DR. HAMILTON SEAMANS. CR. 1900 1900 Feb. 2 To Mdse., 1 mo., $275 Feb. 8 By Mdse., 1 rno., $190 Feb. 13 To Mdse., 1 mo., 360 March 1 By Cash, 275 March 9 To Mdse. , 430 March 20 By Cash, 260 March 28 To Mdse., 110 March 30 By Cash, 235 246 MODERN COMMERCIAL ARITHMETIC l^. Equate the following account : DR. J. P. WORTH. CR. 1900 II 1900 Jan. 5 ToMdse., 300 Jan. 9 By Cash, $260 Jan. 23 To'Mdse., 250 Jan. 30 By Cash, 180 Feb. 8 To Mdse., 175 Feb. 13 By Note, 1 mo., 350 no int., Feb. 37 To Mdse., 400 March 8 By Cash, 335 OPERATION Due Horns Term Interest Paid Items Term Interest Jan. 5 Jan. 23 Feb. 8 Feb. 27 $ 300 250 175 400 67 da. 49 da. 33 da. 17 da. $20.10 12.05 5.78 6.80 Jan. 9 Jan. 30 March 13 March 8 $ 260 180 350 335 63 da. 14 da. Oda. 5 da. $16.38 2.52 0.00 1.68 $1125 1125 $44.73 20.58 $1125 $20.58 $24.15 $24.15 -j- 6 = $4.03, balance due March 13. EXPLANATION. There is no balance in this account, the two sides being equal, but there is an interest balance of $24. 15 on the Dr. side, which shows that that amount should be paid or added to the Cr. side. But the interest as here reckoned is at 36 % - Since the true rate is $%, divide $24.15 by 6, and the quotient will show the interest balance due March 13, 1900. 15. Equate the following account, interest at 4% : DR. L. M. BEEKMAN. CR. 1900 1900 Feb. 3 ToMdse., 8250 Feb. 6 By Cash, $170 Feb. 21 ToMdse., 160 Feb. 26 By Cash, 190 March 7 To Mdse., 300 March 14 By Mdse., 400 March 31 ToMdse., 270 Apr. 4 By Cash, 200 Apr. 9 To Mdse., 400 Apr. 25 By Cash, 420 ACCOUNTS CURRENT 433. A statement of a running account showing the debits and credits and the cash balance, with interest or discount to date, is called an Account Current. ACCOUNTS AND BILLS 247 434. Adjusting an account is finding the cash balance due at a given date. 435. Theoretically, all sums due draw interest, and all sums paid before they are due are subject to discount. Most retail dealers do not charge interest on the items of a running account, but the balance of a closed account draws interest from the date of the last item. Custom or agreement between wholesale dealers determines whether the items of a running account draw interest. It is customary to charge interest on such items after a certain term of credit. 436. Equating an account is finding at what date the bal- ance is due. Adjusting an account is finding the balance due at a given date. EXAMPLE. Find the balance due on the following account June 1, 1900, interest at 6%. DR. R. H. KNAPP. CR. 1900 1900 March 6 To Mdse., 1 mo., $350 March 21 By Cash, $200 March 29 ToMdse., 2 mo., 400 Apr. 16 By Mdse., 1 mo., 360 Apr. 7 To Mdse., 200 May 3 By Cash, 270 Apr. 25 ToMdse., 2 mo.. 250 May 26 By Mdse. , 1 mo. , 300 OPERATION Due Items Term Int. Disc. Paid Items Term Int. Disc. April 6 May 29 April 7 June V 25 $ 350 400 200 250 56 da 3 da. 55 da. 24 da. $3.27 .20 1.83 $1.00 Mar. 21 May 16 May 3 June 26 $ 200 360 270 300 72 da. 16 da. 29 da. 25 da. $2.40 .96 1.31 $1.25 81200 $5.30 $1.00 $1130 $4.67 $1.25 $1200 + $5.30 $1 = $1204.30. $1130 + $4.67 $1.25 = $1133.42. $1204.30 $1133.42 = $70.88, balance due June 1, 1900. EXPLANATION. Find the interest on each item from the day it is due to the day of settlement. If an item falls due after the date of settlement, it should be discounted, and the discount should be taken from the amount due at the date of settlement. The sum of the items, plus the interest, less the discount, is the total amount of either side of the account. 248 MODERN COMMERCIAL ARITHMETIC PROBLEMS 1. Find the balance due May 1, 1900: DR. A. L. KINNEY. CR. 1900 1900 Feb. 2 To Mdse., 2 mo. 420 Feb. 5 By Mdse., 1 mo., 1360 Feb. 21 To Mdse., 1 mo. 250 Feb. 24 By Cash, 240 March 3 To Mdse., 1 mo. 160 M arch 16 By Cash, 150 March 20 To Mdse., 1 mo. 380 March 27 By Mdse. , 1 mo. , 400 March 24 To Mdse., 1 mo. 300 Apr. 17 By Cash, 200 2. What is the balance due June 1, 1900? DR. F. A. SAYRE. CR. 1900 1900 March 6 To Mdse., 225 March 19 By Mdse. , 2 mo. , 350 March 21 To Mdse., 500 Apr. 6 By Cash, 400 Apr. 14 To Mdse., 340 Apr. 25 By Mdse., 2 mo., 650 May 5 To Mdse.. 1 mo., 720 May 11 By Cash, 450 May 23 To Mdse., 1 mo., 400 May 25 By Cash, 300 3. What is the balance due Sept. 1, 1900? DR. J. H. HADLEY. CR. 1900 1900 June 2 To Mdse. 150 June 7 By Cash, 100 June 20 To Mdse. 270 June 29 By Cash, 175 July 11 To Mdse. 500 July 6 By Mdse., 1 mo., 600 July 23 To Mdse. 120 July 27 By Cash, 100 Aug. 4 To Mdse. 310 Aug. 1 By Mdse., 250 Aug. 22 To Mdse. 230 Aug. 29 By Cash, 200 4. What is the balance due Oct. 1, 1900? DR. C. F. CHASE. CR. 1900 1900 July 7 To Mdse., 350 July 2 By Mdse., 1 mo., 400 July 28 To Mdse., 540 July 17 By Mdse., 1 mo., 250 Aug. 9 To Mdse., 200 Aug. 1 By Cash, 100 Aug. 29 To Cash, 100 Aug. 15 By Mdse., 1 mo., 500 Sept. 8 To Cash, 400 Sept. 5 By Mdse., 1 mo., 350 Sept. 26 To Cash, 200 Sept. 21 By Mdse., 1 mo., 220 ACCOUNTS AND BILLS 249 ACCOUNT SALES 437. A statement rendered by an agent, showing his sales for his principal, his charges against the principal, the amount previously remitted (if any), and the amount due at the equated date, or the amount due at a given date, is called an Account Sales. 438. The agent's gross sales constitute the credits of the account, and his charges constitute the debits. 439. The agent frequently guarantees the quality of the goods he sells for his principal, for which he is allowed a com- pensation called Guaranty. Guaranty, like commission, is computed at a certain rate per cent on the sales. 440. The agent's charges include commission, guaranty, freight, cartage, storage, insurance, etc. 441. An account sales may be rendered simply as an equated account or as an account current. That is, it may show the balance due on the equated date or on a given date. Items of freight, cartage, storage, and insurance are ren- dered as due on the date the agent paid them. Commission and guaranty are sometimes considered as due on the date the account is rendered, sometimes on the date of the last sale, sometimes on the date of each sale, sometimes on the average date of the sales, and sometimes on the average due date of the sales. When sales are made on credit, the average date of sales is not the same as the average due date of the sales. An account sales is equated or adjusted like any other account. PROBLEMS 1. W. S. Davis, Chicago, 111., sold lumber for Aldridge & Bro., Milwaukee, Wis., as follows: Jan. 3, 1900, 24000 ft. hemlock at $13 per M; Jan. 9, 59000 ft. pine at $24 per M; Jan. 26, 18500ft. chestnut at $42 per M; Feb. 5, 27300 ft. oak at $40.50 per M; Feb. 13, 35700 ft. pine at $23 per M. The agent paid for freight, on Jan. 2, $275 ; $62 for storage, 250 MODERN COMMERCIAL ARITHMETIC on the date of the last sale; Jan. 29, he advanced $2500. His commission was 4%, due on the average date of sales. He rendered his account sales Feb. 17, 1900. Reproduce the account. 2. Find the balance of the following account sales, and when due by equation. Consider the entire commission due on the date of the last sale: New York City, Nov. 7, 1900. Account Sales of Apples, For % of WILSON & CO., Buffalo, N. Y. ByJ. C. FOWLER. 1900 Oct. 5 SALES (CR.) 240 bbl. Snow, @ $2.10, cash, 10 350 bbl. King, @ $1.85, 1 mo., 16 180 bbl. Greening, @ $1.50, 1 mo , Nov. 7 400 bbl. Baldwin, @ $1.60, cash, Total, Cr., Oct. Nov. 1 15 23 6 7 CHARGES (DR.) Freight, Cartage, Cash advanced, Storage, Commission and guaranty, 3$>, $525 48 800 40 00 00 00 00 Total, Dr., Net proceeds, Due , 1900. 3. Find the net proceeds and when due of the following account sales : 1902 SALES (Cn.) July 3 25 bbl. Pork, @ $12.80, 30 da., July *>$ 45 bbl Pork (a) $12 60 cash Am? 40 bbl Pork @ $13 00, 20 da., Auer. 17 30 bbl. Pork, @ $12.85, 10 da., Tnfal fV CHARGES (DR.) July 1 Freight, $ 71 40 July 1 Cash advanced, 250 00 Commission (due ), 4$>, Total, Dr., Net proceeds, PARTNERSHIP PARTITIVE PROPORTION 442. Partitive Proportion is the process of dividing a num- ber into parts proportional to two or more given numbers. PROBLEMS 1. A worked 3 days and B 4 days, for the same daily pay. Altogether they received $14. What was the daily wages of each? 2. Two men performed a piece of work for $40. The first agreed to take $3 for every $5 received by the second. How much did each receive? 3. Divide $60 into parts proportional to 2, 4, and 6. 4. Divide $216 into parts which shall be to one another as 5, 6, and 7. 5. Divide 750 into parts proportional "to 10, 15, and 25. 6. A, B, and C put their sheep into one flock and agreed to sell them at a common price. A put in 40 sheep, B put in 70 sheep, and C put in 85 sheep. They sold the flock for $1170. What did each man receive? 7. A has $500, B $600, and C $400 invested in a business. What fractional part of the gain ought each to receive? If the whole gain is $150, what will be A's share? 8. Two men engage in business. A puts in $150, and B puts in $270. If they gain $210, how much should each receive? 9. Three men invest the following sums in a store: $1200, $1400, $1800. If they gain $550, what sum should each receive? 10. Divide 75 into parts proportional to % and . NOTE Fractions to be compared must have a common denom- inator. Then they are to each other as their numerators. and J are equivalent to f and |, and are to each other as 3 and 2. Therefore, divide 75 into parts proportional to 3 and 2. 251 252 MODERN COMMERCIAL ARITHMETIC 11. Divide 72 into parts proportional to and \. 12. Divide 470 into parts proportional to 3 and 4. 13. Divide $1105 into parts proportional to f , f , and . 14. Divide $373.10 into parts proportional to 1475, 1325, and 2530. 15. Divide $405 into parts proportional to f , f , and f . PARTNERSHIP 443. An association formed by two or more persons invest- ing capital in a business and agreeing to share the gains and losses of the business is called a Partnership. 444. The persons that form the association are called Partners. Collectively, they are called a Company, Firm, or House. 445. The capital invested may be money, other property, or labor. 440. The gains and losses of a partnership are shared in proportion to the value of the capital invested and the time the capital is employed. 447. Eesources, or Assets, consist of the property of the firm and the debts due the firm. 448. Liabilities are the debts of a firm. 449. The Net Capital is the excess of the assets over the liabilities. 450. The Net Insolvency is the excess of the liabilities over the assets. 451. With respect to their manner of connection with a firm, there may be four kinds of partners: real, dormant, nominal, limited. 452. A Real, or Ostensible Partner, is one who has cap- ital invested, and is simply a partner without restrictions or conditions. 453. A Dormant, or Silent Partner, is one who has capital invested, but who tries to conceal the fact and does not appear to the public as a partner. PARTNERSHIP 253 454. A Limited Partner is one who gives legal notice, by publication, of the limit of his responsibility for the debts of the firm. 455. A Nominal Partner is a partner in name only. He has no capital invested, and allows the use of his name as a partner simply to give prestige to the firm. 456. A real partner is liable for all the debts of the firm. A silent partner is also liable for the debts of the firm, but he cannot be held responsible unless his connection with the firm is known. A limited partner is liable to a limited extent. A nominal partner is liable for the debts of the firm to all persons who have trusted the firm because such partner was a member of the firm. If persons are deceived by a nominal partner, the partner should pay for the deception. EXAMPLE. A, B, and C formed a partnership. A fur- nished $4000 of the capital, B $6000, and $8000. If they gained $1440, what was each partner's share of the gain? OPERATION $4000 + $6000 + $8000 = $18000, total capital yVoVo or I = A's share of the capital TYoVV or | = B's share of the capital T\V) through London at 4.89, there buying exchange on Paris at 25.19 fr. a pound sterling? MISCELLANEOUS KEVIEW PROBLEMS 275 85. From the following balances determine the gains and losses, the resources, liabilities, and present worth : Dr. Cr Proprietor (investment) 35000 Merchandise $8250 6480 Cash 9560 7845 Bills receivable 1550 Bills payable 780 880 James Hill . . . . 425 150 Wheeler & Wilson 630 840 Merchandise on hand per inventory, $4125. 86. A certain stock pays 10%. At what rate must it be bought to yield 6 % on the investment? 87. What single discount is equivalent to a trade discount of 10, 15, and5%? 88. What premium must be paid to insure a cargo of 4880 bu. of wheat, valued at $1.04 per bushel, at l-J-%, the policy being for only f of the value? 89. Find the cost of oil cloth for a hall 8f yd. long and 14 ft. wide, at 90^ a square yard. 40. Find the interest on $820.45 from June 17, 1889, to April 13, 1892, at 4%. 41. 4-ft. wood piled 5-J- ft. high requires how many feet in length of the pile for 2-J- cords? 42. Eeduce 3|% to a decimal. 48. A room 18 ft. by 16 ft. is carpeted with carpet f of a yard wide, and the smallest possible number of yards of the carpet is used. Find (a) the number of breadths; (b) the number of yards. 44' What sum will amount to $354.09 in 7 mo. at 3% per annum? 45. How many brick in a pile 16 ft. by 6 ft. by 4 ft., each brick being 8 in. by 4 in. by 2 in.? 46. Eequired the exact interest on $146.73, for 23 da., at 5 % per annum. 276 MODERN COMMERCIAL ARITHMETIC 47. A note of $285, bearing 6% interest, given June 17, 1891, has endorsed upon it a payment of $100, March 4, 1892. Find the sum due on the note Nov, 1, 1892. 48. If a grocer sells coffee that costs him 26-J-0 per pound and 32^ a hundred for freight, for 36^ per pound, what is the gain per cent? 4-9. If the average yield per bushel of seed is 14 bu. 1 pk., how much is the yield from 7 bu. 3 pk. 2 qt.? 50. Find the loss on 26 shares of stock bought at 101, and sold at 87, brokerage -J-% both for buying and selling. 51. Eequired the cost of 24 3-in. planks, 18 ft. long and 10 in. wide, and 35 pieces of 2 in. by 4 in. scantling 18 ft. long, at $20 per M. 52. A commission merchant sold 2140 bu. of oats at 39^ per bushel, paid $47.60 freight, and retained 2% commission. How much did he remit to the consignor? 53. What is the difference in weight, expressed in avoirdu- pois pounds, between 300 Ib. Troy and 300 Ib. avoirdupois? 54. An importer receives a bill of goods of $575, pays a duty of 45%, and sells them at a gain of 20%. The price paid by the purchaser is what per cent of the exporter's price? 55. A man bought a house for $4200, paid $640 for repairs, and rents the place for $50 per month. If he pays $115 taxes, what is the per cent of income? 56. If a merchant marks goods 50% above cost, what dis- count from the marked price can he give a customer and make a profit of 33%? 57. If 6% bonds are selling at 87, how much money must be invested in them to secure an annual income of $750? 58. An agent has $23150 of his principal's money and is instructed to buy oats at 48^ per bushel, with a commission of 5%. How many bushels should he buy? 59. When N. Y. C. 4^'s are at a premium of H^-%, what sum must I invest to secure an income of $720? 60. I bought a house for $750, and 2 yr. 9 mo. afterwards sold it for $900. If I paid taxes amounting to $29.17, what was the annual rate per cent of gain on the money invested? MISCELLANEOUS REVIEW PROBLEMS 277 61. How many feet of lumber are required to make a box 4 ft 8 in. by 3 ft. 6 in. by 2 ft. 4 in.? 62. If there is a duty of $1.25 per gallon, and 45%, on var- nish, at what price must it be sold per gallon to gain 33%, if the cost in London is $2.11 per gallon and there are no freight charges? 68. If bell metal is composed of 78 parts copper and 22 parts tin, what weight of each of these metals will there be in a bell that weighs 900 lb.? 64. A lot 60 ft. by 150 ft. was sold for $500. What was the price per acre? 65. If the water from a spring yields Q^% of its weight in salt, how many tons of water will be required to make 1000 lb. of salt? 66. What is the rate of income on an investment in 5 % bonds at 80%? 67. What is the cost of 3130 lb. of coal at $5.25 per ton, and 1820 lb. at $6.90 per ton? 68. If your standing in attendance at school is marked 88 % and you were absent 9 da., how many days of school were there? 69. Find the proceeds of an interest bearing note for $186 given for 3 mo. and discounted the same day it was made, interest and discount being 6% each. 70. If a man bought stock at 2% above par, and sold it at 7% below par, what per cent did he lose? 71. The valuation of the taxable property of a town is $498700 and the tax to be raised is $5850. What will be the tax on $5000? 72. I bought shoes at $2.40 per pair. At what price must I mark them that I may allow a discount of 25% and make a profit of 20%? 78. A man bought a lot for $400 on these terms: $100 cash and the balance in monthly installments of $20, with 6% interest on the part unpaid, interest payable with every installment. What was the total amount paid for the lot? 74> Four men formed a partnership. A put in $12800, B 278 MODERN COMMERCIAL ARITHMETIC put in $14000, C put in $11900, and D put in $15000. After 9 mo. A drew $2800, and 10 mo. later put in $4600. After 10 mo. B added to his investment $5200, and C withdrew $6100. They paid $8425 for labor, $1750 for repairs, $1450 taxes. In 2 yr. they were worth $68000. What was each partner worth? 75. I bought, through an agent, 5000 bu. of corn at 580, commission 2%. The agent sold the corn at 640, commission 2|%, charges $42.50, and remitted the balance by a draft purchased at f % premium. What was my gain? 76. A man bought goods for $2560. He paid $21.50 insurance, $75.50 cartage, and sold them for $2975, allowing the agent a commission of 4% for selling the goods. What was the gain per cent? 77. In writing on the typewriter the letter a was struck 33520 times, b 13080 times, e 25160 times, c 15260 times, and j 2180 times. What was the per cent of use of each letter? 78. A grocer has teas worth 140, 180, 250, 320 per pound. In what proportions can he mix them so as to make the mix- ture worth 220 per pound? How many pounds of the mixture must he make to use up 125 Ib. of the 180 tea? 79. A dealer bought 300 casks of vinegar, each containing 45 gal., at 100 per gallon. He paid $1 apiece for the barrels, 10 per gallon freight, 100 per barrel cartage. He sold it at 12^0 per gallon, receiving 900 for each barrel, and paying 7|% commission for selling. What was his total gain? 80. On the following note these payments were made : Feb. 11, 1902, $240; March 18, 1902, $375; May 20, 1902, $260: $1500.00. Chicago, 111., Jan. 2, 1902. Six months after date, for value received, I promise to pay J. K. Welsh, or order, fifteen hundred dollars, with interest at 6 per cent. THOMAS H. BEAMAN. What was due on the note Dec. 24, 1902? ANSWERS Art. 17, p. 13 8. 10343279 Art. 81, p, 24 1. 44782 9. 8605464 1. 2207 2. 488943 10. 8179692 & 4883 S. 4201420 11. 9083787 J. 716 4. 4309104 12. 10170524 4. 932 5. 3882493 IS. 8638249 5. 6178 6. 384929783 14. 9632366 6. 470597 7. 397334694 15. 11443360 7. 242202 8. 406274798 16. 8627036 5. 89787 9. 423007285 17. 10084434 9. 222071 10. 445609793 18. 9816995 10. 293253 11. 549333911 19. 10046349 11. 14722007 12. 446378793 20. 10647803 12. 3049258 IS. 659402301 21. 10755906 15. 15812417 14. 878654438 U. 2329188 15. 781811566 Art. 30, p. 22 15. 1835007 16. 766652422 16. 7175118 1. 643 17. 808479253 18. 901317663 19. 690428361 2. 4016 3. 3545 Art. 34, p. 25 1. 1480 20. 8238138 4. 33243 & 688 21. 7774858 22. 7547458 5. 15211 6. 29312 3. 5156 4. 76628 23. 9165011 7. 23811 5. 17472 24. 8615227 *. 20680 6. 63087 25. 8725835 9. 30765 7. 4100 26. 7131512 27. 9288411 10. 186999 11. 145408 12. 102527 8. 23303 P. 18366 10. 64369 Art. 22, p. 18 15. 87099 11. 7459 1. 9147136 14. 48641 12. 11401.34 2. 7422902 15. 17034 13. 6871.45 3. 9934025 16. 26682 14. 9434.93 4. 8722928 17. 20322 15. 14667.20 5. 8988061 IS. 26428 16. 45360.45 6. 7391684 19. 90314 17. 116104.28 7. 7553772 m 115853 18. 140.73 379 880 AtfSWEBS 19. $861.11 J. 2666 4. 42756 go. $13196.43 4. 2044 5. 257796 Art. 37, PC 27 5. 6392 6. 3922 ft 126468 7. 68962 1. 183 7. 3886 S. 120411 S. 210 S. 3420 9. 4503 5. 195 9. 2668 10. 125944 4. 240 10. 1015 11. 196315 5. 221 11. 1702 12. 25245 ft 247 12. 2544 13. 16536 7. 238 15. 1950 14- 125008 o oon o. l\) U. 3901 15. 50996 9. 304 15. 5952 16. 633825 10. 306 11. 285 16. 3293 17. 3404 Art. 44, p. 31 10. 288 IS. 3496 1. 198 15. 361 19. 4898 0. 286 14. 252 $0. 6072 3. 1485 15. 255 836 16. 238 Art. 42, p. 30 5. 3465 17. 216 1. 5382 ft 2706 IS. 342 g. 14484 7. 7909 Art. 39, p. 28 5. 21546 4. 42490 8. 2893 P. 3916 1. 35100 5. 180852 10. 10373 g. 8432000 ft 394632 11. 2827 3. 712400000 7. 77868 lg. 7095 4. 46800000 S. 323228 15. 14982 6. 1736000 9. 95988 14. 27280 ft 56000000 10. 141245 15. 78716 Art. 40, p. 29 11. 124323 1ft 33682 1. 86625 g. 6019936 n. 2313396 15. 1783800 Art. 47, p. 33 3. 56939 14. 3382260 1. 327 4. 6774792 15. 1878874 0. 444(8rem.) 5. 3059625 16. 484956 3. 538 6. 74664 17. 351663 4. 144 7. 867456 18. 5344924 5. 4739 8. 217665 19. 4591947 6. 43785 (1 rem.) 9. 359074 gO. 649540 7. 21318 /0. 271466 Art. 43, p. 31 8. 317 5. 327 Art. 41, p. 30 1. 3068 10. 514 1. 1428 g. 6608 11. 1006 g. 1481 5. 21046 12. 428 AHSWERS 281 IS. 9512 7. lift 14. 11080 (398 rem.) 8- 1TW 15. 3323 (715 rem.) 16. 1921 (1771 rem.) _20 2-jY^r 17. 208 (1846 rem.) 11. 20 18. 63 (7323 rem.) 12. 70 19. 647 (2983 rem.) 15. $6 gO. 1139 (652 rem.) 14. 117 Art. 48, p. 33 15. 255f 1. 468 ' Il3 ^33 g. 73 15. 12| 3. 28 (60 rem.) 19. 40 4* 79 (6800 rem.) 5. 63 (875 rem.) gl. 70 ft 420 (70 rem.) 22. 3402 7. 537 (31936 rem.) 25. 15 8. 80016 (7829 rem.) 9. 670 (197356 rem.) 24. 144 T V 10. 5179 (36527 rem.) si Art. 54, p. 36 Art. 73, p. 46 1. $160 1. 25.6 g. A = 90 2. 425.54 3. $24 5. 659.656 4. A = 24 4. 1797.90311 5. $30 5. 1112.928 [6. $75 6. 1610.9902 1 7. $70 8. $25 7. 329.9351 ft 260.53739 9. $60 10. $40 Art. 74, p. 46 11. $6 1. .706 12. A = 25 2. 7.8016 IS. 4360 5. 407.585 1^. 751 4. 206.9985 15. 24 5. 41.8017 6. 3158.1015 Art. 58, p. 38 7. 902.8887 1. 22f ft 16360.4852 *. 2730| S. 40j Art. 75, p. 47 4- 3T 2 T 6 /A 1. 1.162 . 2j- j- 2. .4025 ft 1ft 5. .022032 4. .0960 5. .0001537 ft .000320608 7. .0150294 ft .0279 9. 3.385 10. .08891 11. 10000 12. .01 13. .0001001 14. .0000001 15. .00010001 16. 1 17. 10011.0011001 18. 100000001 19. 25025025 20. 2500002.5025000025 Art. 76, p. 48 1. 2.725 2. 356^ 3. 587.2 4. 60000 5. .000052 6. .0000007 7. 250000 ft .26| 9. 5.0607 10. .156 11. .0135 12. 2.143 15. .00009325 14. .144 15. .00000697 16. .10809 17. .007225 18. .000901 19. 1, 100, 10, 10, 10000, .1, 100, 1, 1, .01 SO. .5, .005, .5, .005, .0005, 50000, .0000005, 50000, 50, 5000000 282 ANSWERS 21. .16, 1600, .00016, 1.6, 16000, 160000000, .0000000016, .0016, 160000, 1.6 22. 3, .000003, 30000000, .03, 30000, .OOOC00003, .03, 300, 3000000, .00000000003 23. .02, 2000, 200000000, .000000002, .02, 200, .000000002, 2000000000, 2000, .00002 24. .005 .00005, 500000000, 5, .000000005, .005, 500, 50000000, .000000005, .005 Art. 7 7, p. 49 2. .64 S. T* 4. .8 5. .8 6. .25 8. 10. jUh x T!L 13. 17. 17 T W<> 18. O. 101 T VoV 22. A 23. .625 24. .8 25. .75 26. .375 27. .5 28. .6 9. .875 50. .1666+ 51. .555+ 50. .666+ 33. .5714+ 54. .1875 55. .5333+ 36. .15 57. .64 55. .35 59. .9166+ 40. .933+ 41. .6428+ 40. .8333+ 45. .09375 44- -032 45. .8875 46. .175 47. .248 48. .2733+ 49. .1666+ 60. .246 51. .24416+ 50. .96875 53. .184 54- 7.125 55. 17.0285+ 56. 14.2857+ 57. 20.0075 55. 25.08 59. 9.0583+ 60. 16.005 61. 5.00125 Art. 78, p. 50 i. V A W 5. 4. W J ' ffi 11 *0. Art. 79, p. 51 3. V 5. * J&6 * W 9. W 5 -w 10. W Art. 80, p. 51 1. 7j 5. 15 4. 47| 5. 8f 6. 8. 13 9. 19Jf 10. 55 JB 11. 341* Art. 81, p. 51 * I I ANSWERS 283 s. rYA, T VA, 11. I! 12. Iff -t I!!!!!' 5BJ4 *' "7. Art. 82, p. 52 ^A, T WA> W. 37i IS. iVAVA, Art. 96, p. 57 5. TT e. m 5. 4}J 7. TTS Art. 90, p. 54 6. 43 s. m i ^v 7. 5* 9. ft J rv 9. it W. Iff .' ,1 01 10. 8} w. Hit 4 1 74^ a-r. 4 . T 1 ^ 5 ; tin . I Art. 88, p. 53 6. lH "' / 7 _j 24 7. tiiii ^- 6 ia *. 144 8. Ifji "' S. 360 9. 6H ' V 4. 84 ^ ' s ^ 18 ' ^ 5. 60 U 6. 11088 IS. 42lHf m Jj 1 ^. 42 Art. 89, p. 53 ^ |{f M. 17H - 072 188^ 7_S "^" 3 3 6 1. II $1 frfl 8% yr t 16 2. in, ui, m, 444- #0 262 5. Ill ift, Aft, Art * 91f P * 65 ^- 189A 1. A 98. 189 . J 29. 306jf 5. ify 50. 1309J 5. |M, |jj, i**, 4. ^A 6507 iff 5. A W- H568f 6. *VV W- 1525A 7. 2 1 54. 672| S. H 55. 45895 57. 1389 284 55. 4835 39. 790 40. 1220 41. 445| 42. 732 43. 492 45. 1113 '40. 244f 47. 5483if 45. 4821 A 42. 39971$ 50. 904f 51. 265011 58. 228381H 53. 41269/? 54. 760f-f 55. 381 f| 56. 1146fj 57. 2264 T \ Art. 97, p. 59 i. ' *. A * A e. A 7. A * A 9. A 10. A 15. l j*. H 17. . J . Mr . ft A ANSWERS 5. 276}$ ^7. 6H 98. 14T 6 ! . 11221 36. 4 57. li ^. n 39. 40. 41. 46. Art. 98, p. 61 1. 120 *A 5. A 4. 3li 5. 49 e. 24 7. 547J 5. If . iiV m 135 ^- A ** fit 15. ^. 17. 22H P. T ?. 1st, 3d, ;2d, Art. 100, p. 64 i. llf wk. ^. 58.24bu. 5. .814| mi. 4- A, 5. 116H ^. $2.02 7. iieH ^. 169.71 P. 1000 Ib. Art. Ill, p. 68 1. 91800, 45900, 22950, 15300, 11475 2. 122400, 81600, 61200, 40800, 30600, 20400, 15300 3. 18400, 12266f , 9200, 6133^, 4600, 3066f , 2300 4. 140, 1400, 14000, 105, 1050, 10500 5. 140, 1400, 14000, 70, 700, 7000 6. 810, 8100, 81000, 4050, 40500 7. 157J, 1575, 15750, 315, 3150, 81500 ANSWERS 285 8. 1300, 13000, 390, 3900, 39000, 5200 9. 1600, 16000, 2133}, 21333}, 6400, 85333} 10. $1320, $880, $660, $440, $330, $220, $165 11. $61.33}, $460, $30.66f, $230, $153.33}, $11.50 12. $16.80, $11.20, $8.40, $112, $84, $5.60, $42, $4.20, $56, $420, $28, $210, $280, $21, $168, $1120, $840 $560 13. $12.80, $1920, $128, $192, $1280, $19.20, $9.60, $2560, $96, $256, $960, $25.60, $64, $51.20, $640, $48 14. $660, $66, $880, $1320, $88, $132 $176 15. $28800, $192, $14.40, $96, $1920, $72, $960, $36, $36000, $720 Art. 115, p. 73 1. $6.64 2. $38.98 3. $93.22 4. $164.52 5. $130 6. $6.88 7. $21.98 8. $10.83 9. $60.30 10. $111.34 Art. 116, p. 73 7. 5 da. 16 hr. 2 min. 1. $17.79 34 sec. 2. $82.92 8. 9 sq. yd. 8 sq. ft. 3. $98.33 47 sq. in. 4. $49.49 9. 23 Ib. 10 oz. 2 dr. 5. $20.05 Isc. 6. $13.10 10. 1 mi. 14 ch. 3 rd. 7. $54.94 31. '*. $1.68 11. 215 rd. 4 yd. 1 ft. 9 in. Art. 11 7, p. 74: 12. 6 bbl. 25 gal. 1 pt. 1. $23.47 7. $22.31 1 gi- 2. $26.06 A $25.11 13. 2 Ib. 6 oz. 5 dr. A $11.71 9. $24.11 14. 7 mi. 67 ch. 3 rd. 4. $29.36 10. $13.73 9 1. 5. $31.20 11. $132.41 15. 5 sq. yd. 3 sq. ft. 6. $101.44 12. $38.68 40 sq. in. IS. $493.01 16. 47 da. 17 hr. 54 14. Cost, $6.22; price, $5.50 min. 17. 10 Ib. 8 oz. 13 gr. 15. $7.17, 23c. 18. 12 cu. yd. 4 cu. 1(5. $18.28 ft. 605 cu. in. 17. Cost, $6.91; price, 19. 24 bu. 3 pk. 4 qt. $6.25 1 pt. 18. $41.81 80. 46 Ib. 11 oz. 8 19. $72.77 pwt. #0. $38.98 81. 1672 in. 81. 2625 Ib. 88. 17174ft. 0. $10.03 83. 16336 gr. 2S. $29.70 84- 15460 sec. &#. 35c 85. 8476 sc. 25. $82.67 86. 3319 in. 87. 4082 pwt. Art. 160, p. 85 28. 2906 min. 1. 21 rd. 2} yd. 8 in., 29. 1321 sc. or, 21 rd. 2 yd. 2 SO. 57620 sec. ft. 2 in. SI. 72920 sec. 8. Imi. 117 rd. lyd. 32. 191 gi. 1 ft. 5 in. 33. 955 pt. S. 10 bu. 2 pk. 6 qt. 34. 166694 cu. in. 4- 39 gal. 3 qt. 35. 165 pt. 5. 2 oz. 14 pwt. 2 gr. 36. 518 gi. (5. 2 Ib. 8 oz. 4 dr. 37. 161 pt. 2 sc. 12 gr. S8. 147 cu. ft. 286 ANSWERS 39. 466 gi. 8. 120 sov. 8s. 3d. 3 23. 4 gal. 1 qt. 1 pt. 40. 698 dr. far 1 gi. 41. 6 ft. 7^ in. 9. 162 sov. 16s. 7d. 24. 167 bu. 1 pt. 42. 35 rd. 9 ft. 2 in. 3 far. 25. 5 yr. 8 mo. 19 da. 43. 14ft. 2. 28 in. 20. 79 sov. 2d. 1 far. 9 hr. 58 min. 44- 4 oz. 10 pwt. 22. 1325 sov. 7s. 9d. 45. 3hr. 1 far. Art. 163, p. 90 46. 9 oz. 2 sc. 17.6 gr. 12. 120 sov. 9s. 4d. 2. 2 yr. 9 mo. 7 da. 47. 11 cu. ft. 1404 cu 2. 3 yr. 9 mo. 11 da. in. Art. 162, p. 89 3. 1 yr. 8 mo. 15 da. 48. 2 pk. 1 qt. if pt. 49. 3qt. 1 pt. 2.08 gi. 2. 26 Ib. 4 oz. 7 pwt. 4. 2 yr. 1 mo. 16 da. 5. 3 yr. 3 mo. 16 da. 50. 3 sq. ft. 108 sq. in. 51. 3 qt. 1 pt. 11 gr. 2. 250 rd. 1 yd. 1 ft. 6. 186 da. 7. 93 da. 52. 16 sq. yd. 5 sq. ft. 106. 2 sq. in. 7 in. 3. 103 gal. 3 qt. 8. 98 da. 9. 179 da. 53. j-$? bu. 4. 11 cu. yd. 10 cu. 20. 156 da. 54. .019375 gal. ft. 524 cu. in. 5. 1 Ib. lloz. 13 pwt. Art. 164, p. 91 56. sij Ib. 4gr. 2. 37.236qt. 57. Tinnrlb. 6. 7 2. \^ Ib. 58, .02074+ cu. yd. S9. j"4o "o"o~ mi. 7. 13 8. 96 bu. 2 pk. 6 qt. 3. 10 Ib. 7 oz. 12 pwt. 2gr. 60. .0001475 Ib. 9. 2 da. 20 hr. 29 4. 1 Ib. 2 oz. 11 pwt 61. -rJ-g- gal- min. 24 sec. 16 gr. 6#. TIT o"ir mi. 20. 4 mi. 286 rd. 1 5. 54 + 63. .0041b. yd. 1 ft. 2 in. 6. 51yV 64. .003645 Ib. 22. 1 bu. 2 pk. 3l qt. 7. 496 bu. 65. . 38611+ Ib. 66. .528125 da. 13. 8 Ib. 3 dr. 14 gr. 8. 137J bu. 9. 7iiy gal. 67. .6875 gal. 24. 42 da. 6 hr. 2 min. 20. 2872f gal. 68. .64lf sov. 69. .703125 bu. 48 sec. 25. 189 rd. 4 yd. 7 in. Art. 170, p. 93 70. .727083jsov. 26. 41 27. 37 gal. 2. (a) 12X18, 24X9 Art. 161, p. 87 18. 52 mi. 17 rd. 2 ft. 2 in. (c) 14X21, 28xlO| (d) 16X22, 32X11 1. $124.07 19. 8 mo. 26 da. 18 4. 7X9J, 4X14, 2. $398.25 hr. 34 min. 32 sec. 3^X21 3. $105.99 20. 88 Ib. 5 oz. 4 pwt. 5. 7X11, 5^X14 4. $82.43 16 gr. 6. 5^X8, 4X11, 5. $169.06 21. 42 mi. 186 rd. 5 2fxl6 6. $65.06 yd. 2 ft. 3 in. 7. 10fx5i, 4Xl4f, 7. 31 sov. 7s. Id. 3 22. 89 A. 131 sq. rd. 8X7J, 5^X11, far 27| sq. yd. 16X3|, 2fx22 ANSWERS 287 8. 5^X8, 4X11, part; 3d, 2 parts; Art. 188, p. 103 2fxl6, 2X22 9. Same as problem 4 10. 7^X4, 2|X10|, 4th, 4 parts 3. 5. 2, 2, 20, or 5, 4, 4, 20, or 5, 6, 6, 20 1. 28$ sq. yd. 2. 10 A. 150 sq. rd. 3. 96 sq. rd. See problem 3, 4. 3, 3, 3, and 12 lb. respectively 5. lst,641b.;3d,81b. 6. 240 lb. at 40^, 240 4. 283^ sq. ft. 5. USAVrsq. rd. (5. 17 A. 37ilf sq. rd. 7. 6A. 6ix6j, 4jx9^/ lb. at 55^, 80 lb. at 8. 1512 sq. ft. 8^X5, 3|xl3, 65^, 400 lb. at 75^, 720 lb. at 85^ 9. 714 sq. yd. 10. $14.85 6ix6f, 4^X10, 7. 5, 6, and 5 parts respectively 21. 5| sq. rd. 12. 1638 sq. ft. 9jx5j, 3^X14, 8. 10, 10, 10, and 29 15. 64{f rd. 7X7, 4f XlOi parts respectively 14. 4736 sq. in. 14X3|, 2^X21, See problem 7, 9. 19 of 1st, 39 of 2d, 285 of 3d, 25 of 4th 15. 606| sq. yd. 12X6,4^X16, P. 99 Art. 189, p. 105 9X8, 6X12, 18X4, 3X24, 1. $.8115 2. $1516.69 2. $2160 3. $5000 123X63, 4jXl6?, 3. $44622.09 4. $2052 93X83, 6^X12^, 4. 130 5. $1400 19X4 S ,3 1 6 X25 5. $111.27 6. $10666f 11. 8X4|, 3X12|, 6. 33.75 7. $2666.67 12X3L 2X19 7. 197.21 o ov/ i ti 9. 98^ Art. 190, p. 106 sheets in the table large enough. 10. W, ^ part 11. $72.30 1. 128 rd. 2. 151.4004 A. 13. 22x32 U. 25X40 12. $16.61 13. 190.40 3. 19yd. 4. $918 15. 32X44 14. 41.89 qt. 5. 63 T \ ft. Art. 172, p. 96 15. 6178.92 ft. 6. 544 sq. ft. 16. 1608 lb. 7. 10164 sq. yd. 1. 74i/ 17. 2202.07 francs, 8. 64 sq. yd. 2. $.0341 1781.97 marks 9. 38|f rd. 3. $.5996 18. $286.80 10. 1144 sq. ft. 4. $.704 19. $605.31 5. 47Jc 20. $22.30 Art. 192, p. 108 21. $32.82 1. 1152 sq. ft. Art. 173, p. 98 22. $.60 2. 3180 sq. yd. 1. 50 lb., 125 lb., 50 05. 23 bu. 3 pk. 7 qt. 3. 2294 sq. rd. lb., 50 lb. 84- $407 4. 7475 sq. rd. 2. 1st, 2 parts; 2d, 1 25. $11.20 5. 3182 sq. ft. 288 ANSWERS 6. 15435 sq. yd. 7. 11684f sq. yd. 8. 65ft. 9. 19200 sq. ft. 10. 832J sq. ft. 11. 21-rV A. 1*. 26f rd. .13. 8|4 ft. 14. 8 rd., 6 rd., 4| rd. 15. 203 sq. rd. Art. 193, p. 109 1. 266 sq. ft. S. 750 sq. yd. 3. 3755 1 sq. yd. 4. 118572 sq. ft. 5. 3801 sq. rd. 6. 20 rd. 7. 14^V A. A 600 sq. rd. 9. $6776 10. 7l|rd. Jl. 6V A. IS. 28j sq. ft. JW. 384 sq. ft. U. 494 sq. in. Art. 198, p. 110 1. 65 sq. ft. S. 174 sq. ft. 3. 186ft sq. yd. 4. 477.12sq. ft. 5. 93f sq. ft. 6. 166.32 in. 7. 7ift. 8. 10.4ft. 9. 43. 0119 sq. ft. 10. 1731. 197 sq. yd. 11. 21.217 sq. ft. IS. 2519. 134 sq. ft. 13. 2992 sq. ft. 14. 3330.84 + sq. yd. 15. 1236. 077 q. ft. Art. 203, p. 112 1. 87.96 ft. S. 15.597yd. 3. 17.825 yd. 4- 43.98ft. 5. 10.822yd. 6. 131.947ft. 7. 15.119yd. 8. 43.982rd. 9. 39.152rd. 10. 72.78ft Art. 205, p. 113 1. 198.9f sq. ft. S. 8148.48 sq. rd. 3. 286.478 sq. ft. 4. 17.104sq. ft. 5. 50.265 sq. ft. 6. 53.794 sq. yd. 7. 91. 987 sq. yd. 8. 5674.515 sq. ft. 9. 254.469 sq. rd. 10. 452.39 sq. ft. 11. 452.39 sq. rd. IS. 804.249 sq. yd. 13. 17A.107.44sq.rd. 14. 50A.148.70sq.rd. 15. 33.183sq. ft. 16. 151.83 sq. ft. 17. 3 sq. ft. 70.65 sq. in. in. 18. 97.482 sq. ft. Art. 206, p. IU 1. $12.80 2. 44 sq. yd. 3. Floor, etc., 105 J sq. yd. ; walls, 59j sq. yd. 4. 108i sq. yd. 5. $27.11 6. $39.60 7. 9350 sq. yd. 8. 532 9. 1344 sq. yd. 10. 464 sq. yd. 11. $670 12. 36f sq. yd. 13. 145J sq. yd. 14. $114.64 15. $4833.89 16. 3019. 077 sq. ft. 17. 472$ sq. yd. 18. 21 f squares 19. 160 sq. yd. SO. 1020 sq. ft. Art. 208, p. 115 1. 12 S. 15 3. 10 4. 25 5. 12 6. 104yd. 7. 541^ yds. 8. 23J Art. 209, p. 117 1. $61.60 * 72 or 69. 80 3. Lengthwise, $64.80; crosswise, $64 4. 49yd. 5. (a) 186 yd. cross- wise, (b) 50 yd. cross- wise, (c) 136 yd. length- wise, (d) 21 1 yd. cross- wise, (e)37j yd. cross- wise 6. (a)9iyd., (b)ljyd., (c)8yd., (d)3jyd., (e) 4f yd. ANSWERS 289 7. (a)86f yd., Art. 219, p. 122 7. 1481^ gal. (b) 77 yd., 1. (a)H>U, . 63.9744 bbl. (c) 26 yd., (b) 5i, 9. 18|| gal. (d) 32 yd., (c) li, 20. 1.652 gal. (e) 83j yd. 22. 15.9936 gal. Art. 213, p. 118 (e)7j 2. $23.63 12. 239f*gal. 25. 82.318 bbl. 1. (a)2304cu. ft, 3. 96 ft. (b)784cu. ft, 4. 4$ ft. Art. 225, p. 126 (c)2560cu. ft, 2. 115.2 bu. (d) 1233 ou. ft. Art. 222, p. 123 2. 65| bu. 2. 200 cu. ft. 2. 20 3. llS^bu. 3. 1344 cu. ft. 2. 10i 4. $76.56 4. 5939.886 cu. ft 3. 15 5. 800 cu. ft., 25 ft. 5. 1200 cu. ft. 4. 25 6. 4^ ft. 6. 2781. 173 cu. ft. 5. 32 7. 117.81 cu. ft,883.57a 7. 114648.596 cu. ft. 6. 11 gal. 7. 6 8. 6.565 ft Art. 214, p. 119. 8. 15j 9. 340 bu. 2. (a) 197.92 cu. ft, 9. 26| 20. Sll^bu. (b) 298.414 cu. ft., 10. 42 22. 47Jbu. (c)326.726cu. ft, 22. 46f 22. 87j bu. (d) 982.62 cu. ft 22. 162 13. 1 H ft. 2. 2513.28 cu. ft. 13. 106f 14. 62fcwt 3. 436.72cu. ft. 24. 16 15. 14f T. 4. 26507.25 cu. ft. 15. 12 16. 10 T. 5. 30.968 cu. ft. 2S. 18| 27. $162.85 6. 8.84ft 27. 10 2. 67|J ft. Art. 217, p. 120 18. $5.53 29. 640 cu. ft., 4f ft 2. 1050 cu. ft. 29. $63 0. 194$ bu. 2. $59.05 3. $36.27, cost of dig- ging; $58.83, cost of wall ; 10^ 5 j cd. 20. $25.05 2. $130.65 22. $32.82 23. $21.20 Art. 231, p. 132 2. 24 2. 75 4. 2404 cu. ft. Art. 223, p. 125 3. 206 4. 1234 5. 12 1 cd. 6. 27720 2. (a) 808. 57 5. 2.645 7. Cost of digging, (b) 20.09 6. 1.414 $93.33; cost of lay- ing, $64.15; Bffy cd. 2. 176} ibu. 7. 126 5. 1626.57 M 8. 4838 4. 84|bu. 20. .968+ 9. 2592 cu. ft. 5. 807?^ gal. 22. 48.989rd 10. $148.78 6. 79 ; bbl. 22. 125 290 ANSWERS 13. 38809 P. 136 S. $18, $21.60 14. $397.60 1. 176. 715 sq. ft. 3. $63 15. 26.92 ft. 2. 257.8 sq. ft. 4. $47.40 107.7 ft. 3. 24f ft. 5. .75 Art. 232, p. 133 4. 390H bu. 6. 25 1. 419.98 sq. ft. 5. 8 T 5 g-cd. 7. 1200 S. 9 A. 47.914 sq. rd. 6. 132.53 cu. in. 8. $1440 S. 1200 sq. ft. 7. 164.93 sq. ft. 9. 500 4. 2 A. 111. 155 sq. rd. 8. $320.89 10. .25 5. 14.273 rd. 9. $392.89 11. .125 6. 1664.45 sq. ft. 10. 2035.75+cu. ft. 12. 40.48 7. 45.13 rd. 11. 21120 IS. 5, 500 8. 31.7rd. 1*. 83^ yd. 14. 625 9. 3.19 ft. 15. 8750 Ib. 15. 5lf 10. 309.74sq. rd. 14. 168ft. 16. 252 11 Circle, 795.77 sq. 15. 2361.28ft. 17. .12 yd. ; square, 625 16. 1060 cu. ft. 18. .92 sq. yd. 17. 238H toads 19. 1.12 IS. 13.541ft. IS 47.1! bbL #0. .30 13. 88.622 ft. 19. $6!55 SI. .08 14. 6.684ft. 0. 130| sq. yd. 0*. 618.75 15. 9 A. 137. 9 sq. rd. SI. 72j-Jcd. 23 86 16. 17 ft. 8 in. SS. 48486 & $51 17. 191.33 sq. yd. 0*. 63ft. 05. 301 18. 110.84 bu. 5016 ft., 209 posts Art. 251, p. 142 19. 6196.77 sq. ft. 25. 168|bu. 1. 86.4, 446.4, 273.6 SO. 239.24 sq. ft. 26. (a) 288 ft. S. $294, 994, 406 Art. 234, p. 135 (b) 1104 ft. (c) 2520 ft. 3. $548.10, 1388.10, 291.90 1. 9yd. (d) 120 ft. 4. $632.875, 1395.375, S. 25 ft. (e) 1877J ft. 129.625 3. 7 A. 56 sq. rd. (f) 3645 ft. 5. $158.77, 696.97, 4. 31.112ft. (g) 2862 ft. 379.43 5. 26.076 ft. (h) 1842.4 ft. 6. $171.10, 855.50, 6. 19.31 ft. (i) 526 sq. yd. 513.30 7. 75ft. (j) $27.84 ' 7. $699.45, 1632,05, 8. 22500 sq. ft. 27. (a) 1780 cu. ft. 233.15 9. 105.47ft. (b) 13ft cd. 8. $212.87, 851.47, 10. 18.7 ft. (c) 815 A- loads, 425.73 It 1.414ft. (d) 112 sq. yd. 9. $8517.25, 18251.25, IS. 1536 sq. ft. (e) 748 sq. ft. 1216.75 13. 150 ft. 14. 28.42 ft. , Art. 235, p. 139 10. $832, 2080,, 416 11. 105, 945, 735 15. 470.3 sq. ft. 1. 20, 182J, 504 IS. 354, 1298, 590 ANSWERS 291 13. $42732, 2991.24, 21. 97 T V$ Art. 255, p. 149 2136.60 00. .0005 jg$ 1. 1800 14. 3924.48, 8596.48, 2S. 53^$ 0. $284.88 747.52 04- ^\% 5. 882 T 6 T 15. $398.95 16. 6995. 08^ Art. 253, p. 145 4. $497.14 5. $876.46 17. 8525 1. 62400 0. 1200 18. $330.13, 1251.43, 0. $135.625 7. $404.53 351.17 5 1416 8. $122.74 VJ. $397.39, 1147.19, 4 ll 9. $275.85 352.41 5. 4060.15+ 10. 2100 20. 1301.44, 5949.44, 6. 1650000 11. 1909.556 Ib 3346.56 7. 111854 12. 140.2yd. 21. 546 S 12g 13. 110.731b. 22. 6 T. 716 Ib. 9 .0378$ 14. 400 2S. 11.28 gal. 10. 93000000 15. $1030.93 24. 87 T. 25, 5.46^ yd. 26. 18.306 Ib. 11. $35812.50 12. $1172958.46 IS. $13861111.11 Art. 256, p, 152 1. $1063.59 27. .1681b. 14. $27500 0. llyr Art. 252, p. 143 15. 5170H- gal. 10. 2857lf Ib. 3. 12if $ 4. $3.834 1. &>\% 17. 400000 gr. & ^TT 9 ""$ 3. 75$ 18. $9166.67 19. 1578if Ib. silver, 6. 5.48fr* 7 $140. 62^ 4- 11$ 1263 T 3 lb tin, & HTT$ 5. 60|f $ 16736 jf Ib. copper 9. 13H^ 6. 150$ 20. 326^- Ib. 10. 3lJ$ 7. 25$ 11. 25^ 8. 133j$ Art. 254, p. 147 10. 3.384$ 9. 75iW# 1. 888f 15. $7043.75 fe/. .06ff$ 0. 1600 14. 40$ 11. 3333|$ 3. $26 15. 63 T \$ 10. 4$ 4- 446y 6 o 3 3 16. 16f? IS. 242 |f $ 5. 64.224+ 17. 6f$ 14. 3?A $ 6. $1219.23 IS. 60^ 15. 1580$ 7. $1311.69 19. $40.19 16. 99.156+2 8. 757.696+ 00. $4 17. 2.414+$ 9. $469.56 01. 4$ 1*. 831J.11 10. $1193 74 00. 6i$ loss 19. .004985+%, ' 11. 684 2S. $3.4l| .006186+$, 10. $1600 04. $3 .007123+$ 13. 25 1 V T lb. Art. 2 71, p. 158 20. 7.84$, 32H$, 14- 17.8875 ft. 1. Com., $19.70; 59|$, 6.641b. 1C. $9259.26 cost, $895.30 292 ANSWERS 2. Com., $30. 21; 18. Com., $17.06; 10. (a) $94.08 cost, $984.21 guar., $3.41; (b) $436.30 3. Com., $8.10; proceeds, $320.78 (c) $71 52 guar., $.81; 19. $2653 (d) $449.75 cost, $332.91 20. $14610 (e) $948.49 4. Com., $37.96; 21. $661.68 (f) $2383. 78 guar., $12.65; 22. $1123.09 (g) $2991.96 cost, $1316.01 2S. $2932.50 (h) $318.05 5. Com., $70.81; 24. $7125.00 (i) $499.80 guar., $19.31; 25. $5975.65 (j) $636.53 cost, $1398.75 26. $2418.33 11. (a) 60% 6. Com., $35.28: 27. $1467.72 (b)38f% guar., $5.04; 28. $11000 (c) 43f % cost, $544.34 29. $7266.67 (d) 20% 7. Com., $86.25; SO. $18150 (e) 68|% guar., $40.02; SI. $5700 (f) 108 % cost, $2127.27 32. 2ff % (g) 66f% 8. Com., $13.11; 33. 5% (h) 13 \% guar., $8.74; S4. \% (i) 100% cost, $472.67 35. 6^"5" % (j) 5^4% 9. Com., $9.14; 36. Com., $374.56; 12. Firstl^ cost, $313.94 guar., $46.82; Art. 276, p. 165 10. Com., $30.35: proceeds, $4233.62 1. has, aas, dhm. proceeds, $836.90 37. $6320.05 hem, aos, ms, no, 11. Com., $2.42; S8. $2781.13 his, nnn, dns guar., $.24; 39. $479.15 S. $1.55, $5.70, $.99, proceeds, $94.09 40. $1418.27 $2.18, $3.65, $4.25, n. Com., $1.28; 41. $3515.24 $8.00, $7.20, $12.75 guar., $.32; 42. $5809.15 3. hid, m h n, had, proceeds, $62.64 43. $99.25 amn, dso, snh IS. Com., $123.98; proceeds, 44. $14070.99 45. Second agent by hyyy iyv, hsld $2975.62 \% Vl Q V 14. Com., $91.27; Art. 2 74, p. 162 X V II D J r ' wcc guar., $27.88; proceeds, 1. $49.94 (b) $1706.65 2. $445.50 e r 15. Com., $1523.44; 3. $708.75, $725.76, hms proceeds, $4489.06 $696.15 h a a 16. Com., $520.96; 4. $23.60 ,^ a d y guar., $71.04; 5. $489.60 ' h i s proceeds, $1776 6. $10 .has 17. Com., $132.62; 7. $7.35 ' w r i guar., $18.42; 8. $351.47 /f) a yy proceeds, $534 26 9. 1% ' h a s ANSWEB8 293 hoy I 1st, $750.66; 94. 8V hs s 2d, $1154.60; 55. $53.75 /u\ *y 8d, $1867.82 36. $2.20 * ' wsp 6. 1st, $3302.18, pre- 57. $4.71 r.v hay miums ; 55. $10.08 W hws 2d, $3923.60, pre- 50. $16.59 / -\ h d 7 miums ; 40. $37400 w o li Art. 280, p. 166 8d, $6347.25, pre- miums; $4617 P. 181 1. $1.46 1. $598.93 received 0. 61? 0. $354.63 P. 179 5. 78? 3. $996.07 1. $9.52 * $15.23 4. $160.90 2. $2.28 5. $4.77 5. $70.89 S. $5.34 6. 82.53 . $61.84 4. $10.56 7. $5.37 7. $207 5. $1.51 8. $9.70 8. $113.54 6. $1.19 0. 84? 0. $13.36 7. 63? 10. 94? ;0. $19.60 8. $14.06 11. $2.47 P. 171 0. $13.91 10. $8.24 10. $17.44 13. $3.15 /. $41.56 $27126.93 5. $189776.67 11. $161.97 12. $7.16 15. $17.22 14. 75? 15. $1.74 16. $1.84 4. $6.48 14. $74.62 17. $1.05 5. $18.23 15. $29.93 18 S3 32 5. $3053.81 16. $267.38 10. $2.23 7. $7275 5. $3690.93 0. $100260, loss to 17. $20.63 W. $33.69 19. $1.29 00. 44? 21. $1423.81 00. $1001 Co. ; $43615 loss to owner under aver- 00. $2.42 21. $144.87 05. $278.05 94. $317.94 age clause 10. $22. 50 per $1000 11. 44f yr. 00. $16.13 05. $4.22 04. $2.77 05. $279.14 26. $829.57 07. $419.22 12. 8.79^ 13. $700.62 and $774.38 05. $37.66 26. $1.39 07. $58.09 28. $618.13 00. $284.59 50. $116.25 P. 176 05. $66.82 SI. $786.97 2. $91.06, premium: 00. $160.09 50. $574.60 $820.06 loss 50. $7.55 55. $1827.93 3. Receive $2000; SI. $2.42 54. $68.22 would have re- 50. $26.41 55. $14800.46 ceived $1703.34 33. $12.91 86. $844.29 $94 AKSWEBS 37. 1207.84 38. 259.73 39. 359.11 M. 674.66 1. S. 1.88 10.37 5. 22.64 6. 8.62 7. 97.51 S. $57.99 9. 146,41 10. 25.59 11. 24.29 ^ 6.21 15. 6.91 14. 175.07 J5. 17.93 16. 2.90 17. $10.35 18. 5.84 19. $231.30 m $66.18 84. 2.63 8.16 12.24 85. $19.42 26. 1.11 27. 1.51 2S. 4.86 89. $6.87 30. $7.70 P. 31. 56^ 52. 1.49 55. 3.05 5 $4.29 25. #1.82 50. $5.61 7. $1.13 $8. 11.19 59. 34.96 40. 31.85 P. 185 & 17.59 5. 6.60 4. $.18 5. 62.48 0. 35.93 7. $19.66 5. $6.26 9. $2.63 10. $8.67 P. 186 1. $51.75 5. 24.10 4. 156.60 5. 40.64 0. 14.53 7. $29.98 5. 9.59 9. 25.28 10. 27.67 Zl. 59.85 18. 139.50 *5. 3.18 14. $4.29 15. 14.73 16. $9.59 17. 9.24 15. 6.55 19. 1.51 20. 3.63 '1. $22.09 88. 40.4 25. 66.22 84. 126.02 25. 79.04 26. 259.72 27. $225.16 88. $115.77 89. $73.37 50. 131.84 51. 246.02 58. 172.83 55. $21 54. 261.36 55 30.76 56. 12.22 37. 61.57 55. 108.25 59. 55.87 40. 152.64 P. 187 t $2090, $663.90 & 22=50, 740.50 3. 3.05, 139.05. 4. 9.11, $216.11 5. 39.25, 1731.25 0. $45.35, 2091.35 7. 1.76, 252.76 8. $3.24, 238.24 P. 94.46, 2281.46 10. 14.01, 528.01 11. 54.86, 396.86 18. 31.59, 665.59 13. 74.29, 812.29 14. 51.15, 1286.15 15. 133.73, 3083.73 16. 80,38, 1045.68 17. 6.98, 439.98 18. 23.48, 1603.48 19. 64.17, 939.17 80. 41.86, 427.86 81. 43.59, 958.59 82. 110.80, 1133.80 8S. $238.66, $3845.66 84. 59.75, 678.75 85. 54.17, 839.17 86. 2.31, 317.31 87. 3.44, 430.44 88. 10.58, 1390.58 25. 14.12, 1039.12 50. $7.24, $723.24 31. $10.03. 834.03 ANSWERS 296 51 $3.74, $909.74 S3. $3.84, $317.84 34- $8.06, $431.06 P. 188 35. 45?, $206 45 86. $3.56, $344.56 37 $10.53, $1227.53 38, $11.82, $1098.82 99. $4.75, $425.75 40. $3.77, $541.77 41. $8 04, $681.04 42. $2.03, $740.03 43 $1.62, $298.62 44. $4.02, $313.02 45. $5.32, $278.32 46. $7.23, $955.23 47. $49.90, $3120.90 48. $34.44, $4287.44 45. $29.37, $1435.37 50. $36 72, $2386.72 Art. 328, p. 188 1. $4.80 2. 4 mo. 3. % 4. $374.40 5. 6^ 5. 8 mo. 7. $360 8. 4 yr. 2 mo. 10. 8 mo. 5 da. 11. 16-Jyr., 20yr.,12j yr., 33j yr. r 14f yr., 10 yr. 12. \% 13. 3 yr. 4 mo. 27 da. 14. $537.04 16. 1 yr. 2 mo. 22 da. 18. $904.98 10. 22f yr. Art. 335, p. 191 5. Due date, April 8 ( 1. $147.82 8728.44 2. $43.33 Due date, May 13 3. $77.50 $599.36 4. $154.25 . Due date, June 7, 5. $138.06 1900, S297.44 6. $149.34 6. Due data, Oct. 3> 7. $139 1898, 551278.85 8. $589 88 7. Due date, July 4, S. $330.80 1898, $381.30 10. $368.02 8. Due date, Pec. 28, 11. $178.05 $234.72 12. $533.98 9. Due date, Nor. 29, 13. $453.06 $4217.90 10. Due date, Dec. 30 Art. 337, p. 192 $551.01 1. $129 11. Due date, Dec. 1, 2. $132.55 $348.21 5. $74.64 12. Due date, May 8 4. $883.15 $438.32 5. $144.29 'ft $697.97 Art. 361, p. 202 7. $159.27 1. $292.40 5. $469.22 2. $289.04 3. $595.51 Art. 338, p. 193 4. $367.29 1. $681.88 2. $1002,05 Art. 362, p. 204 3. $1002.80 1. $1549.55 4> $314.36 2. $527.86 5. $377.95 3. $1213.01 6. $4097.36 4. $174 7. $1243.53 5. $1353.51 5. $1622.84 9. $1265.59 Art. 363, p. 206 10. $1594.59 11. $141.87 1. $1051.92 2. $1912.37 IS. $181.81 3. $219.40 13. $1353.14 4- $853.05 14. $380.25 5. $1597.33 15. $642.21 6, $312.86 7. $934.37 Art. 359, p. 200 *. $1735.54 1 $269.08 9. $623.72 2 $512.32 10. $1373.82 296 ANSWERS Art. 366, p. 208 1. 8807.69 2. $1193.89 3. Present worth, $1204.95 ; discount, $12.05 4. Present worth, $578.37; discount, 847.43 5. Present worth, $2928.47; discount, $155.5? 6. Present worth, $196.94; discount, $18.31 7. Present worth, $1016.15; discount, $11.85 8. Present worth, $1906.25; discount, $228.75 9. Present worth, $599.16; discount, $116.84 10. Present worth, $867.59; discount, $69.41 11 $1245.02 12. $3556.60 13. $1266.98 14. 4^ gained on each by buying on time 15. Time offer $23.81 better 16. $2500 17. $15.31 gain 18. 19.23^ 19. Cash offer is $11. 54 better SO. $6124.95 21. $8333.33 23. $110.57 4. $3162.98 Art. 374, p. 213 Art. 390, p. 218 Disc. Proceeds 1. $121.82 1. $ 8.24 $ 184.22 2. $71.33 2. 1.56 223.44 #. $399.38 3. 5.00 436.45 4. 229.16 4. 6.59 253.41 5. $131.84 5. 2.37 709.63 6. 3.65 452.35 Art. 395, p. 22S 7. 4.52 577.19 1. $851.70 8. 4.30 326.59 & $1283.20 9. 6.56 485.72 3. $3803.04 10. 3.98 314.02 4. $1571.57 11. 10.41 640.26 5. $9709.70 12. 13.07 527.68 6. $3260.92 13. 17.79 1142.21 7. 1st, $1.002 ;2d, $5 14- 6.19 475.93 3d, $850 15. 1.23 327.77 8. $2506.27 16. 19.65 1280.85 9. 32^ 17. 6.23 731.77 10. $4491.02 18. 2.70 423.30 Art. 398, p. 224 19. 4 69 375.31 1. $2305.75 20. 7.13 534.65 2. $18227.19 21. 4.20 835.80 S. $6580 22. 2.21 234.79 4. 753.40 Art. 385, p. 216 5. $1592.93 1. $337.17 6. $1692.42 2. $246.25 7. $3780.52 3. $471.83 8. $1260 4. $594 9. $537.46 5. $845.75 10. 2522.52 6. $1188 Art. 401, p. 225 7. $397.33 1. $199.50 8. $357.60 2. $177.75 9. $512.20 3. $174.12 W. $173.98 4. $1779.75 il. $227.70 6. $2333.45 *. $147 7. $580.58 Art. 388, p. 21 7 8. $2380 1. $197 Art. 410, p. 228 2. $345.62 1. $2044.60 3. Proceeds, $346.50, 2. 264 sov. 6s. 8d. lack, 3^f ; amount 2 far. due, 1352.28 J. $606.43 ANSWERS 297 4. $6937.98 10. June 13 8. Beekman, $480; 5. $4523.75 11. March 16 Hadley, $320; 6. 937 sov. 6s. Id. 12. Balance, $220; Perry, $440 3 far. Dec. 27, 1899, S. $1500, $1000 7. $3497.40 $225.68 4. Watson, gain 8. Cost, $4918.38; 13. Feb. 19, $216.47 $1292.31; P. W., face, $4900 15. $1.43, interest due 56892.31; Barnes, 9. $20281.40 April 25 gain $1507. 69; 10. $38528.44 Art. 420, p. 232 Art. 436, p. 248 1. $157.59 P. W., $7807.69 5. A, $5400; B, $8100; C, $6750; D, $9450 1. $5.23 8. $410.80. 8. $41.21 S. $159.64 6. Gooding, $22193.68; S. $899.88 4. $22.38 Spencer, $17966.32 4. $186.80 Art. 441, p. 249 7. Ha wes owned yV ; P. 233 1. $288.86 8. $586.65, due Dec. 12 Gross, T 5 s ; Ha wes received $23333 J; 8. $3607.50 3. $455.56 4. $811.42 J./Q 1. $1417.48 S. $1399.40, due Aug. Cross, $16666| ; Hawes received $26666f 5. $2116.80 20 9. Martin, $190328; 6. $131.21 Art. 442, p. 251 Gould, $1480.33; 7. $206.79 1. A, $6; B, $8 Towne, $1776.39 8. $767 64 8. 1st, $15; 2d, $25 10. Howe, $470.14; 9. $232.23 3. 10, 20, 30 Benton, $428.72; *ft $323.72 4. 60, 72, 84 Ward, $351.14 Art. 431, PC 240 5. 150, 225, 375 11. Bush, $104.35; 1. Feb. 25 6. A, $240; B, $420; Austin, $208.69; 2. March 11 C, $510 Fox, $86.96 3. May 30 7. A, A; B, T^; C, 12. Johnson, $1033 J; 4. July 6 3*3-; A's gain, $50 Chapin,$466| 5. Aug. 9 8. A, $75; B, $135 13. Wright, $13645. 18 ; 6. Sept. 4 9. $150, $175, $225 Greene, $11574.01; 7. Oct. 19 10. 45, 30 Bates, $14590.15; 5. Oct. 2 U. 45, 27 Thompson, 18. 210, 60 $6190.66 Art. 432, p. 242 13. 225, 400, 480 14. Randall, $7865.89 1. Dec. 21 74. $103.25, 92.75, Chapman, #. May 25 $177.10 $8125.92 Holt, 3. Aug. 6 15. $120, $135, $150 $6508.19 5. Jan. 4, 1901 5. March 27 Art. 456, p. 253 Art. 471, p. 259 7. March 23 1. Jones, $980; 1. $288 . Feb. 4 Smith, $700; 2. $460 . June 2 Brown, $1260 S. $801 298 ANSWERS 4. 8206 Art. 490, p. 268 11. \% 5. , 5i# 1. 8208.13, 817.68, 12. 2299. 12 sq. ft. 6. 2-gft> 8273.56, 85.51, 13. 24856^ mi. 7. 6 \ % above par 83.51. 81.46,' 82.05, 14. 213 T. 1650.3 ?b. 8. 814328.13 82.59, 852.61, 15. 81368.89 9. 830220.25 825.20, 8156.38, 16. 33.941 rd. 10. 85773.75 8104.06 . 7. 42 57 rd. 11. 88385 2. Rate, .0053; 18. 11.653 rd. 12. 81020 865.69, 824.17, 19. 14.142 in. 13. 812578.13 838.56, 8131.92, 20. 842857.14 u. 810046.25 8228.56 21. 20.67^ 16. 56 3. (a) 818.10 22. Increased 855 17. 28 (b) 87.31 23. 81173.54 18. 48 (c) 85.90 24. 84485.61 19. 8^V premium (d) 817. 28 25. 15033 T 3 T Ib 21. 8896 (e) 86.48 26. 2 yr. 9 mo. 18 da. 22. 8320 4> .00745 27. 720.28 23. 8672 5. .008065 28. 1.135% loss 24. First $17.50 6. 86403 29. Gain, 82405; P 25. Increased $> 7. 8116.47 W., 810405 26. 135 8. 822.97 30. 834.24 27. 28. 814445 820146.88 Art. 506, p. 270 32. 8513.75 8647.07 ?9. 840026.25 1. $835.80 AQ-f O 33. Oct. 3 SO. 834200 2. 5fol4 34- (a) 81021. 36 91. 1% 3. i 88828 (b) 81021.10 33. 5~t % 4- $tt 35. Gain, 82355; 94 1\% 5. 88350.70 resources, 89225 35. Latter \\% 6. 8188.70 liabilities, 86870 142f 7. 867.13 P. W., $7355 88. 96 8. 815480 36 166f# 14f% 9. $659.32 37. 27.325% 40. 10. 8840 38. 847.58 Art. 474, p. 264 P. 272 39. 836.75 1. 8 40. 892.62 2. $292.95 $4315.72 3. 19 T, 57lf Ib. . 3515.6 Ib. 41. 14 T 6 T ft. .0375 3. 4. $86.47 $4900 5. 32400 Ib. 833.10 43. 44- 8 breadths, 42-| yd. 8348 Art. 489, p. 267 6. 10 rd. 3 yd. 2 ft. 45. 10368 L 882.40 7. 4ftf 46. 46^ 2. 8260.63 8. 1737.59 gal. 47. 8205 $. .0074, 822.05, 9. 81000 48. 34.22% $272.84 10. 8119.65 49. Ill bu. 1 pk.2J qt. ANSWERS 60. $390 51. $30 52. $76613 53. 58} ib. 54. 55. 57. $10875 55. 45932jf 59. $17840 0. 5.86^ 01. 70$ it. 6*2. $5.75 6. 702 Ib. copper, 198 lo. tin. 64. $2420 65. 7 T. 1384 Ib. 66. {% 67. $14.50 68. 75 69. $185.96 70. 8.8 T \# 71. $58.65 n. $3:84 75. $412 7& A, $17671.8)- B, $23557.24 C, $7933.85, D, $18837.10 7J, $100.38 76. 7.48?* 77. (a) 37.57^ (b) 14.66^ (e) (c) 17. 1 (j) 2.4 75. 51b. at 14?; 3 Ib at W; 4 Ib. at 25^: 4 Ib. at 32^, 666| Ib. 79. $15.94 80. $673.96 T.. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO SI.OO ON THE SEVENTH DAY OVERDUE. 2 1934 MAR 26 1934 301049 +< 20Dec'59JO REOD LD DEC 6 - 1959 LD 21-100 V8 '1 7244 X . / '