Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/commonschoolaritOOeatorich THE COMMON SCHOOL ARITHMETIC 1NINO ANALYSIS AND SYNTHESIS ; APAPTEU TO THE BEST MODE OF INSTRUCTION IN THE ELEMENTS OF WRITTEN ARITHMETIC. BY JAMES S. EATOX, M. A., IKiTBrCTOR I!» rHILLlPS ACADEMY, AXDOVER, AND AUTHOR OF "EA*Y I.MIOJI IIT Mi.STAL ARITHMETIC," A>D "A TREATISE OX WRITTEN AH* ruMETlC.- [fc BOSTOWA $) S&. 1630. jSt BOSTON: 1 AGGARD AND THOMPSON, 20 COKNHILL. SAN FRANCISCO: II. II. BANCROFT * COMPANY. 18 • • . • Entered, according to Act of Congress, in the year 1863, By JAMES S. EATON, M. A., In the Clerk's Oflico of the District Court of the District of Massachusetts EDUCAT1UW DKPT A .V D O V r. R : ILBCIEOirriD A5D P K I Jf T X D JB T W. r. U E A P K K. PB K FACE Thf.rk is a largo class of pupils whose limited time renders it lible for them to pursue an extended mathematical course. The author, in accordance with his original intention to prepare a oki in Arithmetic, has now endeavored to adapt this work to tho wants of this class of pupils. With this purpose in view, the simple, elementary, practical prin- ciples of the science are more fully presented than in his larger work, while the more intricate and less important parts have been I more briefly or entirely omitted. A corresponding change in the character of the examples has also been made. As in the irork, so here, constant attention has been paid to the brevity, simplicity, perspicuity, and accuracy of expression ; and no effort has been spared in the endeavor to render the mica! execution appropriate and attractive, nitions, tables, and explanations of signs have been distrib- through the book where their aid is needed, to enable the pupil to learn them more readily than when they are pre ivfly. ily all the example! have been prepared for this book, and are different from those of the larger work ; still, to secure uniform- ity of language (a matter of great importance, as every experi -. the leading example! in the several subjects, the definitions and rules, with few exceptions, have been intentionally 1 with but little modification. 6*349529 iv PKEFACE. Articles on United States Money, Percentage, Stocks, Custom- House Business, and Exchange have been prepared for this book ; and all the principles requisite for a practical business life have been presented in a simple, intelligible, attractive manner, and with sufficient minuteness and fullness and a due regard tc logical arrangement Brief, suggestive questions have been placed at the bottom o( the page, designed in no way to interfere with the fvw t original questioning which every teacher will adopt for himself, but merely to aid the young and inexperienced pupil in fixing his attention upon the more important parts of the subject Here, as in the larger work, some of the answers to examples have been given to inspire confidence in the learner, and others are omitted to secure the discipline resulting from proving the oper- ations, a discipline and a benefit which the pupil should not forego nor the teacher neglect Fully appreciating the favor which has been bestowed on his other works, the author sends this forth, hoping it may commend itself to the approval of committees and teachers, and that it may be found adapted to contribute in 6ome measure to the happiness and improvement of the class of pupils for whom it is deagned. A Key, containing the Answers not given in this book, is published for the use of Teachers. Phillips Academy, Andovek, ) April ld' } 18G2. \ CONTENTS SIMPLE NUMRERS. Definitions Notation and Numeration . . French Numeration Table . . - in French Numeration . Numeration Table . . Exercises in English Numeration PAOE . 7 PAOK Roman Notation 15 K\« irises in Roman Notation . . 10 Addition 17 Subtraction 24 Multiplication SO Division 42 REDUCTION OF COMPOUND NUMBERS. Definitions 58 English Money 69 Weight 62 Apothecaries' Weight 63 Avoirdupois Weight 64 Cloth Measure 66 Long Measure 67 Chain Measure 68 Square Measure 69 Solid Measure 71 Liquid Measure 74 Dry Measure 75 Time 70 Circular Measure 77 Miscellaneous Table 78 Examples in Reduction .... 79 GENERAL PRINCIPLES. Definitions 80 Factoring Numbers 81 Greatest Common Divisor Least Common Multiple COMMON FRACTIONS. General Principles of Fractions . 92 Mixed and Whole Numbers Re- duced to Improper Fractions . 93 Improper Fractions Reduced to Whole or Mixed Numbers . . 95 Fraction Reduced to Lower Terms 95 Fraction Multiplied by an Integer 96 Fraction Divided by an Integer . 98 Fraction Multiplied by a Fraction 100 CMMcttng 101 Fraction Divided by a Fraction . 104 Complex Fractions made Simple . 107 Common Denominator 108 Common Numerator 110 1* To Reduce a Fraction of a Higher Denomination to one of a Lower 111 To Reduce a Fraction of a Lower Denomination to one of a Higher 112 To Reduce a Fraction of a Higher Denomination to Whole Num- bers of Lower Denominations . 113 To Reduce Whole Numbers of Lower Denominations to a Frac- tion of a Higher Denomination 114 Addition of Fractions 116 Subtraction of Fractions . . . .119 Miscellaneous Examples .... 121 Analysis 122 VI CONTENTS. DECIMAL FRACTIONS. PACE Definitions 128 Decimal Numeration Table ... 129 Notation and Numeration ... 131 Addition 132 Subtraction 133 Multiplication 134 Division 136 Circulating Decimals 137 PAG) Common Fractions reduced to Dec- imals 138 Integers of Lower Denominations Seduced to the Decimal of a Higher Denomination .... 140 A Decimal of a Higher Denomina- tion Reduced to Integers of Low- er Denominations 141 UNITED STATES MONET. Definitions and Table 144 Reduction 140 Addition 147 Subtraction 148 Multiplication 148 Division 148 Practical Examples 149 Aliquot Parts of a Dollar .... 152 Bills 154 Miscellaneous Examples .... 156 COMPOUND NUMBERS. Addition 158 Subtraction 162 Multiplication • 166 Longitude and Time 171 Division 172 Duodecimals 176 PERCENTAGE. Definitons and Problems . . . . 1S3 Interest 1«7 Partial Payments 194 Problems in Interest 203 Compound Interest 206 Discount 209 Banking and Bank Discount . . 211 Insurance 214 Stocks 216 Commission and Brokerage ... 219 Taxes Ll'l Custom-House Business .... 224 Exchange 227 Equation of Payments 232 Profit and Loss 242 Partnership 248 MISCELLANEOUS. Ratio 254 ; Application of Square Root Proportion 256 Simple Proportion ...... 257 Compound Proportion 263 Alligation Medial 268 Alligation Alternate 268 Involution 274 Evolution 276 Square Root 277 Cube Root Application of Cube Root Arithmetical Progression Geometrical Progression Annuities Permutations .... Mensuration .... Miscellaneous Examples 282 286 291 292 295 298 300 301 307 ARITHMETIC. ttcle fl. Arithmetic is the science of numbers, and the art of computation* A NiMi.ri; ifl :i unit or a collection of units, a unit being one, i. e. a single thing of any kind ; thus, in the number six tin- unit is one ; in ten days the unit is one day, 2. All numbers are concrete or abstract. A Concrete Number is a number that is applied to a par- ticular object ; as six books, ten men, four days. An Abstract Number is a number that is not applied to any particular X)bject ; as six, ten, seventeen. 3* Arithmetic employs six different operations, viz. Notation, ■u, Addition, Subtraction, Multiplication, and Division. NOTATION AND NUMERATION. 4r» Notation is the art of expressing numbers and their relations to each other by means of figures and other symbols. •5. NUMERATION is the art of reading numbers which have 1 by figun Art. I. What is Arithrmtir ' What is a Number? A Unit? %■ Wh»t Is s inber? An Abstract Number? 3. How many operations in Arith- metic? What »r« the> •» 4. What is Notation? *. Numeration? NOTATION AND NUMERATION. 6. Two methods of notation are in common use : the Arabic and the Roman. 7. The Arabic Notation, or that brought into Europe by the Arabs, employs ten figures to express numbers, viz. : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. These figures are called digits, from the Latin digitus, a finger ; a term probably applied to figures from the custom of counting upon the fingers. 8. The first Arabic figure, 0, is called a cipher, naught, or zero, and, standing alone, it signifies nothing. Each of the remaining nine figures represents the number placed under it, and for convenience in distinguishing them from 0, they are called significant figures. 9. No number greater than nine can be expressed by a single Arabic figure, but by repeating the figures, and arranging them differently, all numbers may be represented. Ten is expressed by writing the figure 1 at the left of the cipher; thus, 10. In like manner, twenty, thirty, forty, etc., are expressed by placing 2, 3, 4, etc., at the left of ; thus, 20, 30, 40, 50, 60, 70, 80, 90. Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety. 10. The numbers from 10 to 20 are expressed by placing the figure 1 at the left of each of the significant figures ; thus, 11. 12, 13, 14, 15, 16, 17, etc. Eleven, Twelve, Thirteen, Four'.een, Fifteen, Sixteen, Seventeen, etc. In a similar manner .all the numbers, up to one hundred, may be Written; thus, 21, 36, 66, 98, etc. Twenty-one, Thirty-six, Sixty-six, Ninety-eight, etc. 6. How many methods of Notation? What? 7. How many figures in the Arabic Notation? What caJled? Why? 8. What is the first figure, 0, called? The others? Why? 9. The largest number expressed by one figure? Ten, how expressed? Twenty? 10. Numbers from ten to twenty, how expressed? NOTATION AND NIMEKATK ».\. S 11. One hundred \s i by placing the figure 1 at the of too ciphers; thus 100, In like manner two hundred, three hundred, etc., are written; thus, 200, 300, GOO, 800, Two hundred, Three hundred, Six hundred, Ei<;lit hundred, etc. 1£. The other numbers, up to one thousand, may be ex- ed by putting a significant figure in the place of one or cadi of the ciphers in the above numbers ; thu*, Two hundred and three, expressed in figures, is 203, hundred and eighty, expressed in figures, is 680, >.'ine hundred and ninety-eight, expressed in figures, is 998. 13. The place of a figure is the position it occupies with iice to other figures; thus, in 436, the G, counting from the right, is in the Jirst place, 3 is in the second place, and 4 in the third place. The figure in the Jirst place represents simple units, or units of the first order ; the second figure represents tens, or units of the second order; the third, hundreds, or units of the third order ; the fourth, thousands, or units of the fourth order, etc. ; thus, in the number 3576, the 6 is 6 units of the first order; the 7 tens is 7 units of the second order ; the 5 hundreds is 5 'units of the third order, etc. 11. From the foregoing it will be seen that each significant figure has two values ; one of which is constant (i. e. always the same), the other variable; thus, in each of the numbers 2, 20, and 200, the left hand figure is two ; but in the first it is two t/)iits ; in the second, two tens; and in the third, two hundreds. The former of these values is the inherent or simple value, and the latter is the local or place value. It5. It is also evident that the value of a figure is made ten fold by removing it one place toward the left; a hundred fold by removing it two places, etc. ; i. e. ten units of the first order 11. One hundred, how expressed? Two hundred? 13. Other uumhe 8, how Md! 13. What is the place of a figure? What does the flgui* in the eond place? Third? 14. How many, and what value* ~ liaa a figure? 15. How does moving a flguru towards the left nflwet its vaiu«' 10 NOTATION AND NLMEKAT: make one ten, ten tens make one hundred, ten hundreds make one thousand, and, in short, ten units of any order make one unit of the next higher order, 10. The cipher, when used with other figures, fills a place that would otherwise be vacant ; thus, in 206 the cipher occupies the place of tens, because there are no te?is expressed in the given number. 17. The figures of large numbers, for convenience in read- ing, are often separated by commas into periods or group-. There are two methods of numerating: the French and the English. By the French method a period consists of three figures ; by the English, of six. The French method is most convenient, and principally used in this country. 18. By the French Method of Numeration the first or right-hand period contains units, tens, and hundreds, and is called the period of units ; the second period contains thousands, tens of thousands, and hundreds of thousands, and is called the period of thousands ; etc., as in the following FRENCH NUMERATION TABLE. i il 1 i i ■,- 1 4 Is &% GJ «§ Ml smosS OT o =3 c— 5 ~ • o^3 t-o'C t^os t-oc t-oc uoS 13 o hc khc wes w#3 wSa wSS »Ss 2 8, 7 6 9, 5 4 0, 7 6, 4-7 6, 1, 8 4 3. 7th period, 6th period. 5th period, 4th jieriod, 3d period, 2d period, 1st period, C^uiutillions. Quadrillions, Trillions, Billions, Millions, TIioumuius, Uniw, 10. For what is the cipher used? 17. How many methods of numerating? What are they? Which is generally used in this country? 1*. Name the diff- erent periods in the French Numeration Table. Repeat the table. NOTATION AND TION. 11 10. The value of the figures in thi> table, expp— ed in . i- twenty-eight quintillioo, Mven hundred and sixty-nine quadrillion, live bandied and forty trillion, seven bandied and mx billion, four hundred and s<.vcnty-HX million, one thousand, tight bandied and fortj-tl 1 10 iiKADixo of a number consists of two distinct pro ■ nmting the order of the placm, beginning >\t the light hand ; thus, Mnu, hnndreds, etc., M in the Numeration Table; and, second, n aim of the figures, btgfnaiBg at the left, as above. To distil these processes, the first may be called numcratiny, and the second reading, the Dumber. 20. The table can be extended to any number of places, adopting a new name for each succeeding period. The periods above quintillions are sextillions, septillions, octillions, nonillions, lions, undecillions, duodecillions, etc. 21. To numerate and read a number according to the French method : BULK. 1. Beginning at the right, numerate and point off the number into periods of three figures each. 2. Beginning at the left, read each period separately, giving the name of each period except that of units. Exercises in Numeration by the French Method. 2S. Let the learner read the following numbers : 1. 24 11. 7,435,720,597 * 2. 357 12. 74,090,007.407 3. 4,649 13. 297,999,399,089 4. 95,679 14. 6,137,731,975,468 5. 549,:> I 7 15. i:».719,456,972,145 6. 5,745,328 16. 457,71 iU:;o,958,083 7. 52,073,712 17. 8,125,945,654,315,756 8. 213,967,184 is. 57,968,568,194,437,978 9. 4,674,925,178 19. 867,942,148,866,145,816 10. 48,404,876 20. 3,593,047,671,350,486,950 19. What is the value of the number expressed in the (able? Heading a nutn- i-ists of how many processes? What are they? 80. What are the names of periods above Quintillions? 21. Rule for numerating and reading a number by the Frenoh method r 12 NOTATION AND N [OK. 23. To write numbers by the French method : Rcle. 1. Beginning at the left, write the figures belonging to the highest period. 2. Write the figures of each successive period in their order, filling each vacant place with a cipher. Exercises in French Notation and Numeration. 2-1:. Let the learner write the following numbers in figures, and read them by the French method : 1. Two units of the third order and five of the fir Ans. 205. Note. Since no figure of the second order is given, acipha- is written in the second place 2. Six units of the fourth order, three of the second, and eight of the first. An*, <*.'>38. 3. One unit of the seventh order, three of the sixth. s«ven of the third, and two of the second. An 720. 4. Five units of the fifth order and three of the fourth. 5. Six units of the fourth order and one of the third. 6. Two units of the eighth order and three of the sixth. 7. Nine units of the ninth order, six of the fifth, one of the second, and three of the first 2«5. Express the following numbers in figures by the French notation : 1. Three hundred and fifty-six. Ans. 356. 2. Six hundred and fifty-three. Ans. 653. 3. Five hundred and sixty-three. Ans. 563. 4. Three hundred and sixty-five. 5. Six hundred and fifty-one. 6. One thousand, six hundred and fifty-one. Ans. 1,651. 7. Forty-two thousand, five hundred and fifty-four. 8. Eight hundred sixteen thousand, and two hundred. 9. Six million, one hundred four thousand, two hundred and seventy-six. Ans. 6,104,270. S3. Rule for wriUng numbers by the French method? NOTATION AND NUMERATION. 13 10. Three bandied six thousand, five hundred and two. 11. Nine hundred i'oriy-MX million, live hundred fourteen thousand, nine hundred and twenty-five. 12. Six billion, fifteen million, seven thousand, and four hun- dred Ans. G,0 15,007,400. 13. Five million, six hundred fifty-one thousand, four hundred and six. 14. Seventy-four million. 15. Sixty-three million, fourteen thousand, and seven hundred. 26. By the English Method of Numeration, the first period contains units, tens, hundreds, thousands, tens of thou- sand-, and hundreds of thousands, and is called the period of units; the second period contains millions, tens of millions, hun- dreds of millions, thousands of millions, tens of thousands of millions, and hundreds of thousands of millions, and is called the period of millions ; etc., as in the following ENGLISH NUMERATION TABLE. L *| l a „~ . 4 o | g i ^ © | S -*- 9 © «e 5? T •— 5 'o 5 3 g S 3 a E 3 c g g •- 70647 G, 00184 3. lod, "il period, 1st period, liillious, Millions, 20. By the English numeration what figures arc in the first period? Second period' Third? Repeat the tnb!e. 2 14 NOTATION AND NUMERATION. 27. The value of the figures in this table, is twenty-right trillion, seven hundred sixty-nine thousand five hundred ;md forty billion, seven hundred six thousand four hundred and ;itv-six million, one thousand eight hundred and fbrty-thn -e. 28. The names of the figures and their values are the same in the two tables for the first nine places from the right, After which they are alike in value but different in name. A trillion by the English method is much more than by the French. 29. To numerate and read a number according to the English method : BULK. 1. Beginning at the right, numerate and point off the number into periods of six figures each, 2. Beginning at the left, read each period separately, giving the name of each period except that of units. Exercises in Numeration by the English Method. 30. Read the following numbers : 87,658765,G47596 95467,694164»745I 47,G78600,709050,359G9l 31. To write numbers by the English method : Kile. 1. Beginning at the left, write the figures belonging to the highest period. 2. Write the figures of each successive period in their order t fitting each vacant place with a cipher. Exercises in English Notation and Numeration. 32. Write the following, and read by the English method : 1. Five units of the eighth order, six of the seventh, two of the fourth, and one of the third. Ans. 56,002100. 27. What number is expressed by the table? 28. Are the names of figure* alike in the French and English tables? Their values, alike or unlike? 29. Rule for numerating and reading a number by the English method ? 31. Rule for writing a number by the English method? 1. 4. 2. 8581 5. 3. 7.17 6 NOTATION AND Nl Mi: RATI ON. 1 6 2. Nino units of the fourteenth order, two of the twelfth, three of the eleventh, >ix of the eighth, nine of the sixth, two Of the tilth, and three of the fourth. Am 90^30060,928000. 3. Two units of the ninth order, six of the sixth, one of the fifth, two of the third, ;id. ami live of the first. 33. Express the following numbers by the English Notation : 1. Seventy-two million, six hundred thirteen thousand four hundred and forty-six. Ans. 72,613446. 2. Five hundred seventeen billion, three hundred twenty-two thousand one hundred fourteen million, eight hundred forty-one thousand nine, hundred and sixty-nine. 3. Two hundred and ten billion, and six thousand. Note. These and other exercises will be varied and extended by the teacher as circumstances may dictate. 31. The Roman Notation, or that used by the ancient Romans, employs seven capital letters to express numbers, viz.: I, Y, X, L, C, D, M. One, Five, Ten, Fifty, One hundred, Five hundred, One thousand. All other numbers may be expressed by combining and re- peating these letters, 3»>. The Roman Notation is based on the following princi- ples : 1st. When two or more letters of equal value are united, or when a letter of less value follows one of greater, the sum of their values is indicated; thus, XXX stands for 30, LXV for 65, CC for 200, MDCLXVII for 1667. 2d. When a letter of less value is placed before one of greater, the difference of their values is indicated ; as, IX stands for 9, XL for 40, XC for 90. 3d. When a letter of less value stands between two of greater value, the less is to he taken from the sum of the other two ; as, XIV stands for 11, XIX for 19, CXL for 110. 3». How many and what characters are employed in the Roman Notation* a!ue of each? 35. What is the first principle in Roman Notation t [I Third? 16 NOTATION AND NUMERATION. 4th. A letter with a line over it represents a number one thousand times as great as the same letter without the line ; thus X Staodfl for ten, hut X stands for one thousand times ten, i.e. ten thousand ; M stands for one thousand, but M for one thousand tvnes one thousand. TABLE OF ROMAN NUMERALS. I 1 XVI 16 cccc 400 II 2 XVII 17 D 500 III 3 XVIII 18 DC GOO IV 4 XIX 19 DCCCC 900 V 5 XX 20 M 1000 VI 6 XXI 21 MI) 15QQ VII 7 XXIV 24 MDC 1G00 VIII 8 XXV 25 MIX LXV 1GG5 IX 9 XXIX 29 MDCCXLIX 17-19 X 10 XXX 30 MDCCCXV1 1816 XI 11 XL 40 HDCCCLXH 1862 XII 12 L 50 V 5000 XIII 13 LX L 50000 XIV 14 XC 90 C 100000 XV 15 c 100 M 1000000 Exercises in Roman Notation. 36. Kx press the following numbers by letters : 1. Twelve. 2. Eighteen. Ana. XII. Ans. XVIII. 3. Twenty-nine. 4. Ninety-nine. 5. Two hundred and eijrhty-four. 6. One thousand four hundred and forty-six. 7. One thousand six hundred and forty-four. 8. The present year, A. D. . Note. The Roman notation is very inconvenient for Arithmetical oper- ations, and the Roman numerals arc now seldom used, except for number- ing the pages of a preface, the divisions of a discourse, and the sections, chapters, and other divisions of a book. 33. What is the fourth principle in Roman Notation? 36. Are Roman nume- rals much used in arithmetical operations? Why? For what are they used? ADDITION. 17 37. Boridei the Arabic and the Roman figures, there are is marks used to indicate that certain operations are to be performed, sneh, e. g., a> the sign of addition, -\-\ the sign of rfion, — ; etc. These rigQfl will be given, and their usea explained when their aid is ne< ADDITION. 38. Addition is the putting together of two or more num- bers of the same kind, to find their sum or amount. The sum or amount of two or more numbers is a number which contains the same number of units as the two or more numbers put together; thus, 7 is the sum of 3 and 4, because there are just as many units in 7 as in 3 and 4 put together ; for a like reason 11 dags is the sum of 2 days, 4 days, and 5 days. Ex. 1. James has 4 marbles, John has 5, and Henry has 3 ; how many marbles have they all ? To solve this example, add the numbers 4, 5, and 3 : thus, 4 and 5 are 9, and 3 are 12; therefore James, John, and Henry have 12 marbles, Ans. 2. How many are 3 and G ? 6 and 3 ? 2 and 5 and 7 ? 39. A Sign is a mark which indicates an operation to be performed, or which is used to shorten some expression. 40. The sign of dollars is written thus, $ ; e. g. $2 repre- sents two dollars ; $10, ten dollars, etc. 41. The sign of equality, =, signifies that the quantities he- tweeo which it stands are equal to each other; thus, $1 = 100 cents, i. e. one dollar equals one hundred cents. 37. What characters are used in Arithmetic besides the Arabic and Roman ' For what ' as. What is Addition? Sam or amount? 39. A sign? 40. Make the eign of Jollari on the black-board. 4 1. Hake the sign of equality. "What does it meant 2* 18 ADDXTK 42. The sign of addition, 4-, called plus, denotes that the quantities between which it stands are to be added together ; thus, 3-|- 2 = 5, i. e. three plus two equals five, or three and two are five. 43. Three dots, thus, .*. , are the symbol for therefore, hence, or consequently ; thus, 2 + 3 = 5, and 3 + 2 =5, .-. 2 -f- 3 = 3 + 2, i. e. therefore the sum of 2 and 3 is equal to the sum of 3 and 2. Ex. 3. William paid $4 for a pair of skates, $3 for a sled, and SI for a knife ; what did he pay for all ? -S3 + S1=$8, Ans. 4. What is the sum of SG + S3 ? S"> -f $2 -f $8 ? 5. What is the sum of 4 + 6 + 2 + 3? 3 + 5 + 8 + 2? 44. To add when the numbers are large and the amount of each column is less than 10. 6. A manufacturer sold 125 yardi of cloth to one merchant, 342 to another, and 231 to another; how many yards did he sell in all? Ans. 698. Haying arranged the* numbers so that units operation, stand under Emits, ten ander tens, etc., add the 1 2 5 units ; thus 1 and 2 are 3, and 5 are 8, and 3 4 2 set the result under the column of units. Then 2 3 1 add the tens ; thus, 3 and 4 are 7, and 2 are 9, Sum fi9 8 set ^ 0%vn tne result > **d so proceed till all tho columns are ad fort bone, >231 for a chaise, airj )t a harness ; what did lie pay for all? Ans. $388. «M. To add when the amount of any column is 10 o * more. 18. Add together 27, 93, and 145. Ans. 265, Having arranged the numbers, add the column operation. of units ; thus, 5 and 3 are 8, and 7 are 15 units 2 7 ( = 1 ten and 5 units). The 5 units are placed 9 3 under the column of units, and the 1 ten is added 1 1 5 to the column of tens ; thus, 1 and 4 are . r >, and Ans. 2 6 5 9 are 14, and 2 are 16 tens (= 1 hundred and 6 tens). The 6 tens are set under the tens, and the 1 hundred i> added to the 1 hundred in the third column, making 2 hundreds to be set under the third column. 19. 20. 21. 22. 276 748 4681 36487 483 249 73 62 10 4 6 2 874 838 8428 38420 Ans. 16 3 3 1835 20471 85369 23. 24. 2,3. 26. 417 246 387 1 34827 819 385 1920 5148 234 274 4208 97604 846 961 3 186 27 72 1 249 8004 86129 Ans. 3 3 7 27. 28. 29. 80. 46723 4 6 28 327 3 5 i 943 > 56948 784 46 4876 and 426. 98 643 31. Add 8467, 82, 946, 18845, Ans. 18766. All 6 1287, S 12, 8694, 32, and 46872. 8, '.'7. 4T.82, 3800, and 47289. 84. Add 884, 16842, 31, 87, 6294, and 8274. 20 ADDITION. 46. The examples already given embrace all the prin- m in addition. Hence, to add numbers, Rule. Write the numbers in order, units under units, tens under tens, etc. Draw a line beneath, add togtther the fg tires in the units' column, and, if the sum be less then ten, set it under that column ; but, if the. sum be ten or more, write the units as before, and add the tens to the next column. Thus proceed till all the columns are added. 47. Proof. The usual mode of proof is to begin at the top and add downward. If the work is right, the two sums will be alike. n: I, By this process, we combine the figures differently, and hence shall probably detect any mistake which may have been made in adding ujimird. ILLUSTRATION. Tn ■**« *P"*** WG «*! 2 ««"* C ■* Ex 35 8, and 7 are 15, and 4 are 19, etc; bat in adding downward, we saw 1 and 7 an 11, and 6 are 17, and 2 are 19, etc., thus obtaining the same result, but by different 4 8 2 9 7 nig** ***** If we do not obtain the same result by Sum, 2 4 9 5 7 9 the two methods, one operation or the Proof, 2 4 9 5 7 9 other is wrong, perhapfl both, and the work must be carefully performed again. te 2. Tn adding it is not desirable to name the figures that we add ; thus, in example 35, instead of saying 2 and 6 are 8, and 7 are 15, and 4 are 19, it rter, and therefore better, to say 2, 8, 15, 19 ; setting down the 9, say 1, 6, 10, 19,27 36. What is the sum of 8432, 42698, 34, 1892, 70068, 5142, and 68742? Ana, 197008. 37. What is the sum of 2468, 13579, 276, and 4_ 38. What is the sum of 3406, 872, 6.V11, 2, and 17 ? 39. What b the sum of 3910, 4, 876, 27, and 89462 ? 46. If the amount of any column is ten or more, where is the right-hand figure of the amount writteu ? What is done with the left-hand figure? Repeat the rule for Addition. 47. How is Addition proved? Why not add upward a second time ? Is it desirable to name the figures as we add them? ADDITION. 21 Ex. 40. n. 42. 43. 51000 20404 21153 31201 1 1 G08 44346 25000 22222 880 80 • 40 15 000 6 6 6 6 6 4 9 l 8 I 9 0000 5 5 5 5 5 5 5 5 5 5 l 2 B 95000 5 4 4 1 5 3o -2643 •II. HOW many are 876 -f 9287 + 69842 + 7700 ? Ans. 87705. 45. How many are 3G904 + 21G -f 8942 + 47 ? Bow many are 18 + 4 + 7G984 -f 327 + 1 1 ? 47. 84G + 972 + 84 -f 300 = how many ? Ans. 2202. 48. 2468 + 98G7 + 37428 + 278 = how many ? 49. 3004 + 6094 -j- 87642 -f 36 = ? Ans. 9G77G. „ 168 + 13579 + 100 + 6042 + 187 + 19 =? 51. Add four hundred and sixty-two; three thousand two hundred and fourteen ; seventy-nine thousand six hundred and hfty-nine; and two hundred and eighty-four. Ans. 83619. Add four hundred and iif'ly-six ; eight thousand, four hun- dred and BeVenty-tWO j fifteen thousand, seven hundred and twenty-one ; forty-three million, seven hundred and thirty-three thousand, eight hundred and fifty-nine; and ten. 53. The population of England in 1851 was 16921888; of and, 2888742; of Wales, 1005721; of Ireland, G515794. What was the population of Great Britain and Ireland? B I. England and Wales contain about 55100 square miles; Scotland 29600 j and Ireland, 32000 ; what is the area of the British Islands.? Ans. 116700 square miles. l)y the census of 1860, the number of inhabitants of Maine, was 628276; of New Hampshire, 826072; of Vermont, 815116; of Massachusetts, 1231065; of Rhode Island, 174621 ; of Connecticut, 4G0151 ; what was the population of New Eng- land? Ans. 3135301. 5G. The area of Maine is 35000 square miles; N. II., 8030; •no; Mass., 7250; R I., 1200; Ct, 4750. What fa the I & New England ? 22 ADDITION. *<7. In 1850 the population of Maine was 583169; of New Hampshire, 317076; of Vermont, 514120; of nfirmrnnni 14; of Bhodfl Island, 117 Connecticut, 370702; what was the population of these *ix Stales in 58. A merchant, commencing business, h;i. In a year be gained $2475; irnsd he worth at the end of the year? 59. In one year a l id a pair of oxen for $125, two cows for $75, three swine for $'.» p for $120, and a bone for $156 ; what did lie receive for all ? CO. On Monday, a merchant .-old goods for $:V>7, on Tuesday, rhnrsday, lor S3 18, on Friday, I I on Saturday for $316; what was the value of the goodfl .-old during the week ? 61. In 1850 the population of New York : of Philadelphia, 840045 ; of Baltimore, 169054; of IV 5881; of New Orlea sad of Cincinnati, 11543$; whal number of inhabitants in these six cities in 1850? .'. In the middle of the nim •:■ the population of l ; of Paris, 1053897 j tan* tinople, 786990 ; c ~ ; of Vienna. 4771 of Berlin, 441931 ; and of Nap! tbe popu> 63. In 185 1 ion of the (Jailed Stat- ,i>out 1876; of Britain and 116181427882145; of Fr; -3170; of R. 88000; and of Aosftri what wai the population of these five count] ,. The population of North America i> about 30257819; of South America, 18878188; of Europe, 265368216; of A of Africa, 61688770, and of Oceanica, 23444- what is about the population of the globe ? Ans. 1038803745. of the American army for li commencing in 1812. was $12187048, $19906362, $20008306, ;<>0, and #16475412 ; what was the cost for five \ to D $4082, to E $207, to F $54, imd to G $1353; how much does he owe? ABDRXO I 68. 69. 70. '98 5 9 28738 . 5 8 7. 7 9 819 9 03 a i 1 l 8 5 8 2 7/. | 55555 4 l 2 93977 27579 1 2 6 7 7 505 04 11111 2 ! 7 6 1 4 7 80 1 56 6 88888 I 1 9 1 1 8 7 1 8 8 47 69 5 5 i ; 6 i 2525] I 6 9 ;98 2 12 7 4 1 2 1 6 $ 6 i i 90 28 5 I 00 i i 98G95 72869 •8 78 2 5 D 6 5 6 1 27 121 4 5 2 2 7 2 8 90.^ 46^ 1 6 5 92672 6 2 12 8 2 7 •2177 8 j 6 7 74279 6 l • 2 5 1 2 7 7 6 1 5 2 2 4 7 2 5 5 2 9 l 267 76592 27248 47214 73017 15172 47510 23 71. In January there are 31 days, in February 28, in March 31, in April 30, in May 31, in June 30, in July 31, in August 31, in September 30, in October 31, in November 30, and in mber 31 ; how many days are there in a year ? 7 J. A gardener has 3476 apple trees, 847G pear trees, 5684 peach trees, 1845 plum trues, 4680 quinee trees, and 9 187 orna- mental trees; how many trees are there in his nursery? 73. The first of three numbers is 4768, the second is 8942, and the third is as much as the other two ; what is the sum of the three numbers? 7 1. I have $376 in one bank, $1078 in another, and in anoth- much as in both of these ; how much money have I in the three banks ? . An army consists of 276450 infantry, 14875 cavalry, i artillery men, and 127462 riflemen; what is the number of men in the army ? A carpenter engaged to build 1 houses, the first I eood i'nv e third t . and the fourth for $12469 ; what shall he receive for the lour bous< 24 SUBTRACTION. SUBTRACTION. 48. Subtraction is taking a less number from a greater number of the same kind, to find their difference. The greater number is called the minuend; the less nun is called the subtrahend ; and the result is called the differ- B or remainder. Ex. 1. Arthur had 7 apple-, hut be has given 4 of them to Mary; how many apple.- has lie : 18, 3j because 4 apple- taken from 7 apples leave 3 apples. 2. John having 17 marble-, lost 7 of them ; how many had he left ? 4:9. The sign of subtraction, — , called minus, signifies that the number after it is to be taken from the number before it; thus, 7 — 4 = 3, i. e. seven minus four, or seven diminished by four, equals th: 3. How many are 10 — 6? Ans. 4. 4. How many are 12 — 8? 12 — 4? 1G — C? I e. When the numbers are small, the subtraction is ren as tnnt place ' s occupied l>v Minuend, 2 ; but we can borrow one of the Subtrahend, 4 3 8 G hundreds an(1 geparate the one Kiinder, 16 4 hundred into 9 tens and 10 units ; then, putting the tens in the place of tens and adding the 10 units to the 2 units, we can sub- tract 8 from 12, 3 from 9, and 4 from 5. Note 2. This process will probably bo more readily understood by the young learner than the second method given in the rule, though the latter, being thought more convenient, is usually adopted. 29. 30. 31. From 8702 4003 870000 Take 2465 1876 324872 32. From 804 take 567. Ans. 237. 33. From 4687 take 2398. 34. From 87062 take 36981. 35. Subtract 2437 from 8064. Ans. 5627. 36. Subtract 160874 from 4769872. 37. Subtract 3768942 from 7000000. 38. Take 87406 from 95472. Ans. 8066. 39. Take 2704698 from 8749206. 40. How many are 3642 less 1468? Ans. 2174. 41. How many are 87649 less 24065 ? 42. 8749 — 3684 = how many ? Ans. 5065. 43. 7248 — 2943 = how many ? 44. The difference between two numbers is 365 and ^he greater number is 876 ; what is the less? Ans. 511. 45. What number added to 3876 will give 7469 ? 46. What number taken from 8742 leaves 3748 ? 53. What is there peculiar iu Ex. 28? Explain the process. 28 SUBTRACTION. 47. The sum of two numbers is 8629, and the less of the two numbers is 2G89 ; what is the greater? Ans. 5940. 48. The sum of two numbers is 8426, and the greater is 7162 ; what if the less ? 49. From fourteen million, eight hundred and sixty-two thou- sand, three hundred and twenty-five, take six million, six hundred and eighty-six thousand, two hundred and fourteen. Ans. 8176111. 50. From seven hundred and thirty-three thousand, six hun- dred and fifty-four, take two hundred and twenty-seven thousand, five hundred and fifteen. 51. How many years from the discovery of America by Co- lumbus in 1492 to the birth of Washington in 1782 \ 59. How many years have elapsed since the d i scove ry of America in 1492 ? 53. By the 06BSM of 1860, the number of inhabitants in Massachusetts was 1231065, and the number in Vermont was 315116 ; how many more in Massachusetts than in Vermont? 54. The population of the United Stat.- wm 23191876 in 1850, and 17063353 in 1840; what was the increase in ten years ? 65. The area of New England is 64230 square miles and the area of Maine is 35000 square miles; what is the area of the other five New England St:r 56. About 56619608 bushels of corn were raised in Ohio in 1850, and 73436690 bushels in 1888 ; what was the increase? 57. Bought a paper mill for $15475, and sold it for $17925 ; what did I gain ? 58. How many are 876942 — 468279 ? 59. How many are 742006 — 387429 ? GO. How many are 820654 — 260408 ? 61. Washington was born in 1732 and died in 1799 ; at what age did he die ? 62. A merchant sold goods to the amount of $4276, and thereby gained $1142 ; what did the goods cost him? 63. A farm was sold for $3462, which was $876 more than it cost ; what did it cost ? ADDITION AND SUBTRACTION. 29 84 The distance from the earth to the sun is about 95000000 miles; the distance to the DMOO is about 210000 miles. How much further to the sun th:in to the moon ? 65. Methuselah died at the age of 969 years, and Washington at 67 ; what was tin* ditlerenee of their ages? C6. Mr. Hale, owing a debt of $3762, paid $24S6 ; how much remained unpaid ? LMPLES in Addition and Subtraction. 1. From the sum of 7G and 92 take 14. Ans. 154. 2. From the sum of the three numbers, 876, 493, and 91G, take the sum of 842 and 397. Ans. 1046. 3. I owe 3 notes, whose sum is $600. One of these notes is for $150, another for $200 ; for what is the third one ? •1. My real estate is valued at $4500 and my personal prop- erty at $2596. I owe to A $600, to B $1358, and to C $318 ; what am I worth? ■ Ans. $4820. 5. Bought a barrel of flour for $9, four yards of cloth for $2, and 8 pounds of sugar for $1. In payment I gave a ten and a five dollar bill ; what change shall the merchant return to me ? 6. Mr. Fox, owning 3762 acres of land, gave 563 acres to his oldest son, and 072 acres to his youngest son; how many acres had be remaining? 7. The area of Maine is 35000 square miles ; N. IT., 8030 ; B000 ; Mass., 7250 ; R. I., 1200 ; Ct., 4750. Which is the greater, Maine or the rest of N. E. ? How much ? 8. Gave my note for $8465. Paid $1300 at one time, and $575 at another; how much do I still owe? Ans. $1590. 9. Mr. T., opening an account at the Andover Bank, deposited $187 on Monday. $362 on Tuesday, $580 on Thursday, and OH Friday. On Tuesday he withdrew $67, on Wednesday OO Friday $350, and on Saturday $125; how much re- mained on deposit at the close of the week? Ans. $1049. 10. A traveler who was 875 miles from home, traveled to- ward home 141 miles on Monday, 127 miles on Tuesday, 156 miles on Wednesday, and 157 miles on Thursday; how far from home was he on Friday morning ? 3* 30 MULTIPLICATION. 11. From the discovery of America by Columbus in 1492, to the settlement Of Jam^town in 1G07, was 115 years, from the K -ill. -incut of Jamestown to the Declaration of Independence in 177*'.. was 1G9 years, and from the Declaration of Independence to the present time (1862) is 86 years. Methuselah died at the age of 969 years; how much longer did he live than from the discovery of America to the year 1862 ? j 1 2. Four men, A, B, C, and D, commencing business together, famished money as follows: A ? 18475] B, $8475] C, $2850| and D, $4500. At the end of a year they closed business, hav- ing lost §o'22o ; how much money had they to divide between them? MULTIPLICATION. 54. Multiplication is a short method of adding equal numbers ; that is, multiplication is a short method of finding the sum of the repetitions of a number. Or, Multiplication i- a short method of finding how many units there are in any number of times a giv the number to be repeated The Multiplikr is the number which shows how many times the multiplicand is to be taken. The Product is the sum of the repetitions, or the result of the multiplication. The Multiplicand and Multiplier are called Factors. Ex. 1. There are 7 days in 1 week; how many days in 4 wee' This example may be solved by addition ; thus, 7 — |— 7 — [- 7 — |— 7 r= *J3 ; or more briefly, by multiplication ; thus, 4 times 7 are Yns. 54. "What is Multiplication? Another definition? What is the Multiplicand? Multiplier? Troduct? What are the Multiplicand and Multiplier called? MCLTIPLICYTION. 31 •!«>. Tlie pupil, before advancing further, should learn the MULTIPLICATION TABLE. following Once Twice Three limes Four times Five times Six times 1 b l 1 a: 1 are 3 1 are 4 1 are 5 1 are 6 2 I 1 •1 2 6 2 8 2 10 2 12 3 6 9 3 12 3 ]:, 18 4 4 1 8 4 19 ■1 16 1 20 1 24 5 5 10 5 15 5 20 25 S 30 6 a 19 6 18 8 24 6 30 8 36 7 7 7 1 l 7 2] 7 28 7 35 7 42 8 8 8 16 8 84 8 32 8 40 48 9 9 9 18 9 27 9 86 9 45 9 54 10 10 10 20 10 30 10 40 10 50 K) 60 11 11 11 89 n 33 11 44 11 55 11 66 12 12 19 u 12 36 12 48 12 60 i 12 72 Seven times Eight times Bm times Ten times Eleven times Twelve times 1 are 7 1 are 8 1 are 9 1 are 10 1 are 1 1 1 are 1 2 2 14 2 16 2 18 2 20 2 22 2 21 3 21 3 24 3 27 3 30 3 33 3 36 4 88 4 82 4 86 4 40 1 44 4 48 5 35 5 40 5 45 r> 50 5 55 5 60 G 49 6 48 6 54 6 60 6 66 6 72 7 7 T>6 7 C3 7 70 7 77 7 84 8 5G 8 64 8 72 8 80 8 88 8 96 9 G3 9 9 81 9 90 9 99 9 108 10 To 10 80 10 90 10 100 10 110 10 120 11 77 11 88 11 11 Hit 11 121 11 132 12 84 19 96 12 108 12 120 12 132 12 144 j 2. How many are 8 times 3? 3 times 8? 6 times 4? 4 times 6? 7 times 7 ? 5 times 9? 3. How many are 9 times 7 ? 9 times 11? 8 times 6? 6 times 12? 12 timet 6? 9 times 8? •1. If I deposit SIO a month in a MVlDgS hank, how many dollar* shall I deposit in 4 months? In 7 months? In o months? In 12 months? 82 MULTIPLICATION. 5. When wood is worth SG a cord, what shall J pay for 3 cord~? 5 cords ? 8 cords ? 11 cord? ? G. In one year there are 12 months, how many months in 2 years? 4 years? 7 years? 12 year-? 7. If I study 11 hours in a day, how many hours shall I study in days? 5 days?. 8 days? 11 days? •56. To multiply by a single figure. 8. In one bushel are 32 quarts ; how many quarts in G bushels? by addition-. bt nuLTi plicatiox. In G bushel* there are, 3 2 3 2 lently, 6 times m many 3 2 6 quarts as in 1 bushel, and 3 2 -p , i~o~o tne numoer °f quarts in G 3 2 lrcKluct > '• bushel* may be obtained 3 2 by adtlinfj, as in the margin; or, more briefly, 3 2 by multiplying ; thus, G times 2 units are 12 S„ m i o o units = 1 ten and 2 units; write the 2 units UIIJ, X J — . . in umt> place, and then say tlDQ are 18 tens, which, increased by the 1 ten previously obtained, make 19 tens = l hundred and 9 tens, and these, written in the place of hundreds and tens respectively, give the true product. Hen RULE. Writs (he multiplier under the multiplicand, and draw a line beneath ; multiply the units of the multijilicand. set the iinits of the product under the multiplier, and add the QP»y, to the product of the tens, and so proceed. 9. 10. 11. Multiplicand, 427 1347 1064 Multiplier, 2 5 8 Product, SJ1 6 7 3 5 85 12 11 13. 14. 15. 8423 5436 26493 76489 7 9 3 4 5 8 9 G 1 5G. Which flgure of the Multiplicand is multiplied first? Where are the uuits of the product written? What is done with the tens? Eepeat the ruJe. Ml I/TII'LK'ATIOX. 33 16, 17. 18. 141 4787243 3424270 6 9 7 2 16 2 5 2 88969890 •57. To multiply by two or more figures. 19. How many (marts in 46 hu.-dn-l-? operation-. First multiply by 6,m though G w<>rn Multiplicand, 3 2 the only figure in the multiplier ; then Multiplier, 4 6 multiply by 4, and set the first figure of - q a tl ns product in the place of tens ; for . 2 g multiplying by the 4 tens is the same as multiplying by 40, and 40 times 2 units Product, 14 7 2 are 80 writs = 8 tens; i. e. the product of units by tens is tens. Having multi- plied by each figure in the multiplier, the sum of the partial products will be the true product. Note. So much of the product as is obtained by multip/ying the whole multiplicand by one jiyure of the multiplier is called O partial product ; thu>, in the 19th example, 192 is the first partial product and 128 tens is the second. •58. Similar reasoning applies however many figures there may be in the multiplier. Hence, Rule. 1. Set the multiplier under the multiplicand and draw a line beneath. 2. Beginning at the right hand of the multiplicand, multiply the multiplicand by each figure in the multiplier, setting the first figure of each partial product directly under the figure of the multiplier which produces it. 3. The SUM of these partial products will be the true, product. •59. Proof. Multiply the multiplier by the multiplicand, and, if correct, the result will be like the first product. Hon. This proof rests on the principle, that the order of the factors is immaterial; thus, 3X4 = 4X3 ; 5X2X7 = 2X"X5. 57. Which figure of the multiplier is first employed? Where is the first figure of etch partial product written? What it a partial product ? 5R. Rule for multiplying by two or more figures? 59. Proof? Principle? 34 MULTIPLICATION. Kx. 20. Multiply 5236 by 2413. OI'EItATIOX. PROOf. Multiplicand, 5 2 3 G 2 4 13 Multiplier, 2413 5 2 3 6 15 7 8 TT~4~7& 5236 7239 20944 4826 10472 12065 Product, 12634468 =r 126344GB 22. 12474893 7* Multiplicand, Multiplier, 21. 2640873 4622 34678 54 24. 54327 32 1 2,5. 26. 8645 357 9 4 6 3 2 1 I 27. Multiply 4276 by 356. Ant, 1522256. 28. Multiply 5463 by 248. 29. Multiply 4628 by 336. 30. Multiply 3874 by 846. 60. The sign of multiplication, X> signifies that the two numbers between which it standi are to be multiplied together; thus, 6 X 5 = 30, i. e. six multiplied by five equals thirty.; or, more familiarly, six times five are thirty. 31. How many are 726 X W ? An<. 19602. 32. How many are 4628 X 554? Ans. 2563912. 33. 3648 X 36 = how many? Ant, 131328. 34. 4275 X 54 = how many? Ans. 230850. 35. 3759 X 8463 =? Ans. 31812417. 36. 53642 X 63 = ? 41. 37642 X 57 = ? 37. 4020 X 524=? 42. 37942 X 386 = ? 38. 8726 X 463 = ? 43. 27403 X 584 = ? 39. 7692 X 356=? 44. 36008 X 412 = ? 40. 2146 X 179 = ? 45. 81650 X 789 = ? GO. ijign of multiplication, what does it signify? Ml'LTIPLK 35 4G. If 37 men do I pSeM <>f work in 20 days, in how many .. ill 1 man do the MUM work ? •17. Whal Kl the vain.- of .'-7 land, at $43 per acre? 48. If a bone CBO travel 41 miles per day, how lar can ho 1 in 17 d;i 49. I low many yard* of cloth in 29 pieces, if each piece con- 81 yard- ? 61. To multiply by a composite number. A Compositi: NUMBEH El the product of two or more num- . thai l"» m a composite number, whose factors are 3 and 5; and 12 is a composite number, whose factors are 2 and G, or 3 and 4, or 2. 2. and 3. It will 1).« observed that a composite number may have several sets of (acton* 50. If 35 men have $37 each, how many dollars have they all ? The 85 men may be OPERATION. 35 = 5 X 7. Multiplicand, S S 7 Lai Factor of Multiplier, 5 2.1 Factor of Multiplier, 7 Product, $ 1 2 9 5 51. Multiply 3G7 by 1G8. FIRST OPERATION. 168 = 8 X 7 X 3. Multiplicand, : Factor of Multiplier, nd Factor of Multiplier, separated into 7 groups of 5 men each. Now 1 group of 5 men will have 5 times $37 = $185, and if 1 group baa $185, evidently 7 groups will have 7 times $185 = $1295, Ani. ; i. e. 7 times 5 times a number are 35 times that number. Ans. 61656. 3G7 8 293 6 7 SECOND OPERATION. 168 = 4 X 7 X G. 3G7 Third Factor of Multiplier, Product, 20552 3 61656 = 1468 7 10276 6 G 1 G 5 6 01. What || a o.nij'UMtf uuinbor! ne Mt of factor..' May a composite number bare more than 3G MLLTII'LICATION. Several other sets of factors of 1 C8 may be used, and give the same product. Every similar example may be solved in like manner. Hence, Rule. Multiply the multiplicand by one of the factors of the multiplier, and that product by another factor, and s<> on until all the factors in the set have been taken ; the last product will be the true product. 52. Multiply 743 by 42, i. e. by 7 and 6. Ans. 3120G. 53. Multiply 3467 by 56. 54. 839 X 54 = how many ? Ans. 45306. 55. 7869 X72 = ? 56. 469876X81=? 57. 478969 X 1728 = ? Ans. 827658432. 58. 5387469 X 96 = ? 59. 987462 X 49=? 62. To multiply by 10, 100, 1000, or 1 with any number of ciphers annox**!: 1 J r i. E. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number so formed will be the product. i ::. The reason of the rule is obvious. Annexing a cipher removes each ii-ure in the multiplicand one place toward the left, and thus its value is made ten fold (Art. 15). 60. Multiply 74 by 10. Ans. 740. 61. Multiply 869 by 10000. Ans. 8690000. 62. Multiply 4698 by 1000. 76984 X 100000 = ? Ans. 7698400000. 64. 59874 X 1000000000 = ? 63. To multiply by 20, 50, 500, 25000, or any simi- 1 ir number: Rule. Multiply by the significant fgures, and to the product annex as many ciphers as there are ciphers at the right of the significant fgures of the multiplier. 61. Rule for multiplying by a composite number? 62. now is a number mul- tipledbylO? By 100? Why ? 63. How is a number multiplied by 20? Why? Ml LTIl'UCATIoN. 87 Si Multiply NM l.y 30. Ans. 22G80. 1 1 iov. TUl ifl upon the principle of Art. Gl. The 7 5 G factors of 30 are 3 and 10. Having multi- 3 plied by 3, the product is multipled by 10 22 680 h >' annexin S ° ( Art - c -)* 66. Multiply 743 by 3500. OPERATION. 7 4 3 7 The factors of 3500 are 7, 5, and — — 100, .-. multiply first by 7, then by 5, inn then annex two Ctpb IT . Product, 2 6 5 67. Multiply 84G93 by 480000. Alts. 40652640000. 68. 87G9432 X 7200000 = ? • 69. 94684235 X 49000000 = ? <»-l. To multiply when there are ciphers at the right of both multiplicand and multiplier: RULE. Multiply the significant figures of the multiplicand by those of the multiplier, and then annex as many ciphers to the product as there are ciphers at the right of both factors. 70. Multiply 8000 by 900. operation. The factors of 8000 are 8 and 8 00 1000, and those of 900 are 9 and 9 100. Now. as it is immaterial in what a„. -Vn'nTwTA order the factors arc taken (Art. 59, Ans. < 1 XT . i • i o i n .i l Note), first multiply 8 by .'. then mul- tiply this product by 1000 (Art. G2), and this" product by 100. 71. Multiply 730000 by 2900. OPERATION. 730000 2900 G5 7 146 Product, 2117000000 64. Kule when there are ciphers at the right of both factors? The reason? 4 38 MULTIPLICATION. 72. Multiply 840 by 2700000. Ans. 2268000000. 73. 7G93000 X 569000 = ? 6«5. To multiply when there are ciphers between the significant figures of the multiplier: Rule. Multiply only by the significant figures of the mul- tiplier, talcing care to set the first figure of each partial prod- tict directly under the figure of the multiplier which gives that product. 74. Multiply 5728 by 2004. Thi* is only carrying out the operation. principle (in addition) of setting 5 7 23 units under units, ten- under tens, 200 4 etc The 2 of the multiplier is 2~2lT92 2000 ' Jind 2000 timea ;) tan G000 > IIAaq •"• tM< * f of the partial product should be written in the thousands' Product, 11468892 v \. u .,. m j. ... directly under the 2 of the multiplier. 75. Multiply 3724 by 4008. Ans. 14925792. 76. 698427 X 420006=? 77. 7800076900 X 2008040000 = ? 66. To multiply by 9, 99, or any number of 9's: IU-i.r. "S many 0'* to the multiplicand as there are 0's in the multiplier, and from the number so formed subtract the multiplicand ; the remainder will be the product sought. 78. Multiply 234 by 99. OPERATION. 2 3 4 = 100 times the multiplicand. 2 3 4= 1 time the multiplicand. 2 3 16 6= 99 times the multiplicand, Ana. 79. Multiply 3746 by 999. Ans. 3742254. 80. Multiply 427 by 9999. C3. Kule for multiplying when there are cipher* between the significant fig- ures of the multiplier? The reason? 66. To multiply by 9? By 99? Rule? Reason? MLi/nri.. 39 157. To multiply by 18, 14, 15, 1G, 17, ct l; ; Multiply by the right-hand figure of the multiplier, get the produrt und.-r (he multiplicand, i I i ikther TO add. 81. Multiply «M by 17. operation. The 1989 fa 7 times 42G, and the 420, 4 2 6 standing one place further to the left, is 10 2 - timet 486 (Art 15), .% their sum is 17 7 848, Am **" ; ~ 82. Multiply 849 by is. By l 1. By 1G. In a similar manner multiply by 102, 1005, 10009, etc. 83. Multiply 24G3 by 102. OPERATION. 2 4 G3 = 100 times 24G3. 492C = 2 " 25 122 = 102 " " Ans. 84. Multiply 3248 by 104. By 1004. By 1008. 68. To multiply by 21, 31, etc. : Rule. Multiply by the left-hand figure of the multiplier, set the product under the multiplicand, one place further to the left, and add. 85. Multiply 324 by 21. SHORT METHOD. COMMON METHOD. 324 324 648 21 6 8 4, Ans. 324 648 6 8 4, Ans. 86. Multiply 34264 by 81. By 41. By 61. Iq like manner multiply by 201, 301, 6001, etc. 87. Multiply 4237 by 501. Ans. 2122737. 88. Multiply 342G5 by 801. By 4001. By 30001. C7. To multiply by 13? By 15? By 102? By 1005? Reason? GS. To mul- Uply by 21? By 31? By 601? Reason? Why better than the common nwtliodf 40 MISCELLANEOUS EXAMPLES. Miscellaneous Examples in Multiplication. 1. What cost 11 pounds of beef at 9 cents per pound ? Ans. 99 cents. 2. What cost 98 tons of hay at $15 per ton ? Ans. $1470. 3. In one hogshead of wine are G3 gallons ; how many gal- lons in 75 hogsheads ? 4. In a certain house arc 75 rooms, in each room four win- dows, and in each window 12 panes of glass; how many panes of glass in the hou 5. The eftfth, in its annual revolution, moves 19 miles in a second; how far will it m<»ve in an hour, there heing CO seconds in a minute, and 0<> minutes in an hour? G. Light nfoffl I92CO0 miles in a second; how far will it move in an hour? 7. I low many yards of cloth En 10 bales, each bale containing 25 pieces, and each piece 2 1 yard-? 8. If 12 men do a piece of work in 7 days, in how many d can 1 man do 5 times as much work ? '.'. Multiply forty-three million, seven hundred and four thou- ,! hundred and sixteen, by forty-two thousand and eight. 10. A man booghl 21 city lots at $3G5 each; what did they all GOSl him ? 11. Multiplicand = 4632 j multiplier = 4008 ; product = ? 12. Multiplier = 3333; multiplicand = 4444 ; product = ? Examples n the Foregoing Principles. 1. Two men start from the same place, and travel in the same direction, one at the rate of 5€ miles and the other 75 miles per day, how far apart are they at the end of 43 day-.? 2. Had the men named in Ex. 1 traveled in opposite direc- tions, how far apart would they have been in 5G da 3. Bought o$ tons of hay for $G00 and sold it for $12 per ton ; did I gain or lose ? How much ? 4. Bought 25 horses for $125 each, and 14 pairs of oxeix at a pair ; what did I pay for all ? ■LLAHBOin EXAMPLES. 41 Bought 5G barrels of flour at :-rcl. and in pay for COrdfl of r cord, and the rati in UK how mnefa money did I | SCO for I and sold the flock for $425 ; did I gain or lose? How much? 7. A tanner sold 56 boahels of wheal at $2 per bushel, for which he received 40 yards of cloth at $2 per yard, and the balance in money ; how much money did he n OB1T1 ''■ 8. A merchant bought 84G barrels of Hour for $7191 ; he sold 526 barretl r barrel, and the remainder at $8 per barrel ; did be gain or lose \ Sow much? Ana. Gained S103. 9. A man's iocOJDM i- (1572 a year, and his expen-es a; a day; what do in a year of 865 days? Ans. $180. 10. Bought 18 tons of iron at $39 a ton, and 27 tons at $41 ; what >hall I L r ain by Belling the whole at $13 a ton? 11. A drover bought a herd of 33 oxen, pajing as many dol- lars for each ox as there were oxen in the herd. He paid $500 in money, ami gave his note for the balance ; what was the size of the note ? 1 •_>. How many are 8 + 2 X 7 — 3 X 5 ? Ans. 7. 13. How many are 9X7+3x5-12? Ans. G6. 1 1. How many are 48 — 3 X 6 — 4 ? Ans. 2G. 1 ■">. The factors of one number are 20, 14, and 23, and of another 1C, 8, and 7; what is the difference of the two num- Ans. 5544. 1G. The President of the United States receives a salary of 1 > a year ; what will he save in a year of 3 Go days, if his expense! are S.")() a day ? 17. A man having a journey of 313 miles to perform in 6 day-;, travels ."> 1 miles a day for days : how far must he go on 18. A many sold three farms - for the first he reeeiv- for the second, than for the first, and for the third, he ta much as lor the other two; how much did he r the three fail: 19. What shall 1 pay for 25 horses, at $75 each, and 12 oxen, at |54 each? 4* 42 DIVISION. 20. If a teacher receives a salary of $800 a year, and pays $210 a year for board, $75 for clothing, $o0 for books, and $100 for other expenses, how much will he save in 3 years ? DIVISION. 60. Division is the process of finding how many time3 one number is contained in another. The Dividend is the number to be divided. The Divisor is the number by which to divide. Tin- QtJOTll m \$ the number of times the dividend contains the divisor. If the dividend does not contain the divisor an exact number of times, the part of the dividend whieli is left is called the Ki:mai.\[)!i;. Notk. The remainder is always of the same kind as the dicidend, because it is a part of the dividend. Ex. 1. How many oranges, at 4 cents each, can be bought for 12 cents? An>. As many oranges as there are times 4 cents in 12 cents; •1 eentfl are contained in 12 cents, 3 times; .*. 3 oranges, at 4 cents each, can be bought for 12 cents. 2. How many apples, at 2 cents each, can be bought for 10 cents ? A D& A< many as there are times 2 cents in 10 cents, or as there are times 2 in 10, viz. 5. 70. The sign of division, ~-, indicates that the number be- fore it is to be divided by the number after it ; thus, 8 -f- 2 = 4, i. e. 8 divided by 2 equals 4, or 2 in 8, 4 times. 8. How many are 6 -v- 2 ? Ans. 2 in 6, 3 times. G9. What is Division? What the Dividend? Divisor? Quotient? Remain- der? Of what kind is the remainder? 70. The sign of Division, what does it indicate? DIVI 43 In the same manner, let the pupil ftTphrifl and recite the fol- lowing DIVISION TABI 1 -5- 1 =1 2-4-2=1 3-4-3 = 1 4-4-4 = 1 2 - h 1 =2 4-4-2 = 2 6-4-3 = 2 8-4-4 = 2 3 - h 1 =3 64-2 = 3 9 -4-3 = 3 12 -4- 4 = 3 4- -1 = 4 8-4-2 = 4 12 -h 3 = 4 1G-4- 4 = 4 - 1 =5 10 ~ 2 = 5 15 -4-3 = 5 20 -4- 4 = 5 6- -1=6 12-4-2 = 6 18 -4- 3 = 6 2 1 -- 4 = 6 7 - -1=7 14-4-2 = 7 21 -4-3 = 7 2s -s- 4 = 7 8 - - 1 =8 16 -v- 2 = 8 2 1 :- 3 = 8 h 4 = 8 9 - -1=9 18-4-2 = 9 27 -4- 3 = 9 36 -4- 4 = 9 5-^-5 = 1 6-4-6 = 1 7-4-7 = 1 8-4- 8= 1 10 -4- 5 = 2 12-4-6 = 2 U-*- 7 = 2 16 -4- 8 = 2 15 -4-5 = 3 18 -j- 6 = 3 21 _f- 7 = 3 24 -4- 8 = 3 20 -4- 5 = 4 24 -j- 6 = 4 28 -4- 7 = 4 32 -4- 8 = 4 25 -4- 5 = 5 30 -4- 6 = 5 35 -4- 7 = 5 40 -4- 8 = 5 30 -4- 5 = 6 36-4-6 = 6 42 -4- 7 = 6 48 -4- 8 = 6 35 -4- 5 = 7 42 -4- 6 = 7 49 -4- 7 = 7 06 --8 = 7 40-4-5 = 8 48 + c = 8 56 -4- 7 = 8 64-^8 = 8 AS -:- 5 = 9 54 -4- 6 = 9 63 -f- 7 = 9 72 +8—9- 9 -r- 9 = 1 10-4-10=1 11 -4-11 = 1 12-4-12 = 1 18-4-9 = 2 20 -j- 10 = 2 22-=- 11 = 2 24- -12 = 2 27 -4- 9 = 3 : 10 = 3 33-4- 11 = 3 36- -12 = 3 36 -4- 9 = 4 40-4-10 = 4 44-4-11 =4 -12 = 4 45 -4- 9 = 5 50-4-10 = 5 55-4-11 = 5 60- -12 = 5 :- 9 = 6 00-4-10 = 6 66 -4- 1 1 = 6 72- -12 = 6 63 -4- 9 = 7 70 -h 10 = 7 77 -4- 11 =7 84- -12 = 7 :- 9 = 8 80-4-10 = 8 88-4-11=8 96- 12 = 8 81-4-9 = 9 90 -r- 10 = 9 99-4- 11 = 9 108- -12 = 9 \. 32 are how many times 4? 8 ? 2? 16? - are how many times 4? 6? 12? 8? 3? 16? G. 36 are how many times 12? 6? 9? 3? 4? 2? 7. 40 are how many times 8 ? 4 ? 2 ? 10 ? 5 ? 20 ? 44 DIYlSiU.W 71. Division is also indicated by the co/on ; thus, 8 : 2 = 4. Also by writing the divisor before the dividend, with a curved line between ; thus, 2) 846 , or thus, 2 ) 8 4 C ( , the quotient to be placed under or at the right of the dividend, and separated from it by a line. Also by writing the divisor under the dividend, with a line between ; thus, $ = 3; i. e. 6 divided by 2 equals 3 ; or, more familiarly, 2 in G, 3 times. Ex. 8. IIow many are §? Ans. 2 in 8, 4 times. The fourth mode of indicating division gives the the following compact and convenient DIVISION TABLE. 1 | = 1 1 = 1 * = 1 * = 1 2 * = 2 $ = 2 1=2 ¥ = 2 3 5 = 3 = 3 V = 3 ^3 4 | = 4 = 4 V = 4 Y = 4 5 If = 5 ¥ = 5 V- = 5 V = 5 G ^G ^G V = c V = c 7 V = 7 y. = 7 V = 7 = 7 8 = 8 -^8 -8 ¥ = 8 9 Jj/i = 9 = 9 ^ = 9 :^9 V = 2 V=3 V = 4 V = 5 V = 6 V = 9 f=1 1 = 1 8 = 1 l:: = i n = ) >,* = 2 = 2 ¥■ = •2 « = 2 .!i'=2 V = 3 ^ = 3 V=3 ?8 = 3 ^3 3,1 = 4 •v = -* ■V = 4 « = 4 n = 4 V = o ^ = 5 ¥ = 5 « = « if = 5 ^ = G V == 6 V = « H = 6 It = d V- = " ¥ = 7 = 7 ;:: = 7 il = < V = 8 = 8 = 8 = 8 ?? = » "A s= 9 = 9 = 9 H = 9 « = 9 «== 8 '5 = 3 H = 4 ff = 5 H = 6 tt==7 |} = 8 = 9 71. Second sign of Division, what is it? Third mode of indicating Division,' what is it? Where is the <;uotieut to be written? Fourth method, whaf How are the dividend and divisor written in the second Division Table? division*. 45 Ex. 9. How many 6, or « « ? Ans. 4. 10. How many arc 46 -:- 8, or -* g ft ? 1 1. How many are GO -:- 1 1, or f f ? 12. How many are 84 -f- 12, or j 1.:. How many are 68 — 9, or «„ Ans. 7. 1 1. How many are i-S -f- 6, or ^ ? 1">. How many air 77 «+■ 11, or \\? 1 «'•. How many arc 72 -f- 8, or ^ ? 17. How many an- H I 12, or ^ ? Ans. 8. 18. How many are 88 -+■ 8, or ^ ? 19. How many an- 72 -f- 12, or ^ ? 72. When the dividend is large the division may be per- formed in two ways, as follow- : 2a Divide 1384 by 4. nnsT operation. Having written the divisor and divi- 4)1384(346 dend as in the margin, we first inquire 1 2 how many times 4 is eontained in 13, 7~7> (the fewest figures at the left of the dividend that will contain the divisor,) and find the quotient to be 3, which 2 4 tve set at the right of the dividend. 2_4 V,\- then multiply the divisor by the q quotient, 3, and set the product, 12, under the 13 of the dividend, and sub- tract it therefrom. To the remainder, 1, we annex 8, the next figure of the dividend, find then inquire how many times the divisor is contained in 18, the second partial dividend; there- suit, 1. we set as the second figure of the quotient, and then multiply, subtract, annex, etc., as before, until all the figures of the dividend have been taken. the 13 of the dividend ia hundreds, the 3 of the quo- tient is also hundreds ; since the 18 is tens, the 4 is also tens ; and, iinircrsalhj, any quotient figure is of the same order as the right-hand figure of the dividend taken to obtain that quotient figure 73. How many ways to perform Division* Of what order is any quotient 46 DIVISION. The foregoing operation is called Long Division, but the work may be much shortened by carrying the process in the mind, in- stead of writing it ; thus, second orERATiox. having written the divi- Divisor, 4) 1384 Dividend. % 80r and dividend u Quotient, 3 4 6 fore, Bay, 1 in 13, 3 times and 1 remainder; set the quotient, 3, under the 3 of the dividend, and then, innu/i/dng the remainder, 1, placed before the 8, say, 4 in 18, 4 times and 2 remainder ; set down the 4 as the second figure of the quotient, and imagine the 2 set before the next iigure, and so proceed. This operation is called Short J)lvisiun, which is usually adopted when the divisor is so small that the process may be readily carried in the mind. Hence, 73. To perform Short Division : Rule. Divide the left-hand figure or figures of the dividend, (the fewest figures in the dividend that will contain the divisor,) and set the quotient under the right-hand figure taken in the divi- dend ; if anything remains, prefix it mentally to the next figure in the dividend, and divide the number thus formed as before, and so proceed tUl all the figures of the dividend have been employed. Ex. 21. Divide 248G4 by 8. OPERATION. Divisor, 8)24864 Dividend. Quotient, 3 10 8 22. Divide 3240 by 2. Ans. 1623. 23. Divide 1326 by 3. Ans. 442. 1 1. Divide 72345 by 5. Ans. 14 169. 2& Divide 3283 by 7. Ans. 469. 26. Divide 59684 by 4. Ans. 14921. 27. Divide 69545 by 5. Ans. 13909. 28. Divide 36945 by 9. Ans. 4105. 29. Divide 27512 by 8. Ans. 3439. 7». What is the first method of Division called? What the Second? When i» Short Division employed? 73. Rule for Short Division? division. 47 30. Divisor, 8) 7641 28 Dividend j 1 8 GJ_2 QBPtfenlj 9 5516 7 1214 32. 33. -I. G ) 3 2 4 9 6 2 ) 1 48G50 B I 4 5 8 2 8 9 2 7 74. When then' i> no remainder, as in the first thirty-four example, the division is complete. The dividend is then said divisible by the divisor, and the divisor is called an exact dir>> When there is a remaindi t\ as in Ex. .35, the di\ -i.-ion is in- complete, and the dividend is said to be indivisible by the divisor. 35. Divide 2781 by 8. OPKUATION. Divisor, 8)2781 Dividend. Quotient, 3 4 7 ... 5 Remainder. 36. Divide 3654 by 4. Divide 72584 by 5. 38. 86471 -f- 3 = how many ? 39. 40505 ~ 7 = ? 40. 476589-^9=? 41. 987654 -J- 12 = ? 1l>. 334523-^-11=:? 15. In one week there are 7 days; how many weeks in 255 Ad-. 36 weeks, Rem. 3 days. 1 L llnw many barrels of flour, at $0 a barrel, can be bought for S7:-<»? ■15. It' 6 shillings make a dollar, how many dollars are there In 27.3G -hillmgs? 46. It 1 weeks make a month, how many months are there in 69 i \\. 74. Wbtl i- t lie division complete? When is one number divisible by an- other! What is an exact divi«or! When ia one number %ndivi$ible by another! >aoticnts, Rem. 913, 2. 14516, 4. 28823, 2. 5786, 3. 48 mvr?iox. 75. When the divisor is large, the operation is usually performed by Long Division, as follows : Ex. 47. Divide 2875 by 23. operation. This operation is like the first 2 3 ) 2 8 7 5 ( 125 operation in Ex. 20. The partial 2 3 dividends are 28, 57, and 115 ; the .- j successive quotient figures an- I, 2, a g and 5, and these quotient fig - multiplied into the divisor, give 23, 11^ 46, and 115 for the successive prod- * 1 J ucts or subtrahends and the last product, 115, taken from the last dividend, 1 15, leaves no remainder; .-. 125 is the true quotient. Ihnce, 76. To perform Long Division : Kri.i:. 1. Write the divisor and dividend as in sliort division, anil draw a curved line at the fight of the dividend. 2. Divide tl nwmb r of figvru in the left of the divi- dot'! thai vill tht divisor, and set the result as the first ft jure of the quotient at the right of the dividend. 3. Multiply the divisor by the quotient figure, and set the product under that part of the dividend t<- 77. Division is (he reverse of multiplication. In multiplica- tion the two &Ctor li. ami the product is required j in division the product and one factor are given, and the other f ;l .-- - required. The dividend ia the product, and the diruor and quotient are tlu* factors ; thus, IN Mil 1 Ii'LICATIOX. IN IMVISIOIf. Factors, Product. Dividend, Divisor, Quotient 5x 4 = 10 20 -f- 5 = 4 Or, 20 -J- 4 = 5 Hence the 78. Proof. Multiply the divisor by the quotient, and to the product add the remainder ; the sum should be the dividend. 48. Divide 2537 by 53. OPERATIOX. 53)2537(47 2 1 2 417 371 46 PROOF. 5 3 Divisor. 4 7 Quotient. 371 212 4 6 Remainder. 25 3 7 Dividend. 49. 21 )864(41 84 24 21 50. 87)3659(42 348 179 174 3 5 51. A flock of 1728 sheep were divided equally in 9 different lAttarea, how many ibeep were there in each pasture? 77. What is said of Division and Multiplication? In Multiplication what is ' What required ? In Division what is given? Required? 78. How is :i proved? 5 Quotients. Rem. 1509, j6. 2615, 16. 10175, 24. 5075, 43. 10475, 7. ii 11237, 57. 1090, 124. 50 DIVISION. . Divide 46782 by 31. 53. Divide 47086 by 18. 54 Divide 468074 by 46. 1 10068 by 67. ! -h 83 = how manv ? 57. 9 ;7048-^99=rhowmauy? + 78 = BOW many? 59. 276984 -*- 254 = ? CO. 376958-4-84$=? 81. 876598-4-427=? 61 469873^789=? 9 -s-803 = ? 64 896842 ~ 548 = ? ,:*2^45=r? \ ~ 4893 = ? 5=? 68. Divide four hundred eighteen thousand, six hundred and •€ight, by tw.nty-four. An?., Quo. 17443, Rem. 16. 69. Divide two hundred one thousand, live hundred and ninety-live acres of land, into twenty-three equal parts. 7<». A railroad that cost f divided into 7153 equal shares ; what was the cost of etch share? 71. A farmer raised 2001 bushels of wheat on 87 acres of land ; how many bushels did he raise per acre ? 72. In how many days will a ship sail 34">6 miles, if it sails 1 I 1 miles per day? A farmer raised 4088 bushels of corn, his crop averaging 56 bushels per acre; how many acres did he plant ? 7 1. A drover paid $3175 for 29 oxen; how many dollars did he pay for each ox ? 7 "'. The product of two numbers is 10707, and one of the numbers is 129 ; what is the other number? The earth, in its revolution round the sun, moves about 1641600 miles in one day ; how far does it move in one second, there being 86400 seconds in a daj \ 77. Divide §1064 equally among 8 men. Ans. S133. PI VI 51 79. To divide by a composite number. . 78. Divid equally among 85 men. on z 7 X 5. The 85 men may ictor, 7 ) $ 1 8 5 5 Dividend. be separated into 1 M Emctor, 5)S265 1st Quotient, - r ", u '" 'f * .*" ' ^ each, riicn divid- $5 3 True Quotient. fog by 7 giv< for each group, and dividing the $2G."> by 5 gives S">3- for cacli man. Vhcn a eomposito numl>er is made up of different sets of far- tors, os in Ex. 79, it is immaterial which sot is taken. It is also immaterial in what order the factors arc tuken. 79. Divide 10G5G by 288. 288 = 4 X 6 X 12 = 6 X 6 X 8 = 8 X 3 X 12, etc. FIRST OPKRATIO.V. 4) 10656 BECOJCD OPERATION. G ) 1 6 5 6 664 1 2 ) 444 37 C) 1776 8)296 "77 From these examples we have the following Kile. Divide ike dividend by one factor of the divisor, and the quotient so obtained by another factor, and so on till all the the set have been used. The last quotient will be the true quotient. 80. Divide 1551 by 81. Divide 31794 by 42. Divide 47986 by 5G. 88. Divide 24840 I M. Divid." 7665 by L05, Divide 1064 by 5G. Divide 1984 by 64 87. Divide 8321 by 81. 88. Divide 18723G by 252. 89. Divide 1255872 by 192. i»o. Divide 1865 by 105. 91. Divide :>:5.V> by 315. 92. Divide GG99 by 281. 93. Divide 8822 by 294. 94. Divide s:>i}8by504. 95. Divide 7245 by 315. 7'J. Bali for dividing by a Composite Number? la it material which factor of" the divisor is used flrkt? 52 DIVISION. 80. In dividing by the factors of the divisor, there may be a remainder after either or each of tin* divisions. Should the learner find a difficulty in determining the true re- mainder, he has but to remember that it is always of the same kind as the dividend (Art. G9, Note). 96. Divide 86 by lh OPERATION. 7 ) 8_6 In this example, as 86 is the true dividend, 2 is the true remainder. In this example, as 23 is only one fourth of the true dividend, so the remainder, 2, is only one fourth of the true remainder; .•.2X4=8, true remainder. By the explanation of exara- 7 Tg~7 5 j^ eni pies 96 and 97, we see that 5 is ' — " " one part of the true remainder, Quotient, 1 % . . . 8 Rem. an( j tbmt 5, the second remain- der, multiplied by 6, the Ant divisor, is the other part; i. e. 5 -\-3 X & = 23, is the true re- mainder. The same species of reasoning applies when there are more than two divisors. Hence, To obtain the true remainder when division is per- formed by using the factors of the divisor: Rite. Multiply each remainder, except that left by the first division, by the continued product of the divisors preceding that which gave the remainders severally, and the sum of the products, together with the remainder left by the first division, will be the true remainder. Note 1. "When there are but two divisors and two remainders, the rule 80. Rule for finding the true remainder when the factors of the divisor ar« used separately? The reason? What is meant hy a continued product? 3)12... 2 Rem, Quotient, 4 Divide 9! OPERATION. 7)23 28. Quotient, 3 . . . 2 Rem. 98. Divide 5S7 bj OPERATION. 6)527 DIVISION. 53 only requires the addition of the Jirst remainder, to the product of the firtl and second return Rbl I 2. When thnf or more factors are multiplied toget h er , the product is called a continued product. Quo. 52, Rem. 14. I 1:1 1. KIM \IM 4= lei Rem. < 5 = 10 = 2d Rem. X 1st Div 14 = True Rem. TRUE REMAINDER. 3 = 1st Rem. 99. Divide 1884 by 35. OPERAT1 -5X7. 5)1834 \ 7 ) 3 6 6 ... 4, 1st Rem. Quo.,~~5~2 ... 2, 2d Bern. 100. Divide 18328 by 385 oil i: vriON. = 5X7X11. 5)18328 7)3 6 65 ... 3, 1st Rem. 11) 5 2 3 . . . 4, 2d Rem. Quo., 4 7 ... 6, 3d Rem. 101. Divide 5273 by 42. 42 = 6 X 7. 102. Divide 4G987 by 504 103. 104. 105. 10& 1<)7. 437298-^-54 = ? 21C349-t-64 = ? 8411 -5-72=? ,,7_j_45 = ? 65947 -f- 25=? 4X5= 2 = lstProd. 6X7X5 = 210= 2d Prod. 2 3 3 = True Rem. Ans. 125 and 23 Rem. ng the factors of the divisor. Am. 88 end 115 Rem. 108. 6842 -v- 49=? 109. 7829 -f- 35 = ? 110. 8748 -#- 42h± t 111. 4629-^81 = ? 112. 3643 «)L 48 = ? 81. To divide by 10, 100, 1000, etc. : Rule, Cut off] by a point, as many figures from the right I of the dividend at then are ciphers in the dirisor. The • s at the left of the point arc the quotient, and those at the right are the remainder. 1 13. Divide 75G by 10. Ans. 75.6, i. e. 75 Quo., 6 Rem. 81. Rule for dividing by 10? By 100? 5* 54 DIVISION. Note. The reason of the rule is obvious. By taking away the right- hand figure, each of the other figures is brought one place nearer to units, and its value is only one tenth as great as before (Art. 15), and .'. the whole is divided by 10. For like reasons, cutting off ttco figures divides by 100 ; cutting off Utree figures divides by 1000, etc. 114. Divide 402763 by 10. 115. Divide 7G943 by 100. Ans. 769 and 43 Rem. 11G. Divide 98765423 by 100000. ! Ans. 987 and 65423 Rem. 117. Divide 3078654321 by 100000000. 8£. To divide by 20, 50, 700, or any similar number : Rdle. Cut off as many figures from the right of (he divi- dend as there are ciphers at the right of the significant figures of the divisor, and then divide the remaining figures of the dividend by the significant figures of the divisor. Note 1. This is on the principle of dividing by the factors of the divi- sor ; .-. the true remainder will be found by the rule in Art. 80. 118. Divide 74689 by 8000. Ans. 9 and 2689 Rem. operation. AW divide by 1000 by cutting 8 ) 7 4.6 8 9 off 689, which gives 74 for a Ouotient ~~ 9 2 Rem quotient and 689 for a reiuain- ^ ' der ; then divide 74 by 8, and obtain the quotient, 9, and remainder, 2. This remainder, 2, is 2000, which, increased by 689, gives 2689 for the true remainder (Art. 80). Note 2. It will l>c observed that the true remainder, in all examples /ike thollSth, is obtained by annexing the 1st to the 2d remainder. 119. Divide 07475 by 2400. 120. Divide 74689 by 4200. Ans. 17 and 3289 Rem. 121. Divide 276987 by 3300. 1 22. 769842 -^- 45000 = ? Ans. 17 and 4842 Rem. i«8. 9999999-1-33300=:? 124. 80407080 — 40000 = ? 125. 987654321 -f- 90900 = ? 81. Beason of rule for dividing by 10? 83. Rule for dividing by 20? By LOO? Reason? IIow is tbe true remainder found? DIVIifi 56 Qf Div; 821. The value of a quotient depends upon tlio relative values of the divisor and dividend, and not upon their luU values, us will be seen by the following propo- rtions. (a) If the divisor remains unaltered, multiplying the dividend by any number is, in effect, multiplying the quotient by the same number ; thus, 15-f-3= 5 _4 _4 00-^3 = 20; i. «•. multiplying the dividend by 1 multiplies the quotient by 4. (1>) DMding the dividend by any number is dividing the quotient by the same number ; thus, 24-4-2 = 1 2 3)24 8-^ 2= 4=12-v-3; i. e. dividing the dividend by 3 divides the quotient by 3. (c) Multiplying the divisor divides the quotient ; thus, 3 0-^-2 = 1.-) 3 naltipljiag the divisor by >°> divides the quotient by 3. (d) Dividing the divisor multiplies the quotient ; thus, 40-4- 10— I 5) 1_0 4 -f- 2 = 20=r4X-">; i. e. dividing the divisor by o mult plies the quotient by ">. B3. Does the size of the quotient depend upon the absolute size of divisor and i ; .hi what dOM it clrpoiul ? What is the first proposition? Second r • Fourth? 56 DIVISION. (e) It follows, from (a) and (b), that the greater the dividend, the greater is the quotient ; and the less the dividend, the less the quotient. (f) Also, from (c) and (d), that the greater the divisor, the less is the quotient ; and the less the divisor, the greater the quotient. 84. From the illustrations in Art. 83 we see that any change in the dividend causes a similar change in the quotient, and that any change in the divisor can an opposite change in the quotient. Ilence, (a) Multiplying both dividend and divisor by the same number does not affect the quottini : thus, 12-^3 = 4 _2 2 2 4 -r- G = 4, Quotient unchanged. (b) Dividing both dividend and divisor by the same number does not affect the quotient ; thai, 20 -4- 10 = 2 5)20 5) H) 4 -j- 2 = 2, Quotient unchanged. (c) It follows from (a) and (b), that the operations of multi- plying and dividing by the same number cancel (i. e. destroy) each other ; e. g., If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand : thus, 8X7 = 56, and 56 -J- 7 = 8, the multiplicand. Also, if a number be divided by any number, and the quotient be multiplied by the divisor, the product will be the dividend ; thus, 15 -f- 3 ■= 5, and 5 X 3 = 15, the dividend. 83. What follows from (a) and (b)? From (c) and (d)? 84. Any change in the dividend, how does it affect the quotient? Any change in the divisor, how? First inference? Second? Third? Illustrate. MISCELLA SAMPLES. 6f S3. These general principles may be more briefly 1 as follows: 1-t. Mult iplying the dividend multiplies the quotient ; and dividing the dividend diviiles the quotient (Art. 83, a and b). 2d Multiplying the. divisor divides the quotient ; and dividing the divisor multiplies the quotient (Art. 88, and <1). Multiplying both dividend and divisor by the same num- ber ; or dividing both bij the same number does not affect the quo- ( Art. 84, a and b). Examples in the Foregoing Principles. i 1. Hon 'Many bu>hels of corn at SI per bushel mutt be given for 6 barrels of flour at $7 pel barrel? 2. How many bands of apples at $2 per barrel must be given be 8 cords of wood tl $6 per cord? 3. A speculator bought 640 acres of land at $8 per acre, and void the whole for $3200; how much did he gain by the trans- lation? Bow much per aen 4. Bought 320 acres of land for $1760, and 320 acres more at ^7 pel acre, and sold the whole at $6 per acre; did I gain or How much? Ans. Lost $160. 5. The; expenses of a boy at school for a year are $126 for board) $24 for tuition, $15 for books, $35 for clothes, $10 for railroad and coach fare, and $0 for other purposes ; what will be the expenses of 250 boys at the same rate ? ('.. If :; men build 24 rods of wall in 4 days, in how many ill 5 men build 70 rods? Ans. 7. 7. The product of 4 factors i< 1155; three of the factor! 8, 5, and 7 ; what is the fourth? Ans. 11. 8. How many miles per hour must a ship sail to CTOS Atlantic, 2880 miles, in 12 days of 24 hours each ? ( J. The first of 3 numbers is 6, the second is 5 times the first, and the third is 4 times the sum of the other two; what is the difference between the first and third? W. A more brief *t«totn«>nt of (hew principle fecoad! Third? 68 REDUCTION. ^ 10. Sold two cows at $30 apiece, 3 tons of hay at $20 per ton, 50 bushels of corn for $50, and 10 cords of wood at $7 per cord, and received in payment $200 in money, a plow worth $15, 50 pounds of sugar worth $5, and the balance in broadcloth at $4 jet yard ; how many yards did I receive? Ans. 5. 11. In how many days of 24 hours each will a ship cross the Atlantic, 2880 miles, if she sails 10 miles per hour? 12. If I receive $60 and spend $40, per month, in how many years of 12 months each shall I save $2160 ? Ans. 9. 13. "What ii the value of 27 hogsheads of molasses at j - per hothead? 14. What i> the value of 87 yards of cloth at .ml ? 15. Bought 87 acres of land at | and paid $3150 in cash, and the balance in labor at $240 a year; how many I of labor did it take ? 16. Bought 42 yards of cloth at 15 cents per yard, and paid for it in corn at 90 cents per bushel ; how many bushels did it take? 17. If I take 13729 from the sum of 8762 and 14967, divide the remainder by 50, and multiply the quotient by 19, what is the product ? Ans. 3800. REDUCTION 80. All numbers are simple or compound. A Simple Number consists of but one hind or denomina- tion ; as 2, $4, 8 books, 5 men, 6 days, 10 miles. A Compound Number is composed of two or more denom- inations ; as 4 days and 7 hours ; 8 bushels, 2 pecks, and 5 quarts ; 5 rods, 4 feet, and 6 inches. All abstract numbers (Art. 2) are simple. 86. What is a Simple Number! A Compound Xumbcr? An Abstrac dum- ber, is it simple or conipom.U I A concrete UBUllMT, whrtlit -r simjilo or compound, is often called a Denominate Xm i: 1. All operations in the preceding pages are upon simple num- bers. The several parts of a compound number, though of different denominations, are yet of the same general nature; thus, 2 Iraki, •'* days, and 6 hours are similar quanti ^titute a comjiound number; but 2 md 6 quarts arc DIUKI in thliu xailue, and do not constitute a compound number. S7. Induction is changing ■ number of one denomination to one of another denomination, without changing its value. It is of two kinds, viz. Reduction Descending and Jteduction Ascending. Reduction Descending consists in changing a number from a higher to a lower denomination. duction Ascending is changing a number from a lower to a higher denomination. ENGLISH MONEY. 88. English Money is the Currency of Great Britain. TABLE. 4 Farthings (far. or qr.) make 1 Penny, marked d. 12 Pence u 1 Shilling, " p. 20 Shillings " 1 Pound, " £ g s 14 m 1 Shilling, 1 Pound, £ 1 20 • d. 1 = 12 = 8 qr. s= 4 = 48 = 960 80. REDUCTION Descending is performed by multiplica- tion ; thus, to reduce 15£ to shillings, we multiply 15 by 20, use there will be 20 times as many shillings as pounds. So to reduce 15£ and 12s. to shillings, we multiply 15 by 20, and to the product add the 12s. 86. A Concrete Number, what is it called? 87. What is Reduction? How many kinds of Reduction? What are they called? What is Reductiou Deecend- Itafl 88. What is English Money? Repeat the table. N'J. How is Rcduclion l>c.-c»-ndiiij( performed? 00 REDUCTION. In a similar manner all mob examples are reduced. II 90. To reduce the higher denominatious of a com- pound number to a lower denomination: Rule. Multiply die highest denomination given by the number it takes of the next lower denomination to make one of this higher, and to the product add tfte number of the lower denomination ; multiply this sum by the number it takes of the next lower denom- ination to make one of this ; add as before, and so proceed till the number is brought to the denomination required. Ex. 1. Reduce 11X 17s. 9d. 3qr. to farthings. Eleven pounds = 220s., and the 17s. added make 237s. = 2844d., and the 9d. added give a 11412qr., which, in- creased by the 3qr., give 11415 (jr., the answer. 1 1 1 i o qr., An>. 2. Reduce 6£ 18s. 4d. lqr. to farthings. Ans. G641qr. 8. 1: £ 9s. 8qr. to farthings. Ans. 7155qr. Note. Since there are no pence in the 3d example, there is nothing to add to the product obtained by multiplying 1-v i& ft. Etedoee 27£ 15s. Gd. 2qr. to farthings. 5. Reduce 32£ 8d. 3qr. to farthings. 01. Reduction Ascending is performed by division ; thus, to reduce 4299 farthings to pence, ue divide the 4299 by 4, because there will be only one fourth as many pence as far- things. Performing the division we obtain 1074d. and a remain- der of 3*qr. If we wish to reduce the 1074d. to .-hillings, we divide by 12, because there will be only one twelfth as many shillings as pence, and obtain 89s. and a remainder of Gd. Again, 93. Kepeat the rule. Explain the process in Ex. 1. How are the 2G7 shil- lings obtained? How the 2S63 pence? The lHlo farthings? 01. How is Re- duction Ascending performed? £ 8. 11 17 20 d. 9 2 3 7s. 12 2853d. 4 ION. 61 tO pounds by dividing by 20, givi: Hod • renuunder of 9a. Tims ire Sod that 4299qr. are eqnal to 8qr. Llkl >g ippUoi to all similar mWiplfMli II' I 92. To reduce • number of a lower denomination to numbers of higher denominations: BULK. Divide the given number by the number it tales of that denomination to male one of the next higher; divide the quotient number it takes of THAT denomination to make one of the ///ter. and so proceed till the number is brought to the de- nomination required. The last quotient, together with the several remainders (Art. 69, Note), will be the uns> 93. Redaction Ascending and Reduction Descending prove each other. Ex. 1. Reduce 11415 farthings to pence, shillings, and pound-. araanox. Fi rst divide by 4 to reduce the A ) 1 1 4 1 o qr. farthings to pence; then divide by 1 2 i~2 8 5 3 d -4-3or *^ t0 re( ^ uce P ence t0 shillings ; ' then by 20 to reduce shilli: 2 ) 2 3 7 s. + 9d. pounds, and thus obtain ll£ 17s. 1 l£+17s. & .'Mir., Ana, 2. Reduce 17229qr. to pence, shillings, and pounds. . 17£ 18s. lid. lqr. o. Reduce 6874d. to shillings and pounds. Ana. 28£ 12a, lOd. I 1. Since Ex. 3rd is givm in pence instead of farthings, the first divisor is 12 rather than 4. 4. Reduce 84697qr. to higher denomination-. Etedooa 124G83qr. to higher denominations. G. Reduce 3-17G2 Iqr. to pence, shillings, and pounds. 7. Reduce 8746d. to shillings and pounds. \ to pounds. 'J 1. iapeat the ru!f. Explain the process in Kx. 1. How are the 8qr. olr: Tli.lTr.? TliclU? 03. What is the J'/ - tloaJ 8 REDUCTION. Note 2. The numbers employed in the reduction of a compound num. ber are called a Scale. The descending scale for Reduction Descend' in j and an ascending scale for Reduction Ascending; thiu, in English money the desceudinq scale is 20, 12, and 4, and the ascending scale is 4, 12, ancrs at the left hand of the table, taken in order from the bottom to the top of the tabic, and the ascending scald consists of the same numbers taken in the reversed order, i. e. from the top to tha bottom of the table. In like manner the scale is found in the other tables. TROY WEIGHT. 04. Trot Weight is used in weighing gold, silver, and precious stones. TABLE. 24 Grains (gr.) 20 Pennyweights 12 Ounces make u 1 Pennyweight, dwt. 1 Ounce, oz. 1 Pound, lb. lb. 1 oz. 1 = 12 = dwt. 1 = 20 =: 240 = 24 480 5760 Ex. 1. How many grains in 71b. lloz. 14dwt. 18 OPERATIOX. 71b. lloz. 14dwtl8gr. 12 9 5oz. 20 Ex. 2. Reduce 45954gr. to pounds, ounces, etc. OPERATION. 24) 4595 4 gr. 20) 1914 dwt-f-18gr. 12)9 J 5oz. -j- 14 dwt. 7 lb. +11 oz. Ans.71b. lloz. 14dwt. 18gr. 19 14 dwt. 24 7674 3828 ~> 4 gr., Ans. Note 1. In solving Ex. 1, the several numbers of the lower denomina 93. What is a scale? A descending scale? An ascending scale? What ar» the scales for English money? Where are these scales found? Taken in what order? 94. For what is Troy Weight used? Bepeat the table. Descending scaler Ascending? [0», dons are added vimtallif, and only the i rittm ; thus, 12 ti are 84, and the llos. a. 1 >/. Then multiplying tin- Moa I Ling the 1 -l 1 1 w r . . If] idwt. Finally, in multiplying the 19U dwt. by M, list multiply by 4, adding in tin- ISgr., and then multiplying uitl adding thi results wt hm\ wtr, •_\ if any divisor is so large that the work is lvdono by Short l)ivic taken upon the llnH and the work done by Long Division, setting down only the results. I I.»w many grains in 101b. 8oz. 19dwt.? Ans. 964a6gr. 4. Reduce 386'Jogr. to pounds, etc Ans. 61b. 8oz. 12dwt. 7gr. ft. Bedaee 87942gr. to pounds, ounces, etc lb. 8oz. 6dwt 16gr. to grains. 7. Bow many BpoOQtj each weighing 2oz. 8dwt. 20gr., can b« made from 21b. Ooz. 6dwt of silver? Ans. 12. 8. A jeweller made 8oz. 16dwt. of gold into rings wbicb weighed 8dwt 16gr. eacb ; bow many rings did be make ? APOTHECARIES' WEIGHT. 93>. Apothecaries' Weight is used in mixing or com- pounding medicines ; but medicines are bought and sold by Avoirdupois Weight. TABLE. 20 Grains (gr.) 3 Scruples 8 Drams 19 Ounces make 1 Scruple, sc. or 9 " 1 Dram, dr. or 3 " 1 Ounce, oz. or 5$ 1 Pound, lb. or lb oz. lb. 1 1 = 12 sc. gr. dr. 1 BS 20 1 = 3 = 60 8 = 24 = 480 96 ss 288 = 5760 :: 1. The pound, it are equal, but the ounce, and grain, in Apothecaries' and Troy • differently subdivided. 9*. In solving Ex. 1, what is done with the numbers of the lower denomina- te 2, how is the work done? 95. For what is Apothecaries' W. Iftt Used? Repeat the tabic. Descending scale? Ascending? What denomination:! •f Apothecaries' Weight ars like thow of Troy Weight? Bl AUCTION. I.x. 1. How many scruples .2. In 13619 how many in 41b 8§ 53 29 ? pounds, oun< VTIOX. 4 lb 8§ 53 29 3 ) 1_3 G 1 9 11 8)4533 + 29 8 § It)£SB + 5 5 TT^3 4lb+8 5 3 13 6 19, Aim. Aim. 41b 6j 53 3. Reduce Coz. 3dr. lsc. 19gr. to grain-. . 3099gr. 4. Redact I tins to pounds, ouiv 21b. 9oz. 2dr. 5. \l tmce 876943 grains to higher denominations. <". Bednee 27. , lr. 2sc L5gr. to grains. 7. How many pounds, ounce-, etc., of medicine will an apoth- ecary use in preparing 974 prescriptions of 15 grains each? Ans. 21b. 6oz. 3dr. lsc lOgr. AVOTBDUPOIS WEIGHT. 1H>. AyontDUPOn WUQl 1 for weighing the coarser articles of DK h afl bay, eotton, tea, sugar, copper, iron, etc. TABU. 16 Drams (dr.) make 1 Ounce, OZ. 16 Oqdc l Pound, lb. 2.'» ; l Quarter, qr. •l < 1 HundredWei^ht, cut. Weig 1 Ton, t. oz. dr. lb. 1 = 16 qr. 1 = 16 = iwt. 1 = 27, = 400 = G400 t. 1=4 = 100 = 1C00 = 25600 1 = 20 = 80 = 2000 = 32000 = 512000 ■ voirdnpoU Weight | RE1' 05 m formerly to consider 281b. a quarter, 1 12 lb. a hundred weight and UU-iOlb. a tou ; but now the twu ' « 1 League, 1. G0^ Statute miles, nearly, " 1 Degree on Circ. of the Earth, 1° 3 GO Degr " 1 Circumference, circ. in. ft. 1 as b. c. 3 yd. I = 12 = 36 rd. 1 = 3 =z 3G = 108 fur. 1 = t>\ — 16J= 198 = 594 m . l = 40 = 220 — 6G0 = 7920 = 237G0 1 =z 8 = 320 = 17G0 =± 5280 = 63360 = 190080 : 1. The earth not being an exact sphere, the distance round it iu different directions is not exactly the same By the most exact measure- ments made, ■ degTM is a little less than 69£ miles. Note 2. The barleycorn is but little used. I 3. The 3 before miles iu the table is not a part of the scale. Ex. 1. How many rods in Ex. 2. Reduce 2710rd. to 8m. 3fur. 30rd. ? higher denominations. OILKATI OPERATION. 8 m. 3fur. 30rd. 40)27 1 Ord. 8 6 7 fur. 40 8 ) 6 7 fur. + 30rd. 8 m. -|-3fur. J710 rd., Ans. Ans. 8m. 3 fur. 30rd. 3. In 1yd. feft. 8in. how many barleycorns ? Ans. 528. 08. I Long Measure used? Table? Scale? A degree upon the earth, how long? 68 II EDUCTION. 4. Reduce 473b.c. to higher denominations. 5. The distance through the center of the earth is about 7912 mil.- ; how many rod- is it? 6. Thfl distance round the earth is about 8000000 rods; how many miles is it? CHAIN MEASURE. 99. Chain MbASUBI \t used by engineers and surveyors ir measuring roads, canals, boundaries of fields, etc TABLE. 7 tVa Inches (in.) make 1 Link, li. 25 4 10 8 Lii. Rods Chains Furlongs 1 Rod, Perch, or Pole, rd. 1 Chain, ch. 1 Furlong, fur. 1 Mile, m. li. in. 1 = 7flfc = 25 = 198 = 100 = = 1000 = : = 8000 = 63360 m. 1 fur. 1 = ch. 1 10 80 rd. 1 = 4 = 40 = 320 Note. To measure roads, etc., engineers often use a chain 100 feet long l \. i. Reduce 5m. 8ch. 3rd. 1511 to links. 7fur. •2. Seduce 47890 links tc higher denomination-. OPKRATI' 5 m. 7fur. 8ch. 3rd. 15 li. on 2 5 ) 4 7 8 9 li. 8 4 7 fur. 10 47 8ch. 4 4) 1 9 15rd. + 1511 10)47 8ch. + 3rd. 8 ) 4 7 fur. -f- 8 ch. ."> m. -j- 7 fur. I915rd. Ans. 5m. 7fur. 8ch. 3rd. 15 li 95 90 3830 4 7 8 9 li.. Ans. 09. For abatis Chain Measure used? Table? Scale? Note? tion. r,o 8. Jn fiftir. 2rh. 8r& 1811 bow many links ? An*. <-293. •1. Bt dace 8879 links to bigbet denominations. 4ns. 3fur. 8ch. 3rd. Hi. ; M 17m. Slur. 5dfc 2nl. 2 Hi. t<> link-. Efodace l.">17."> links t<» higher denominations. Prom Boston to Anduver if 23 miles; how many links is it ? Prom Boston to Fitchburg is 400000 links ; how many miles is it ? 9. The distance round a field is 7fur. 6ch. 3rd. ; what will it to fenoe the field at $2 per rod? 10. How many miles, etc., in 637482 links? SQUARE MEASURE. 100. Sqtabe Measure is used for measuring surf TABLE. 144 Square Inches (sq. in.) 9 Square Feet 80] Square Yards or) Square Feet j 40 Square Rods make u m U 1 1 1 1 Square Foot, Square Yard, Square Rod, Rood, sq. ft. sq. yd. sq. rd. r. I Roods m 1 Acre, a. f>l<> Aeres u 1 Square Mile, sq. m. (a) Also in Chain Measure, 10000 Square Links or) 1 6 Square Rods ) make 1 Square Chain , sq. ch. 10 Square Chains M 1 Acre, a. §q. rd. sq.ya. 1= sq.ft. 1= 9= sq.in. 144 1296 1 = 30J: 272J= 1= 40= 12l0z 10890= 1568160 fq . m . 1= • 4= 100= 10= 43560= 6272640 £=$40=2560=102400= 3097600=27878400=401 44*3 nriii^' land, surveyors use ■ 4-rod chain composed of 100 .films the halt'-ehain of 50 link^ is mi loo. I r what is Square Mcasurt used? Tabl** Scaler Table in Chain Measure ' Note* 70 DEDUCTION. Ex. 1. In 2sq. m. 625a. 2r. 25sq. rd. how many sq. rods ? OPERATION. 2 sq. m. G2 5 a. 2r. 25sq. rd. 640 1905a. 4 Ex. B. Reduce 804905sq. rd, to higher denominations. OPERATION. 7 ii2 2r. 40 40) 304905 gg rd. 4) 7 622 r. + 25sq.rd. 6 4 ) 1 9 5 a. -f 2r. m.-j- ( '--~'.'i. Ans. 2>q. m. 625a. 2r. 25sq. rd. :'. 1 905 Sty rd., An?. 3. In 14sq. m. 25a. 3r. n and each angle is called a rigid angle. 101. How ia the area of a rectangle or square ascertained? What is said of the angles of a rectangle or square? What is each angle called? REDUCTION. 71 10*2. The area of a rectangle divided by the length will tlic breadth, and the area divided bj tin- breadth will ghr< fr/iy//* ; thus, in Fig. I, !•"> -f- 5 = 8 end i"» -+■ 3 = 5. Sow many square rods in a field that ifl 7 padi wide and 9 rods long AlH 7. How many square rods in a field that is 25 rode Wide and 48 rods long? How many acres? N An-. 7a. ft 8. Aboard containing 30 square ft-ot, is 12 feet long; how wide is it ? '.). A fiower garden containing 300 square feet is 12 feel wide ; how long is it ? 1<>. How many acres in afield that is 20 rods wide and 56 >ng? Ans. 7. SOLID OR CUBIC MEASURE, 103. Solid or Cubic Measure is used in measuring things which have length, breadth, and thickness. TABLE. 1728 Cubic Inches (c. in.) make 1 Cubic Foot, cu. ft. ■J7 Cubic Feet " 1 Cubic Yard, c. yd. 16 Cubic Feet " 1 Cord Foot, c. ft. 8 Cord Feet or) u - ^ , 128 Cubic Feet / X ^"^ °* cu. ft. c. in. p. v.i. 1 = 1728 1 = 27 = 46656 Note 1. The scale in this tabic only includes 1728 and 27; the nthpr numbers arc irregular. Transportation companies often estimate freight, especially of [ticks, by the ipaCC OCCOpied, rather than by the actual weight In itimate, from 25 or 30 to 150 or 175 cubic feet are called a ton. This i> railed arbitrary weight, and it varies with different transportation compa- :i 1 lotnewhai according to the risks of carriage. The Boston and •isiders a thousand of bricks a ton, whereas tho BOtaal weight is more than two tons. Again, a horse is estimated at 30001b-, brsadtl of a rectangle found when the area and length are known' Bow tlic length, when the area ami hrendth are known! 103. For what is Solid Measure u*e 7 ~~34c. yd. + 2 Gcu. ft. 2 04 68 Ans. 34c. yd. 26cu. ft. 9 4 4cu. ft., An<. 3. In 3c. Gc. ft. 15cu. ft. 156c. in. how many cubic inches ? Ans. 855516. 4. If 40cu. ft. make one ton, how many tons, cubic feet, etc. In 889664 cubic inches ? 104. A body like Fig. 2 is called a prism. Each side, ai D Fi S- 2 ' C f i G E F A B C D or A B F E, is called a face of the prism. If eacl _ _. '_ > ' ' / A /- /• ! ' f\ >\ ! ' A " % v\ ) o B d b g e fa / 104. What is a body like Fig. 2 called? What is one side of the prism called 73 nn?le of the faces is I rim is rectangular, and the priMD To determine the jular prism, first find the area of the upper A li C D, as in Art. lOlj then going from A, B, and C downward 1 inch to a, 1>, and C, tad pacing a plane through a, b. shall cut off 1"> solid inches, i. e. 5 X 3 X 1 solid inches, So if a plane be passed through d, e, and f it will cut », or 5 X 3 X 2 inches etc. ; i. c. the continued product of the numbers expressing the length, breadth, and depth, will give the solid contents of the prism. 103. So also, the solid contents divided by the area of the top face will give the depth; the contents divided by the area of one end will give the length ; and the contents divided by the area of one side will give the breadth or width. "What are the solid contents of Fig. 2 ? . ">. How many cubic inches in a rectangular prism or block of wood which is 12 inches long, 8 inches wide, and 6 inches thick? An, 12X8X6 = 57(3. G. How many cubic feet in a room which is 18 feet long, 15 wide, and 9 feet high? 7. A rectangular block of marble which contains 96 cubic feet, is 8 feet long and 4 feet wide ; how thick is it ? Ans. 3 feet. 8. A grain-bin which holds 21 cubic feet of grain is 3 feet deep and 2 (eel wide; how long is it ? 9. A lady's work-box contains 480 cubic inches; it is 12 inch- es long and 5 inches deep | how wide is it ? 10. In a pile of wood 1G feet long, 4 feet wide, and 6 feet high, how many cord- ? Ans. 3. 11. If a load of wood be 8 feet long and 4 feet wide, how high must it be to make a cord? 12. My bedroom ia 1"> feet lon«r, 12 feet wide, and 9 feet : in how many minutes shall I breathe the room full of air, if 1 breathe 1 cubic foot in 2 minutes? 10J. When is a prism rectangular? When is it a cube? How arc the con- fa rectangular prism found? 103. How the depth, length, or breadth, if we know the content* of tha body and the area of one faco? 7 7 i REDUCTION. LIQUID MEASURE. 100. Liquid Measure is used in measuring all liquids. The U. S. Standard Unit of Liquid Measure is the old English wine gallon, which contains 231 cubic inch TABLE. 4 Gills (gi.) make 1 Pint, pt. 2 Pints ■ 1 Quart, qt. 4 Quarts u 1 Gallon, gal. pi. gi. at 1 = 4 gal. I', ss I a 8 1 = 4 = 8 = 32 Note 1 . It has been customary to measure milk, and also beer, ale, and ntln t malt liquors, by beer measure, the gallon containing 282 cubic inches, but this custom is fast going out of Note 2. Casks of various capacities, from 50 to 1 50 or more gallons, are indiscriminately called hogsheads, pipes, butts, tun-. Ex. 1. In Ggal. 3qt. lpt. Ex. 2. Reduce 222 gills t* 2gi. how many gills ? gallons, quarts, etc OPERATION. OPERATION. ft gaL 5qt lpt 2gL 4 ) 2 2 2 gi. 4 2)5j>pt +2gi. 2 7qt. 4)2 7qt. + 1 pt. G gal. + 3 qt. 5 ^P t Ans. Ggal. 3qt. lpt. 2gi. 2 2 2gi., Ans, 3. Reduce 8gal. 2qt lpt 3gi. to gilK Ans. 279gi. 4. Reduce 7496 gills to higher denominations. 5. How many demijohns, each containing 2gal. lqt. lpt. 3gi. may be tilled from a cask which contains 98 gallons and 3 quarts? 6. How many gallons of molasses in 24 jugs, each containing 2gal. 3qt. lpt? 106. For what is Liquid Measure used? Table? Scale? Notel? Note 2? RID1 75 DRY MEASURE, 107. I>r.\ Mia -i i;i. i- need in measuring grain, fruit, pota- TABLE, I Pints (pt.) in:ikc 1 Quart, qt. 8 Quarts " 1 P< pk. t Peeka " 1 Bushel, boah. qt. pt. pk. 1=2 i.ush. i = 8 = u; I = 4 = 32 = <;i . The bushel ineiisnrc is 18$ inches In diameter and 8 inehc- nnil contains ft Utile lesfl th:m S150| ■oHd inches, or nearly 9J wine gallons. 1. In Sbuah. 8pL 7qt Ex. 2. Reduce 2 jo pints to lpt. how many pints ? OPERATION. 3bu>h. Spk. 7qt lpt. hnnhoto pecks, etc. OPERATION. 2)25 5})t. 4 15 pk. 8 1 2 7 qt. 2 8) 127qt. + lpt. 4 ) 1 5 pk. + 7qt. 3 bush. + 8plc .Ajis. 3bush. 3pk. 7qt. lpt. 2 5 5 pt., Ans. 3. Reduce 8bush. 2pk. 3qt. lpt. to pints. Ans. 55 lpt. 1. Reduce 7893pt to higher denominations. 5. Redact 4698pt to higher denominations. 8. How many pints in 15bu>h. 8pk. 6qfe lpt.? 7. How many pints in 2 -Ibush. Ipk. 7qt lpt.? 8. What is the e bosh. 2pk. of graas seed at $1 n 9. Reduce 345G9 pints to higher denominations. I leduce 63 bush. 2pk. 7qt. lpt. to pints. 107. For Wtttl IN Mftdf Table? BOfttet What are the ilinu n. Of the bushel measure? How uiauy cubic inches dues it coutaiu ' llow many *«ue gallon*? 76 REDUCTION. TIME. 108. Time is used in measuring duration. The natural divisions of time are days, months (moons), seasons, and roars. The artificial divisions are seconds, minutes, hours, weeks, etc TABLE. CO Seconds (sec.) make 1 Minute, m. 60 Minute " 1 Hour, h. 24 Hours « ID d. 7D f l 119 wk. 1 W " 1 Lunar Month, 1. m. 13 Months, 1 Day, and 6 Hours " 1 Julian Year, J. yr. 12 Calendar Months (=865 or) i), 1 Civil Year, cyr. 100 1 make 1 Century, C. m. SCO. h. 1 = d. 1= 60 = 3600 wk. 1 = 24= 1440 = l.m. 1 = 7 = 168= 10080 = 604800 1 = 4 s= 28 = 672 = 40320 = 241. 13 T f , = 52J« , = 365| = 8766 = 525960 = 31557600 J.yr. 1 = Notb 1. The twelve calendar months have the following number of days: January (Jan.) has 31 days; February (Feb.), 28 (in leap year, 29) ; March (Mar.), 31 ; April (Apr), 30 ; May, 31 ; June, 30 ; July, 31 : Au_rust (Aug.), 31 ; September (Sept.)i 30; October (Oct.), 31 ; Xovcmbet (Nov.), 30 ; December (Dec), 31. Note 2. The number o^ days in each month may be easily remembered by committing the following lines : " Thirty days hath September, April, June, and November; All the rest have thirty-one, Save the second month alone, "Which has just eight and a score Till leap year gives it one more." Note 3. A solar year, i. e. a year by the sun, is very nearly 365 days, 5 hours, 48 minutes, and 50 seconds. 103. For what is time used ? What are its natural divisions? Artificial divi- sions? Table? Scale? What are the names of the calendar months? How many days in each? Length of a solar year? TIOX. 77 Ex, 1. Reduce 3wk. •m. to mm OPl RATIO 6d ;!ul, 6d 23b. 59m. _7 2 7d. _ 2 _1 f3 1 J. Reduce 40319m. to orii! crioy. 6 ) 4 3 19 m. 2 1 ) OJTl h. -f 59m. 7)2 7<1. + 23h. k. + Od. Ans. 3wk. Cd. 23b. 59m. 1 h. GO ; 1 9 m., An>. : face lwk. 4d. ICh. 8m. to minutes. Am. lG808m. 1. EKedo seconds to higher denominations. luce 8659G000 10*J. For wliat is Circular Measure u- Scale? 78 REDUCTION. Note. A Circle is a figure bounded by a curved line, all parts of the curve being equally distant from the center of the circle, Th* aaoi is the curve which bounds the circle. An Aft is any portion of the circumference, as A B or B D. An arc equal to a quarter of the circumference, or 90°, is called a qmattnmU A luul line drawn from the center to the circumfcr- asCAorCB. A Diameter is a line drawn through the center and limited by the curve, as A D. Ex.1. How many second Ex. 2. Reduce 632931" to in 5s. 20° ;.r '? higher denominations. on OPERATION. 5s. 25° 48' 54". 60) 6 3 2 9 3 1 30 175° 60 10548' 60 )10548'+54" | l 7o°-f48' 5 s. + 2 5 60 Am. 5f.25 4ff 632934 . ;;. Be ;7"to second. Am. 1047847* \* how many cirrumt'er tc. ? 5. Ill o quadrants, 10° s' ."/' how many seconds? 6, Reduce 984627" to oaadram% degrees, etc. MISCELLANEOUS TABLE. 1IO. This table emhracos a few tnnis in common use, and may be indefinitely extended. 12 Single things make 1 Dozen. 12 Doz .. 1 Gross. . 12 Gross u 1 Great Gross. • 20 Single things u 1 Score. 24 Sheets of paper a 1 Quire. » 20 Quires u 1 Kcam. 196 Pounds m 1 Barrel of Flour. 200 Pounds a 1 Barrel of Beef or Pork. 109. What is ft Circle? Circumference? Arc? Quadrant? Radius? Diameter? 79 I. How many dosCU bottles each bottle holding lqt lpt. will be sufficient to 1m. ti! lqt lpt. of win- How man. : papf t in o nams, 18 quires, and 23 Ml-( I I.I.AM...I - I'.XAMll.l S IN RKOICTION. l. Reduce ¥?£. 14a, 6U 8qr, to farthings. i*. Reduce 18busa. 8pk. T«jt . lpt. to pints. 3. RedoOt 7t. 1 lewt. L>qr. 121b. 8../. (Mr. to drams, •1. Hbw many tons, etc., in 57 1 «'»'J2 ounces? 5. Reduce 1 •"> 7 7 < » IS seconds to minutes, hours, etc. «',. Reduce 84888 grain* to icruples, drams eta 7. Redno bXT to seconds. 8. Reduce 3m. 5 fur. 7ch. 2rd. 20 li. to links. '.'. Reduce 14 lb. 7oz. 15dwt 88gr. to grains. 10. Reduce 6ft 4J .')5 IB 6gr. to grains. 11. Reduce 27)48 equate inches to higher denominations. IS. Rednee 4 1 1 nails to qu art ers and yards. 18. li. .luce 7432 farthing-- to pence, etc. 14. Reduce 18469874 drams, Avoirdupois, to ounces, etc 15. Reduce 54896 grains to pennyweights, etc. 16. R e d nee ssq. m. 25a. 3r. 84sq. rd. to square rods. 17. Reduce 8c yd. 1787c in. to cubic inch 18. Reduce 4sq. yds. to Bquare Inches. 19. Reduce Igal. lpt. to gills. 20. Reduce 2wk. 6d. 8h. lCsec. to seconds. 21. Reduce 4m. 7mr. 39rd. to rods. L , L , . Reduce 3795 rods to furlongs, etc. 23. Reduce 17yd. 2qr. 3na. to nails. Reduce 10881 links to miles, furlongs, etc . Reduce 6598 pints to quarts, pecks, etc Reduce 4868294" to higher denominations. 87. Reduce 4680 gills to higher denominations. 28. Reduce 195261 cubic inches to feet and yards. Reduce 310556 square rods to rood-, acres, and miles. i:. This subject will receive further attention in the atti< lea on Frao- tious. 80 DEFINITIONS AST) LAL PRINCIPLES. DEFIXITIOX8 AND GENERAL PRINCIPLES. 111. All numbers are even or odd. An Kvi:n NuMBJtB is a number that is divisible by 2 (Alt 71); as 2,4,8, It An Odd Number is a number that is not divisible by 2 ; as 1, 3, 5, 11, 1.'. 112. All number* are prime or composite. A Pkimi: Ni m: number that is divisible by no whole number except its, r/ as 1, 2, 3, 5, 7, 11, 19. Nou 1 Tuo is the only even prime number, for all even numbers art divisible by 2. Xotk t, Two numbers arc mutually prime (i. e. prime to each other) when no whole number but one will divide each of them ; thus, 8 und 9 are rau- tually prime, although neither 8 nor 9 is absolutely prii: A ( iMFOfi] i I NiMiiKR is a number (Art. Gl) that is divisible by other nun. i 0M| thus, G is oomp il is divisible br 3 and b/3j 1- b compotito» bncigm is OOBlpO iuse it is divisible by ."> and ."). Note 3. A composite BMtbtf that is composed of any number of equal factors is called a power, and t ;.ic tailed the roots of the .\ huh canals 3 X 3 is the ser otul power or square of 3, and 3 is the second or square root of 9 ; CI, wh; 4X4X4, is the third power or cube of 4, and 4 is the third or cube root of 64. ; | 4. The power of a number is usually indicated by a figure, called an index or exponent, placed at the n\>ht and a little al>ove tho number; thus, the second power or square of 4 is wriucn 4*, which equals 4 X 4 = 16; tho tit ird power or cube of 4 is 4 8 , which equals 4 X 4 X 4 = 64. Note 5. A rtxf may be indicated by the radical tig*, -J ; thus, «/9 indi- cates the second or square root of 9, which is 3. So 8 >/8 indicates tin- third or cube root of 8, which is 2. The s-fare root of a number is one of its tico nutors ; the cu!>e root is one of die three equal factors of the number. I 6. Every number is both thejir> I thejirst root of V 111. What is an Even Number? An Odd Number? 113. A Prime Number? What is the only even prime number? When are numbers mutually prime? : is a Composite Numbi : . f I ' How is a power indicated? A root? A number is what power of itself ? What root? PUB. 81 roinra N 113. The I\\< ; number are those numbers u 1 continued product u the number; thus, 8 and 7 are the factors Of J 1 ; o and 6, or 3, 8, :i:id 8 ;iiv the factors of 18; etc :v iiuiiiImt is I t'.u t..r of itself, the other fm-tor being I. Th« prim mdon of a number are those prime numbers continued product is the number ; thus, the prime fecton Of 12 ;nv 2, 2, end 8 ; the prime f'aetors of 3C are 2, 2, 3, and tc Note 2. Since 1, as a factor, is useless, it is not hero enumerated. 19 1. To factor a Dumber is to resolve or separate it into its meters. Id resolving a Dumber into its factors, The following facts will be found convenient : (a) Every number whose unit figure is 0, or an even number, u, and .*. divisible by 2. (b) Any number is di\ isilde by 3 when the smn of its digits (Art. 7 ) is divisible by 3 ; thus, 4257 is divisible by 3 because the .Mini of it> digits, 4 + 2 + 5 + 7 = 18, is divisible by 3. (e) Any UQmber is divisible by 1 when 1 will divide the num- qprODBOd DJ me tl/BO righl-Itaud figures ; thus, 4 will divide oL\ .-. it will divide lo.Vl. (d) Any number whose unit figure is or 5 is divisible by 5 ; I, 1740, 85, 8 1075, etc. (e) Any eri n number which is divisible by 3 is also divisible by C; tin: divisible by 3 and .*. by G. Note 1. For 7 no general rule id known. (f) Any number is divisible by 8 when 8 will divide the num- by the three rigid-hand fgures ; thus, 8 will divide . . it will divid- 113. What are the Factors of a number? Is a number a factor of itself* What are \)w prime factors of a number? 114. What is it to factor a num- H 4? 0? 61 YVhut is said of 7/ I . aiubor ia divisible- ' 82 DEFINITIONS AND <.iL.NJiiL.VL PRINCIPLES. (g) An)' number is divisible by 9 when tlie nM of its digits is divisible by 9 ; thai, 7140 is divisible by B the sum digit*, 7 + 1 + 4 + 6 = IS, u divisible by 9. (h) Any Dumber ending with is divisible by LO. (i) Any number is divisible by 1 1 when the sum of the digits in the odd places is equal to the sum of the digits in the even place- ; llso when the difference of these sinus is divisible by 11; thus, 8129, in which 9 + 1 = 2 + 8, is divisible by 11; also 5280714, in which the sum of the digits in the odd places, 4 + 7 + 8+ 6, differs from the sum of the digits in the even places, 1 + + 2, by 22, a number divisible by 11. (j) Any number divisible by 3 and also by 4, is divisible by 12 ; and, generally, any number that is divisible by each of several numbers that are mutually prime, is divisible by the product of those numbers; i: divisible by 2, 3, and 7, separately, and .*. 84 is divisible by 2 X 3 X 7 - 108 is divis- ible by 4 tad 9, and .-. by 4 x 9 = Note 2. Every jtrime number, but 2 and 5, has 1, 3, 7, or 9 for its unit figure. Por further aid in determining t! ..; of numbers, we the follow TABLE OF TRIMi: NUMBERS FBOM 1 TO 997. 1 41 101 167 313 467 j r 911 2 103 17: 1 . 57J 647 919 3 L07 i;.' 25 1 577 929 5 109 181 L19 751 937 7 113 191 421 757 853 941 11 Gl 127 193 349 l.il 599 701 857 947 13 G7 131 197 271 3,33 859 17 71 137 L>77 359 521 863 19 73 211 281 613 691 787 877 23 149 283 373 541 617 701 881 29 83 151 227 293 457 017 G19 883 983 31 89 307 383 4G1 557 C31 719 811 887 991 37 97 233 311 389 4G3 5G3 641 727 821 907 997 H4. \That number is divisible by 8? By 10? 11? 12! General principle? i fanaarsM, 83 llt*i. A m ii something to be domj or, it U ■ • tion which requ Sutisfiof ■ problem con- ■ •?' tin- operations nece^ary for finding the answer to the question. To anVi ■ problen i the operations for findio s*er. 110. PbOBLKK 1. To resolve or separate a number its prime factors : fa the given number by ang prime number greater titan one, that will j prime n um b er greater than one that will divide it, oati m wi ftfl //<« nl m prime. The several divieart and /<«; quotient will be the prime factors taught, . 1. What arc the prime factors of 30? Ans. 2, 3, and 5, on It is immaterial in what order the prime fac- o \ j I tors are taken, though it will usually be most convenient to take the smaller factors first 5 2. What are the prime factors of 24? Ans. 2, 2, 2, and 8L .".. Resohre M into its prime factors. Ans. 2, 2, 3, and 7. 4. Resolve 375 into its prime factors. Ans. 3, 5, 5, and 5. 5. What an' the prime factors of 34C5 ? • ".. What an- the prime factors of 19800? 7 \\ : • are MM prime factors of 1 140? 8. What arc the prime factors of 3150? 9. What are the prime factors of 2310? Id. What arc the prime factors of 1728? 11. What are the prim.- fedora of 1800? 12. What arc the prime factors of 2448? L& What are the prime factors of 4824? 1 1. What are the prime factors of 3048? 16, What are the prime factors of 8696? 16. What are the prime factors of 72G4? 17. What are the prime factors of 507.3 ? 115. What is a Problem' The snlutim of & problem? What is it to $olv* » problem* 11G. Jwule for fiuuiiitf th« prim* factors of a b umber f 84 IIMTIUNS AND GENERAL I'KINCII'LES. 117. If a number has composite factors, they may be found by multiplying together two or more of it- pf -s; thu<, the prime factors of 12 are 2, 2, and ;j. and the composite factors of 12 are 8 X 2, 2 X 8, and 2 X 2 X ^, i. c. the composite fac- tors of 12 are 1, 6, and 12. CKKATEST COMMON DIVISOR. UK A Common Divisor of two or more numbers is any number that trill (/icicle each of them without remainder ; thus 8 is a common divisor of 12, 18, and 30. 119. The Greatest Common Divisor of two or more numbers is the greatrst number that will di\ ide each of them without remainder; thus 6 is the greatest common divisor of 12, 18, and 30. Note. A divisor of a number is often called a measure of the number, also an aliquot part of the number. 120. PBOBUU 2. To find tb -t common divi- sor of two or Lumbers. Ex. 1. What il ; lIUBOa dfrrioorof 18, 80, and 48? km. 2x3 = 6. KW. We see that 2 and 3 are 18 = 2X3X3 fact- to all the 30 = 2x3x5 numbers, and, furthermore, 48 = 2x3x2x2x2 the? ire die only common factors; hen<-e their prod- 2x3 = 0, is the greatest common divisor of the gives numb' r -. 2. What is the greatest common divisor of GO, 72, 48, and 84? Ana. 2X 2x8 = 12. orEn.\TioK. Although 2 is a factor GO = 2 X 2 X 3 X 5 more t/ian twice rn some of 72 = 2X2X2X3X3 the given numbers, y 48 = 2 X 2 X 2 X 2 X 3 it is a factor only twice in 84 = 2X2X3X7 others, we are not at liberty to take 2 more than twice 117. Composite factors, how formed? 118. What is a Common Divisor? 110. Greatest Common Divisor? Other Banff for divisor/ DEFINITIONS I ffBRAL PRDICBFUHI. 85 in finding the | mm divisor. The same remark np- j.lio to ofher metoru, Heme, i 1. nVfOJM todl numher into its prime factors. and the continue! product of all the prime factors that are common to all the given numbers will be the common divisor sought. 3. What i> the greatest common divisor of 24, 40, 64, 80, 9G, Lf0,and i Am. 2X 2 X 2 = 8. 1. Find the greatest common divisor of 15, 4."), 75, 105, 135, 150, and MO. Ana. 15. Find the greatest common divisor of 25, 45, and 70. Ana. 5. C. Find the greatest common divisor of 24, 3G, and (5 1. Am-. 1. 7. Find the >mmon divisor of 24, 48, 72, and 88. 8. Find the .lninon divisor of 45, 75, 90, lo\"), 150, and 180. '.>. I have three rooms, the first lift. 3in. wide, the second 15ft. 9in. Wide, and the third 18ft. wide; how wide is the widot carpeting which will exactly fit each room? II<>w many idtha will be required to cover each room? 1st An*. 27 inches. 191. When the given numbers are not readily resolved into their prim.' factors, their greatest common divisor may be more ea-ilv found by Kri.r. -. Divide the greater of two numhers If) the less, and, if there be a remainder, divide the divisor In/ the remainder, and continue dividing the last divisor by the last remainder indil not/t- ing remains ; the last divisor is the greatest common divisor of the two numbers. If more than two numbers are given, find the greatest dfa . \hen of this divisor and a third uomher. and so on until all the numbers have been taken ; the last divisor will be the ■r sought. Kir lindingthe greatest common divi.«or of two or more number** 181. Second rul« for finding gieakst common divisor? 8 86 DEFINITIONS AND GENERAL PRINCIPE: 10. What is the greatest common divisor of 11 and 20? OPERATION. 14)20(1 14 6)14(2 12 Ans. 2)6(3 "o Before explaining this operation, four principles may be stated, viz. : (a) Every number is a divisor of itself, the quotient being one ; thus, 3 is contained in 3 once; 7 in 7 once. (b) If one number divides another, the 1st will divide any number of times the 2d; thus since 3 divides 12, it will divide 5 times 12, or ant/ number of times 12. (c) If a Dumber divides each of two numbers, it will divide their sum and also tl < ncc ; thus, since G is contained Jive times in 80, and twice in 12, it is contained 5 — |— 2 = 7 times in 30 + 12 = 12 ; and ."> — 2 = 3 times in 30 — 12 = 18. (<1) Not only will the common divisor of two numbers divide their difierenee, but unless one of the numbers is a diviSOf Of th divide what remains after one of the num- bers has been taken from the other as many tim« liblej thus, the gr a nt e e! divisor of 6 and 22 will divide 22 — 3 x 6 = 4. 12*}. It may now be shown, 1st, that 2 is a common divisor of 1 \ and 20, and 2d, that it i> their (jreatest common divisor. First, 2 divides 6, .-. (Art. 121, b) 2 divides 6 X 2=12, and (Art. 121. e) 2 divides 2 + 12 = 14; again, since 2 divides 6 and 1-1 (Art. 121, e) it divides 6 -f- 14 = 20 ; i. e. 2 divides boih 1 1 and 20. Second, The greatest divisor of 14 and 20 (Art. 121, c) must divide 20 — 14 = 6, .\ it cannot be greater than 6; again, the greatest divisor of 6 and 14 (Art. 121, d) must divide 14 — 181. First principle? Second? Third? Fourth? 123. Explain why 2 is a common divisor of 14 and 20. Why it is their greatest common divisor. :.ni:kal i-kincii'LES. 87 G X 2 s= 2, .'. :' :nmon divisor of* 14 and 20 COSUUH exceed 2, ami, ai it ha- been i»i«-\ iou-1 v shown that 2 is a divi.-or ot' 1 J a:i.l 20, ft U (**• r (jreatest conn/ion i/irisor. imilar explanation b applicable in all oi 123. It will be seen that, in finding the common divisor of 1 20, \\i are led to lind the divisor of G and 14, then of 2 ; i. e. in any example we geek to fiml the measure of the remainder ami divisor, then of tin- next remainder and divisor, on, until the greatest measure of the last remainder ai '1 the divisor which gave that remainder is found, and this measure will he the greatest common divisor of the two given numbers. Thus the question becomes more and more simple a- each suc- tep in the operation is taken. 11. "What is the greatest common divisor of 34.32 and 4760 ? opehatiox. The plan of the operation in Ex. 10 requires more and more time than this in 1 1, though the principle and the reasoning are pre- cisely the same in both. In Kx. 1 1 we first divide. 4760 by 8432, and obtain 1 for quotient and 1328 for remainder; then divide by 1328, obtaining 2 for quo- tient, and 77G for remainder ; and so proceed, dividing the last divisor by the lasi mainder, as directed in Rule 2, until the remainder : The hivt divisor, 8, is the common divisor of 3432 and 47G0. 12. What If the greatest common divisor of 1430 and 3549? Ans. 13. 13. What is the greatest common divisor of 3G40 and 5733 ? 1 I. What is the greatest common divisor of 1440 and 86 What is the greatest common divisor of 2520 and G2J7 ? 2656 Quotients. X 1 = = 2 X X 1 = = 1 X X2 = = 2X X 6 = = 2X ■ 4760 3432 1328 776 77G 552 448 208 104 96 8 1G 1G 193. Explain the opuratiwu lu Ex. 11. 88 DEFINITIONS AND GENERAL PRINCIPLES. 16. What is the greaftaal common divisor of 1G, 21, and 3G ? mm on i:a riON. 8ECOND OPERATION. J 6 )2 1 (1 24)8 6(1 l B 24 8)16(9 12)24(2 ] 6 2 1 Again, 8)3G(4 .In, 12)1G(1 32 1^ Ans. 4)8(2 Ans. 4 ) 12(3 & il In wiring Kw I6j We tirM find the divi-or of 16 and 24, viz. I then find the divi-ur of 8 and 3G ; or first Bud the di- risor of 24 and 12, and t 1i*-ti < >t* 12 Hid 16 j <>r we might firrt find the divisor of 16 and 3G, and then of that divi- ■nd 2 i. 17. What is the gr otto a t common divisor of 84, 9G, 111. and 1 S. What i> the gWHiO B t common divisor of 77, 105, and 140? 19. What is the greatest common divisor of 9 and 16? Ans. 1. What if Um groataat common divisor of 9, 12, and 20? LEAST COMMON MULTIPLE. 1*2 1. A Multiple of a number is any number which is divtti b l e by that number; thus, 1"> is a multiple of 5 and also of 3 ; 21 is a multiple of 7 and of 3. Note. Every number is both a divisor and a multiple of itself. 125. A Common Multiple of two or more numbers, is any number which is divisible by each of the given numbers ; thus. 48 is a common multiple of 4, G, and 8. 193. How is Ex. 16 solved ? 124. What is a Multiple of a number? 125. A Common Multiple of two or more numbers? B&AL PRIN0IP1 126. The Li.ast Common IfUtTtFLl of two or mmv mun- is the least number that II divisible by . adi of the . number-; thttS, 2 1 if the least common multiple of -1, C, and 8. n. There is no such thing as a least common divisor, or g] in multiple. 1*J7. Problem 3. To find the least common mul- tiple of two or more numbers. 1. What is the least common multiple of 20, 24, and 36 ? An-. 2X2X2X3X3X5 = 360. operation. Since 3 GO contains all the 20 = 2 X 2 X 5 factors of 26; 84, lad 86, re- 24 = 2X2X2X3 spectively, it, evidently, is di- 36 = 2x2x3X3 visible bv each of those num- bers. It is also evident that no number less than 360 will contain 20, 24, and 36, tor if one Of the 2's in the common multiple were omitted, it would not contain 21 ; if one of the 3's, it would not contain 30; and if the 5 were omitted, it would not contain 20. Similar reasoning applies in all examples. Hence, Kile 1. Resolve each number into its prime factors, and the nied product of all the different prime factors, each taken the greatest number of times it occurs i/i either of the given num- bers, will be the least common multiple. 2. What II the least common multiple of 12, 16, 20, and 30? Ans. 2X2X2X2X3X5 = 240. 3. What is the least common multiple of 22, 33, and 55? 4. What ia the least common multiple of 16, 36, 40, and 48? What ifl the teas! common multiple of 20, 30, 50, and 80? What is the least common multiple of 15, 25, 45, and 7. What is the least common multiple of 85, 50, 75, and 8. What is the least common multiple of 21, 36, 48, and 64 ? :>. What is the least common multiple of 72, 80, 81, and 96? 10. What is the least common multiple of 42, 49, 72, and 88? The Least Common Multiple? May numbers have a least common diet' visor? Greatest common multiple.' 1SV* liulc for finding the least common multiple? Keasou? 8» 90 DEFINITIONS AND GENERAL PRINCIPLES. 1£8. The same result is sometimes more easily attained by Rule 2. Having set the given numbers in a line, divide by any runn: uumbtr (hat will divide two or more of them, and set tin' quotients and undivided numbers in a line beneath ; proceed with this line as with the first, and so continue until no tico of the fog cam be divided by any number greater than one ; the con- tinued product of the divisors and numbers in the last line will be illiple sought. The second rule may be illustrated by the example already employed in explaining the first rule, viz.: What i.s the least common multiple of 20, 24, and 36? Ans. 2X2X3X0X2X3 = 360. operation. If the process by the 1st rule be examined it will be seen that the fac- "' ) fo 1~2 To* , " r ~ ' s * u,m, l ' ,mi, '* i m ,M< -' .-' Vt n - - imm' ! M 8 i- taken but 3 o ) 5, 6, time- in finding the multiple, it i| re- 5 ? 2, 3 jeeled 1 times. l>y the 2d rule, a! i- rejected 1 times, viz. twice in the 1st division by 2 and twiee in the 2d di\i>ion by 2. The learner may think ted thrcf times in each of the two fir-t y link 2. 5 = 5* 8 ) 1 , 1 6, 2 4, 3 2, 4 8 1U = 2X2X2 X 2 B 8,12,16,24 LM = 2X2X2X3 ; ' ' ' ' ^2X2X2X2X2 £ > 5 > 4 ' b ' °» 1 l 48 = 2X2X2X2X3 2 ) 5, 2, 3, 4, 6 3 )5, 1, 3, 2, 3 5, 1, 1, 2, 1 »^ . — . 138. Second Rule? Explanation? i.UAL l'KIX< Il'LES. 91 ; in. iplo, wliich Ls tho samo in tho two rules, is most rcudily .ved by the BrM operation. What i* the least common multiple of 30, 40, 45, ami 75? 18. What i- the smallest Rffl of money with which I can buy ach, cow- /h, QT iheep at $8 each, using tame sum in each case? Ans. $G00. 1 1. 1 have t wine measures; the first bolda 1 quarts, the and 5 quarts, the third 6 quarts, and the fourth 8 quarts ; what of the smallest cask that can be exactly mea-und by means of each of these measui -. 120 quarts. |5. What is the least common multiple of 10, 15, 45, 75, and In BoWlDg Ex. 15, it is evident tliat 10, 15, and 45 may at on nek OUt, for each of these numbers is a in< a- l .-. whatever multiple of 75, and 90 is found, it, inly, mint be a multiple of 10, 15, and 45; hence, the question fo reduoed to this: What ■ the least common multiple of 75 ami 90? Many other abbreviations of this and other rules m:iv be p ffhcSX l l, but a delicate perception of the relations of numbers, and a skillful npplica- tlOfl of principles, will much mere faeilitate the progress of the learner than uux let of formul rules. (a) If the numbers are prime, or even mutually prime, their product is their least common multiple. 10. What is the least common multiple of 9 and 10? Ans. 9 X 10 = 90. 17. What is the least common multiple of 8, 9, and 25 ? (b) The least common multiple of two numbers is equal to their product divided by their greatest common divisor. 18. What is the least common multiple of 12 and 20? The amnion divisor of 12 and 20 is 4, ami The least common multiple is 12 X 20 -f- 4 = GO, Ans. 19. AVhat b the least common multiple of G3 and 72? < onunon multiple of 33 and 77 ? w oolvcd? What of other abbreviation*' locust common multiple of mutually prime numbers? Of tico numbcra? !>- COMMON rKACTIOMl COMMON FRACTIONS. 129. A Fraction is an expression representing one 01 more of the equal parti of a unit. Note. A unit, or any other whole number, is often called an Integer; > called an Integral or Entire dumber. 130. A Comox or Vuloajb Fraction is expressed by two Homo above and the other below a line; thus ^ (one half), | (two fifths), etc (a) The number below the line shows into how many parts the unit is divided, and is called the Denominator, because it ninates or gives name to the part> ; thus, if a unit is di\ into 3 equal part-, each part is one third; if into 8, each part is one eighth ; etc? (It) The number above the line is called the Numerator, rates or numbers the parts tc (<•) The numerator and the denominator are the Terms of the fraction. 131. A fraction is nothing more nor less than uncr- l division, i. e. division indicated but not performed, the fwmerator being the di tnd the tfar the divisor. Ilence, (a) The value of a fraction is the quotient of the nu- merator, divided by the denominator ; thus, ^ = 12 -f- 4 = 3 : and. .-.. (b) Any change in the numerator causes a like change in the value of the fraction, and any change in the denomi- nator causes an OPPOSITE change in the value of the fraction (Art. 84). These principles are developed in the following Problems. 129. What is a Fraction? Other names for a whole number? 130. A Com- mon Fraction, how expressed? Number below the line, what called? Why? Number above, what called? Why? Terms of a fraction, what? 131. A frac- tion, what is it? Value of a fractiou? What follows? com 93 132. A Proper Pa \< HOI b OM "hose numerator is less than the denominator; a-. v. fo fa. 133. An [MPBOPBB I kaction is one whose numerator f/u, ils or exceeds its denominator j a-, ;}, i. >,, '{']. An improper fraction equals or exceeds a unit; hence its name, DfPSOPEB Hon. 13 I. A SrjfPLl I i: W riOH has but one numerator and one tinator, an 1 i< either /jrojoer or improper 13»>. A ( loXPOl "N"i> 1'i:\mina- lof ■ drriior (Art. l.si ). the reduced to an equivalent whole or mixed number by the following Kite. Divide the numerator by the denominator ; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quot % Etednoe (| tO I wholo or mixed number. Ans. 3, 1 ,. Bedooe H t<> ■ whole or mixed number. Ans. 3. 4. Reduce JJ^ to a whole or mixed number. Ans. 2}f. 5. I _.•;*• to a whole or mixed number. Ans. 26g* ff . 9. Reduce *£. 7. Reduce f\ 10. Reduce §|. 8. 1 11. Reduce i£f&. Problem "-. 141. To reduce a fraction to its lowest terms. !. Reduce j| to its lowest terms. Ans. J. Dividing both terms of a frao run omunov. tionby any number does not alt ei 3| = }} = 3, Ans. the value of the fraction (Art. 84, b, and 131) ; .'. dividing ead| term of || by 8 L r ivrs the equal fraction M ; then dividing eacjb term of this remit by 1 gaVee ,\ and m 3 and 4 are mutually prune (Art 1 12), |§, in it- bweet terms, equals f. In this operation both terms of bkcov :iok. the traction || are divided by their 12)5§ = 2> An<. itest common divisor, 19 (Art. 119), and thus the fraction is re- duced :tt oner tO LtS lowest tflfflDS* 11cm i to. Rule for reducing an improper fraction to a whole or mixed number? 96 COMMON FRACTIONS. Rule 1. Divide each term by any factor common to themj then divide these quotients by any factor common to them, and so proceed till the quotients are mutually prime. Or, Rule 2. Divide each term by their greatest common divisor. 2. Reduce §£ to its lowest terms. Ans. §. 3. Reduce $£ to its lowest terms. Ans. |. 4. Reduce fa to its lowest terms. Ans. \. 5. Reduce tW& to li3 lowest terms. Ans. ft. G. Reduce^. 11. Reduce i£ft. 7. Reduce $?. 12. Reduce ffo. 8. Reduce | 13. Reduce §§*. 9. Reduce £f £. 1 L. Induce ^%. 10. Reduce gj}$. 15. Reduce |||. Problem 1. 142. To multiply a fraction by a whole number. Ex. 1. Multiply ft by 3. Ans. ft or £. It is just as evident that 3 first operation, times ft are ft as that 3 times 2 ft X 3 = ft, Ans. cents are 6 cents, or that 3 timet S are 6 ; i. e. when the numerator is multiplied by 3 the fraction represents 3 times as many parts as before, and each part continues of the same size ; .*. the frac- tion is multiplied by 3. If the denominator is divided secon-d operation. by 3, the fraction represents just ft X 3 = $, Ans. as many parts as before, but each part is three times as great, and .-. the whole fraction is three times as great. Hence, Rule 1. Jfultiply the numerator by the whole number. Or, Rule 2. Divide the denominator by the whole number. Note 1. The correctness of Rule 1 is also evident from Art. 83 (a), and Art. 131. Rule 2 also depends on Art. 83 (d). ■ 141. First rule for reducing a fraction to its lowest terms? Second rule? RatMNfcl 142. First rule for multiplying a fraction by a whole number? Why? Second rule? Why? Another reason ? COMMON FRA< CIONS. 97 2. Multiply T \ bj 3. Ans. fa or $. ferable in this and all similar examples, because it ghrti tho faction In smaller terms. 3. Multiply 4 7 5 by 5. Ans. $. 4. Multiply i\ by 11. Ans. f or 1?. 5. Multiply yV by 4. 1 3 tX 4 — ^, by Rulel; or, VVX4 = |-,byRule2. Note 3. The first rule is preferable for this and all similar examples, ba- cause the second gives a complex fraction. 6. Multiply ft by 4. Ans. |f or — . 7. Multiply rf > by 6. Ans. $?. 8. Multiply ^ T by 4. Ans. §$. 9. Multiply fc$ by 3. Ans. §f. 10. Multiply 1$ by 5. Ans. f|. 11. Multiply ft by 4. 12. Multiply f s by 5. . Multiply rttrby 15. Ans. ^V 14 Multiply ^ by 15. 15 = 5 X 3. A X 5 = f ; and f X 3 = V, Ans. i 4. We may here, ns in whole numbers (Art. 61 ), use the factors of the multiplier, and in using these factors we may apply the 1st or the 2d rule, or both. 15. Multiply |i by G6. 66 = 6 X H- « X 6 = t| ; and « X 11 = W> Ans. 1G. Multiply || by 42. Ans. i|i. 17. Multiply v 6 \ by 84. Multiply iff by 44. (a) If we multiply a fraction by its denominator, the product will be the numerator. 19. Multiply | by 8. Ans. J X 8 = { = 7, by Rule 2. 20. Multiply || l.y 44 14». May the factors of the multiplier be used? What is the product if a frac- tion is multiplied by its denominator? 9 08 COMMON FRACTIONS. (b) To multiply a mixed number by an integer : Multiply the fractional part and the entire part separately, and add the products together ; or, reduce the mixed number to an improper fraction (Art. 139), and tJien nmlti]>ly. 21. Multiply 3$ b y 5 - Am. 19. First multiply $ by 5 and the product is 4 ; then multiply 3 by 5 and the product is 15. These partial products added give 15_|_4— 19 for the true product. Or, first reduce 3| to *£ and then multiply by 5 and the product is 19, as before. 22. Multiply 8? by 9. $X9 = 3£;8X9 = 72; tind 72 + 8f = 75$., Ant 23. Multiply 9W by 12. Am. 113 t \. 24. Multiply 18| by 20. 25. Multiply 23 £ by 7. Problem 5. 143. To divide a fraction by a whole number. Ex. 1. Divide § by 4. Am. jj or ^. It is just as evident that one fourth first operation of | is § as that one fourth of 8 cents | -i- 4 = § , Ans. is 2 cents, or that one fourth of 8 is 2 ; i. e. when the numerator is divided by 4 the fraction represents only one fourth M many porta as be- fore, and each part continues of the same size; .-. the fraction is divided by 4. If the denominator is multipled second operation. by 4, the fraction represents just as f -T- 4 = ^r, Ans. many parts as before, but each part is only one fourth as great y and .*, the whole fraction is only one fourth as great. Hence, Rule 1. Divide the numerator by the whole number. Or, Rule 2. Multiply the denominator by the whole number. Note 1. These rules may also be explained by Art. 83 (b) and (c). 143. How is a mixed number multiplied by an integer? Another way? 143. First rule for dividing a fraction by a whole number? "Why? Second rule? Why? Another explanation? COMM'»\ 99 2. Divide \} by 2. Ans. W h y Rule 1 I hi by ™* 2. initio. Why? i, Divide 1 1 by 6. Ans. 2 V 1. Divide H h Y n - Divide H 1} y 25 - j by 12. 7. Divide tf by 4. H-M = ^|,byRulel; or, Note 3. Tlic 2d rule is preferable in this example. Why T B, Divide \l by 5. An?. ^. 9. Divide fig by 11. Ans. fft- 10. Divide ]',' by 6. 11. Divide 3$ by 4. 12. Divide & by 20. 20 = 4 X 5. &-*-* = A, and ft -f- 5 = T * y , Ans. 4. See Art 142, Note 4. 13. Divide \§ by 85. 35 = 5 X 7. M* *=*V ■■* A-*- 7 = y?x» Amu 1 1. Divide $J by 18. Ans. W 15. Divide ^ by 14. Ans. fo. Divide a?± by 44. i To divide a mixed number by a whole number. 17. Divide 23$ by 1. Ans. 5f. Fir-t divide as in Art. 71, 4 ) 8 Ex. 35, and obtain the quo- Qoo,T7. . 3* Rem. 1 i, ' nf ' 'I; a »« l , ' 1 " remainder, 31. Then reduce 3£ to the 3 J = V» a 11 * 1 V + 4 = f, improper fraction, V-, divide ,•. 5-f-f = - r >.',. An* it by 4, and add or annex tlie raanltj |, to the partial quo- 5, and we have .">$ for the true quotient, 143. May tin- factor- <>f Hm dlllWt be used separately? A mixed number, now divided by an integer? 100 COMMON FRACTIONS. 18. Divide 27$ by 6. Ans. 4f. 19. Divide 17g by 9. Ans. 1|J. 20. Divide 65^ by 8. 21. Divide 5| by 7. 5i = 4£; ^--7 = fi,Ans. Note 5. In Ex. 21, the dividend is less than the divisor; hence the quo- tient is a proper fraction. 22. Divide 7^ by 9. Ans. f $. 23. Divide 5f by 11. 24. Divide $6| equally between 9 boys. Problem 6. 144. To multiply a fraction by a fraction. Ex. 1. Multiply $ by $. Ans. &. To multiply $ by £, 1st, * X 3 = f (Art 142, Rule 1) ; but the multiplier, 3, is 5 times $, .*. the product, $, is 5 times the product sought; hence, 2d, f -?- 5 = & (Art. 143, Rule 2) is the product sought ; i. e. ?X& = &. Hence, Rule. Multiply the numerators together for a new numerator, and the denominators for a new denominator. 2. Multiply y\ by $. An?. ^. 3. Multiply & by J. Ans. 4. Multiply f'by J z . Ans. AV 5. Multiply \% by -ft. 6. Multiply tf by £$. (a) To multiply by a fraction is only to multiply by the numerator, and then divide the product by the de- nominator. In Ex. 7 we multiply £$ by 5, and obtain *f- (Art. 142, Rule 2), and then ^ divided by 6 gives ? (Art. 143, Rule 1), the result sought. 144. Rule for multiplying one fraction by another? Reason? To multiply by a fraction, what is it? What principles in the operation in Ex. 7? COMMON FRACTIOH1. 101 Multiply H by f. 2 7 In t/tis rimpU operation it rm wiiole principle LINO. To i-'inrrl (i. <•. ftriJb owf, or reject) any fac- tor of a Dumber, is to divide the number by the rejected i'aetor; thus 85 is the saint* as 5 X 7, and if the 5 is canceled, there will remain <>nly 7, which is the quotient of 3o divided by J. 8. Multiply if by ft. 4 2 ** v **_ 8 A 5 9 The 8th example is solved on the same principle as the 7th. 12 X 14 ,. , . It may be written thus, — — -, which is the same as oo X *' 4 y 3 y 2 y 7 - - -, and then canceling 3 and 7, i. e. dividing both 5X7X9X3 numerator and denominator by 3 and 7 (Art. 84, b, and 131) we _ 4X2 8 5X9 45 Note. There can be no difficulty in canceling so long as we remember the simple principle, that it rests upon rejecting equal factors from dividend and divisor (Art. 84, b). The process is only to strike out or cancel the same factors from numerator and denominator, and it often saves much labor. It can be profitably applied whenever the product of two or more numbers ia to constitute a dividend, and the product of other numbers is to constitute a -, provided that there are equal factors in the dividend and divisor. Multiply || by f $. 2 5 In this example, cancel 23 4& &$ 10 with 46, giving 2 in the nume- * <20 = 1~7' S " rator; and th.-n cam-el 5 in L\> and 85, giving 5 in the numera- tor and 17 in the denominator. 144. Explain Ex. 7. On wh»t principle* does canceling rest? When should It U applied? 9* 102 COMMON FRACTIONS. 10. Multiply f| by ft. Ans. ft. 11. Multiply §| by §}. 12. Multiply $$ by J|. 13. Multiply tfft by «. (b) In canceling 3 and 5 in Example 14, we obtain the quotients 1 and 1 in the numerators, and whenever an entire term cancels we obtain 1 to place instead of the teen ctnodod; but since 1, as a multiplier or divisor, is valueless, there is no need of retaining it under any circumstances, except where all the numerators are canceled ; in such a case, 1 t* the true MMMT- ator, and must be retained. 14. Multiply ft by ft. 1 1 $ t> 1 K 5 4 15. Multiply }*t by ftV 1 m v n i 5 | 16. Multiply V by V- 5 4 17. Multiply f f by *fo. 18. Multiply || by ft^ 19. Multiply | J by Jf. Ans. 0. 20. Multiply ff by JJ. Ans. f> 21. Multiply ft^ by Jf. (c) To reduce a compound fraction to a simple one. 22. What part of an apple is $ of } of it? Ans. Jj. If £ of an apple be divided into 7 equal parts, one of those parts will be ft of the whole apple; and if } of £ is ft, then | 144. la canceling when should the quotient 1 be retained? COM lOo of $ will be ft, and 9 of ; » l will be Hi i. e. « ampmmdfi* >/e one by the rult- f>r multijilyint/ a frac- Multiply | by V). i. a. reduce | of £ to a staple frac- tion. 2 I. Redace 9 of § of \$ to a simpl.- fraction. Ans. }$g. Reduce 3 of ^ of \\ to ■ simple fraction. 16, What is I of g of ^ of f of $ of f of I of f ? £x|xfx44*« 27. 1 v< (luce § of f of -ft of I to a simple fraction. 28. Reduce $ of }? of tf of ft to a simple fraction. 29. What co.^t J of B yard of cloth at * of ■ dollar per yard? An-. I of a dollar. 30. If a man builds $ of a rod of wall in a day, how much will he build in £ of a day? 31. A man owning g of a farm sold g- of his share ; what part of the farm did he sell? (d) To multiply ■ whole Dumber by a fraction. At $8 a barrel what will ^ of a barrel of flour < Aih. nasi oMnunosr. If a barrel costs $8, then 1 4) $ 8, Price of 1 bid. fourth of a barrel will cost \ of ZTZ „ t c , 1 1 1 $8, viz. $2, and 3 fourths will Costof|bbl. |2=$6>Ana. $ 6, Cost of % bbl. SECOND OI M.KATION. ■.'.*«. A « , r»,,» ^ Q ^ . e .. , . If lbbl. costs $8, then 3 bbl. $ 8, Price of lbbl. ^ ^ 3 ^ ^ = §2 u ;mil sincv I of :; i.i)i. u the same 4 ) $2 4 , Cost of 3 bbl. of lbbl. we divide the cost of ~$~c, Cost of \ bbi. ■*•* *7* :m ; 1 ■• ,iml th ;'. "* of ^ of a barred, viz. $6, which is imc result as by the fir-t operation. 144. How in a compound fraction reduced to a pimple one? How many wayt to multiply an integer by a fractiou ? First method f Second • 104 COMMON FRACTIONS. 33. Multiply 24 by |; i. e. find £ of 24. Ans. 15. 34. If an acre of land costs $45, what will $ of an acre cost? 35. What is the value of £ of a buahfll of clover seed, hi ^7 per bushel? Ans. $5J, (e) To multiply a mixed number by a fraction or. mixed number: Reduce each factor to the form of a fraction and then multiply the fractions togeth • 36. Multiply 22 by 1$. 2|Xlt = V'X* = H = 4iJ > Ans. 37. What cost 2| yards of cloth, at %\\ per yard ? Ans. $3f. 38. What cost \\ cords of wood, at $6£ per cord ? 39. How many square rods of land in a garden that is G§ rods long and 5 J rods wide? Problem 7. 145. To divide a fraction by a fraction. Ex. 1. Divide \ by f. Ans. \%. To divide \ by $, 1st, § -r- 5 = ft (Art. 143, Rule 2) ; but the divisor, 5, is 7 times $, .*. (Art. 83, f ) the quotient ft is only } of the quotient sought ; hence, 2d, ft X 7 = Jf (Art. 1 42, Rule 1) is the quotient sought ; i. e. 1-^ = | Xi = H. Hence, Rule. Invert the divisor, and then proceed as in multiplica- tion (Art. 144). The rule may be otherwise explained as follow.- \ First, To divide by any number is the same as to multiply by its reciprocal (Art. 138). Thus, 12 -j- 4 = 3, and also 12 X J = 3. Again, $ — 4 = ^, and also ^ X i = ^J i e. dividing by 4 144. Rule for multiplying a mixed number by a mixed number? 145. Rula for dividing a fraction by a fraction? Reason? Second explanation? COMMON FRACT1<» 105 anor: thus, £ -f- $ = o -+- 11. Divi ft. Ans. I. 12, Divide ?$by ft. Ana, 12. 145. How is the dhrMra nominator* are alike? 106 COMMON FRACTIONS. 13. Divide j\ by yV Ans. }. 14. Divide )} by ft. 15. Divide |f by #f 16. Divide §£§ by f]f 17. Divide m b 7 Mf • Ans - HI = I- 18. Divide |}} by */&. (b) When the numerator and denominator of the divisor are respectively factors of the corresponding terms of the dividend, as in Ex. 19, it is best to divide numerator by numerator, and denominator by denominator. This mode is true in all examples but not always convenient. Why true ? Why not convenient? 19. Divide W by } Ans. ft. 20. Divide T f j, by AV 21. If | of a yard of cloth cost ^ of a dollar, what costs 1 yard ? 22. If I earn -ft of a dollar in g *f a day, what shall I earn in ] day ? 23. If I pay § of a dollar for J of a bushel of corn, what shall I pay for 1 bushel ? Ans. $1£. (c) To divide a whole or mixed number by a fraction or mixed number: Reduce divisor and dividend each to the form of a simple frac- tion, and then divide by the rule already gic> n. 21. Divide 82 by 3^. 8}-*-3} = V-*- j = j = 2J, Au<. 25. Divide 8 by 3 j. 8-s-8i = f-5-¥ = fX A = « = 2A»Ans. '26. When 3^ lb. of beef cost 43| cents, what is the price per pound? Ans. 12£ cents. 27. B traveled 19}£ miles in 5} hours ; how far did he travel per hour ? 28. B traveled 19} \ miles, going at the rate of 3§ miles per hour ; how many hours did he travel ? 145. Mode of dividing when the terms of the divisor are factors of the term* *>f the dividend? To divide a mixed number by a mixed number? srs. Iu7 Pkoblem 8. 110. To reduce a complex fraction to a simple one. a 1. The complex fraction | equals what simple fraction? operation required is only to divide a fraction by a frac- I tion ; thus, ! = £-j-f = $Xf = iM. Hence, IJn.i:. First, if necessary, reduce the numerator and denomu of the complex fraction each to a simple fraction ; then the fractional numerator by the fractional denominator (Art. 11.".)." Note. A complex fraction may also be made simple by multiplying each term of the complex fraction by the least common multiple of their denominators; thus, in Ex. I, the least common multiple of the two de- nominators, 4 and 7, is 88, whose factors are 4 and 7. Multiplying the numerator, J, by 4, gives 3 (Art. 142, a), and multiplying 3 by 7, the other factor of the multiple, gives 21 for the numerator of the reduced fractioa In like manner, multiplying the denominator, £, by 7, and that product by 4, gives 20 for the denominator of the reduced fraction. Ex. 2. Reduce —— to a simple fraction. •ft 3. Reduce j— to a simple fraction. Ans. \. 4. Reduce ^-. 8. Reduce -A 5. Reduce *& 9. Reduce **♦£!*. G. Reduce •£ 10. Reduce 6J 4| $ H. Reduce^. !»'''• l Dm reducing a complex fraction to a simple one? Keason? An* otber mode! \ 08 COMMON FBAOnOHfc 12. Reduce - to a simple fraction. 7 = $ -f- G = ^, Ans., by Art. 143, Rule 2 ; or, G | = | X \ = ^, Ans., by Art. 84 (a) and Art. 142 (a). 13. Reduce ^ to a simple fraction. g 14. Reduce - to a simple fraction. j=l x §= P Ans -> by Art - 115 ( c >- 15. Reduce *° f *° f * 0f * to its simplest form. Ans. 1. V ot t\r ot 4 ot I Problem 9. 1 17. To reduce fractions that have not a common denominator to equivalent fractions that have a common denominator. Kx. 1. Reduce % and (j to equivalent fractions having a com- mon denominator. Ana, 1} and \'{. operation. Multiplying both terms of each fraction 2 7 14 by the denominator of the other fraction o X =-= rr will not alter the value of either fraction f Alt. 84, a), hut it will necessarily make r o i- the denominators alike, for each new de- _ x - =. — nominator is the product of the two given ' 3 « denominators. Similar reasoning applies, however many fractions are to be reduced. Hence, Rule 1. MultijAtj all the denominators together for a common denominator, and multiply each numerator into the continued product of all the denominators, except its own, for new numer- ators. 147. Common denominator, how found by Rule 1? How the numerators? Explanation? COMM"X 1KACTI0NS. 2. Reduce J, J, and £ to equivalent fractions having i com- mon denominator! OPKllATIOX. 4x7x9=-"- -"*-' common denominator, 8 X 7 X 0= 189, 14 numerator, f>X4xO= ls,, < M Domerator, 1x4x7= 28, 8d Domerator; , h Md | = }||, igS, and #„ Ana, a Reduce 3, 3, and *}. Ans. £fo 133, and ^. i. Redoee {, 3, and ?. An 5. Reduce §, |, *, and J. Ans. ^| J, 338, kit ■«« G. Badnc tod ». 11. Reduce .*., |, <. and 1 \. 7. Redoee /•,. 12. Redoee }, ;■. i. and T 7 T . 8. Reduce fo t \, and T 4 j. 13. Reduce |, ;,. », and 9. Reduce §, $, and ,',. 1 L Redoee $\, t> 4 ., ;1 ',, and 10. Redoee ( l 5. Reduce ^, ^, 2 2 5 , and (a) The foregoing rule w ill always give a common denomina- tor, but not always the least integral common denominator; this, ever, may always be effected by Kile 2. Reduce each fraction, if ?iecessan/, to itt lowest (Art. ill). Find ike Uaet common multiple of the de- nominatort (An. 127) for a common denominator* Divide tin's multiple, by each given denominator* and multip ly tike a rt for )/< ir timni rators. Notk 1. Kadi of tli bonded OB the principle that multiplying l>oth terms of ■ fraction by tlie same numher does not alter its value. l & Redoee |, |, and 1 7 ,. operation i:v Tin: BBCOVP 1:11.1:. 2X2X3X2= 24, least e.»m- mon multiple of denominator-, ^X3= 9, 1st numerator, ^X5 = 20, 2<1 Domerator, ; X 7 = 14, 3d numerator; .-. h| 9 and &=:&,&, end \\, a i » 7 KuU« for finding the least common denominator? Kale for finding tu« numerators* Principle' 10 a 2)8' 6' V 12 2)4, 3, 6 8)2, 3, 3 110 COMMON FRACTIONS. 17. Reduce A, f, f, and |. Ans. |g, f g, ft Jg. 18. Reduce &, A> Ai and fo- ld. Reduce At A.M. "ndl J- Kote 2. The first clause of Rule 2 is omitted by many authors, but ita necessity is apparent from the following example : 20. Reduce f , A> and A to equivalent fractions having the least common denominator. Disregarding the first clause of the rule, we find 72 to be the least common multiple of the denominators, and the fractions §, A> and A» reduce to fa, fa, and fa; but, regarding the lint clause, we have |, A> and A = !> 4> and £ = A* i 4 s> an<1 which have a common denominator less than 7 $. 21. Reduce A> |, A. and $• Ans. $$, $&, J J, and J J. 22. Reduce f , fa, A> ^d A- 23. Reduce A. H» A. and fa. 24. Reduce iJ, JJ, A. ">d «. Note 3. In this and the following problems, each fraction should be in its simplest form before applying the rule. 25. Reduce § of J and -i * 5 s §of | = |; i^ = ¥-f.« = S; but § and ^ = A and ?£, Ans. 26. Reduce J of f, 2J, ^-, and A- 27. Reduce |, J of g , and A* Remark. The numerators, as well as the denominators, of fractions, may be made alike by reduction ; thus, § and £ are equal in value to fa and \$ ; also 4 and A = H an d H » a ^ s0 4, Ai an d $■ = |J, 31, and f | ; etc The process is simple, but of little practical importance, and therefore seldom presented in Arithmetic. 147. May the numerators of fractions be made alike.* How 7 \CTIONS. Ill KI.I.U 10. 118. To reduce a fraction of a higher denomination to a fraction of a lower denomination. 1 . Reduce J of a penny, to the fraction of a farthing. 1 pennj i< equal to I fiurthingt, so any fraction of a penny will be 1 lime- as great a fraction of a farthing; .*. £d.= 4 times 2. Reduce ^ of a shilling to the fraction of a farthing. La is equal to 12d., so ^s. = 12 times ^d. = }d., and id. = 4 times jqr. = f qr., Ans. Hence, BULB. Multiply the fraction by such numbers as are neces- sary to reduce the given to the required denomination. 3. Reduce J s s. to the fraction of a farthing. ^s. (= ^d. X 12) = Jd. ( = $qr. X 4) = W-, Ans. ; or, 7 X 12X4 _ 7X^X4 _28 36 W~3 3 qr -' AnS -' ** bet0re ' Note 1. The sijrn of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 8, M -i-n is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from fWtHffg 20 in 240, and the 3 from canceling 4 in 12. 4. Reduce 5 J 5 of a ton to the fraction of a dram. 7X20X*X25 X 16X16 44800 m~ ~w~i = "T- dr " Ans - 5. Reduce \\ of a rod to the fraction of a barleycorn. 11 6 10 X 16* X 12 X 3 _ 10 X 3 3 X tt X 3 __ 1980, . -$ix 2 ~ 7 b - c -> An8 - 7 n 2. In the first statement of Ex. 5, the 16 J, in the numerator, is equal to ^, and, in the second statement, the 33 is retained in the nuuiera- IOC in the dividend, and the 2 is put in the denominator as a fao. tor in the
  • . Reduce .,t 3 of an acre to the fraction of a square yard, 16. Reduce ^yd. of cloth to the traction of as inch. 17. Reduce ^circ. to the fraction of a second. 18. Reduce fy of a ton t<> the traction of an ounce. 19. Reduce 3f a shilling. Jtfqr. (= *£&. + 4) = {d (= («. ~ 1 1) — B ^8., Ans. ; or, . Reduce igfigr. to the fraction of a pound, Apothecaries! ht. Ans. T far 7. Reduce 1 § a gr. to the fraction of a pound, Troy Weight. 8. Redu to the fraction of a day. 9. Reduce ^in. to the fraction of a yard, Cloth Measure. 10. Reduce l§&see. to the fraction of a week. 11. Reduce ±9 2sf l "i. t0 tne fraction of a yard. 12. Reduce V^ links to the fraction of a furlong. 13. Reduce 'Y^yd. to the fraction of an acre. Ans. % ^ v . 14. Reduce J ffi seconds to the fraction of a sign. 1"». Reduce } Jj gills to the fraction of a gallon. Problem 12. 150. To reduce a fraction of a higher denomination to whole numbers of lower denominations. 1. Reduce J£ to shillings and pence. Ans. 3s. 4d. (= Js. X 20) = i^s. = 3 Js. ; again $s. (== $d. X 12) = 4d. ; .-. J£r=r3s. 4d., Ans. Hence, Krir. Reduce the gi ion to a fraction of the next lower denomination (Art 1 18) ; tkem\if the Jraetiem is improper, reduce it to a whole or mixed number (Art. 14<>). If the result is 150. Rule for reducing a fraction of a higher denomination to Integer! o/ lower denominations? Explain.' 10» 114 COMMON FRACTIONS. a mixed number, reduce the fractional part of it to the next lower denomination, as before, and so proceed as far as desirable. Note. If, at any time, the reduced fraction is proper, there will be no whole number of that denomination. 2. Reduce %\£ to whole numbers of lower denominations. J|£ (= if s. X 20) = f f s. = 4^8. ; fas. (= ^d. X 12) = $d., a proper fraction ; £ d. (= f qr. X 4) = 3qr. ; ,\ J J £ = 4s. Od. 3qr., Ans. 3. Reduce fa of an acre to lower denominations. . lr. 17rd. 18yd. 1ft. 50$in. 4. Reduce fa of a furlong to rod-, yard. Ans. J8rd. 3yd. 2ft. 5. Reduce $ of a week to days. C. Reduce §|$ of a rod, L ure, to yards, etc 7. Bedace HI8J °^ a cucumference t<> signs, etc 8. Reduce fa of a ton to hundred • 9. Reduce £||tt> to oun< < ~. drams, Bcrn| 10. Reduce ffflfacxrc. to ■ 11. Reduce }} of a civil year (865 days) to days, etc 12. What is the value of fa« 4 % of a pound Troy ? 13. What is the value of tf of a but 14. What is the value of £$ of a gallon ? 15. What is the value of fa of a pound. Apothecaries' Weight? 16. Reduce fa of a mile to furlongs, chains, etc. 17. Reduce ^ of a cord to cord feet, cubic feet, etc 18. Reduce fa of a yard to quarters, nail-, etc. Problem 13. 151. To reduce whole numbers of lower denomina- tions to the fraction of a higher denomination. Ex. 1. One farthing is what part of a penny? Ans. \. Since 4 farthings make a penny, 1 farthing is \ of a penny. 2. Six pence and 1 farthing are what part of a shilling ? 6d. -L- lqr. = 25qr; and Is. = 48qr. ; .*. 6d. and lqr. z= ||s., Ans. co.v 11' To determine what part one thing is of another, considered as .> or whole t!,in>j. the part is always made the numerator of a {•on, and the unit or whole thi/t;/ is j>ut for the denominator ; tfcos, the traction -i t expresses the part that o miles is of o miles. comparison can be made, the part and the whole most be of the same kind or denomination ; thus, :5 pecks is not g of 5 bushels, hut, reducing the 5 bushels to 20 pecks, we hav. :; - equal to J' of 20 peeks, i. e. *?$ of 5 bushels. Benee, Kri.i: 1. Unhirr the given quantity to the lowest denomina- ' /• a mssurofcr ; and reduce a unit of the higher denomination to the same denomination as the numerator, 3. Reduce Grd. 5ft. 9in. to the fraction of a furlong. Grd. 5ft. 9in. = 1257in. and lfur. = 7920in. .-. Grd. 5ft. 9in. = }$S& fur - = sV&far., Ans. 4. Reduce 7oz. 4d\vt. to the fraction of a pound. Ans. £. 5. Reduce 9 rods, 1 foot, and 6 inches to the fraction of a furlong. 9rd. 1ft. Gin. = 1800in. and lfur. = 7920in. ; .-. 9rd. 1ft. Gin. = ^§^fur. = ^ur., Ans. (a) In Ex. 5, Cin. = ift. ; lift. = JyA = ^rd. and 9-rVrd. = tj'Y'rd. = .Afur., Ans., as by Rule 1. Hence, Ivii.i: 2. ]>;>■■'>n, annex the to the given number of that denomination, and so proceed as far as necessary. 1. This rate is frequently preferable to the 1st, because it enables ■f, t<> use smaller numbers and giTSf the result in lower terms. l">l. !: ■ lor MdadBf the lower denominations of a compound number to a fraction of a higher .1. nomination ? Explanation? Principle? Second rule for reducing integers of lower denominations to the fraction of a higher denomi- nation? Explanation .' Why preferable to Rule 1? 116 COMMON F HAITI > 6. Reduce lr. 2sq. rd. 20sq. yd. laq. it. 72aq« in. to the fraction of an acre. Ans. x * 5 . 7. Reduce 4oz. Cdwt. 9ggr. to the fraction of a pound. Ana, Note 2. In Example 7, by Rule 1, reduce 4oz. 6dwt. 9$gr. to Jifths of a grain for a numerator, and lib. to fifths of a grain for a denominator. How shall it be done by Rule 2 ? Which mode is preferable I Wliv I 8. Reduce lpk. 3qt. lpt to the fraction of a bushel. 9. Reduce (>s. 20° 20' 30" to the fraction of a circumference. 10. Reduce lm. 2fur. llrd. 2yd. lit 2}b. c. to the fraction of a league. J 1. Reduce Iqr. 2na. ^in. to the fraction of a yard. 12. Reduce 3wk. 6d. Uh. 27m. to the fraction of a Julian year. 13. Reduce lqt lpt lfgi to the fraction of a gallon. 14. Reduce 4 cord feet, 12 cubic feet, and 1982] cubic inches to the fraction of a cord. Ans. g. 15 Reduce 3oz. 4dr. Itc lOgr, to the fraction of a pound. 1G. Reduce 4 fur. 6dL Sid 20li. to the. fraction of a mile. 17. Reduce tlcwt 1 1 lb. loz. 12fdr. to the fraction of a ton. 18. Reduce 3 bushels, 1 peck, 4 quarts, and 1 pint to the fraction of a bushel. Ans. *j / . Note 3. Sometimes, as in Ex. 18, the number called the part is greater than the unit with which it is compare iul to the unit. Problem 1 1. lotj. If numbers of the same kind are added together, their sum will be of the same kind as the numbers added; thus, 3 books -f- 4 books = 7 books ; 3 hats -\- 4 hats = 7 hats ; and for a like reason, £-[-$ = £; ^ _L- ^ = ^ etc., etc. (a) Numbers of different kinds cannot be united by addition ; thus, 3 hats -f- 4 books are neither 7 hats nor 7 books ; so £ -f~ % are neither $ nor $ ; but numbers that are unlike may some- times be made alike by reduction, and then added ; thus, l+t = tt + H (Art. 147) = «. (b) Again, 2bush. -j- 3pk. are neither 5bu»h. nor 5pkl ; but 2bush. == Spk., and then 8pk. -f- 3pk. = llpk. ; so Shush, -j- co.v 117 i\>k. nt neither }bush. nor 5 |»k. ; lmt Shush. = 8nk. (Art. 148), and then ;pk. -f- ^pk»= Vpk- H" To vdd fraction Ettn 'nee the fractions, if necessary, fret to the same ,'natio/i, then to a common d> . rhich write m 0/ the new numerators over the common denominator. 1. Add (■-, end ,\ together, Ans. kher. Ans. [g. 3. Add Jv, Jv, ,'V, and }? together. Ans. ?f = 2ft. 1. Add J and J together. Ans. V = *t = U- 5. Add ^, ^ 5 , -ft, and ^ together. Ans. 1$. C. Add ,. fa and /:, together. 7. Add §3, &, ^ Hi Mrf A together. 8. Add together ft, t 7 *, if, it, and ft. Ans. 2£. Add together 3 * ff , Jg, Jf, £g, and ft. 10. Add together gg, gg, gg, and |g. 1 1. Add together T | F , ft^, ^ft, T *w »"<* iW- 1 _'. Add together J, g, f , and £. Ana. 4$. 13. Add together gg, *g, *g, and gg. 14. Add together g, ft, and ft. -* § + ft + ft = it + fg + ig(Art. 147,Rule2) = = 1*1 Ans. 1"». Add together g and g. 3 + I = H + M (Art. 147, Rule 1) = §g = ljg, Ans. 16. Add together ft, |i tnd $. 17. Add together ft, ft, and §. + A + i = i + i + 3=* = li,Ans. 18. Add . and gg. 19. Add g of g to g of *g. } + * = J, Ans. 20. Add g of ?g to g of jjg. 21. Add ' S ' h to J of |. Ans. ft. to 8 x A- 15*. Rule for adding fraction*? Can nnlike number§ be added? Of what kind i» the sum of two or more numbers? 118 COMMON FRACTK' 23. Add }§. to $d. |8. + §d. = Vd. + H=Ud. + i8d. = «d. = 5A^Ans., or, f s. + §d. = f s. -f- As. = fgs- + ^s. = |^s., 2d Ans. = 1st Ans. 24. Add fgal. to £qt. Ans. f $qt. or fjgal. 25. Add together ^bush. $pk. and £qt. 26. Add together $ton fcwt. and $qr. (c) To add two fractions that have a common numer- ator : Multiply the sum of the denominators by either numerator, and place the product over the product of the denominators. 27. What is the sum of ? and | ? 1-1-1 — ^+1 — 15 3 3_3xl5_45 7~ i ~8~~7X8~56 ; •' , 7 i "8~" 56 ~56' AnS * 28. What is the sum of £ and £ ? £f = l/n A* 19 - 29. What is the sum of f and ft ? (d) To add mixed numbers : Add the sum of the fractions to the sum of the integers. 30. What is the sum of 3f and 4} ? |+|=»» + H==tt = lA5 3 + 4 = 7; .-. 3* + 4| = 7 + 1 A = 8ft, Ans. 31. What is the sum of 5|, 3$, and 12$£___JLns. 21 }f- 32. What is the sum of 18ft, 5|, and 24|? 33. What is the sum of 15$, 24, 7*, and ^ ? _ 34. What is the sum of 3^, 6&, 4^, and 8 35. What is the sum of £ of $ of 6}, ^, and 41 ? 36. What is the sum of |t, 3f , 6|, and J of J ? 37. What is the sum of 3 -ft, 4&, 8^, and 25 ? 38. How many are 8$ -f 3£ -f- 8£ + 14 ? 153. Mode of adding two fractions that have like numerators? Mode of adding mixed numbers? COMMON >NS. 119 Problem 16, Itl3. To subttf fraction from a greater : r. P repare the fractions as in addition, and then the difference of the numerators over the common denominator. 1. From A take tV A — A = A=i» An *- 2. From tf take J 7 . Ans. J 7 . 3. From J§ take fc 4. From £? take J$. 5. From || take J|. From ^ take 7. Take ft from £g. An?. ft. 8. Take §$ from |f. Take }| from §1. 10. Take fft from /, 11. From $ take §. (a) t — # =H~ « = A. Ans - ( See Art - 152 > »)• 12. From | tai. Ans. £$. 13. From ^ take ft. 14. From J | take |. « — § = §f — A = * An - 15. From ft take f. Ans. ^. 16. From \l take ^. 17. From ij take | LR From || take^. fi — A = A — A = AV — A 1 * = A 4 * = A. Ans. 19. From 3 7 3 7 ff take ^fo. A..<. |J. 20. From fft tak e ^. 31. From J of | take £ of *. |X*—*X* = f—f =»—»—«, Am. . From J of # take $ of -ft- Ans. ||. From ?of \$ take j| of [%. •ji. From | of >.■; take ft of From A of & take & of Of | take $ of vVof §. I for subtracting one fraction from another? Uow are toe fractions prepared in addition ' 120 COMMON FRACTIONS. 27. From ?| take If. Ans. U. ^ = ^ = a^-j-J^. = ^x 1 3 j = 5; Complex fractions 7 a \ reduced to simple g | = * = V + ¥ = i(ArLH5,b);j ones. f — * = « — « = «. Ans. 28. From ^ take ^|. Ans. H = 1 H- 29. From -f take |. 30. From }s. take £d. (See Art. 152, b). (b) |3. — Jd. = Vd— Jd. = JJd. — Ad.= jfd. = llJd.,AM. or, fs. — £d. = }s. — ^s. = ;£frs. — ,^8. = **&s., 2d Ans. 31. From §qt. take £pt Ans. ^.(jt. or l£pt 32. From § ton take $cwt. 33. From $ acre take f rod. Ans. i|f$a. or 67$§rd. Note. The answer to these examples may be in any denomination of the table. 34. From \ of a week take $ of an hour. (c) To subtract when the fractions have a common numerator : Multiply the difference of the denominators by either numerator, and write the product over the product of the denominators. 35. From £ take f. 1 1_ 8 — 5 __ 3 4 4_ 4X 3 _12_ 3 5 8~~5 X8~40 ; *'* 5 8~~ 40 ~40~"l0' nS * 36. From $ take yV Ans. ?f 37. From £ take $. (d) To take a mixed number from a whole or mixed v number. 38. From Ci take 6$ — 2$ = 4£, Ans. (See Art. 152, d). 153. Mode of subtracting when the fractions have a common numerator? COMMON FRACTIONS. 121 89. From 8^ take 2^. In Kx. 38, take $ from J, and 2 units from G units; but in 19 we cannot take ^ from -ft, .\ reduce one of the 8 units io \ {, and add it to the .,»,, making \ f, and then take the -ft from $ ;•. and the 2 unitfl from the remaining 7 units. 1»>. From 9£ take 3|. 9$ — 3$ = Si — 3| = 8|§ — 3^ = Hh Ans. 41. From 12$ take 4g. 42. From 9 take 5?. Ans. 8ft 43. From 8 take 2§. Miscellaneous Examples in Fractions. 1. Multiply j 3 r by 5. Ans. \%. 2. Multiply ft by 6. 3. Reduce £§ to its lowest terms. 4. Add 8fV to 6H- - il.tract I8)| from t5tf G. Reduce 23 § to an improper fraction. 7. Reduce K to a fraction wboM denominator is 27. 8. Reduce 9 to 6 fractional forms. 9. Divide £| by ft. 10. Divide V by V. 11. Divide W by f. 12. Reduce | dt' a day to hours, minutes, and seconds. 1 8. Eteduce !pk. 5qt. lpt. to the fraction of a bushel. 1 t. .Multiply ,S£by 10. 16 Divide 9| by l. ia Divide || by 9. 17. Divide 18 by $. 1 8» B 'A to a mixed number. 19. Keduee LJ|I to a whole number. Multiply || by | J. 21. Reduce $ of J of if to a simple fraction, in g. 133. Mode of taking a mixed number from a whole or mixed number 1 11 122 • FRACTIONS. '<. Reduce fo *> and $ to equivalent fractions thai haw common denominator. S I. Reduce \%, ££, and $J to equivalent fractions having the least common denominator. 25. Reduce ^ and § to equivalent fractions having a common numerator. 26. Reduce $, -ft, and $$ to equivalent fractions having the least common numerator. 4| 27. Reduce — to a simple fraction. 28. Add & to ft. 29. Divide ft by 4. 30. K< (luce § of a gallon to the fraction of a quart 31. Reduce $ of an hour t<> the fraction of a ireek. 32. Reduce f-° F jAr A° r v t0 its snnpkd 6nn- f of 3 of A of i 7 * 33. Multiply §£ by 33. 31. Multiply 25 by $. 35. Multiply 25 by jj. 86. Divide /■, by |. 37. Add §£, is., and $d. together. 38. Subtract $ of a gill from $ of a gallon. 39. Add fV, VV, fV ^3, and ^ together. 40. From §§ take §g. •11. Five gallons, 3 quarts, 1 pint, and 8 gills, are what part of 1 gallon ? (See Art. 151, Note 3). 42. Three pecks are what part of 3 pecks? Examples in Analysis. 1»*1. We analyze an example when we proceed with it, step by step, according to its own conditions, without being guided by any particular rule. Ex. 1. If 4 tons of hay cost $48, what will 7 tons cost ? Solution. If 4 tons cost Sis, then 1 ton will cost \ of $48, which is $1S : and if 1 ton cost $12, then 7 tons will 00 times $1*2, which is $84, An-. COMM<>\ 123 2. What k llM value of 12 Mm of land, if 8 MTM cost $81? Ans. $324. What i> the cost of 1G barrels of flour, if 3 barrels cost I. If a man can cut 8 cords of wood in 4 days, how much will M cut in 7 days? 5. If 1 ton of hay costs $15, what will 1, of a ton cost? Solution. One ton costs $15 ; /. J of a ton costs £ of $15 = $3, and $ cost 4 times $3 = $12, Att. f>. What is the value of $ of an acre of land, at $40 per MN ? Ans. $35. 7. If f» men mow 12 acres of grass in a dav, how many acres will they mow in £ of a day ? 8. II" a man cradle 18 acres of wheat in 9 days, how many ..ill he cradle in ~> d 9. Paid $6 for J of a yard of velvet ; what was the price ■ird? Solution. Since $6 were paid for £ of a yard, \ cost £ of $6 = $2, and .*. |, or a whole yard, cost 4 times $2 = $8, Ans. 10. If I of a yard of ribbon cost G3 cents, what will a yard co«t ? 11. If § of an acre of land cost $75, what is the price per acre ? 1 2. If 234 bushels of potatoes grow on $ of an acre, how many busbelfl will grow on an am? 13. If J of a farm cost $4300, what cost $ of it? rnoN. If J cost $1200, Iheli } costs $ of $4200 = $1400, and | cost 4 times $1400 = $5600. Now the whole farm 100, .-. \ of it costs \ of $5600 = $800, and $ cost b times $800 = $4000, Ans. II. If § of a cord of wood are bought for $3£, what will J of !•">. If ,;, of | ihSp aie W0H . what is the value of \ of h 16. If \ of the distance from A to B is 32 miles, what is ft of the fjjatance from A to B? 124 common nucnoHS. 17. If 3 men build # of a rod of wall in an hour, how many rods will 4 men build in G hours? 18. If 6 men can do a piece of work in 3^ days, how long will it take 4 men to do the same work ? 19. What cost 61b. of sugar, at 8$c. per lb. ? 20. What shall I pay for 16Mb, of rice, at 4c. per lb.? 21. Bought 41b. of raisins, at 12£c. per lb., and paid for them in Bggft, at 16§c. per dozen ; how many dozen did it take? 98. What cost 12£lb. of pork, at 6c. per pound? 23. If f of a bushel of wheat co>t $1}, what is the cost of 12^ bushels? 24. If 7bl)l. of flour ( what will 3.U)bl. cost? 25. If 2 1 cords of wood will pay for 27 gallons of molasses, how many cords will pay for 4 timet -" gallons ? Ans. 4 times 2^ cords, viz. 9 cords. 26. What cost 12) yards of silk at si | per yar.1 ? 27. How many times will a wheel that is 9 feet in circumfer- ence turn round in running 20.\ miles? 28. How many cubic feet in a box that is 6^ ft. long, 5£ft. wid •, and 3§ft. deep ? Ans. 117. (See Art. 104). 29. How many bottles containing 1^ pints each are requited to bottle 21 gallons of wine? 30. What costs a farm of 7o\ acres at $96^ per acre ? 31. If it costs $8| to carry 13cwt. 3qr. o| lb. 8* miles, how far Ban the same be carried for $16^? 32. Bought ^ of a 20-acre lot, and sold J of the part pur- chased ; how mueli had I remaining? 33. If 3 ^ bushels of oats will sow an acre, how many bushels will sow 7£ acres ? 34. A staff 3ft. long east a shadow £ of a foot at 12 o'clock ; what is the length of a shadow cast by a steeple 125ift. high, at the same time? 35. If a staff 3ft. long casts a shadow of £ of a foot at 12 o'clock, what is the bight of a steeple that casts a shadow 31|ft., at the same time ? 36. Sold a watch for $43?, which was I of its cost; what was its c<> 1 25 . many pounds of butter in 24 firkins containing 33.J, lb. Hid what IS it worth at } of a dollar per pound ? 38. If 6 mfl number, what is 5} times that number? Am. 44. \\ then ', is | of •*., which is 2, and | are 4 times 2 = 8. Since s i> the number, 5} timet the number will be 'VI timea 6 — l i, Ana* If 12 is $ of some number, what i< 7| times that number ': 40. Fifteen is § of how many times 10? .1. Analysis. If 15 i- |, then !; is ^ of 15 = 5, and | are 8 £mei 5 = 40. Now 40 is 1 times 10 ; .*. 15 is § of four times 1". A 11. Twenty-four is -ft of how many times 2? Ans. 22. Thirty-five is | of how many times 5 ? 43. Seven ninths of 72 are £ of how many times 7 ? Ans. 10. Analysis. Oi.v. :. : nth of 72 is 8, and # are 7 times 8 = 5G ; La |, thru I \< \ of 56, which is 14, and £ are 5 times 14 = 70. Now 70 is 10 times 7 ; .-. $ of 72 are § of *e/t times 7, Ans. 11. Three eighths of 40 are $ of how many times 5 ? Ans. 7. 45. Seven eighths of 48 are f J of how many times 8? 40. Six fifths of 30 are | of how many sixths of 24 ? Ans. 8. Analysis. One fifth of 30 is 6, and § are G times 6 = 36 ; if 36 is g, then \ is £ of 36 = 4, and | are 8 times 4 = 32. of 2 1 El 1. and 4 is contained 8 times in 32 ; .-. § of 30 are | of eight sixths of 2 I, An*. eighths of M are $ of how many thirds of \ 48. Four sevenths of 86 are y C) of how many eighths of 1" '< Of the inhabitants of a certain town, J are farmer-. | me- chanics jVj manufacturer-. \ students and professional men, and the remainder, cambering 246, are engaged in various occupa- tion^. What i- the population of the town? An- What would be the population of the town mentioned in Ex. 49, all the conditions remaining the same except that 240 shall 1 Ans. 1 C 4«>. 11* 126 COMMON FRACTIo 51. A certain room is 16£ft. long, 15ft. wide, and 9ft. high; how many square feet in the walls ? Ana, T>G7. (See Art 101). 52. What would be the cost of carpeting the room mentioned in ESx. 51, the carpet being lyd. wide, and costing $1£ per yd.? 53. A merchant bought 48^1b. of butter of one customer, 28$ of another, 25^ of another, and oGfa of another ; how many pounds did he buy, and what was the cost of the whole at I5& per pound? 54. In a certain school ^ the scholars study arithmetic, £ algebra, fa geometry, and tin* remainder of the lehool, viz. 14 scholars, study surveying; how many scholars are there in the school ? Ans. 84. 55. How many scholars would there be in the school men- tioned in Ex. 6 1, if only seven scholars studied surveying? 5G. A fox has 1 G rods the start of a hound, but the hound runs 22 rods while the fox runs 20 ; how many rods will the fox run before the hound overtakes him? 57. A fox has 18 rods the start of a hound, but the hound runs 25 rodfl while the fox runs 22 ; how far must the hound run to overtake the fox ? 58. A boy being asked how many doves he had, replied that if he had as many more, | as many more and G doves, he should have 5G; how many doves had he? 59. A boy being Mked Imw many lambs he had, replied that if he had twice a^ many more, | ftfl many more and 5£ lambs, he should have 30 ; how many lambs had lie ? GO. If 2 be added to each term of the fraction j, will the value of the fraction be increased or diminished ? Ans. Increased by fa. 61. If 2 be added to each term of the fraction $, will its value be increased or diminished ? Ans. Diminished by fa. 63. If 2 be added to each term of the fraction ■{*-, will its value be increased or diminished? Ans. Neither. What principle is involved in the last three examples? How would the values of the several fractions in the last three exam* pies be affected if 2 were subtracted from each term ? 127 A merchant owning § of a ship, sold $ of his shan for $3000; what was the value of the ship? Ans. $12000. 54. A «an tlo a piece of work in 6 days, and 11 in 12 days; in what time can A and B together do the work? A. B, an.l (' can do a piece of work in 1 day-; A and B lo it in 5 days; in what time can (' do it? Bought a pair of oxen and a hone for $180. The oxen of the price of the hone ; what was the price of the l.< Bough! a pair of oxen and a hone for S17o, and a n of the price of the bone. The bora ai much as Leo ; what was the price of the wagon? Ans. $45. Six men are to be clothed with cloth thai is Hyd. wide. Now if it takes 2§yi/it or sepuratri.r. pi before the decimal; the first figure at the right of the point is (ruths; the second, hundredths; the third, thousandths; etc.; thus, .6 = ^, .06 = T $s, .0QG = TJ foj 5y etc., the figures in the decimal decreasing in value from left to right, as in whole num- Art. 15). l",r». What is a Decimal Fraction? Decimal, from what derived? What i-< usually meant by the word decimal? 156. A Common Fraction, what is its denominator? Are the principles of common fractions applicable to decimals? 157. Is the denominator of a decimal usually expressed? 158. How is a decimal traction distinguished from a whole number? What is the first figure at the right of the point? Secoud? Third? !•?!). since whole numberi ind decimal fraction! both aae by tlu' Mime law fVdii) lrit to right, they may be r in the same example, and numerated a-s in the following NUMERATION TABLE. 9 a «r . Hi? w r— 3 I '•— • C» 2 » —IS ■ "= : -r ~ = ~ — S^-tSu i? / 3 *l *? 'S •§ ^ •§ c s - fi — A c 8 4 7 1. 8 8 8 8 7 2 8 4 3 2 1 6 1GO. A whole number and decimal fraction written together, m in the aboTe table, form a mixed number. The integral part is numerated from the decimal point toward the left, and the fraction from the same point toward the tight, each figure, both in the whole Dumber and decimal, taking i/s inline and value by its distance from the. decimal point, Hence, 101. Moving the decimal point one place toward the right, mul tipb'e i the number by 1<>; moving the point two places mul- tiplies the Damber l»y 100, etc, AJfO moving the point one to the leftjdwidet the number by lOj moving the [joint two divides by LOO, etc. 1<»£. la reading a d.cinial. we may give the name to each figure separately, or we may read it as we read a whole Dumber, and (jirc tin- name of the right-hand figure only; thus, the expres- sion £8 may be r.ad {- and r §y, or it may he read ffc, for ^ and , 150. "^ imcnitiuii liitilo. 100. What h a mixed Dumber? Wliic-h wejr to the Integral part numerated 1 VTbJeb way tbe decimal? What M of a Iftfre I ir, i. i orina; tbe d< i<>i»it :.» t'i<- right eflbel f a namber? How mortog it to tbe i 1 1 . _' In decimals, as in whole numbers (Art. 1C), ciphers arc used to till placet that would otherwise be vacant. 4. Write the decimal >ix hundred and forty-one thousandths. Decimal five hundred and eighteen teu-thou.-andihs. •'.. Bight hundred ami decimal eight thousandths. 7. Six thousand and decimal six miliionths. 8. Nine hundred and thirty and eight tenths. '.'. Decimal two hundred and forty->ix ten-millionth*. 1<>. One thousand and decimal two hundred-thou>nndths. 11. Eleven and eleven ten-billiontbs. 12. Six hundred ami sixteen and sixteen trillionths. 13. Ten thousand and decimal four ten-thousandths. I 1. Decimal three hundred twenty-live thousand, four bun- dred and eighty-seven hundred-millionth-. 1GS. Write the following numbers in words, or read them orally : 7. 8694.876942 8. 760.4071 9. 4004.40040004 10. 839 ;:;;; 11. 46,00046482 12. 8769.27642! What uncertain- -:« in nailing iiu'm «I iiiiihImt- ' ll'»w cau thin ainl.i-uity be a\ oiilcl ' lor what are ciphers u.*ed iu tin 1. i2A ■2. ::.789 1. 10045 6. 1.8< 132 DECIMAL TRACTIONS. Note 1. Addition, subtraction, multiplication, and division of decimal fractions are performed precisely ii the ihjm operation! in whole nam no further explanation l»eii , except to determine the place of the decimal point in the several results. Note 2. The proofs arc the same as in whole num!>ers. Problem 1. 109. To add decimal fractions: Rule. Place tenths under tenths hundredth* under hun- dredths, etc.; tJien add as in whole numbers, and place the point in the sum dinctlij under the points in the number* added. Ex. 1. 3 6.4 7 8 8 4.9 2 G 2 8.0 4 7 2. 84 8.842 3 8 7.6 4 G 9 8 4.2 8 5 ft. 5 G 4.9 8 7 4 2 6 1 2.8 G 5 3 9 8 7 4.8 2 7 6 4 1 Sum, 1 4 9.4 4 5 1 7 2 8.7 7 3 1 4 8 2.6 8 4 5 7 Proof, 1 4 9.4 4 5 1 7 2 8.7 7 3 1482.680457 4. 5. 8 7 2 1 4 3.8 7 2 9 34820 9.1 534268741. \ 24 10.40 2* 2 7o.i 23 7 9 1 8 4 2.2 1 G 3 9 l 2 9.8 7 251842172 841.3 60498 86512 3.7 1942 7 2 4 3 1 0.0 6 8 4 3 31981 7.0 5841628347 6. Add 42.76, 934.247, 27.862. Ans. 1004.8G9. 7. Add 3.54G, 44.8693, 2.8769, and 784.68728. 8. 872.34, 6789.3274, 22.987, and 346.4 2. 9. Add 3582.47, 62.84693, .47249, ami 7.458. 10. Add five hundred and decimal six thousandths; forty-five millionths ; eighty-four million and decimal twelve millionth* ; seventy thousandths ; and decimal three hundred and fifty-four hundred-thousandths. Ans. 84000500.079597. 11. What is the sum of one thousand two hundred twenty-six. 168. How are addition, subtraction, multiplication, and division of decimals performed? Proofs? 1G9. Rule for addition? The point, where placed? fi.nl decimal one hundred :m figures in the subtrahend tlian in the minuend, the deficiency may be supplied hy annexing ciphers, or sup- tnem annexed, to the minuend (Art 164). 1. From 65.8487 take 24.3869. Ana, 11.1618. :>. From 1684.469 take 968.8749 6. From 9846.2764 take 5427.9824. 7. From 21 W.6872 take L724.19 ■ one thousand eight hundred seventy-six and deci- mal three hundred sixty-lour thousandths, take eight hundred D and decimal three hundred and three thousandths. Ana. 1060.061. 9. From U-n take six millionth.*. ]<». A man owned eightj-eeren hundredths of a railroad and •old forty-eight hundredths of it; what part of the road did be still own ? ■ :<». I I for subtraction of decimals? When the number of «lccim:i: in the subtrahend exceeds the number of urcimal plftQM in thfl BUM If 134 bECIMAL FRACTION. Problem 3. 171. To multiply one decimal by another: Rule. Multiply as in whole numbers, and point off as many figures for decimals in the product as there are decimal places in both factors, counted together. ; Ex. 1. Multiply .48 by .26. OPERATIOX. PROOF. Multiplicand, .4 8 Multiplier, .2 6 .48 288 208 96 104 Product, .1248 = .12 48 (a) If the number of figures in the product is less than tho number of decimal places in the two factor-, the deficiency must be supplied by prefixing ciphe r* to the product, as in Ex. 3. 2. 3. Multiplicand, 2 6.2983 .:: 2 Multiplier, 8^ .2 3 1051932 ~9~6 2 I 03864 64 Product, 2 2 0.9 5 7 2 .0736 Note 1. The reason of the rule for pointing the product will Ik) ol>\ if wt change die decimals to the form of eotnmon fraction! and then per* form the multiplication ; Thus, .48 x .2*6 = iW, X A«j = flflftfr = .1248, as in Ex. I. Again, .32 X .23 = ^ X A\ = i lllu = -°"36, as in Ex. :). N«»tk 2. The reason of the rule for pointing the product may be ex- plained in another maimer, as follows : The smaller the factors arc, the smaller is the product. Now, by trial, wo know that 32X2 3= 73G; .-., dividing one factor by 10 (Art. 1G1 ), we have • J 2X 2.3 = 7 3.6 = A °f tnc previous product ; dividing again by 10, 3 2 X .2 3 = 7.3 6 = tV of the 2d product ; dividing the other factor by 1 0, 3.2 X. 2 3 = .736 = j\y of the 3d product ; dividing again by 10, .3 2 X .2 3 = .0 7 3 6 = tV of the 4th product ; dividing again by 10, .03 2 X 23 =.00 736= A of the 5th product; and so on to any extent .3 2 5 9 .00 00 2 5 i e a 6518 .0 000081475 Ans. .208754. Ans. 156.915. Ans. 182 Aik .0082824. Ans. .0000204. 4. Multiplicand, Multiplier, .5 l 1 6944 21180 Product, 2 2 8.7 4 4 & Multiply .5642 by .37. 7. Multiply 34.87 by 4.5. 8. Multiply 2769 by .84. Multiply .2436 by .034. 10. Multiply .0068 by .003. 11. Multiply 86.874 by .5421. 12. Multiply .14687 by .00054. IS. Multiply .17288 by .11403. 11. Multiply .00869 by .24683. 15. Multiply 8.756 by 10. Ans. 87.56 (See Art. 161). 16. Multiply 356.4 by 100. Ans. 35640. 17. Multiply 9.8765 by 1000. 18. Multiply 348.09 by 100000. 19. Multiply 286.487 by 100000. 20. Multiply 37438 by 100000. 21. Multipfy 4.68 by 20. Ans. 93.6. In Ex. '21 multiply by the factors of 20, viz. 10 and 2; i. c move the point oik- place to the right, and then multiply by - , . 22. Multiply 86.42 by 60. Ans. 2185.2. 28. Multiply 472.8 by 800. An-. 878240. 24 Multiply 86.74 by 300. Multiply 54.26 by 406000. 26. Multiply throe hundred and fifty-six thousandths by one hundred and forty-five ten-thoii>andth>. An-. .y t\venty->ix ten-millionth-. Multiply eight hundred and forty-two thousandths l.\ hundred thousand. 171. Kuie for mttfttpllOStton of decimals' - re are not tiguns b in t he product? PWMOM Of t Iio rule for pointing the prod;. explnn:e 136 DECIMAL FRACTI" Problem 4. 173. To divide one decimal fraction by another : Rule. Divide as in whole numbers, and point off as many figures for decimals in the quotient as the number of decimal places in the dividend exceeds those in the divisor. Ex. 1. Divide .625 by .25. OPERATION. rilOOF. .2 5 ).G 2 5 (2JS £6 Divisor. 5 2.5 Quotient i % 6 125 125 50 2 5 Dividend. 2. Divide 1.257* Ans. .508005. 3. Divide 8.4364* .06. Ans. 140.608018. (u) If the number of figWQf 10 the QOOtlont i.> leu tlian the excess of decimal places in the dividend over those of the divi- sor, supply the deficiency by prefixing ciphers to the qu o tie n t , 4. Divide .000744 by .62. Ans. .0012. Notk I, The dividend is a product, the divisor and quotient being the .'t. 77) ; hence the rule tbt pointing the quotient. Note 2. The rule for determining the place of the point in the quotient may alio he explained by changing the decimals to the form of common fraction* and performing the Hviaiea ; thus, -•25 = T ^V-i 2 A = ?3 = 2.5. m 3. By attending to the relative size of divisor end dividend (Art. 83), we have another mode of fixing the place of the decimal point in *.he quotient ; thus, G 2 5 -f- 2 5 = 2 5 ; .*. , by dividing the divhliml by 10 (Art. 161), we have 6 2.5 -f- 2 5= 2.5 = tV of the preceding quotient ; dividing again by 10, 6.2 5 -I- 2 5 = .2 5 = yV of the 2d quotient ; dividing again by 10, r 25 =.0 2 5 = iV of the 3d quotient. Now dividing the divisor by 1<>, ,6 8 5 r 2.5 = .2 5 = 10 times the 4th quotient ; dividing again by 10, .6 2 5 -r .2 5 = 2.5 = 10 times the 5th quotient; and so on to any extent. 17 4. Rate for dividing decimals? "What is said of ciphers la the quotient? Reason of the rule for pointing the quotient? Second explanation? Third? DECIMAL IKAtTI Divide 88.742$ bj . Ant, L6497. Aas. .30. 7. Divide .000975 by .15. to 17.472 by .48. 1.77 11 bj '■■-Ml. ia Divide 08.794 by 1l\:;. . (h) If there tie more decimal places fa the divisor than in the dividend, the number may be made equal by annexing one or more ciphers to the dividend. The quotient will then be a whole number ; thus, L6 -~ .18 = 4.50 -*- .18 == 25. 11. Divide 8647 by .125. Ans. 2917G. 12. Divide 90821.6 by 8.642. Ans. 24800. 1.;. Divide 72 by .064. (c) If there fa a remainder after all the figures of the divi- dend have been used, the division may be continued by annex- iphera to the dividend. Bach cipher annexed bacon decimal place in the dividend. In some examples this operation may be continued until there i- no remainder, but in others there will necessarily be 8 remain- der, however far the operation may be continued. This latter dam of examples give- rise to circulating decimals ; thus, .7 — .9 = .7777, etc. Again, .8 ~ .11 =.727272, etc In the first of these example-, the figure 7 will be repeated perpetually, and in the Second example, the figures 7 ami 2 will be repeated in like manner. Whenever the remainder consists of the -aim- figure or figures a- any preceding dividend, the quotient figures will begin to repeat It may be remarked, however, that, if the divisor contains no prime factors but 2'fl and .Vs, the divison can .G412 by 400. Ans. .009103. 27. Divide 56.487 by 8000. 28. Divide :5G.49 by 600. Divide three thousand eight hundred end fifty-three hun- dred-thousandths by thirty-two millionth*, An-. 1204.0625. 30. Divide eighty-four and eighty-four hundredths by forty- eight thousandths. Problem 5. 173. To reduce a common fraction to a decimal. Ex. 1. Reduce £ to a decimal fraction. I X 100 = a$ft = 75 ; and 75 -f- 100 = .75, Ans. If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand 172. For what is the sign -j- sometimes used? DECIMAL PBACTIOl 139 in the above example, | is multiplied by 100 nnexing two riphen t<> the numerators the fracti then reduced to the whole number 7"», and, finally. 7"> ii divided by LOO by placing the decimal point before the 7'> j .-. $ = .75. Hen Rl i r. Annex one or more ciphers to the numerator and di- vide Uie result by the denominator. cottinuinrf the operation until Is no remainder, or as far as Is desirable. Point off us man;/ decimal places In the quotient as there are ciphers am to the numerator. •J. Etedoce $ to a decimal fraction. I X 1000 = a^- a = 375 ; and 375 h- 1000 = .375, Ans. 5. Reduce fa to a decimal. Ans. .4375. 4. Reduce- H to a decimal. Ans. 1.140625. 5. Reduce $$ to a decimal. 6. Reduce fa to a decimal. Ans. .5833— (-. 7. Reduce £ to a decimal. Ans. .3333+. 8. Reduce 9 to a decimal. Ans. .428571+. 9. Reduce I, I I ||, l if ^ and *J to decimals. 171. Every decimal fraction is a common fraction, and, if nominator be written, it will appear as such. It may then [need to lower terms, or modified like any other common fraction* This proves the rule in Art. 17.'5. 10. Reduce .48 to the form «>f a common fraction and then to it> lowed term-. .48 = ^*0 a= tt> Ant, 1 1. Reduce .125 to its lowest terms. •125 = A 2 A = M = A = b Ana, 12. Reduce .17 to the form of a common fraction. Ana, tVj. 13. Reduce .27 ■ KX)25, and ,00 14 J.8. 2.8 = f§ = V> An<. 15. : \ and 2.0 1 17.1 I:,',, for rtdndBf a common fraction to a decimal' Explanation* .•imal al?o a common fraction? How is this made evident' How iuli' in Ait. 173 be proved MM 140 DECIMAL FRACTIO Problem C. 175. To reduce whole numbers of lower denomina- tions to the decimal of a higher denomination. Ex. 1. Reduce 2pk. ?x\t. to the decimal of a bushel 1st. 3qt. = gpk. = .375pk. ; .-. 2pk. and 3qt. = 2.37."ipk. 2d. 2.375pk.= ^Y- J '• bush • = •' V • ,;j7: ' ,,l, ^ 1 •■ Ans. The principle is the same as in Art. 17.5. Hence, Hull. Hearing annexed one or more ciphers to the lowest de- nomination, divide by the number it takes of that denomination to make one of the nest hiyJa r, and annex the quotient as a decimal to that next higher ; (hen divide the result by the number it takes of THIS denomination to make one of the next higher, and so continue till it is brought to the denomination required. 2. Reduce 9s. Gd. 3qr. to the decimal of a pound. on: RATION*. 3.0 qr. 4 12 20 <..7 5 OOcL 3qr. = .75d. ; C.75d. = .5636s. ; 9.5 6 2 5 s. 9.5C25s. = .478125£, Ans. .4 7 8 1 2 5 £, Ans. Note. In dividing by 20 to reduce the decimal of ft pound, and in all similar examples, we may point off the in the divisor, and then divide by 2, but in Mich u case the point in the dividend must be moved one place t< the left, for by so doing both divisor and dividend are divided by 10, and .-. the quotient is unchanged (Art. 84, hj. 3. Reduce 2ft. 9in. lb. c. to the decimal of a yard. OPERATION*. 1.0 b. c. In this example 9.3 3 3 3 3 3 -f in. t, "' n ' wil - be a re " 3 12 3 mainder, however far .9 2 5 9 2 5 + yd., Ans. ried. 4. Reduce 3cwt. 2qr. 201b. 8oz. to the decimal of a ton. 175. Rule for reducing the lower denominations of a compound number to the decimal of a higher denomination? Principle? Mode of dividing when the divisor is 20, 40, etc ' When the divisor is a mixed number? QIAL i B \< ii' 1 11 .../.. LSdwt I8gr, to the decimal of a pound, Troy ht. Ans. .303125. imal of B pound. 7. Reduce 5yd, 8ft. Sin. to the d.-eimal of :i rod, Long ire. on Since one of the di- 1 2 3 2 1 1 J... 00 ft. 5.8 3 3 3 + yd. 2 11.6666 + half yd. 6.0in. ra, in this example, U 51, both divix.r and dividend air reduced to halve*. The feel and inches are more than a half yard : ,\ the gum \ 06 _j_ rods " Ans> Of the - given numbers il 1 more than a rod. 8. Reduce 3s. 15° 30" to the decimal of a circumference. Ana. .291689+. Reduce :M. Oh. 18m. fieec. to the decimal of a week. 10. Reduce 2qt. lpt. lgi. to the decimal of a gallon. 1 1. Reduce War. Boh, Sid. lOli. to the decimal of a mile. 1 J. Reduce 8cu. ft. 144c. in. to the decimal of a cubic yard. U educe 3r. 2rd. 20yd. to the decimal of an acre. 1 t. Reduce 5fur. 30rd. 5yd. 1ft. 9in. 2 b. c. to the decimal of a mile. Probum 7. 176. To reduce a decimal of a higher denomination to whole numbers of lower denominations. Ex. 1. Reduce . 12.si:?o£ to .-hillings, pence, and farthings. OI'KKAI ! £. 1 '2 s 1 This article is the reverse of Art. 2 175; .-. litvt multiply by 20, because ") 2 5 s then; will be 20 times a- many shil- , 2 luuHi aa pounds. Pot alike reason, multiply the fractional part of a shil- i 500d. Kngby 12, to reduce it to pence, etc. 4 After having fixed the decimal point ,i ( , r . in the several pr o d uct s , the lty the number it takes of the next lower denomination to make one of this, and so proceed as fir as necessary. The several numbers at Ute left of the points tvill be the answer, 2. Reduce .984375 of a bushel to pecks, quarts, and pints. Ans. 3pk. 7qt. lpt. 3. Reduce .40625 of a gallon to quarts, pints, and gill>. 4. Reduce .902288 of a lunar month to weeks, days, hours, minutes, and second-. An-. $w. id. C>h. 80m. 1 ">.12'J0sec 5. Reduce .90625 of a yard to quarters, nail- 6. What i> the value of .37.">° ? Ans. 22' 30". 7. What is the value of .375 of a ton ? 8. What is the value of .4658 of a pound, Troy Weight ? 9. Reduce .3587 of a mile to furlongs, rods, yards, etc. 10. Reduce .5621b to 5, 3, etc Miscellaneous Examples in Decimal Fractions. 1. What is the cost of 6.251b. of beef, at 12 cents per pound? Ans. 7oc. 2. Bought 4.5 tons of hay, at $12.50 per ton; what was the cost of the whole? An-. $56. 3. What i- the raloe of 8 acres of land, at $62.50 per acre? 4. Paid >">00 for 8 acres of land; what was the priee per acre? 5. Paid $500 for a piece of land at $02.50 per acre ; how many acres were bought ? 6. Bought land at $62.50 per acre, and sold it again at $75 per acre, thereby making $100 ; how many acres were bought? 7. Bought 8 acres of land at $62.50 per acre, and sold the lot |600 ; was there a gain or a loss? How much total? How much per acre ? J 76. Rule for reducing a decimal of a higher denomination to whole num- bers of lower denominations? Explanation? m:< imal IRA4 ii' 143 B, v> '!.•• Br. 20rd. of hud, aft $40 per acre ? : St. I5cwt lojr. 12fclb. of coal, at $G per ton? in. What oosi 12.25 cords of woo. 1, at $6 per cord ? 1 1. What cost 7} oorda of wood, at $64(5 per oord ? 12. What will it cost to build 24m. Sfbr. 20rd of railroad, at $3775 per mile ? 18. A rectangular field is 40.5 radi long, and 30.5 rods wide ; wliat will it cost to build I wall around it, at $1 per rod? 1 1. What coal 5yd. Bqr, 2na. of cloth, at 16c. per yard ? 1"). How much land in a rectangular field that is -10.5 rods i rods wide ? in. Whal would 16 hales of cotton cost, each bale weighing tftewt, at $10.50 per cwt? 17. What 0081 .825 of a ton of coal, at S7 per ton? 18. What OOat .825 CWt Of coal, at ^7 per ton? l'.t. What is the value of .25 of a ton of hay, at 2£ 5s. Gd. lqr. per ton ? 20. What is the value of .7") cwt. of hay, at 2£ 5s. Gd. lqr. per ton ? 21. Taid 3£ 9s. 6d. lqr. per acre, for 5a. 2r. 15rd. of land ; what was the entire 606t \ 22. If 365^ days make a year, how many days, hours, etc., arc there in .785 of a year? 23. What is the cost of 3 pieces, of cloth, the fir>t containing id-, at $2.25 a yard; the second, 12*5 yards, at S*>.50 a yard ; and the third, 8.8 yards, at $8.25 a yard? 24. A threi -sided plat of ground is inclosed by a railroad on tide, and highways on the other two sides ; the side next the railroad is 4.1 rods long, and the other two sides are respectively •1 rods and .i) of a rod in length ; what is thecostof fencing this pint, the fence costing S3.75 a rod? It a boat sails 8.7."> miles an hour, how far will it sail in 8. 1 1 1 < Sow many bins, each holding 37.5 bushels, will be filled with 1687.5 boaheli of grain? 27, How many coats, each requiring 2.75 yards of cloth be made from '.W>.7-'> yards? 1U UNITED STATES MONEV. 28. In how many days will a man earn $20.12."), if ho earn $1.76 a day? 29. How many square feet in a board which is 18.25 feet long and 2.8 feet pride! 30. Bought a load of straw that weighed It. 2cwt. 3qr. 12£lb., at $8 a ton ; what shall I pay for the load ? 31. Paid $7,175 for 35 gall. 3qrt. lpt. of vinegar; what was the price per gallon ? 32. If a pole 12.5 feet long casta a shadow 3.125 feet at 12 o'clock, what la the bight of a steeple that casts a shadow 33.28125 it the same time? 33. What n the cost of carpeting a room that is 16.5 feet long, and 15 feel widi •. thr carpet costing, $1.25 per square yard? UNITED STATES MONEY. 177. Tnited States Money, sometimes called Federal Money ) is the currency of the United States. TABLE. 10 Mills (n i.) make 1 Cent, marked c. 10 Cents u 1 Dime, M d. 10 Dimes u 1 Dollar, CI $ 10 Dollars H 1 Eagle, Cents. II Mills. e. Dimes 1 — 10 Dollars. 1 = 10 sa 100 ,,dc. 1 ~~~ 10 = 100 — 1000 1 = : 10 = 100 sa 1000 as lOOoo The terms eagle and dime are seldom used in computation ; - and dollars being road collectively and called dollars, and dimes ami cents being called cents ; thus, 3 eagles and 5 dollars arc called $35, and 4 dimes and 3 cents, are called 43 cent-. 177. What is United States Money? Bepeat the Table. Are the terms eagle and dime much used? I ■!."> 17**. Tl ■ n - ■ of the United S upon rules for i in tbil .I al-o manv example*, have ah< I : but th>- importance of the mbject ju.-ti: ion of it. 170. A ■■il'l. silver, or other metal, stamped by authority of tin- Genera] Government, to be need ai money. 1**0. The coins aothorixed by our Government, and stamped at the l'. S. Mint, arc the following: Gold. Silver. Doulilc Eagle, 0,6ft Dollar, $1.00 10*00 Half Dollar, Halt Quarter Dollar, .25 Quart' Dime, .10 -Dollar 1' Half Dime, .05 One Dollar, L00 Three -Cent Piece, . of Copper and Nick< I, ant, .01 181. Gold and aihrer, for <•< >ii. hardened by b I with harder and cheaper metals. These cheaper m when combined with the gold and silver, are called alloys. 183* Carat is a term used in indicating the purity or fine- ness of gold. If a piece of metal is pure gold it is said to be 24 carats fine ; if r. ] of it are gold, and the remaining ^ is alloy, . 1S3. The ttamdard purity of gold and silver coin at the DUN metal and ^ alloy. The alloy in >il- iin i> pore copper. The alloy in gold coin is copper and . silver not l the oopp (a) The new of 88 part.- of copper for 12 118. 0* 1 .iirnncy of the U, 8. band! 179. Wliat is a coin' 180. What gold coins nre authorized by our Government ' What rflr«f : Mdl! 181. What is aI!o\ ? lor wliat used? 183. For irat used? Pure gold is how many carats tine? 183. Wlmt •ity of gold and silver roin' What is the alloy for diver' What for gold? What part of the new cent is nickel? 13 40 IXITED STATES MON! m 1. The copper cent is still in use, but is no longer coined at the U. S. Mint. Note 2. The mill is not coined. Note 3. Other pieces of money, u the 50-doOw gold piece, the half ■ad quarter dollar gold pieces, arc in u>e to KMM extent, but arc not legal coin. pk 4. The greater part of the money in general use, OOMbtl offtsafe Lilh, which are niu-li more convenient fat most purposes than gold and silver. 184. The weight of the en s ltijik, Troy. The silver dollar weighs 412^ grains, but the smaller coins are not so heavy in proportion to their value; thus, the lialt* dollar weighs only 192 grains; the quarter, only 00 grains, etc The new cent Wtighfl 72 grains. Note. These standards of weight and purity arc regulated by Con- gross, and* may l>c changed at any time. 18t5. In this currency, the dollar \> the unit, cents and mills being deeimait of a dollar; thus, $8.62 represent! three dollars and BlXty-tWO itfl iour dollars, eight cent-, and live mills, etc. Note. Figures at the right of the third decimal place, npmtol j«irtsof mills; thus, $5.3627 = 5 dollars, 36 cents, i> mills, and t 7 o" of a mill. REDUCTION. 186. The reduction of U. S. Currency is very simple. Dollars are reduced to cents by annexing two ciphers (Art. 62), and to mills by annexing three ciphers ; thus $ I = 400 cents = Dollars and cents are reduced to cents by removing the deci- mal point; thus, $3.56 = 356 cents. Dollars, cents, and mills 1^3. Is the mill coined? What of other pieces of money? What of paper money? 184. What is the weight of the eagle? Of the silver dollar? Half dollar? By whom is the standard of weight and purity fixed? 185. What is the unit in this currency? What are cents and mills? What are figures at the right of the third decimal place? 18G. How are dollars reduced to cents? Jlow to BriBsT How are dollars and cents reduced to cents? How dollars, cents, and mills to mill? ' 117 are reduced to mitts In the Bam thus, $5,468 =r mills. :. Reduo Aii ;it<. •j. Reduce $8446 to <•■ An . nte. .:. i: ; : .).45G to mills. Acs. 8436 mills, l. Reduce $488 to cents. To milN. ... ft dncc |6J i To mitts. Reduce $1.87G to mills. 187. Ctfnrt are reducrd to dollars hy pointing off two d> <{- mal placet (Art. 81). Mills are reduced to dollars by pointing off three decimal plans; thus, 37G8 cents = $37.68 ; 37G8 mills = So.768. 7. Reduce 5G4 cents to dollars. s. Reduce 8692 mills to dollars. 9. Reduce 87G94 cents to dollars. 1<>. Reduce 76848 mills to dollars. Am. $5.64 >3.G92. I**. Addition, Subtraction, Multiplication, and Division of U, S. currency, are performed precisely as the correspond- ing operations in I>< dmaX Fractions. ADDITION. $ 8 7 6.5 4 2 3 9 7. 1 8 8 6 7 9.8 2 1 3. 6 4 8 7.3 3 4 2 9 6.8 7 •1 4.98 Ex. 1. $ 7 5.5 6 4 2 4.8 7 6 9 6. 1 45 Sim,. s85 1. Paid $87.50 for a horse, $145.25 for a pair of - ?1 L25 for a wagon, ami $1~>.75 for a cart ; what did I pay for all? An-. $292.75. lit a hat for $ !. .75, a vest for $5.25, ami a pair of boots for $."» ; what did I pay for all ? . How arc cents reduced to dollars.' ll<>\v mills to dollars? 188. How are Addition, Subtraction, Multiplication, nnd Division of V. S. Money per- formed? 148 VNITi:i> STATES MONEY. SUBTRACTS )\. From Take Kx. 1. $ 4 8 7.9 6 1 $2 6 8,7 8 8 2. ■;.8 7 $47.43 3. $8 6,4 8 5 $ 4 4.3 6 8 Ans. $2 1 9.1 7 6 4. A man who owed SG99.GO, paid $1C4.60; how much <1M he -till owe? An 5. Bought a farm for $3G84.7"'. and Stock and tools for the farm fur $1 L7&25 : how much more did I pay for the farm than for the stock and tools ? MULTIPLICATION. Ft 1 Ft 5 Multiply $3 4 8.7*65 $3684.375 By 2 5J 2^1 7 13 950 80 1743825 3. 697530 $4386.942 Ans. S88586.310 369 4. If 12 gentlemen have $7497.84 apiece, what sum have they all ? Ans. $89974.08. 5. If 45 persons deposit $346.25 each in a savings bank, how many dollars arc deposit DIVISION. K\. l. I; $225 are divided equally between 27 men, what sum will each receive ? OPERATION. L> 7 ) $ 2 2 5 ( $ 8.3 3 $, Ans. Dividing 225 by 27, g 2 1 6 8 for quotient and 9 for re- ^ mainder. Annexing ciphers g •• and continuing the division, as in Decimal Fractions 9 (Art. 172, c), we obtain 8J $8.33£ for the share of each o man. KONBY< 14'J 2. Divide $69 I 18 men. Ans. $3852.52. ,40 into 21 equal pari PRACTICAL Examples. 1*9. To find the cost of a number of things when price of one thing is given. 1. If applet arc worth $2.50 per barrel, irbat are 3 barrels worth ? Three barrels are worth 8 times as much as one barrel, .'. 3 barrels are worth S*-\50 X 3 = §7.50, An, 1 1 dice, BULK. Multiply the price of one by the number. 2. What is the cost of 9 barrels of flour, at $7.75 per barrel? Ans. $G9.75. Bought 25 sheep, at $&88 each ; what was the cost of the flock? 1. Bought 18 yards of broadcloth, at $3,875 per yard; what WWi the cost of the piece ? 5. What is the value of 75 acres of land, at $37.50 per acre? IOO. To find the price of an article when the cost of a given number of articles is known. 6. When eight cords of wood arc worth $44, what is the value of 1 cord ? If 8 cord< are worth $44, one cord is worth £ of $44; and ■.."»<». Ana. Hence, Rule. Divide the cost by the nu> 7. U 2 1 y.uds of broadcloth cost $93, what is the price per yard F s7£. 8. Bought 37 pounds of butter for $8.51, what was the price ? Ana, 23c 189. How is the cost of a number of things found when the price of one ii kuowu? 190. How tli.- price of oue when the cost of a number is known! 13» 150 i XITED STATES MONEY. Note. Price is, appropriately, the sum asked for one article; thus, when any one asks a flour dealer the price of flour, he is understood to ask what lie must pay for a single barrel, not fifty barrels, nor half a barrel, nor any quantity except one barrel. Hence we distinguish between price and cost, or \>rice and value. '». Bought 350 bbls. of flour for $302 G ; what was the price ? 10, Bought a farm containing 1 2.3 acres for $6843.75; what MM the price per acre? 191. To fnul the quantity when the cost of the quantity ami the price of ono arc gi\ 1 1. At $G per ton, how many ton- of 00*1 can I buy for $21 ? I can buy as many tons a9 $6 is contained timet fin $84 ■ illl( l :- $G = 4, .-. I can buy 4 Dons. 1 1 nee, r. Divide the cost by the price of one. 12. At >:; per yard, how many yards of cloth can be bought for $546? An, 182. .50 per acre, how many acres of land can be bought for $ i 11. At 5G cents a pound, how many pounds of tea may be bought for $25.20? 15. A drover bought oxen at $62.50 eaehj how many oxen did he buy for $1562.50? 192. To find the cost of articles sold by the 100 or by the 1000. l& At S1.50 per 100 feet, what will 842 feet of timber cost? Had the price been $4.50 per foot, the cost would have been $4.50 X 342 = $1539; but unee the price U $4.50 per hundred feet, the true multiplier is one hundredth part of 342, viz. 3.42, and the true cost is 0, Ans. ,50 x 3.42 = $15.39. • 90. Meaning of price T Difference between price and cost, or price and value? 191. Rule for finding the number of things when the cost and price are known* 10*. BxpbteEx.lt 3.4 2 1800 1350 151 1 the price been $4.5n pet thousand feet, the true multi- would ha\ B48, and ti. iM have Im.-h $4.50 X .3 12 = 81.530. Hence, Ki : the quantity to hundreds and decimals of a hundred, or to thousands and decimals of a thousand, as the example may require ; tin n multiply the price by the quantity, and point the product as in multiplication of decimal* (Art. 171). ml. C is used to indicate hundreds, and M to indicate thousands. 17. What cost 1200 feet of boards at $2.10 per C? Ans. $25.20. 18. What cost 12514 feet of timber, at $13.50 per M p Aus. $1G8.939. Note 2. In business transactions the answer to Ex. 18 would be called 8163.94. In the remaining examples in V. S. Money, the mills in the answers will he omitted if less than 5, and one will be added to the cents if the mills are 5 or more. 19. What cost 20000 shaved pine shingles, at $6 per M? 20. What cost 13725 bricks, at $6.50 per M ? Ans. $89.21. (a) To find the cost of articles sold by the ton. 21. What cost 24401b. of hay, at $18.50 per ton ? OPERATION*. First divide by 2000 (i. e. point off three decimal places and divide by 2), to reduce the WQlght to tons and deci- mals of a ton ; then multiply by the price. In multiplying, the 50 cents may be used decimally, or the common fraction, $2 2.5 7, Ans. \, may be D8M, U in the operation. 22. What cost 58481b. of coal, at SG.25 per ton ? Ans. $18.28. 193. Rule for finding the eost of articles sold by the 100 or 1000. For what U C used? M I \\ i timliug the coet of articles sold by the tour 2 ) 2.4 4 1.2 2 18* 61 976 122 163 UXITl.b i-Y. 193. To find the cost or value of any Dumber of articles when the price is an aliquot part of a dollar. TABLE OF ALIQUOT TARTS OF A DOLLAR. 50 cents = 1 of a dollar, 20 cents = 4, of a dollar, te = I of a dollar, 1 G J cents = | of a dollar, 25 'cents = \ of a dollar, I2| cents = \ of a dollar. 23. "What cost C I !' cloth, at 87) cents per yard ? OPERATION. $G_4 = costof Gh.l. :.t SI. 3 2 = cost of G4yd. at 5 c, or h of 1C = cost of G4*vd. at 2 5 c. or _8 sb cost of C4yd. at _l_2$c., or £ of 25c. Ans. $ 5 G = cost of G4yd. at 8 7 J c. The cost at $1 is evidently as many dollars as there are yards ; the cost at 50c. is half as much as at $1 ; the cost at 25c, half as much as at 50c; and the cost at 12$c, half as much as at Then the cost at 50c, at 25c, and at 12£c, added, gives the cost at 87 £c This process is uuially called Practice, for which we have the following Rule. Take such aliquot parts (Art. 119, Note) of the num- ber of articles as the price is of | 2 1. What cost 48 barrels of apples, at $3,371 per barrel? OPERATION. $4 8 = cost at SI. _3 8144 = cost at $3. 1 2 =rcot! a; .:' c. or \ of G = cost at .12^ c, or £ of 25e. Ans. $ 1 G 2 = cost at S3.3 7 } c 25. What cost 24 barrels of flour at $6.33$ per barrel? Ans. $152. 193. Rule for finding the cost when the price is an aliquot part of a dollar? What is this process called? Name the most convenient aliquot part* of a dollar. 153 26. What co..•. per bushel? 1 i<> pair- per pair? ich? 194. To find the cost when the number of articles is r Pepcid by a compound or by a mixed number. 81. What oori 9a. 3r. 20nl. of land, at $40 per acre ? OPEKATI $4 0, price per u« 9 $360 = cost of 2 = cost of 2r., or },:\. 10 = cost of lr., or £ of 2r. 5 = cost of 20rd., or frr. $ 3 9 . '> = shares of railroad stock, at $108.50 per share ? OPERATION. S 1 8.5 0, price per share. 8J ss lis.nn seosJ of 8 shares, 5 1.2 ."> = cost of | share, 2 7.1 3 = co-t of | share, $ 9 4 9.3 8 = cost of 8$ shares, This process is also called Practice, and may be stated thus: Multiply the price by the entire number of articles, and to this ct add such aliquot purtt of the price as the fractional part of the number is of a unit. 33. What oust 3t. 16ewt. Iqr. 201b. of hay, at $16 per ton? 1.16. 84. Whai 5c. ft, 8ca.fi. of wood, at $6 per cord? • . What cost 24| acrei of land, at $48.72 per acre? «'JJ. Mtag the cost when the number of articles ii expressed by uml or by a mix<.. u there are dollan io the cost of the cloth, and 5 times 9.5 = 47.5, or 47. 1 , number of pounds of batter required, I Dividing $9.50 by 20c. will give 47.5, or 47*, the same result a> before* Thii exchanging of goods is usually called Barter. The examples are solved by Analysis. . How many pounds of sugt»i\ at \'2\e. per pound, may be j-lit for 3 bushels of corn, at 87 f,c. per bushel ? Ans. 21. 38. How many cords of wood, at $5.50 per cord, shall be given in exchange for a barrel of flou**, at $7.50, and 5 yai cloth, at $2.35 per yard? BILLS. 190. A Bill of Goods is a written ftatement of articles sold, giving the price of each article and the cost of the whole. Find the cost of the several articles, and the amount or foot- ing of each of the following bills. (1.) Boston, Jan. 1, W Mr. Abel Snow, Bought of Jonx Ada.m,«, 2 5/5. N. O. Sugar, at' 9c. 4 0lb. M iar, 18?e. 6 lb. Cheese, « 12 Sib. But « 2:; 4 lb. Raisins, " 1 5 1 . 2 lb. Cream Tartar, « 4 5 c. $ 1 3.8 4 Received Payment, John Adams. 195. What is Barter? How are examples in barter solved? 196, What is a Bill of goods? IW'ITLD ST A. New York, Jan. 15, 1.^ mi. Bought of J auks Phil: •adcloth, at ."> 10 Broadcloth, u 7 y ■>. I xrf, " 1 .2 5 A yd. Black Satin, " 4.5 Received Payment, .Jami> $ 9 2.2 5 i Phillips, 7fy K. Low. Philadelphia, Mar. 1, 18G2. - I EWART, L861. To Holt, Wilder & Co., Dr. June Oct. 5. 12. To u M M G JFefater'* Dictionaries, at 1 2 /)ay's Algebras, 8 6 ftftom " 9 ^o//o A'A/c*, SG.OO 1.5 2.5 Received Paymrut. $8 5.5 S. Dam For Holt, Wilder & Co. (1.) On Mar. 1, 18G2. LP. -ll WETT, 18»'.i. Samuel Palmbb, Dr, Apr. & ZW6750J 4 $!2£0perM. $209.38 •■ 175 M. May IS " "'ink, " 2 5.0 0^erJ/. $3 3 8.8 8 - 'I. May :». By 3 T. Rednce 54598 cents to dollar-. 11. Reduce 47689 mills to dollar. 12. Bfy farm cost $3725 and my house cost $1862.75 ; how much more did the farm OOSt than the hoi, 13. A gentleman bequeathed S750 to each of his 3 sons, and $500 to each of his 4 daughters ; how much did he bequeath to his children ? 14. Paid $16.50 for a coat, $4.25 for a vest, $5.75 for a pair of pants, $3.50 for a hat, $4.37^ for a pair of boots, and $12,621 for other articles ; what did I pay for all ? 15. Divide $113.7"> equally between 7 men. 16. Paid $68.75 for flour, at $6.25 per barrel; how many barrels did I buy? 17. How many yards of lace, at 62 £c. per yard, may be bought for $3.75 ? 18. What cost 8725 feet of boards, at $12.50 per M? 19. What cost 8248 lb. of coal, at $6 per ton ? 20. What cost 3a. 2r. 20rd. of land, at $48 per acre ? 21. How many pounds of sugar, at 12£c per pound, will pay for 12 dozen Qgp$ at 16^c. per dozen ? l.~,7 My rea] estate ia worth i and my persona) i 50 ; trKat am I worth ? 23. an 1 carriage) how far may I 2 1. A dnr? r bought sheep al $8^7] per head and aold them per head, and gained $87.50 by the transactions ; how many ibeep did he buy ? Bought 100 sheep at $& 875, and sold them again at ha! was the gain per head and total? Bought 20.5 tons of hay at $12£75 per ton; what was tin- coal of ilit* whole? What Si the value of 67.75 acres of land at $02.50 per acre? 28. Paid £1231.375 for 67.75 acres of land; what was the price per i 29. Paid $423 4.375 for a piece of land at $G2.50 per acre ; how many acres were bought? Bought land at $62.50 pet acre, and sold it again at pes acre, thereby making $846,875; how many acres were lit? 81. Bonghl 87.75 acres of land at $62.50 per acre, and sold the lot f<>r $5081.25 ; was there a gain, or loss ? how much total and per acre ? Bought 856.251b. of wool at 37 lc, which was manufactured into cloth at an expense of $62.50 ; for what sum must it be Bold to gun $87.50? Bought 1 1.75yd. of sheeting at i4 cents per yd. ; what wai the coat of the piece ? 84 What would 7$ bales of cotton cost, each bale weighing L75 per cwt.? 35. What cost l-lyd. 2.jr. 3na. of cloth at $4.67 per ell French. the ell French being 6qr. ? 86. Bought lbbl. Hour at $12.50, Sbush. corn at 8? 11). sugar at 8Jc, 3gal. molasses at 37£c, 211». tea at r.2.[c., Mb. at lie, t51ba> rice at l]e< and 41b. butter at 22c.; what the cost of the whole? Ans. $21.76. What cost :;i L5cwt 2. ( r. 12.U1). coal at $9.75 per ton ! 14 158 coMrorxD numbers. 38. What will be the expense of papering a room that is 20 feet long, 15 feet wide and 8.5 feet high, a roll of paper being 8 yards in length and £ of a yard in width, and costing G2£c. per roll ? 39. Bought 133.5yd. of broadcloth at S3. 25, and sold 88yd. of it at $3.88$, 50yd. at $3,875, and the remainder at $3.60; Low much was gained by the transactions ? COMPOUND N I M1JERS. ADDITION. 107. A Compound Number is composed of two or more denominations (Art. 8G) which do not usually increase decimally from right to left; consequently, in adding the different denom- ination-, we do not carry one for ten, but for the number it takes of the particular d enomina tion added,. to make a unit of the next higher denomination ; thus in adding Sterling or English Money, oarry 1 for 4, 1 le 4qr. make Id., 12d. make nl •Ji>s. make 1 1". Ex. 1. Add 6£ 7s. 9d. 3qr., 5£ 12s, lid. 2qr., 27£ 18s. lOd. 3qr., and 19£ 14s. 8d. lqr. operation. Baring arranged the numl ■£ s. d. qr. a> in tin 4 margin, the amount of 6 7 9 3 the right-hand column is 9qr. = 5 12 11 2 2d. and lijr. Upon the lame 2 7 18 10 3 principle Bfl in addition of sim- 19 14 8 1 pie numbers, the lqr. is set under c ,«, rTa T~a a 7 the column of farthings and the hum, o y 1 4 4 1 .. . , ° . 2d. are added to the pence in the mple, making 40d. = 3s. and 4d. Setting the 4d. under the the column of pence, add the 3s. to the shillings in the example, making .Vis. = 'J£ and 14s., and so proceed, uutil all the columns are added. 197. Do Compound Numbers increase decimally? Explain Ex. 1. ADDITK 19$. The j of procc: ihc samo as in addition of simple numb To odd oompound numbers, » lit- 1 -i. Write the numbers so that each denomination shall orrupy a separate c. oz. 2 3 4 1 - 40 30 6 14 2 20 8 3 3 6 G 20 30 4 6 2 10 8 5 1 2 20 30 15 ;{ 18 3 10 12 8 3 2 13. 28 45 4 6 3 18 6 14. circ. s. o / // yd. qr. na. in. 2 8 20 40 50 3 3 3 2 1 4 12 18 20 8 2 3 1J G 6 25 50 7 6 3 10 4 9 29 49 59 7 12 2 1G. fur. rd. vl. ft. in i. b. c. yd. ft. in. 1 3 2 10 1 4 2 4 2 4 4 2 4 2 3 1 7 3 G 5 6 2 5 6 1 3 4 2 2 7 4 2 rd. 4 2 7 7 21 3 U 1 or 7 2 1 2 10 2 or 3 1" 2 6 Note 2. A fraction occurring in the amount may sometimes be reduced to whole numbers of other denominations ; thus, in Ex. 15, the half yard equals 1ft. and 6in. ; the Gin. ]>ut with the 4in. make lOin. and the 1ft. put with the 2ft. make 3ft. or 1yd. Oft., and, finally, the 1yd. pot with the 1yd. in the original amount gives 2yd. The answer, when reduced, may contain h denomination higher or lower than any in the given example ; higher, as in Ex. 16 ; lower, as in Ex. 17. 199. What may be done with a fraction in the amount? Explain Ex. 15. Ex. 16. Ex. 17. May the answer contain a higher or lower denomination than the example? How? ADDITI> 101 17, a. r. rh. 2pk. lpt.; in another, 28bush Gqt. ; and in another, 75bosh. lpk. oqt. lpt. ; how much wheat did he in Ihe l BeMe ? A planter sold cotton at rarioOS tim<'<, as follows: 2t. :. 2qr. l-' Ul>.. 6t lewt Iqr. Gjlb., 8t I9ewt 8qr. 18|lb., 16t.6ewt.8qr. 12111.., and 16t8qr. 181b. ; what did he sell In all? 2 1. What h the nun of l la. Jr. 80rd 85yd. 8ft. 72in., 87a. tad. 80yd. 6ft. 86ul, 50a. lr. I8rd. 25yd. 2ft. 108in., and . 85id 25yd. 8ft. 72m.? 25. A- id 8ci . 2circ Us. 2.".° 20' SO", 5circ. 4«. 8 . and 6eirc 10s. 1"° lo' io" together. traveled 85m. 6fbr. 18rd, 5yd. in one ihy, 42m. !. the next day, 87m. 5for, 82rd. 4yd. the next, and 15m. 7fur. 2 Ird. 8yd. the next ; how far did he travel in the 4 da; 27. A blacksmith bought It. 18cwt 8qr, 201b. of iron at one time, St 15ewt 8qr. 121b. at another time, St. 6cwt Itjr. 181b. at another, sad 8t ScwL 2qr. 101b, at another; how much did iv in all } 11- L62 COMPOUND NUMBERS. SUBTRACTION. 200. The principle is like that of subtraction of simple numbers. Hence, To subtract compound numb* Rule. 1. Write the less quantity under the greater, arranging the denominations as in addition, 2. Beginning at the right, take each denomination of the subtrahend from the number above it, and set the r beneath. 8. If any number of the subtrahend is greater than the num- ber above it, add to the upper number as many as it takes of that denomination to make one of the next higher, and take the subtra- hend from the sum ; set down the remainder, and, considering the number in the next denomination in tlie minuend one less, or that in the subtrahend one greater, proceed as before. 201. Proof. As in subtraction of simple numbers (Art. 53 ). Ex. 1. From 8£ 6s. 9d. 3qr. take 2£ 1 . 5d lqr. OPERATION. £ 8. d. qr. Miii., 8 6 9 3 Only the Island 2d sections of .!>., 2 4 5 1 Ihfl rule apply to this example. Km., 6 2 4 2 Proof, 8 6 9 3 2. From 9£ fe. LOd. lqr. take t£ 17a. 2d 8qr. operation. As 8qr« cannot he taken from* £ s. d. qr. lqr., borrow OOQ of tlie 1<>«1.. r«- Min., 9 6 10 1 it to farthings and add it to Sub., 2 17 2 3 the lqr., giving 5qr.j then aaj) R em# (j 9 7 2 tlu!n ""l r " I, ' :iu ' -'l r ' ^' ou '> * as one of the lOd. has been em- Proof, 9 6 10 1 ployed, say 2d. from 9d., or, « hat is practically the same, 3d. from lOd. leave 7d.. and so proceed through the example. 200. Rule for subtracting compound numbers? Principle? 201. Proof f Explain Ex 1. Ex. 2. IN The form of the minuend ni:iv 1 1 and the work per- formed U follow. (All. D SECov £ 8. d. qr. £ 8. d. qr. Min., 9 G10 1)(8 2G9 5 «- Sub., 2 17 2 3 j — ( 2 17 2 3 K. -in., G9 7 2 = G 972 3. 1. t. cwt. qr. 11). oz. dr. lb. oz. dr. sc. gr. From 12 8322 G 1 5 6 4 3 118 Take 3 19 2 18 8 12 2 3 C 2 1 2 Bern., 8 9 1 3 14 3 4 4 2 6 f, 12 8 3 22 G lo 5. G. yd. qr. na. in. 1. m. fur. rd. yd. ft. in. From 16 12 1 62427*518 6 3 12 222 35 225 8. a. r. rd. yd. ft. in. pal. qt. pt -i From 6 I ■_' 5 3 4 134 14 2 3 Take 13 39 5 8 140 5312 '.'. 10. lb. oz. dwt. gr< c. eft. cu.ft. cu.in. Min., G 5 15 25 4 15 1727 S.il.., 3 10 1 2 y 2 3 4 7 5 169 2 7 2 23 6 5 15 22 11. 12. bmh. pk. <]t. pt. wk. d. li. m. sec L 6 1 3 4 23 45 2 4 8 7 1 1 G 1 G 3 4 5 201. Kxplain 2d operation in Ex. 2. 1G4 COMPOUND XiMl : 202. Sometime*, as in the following exai is neces- sary to borrow two of the higher denomination of the minuend in-trad of one; but in all such cases we must carry too to the in-xt term of the subtrahend ; i. e. ice must PAT at much as tee no into w. 13. rd. yd. ft. in. b.c. rd. yd. i i a o a 8 Take 3 5 2 8 2 1 ) ( l l 2| — \ ft. in. b. c. 4 17 4 a 8 2 Rem., 7 5 2 9 2 = 7 5 2 9 2 Proof, 120261 = 10104174 From Take a. 7 1 r. rd. 2 3 39 30 ft. 5 8 14. in. 124 143 Rem., 5 1 39 29^ 5 1 25 Proof, Min., Sub., 7 m. a 2 ;o. fur. rd. vd. ft 3 7 1 b b a 124 12 4) (6 1 7 1 1 8 ) ~ X 1 3 3 L It in. 8 60 9 196 9 3 8 148 = 5 1 39 30 1 53 = 6478G0 9 19G 1G. circ. deg. m. fur. rd. yd. ft. 7 1 1 2276933951 Krim 3 G 5 2 Proof, G 3 7 1 30ft. To find the time between two dates. 17. What i- the difference of time between July lo, 1857, and Apr. 25, 18 Am. lyr.Dm. 10d. FIRST OPERATION*. SECOND OPERATION'. vr. in. (1. vr. m. d. Min., 18 62 4 2 5) (186 1 3 24 Sub., 18 57 7 15 } or X 1856 6 14 Rem., 4910= 4910 202. What id said of borrowing tico? Explain Ex. 13. 203. How many modes of finding the time betweeu two dates? What are they? <\irlirr fn> mth. In tin .:ioii, the number of the : the month, is used ; in the iscood, the Dumber of i dayi net Aow dapmi lines the commencement of the cinis- ra, is used. cm give the same rc.-ult, l.ut As yirs/ is 1 I. How long from the battle of Waterloo, June 18, 181 the death of Napoleon, May .">, 1mm ? An-. 5yr, lOitt. I7d. 19. How long from tin.' battle of Lexington, Apr. 19, 177.'». to tin- surrender of Cornwallis, Oct 19, 1781 ? I low long from tin* inauguration of Washington, Apr. 30, tie of New Orleans, Jan. 8, 1815 ? •J 1. How long from tin- 1 >< claration of Independence, July 4, I77t>, to the present time? Daniel Webster was horn Jan. 18, 1782, and died Oct. ge did he die? ote given July 6, U | aid Sept 9, 1861 ; hovr vraa it on interest? J 1. Find tin- time from Apr. 4, 1857, to Dec. 12, 18< Find tin- time from Dec. 16, 1839, to Mar. 26, 1848. 26. Find the time from Nov. 13, 1816, to May 12, 1841. •J 7. Find the time from June 21, 1842, to Feb. 20, 1860. BZAXPLBfl EH Ai»i»itk»n and Subtraction. I. A fanner raited I50tmah. 8pk, 4qt of oats. Having sold BObosh. -jik. and BSed 27bosh. lpk. lijt., how many ha> lnj remaining? Ana. 7Sbosh. Baring a journey of 127m. 4fur. lOrd. to perform in 3 I travel 48m. 2fbr, 6rd the nml day, and 54m. 4rd the 1 ; how tar hum I travel on the third d I have one piece of land containing 17a. 8r. 25rd. and another contain. '•. l.">rd. ; how much land shall I have • Uing 87a. 4. From the mm of 8bnsh. 8pk. 2qLlpt.and lObnsh. 2pk. 7u>h. 1 j>k. 8qt lj't. and 49boah. Spk. 2qt l; . Idbnah. Iqt 203. Wliich i> preftnblet How many days rrc considered a mouth! 166 COMPOUND NUMBERS. 5. From the sum of 5rd. lyd. 2ft. 4in. lb.c. and 4rin. 2h.<\, take the difference between lOrd. 5yd. 2ft. Tin. 2b.c. and lid lyd. 1ft. Gin. An?, lb.c G. From a piece of silk mot Wiring 10 yd. lqr. 3na. 2in., there were cut 3 dresses, the first measuring 15yd. 8qr. lna. lin., the second 14yd. 3qr. 3na. lin., and the third 14yd. 2qr. 3na. 2in. ; what remnant remained? 7. B sold an ox which weighed 16cwt lqr. lolb., and 2 cows thai weighed 6cwt lqr. 10 lb. arid 5cwt 3qr. 201b.; also 2 swine, that, weighed 4e\vt. 3qr. 181b. and 3cwt. 3qr. 21 lb. How mucli more beef than j>ork did be tell? 8. If from 2 casks of wine, containing 63gal. 3qt. Ipt. ogi. and 56gal. 2qt. 2gi., there be taken 75gal. 2qt. lpi. ogi., how many gallons, quarts, etc., will remain? '.'. From a mass of silver weighing 471b. 8oz. 16dwt. 22gr., a >ihersmith made 48 spoons weighing 71b. 8dwt. M .nr. and a Cake-basket weighing 31b. 6o*. Bdwt l"»L r r. ; how much silver remained in the mass? MULTIPLICATION. *<20 I. In the multiplication of both simple and compound numbers, the multiplier is always and necessarily a simple abstract Dumber; for, to attempt to multiply by a concrete num- ber, e. g. 4 miles times 10, is, in the highest degree, absurd, though it is perfectly proper to >ay 10 times 4 miles. The pro- duct is of the same kind as the multiplicand; for repeating a number does not change its nature, £03. The principle is the same as in multiplication of simple numbers. Hence, To multiply a compound by a simple number, Rule. Multiply the lowest denomination in the multiplicand, divide the product by the number it takes of that denomination 204. "What is the multiplier in all cases? What the product? 205. Rule? Troof ? Explain Ex. L MlLTIl'LICATI 1. Gd. 3qr. per yard ? 23. How much wine in o casks containing 28gal. 3qt. lpt. 2gL each? 24. Multiply 9m. 7fur. 8ch. Bid 15 H Sin. by 8. Multiply Scire 5a, 85* K 2.V by 9. £06. To multiply by a composite number : BULB. Multiply by the factors of die multiplier (see Art. 6l). 205. Explain Ex- 14. 206. Rule when the multiplier ia composite? 169 26. Multiply 111.. *oz. lGdwt. 20gr. b\ 11). oz. dwt. gr. Multiplicand, 4 8 16 20 l~t Factor of Multiplier, 8 Partial Product. 37 10 14 i~6 2d Factor of Multiplier, 9 net. 34 1 12 »7. Multiply 7£ 6s. 8d 2qr- by 54. Multiply 8bash. 8pk. 6qt lpt by 81. 29. Multiply Gib 4£ 75 29 6gr. by 49. 207. To multiply when the multiplier is large and not composite. 30. Multiply 8i 4cwt. 2qr. 61b. 8oz. 4 a = 3£sec. of time, and for 5h. 8m. 1 l£sec. Ans. a like reason 8! of longitude give 8sec. of time, which added to the 3&sec previously obtained, give lljsec, and, final h . of longitude give 4 times 77 = 308m. = oh. 8m. of time; .'. the difference in time between London and Washington is 5b. 8m. lllsec., and as London is farther east than Washington, the hour of the day is later in London than in Washington, i. e. it is 8m. ll£sec past 5 o'clock in the afternoon at London when it is noon at Washington. Hence, RULE. Multiply the difference of longitude, expressed in degrees, minutes, and seconds, by 4, and the product trill be (he nee in t> ued in tninutes, seconds, and SOths of 208. Ilow far does tlM MB appear to move in one hour' Wliich way' Give the table of longitude and time. Rule for finding difference in time of two placet <• Iragttn4i <>t m 172 COMPOUND NUMBERS. ik. 1. The place most easterly, has its hour of the day, at a piven moment, latest ; i. e. the day bcgUM first, noon comes first, and the daj closes first at the place most easterly. 39. The longitude of Boston ifl 71° 4' 9" west, and that of liin-ton is 77° 2' 48" west ; what is the difference in the time of the two places, and what time is it in Washington, at 3 o'clock, P. M., in BosttO ? subtraction, the difference of longitude is found to be 5° 58' 3'.> ', .-. (he ffiAueuee in time is 2."'»m. 54gsec, and at 3 in Boston it is 36m. and 5$sec. pmst 1 in Wellington, Ans. 40. The longitude of Paris is 2° 20' 15" east, and that of New York, 74° 0' from Greenwich ; what is the differ- ence in time in the two places? A •. Ans. Since Paris is in east longitude, and Hew York in west, theii difference in longitude is found by adding 2° 2(/ 15" to 74° (K 3". 41. What is the difference in time between Philadelphia, igitude, and Chicago, 87° 35' west longitude? 42. What is the di:" in time between New Orleans, 90° 7 t longitude? 43. What is the difference in time for 90° in longitude ? DIVISION. 209. Here, as in the three preceding sections, the principle is the same as in the corresponding operation in simple numbers. Hence, To divide a compound by a simple number, Rule. Divide the highest denomination of the dividend, and set down the : if there is a remainder, reduce it to the lower denomination ; to the restdt add the given quantity of that denomination, and divide as before, setting down the quotient and reducing the remainder, etc. 808. Which has the hour of the day latest, the most easterly or most west- erly place I ■ difference in longitude found when one place is in east and the other in west longitude? 209. Rule for dividing a Compound by a Simple Number? Principle? DH it; . l. Divide W£ 7a, 1,1. lqr. !• OPKi: un»\. X s. (1. qr. at of 7)30 7 1 1 nderof2i "~7 g g « a reduced to shillings and added 17-., which, divided by 7, give u qnotient of 6s. 30 7 l l. Proof. and a remainder oi 2. Divide lfur. 9rd. 2yd. Oft. Oin. lh.c. by 5. Aiw. 9rcL iv.i. 2& Gin. 2b.c. 3. Divide 20gal. 2qt. Opt. 2gL by 7. 2gal. 3qt- lpt- 2gu l. Divide 181b. loz. 15dwt 8gr. I 5. Divide 171b. 5J 13 1& 1< G. Divide 80t ocwt. 3qr. 24B>. 12oz. by 8. 7. Divide ( 'yd. 2qr. 8na. by 9. 8. Divide 8wk. Gd. 1 lh. 17m. 57sec by 3. :•. Dfvid 4' 10" by 10. 10. Divide I07gal Iqtby 12. 11. Divide Wbosh. Bpk. 2qt. lpt. by 11. 12. Divide 51c. 7r.it. L5ca.ft. 171Gcu.in. by 4. 13. Divide 16a. lyd. 1ft. 7<>in. by 2. 1 1. Divide 87t L2cwt 8qr. 51b. 10oz. 4dr. by 9. 15. Divide 71a. 8r. 1 trd. 8yd. lit. 72in. 1.;. L& If 9 Bilvez Bpoons weigh Lib. l<»z. 17dwt 12gr., what is thr weigh! of etch spoon? Ans. loz. 17dwt ! 17. If ■ family nee 29gaL 3qt 2gi. of molasses in G months, what i> the avenge per month? lqr. of hay is harvested from 5 acres, what on one acre? 19. . 7cwt 2qr. 101b., what is the I grain-bins contain 148haek Spk. Sqfc lpt of grain, wlia; in? 21. It' a man travel 212m. lfur. 2Grd. 2yd. in 7 days, what rel per da v 10. 15* 174 COMPOUND NUMBERS. 210. To divide by a composite number, we may divide by its factors, as in division of simple numbers (Art. 79). 23. Divide 3411b. Ooz. 12dwt. by 72. lb. oz. dwt. gr. 9 ) 341 12 First divide by 9 and 8)37 10 14 16 then J he 5 U0 Ji ent b ^ 8 > ' and thus by 72. 4 8 16 2 0,Ans. 21. Divide 396£ 2s. 3d. by 25. Divide 725hu>h. Opk. 6qt. lpt. by 81. 26. Divide 3971b llg 75 IB 4gr. by 63. 27. Divide 958m. 5fur. 5ch. 12 li. 5£fin. by 48. 211. When the divisor is large and not composite, set down the work of dividing and reducing. There is no device for rendering the operation easier. 28. Divide 135bush. 3pk. 3qt lpt. by 47. bush. pk. qt pt 4 7)135 3 3 1 ( 2bush. 3pk. 4qt. lpt., Ans. 94 4 1 bush. _4 , g ~ i Having found that 47 is con- 14 1 ' tained twice in 135, multiply 47 by 2, and subtract the product, 2 6 pk. g }, from 135, which leaves a re- 8 maiixlrr of 41 bushels ; reduce 2 i i qt, the H hu-diels to pecks, and add 13 3 the 3 peeks, making 1 67 pecks ; — — then divide the 167 pecks by 47, * ^ *&' and so continue the process till the work is done. 4 7 pt. 47 210. Rale for dividing by a composite number? 211. Method of dividing when the divisor is large and not composite? Is there no easier mode? MYISIOy. L75 29. If 587 yards of cloth 4ft. 2d. l«n\, what is the price per yard ? 30. Divide U89gaL Ipt 8gl by ! :;i. A ftrmer raited B5884bush. 8pL 8qt lpt. of corn on 643 acresof land ; how much was the yield per acre? Suppose a man should travel 10599m. ofur. 14r& 4yd 2ft. 5in. in 813 days, what distance would he travel per day? 33. in 127 days a ship sals lis. 9° 4& 8t in length : i . l'is an area, 1 foot long and 1 ineh wide, end 1" is an area I imli iqoare; In enbfce meiinre V k ■ $oKd, l foot long, l foot wide, and 1 ineh d< ■• , 1 inch wide, and 1 inch a cubic inch ; etc 318. Lot Of now determine the denomination of the product obtained by multiplying any two denominations together. riiiLOSornirALi.Y. iiliarlt. 2 units X 3 units = 6 units, i. e. 2ft. X3ft. = 6ft. 1 M X f\ unit = ft unit, i. e. 2ft. X3' = 6* 2 « X T f* " =rf* " >-e.2ft. X3" = 6" etc. etc -j^unitX tV unit= r | T unit, i. e. 2' X3' = 6" A u X X = T VW " i. e. 2' X3" Estf" rV " X T Ai " =nhi " i-e- 2 X3'" = 6 etc etc. T^-unitX T f* «nit= ^ft^ unit, i. c. r X 3" == G"" tIt " X T?V? " = T¥^ffTJ L e - 2" X 3'" __ 6 ///// lU " X W » W " =mfm " i-e. 2" X 3"" ' nilll/t etc etc. «517. What U 1' in linear measure ? 1' in square measure? 1" in tquara ■M««l»f V in cubic meagre? 1"? Y"1 1""? what . 9 OPERA 6 7 9" 2 r 5" 13 3' 6" 3 10' a" 3'" 2' 9" 2'" 9"" IT.' II ince, to determine the denominmtioB of the product of tu in duodeoim BULB. Add the indict* of factors together, and the t ndex of the product. in length n •"," b breadth] First, 9" X 2=18" = 1' 6"; the 6" we write UOder the seconds, and reserve the 1' to add to the next product, thus, 7' x 2=lf. which Su- ited by the 1/ previ- Ans. 17 4' 9" ft'" 9"" onely obtained gtae W = lft. 3'; the § Ifl writ- •wn, and the 1ft. is carried to the product of the feet, mak- [n like manner we multiply by the V and then by the partial products as in the margin* Finally, • partial product! i- the product aought II 219. To perform Multiplication of Duodecimals, Kii.k. Ihj the ride for multiplication of compound numbers, multiply the multiplicand by each term in the multiplier, and write ■■/is if the second partial products in the order of their values, so that similar terms shall stand in a column together ; the sum of the partial products will be the entire product. 2. 3. Multiplicand, 8 4' G" 4 8' 9" Multiplier, 2 8' 5" 2 3' 7" G 9' 0" 9 5' 6" 2 3' 0" 0'" 1 2' 2" 3"' 1' 4" 1 ()'" G"" 2' 9" 1"' 3"" 9 1' 4" 10'" 6"" 10 10' 5" 4"' 3'"' 1. What quantity of board* will be required to lay a floor long end . 103ft. 10' 6" 2'". 218. Rule for determining the denomination of ft produ. philo- sophically and familiarly. 210. Rula for multiplication of duodecimals? 180 COMPOUND .\ 0. What are the contents of ■ granite block that is Cft. 3' long, 2ft. 4' wide, and lft. 3' thick? Ans. 18ft. 2 y 9". (See Art. 104). G. How many feet of flag-stone in a walk loft. 0' long and 3ft. 4' wide? 7. How many solid feet of marble in a block that is 8ft. 3' long, 3ft. 6' wide, and lft. 4' thick ? 8. How many cubic feet of earth must be removed in digging a cellar 15ft. 6' long, 12fu 8' wide, and 6ft. 8' deep? 9. How many feet in a stock of 8 boards, that are 10ft. 8' long and 10' wide ? Ans. 71ft. 1' 4". 10. How many feet of boards 1' thick can be sawed from a a stick of timber that is 12fu 8' long, 10' wide, and 8' 4" thick, provided no timber is destroyed by the saw-cut? 11. How many cords of wood in a pile that is 18ft. 6' long, 6ft. 8' high, and 4ft. wide? 12. How many square yards of carpeting will cover a room that is 18ft. long and 16ft. 6' wide? 13. Multiply 3ft. 6' 4" by 8ft. 9' 6". Division. 2*20. Division of duodecimals is like division of other compound numbers. Ex. 1*. Divide 24ft. 10' 10" 4'" by 7. Also by 9. OPERATION. OPERATION. 7 ) 2 4 1 0' 1 0" 4'" 9 ) 2 4 10' 1 0" 4" ~2 lP 2" 5" 9"" 4""' 3. 6) 45 4' 1" 6"' Note. When both dividend and divi.sor are expressed as compound numbers, they m:iy be reduced to the smallest denomination in either; after which divide, and the quotient will be units, i. e.feet; thus, 68ft. l(y 8" divided by 2ft. 8' equals 9920" V 384" = 25 Jf, i. e. 25ft. 10', Ans. 320. How ia division of duodecimals performed? How when the divisor is compound 1 ts. 3 6' 2. 8" 4"' 8) 31 6' 8" 8'" misci:i.i.\ 181 4. The area of a floor is 197ft. V 8", and the length of the floor is 1.0ft. 8'; what ii its width? 12ft. 7'. 5. The area of a garden walk is 89ft. 4' and its width is 2ft. 8'* what is its length? Miscellaneous ExAMFLM n RD Ximhers. 1. If I52bush. '"'pk. 3<[t. Ipt. of wheat grow on 9 acres of land, how many bushels grow on 7 acr< 2. A man having 207m. 4fur. 25rd. 1yd. to travel in 6 days, 80m. 8fur. 2 "ml. 5yd. on die first day, and 33ra. 4fur. 20rd. 4yd. on the second day; how far per day must he travel to finish the journey in the remaining 4 da; 3. Multiply 3£ 15s. Gd. lqr. by 857, and divide the product by 167. 4. I have a stock of 9 boards, which are 12ft. 8' long and 10 r wide. With these boards I wish to lay a floor 15ft. in length ; how l I make it ? 5. If 1 cubic foot of water weighs 02 lb. 8oz., and if a cubic foot of granite weighs 2 L times as much, what is the weight of a block of granite 12ft. long, lft. s' wide, and 9' thick? 6. From the sum of 3\vk. 6d. lOh. 20m. 18sec. and 2wk. 3d. 50m. aOsec. take the difference between Cwk. 5d. 8h. 25m. 30sec. and 5wk. 2d. 22h. 18m. 15 7. What ifl the difference in time between Amsterdam 4° 44' oast longitude, and Annapolis 7G° 43' west longitude? 8. When it is noon in Dublin, G° 7' 13" west longitude, it is 10m. and 16 }| S CC. past 8 o'clock in the evening in Peking; what is the longitude of Peking? i tow many days, hoars, eta, from 30m. 20sec. past 3 o'clock. i\ m.. Peh. 8, 1864, to 40m. toseo. past 8 o'clock, a. m., July 4, . reckoning each month at its actual length? LO. Bought 8cwt 8qr. 181b. of sugar at 8£c. per pound, and sold £ of it at 8c. and the remainder at Die. per pound; what gained by the transaction 11. What ifl the value in Avoirdupois Weight of 241b. 6oz. llMv, :,t? 16 182 COMPOUND NUMB! 12. How long a time will be required for one of the heavenly bodies to move through a quadrant of a circle, if it moves at the rate of V 3" per minute ? 1.;. Th« distance from Eastport, Maine, to San Fran< California, is about 27G0 miles. If a man, starting from East* port, travel toward San Francisco for 75 days, at tin- rate of iMm. 3fur. 20rd. per day, how far will he then be from Francisco ? 1 1. A certain island il 75 miles in circumference. A and B, start tag at the same time, and from the same point, and going in tli-- same direction, travel round this bland, A at the rate of 24m. 3fur. lOrd., and B at the rate of L5m. C.fur. 20rd. per d how tar apart air A and 15 at the end of live days? 15. A merchant bought L 25 barrels of floor, at IX 15s. 6cL per barrel, and afterward exchanged the flour for 260 yards of broadcloth, which he sold at 18s. 9d. 3qr. per yard; did he gain or lose, and how much ? 16. How many feet of boards will be required to make 12 boxes whose interior dimensions are 5ft. 6', 4ft. 9', and 3ft. 8', the boards being 1' in tliicki: 17. I low many feel less are required to make 12 boxes whose exterior flimmikm i are like the interior of those in Ex. 16, the boards being of the same thickness? Ana. 111ft. 4'. 18. Wnal is the difference of the capacities of the two sets of boxe ! in I.\. 16 and 17 ! Ans. 122ft. 10'. 19. How many times will a wheel Oft. 8in. in circumference turn round in running from Boston to Worcester, a distance of •1 lm. 4 fur.? 20. How many gallons, wine measure, in a water tank 4ft. Gin. long, oft. 8in. wide, and 3ft. 9in. deep ? 21. If a teacher devote 5h. 30m. per day to 50 pupils, what i< thi average time for each pupil ? 22. If a man, employed in counting money from a heap, count 75 -ilver dollars each minute, and continue at the work 12 hours each day, in how many days will he count a million of dollars? 23. How many pounds of iron in one scale of a balance, will poisa 75 pounds of gold in the other 6cale ? 183 DAG i:. *>*J1. | i. || ■ contraction of per cen turn, a Latin phrase which meant by the kttndnd; thai) ten per cent, of a boshd of corn means tea one-hondredthfl of it ; i. <•. tea parts :v hundred pa per cent, of a sum of m I onc-hundredths of the sum, i.e. $6 out of every $100 ; \*..i I of the words p^r cent., it is eastOnMOyto use this sign, °/ ; t!. it. is written b°/ g ; \\ i>cr cent., ±\°/ - 5jtJ % J. The RATE vvk I IEHT. IS the number for each hundred ; (has, 6J£ is ^...or .06, i. e. G parts for each hondred \ £££3. The Pi oent ot'sj. Tlie hubm raeutl by multiplying $1 by 100, and dividing the prodnd by 1. 1 ; 1G* 186 PERCENTAGE. Rule. Multiply the 'percentage by 100, and divide the product by the base. Note. This rule is the converse of that in Art. 226 ; thus, 25 per cent, of S4 is $4 X .25 = $1 ; and, conversely, $1.00 ~ $4 = .25, i. e. l'j per cent. 2. What per cent of $150 is $18? 1800 -7- 150 = 12 per cent., Ans. 3. What per cent of $300 is $19 ? Ans. 6$ per cent. -1. AVliat per cent, of S350 is $43.75? Ans. 12£ per cent 5. What per cmt. of $340 h $84 ? ft. What per cent of $64 is $16? 7. What pet cent, of $1000 is $5? Ans. £ of 1 per cent 8. B inherited $3500, and in 8 months spent $870 ; what per cent, of his inheritance did he spend? What per cent, had he remaining? Ans. Spent 25 per cent., and had 75 per cent. 9. Outofaca.sk of wine containing 96 gallons, 32 gallons were Irawn; what per cent, of the whole remained in the ca>k ? 10. A merchant having $1000, deposited $650 in a bank; What per cent of his money did he depoafl ? 11. A. teacher baring a salary of $2400, spends $2000 an- nually ; what per cent, of h, does he sa\ Pn< • 228. To find the base when the percentage and the rate are given. Ex. 1. |6 is 3 per cent of what sntn ? '• is 3 per cent, then 1 per i of $6, which is $2, and if $2 is 1 per cent., then 100 per cent i- LOO times $2, which 0;.«. si*) i> "> per eent of $200, Ana, The same result is obtained by first multiplying SC by 100, and then dividing the product by 3 ; thus, $600 -r- 3 = $200, Ans. Hence, Bui B. Multiply the percentage by 1 00, and divide the product by the rate. 227. Rule for finding the rate when the base and percentage are known? What of this rule, and that in Art. 226? 238. liule for finding the base when the percentage and rate are known? 2. $0 ia l per cent, of what .sum? Aim 3. $ per crnt. of "hat sum? 160. 4. $12 ia 7 per cent of what sum? 7i.l2f. <> i^ 16 per cent of what mm? 6. 12 m 8 per cent of what numl • 400. :. of what number? 8. 33 is 1| per cent, of what number? 9. A fanner bought a form tor $27.~>C>, which was 25 per cent: of his pr op er ty; what waa his property? An-. $11024. 10. A man sold 56* geese, which was 28 per cent, of his flock ; how many geeM had he? 11. A merchant having a quantity of flour, bought 600 barrela more, when he found that the quantity bought was 75 per cent of all lie then had ; how many barrels had he before he bought the last lot? Ans. 200. 19. r saves $400 annually, which is 16§ per cent, of ury ; what h his salary? 13. The population of a town was 7G9 greater in 1800 than in 1850, and this was an increase of 20 per cent, on the popula- tion of 1850 ; what was the population in 1850 ? I XT K REST. 229. Interest is money paid for the use of money. Tin- Pkin( ii-al is the sum for which interest is paid. Am ui NT is the sum of the principal and interest. 3ttO. An example in interest is only a question in percentage. The pri ncip al i> th»- bate of percentage (Art. 224 ). the interest is \agt (Art. 228), and the interest on SI for a year is the i ritten fcchnaHj ( A £31. The rate i- usually % /7.#W By far, and a higher rate than the law all n/. In and and most of the United States the legal or rial the Principal' Amount? 230. I. what relation to percentage? What is the base 7 The percentage? The rate* rate fixed I What is usury ? Name tin- legal rate in 6ome : 188 PERc lawful rate is 6 per cent. ; in New York, 7 per cent. ; in most of the Western States, M high SI 10 per cent, by agreement; in Texts, as high as 12 per cent f in California, any rate by agree- ment, etc. On debts In favor of the United States, G per cent. In France and England, 5 per cent. Note. In this treatise, 6 per cent, is understood when no per cent, is mentioned. 232. When the rate is 6 per cent., the interest of $1 for a is 6c. j for 2 yea i ft, 1_< •., etc. ; for 1 month, fa of Gc. = 5 mills or ^c. ; for 2 months, lc. ; 3 months, l£c. ; G months, 3c.; 9 months, 4^c, etc ; for 1 day, ^ of 5 mills = £ mill ; for 2 d $m. ; 3 days, £ra.; 4 days, §m. ; 6 days, gin.; 6 days lm. ; 7 days, l.Wn.; 9 days, l$m. ; 12 days, 2m.; 24 days, 4m.; etc., etc Hence, To find the interest of 81 at 6 per cent, for any time, BULB. Tak$ Gc. (= $.0G) for each year, lc.for each 2 months in the part of a year, 5 mills (= $.005) for the odd month, if there be one, and 1 mill for each 6 days in the part of a mojith. 1. What is the interest of $1 for 3yr. 9m. 18d.? OPERATI $.1 8 = interest of $1 for 3 years .0 4 5 = " u 9 months .0 3 = " " " " 1_8 da $.2 2 8= " " " " 3yr. 9m. 18d., Ans. 2. Wbti is the interest of $1 for 2yr. 5m. 20d.? OPERATION. I = interest of $1 for 2 years. .0 2 5 = " " " " 5 months. .003$ = " « " "20 days. $.1 4 8 £ = « « « " 2yr. 5m. 20d., Ans. 5131. What will be understood when no rate is mentioned ? 233. Kule for finding the interest of SI at 6 per cent for any given time? in 189 With very little practice the ptij.il will, without making n | mentally determine the i- -iv length of time. This hahit is very desirable, as it will ( \ hat ii the interest of Si for 8jrr. lm. I Ans. $.187$. 1 What is the interest of $1 for lyr. 3m. I Ans $.079g. of Si for 4yr. 2m. ' Ans. $.250§. G. What is the intm-t of $1 for 4yr. 5m. 17.1.? 7. What is the interest of $1 for 4yr. 9m. IlM.? 8. What ifl the interest of $1 for lOyr, 11a, 7d.? 9. What ifl the interest of $1 for 2yr. 11 in. 10. What is the interest of SI for lyr. 8m. 3d.? 233. To find the interest of any sum at 6 per cent, for any given time. The interest of $2 is evidently twice as much as the interest of $1 ; so the interest of $3, $4, or $7, is 3, 4, or 7 times (he Bt of SI ; and the interest of S2.25 is 2.25 (i. e. 2 and 26 hundredths) times the interest of $1 ; .*. to find the interest of any number of dollars we have only to find the interest of $1, and then multiply the interest by the number of dollars in the princijuil. 11. What ifl the interest of $2 for lyr. 5m. 9d.? S .0 8 6 $ = interest of Si for lyr. 5m. 9d. 2 $.173 = interest of S2 for lyr. 5m. 9d., Ans. 12. What is the interest of S6.50 for 3yr. 8m. 18d.? = interest of SI for 3yr. 8m. 18d. 11150 1338 $ 1.4 4 9 5 = int -G.50 for 3yr. 8m. 18d., Ans. »33. What is the Note ? 233. Rule for finding the interest of any sum at ■ nt., for auy time' IU-asou? 190 1'KIK 13. What is the interest of $300 for 2yr. 7m. 2 1.1. ? $.159 = interest of $1 for 2yr. 7m. 24 in interest only .'J decimal places in the product will DC pre but if the 4th decimal place is 5 or more, the third place will be increased l»y indth. 21. What is the interest of $225.87 for lyr. ftm. 1 -"„!. ? Ans. $17,505. What is the interest of $85.40 for 2yr. 6m. 9d.? Ans. $5,363. 23. What is the interest of S 150.87 for lyr. 7m. 9d.? 24. What is the interest of $375.50 for 2yr. lm. 8d.? What is the interest of $225.75 for lyr. 5m. 12d.? 26. What is the interest of $84.82 for 2yr. 4m. 18d.? 27. What is the interest of $125.16 for lyr. 11m. 25d.? 28. What is the interest of $658.25 for lyr. 2m. 13d. ? 29. What is the interest of $125 from June 7, 1851, to Feb. 11, 1^ Ans. $20,083. Note 2. Ex. 29 differs from the preceding only in its being necessary to find the time (Art. 203). 30. Find the interest of $154.25 from April 18, 1852, to Jan. 26, 1855. Ans. $25,657. 31. Find the interest of $172 from Aug. 7, 1854, to Sept. 9, 1856. 32. Find the interest of $254 from Nov. 12, 1855, to Jan. 30, What is tin- interest of $132.25 from Nov. 13, 1836, to j. L841? 34. What is thfl int. rest of $100 from March 26, 1841, to *34. What of decimal placet after the third in the Ant.t Explain Ex. 29. 235. To find the interest when the principal is in pounds, shillings, pence, and farthing BULK. Reduce the lower denominations to the decimal of a pound (Art. 175), then proceed at icith dollars and cents, and ! fy reduce the decimal part of the interest back to shillings, pence, and farthings (Art. 176). 'it 3 decimal places in the multiplicand are used. 35. What is the interest of 56£ 10s. 6d. 3qr. for 1 yr. 6m. Ans. 5£ 6s. 3d. lqr. 36. What is the interest of 246£ 18s. Od. lqr. for 2yr. 3m. 15d.? 37. W. rest of 125£ 16s. 8d. 2qr. from Nov. 13, 1861, to March 26, 1863? 236. To find the interest of any sum for any time, at any other rate than 6 per cent.: I { r i . r. . / V rsf find the interest at 6 per cent. ; then divide th is interest by <*,. wiM WtBfim tk$ interest at 1 per cent.; and, finally, multiply the interest at 1 per cent, by the given rate. 88. What is the interest of $124.50 for lyr. 4m. 12d., at 5 per cent.? OPERATION. $ 1 2 4.5 0, Principal. .0 8 2 = Int. of $1 at 6 per cent, for lyr. 4m. 12d. 24900 99600 6 ) $ 10.2090 = Int. of Principal at G per cent. $ 1.7 1 5 = Int. of Principal at 1 per cent. $ 8.5 7 5 = Int. of Principal at 5 per cent.. An-. 39. What is the interest of $9 t lyr. 9m. 18d., at 8 percent.? Ans. $49,284. 235. Rule for casting interest on pounds, shillings, etc. ? How many decimal places in the multiplicand are used? 236. Rule for computing interest at any given : •r. 40. What ia the interest of $256.84 for lyr. 3m. 15d., at 9 per cent. ? 41. What is the interest of 24£ 6s. 8d. lqr. for 2yr. 9ra. 12d\, at 5 per cent ? 84 8qr. 48. Wl,a: u the interest of 150£ 10s. for 2yr. 4m. 6d., at 4$ nt. ? 237. To find the amount of any sum at any rate for any ti. ' find the interest by the preceding rules, and to the interest add the principal. 43. What is the amount of $325.75 for lyr. 4m. 24d., at 6 per cent. ? OPERATION. $ 3 2 5.7 5, Principal. .084 = Int. of $1 for lyr. 4m. 24. eighteen handled and fifty-eight, with in t erest to he paid on the first day of April, A. D. eighteen hundred and fifty-nine, and then ird half yearly, at the office of the Treasurer of the said Tr uste es in Andover. ■ J. S. Pay well. In presence of J. L. Tki man. bn>OB8IMKKT8: April 1, 1859, $58.75 ; October 1, 1859, $1 17> ; Nov. 1, l> | Feb. 1. I860, $100; April 1,1860, 7.614; what wi I ily 1, 1860? Ans. $1065.75. 239. The rule given in Art. 238 is the one adopted by the United States Courts and most of the State Courts; but, when settlement is made within a year after interest commences, it is customary to adopt the following 838. Where is the work performed? Why not in the book? 239. What ml« is usually adopted when the time in a year or IN : r.. 1. Find the amount of the principal from the time <"st commenced to the tun from the time of -payment to the time of settle//, with their interest from U of the principal »vo rule is often used whftterer maybe the timo; hut for Ion;; periodl it is manifestly unjust, for by it the debtor, by merely paging ■ annuaUg at 6 per cent, will In Ian than M 1 hi* entire nd not only so, the person who loans the money will actually ! to the one who borrows. Bos{on, Ma* Iff, 1861. For value received, I promise to pay to Samuel Adam?, on demand, three hundred eighty-seven and y *y dollar.-, with inter- IIf.nky Phillips. nts: July 21, 18G1, $75; Oct. 10, 1861, $125; 24, 1862, $50 ; what was due at the time of settlement^ May K», 1862? SOLUTIOX. Principal, $ 3 8 7.7 5 Latere** of Principal for 1 year, 2 3.2 6 5 Amount of Principal, $ 4 1 1.0 1 5 Let Payment, 7 5. Int. of 1st Payment from July 21, 9m. 24d., 3.6 7 5 2d Payment, 12 5. Int. of 2d Payment from Oct. 10, 7m. 5d., 4.4 7 9 Payment, 5 0. Int. of '3d Payment from Feb. 24, 2m. 2 Id., 0.6 7 5 Bom of Payments, with their Interest, 2 5 8.8 2 9 Sum due May 15, 1862, A $ 1 5 2.1 8 6 58. A note of $2500, dated June 4, 1861, has the following Im Sept 1. 1861, >; Dec. 24, 1S61, S84f. ■ . 18, 1862, $362.63 1 what was doe May 12, 1862? Ana. $821.5*9. vi39. Is this rah jii-t for long periods of time! Why not? 17* 198 240. Many business men, In computing the interest on note.*, adopt the following Kn.r. Find the interest of the principal for a year ; also of each payment made during the year from the time of payment to the end of the year. Then subtract the. sum of the paym together with their interest, from the amount of the principal, the remainder is a new pr inn 'pal. with which proceed for another year, and so on to the time of settlement. 59. A note of $1500, dated July 25, ls.V.), has the following iMDOftJ :. IS, 1859, $100; Jan. 25, I860, $300- 19, I860, $260; Dec 15, I860, $235] Aug. 13, 1801, : what wa> due June 16, If SOLUTI" jut of Principal to July 25, 'GO, lvr., ( $ 1 5 9 0. It Paym $100. Int. of U Puv't t«» July 25,'CO, 10m. 12d.. 5.2 2d Payment, 3 0. lot Of W Payment to July 25, 'CO, Cm., 9. Sum of 1st and 2d PayV. with Int., 4 1 4.2 Ipfl Remainder or 2d Principal, 1 1 7 5.8 Int. of 2d Principal to July 25/61, lyr., 7 0.5 4 8 Amount of 2d Principal to July 25, 'CI, 1 2 4 C.3 4 8 3d Payment, 6 0. Int.of* 1 Pay'ttoJaJy25,'61,10nt, 12.7 5 40) P.-mmn't, 2 2-".. Int. of 4th Pay't to July 25, '61, 7m., 7.8 7 5 Sum of 3d and 4th Pay'K with Int., 4 9 5.G 2 5 Remainder or .°>d Principal, 7 5 0.7 2 3 [nt. of 3d Prin. to June 1 8, '62, 1 Om. I 3 9.7 8 8 Amount of 3d Prin. to June 13, 18C2, 7 9 0.5 1 1 5th Payment, $3 00. Int. of ;»th Pay't to June 13, 'G2, 10m., 1 5. 5th Payment, with its Interest, 3 15. Sum doe at settlement, June 13, 'G2, An-.. $ 4 7 5.5 1 1 "40. Third rule for computing interest on notes? 60. A note oi I ' - i M.iv 15, t859, has the following June 1, IS59, $100; July 7, 1860, $100; 1> ; ' I860, |50 j June 7, 18G1, $100; •; what was due July 15,1863? Ans. $302,011 Note. There is, perhaps, no other operation in Practical Arithmetic in which accountants differ so much as in the mode of computing interest. All the methods are based upon the principles developed in the preceding pages, and it is b* is no plan, universally applicable, which is more and simple than the foregoing'. The solution BftJ usually, however, be much shortened, as in the following Article*. principal advantage arises from the best divisions of timo. Facility in making the best divisions can be easily acquired by practice, and to one having frequent occasion to compute interest the attainment is of great importance. 241. The interest of $1 for 6 days, at 6 per cent., is 1 mill. Tlif interest of $1 lor Ml times 6tl. — G0d. = 2m. is 1 cent. The interest of $1 lor ten times 2m. = 20m. = lyr. 8m. is 1 dime. The interest of SI for ten times 20m. = ICyr. 8m. ifl So the interest i f, or $1000, for the same times, is 2, 3, or 1000 mills, cents, dimes, or dollars. Thus we see that any nuni- ber of dollars expresses its own interest in mills, cents, dim. dollars for the above-mentioned times, and hence, to know the Interest it is only necessary to determine the place of the decimal point. 01. What is the interest of $324 for 93 days ? OPERATION. I... % = ht for 3 Od. , AU , h . ke ex ? ra .P le3 can be _ _ , solved in a similar manner. .10 2 = Int. lor 3d. Henoe> = Int. for 9 3 d., Ans. 242. To compute interest at 6 per cent, for months and <1 IIui.k. Move the decimal point in the j)rincij)al two places to- ifl of different modes of computing interest? What of the best divi- sion of rin . ' Ml 1. Any sum of money expresses it* own interest at • cent, fo* what times? 200 PERCENTAGE. tcard the left, and the result will be the interest for two months or sixty pays. Move the point three places toward the lej), and the result will be the interest for ux lays. Then tufa such muU iij)l< the interest. Proof. Divide the computed interest by the interest of tht principal for one month, and the quotient should be the number oj months ea in the example ; or, divide by the interest for one day, and die quotient should be the number of days. ii: 1. This is the most simple modo of proof, and applies to all rules for computing iutcrest. The Problems in Interest, page 203, furnish other methods of proof. : ■ 2. In computing interest it is customary to consider 30 days a month tad IS nootfai ■ year, and ,\ the computed interest tor 12 times 30 days, or 360 days (i.e. for g|j$ = j§ of a year), is truly the interest for a whole year. Thus, the computed interest for any number of days is ^ too large and it must .'. be diminished by 7*3 of itself to find the true rati As UD ially computed for months and days the difference U slight, and, in course < : is seldom considered ; but in England, and in dealing with the United States Government, it is customary to eompute true interest. What is the i: I for 7 months anf'a month, .*. 1 V, of llio im»TiM of $1, or any other sum, for 1 month. i> tin- interest utf the *am« sum for 9 day-. Iii lik-- mannrr, ,V, of the intnv-t of any sum fur fl r •/ months is the interest of the sa/ne sura for Ws :is many days. 64 What is the intm-t of |7W for 2m. Cd.? OPERATION. $ 7.6 5 = Int. for 2m., f. e. for C Od. 5 = Int. for ^ of CM., i.e. $8.4 1 5 = Int. for 1., Alls. C5. What is the interest of $845 for 6 days? 84.3 mills ss $.845, Ans. G6. Whsi is tbfl interest of $345 for 2 months? 845 cents = $8.45, Ans. • for lyr. 8m. ? Ten times 845 cents = $84.50, Ans. G8. What is fee interest of $845 for lG§yr.? Ten times $84.50 = $845, Ans. Note. The pupil will observe that merely changing the position of •imal point, as in the four preceding examples, BJfoSi the interest of any sum lor G days, for 2 months, for 1 year and 8 months, or for 16J yeur>. G9. What is the interest of $845 for lyr. 10m. Gd. ? OPERATION. $ 8 4.5 = Int. for lyr. 8m., i. e. for 2 0m. 8.4 5 sa Int. for ^ of 20m., i. e. 2m. .8 4 5 = Int. for £ of 2m., i. e. Gd. $ 9 3.7 9 5 = Int. for 2 2m. Gd„ Ans. What is fee interest of $348 for 22 days? 3 ) $ 3.4JS = Int. for_6_0 days. LI 6 r= Int. for 2 day s. .116 ss Int. for 2 days. $ lj 7 6=5 Iut. for "22 days, Ans. «_ — _ — 1 ■ ■ '444. One tenth of thf jnttrest of any sum for any number of months, is the Interest of the same sum for how many day*' Bail Ibff •!• ••> 1 mining the brfpfff of any sum for 6 day*? For 2 months! For lyr. 8in.' Siu.f 202 KNTAGE. 71. What Ki tlie interest of S412 for 5m.? Ans. $10.30. 7 2. What ii ill*. interest of 1 12 for 2m. 22d. ? 574 What ia the interest of $54 for 22d. ? Ans. $.198. 7 I Wbal ifl the interest of $2148 for 3m. 10d.? . Wliat ii the interest I C lyr. K'm. i),].? the interest of $173 for 1 yr. 8m. ? 944. In some Statei interest ii allowed on the annual In- terest of the principal which is due and unpaid, if the note is written " with i n t er e st annually." Sndi exanipl. I may be solved by CO est on th> wkoh time and on each year's interest for the titne it is due and unpaid ; but the fol- towSn '■■ il mode of computing M annual interest " will be of service to the business man. R' >m the principal for the given num- ber < \rs ; on this interest find the interest for half of the years less one, and the months and days ; and this latter in- terest is the J. x CESS OF t for the I time. To this excess add the interest on the principal for the whole time, and the sum is the annual interest for the given timu. 77. What ii the annual interest of $800 for B $8 0, Principal. J8 sb 8imple Tnt. of SI for 5 years. 2 4 0.0 ss Simple Int. of $800 for 5 years. * j .12 = Simple Int. of $1 for 2yr. i. e. for —5- = 2yr. 2 8.8 = Excess of annual over simple Ink of $800 for Syr. 2 4 = Simple Int, of the principal, as above. $ 2 6 8.8 = Annual Int. of $800 for 5yr., Ans. 78. What is the annual interest of $600 for 6yr. 4m. 18d ? Solution. The interest of $G00 for 6 years is $216; the interest of $216 for L of (6 — 1) yr., increased by the months and days viz. 8$yr. 4m. 18d., or 2yr. 10m. 184. ia $87,868, and this is the excess of the annual over the simple interest of SC00 for 6yr. 4m. 18d. To this add the interest of $600 for Gyr. 4m. 18d., viz. $229.80, and we have $267,168, the annual int. 344. Rule for computing annual i • _ ... 20S the annual interest of $402.84 for 7yr. 8m. Gd. ? An-. $_'- 80. What i> the excess of annual over simple interest of $2o0 ... *4. In every example in interest there are four elements or particulars which claim special attention, viz. Principal, Rate, Time, and Interest, any three of which being given, the other can be found. To find the Interest when the Principal, Rate, and Time are I, has, thus tar, been the object of our discussion. Tin- other branches of the subject give rise to the following problems : 216. Problem 1. Principal, Interest, and Time given, to find the Rate. . 1. At what rate per cent, must $300 be put on interc-t to -18 in 2 years? w.ysis. $300, at 1 per cent, will gain $6 in 2 years; .*., to gain $18, the rate BUUt be the quotient of $18 -j- $6=3. Hence, i .v.. DMdi the given interest by the interest of the princi- pal, for the given time, at 1 per cent., and the quotient will be the rate. At what rate per cent must $1 12 be put on interest to gain $21.30 in 3 Jtt Am, 3. If S3G gain S7.5G in 3 years, what is the rate per cent.? 4. If $300 gain $43.80 in 2yi\, what is the rate per cent. ? 345. How many particulars claim attention In an example in interest* What are they' How many of them are given? 910. Object of Prob. 1? Rule? 247. Problem 2. Principal, Internet, and Bale gii to lad the T .1. far what time roust $200 be on interest at B pet 1 cent an $3G? Analysis. $200 in 1 yoar, at C per cent., will pain STJ ; in $86, the time in years must be the quotient of $.3 <'» ~ $12 = 3. Hence, Rule. Divide the given interest by the interest of the prhulpal for one year at the given rate, and the quotient will be the time. I low long must $254 be on interest at 5 per cent, to gain $44.45? ^yr. = «Yr. 6m. 1 low long must $75 be on interest at SO? Ans. 2.63 Jyr. = 2yr, 7m. 18d. 1. How long must $200 be on interest at I Si to nt to $286? faff what tinn- must $72 be put to iutrn-t at ft] pet cent, nount to $87.30? I 'or what time must $1000 be put to interest at 9 per cent. to gain $247.50 ? 7. How long must $100 be on intend at 5 pet | any sum trill double itself at i rate per cr/tt.. diride 100 by the rate, and the quotient will be the in years. 8. In how many years will $50 amount to $100, it being on Interest at 8 per cent? Ans. 12yr. Cm. long will it take any sum of money to double itself on interest si • t.? 1<». In what time will a sum of money triple itself on interest at 5 per cent? 247. Prob. 2? Rule? Rule for finding the time in which auy principal will double at any rate per cent.? ^ils. PftOBUttft. Interest, Time, and B find the Pkiwipal, Ex. 1. What principal, at 6 per cent., will gain $18 in 1 jr. Cm.? Awi.v-n. si. in lyr. Cm., at G per cent., will jrain .-. the principal nui-t hi- the quotient of $18 -~ .09 = Hence, Hl i.k. Divide the given interest by the interest of |1 for the given rate and tinn\ and tk§ quotient will he the principal 2. What principal, at 6 per cent., will gain $13 in 8 Ant. S83& 3. What principal, on interest at 8 per cent, per annum, will 150 BemUannoaUj ? 4. IS endowed a professorship with a salary of $2000 per annum; what sum did he invest at G per cent.? i (a) To the preceding we may add Problem 4. Amount, Rate, and Time given, to find the Principal. Ex. 1. What principal, at 5 per cent., will amount to $110 in Analysis. $1 in 8 yews, ^ ■"> p°i* cent., amounts to $1.10; .-. the principal innst be the quotient of $110-4- 1.10 = $100. ice, \U : 'de the given amount by the amount of $1 for (he and time, and (he quotient will be the principal. 2. What principal, at 6 per cent., will amount to $130.39 in 8 mouths ? Ans. $125.37."-. What principal, at 8 DOT cent., for 3 years, will amount to 10? 1 What is the interest of that sum for Syr. Gm.. at 8 per cent., which will, at the | find time amount to $240? FMtv 4' MM 206 PERCENTAGE. COMPOUND INTEREST. 949. Compound Interest is interest on both principal and interest, the latter not being paid when it becomes due. The principal may be inrn a-< <1 l>y adding the interest to it annually, m ■mi-annually. quarter!; cording to agreement, and the creditor may receive compound Interest without 1> liable to the cl amrj (Art. 231), though he cannot leya/Iy collect it if the debtor refuses to | 250. To calculate Compound Interest : Rti r. Make the amount for the first year or specified time, the PRINCIPAL for the second ; the amount for the second the principal for the third ; and so on. From the last amount subtract the fik^t im:in«:ipal. and the remainder is the com- pound interest. \. I. What is the compound interest on $100 for 3yr. 3m., at 6 per cent, per annum ? OrKRATIOX. $100. Ill Principal. $100 X-0 6= 6. Interest for 1st year. 10 6. 1st Am't or 2d Prin. $106 x-0 6 = _6.3 6 Interest for 2d year. 1 1 % S 1 Ain't or 3d Prin. $ 1 1 2.3 6 x -0 6 = [16 Interest for 3d year. 1 1 9.1 1 6 3d Am't or 4th Prin. $ 1 1 9.1 1 6 X-0 1 5 = 1.7 8 65 2 4 Interest for 3 months. 1 2 0.8 8 8 1 2 4 4th or last Amount. 10 0. 1st Principal. $ 2 0.8 8 8 1 2 4 Com. Int. for 3yr. 3m. Note 1. Find the amount for the years as though there were no months In the given time, and this amount is the principal for the remaining mor.tfis. 9. Compound Interest, what is It? How often may the interest b» com- pounded? May the creditor receive compound interest if the debto- c-nooses to pay? Can lie collect it if the debtor refuses to pay? 250. Rule for computing compound interest » Rule when there are months and days in the given time r 209 2. What la the compound inter. Bbl, it 4 at per annum. -089. 3. What is the compound interest on $500 for 3 years, at 7 snt? • $112.5215. 4. What [fl the amount of $5000 at compound interest, for 4yr. 10m. I2d.? Ans. $6640.< What is the amount of $9000 at CO M pO U nd interest for 3 1 ? Ana. «& the compound interest of $10000 for 2yr. Cm. nt.? Ans. $1606.788. 7. Whftl b the compound interest of $10000 for 2yr. Cm. 18d., at 1 per cent. ? Ans. 8. What is the compound interest of $10000 for 2yr. 6m. 18d., at 8 per cent.? Ans. $2177.216. B 2. Four per cent of any number is |, and 8 per cent is J of G per rent of the same number, but the compound interest of any sum of money at 4 per cent, is fan than | of the compound interest of the same sum for the at., and the interest at 8 percent, is more than J of the interest at 6 per cent., as may be teen by example! 8, 7, and 8. The compound interest at 4 per cent, is less than half the compound interest of the same sum at 8 per cent., because the base of percentage, (i.e. the principal,) after the 1st year, is leu in computing interest at 4 per cent, than in computing it at 8 per cent. ; thus, in computing interest at 4 and 8 per cent, the 1st year the base is the same, and one interest is just half of the other; but the 2d year one base is SI 04 and the other SI 08 ; .'. the interest at 4 per cent, is less than half of that at 8 per cent. 9. What is the amount of $250 for 2yr. 6ra., at 3 per cent. h 6m., compounding the interest semi-annually ? Ans. '$289,818. 10. "What if the interest of $36 for lyr. 9m.. at 2 per cent, per quarter, compounding the interest quarterly? Ans. $5,352. 11. What is the compound interest of $864.75 for 8yr. 8m. at 6 per cent.? Ans. $208.1» 12. What is the compound interest of $327.54 for 4vr. 4ra. . It compound interest at 4 for cent, half as much as at 8 per Wh;' 208 PERri;\i 2«H« Compound interest may be calculated more expeditiously by means of the following TABLE, Stowing the Amount of 81, £1, etc., interest compounded annually at 4, 5, 6, 7, and 8 per cent., from 1 to 20 years. Yr. 4 per Cent. 5 per Cent. 6 per Cent. Cent. 8 per Cent. Yr. 1 1 1.040000 1.050000 1.060000 1.O7O000 1.0801 2 1.081600 1.102500 1.123600 1.144900 1.166400 2 3 I.114M4 1.157625 1.191016 1.991 1.959719 3 4 1.169859— 1.215506+ -177— 1.340796+ 1.360489— 4 5 1.216653— 1.27( 9896— 1 409589— 1.469328+ 5 6 1.265319+ 1.418519+ 1.500730+ -74+ 6 7 1.315932— 1 .407100+ 1.6057M + 1.713824+ 7 8 1.368569+ 1.477 548+ 1.7191964- 0990+ 8 9 1 328+ 1.689479 — 1.838459+ 1 .999005— 9 10 1.480244+ 1.628895— D848 7151 + 2.158925— 10 11 1.539454+ 1.714 1 .898299— 9191889 2.331639— 11 12 1.601033+ 2012196+ . 1 92— 2.518170+ 12 13 1.665074— ' 1.8854 »28+ 2.409845+ 2.719624— 13 14 1.781676+ 1.971 2.260904— -534+ 2.937194 — 14 15 1.800944— 2.078928+ 2.396558+ 2.759032— 3.172169+ 15 16 *1+ 8.181 '164— 3.425943— 16 17 1.947900+ 2.292018+ 2.692773— 3.158815+ 3.700018+ 17 18 2.025817— 8404 1399+ 3.379932+ 3.996019+ 18 19 2.106849+ 2.526950+ 3.025600— 3.616528— 4.315701 + 19 20 2.191123+ 2.653298— 3.207135+ 3.869684+ 4.660957+ 20 Note. The interest is 81, XI, etc., less than the amount in the above table. 13. Wh9i is the compound interest on $600 for 20yr. ? $ 2.2 7 1 3 5 = Int. of $1 for 20yr. taken from the Table. 60m0 $132 4.2 81000 = Int. of $000 for 20yr., Ans. 14. What is the compound interest on $30 for 5yr. 6m.? $ 1.3 3 8 2 2 6 = Amount of $1 for 5yr. .0 3 = Int. of $1 for 6m. .04014678 . 338226 = Int. of $1 for 5yr. $.3 7837278 = Int. of $1 for 5yr. 6m. 30 $1 1.3 5118340 = Int. of $30 for 5yr. 6m., Ana. DI8C01 NT. 15. What is the amount of $50, at 7 per cent, per annum, for impound inter- 7 5 9 3 2 = Amount of Si for 15vr. 50 ;7.9 5 1 6 = Amount of $50 for l-'yr.. An--. 1G. What is the amount of compound interest, for 18 year 17. What h the compound Interest of $75 for , at 8 ent? What is the inter 00 for 9yr. fat, :it 1 per cent. for each G months, compounding the interest semi-annually? An- ;25. 19. What is the amount of $100 at compound interest for 40 year-, at 7 per cent, per annum? Ans. $1497.445. 20. What is the amount of $100 at compound interest for 30 . at G per cent per annum? DISCOUNT. 252. Discount is an abatement or deduction made for the. payment of a debt before it is due. The PRESENT worth of a debt, payable at a future time with- out in < vidently, a sum which, put at legal interest, will amount to the debt at the time of its becoming due. The drbt, men, i- an amount, the present worth is the principal, and the discount is the interest of this principal. Hence, 253. The rule for finding the present wortli is that given in Prob. 4, Art. 248, viz. : ide the given sum by the amount of $1 for the given rate and time. T7te discount is found by subtracting the present worth from the face of the debt. . What is Discount! PraMri Worth ? The debt Is the same as what in Art. C; ttl *53. Rule for finding present worth? Discount! Explain I 18 # 210 PER< L 1. What is the present worth of $37.44, due in 8 mont "What the discount ? OPERATION. Amount of $1 for 8m., 1.0 4) 3 7.4 4 (3 6, Present worth. 81 2 $ 3 7.4 4, Given sum, 6 2 4 8 6.0 , Present worth. 6 2 4 $ 1.4 4, Discount. t What k the present worth of a debt of $100, payable in J «ar, without i \\ 'hat the discount? Ans. Present worth, $94,389+; discount, $5.GG1 — . 3. What is the present worth of %Vi I _ M . 1 1 in. ? Ans. $1122.80. 4. What is the present worth of $14 1 .."»<>, dm- in lyf Ana. $131.32+. ... What h the present worth of $340.87, dm in 2jr. ;— . What is the discount on $456.25, due in 9m. 12d. ? . 7. What U the present worth of $ I tjt. Cm.? WW th.» discount? 8. What is the discount on $315, due in 1 year, at 5 per haw | MJta for $1000,i May 1, 18G3; what int shall I make for payment to- . I86f, money PH t at 10 pV cent. pM anmi is. The interest on the present worth equals the discount on the debt. 10. What is the interest for 6 months on the present worth of a note for $350, due 6 months hence? . $10.19. 11. What is the interest for a year on the present worth of a note for $750, due 1 year hence? 12. I have a note for $436, payable June 21, 1863; what is the worth of the note to-day, May 12, 1863, money being worth 8 per cent, per annum ? 13. What is the discount on $896, due in lvr. 8m.? 14. What is the present worth of $475, due in 2yr. 4m. 12d. ? hank: 211 BANKING AND BANK DISCOUNT. 253 a. A i an Institution, ted bylaw, for the kg and loaning of money, deaDug in exchange, fbrnishing a currency for circulation, et& charter incorporating a bank, defines its privileges and limiti Its powers. The Capital Stock of a bank is the money, paid into the bank in specie by the stockholders, as a basis of buss* i: 1. Banks arc of thrM kinds, viz. : Banks of Deposit, Banks of and Hanks of Circulation. A Bank of hi i and takes care of money, subject to the order of the depositor. A Bank of DueotaU Loani money upon notes, drafts, and other securities. A Bank <>t" Circulation issues its own bills or notes, which are usually re- deemable In coin at the hank which issues them, and, because redeemable in ■ when it matures; or, if there are not so many days in the month, it i the last day of the month ; thos, a one month note, dated on the 28th ary, nominally matures Mfir. 28, and legally m : 31 ; but a one month note, dated on Jan. 28 (except in leap-year) or on Jan. 29, Jan. 30, or Jan 31, nominally matures Feb. 28, and legally Mar. 3. £.121 1>. Entered OO money borrowed at a bank is paid ? the money it borrowed. The interest 1 in advance the face of a note, and retained by the bank as con on for the money borrowed, is called Bank Discount. The money n - •1 by the borrower is called the Proceeds or Avails of the note, and is equal to the face of the note, )em the interest The note is said to be discounted. To find the bank discount and the proceeds of a note, payable at a specified future time, without inter- Kri.i. 1 Find the interest on the face of the note, at the given rate, from the time of discounting to the maturity, and the result will be the discount. 2. Subtract the discount from the face of the note, and die remainder will be the proceeds or avails, .. 1. What i> the bank discount on a 90 days note for $368 ? What are the proceeds ? $ 8.6 8 = Interest for 6 days. 1.8 4 = Interest for 3 days. .18 4 = Interest for 3 days. $ 5.7 4 = Interest for 9 3 days, 1st Ans. $868 — $ 5.7 4 = $ 3 6 2.2 9 6, proceeds, 2d Ans. 2. I have a 6 months ncte for S7G8, dated May 12 ; what will be the avails if I get it diseoun: 8? 253a. A note payable in a number of days, when due? In a number of months, when due? 253b. Interest paid at bank, when? Money received, called what? Rule for finding bank discount? For finding the proceeds of a note? PANKP 213 $7.6 8 = Int. owl for 2 m. 1.5 8 6 — Int.-n-t for 1 2 d- $9.2 1 G = Discount $ 7 G 8 — $ 9.2 1 G = $ 7 5 8.7 8 1 , proceeds, Ans. months and grace from May IS expire Nov. 15. From Sept .15 is -in. 12cL, the time for which the note is inted. What will be the bank discount and what the proceeds on a 4 months note !'• 4. On a 90 days note for $1812, at 7 per cent.? In a G months note for $489, at 5 per cent.? G. A 4 months note for $629, dated Feb. 27, was discounted Apr. I2j what were the proc- 7. What La the difference between hank discount and true dis- count (Art. 252) on an 8 months note for S4G00? m, 1. When a note hearing interest is diseountcd before its matur- ity, the amount of the note at vialurity, rather than its/ace, is the base for .ring. 8. AVI nit are the proceeds of a note for $10000, payable in 6 months and bearing interest, if discounted 2 months before its maturity ? The amount of $10000 for 6m. 3d. is $10305, and the in- ; of $10305 for 2m. is $103.05, which taken from $10305, leaves $10201.95, Ans. 9. What are the proceeds of a note for $6844, payable in 4 months and bearing interest, if discounted 1 month after date ? Note 2. Business men often deduct more than the legal rate of interest for present payment of a bill having a term of credit. 10. What shall I pay on a 6 months bill of $75, if 5 per cent be deducted for cash ? 1 1 . What on a bill of $250, if 8 per cent, is deducted ? 353 c. To find the sum for which a note must be written that the proceeds may be a specified sum. 1. For what sum must a 45 days note be written, that the proceed- may he 1240? M3b. What it Note 1? Not* 2? 2\ \ opKRATiosr. The proceeds of SI for $ 1.0 45 days and grace, are Interest of $1 for 48 days, .0 8 $0.9 !>•/, and .-. ti Proceeds of $1. .9 9 2 the note muct b ' dollars as $0,992 is con- $ 2 4 -*- .9 9 2 = $ 2 4 1.9 3 5, tained times in $240, Ans.] $241,935. Hfl Rule. fhe required proceeds by the proceeds the given rate and time, and the quotient trill be the numb* dollars in the face of the required note, J. For what sum must a 3 months note be given, that the pro- ceeds may be $800 ? 3. A t'armrr sold produce for which 1 .lays note, which he immediately had discounted at the bank. The proceeds of the note were $593.70 ; what was its h 1. A DM n -hant wishes to borrow $1200 i , for 90 days ; what shall be the face of the note, the rate of interest being 7 per cent. ? 2»5 I. CmnUkVOl hi security against loss from the damage or destruction of property by fire, shipwreck, or other spe< casualty ; or from loss of life or health by disease or accident £»?•?. The Premium is the sum the influaDOe, and b usually computed at a en-tain per cent on the sum insured. •s according to the nature, locality, etc., of property, or the age, place of resid«-n rf the p« injured; also according to the length of time for which th<> ity is given. o property is so hazardous, that insurance companies dec-lino taking the risk at any per cent 2.10. The Policy is the writing or record of the contract, i by the insurer to the injured. The policy specifies the nature of the ri>k, and names the hour when it begins and ends. 253 c. To find the face of a note such that the proceeds shall be a specified sum, Rule? 254. What is Insurance? 255. Premium* How computed? Does the per cent, rary ? Why ? 256. What is the Policy ? What does it specify | *2o7. If property ii fully insured th<' owner is tempted to destroy the property, and secure its value from the in-uianec company. To prevent inch fraud, oompeoiej will usually insure the j ,' : only about <| or $ its value, requiring the owner t-t ri-k the reniainder. The same property may be insured .'it d different offices, by consent of the companies inanring it, but not bo th.it the whole .sum insured at the differen t offices shall exceed that per cent, of its value which a single company nstomed to insure. 2*18. To calculate the premium on a givon sum : I v i 1.1:. Multiply the sum insured by the rate per cent., written decimally. k. The insured usually pays :i -ivtn <\\m, say, Si. 25, for the policy, in addition to the premium of a certain per cent, on the aum insured. Ex. 1. What is the cost of insuring $2500 on my house for at 2 per cent., the policy being $1.25 ? OPERATION. $2 5 X -0 2 = $5 0.0 0, Premium. 1.2 5, Policy. $5 1.2 5, Ans. 2. What is the annual premium for insuring a manufacturing establishment in the sum of §75000, at 3 per cent. ? Ans. $$260. 3. In a certain house, the furniture, worth $3400, is insured Tor $ its value at 1$ per cent.; what is the premium ? 4. The Merrimac Mutual Fire Insurance Company have insured $2000 on my house for a period of 5 years, at $ of 1 : what ■ the OMt, the policy being $1.25 ? 1 buy a house for $8000, and get it insured for $ of its value at 3 of 1 per cent. ; the house being boned, what is my What the loss of the insut. , $2040; loss of C 257. 1* property usually insured for its full value? Why not? May it be insured at more than one office? On what conditions? 208 Rule for com- puting premium' Cf p<> if 216 PERCENTAGE. 0. What Li the premium, at \\ per r insuring $70000 on a steamboat and cargo from Boston to Havre ? 7. A cotton factory worth $2f>000, and the machinery and stock worth $35000, are insured for ^ their value at 3 \ what b the premium ? 8. "What i- the annual premium for insuring $6000 for on the life of a man 25 years of age, the rate 1>< ing .97 of 1 cent, annuall Ans. $58.20. What will bfl the annual premium for insuring $8500 for 10 years on the life of a man 30 years of age, the premium STOCKS. 259. The Capital or Stock of a Bank, Railroad, Insur- ance, Mining, or Manufacturing Company, or other Corporation, is the money or other property employed in transacting the busi- ness of the Company. City, State, and Government Bonds are also called Stocks. 260. The capital or stock of a company, is usually divided into a number of equal parts, called shares, and the owners of the snares are called stockholders. 20 1. Shares of stock are bought and sold like any other property. The nominal or par value of a share of stock is a fixed sum (in most companies $100, though in some companies more, and in some, less), but the market value varies, according to circumstances ; as, e. g., if a company is prosperous, and its prospects are good, its stock rises in price ; but if the company has been unfortunate, and its prospects are bad, its stock declines. The abundance or scarcity of money also affects the price of stocks. The price of government stocks also depends upon the state of the country as to peace or war, the prospects of the sta- bility or instability of the govern r . etc. Note. In this work, SI 00 is considered the par value of a share of stock, unless some other sum is named. S39. What is the Capital or Stock of a Company? 260. Uow divided? 261. What i* the par value of stock? The matkei value, how doe* it vary! sto' 217 £62. If a share of stock mO nominal value, it is raid to be at pari tf ll Wlb for DOT*, it (a at a premium, in - par ; if it sells for less, it is crt a discount, or ;»ir. 2623. The interest paid on government stocks, and the - from the business of companies, distributed from time to among the stockholders, are called Dividend** The sums of money occasionally required of the stockholders, to meet the losses or expenses of the company, are called Assess- ments. 26 1. Assessments, dividends, discounts, and premiums are percentages on the par value of the stock as a base. Hence, Thoblem 1. To find an assessment, a dividend, dis- count, or premium : BULB. Multiply the par value of the stock by the rate per . written decimally. Ex. 1. The directors of a manufacturing company, wishing to j" their works, call for an assessment of 5 per cent, on the capital of the company ; what will be the assessment on $15000 worth of the stock ? OPEi: vi ; $ 1 5 ^ ie °P erat ^ on * 3 the same as for q 5 computing interest for 1 year, at any — — given rate. S 7 5 o.o o, Aim. 2. The Boston and Maine Railroad Company paid a dividend of 4 per cent., Jan. 1, 18G1 ; what was paid on 25 shares of its stock ? OPEllATIOX. $100 2 5 First find the value of 25 shares, g2 5 o anc ^ l * ien com P ute the dividend. S 1 0.0 0, Ans. 203. When is stock at pur? Abova par? Below par? 201. What aro dJv. lAcnds* Arrestment*? 864. Rule for computing dividends, assessments, cto.f 19 218 PERCENTAGE. 3. What is the discount on $1400 worth of stock which sells at 30 per cent, below par? Ans. $420. 4. Suppose the New England Glass Co. Stock sells at An advance of 10 per cent., what is the premium on 5 -Inns at S-300 per share ? 265. Problem 2. To find the market value of stock when sold at a premium, or at a discount. I'.x. 1. What is the market value of $5000 worth of stock, at a discount of 5 per cent. ? $5000 9 5 Since the stock sells at a discount of 5 per cent., $1 of the stock sells \ a a a *° r :,,) BeirtB » •• e - tne mar ket value 45000 is .95 of the par value. $4 7 5 0.0 0, At 2. What is the market value of f> shares of Fitchburg Rail- road Stock, at an advance of 2 per cent. ? OPERATION. $100 6 First find the par value of 6 shares, $ (To~0 nnc * l * ien • n( ' n,a5e tne F ,ar va - ue Dv tne I q 2 2 per cent, premium, i. e. multiply the par value by 1.02. 12 * J 600 $ 6 1 2.0 0, Ans. similar reasoning holds in all cases. Hence the Rule. Multiply the par value of the stock by the number which represents the market value of $1 of the stock. 3. What shall I receive for 12 shares of the Andover Bank Stock at l) per cent, premium ? Ans. $1308. 1. What il the market value of 75 shares of Railroad Stock at a discount of 85 per cent. ? 5. What is the premium on 15 Shares of the Western Rail- road Stock, at 18 per cent, advance? '463. Rule for finding the market value of stocks* 219 260. Problem 3. To find how many shares of stock may be bought for a piven sum. I.\. 1. How many >hares of Railroad Stock may be bought for $870, when the market price i- 19 pi r cent betow par? OPERATION. $1 of Stork El WOlMh $8 7 -$-.8 7 = $1000. only 87 cents.-, tfc $1000-f-$100 = 10, Ans. :- .87, viz. si 11 '''*, i- the nominal •k botgjkft. Again $1000 divided by 3100, the nominal value of 1 >har<\ gives 10 tharee, Aus. 2. I lew many -hares of the Western Railroad stock maybe purchased for $575, when it is worth 16 per cent, premium? <>ii i.viiox. $\ of stock is worth $ 5 7 5 -h 1. 1 5 = $ 5 0. . i ">, .-. $578 ■* 1.15 = $5 -4- $ 1 ~ 5, An>. $500, is the nominal value of the purchase. A $500 -4- $100 = 5, the number of shares purchased Hence, Kile. 1. Divide the sum expended by the number represent- ing the marb ( value of SI of the stock, and the quotient is the nominal value of the stock bought. 2. Divide the nominal value of the purchase by the nominal value of 1 share, and the quotient is the number of shares bought. 3. Now many shares of the Exchange Bank Stock, at 2"> pet cent, premium, can be bought for $1000? Ans. 8. •1. I low many shares of Mining Stock, at 12 per cent- dis- count, may be bought for $2200 ? COMMISSION AND BROKERAGE. 267. Commission or Brokerage is the compensation • d by an agent for transacting certain kinds of business, men, e. g. at collecting and loaning money, or buying and selling ion at 2 per cent.? 3. The taxes in the town of B for 1862, are $15000 ; what is the cost of collecting them at \ of 1 per cent. ? Ans. $7 4. My agent has lent for me $2124. His commission is \ of 1 per cent.; what shall I pay him? 5. My correspondent in Paris has bought for me C ball French calico, each bale containing 50 pieces of 30 yds. each, at 25c per yd. ; what Lfl his commission at $ per cent. ? 6. My agent in New Orleans has sold for me 400 pair- of boots at $1.50, 400 pairs of shoes at 75c, and 500 pairs do. at $1 ; what is his commission at 3 per cent., and what shall he remit to me ? 2d Ans. $1358. 269. Problem 2. To find the commission or bro- kerage, when the agent is to take his pay from the sum remitted and invest the balance Ex. 1. Sent my agent in London $5100, out of which he is to take a commission, and invest the balance in goods. What sum will he invest, his commission being two per cent, on the pur- chase, and what is his commission ? $ 5 1 -4-1.0 2 = $ 5 0, Investment. 100 — $5000 = $100, Commission. Since the commission is 2 per cent, on the sum expended, the agent must have $1.02 for every dollar he pays for goods; .*. he *» » 36S. Rule for computing commission? TA.V can invest as many dollar mtnined times in S")100, vi/. 95000, Mid thil lubtracted from the- SJ100 gives §100 for ■mmi»ion. Hi BULB. 1. D m sum by 1 increased by the deci- tsing the rate per cent. of commission, and the fju will be tit,- sum tn lie jfMN slrd. J. The sit// t in Listed subtracted from the given sum will bare the commission. 2. I intrust $10000 to my factor in New Orleans ft>r the juir- of cotton. What sum shall he inve>t after deducting £ per cent, commission for the purchase, and what are hi- i< Ant. S- , '. , -"><».i. > o — , Invf-tiii.-nt ; $49.75-j-. Commission. 8. Sent $10100 to a Boston broker for the purchase of bank stock. The brokerage inr\ per cent, on the purchase ; what does bfl pay for stock, and what is the brokerage ? 4. Sold a quantity of merchandise for my employer for $5000. Alao purchased goods lor him to a certain amount, and, having calculated my commission at 5 per cent, on the sale and 3 per cent, on the purchase, our accounts balanced ; what did I pay for the goods bought ? What was my commission on the sale ? On the purcha •» TAXES. 270. A Tax is a sum of money assessed upon the person, the property, or the income of individuals by the authorities of a town, county, state, or other section of a country, or by the national government, to defray the expenses of government, to construct public works of common utility, etc. 271. A tax on property \~ at a certain per cent, on timafed value of the property. The tax on the person, called the capitation or poll tax, is assessed equally upon all individuals liable to pay a poll tax. A I ii called a pull. 20tK Ku!e when tin.- commission is to be taken from the nun remitted! 270. What is a tax? By whom assessed? For what' 871. How is the tax on property assessed? The tax upon the person, called what? What is a poll? 19* 222 PERCENTAGE. 272. Property is of two kinds, viz. real and personal estate. Real Estate consists in immovable property ; e. g. lands, houses, ?nills, etc Personal Estate consists in movable property, as money > notes, cattle, tools, ba)ik stocks, railroad stocks, ships, etc. 273. An Inventory is a list of articles of property, With their estimated value. 274. The method of assessing taxes is not the same in all its details in the different States, but the essential principles are. In some of the States the tax bill is so made as to show the amount of tax upon the real (Mate and personal property sepa- rately ; in other States no such distinction is made In Vermont, each taxable poll is reckoned as so much property, say $200, and DO separate poll tax is calculated. This shortens the operation of making out a tax lit. In Connecticut, personal property is taxed ju-t twice as high as real estate ; thus, if A pays $80 on a farm worth $1000, then B would pay $f>0 on $4000 at interest. 27»T. In Massachusetts, the assessors are required to as upon the polls about one sixth part of the tax to be raised, pro- vided the poll tax of one individual for town, county, and purposes, except highway taxes, -hull not exceed $2.00 for one year. The remainder of the sum to be raised is apportioned upon the taxable property of the town, county, or state. Hence, To Assess Tax Ki'LE. Ascertain the number of polls liable to taxation, and take an inventory of the taxable property. Multiply the sum assessed upon one poll by the number of taxable polls, and tubftxu t the product from the swm to be raised. Divide the remainder by the taxable property, and the quotient will be the tax upon MitUipfg the taxable property of an individual by the number expressing the tax upon $1, to the product add his poll tax, and the su?n will be his total tax. 273. How many kinds of property? What is Real Estate? Personal Estate? 273. What is an Inventory? 274. Are the details of taxation the same in all the States? What peculiarity in Vermont? In Connecticut? 275. The rule in own of A is to be taxed The real I of the town la valued al §500000 and ike personal at $500000. are 666 taxable polls each of which is assess ed $1.50. "What is the tax of B, whose real I I alm-d at $4000 and y at $8000, and who pays 1 poll tax ? $1..'.<> x 666 = $999, sum assessed on th<- polls. 199 — 999 = $5000, sum to be assessed on the property. $500000 + 300000 = $800000, amount of taxable property. $50 >000 = 6} mills, tax on >00 -f $8000 = $12000, B's taxable property. $12000 x .006J = $75, tax on B's property. > = $76.50, B's entire tax, An*. Note. To save labor, (by using smaller numbers,) assessors frequently take 6 per cent of the inventory instead of the entire valuation ; but the labor may be lessened still more by taking 10 per cent., as in Ex. 2. 2. The town of F, whose valuation is $356400, has 6 taxable inhabitants, A, B, C, D, E, and F, who wish to raise a tax of $1800. The taxes of the several inhabitants are for the number of ]x>lls and the property, as in the following INVENTORY. Names. Number of Polls. Real Estate. Personal ate. Total. 10 per Cent A B C 1) i: F To: 3 2 1 3 3 24875 1*469 2842 1 15860 19933 70405 88460 47628 06486 34867 95280 38460 67090 8491M 15860 5 1800 952$ 3846 6709 8491 1586 5480 12 108554 247846 356400 35640 The tax upon eaeh poll being $1.50, what per ecnt. i> levied on the property, and what is the tax of A, F>, ('. 1). E, and F ? assessors to nve labor* What . t is suggested? What is the object Of the Tabk I Lxpiaiu Ex. 2. 224 PERCENTAGE. la calculating a tax list it is must convenient to form a table showing the tax upon SI, $2, ■. in the percentage column, and then calculate the taxes of the BevenU inhabitants from the table ; thus, in solving Ex. 2, first find the tax raised on all the polls ($1.50 X 12 = $18), and. having deducted this from the total tax, ($1800 — $18 = $1782), divide the remainder by the assumed percentage of the taxable property in town ($1782 -i-35640 = $.05), to find the tax on $1 in the percent* age column. Then lorm the TABLE. Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. 8 8 8 8 8 s 8 8 1 0.05 10 0.50 100 5.00 iooo 50.00 2 0.10 20 1.00 200 10.00 2000 100.00 3 0.15 30 1.50 300 15.00 3000 150.00 4 0.20 40 2.00 400 20.00 4000 ►.00 5 0.25 50 2.50 500 25.00 5000 250. G 0.30 GO 3.00 GOO 30.00 6000 300.00 7 0.35 70 3.50 700 35.00 7000 350, 8 0.40 80 4.00 800 40.00 8000 400.00 9 0.45 90 4.50 900 45.00 9000 450.00 Now to find A's tax from this table : OPERATION. Tax on$9000 = $45 0. " " 5 00= 2 5. " " 20= 1. « " 8 = ■ « 3 polls = •!..*> <> A's total tax = $ 4 8 0.9 In the same manner the tax of B, C, etc., may be found. By the abov soiling tin* tax is found to be 5 per cent, on the per- centage column, or i per cent, on the entire taxable property. CUSTOM-HOUSE BUSINESS. 276. Customs or Duties are taxes levied by the General Government on imported or exported goods, to support the gov- ernment and to protect home industry. 276. What are Customs or Duties? 225 277. All g ht into the (Jailed States from fbrtlgn countries mii-t be landed at Berttdli plaees called />o/7s of tutry. • ach port of entry a custom-house ii tttaMkhed by govern- ment, with offloen t<» ( ■iiin])uii' and coiled the «luti« -. All duties are regulated by government, and are different at :t times aud in different conn' Po lirin^ in iiiLTiluuKlise secretly and without paying dm eaMvilsmw/ylinij, and j>< tie liuLle to punishment if detected. £7S. Tn\'N.vi;K is ■ tax upon the vessel, without reference Cargo, for the privilege of coining into I port of entry. The amount of tonnage' depends upon the size of the i ■ The income from duties and tonnage ifl the n-nnue of the rnment Occasionally, when the revenue from duties and - insufficient to defray the expenses of government, ' taxes are levied, by authority of our national con. upon the pertOO, the property, and the incomes of the inhabitants. £7!>. Dutie- are either ud valorem or sj»cijic. An An vai.okkm I)i rv is a certain percentage computed on the market value of the goods in the country from which they are imported. A Sj i .< ific Duty i> a certain sum per ton, gallon, yard, etc-, without regard to the cost of the article. *£>0. An Invoice is a list of the articles sent to a purchaser or agent, with the pvices annexed. Ad valo,:i.m Dities. 2S1. Problem 1. To compute ad valorem duties: BULK, Multiply the mst of the goods by the given per cent. Ex. 1. What is the duty, ■( •!«> per cent., on 25 cases of French broadcloths, invoiced at $30000? S30000x.-i0 = $1200 0.0 0, Ans. Imported goo Weight r MANGE. Deduct the legal draft, l **V*i etc., from the uiity of set >tltij)h/ the remaind the tin* doty, at 6c per lb., on 300 boxes of figs, weighing 1121b. etch, allowing lib. draft and 151b. tare on each ! 728. 1. What is the duty, at 15c per lb., on -18 cheats of tea, each weighing 66 IK, draft being 1 lb. per bos and tare 4 per cent, on mainder? 5. What is the duty, at 5c. per lb., on 800 bags of coffee, weighing 561b. each, draft being lib. for each 1121b. and tare t. on the remainder? EXCHANGE 08-1. Ex< u \ n . ; i : , in commerce, is a mode of paying debts due in distant places by meAns of drafts or bills of exchange, without the cost or risk of transporting specie. 28o. A Draft or Bill of K\< iiam.i; if a written order or request to one person to pay to another a certain sum of money, and charge the same to the account of the person who makes the request. USfi. The Ma ma or Drawer, of a draft or bill of ex- change \- tlw person who re qo ests another to pay; the Drawkf, person who is requested to pay ; and the Path is the d to whom the drawee is requested to pay the money. *2**7. To explain the operation of exchange and show it* benefits, lro(rs( it j'»r 'itjintut by giving the proper notice to the drawer and the pal indorsers. It" this notice il n<»: given in due time the indorser eease to be boldea for the payment. 9921. The United States annually export to and import from >ods to the value of hundreds of millions of dollars. 1 the imports, and sometime- |he When our export* to a given country, England, ur imports from England, the balance of trade is in ourfa\or: England owes US more than we owe England, and reliant I here wi-h to sell bills drawn on England, for the purpose of eolleeting their duo in England, than wi>h to DOJ for the purpose of paying their debts there, and consequent- ly, the supply being greater than the demand, bills on England will sell at a discount. When the balance of trade i* in favor of England, OUT indebtedness is greater than that of England, and bills on England will sell at 1 premium. This ehan. the price of bills is called the COUESI of K.\i iiam,i; The variation in the price of bills can never be very great, for mer- chants will not pay more for premium than the cost of freight and in-urnnee to transport specie. 991. Bilk of exchange, payable after sight, like promi- ; to I discount for the term of credit, the di — count being Computed on the face of the bill. 993. In tl. | the exchange value of the pound 291. Kor what is a bill protested* By whom? How? 29*2. When should a bill be presented for payment? What is necessary to hold tin 291. When is the balance of trade in our favor? When agai. does this affect the price of bills of exchange? What is the Course of Exchange? MUttf the variation be great? 294. Are time bills subject to di-count? 495. What u the exchange value of the X* What the commercial value? 20 230 PERCENTAGE. sterling is $4.44| and bills of exchange are drawn upon tin. basis, but the intrinsic and commercial value is about 9 per cent, more than the exchange value ; thus, Exchange value of l£ =s $4.44 % Add 9 per cent, as .40 Average commercial value of l£ = $4.84$ ; \;. when exchange on England sells at a premium of 9 per cent, it is at true or commercial par. 29G. Problem 1. To find the cost of a draft or bill of exchange. Ex. 1. $1000. Boston, June 4, 18G2. At Bight, j»:iy John Jones, or order, one thousand dollars, value received, and charge the same to my account. A. Tyler. To M fcwa . Smith ft Dana,) Merchants, Chicago. ) What is the cost of the above draft at 2 per cent discount ? #1000 X .98 = $980, Ans. Since exchange is at 2 per ^ cent, discount, each dollar costs 98 cents, i. e. the bill costs .98 (98 hundredths) of its face. 2. $320. l^ttsburg, Aug. G, 18G2. Sixty days after sight, pay to S. Day, or bearer, three hun- dred and twenty dollars, value received, and charge the same to the account of T. Fox & Co. To Alfred Stearns, ) N' m York. ) What is the cost of this draft at 3 per cent premium ? OPERATION. $320 9.G = premium on $320 at 3 per cent 3 2 9.6 3.3 6 = discount on $320 for 60 days and grace. $ 32~6\2 4 = cost of draft, Ans. 3. What is the cost of a draft on St. Louis for $832'), at 2 per cent, discount ? 4. What is the cost of a draft on New York for $7850, at 1 per cent premium ? n 1. An < 'Miiutry whore it ia 000£. Boston, May 12, 1862. At tight of this first of exchange (second and third unpaid), pay to the order of John Flint, in London, two thou -and pounds sterling ed, and charge the same to my account. David Fay. To Geom Peabodv & Co.,) Jeers, London. j What is the cost of this bill in United State9 money, at 9^ at. premium ? OPERATION. S-l.W$X2000 = $888 8.8 8{f = 2000£. 8 4 4.4 4 $ = premium at \ per cent. $ 9 7 3 3.3 3 $ = cost of bill, Ans. Hence, RULE. First, if necessary, find the value of the bill, at par, in United States money ; then increase or diminish this value as the rale of exchange and the term of credit may require. 6. Stuart, Field, & Co., of New York, bought of J. & P. Smith, a set of exchange, payable at tight for 800£, on Bates, Baring, & Co., London, at 8| per cent, premium. What was the cost in U. S. money ? Ans. $3866. 66$. nl An English coin worth IX is called a sovereign. 7. I wish to pay a debt of 1200£ in Liverpool. Which can : afford, to buy sovereigns at (4.85 and pay 2 per cent, for il and insurance, or buy a set of exchange at 9 J per cent. premium? ^109.73^ by buying the bills. SOO. Rule for fluding the cost of a bill ? What is an inland bill ! A for- eign bill? A sovereign? 232 PERCENTAGE. 297. Problem 2. To find the face of a bill which a given sum in United States money will buy. Ex. 1. When exchange n at 0? por cent, premium, what is the face of a bill on London which I can buy for $4990? 1 £ = $ 4.4 4 1 ; S 4.4 4 f + 9 1 per cent = S 4.« 7 £, cost of 1 £ ; $ 4 3 9 -j- § 4.8 7 1 = 9 0, No. pounds in bee of bill, 2. My agent in Chicago, bought a draft on New York, per cent premium, for $8160 ; what was the face of the draft ? $1+2 per cent = $ 1.0 | - l. $ 8 1 6 -f- 1.0 2 = $ 8 0, Am. I i Rule. Divide the cost of the bill by the cost of a bill for $l t l£, etc., and the quotient will be the face of the bill in dollars, pounds, etc. 3. A Boston merchant bought a draft on Chicago, at 9 per cent, discount, for $.3820 ; what w;ts the face of the draft ? Ans. $u000. 4. Bought a set of exchange on London, at 9£ per cent, pre- mium, for $4168.30 ; what debt in London may be paid by this sum? Ans. 856.5£ = 856£ 10* EQUATION OF PAYMENTS. 298. Equation of ? h the method of determin- ing when several debts due from one perm to another, payable at different times may be paid at one time, so that neither party may aafler 1<>— . The equated fume u the date of payment. The time to elapse before a d^bt becomes due is called the term of credit. The average term of credit is the time to elapse before the equated time, 299. PROBLEM 1. To find the equated time when all the terms of credit begin at the same date. Ex. 1. On the 1st of Jan. A owes B S2, payable in 4 months 207. Rule for finding the face of a bill? 29S. What is Equation of Pay- Uiciits ? What the equated time ? Term of credit ? Average term of credit* EQUATION U; ivable in 8 month- ; what i> the average term vl' and the equated lime of payment? in>D. 4x2= 8 Tlic privilege of ki 8X6 = 48 ing $2 for 4m. is the gv 7T .-nine Sfl tin- privilege of ' — for 8m. ; so 7m., 1st Ans. ,,. is the same Jan. 1 -u 7m. = Aug. 1, 2d Ans. |] fo r 48m. J ... for the two debts, A might hoep Si Tor 56m., but as he has $8 to keep, he may retain it ."»6m., viz. 7m., and 7m. from Jan 1, extend to Aug. 1, the equated time. Hence, BULB 1. Multiply each debt by the number expressing the time to elapse before it becomes due, then divide the sum of the products by the sum of the debts, and the quotient is the average term of credit Add the average term of credit to the date of the debts, and the result is the equated time. Remark. Express each time in months, or else each in days. 8ECOND METHOD. The interest of $ 2 for 4m. = 4 c " SG " 8m. = 2 4 c. Sum of debts = S 8 2 8 c. = total interest. Now the question is, in what time will the interest on the sum of the debts be the same as the sum of the interests on the several debts? This may be found l>y dividing the total interest by the interest on the sum of the debts for 1 month; thus, interest of S8 for lm. = 4c, and 28c. -~- 4c. = 7, number of months fan the ge term of credit, as by the 1st method. Hence, Krir 2. Find the interest on each debt for its term of credit, then divide the sum of these interests by the interest on the sum of ! >(s for one month, and the quotient will be the average term of credit in months. id the equated time as in Rule 1. the sam of the debts for a month, it ii 299. Rule for finding average term of credit ? Equated time? Second method T Explain Ex. 1 by each method. Second Rule ? What is Note 1 1 20 # 234 PERCENTAGE. only necessary to move the decimal point two places to the left and divide by 2 (Art. 241 j, for the interest of $1 is just half a cent a month. Note 2. It is the custom of business men to consider 30 days a month ; also, in computing interest, to neglect the cents in the principal if they arc less than 50, and to add 1 to the number of dollars in the principal if the ctnt> are 50 or more. So the fraction of a day, in equating accounts, is neglected if less than k, and it is counted aa 1 if it is i or more. Note 3. Karh method above given is much used by accountants in averaging accounts, but the second is thought to be the shorter and better method. The second only is given in the following problems, but the pupil will practice upon either or both, as his teacher may ,30, payable in 4m., 75 id 8m., and $3500 in 12m. ; what is the average term of credit and the equated time ? Ill Ads. Average term, 7.08 m. = 7m. 2<».l. 2d Ana. Equated tin, . ft b, M, 1862. Note 4. The decimal of a month may bfl reduced to days by multiply- ing by 30 (Art. 176), or more conveniently by taking 3 days for each teuth and 1 day for each 3 } hundredths in the decimal. 3. $1500, $2100 and $2100 are dM in 4, 8, and 12 months, respectively ; what is the average term of credit ? 300. Problem 2. To find the equated time when all the terms of credit are of equal length, but begin at different times. In solving examples where the terms of credit are equal, it is only necessary to find the average date of the debts, and then to this date add the term of credit. In finding the average date, interest may be computed from the date of the first bill, or from any other date /but it is most convenient to compute the interest from the frst of the month in which the first bill is boitgld, because the time for which interest i< to be computed on the several bills i> thereby most easily will be seen by the following examples. ^J99. What is Note 2 ? Note 3 ? Note 4 ? 300. In finding the average date of debts, interest may be reckoned from what time ? Most convenient time ? Why ? $300 $0.60, Int. for 1 2d. -. * - L8d 3.6 0, " * lm. 6d 100 2.4 0, " « 4m. 2 4 J. KgrATIU.N* CM Tii m which interest on uted, fa Called I . 1. Required the equated time of paying the following bills of gi . boaghl on a credit of C DXHltl 0m Mar. 1-J. Dm. •■ lm. Aj>r. »',, 4m. July -J 1. 2 ) 100)$ 1200 $ 7.2 0, total inter lot on nun of bills tin- 1 in., $6 0-1- 6.0 = 1.2m. = Ira. Gd. Thi> gives the average date of purchase 1 month and G dajs from Mar I, viz. Apr. <>. To this add the term of credit, 6m., and we have Oct 6 for the equated time of pay m '.:. Sinn- the time for which interest is computed tmdudte Loth the of tin- month ami the day of purchase, so lm. and Cd. from M;ir. 1 Lnl :i- ending on the 6th of Apr. and not on the 7th. The same principle holds in the following examples. taxation. The time for interest on the first bill is month- and 12 days, the number of days being determined by the Date OF thk BILL. So the time of the second bill is Oin. : of the third, lm. 6d. ; and of the fourth, 4m. 2 Id. The number of months may be obtained by counting from the local date (e. £• for the fourth bill above, April, May, Juno, July, i. o. 1, 2, 8, 4) and, for conveniont US6, the number of months / the l$fi of the date of the bills, severally. reel of each bill is computed for its own time ami irriUen at the right The aggregate or total interest on the bills (in this example, $7.20) is then divided by the interest of the sum of the bills for 1 month (|6),M in Ex. 1, Art. 2'.' iii-th in the average date of purchase. II< a Kt tack bill from the first Of month in which ■'■'// mm bought to the time ef the pur- • f the bills, severally ; divide the sum of these interests by 300. Focal date, what is if L Number of month.*, how found ! Where set! Rule for finding average date ? Bqnttod 236 l'EK- the interest on the sum of the bills for one month, ctnf porcbaM (and consequently [oated time of pajment) i> greatly changed by buying the smaller bills at the earlier or at the later dates. 301. Problem 3. To find the equated time when the terms of credit are unequal and begin at different lli.' maturity of a note or bill is the time when it becomes due. The process for finding the equated time of payment in this Problem is the same as for finding the average date of purchase in Problem 2, except that the interest is computed to the maturity of the bills severally, rather than to the time of purchase. Hence no new rule is needed. 1. Required the equated time of paying the following bills of goods? Cr. Bills. Int. Om. Feb. 12, 4m. $2 $4.4 for 4m. 12d. 2m. Apr. 15, 6m. 40 17.0 " 8m. 15d. 4 in. June 8, 2m. 3_00 9.4 " 6m. 8d. 2)10 0)$9 $ 3 0.8 0, total interest Int. on sum of bills for lm. = $ 4.5 30.80 -f- 4.50 = 6.84m. = 6m. 25d., the time from Feb. 1 to the average date of maturity, i. e. to the equated time. Now 6m. 25d. from Feb. 1, 1862, gives Aug. 25, 1862, Ans. Explanation. The maturity of the 1st bill is 4 months and 1 2 days from Feb. 1 ; the maturity of the 2d bill (found by adding its term of credit, 6m., to the 2m. 15d. from the focal date, Feb. 1, to the time of purchase, Apr. 15) is 8m. 15d. ; the maturity of the 3d bill, found in like manner, is 6m. 8d. 2. Required the average maturity of the following bills? Jan. 8ra. $2000 Feb. 21, 6ra. 3000 June 6, 2m. 600 300. What is the Remark? 301. What is the maturity of a note or Wllf How doe* Problem 3 differ from Troblem 2' Explain Ex 1. 233 303. PBOBLKM 4. To find the equated timo for paying the balance of an account which has both ami credit entries. Ex. 1. From the accounts of A and B it appears that A owes B And thai B owes A $254, due July 18, $500, due Aug. 15, 475, " Sept. 6, 288, 425, " "18, 012, 46, " Oct. 9, 400, When shall P> pay the balance of $G00? OPERATION. A's Debt*. $24 I I for 18d. 1 7 5 5.2 2 " 2m. 64 425 InklM 4 6 .7 ."» 9 " 3m. M. u " 30, a Oct. 3, H " 21, 0m. July 18, 2m. Sept to. " 18, 3m. Oct. Im. Aug. 15, lm. " 30, ;;m. Oct 3m. •• 21, B's Debts. $500 8 1 a 400 Sum of B's debts = =$1800 Sum of A's debts =$1200 $ 1 2.2 7 1 , Total interest on A's debts from tin' focal >lute. July 1, to maturity, i. o. the in- t er e st that B would ^rain it* A paid the sum of his debts, $1200, on the 1st of Jul v. Tut. .7 5 for lm. 1 "m1. 2.8 8 " 2m. 7.4 " 8m. Sid $ 2 3.5 1 6, Total interest on B's debts from the focal dot*\ July I, to maturity, i.e. the tajteresj A would gain it" B paid the BOB <>f his debts, July 1. From tl e it ippeatt that if each party paid his debts July 1, A would gain $23,516, and B would gain SI 2.271 : ,\ A^ net gain and B's net loss would be $23,516 — SI 2.271 = $11,245. Now as it is proposed to settle by B's paying the balance of the account, viz. $600, it is plain he may keep the I after July 1, until its interest >hall equal $11,245, the loss he would sustain by paying July 1. The interest of $600 for lm. is $3, and $11,245 -t. $3 = 3.748, the time in months. Now 3.7 18m. =3m. 22d. ; .-. the time of payment is 3m. 22utc the interest of each item of the account from rat date to its maturity ; find the sum of the interests on the debit items, also the sum on the credit items, and subtract the less sum from the greater; divide this difference by the interest 304. Explain Ex 1. I J Kule for equating account* which hara both debit and orcriit items » 240 on the balance of the account for one month, and the quo- will he the tune in m "een the focal date and the tied time of settlement, the time to be reckoned forward l the greater interest arises on the greater side of the account, and backward, excluding the focal date, when the greater inter- est arises on the smaller side. Note 1. When the hunger interest arises on the smaller side of tlie account, as in Ex. 2, the rule may require the settlement to be made before some of the transactions have occurred, a result which is obviously impractica- ble, and usually some other time of settlement is more convenient than the equated time. If the settlement is made before the equated time, a discount should be made ; if after, the interest should be added. Ex. 3. When ought A to pay the balance of the following account, and for what sum may he settle June 6, 1863 ? Dr. A in account with B. ( r. 1 $ A]>: To Mdse., 6m 356 By Mdse., 4m. 530 June 18 Mdse,, 4ra. 875 27 Mdse., 6m. 651 July 3 * Mdse., 6m. 433 Julv 15 " Cash, 300 Ufl Am. .June G, 186 1 ; 2d Ans. $171.08. (See Art. 253b.). Note 2. In I 1 is the most convenient focal date, the earliest entry l>cing made Feb. 6. The meaning of the account is, that A three different times, bought merchandise of B to the amount of $356, $875, and $433, severally, the 1st and 3d bills on a credit of 6m., and the 2d on 4m. ; also, that on the 6th of Feb. A sold B merchandise worth $530 on a credit of 4m., on the 27th of May merchandise worth $652 on 6m., and on the 15th of July lie paid B $300 in cash. 4. Required the equated time of settling the following account, and the sum due Oct 4, 1862 ? Dr. A in account with B. Ok 1862. $ 1862. $ Mar. 14 To Mdse., 4m. 452 April 13 Br Cash, 500 May 8 « C 1224 May 21 4m. 1000 •« 20 '•Msc., 8m. 150 Aug. 18 ■ Cash, 192 " 27 • Mi- , 6m. 2496 Sept. 11 " Cash, 5420 June 19 M t-e., 3m. 5724 July 30 Gin. 88 I 1st Ans. Nov. 4, 1862 ; 2d Ans. $3006.89. EQUATION OF PAYMENTS. 241 Note 3. Not unfrequently a business man, in full or partial payment of . £ives his note, payable ba ft givan time without interest. The holder of the note may IndoffM it and gal it diacoonted (See Art* 253b.), thus obtaining money for his own use before the note matures ; or he may pass it to his creditor in payment of his own i a noto may be entered in tin account, as in Ex. 4, and treated in the same way as merchandise bought or sold on credit. 5. When was the equated time of settling the following account, and what was due Nov. 13, 1862 ? Or. Dr. A in account with B. 1861. $ 1861. S Nov. 18 To Mdse., 4m. 800 Sept. 27 By lidaat, 1200 1862. Dec. 12 " Mdse., 4m. 800 April 6 " Mdse., 2m. 350 1862 " 30 | ash, 125 May 15 " Mdse., 4m. 850 May 15 " Note, 4m. 1200 July 18 " Mdse., 625 Oct. 12 " Mdse.,2ra. 200 IM Ans. Apr. 25, 1861 ; 2d Ans. $874.40. 6. "When is the equated time of settling the following account, each item being due at date, and what shall A pay on the 27th of July, 1862 ? Dr. A in account with B. Or. 1861. $ Int. Om. June 20 ToM 986 3.2871 5m. Nov. 16 " Mdse., 152 4.205 1862. 8m. Feb, 26 " Mdse., 110 4877 124S 12.360 1861. lm. July 4 6m. Dec 18 1862. 9m. Mar. 5 By Mdse., " Note, " Mdse., I IN 228 Int. 0.895 7.524 450 20 625 836 29 044 $ l i 1 8 836 2 9.0 4 4 1 2.3 6 9 2)100)412, Balance of acct. 1 6.6 7 5, Balance of int. 2.0 6 ) 1 6.6 7 5 ( 8.0 9 m. = 8m. 3d. June 1, 1861 — 8m. 3d. = Sept. 27, 1860, Iff Ans. $ 4 1 2 + $ 4 5.3 2 (Int. for 1 yr. lOni.) = $ 4 5 7.3 2, 2d An* 7. What would be the equated time of settlement in Ex. 6, if each Ktem were on a credit of 6 months ? 303. What is Note 1' 21 Note 2r Note 8f -1- PERCENTAGE. Proof. Some of the debts are due before the equated time, and some after. The sum of the interests on the former, from t/ieir several maturities to the equated tune, will be equal to the sum of the interests on the latter from the equated time to their several maturities. When the account has both debit and credit itnn*, equate each side of the account, and the interest on the two sides for the time between the respective average dates, and the equated time will be the same, or nearly the same (Art. 299, Note 2). PROFIT AND LOSS. 303. " Profit and Loss," as a commercial term, signifies the gain or loss in business transactions. The rule may refer to the ((hsolute gain or loss, or to the percentage of gain or loss, on the purchase price of the property considered. 304. Problem 1. To find the absolute gain or loss on a quantity of goods sold at retail, the purchase price of the whole quantity being given : Rule. Find the whole sum received for (lie goods, and the dif- ference between this and the purchase price will be the gain or loss. Ex. 1. Bought 1 6 bbl. of flour for $100 and sold it at $7 per bbl. ; did I gain or lose ? How mueh, total and per bbl. ? 2. Bought 24 bbl. of flour for $168 and sold $ of it at $6.75 ami the remainder at $7.50 per bbl. ; did I gain or loss ? II<>w much ? Ans. Gained $6. 3. Bought 3cwt. 2qr. 181b. of sugar for $36.80 and sold it at 8Jc. per lb/; did I gain or lose ? How much, total and per lb. ? 4. Bought 164yd. of broadcloth and 287yd. of cassimere for $1107; sold the* broadcloth at $3 and the cassimere at $2.25 per yd. ; did I gain or lose ? How much ? 305. Problem 2. To find the per cent, of gain or loss when the cost and selling price are given : 30 2. Froof of rule for equation of payments? S03. What is Profit and Lo«i? To -what may it refer? 304. Rule for finding absolute gain or loss? i 243 El 1. Booghl 4bbl. of flour for $32 and sold it at $9.50 pet bbl. ; did I gain or lose? How much percent? $9-3 0, telling prl The gain, $0, is 3 «* = ^ 4 of tfae whole cost, and -j^ $3 8.0 0, whole sum rec\I. n_,iu,-,,l ,<, B dean*! (Art | 8 ., 173), givei .18J ; i.e. the gain is 1S^ per cent of tho $6, whole gain. cost> Q, , r > =.182, Am. Hi i.k. I [nviiig found the total gain or loss by Problem 1, male a common fraction hy icritimj the gain or loss for the rator and the cost of the article for the denominator, and then reduce this fraction to a decimal Bought 501b, of wool for $20 end lold it at 34c. per lb.; did I gain or lose? How much per cent.? Ans. Lost 15 per cent. 3. Bought a ca>e of boots at S 1 per pair and sold them a: what per cent, was gained? 4. Bought boots at So per pair and sold them at $4; what Wit was lost? Booghl goods for $2000, and, la one year, sold the - 2155, out of which paid S'. 1 -") for storage, etc. ; how much per cent, on the first cost was lost? 306. Problem 3. To find the selling price, the cost and gain or loss per cent, being given. . 1. Bought goods for $400; how must the same be sold so as to gain 2"» per cent $400 .2 5 2 This is the same as finding the g00 amount of a sum of money on interest for 1 year at 25 per cent <>.0 = gain. (\ r $ 4 0. $5 0. Am 305. Rul« for finding the per cent, of gain or low! 244 i age. 2. Bought a horse for $150, but it being injured, I am willing to lose 6 per cent. ; for what shall I sell him } $ 1 5 Tlii :iihp m finding Am pr eec m worth of a nm $100 = loss. 1. llin- 95 Hi. of soger tor $2, I bsc 20 par cent oo the ; what was the cost per lb. ? 30S. Proi The selling price of goods, and! the gain <>r loss per cent, being given, to find what would I pr lost per cent, if sold at sorao other price. a pair of oxen for $175 and gained 5 per cent. ; what per cent should I have gained it I had Bold them fo» . f9g = f The proposed price ia J of 10 5 = 120 ?98 = f of the actual Bell- i 2 — 10 = 20, Ans. log price, bot the actual aelL. ing price is 1<>"> per cent, of the cost, .*. the proposed price is $ of 1<»,"> per cent. = 120 per cent, of the cost; hence 120 per cent. — 100 per cent. = 20 per cent, would be the gain per cent, if the oxen were sold for 2. Sold a farm for $5000, and thereby made 25 per cent. ; should I have gained or lost, and how much per cent., if I had sold it for S3500 ? M$$ = fft = ^ ; A of 125 = 87|; 100 — 87* = 12* = loss per cent., Ans. The proposed price is found to be 87* per cent, of the cost, .-. there would be a loss of 12$ per cent, if the farm were sold B500, Rule. Make a fraction by writing the proposed piece for the numerator^ and the actual price for the denominator, then multiply tiki per cent, at which the article is sold by this fraction, and the 'd will be the per cent, at the proposed price. The dijf'< between the product and 100 is the gain or loss per cent, at the j)roposed price. 308. Bute for finding Ioh or gain per cent, when goods are sold at a proposed priN I 246 PERCENTAGE. 3. Sold flour at $7 per bbl. and thereby gained If per cent.; what per cent, should I have gained if I had Bold it at |7.85 ? Ana. 16 per cent. 4. Sold beef at $G per cut., and thereby lost 1 per cent. ; should I have gained or feet, and DOW much per cent., had I sold it at 16.50? 5. Sold a watch for $21, and gained 5 per cent, on the ej had I sold it for $18 should I have gained or lost, and how much per cent. ? 5109. Problem G. To mark jroods so that the mer- chant may fall a certain per cent, on the marked p and yet sell the goods at cost, or at a certain per cent. above or below cost. (a) To sell at cost Ex. 1. II«>\\ shall I mark a coat that cost me $18 so that I may fall 10 per cent, from tlue marked price and yet ieU the coat at 00 W = V J V of $ 18 = WO, Ana. Since I am to fall 10 per cent, it follows that the cost, $1 only iVg = & of the marked price, and if $18 is -ft then will be } of $18 = $:?, and \\\ will be 1,0 times $2 = J i. e. the marked price will !.«• \;> of $18 = $20, An-. Proof. 10 per cent, of $20 = Si', which taken from $20 leaves $18, the cost. Hen Bulb. Makt a fraction by writing 1 00 for the numerator, and 100 diminished by the )>er cent, to be abated for the denomi- nator ; multiply the cost by this fraction, and the product will be the marked price. 2. Bought a case of watches at $28.50; at what price shall I mark them to enable me to abate G per cent., and yet sell them at cost ? Ans. $25. (b) To sell at a certain per cent, above or below cost : 309. Rule for marking jroods so as to fall a certain per cent, and yet sell at •est? To sell at a given per cent, above or below cost ? : IT AND 1. - 17 Bulb. Fintjimd latanttay prim hp ftioMmS; then find the marking juice }»j Pt = cost. $"> — $1 = $4, selling \ .20 $1.00 = loss. W X $4 = $5.33 i, marked pH Paid $1 a pair for a case of boots ; how shall I mark the 90 that 1 may tall 10 per cent, from the marked price and yet make \'2k per cent, on the Paid $8 each for a case of bonnets ; how shall I mark the 10 that I may fall 10 per rent, from the marked price and yet make 5 per cent, on the cost? Miscellaneous Examples in Profit and Loss. 1. Bought 75 pounds of tea for $37.50 and sold £ of it at 18 rent- per j>ound and the remainder at 5C cents ; did I gain or How much ? 8. What per cent, do I gain if I buy boots at $3 per pair and -(11 them at $8.87$? 3. Sold floor at $7.50 per barrel and lost $\ per cent on the fa what should it be sold to gain 12] per cent. ? Paid S3 per yard for a piece of lace; how shall I mark iMie to enable me to fall 1«) percent, from the marked price and yet gain SO per eent. on the eost ? liought hats at S3 per hat and sold them at $2.50; what at on the coal wai lost? >old a watch for $42 and lost 12* per cent, on the c what was the 7. Sold eloth at $9 per yard and lost 10 per cent. ; should I have gained or lost, and DOW much per cent., if I had received $2.12*? >!d him so as to gain 12 ut. ; what did I receive for him? 248 PERCENTAGE. PARTNERSHIP. 310. Partnership is the association of two or more per- sons in business. The company thus formed is called a firm or house. The money or other property invested is called the capital or stock of the company. The profits and losses of the firm are divided among the part- ners in accordance with their interest ID the business. 311. Problem 1. To find each partner's sdmre of gain or loss when their capital is employed equal tim Ex. 1 A and B trade in company ; A furnishes $100 and B $800. They gain $300 ; how shall they share the gain ? A furnishes -rVo°o = h °? tne stock, .*. he is entitled to £ of the gain, viz. $100. For alike reason B's gain is § of $300 = $200. Or we may solve the question as follows : $300 -f- $1200 = .25; i.e. the profits = 25 per cent, of the stock ; .-. A's share of profits = $400 X .25 = $100 B's share of profits = $800 X .25 = $200 Entire profit $300 Hence, RULE 1. Multiply the total gain or loss by each partner's fractional part of the stock; and the products will be the respective shares of gain or loss ; or, Rule 2. Find what per cent, the total gain or loss is of the whole stock, and then multiply each partner s stock by this per cent, written decimally. 319. Proof. The sum of the shares of gain or loss must equal the total gain or loss. 2. A, B, and C form a partnership ; A furnishes $4000, B $5000, and C $6000. They gain $3000 ; how shall the gain be divided ? Ans. A's, $800 ; B's, $1000 ; C's, $1200. 310. What is Partnership? What is the company called? "What is the capital or stock? How are the profits and losses divided among the partners? 311. Rule for finding the shares of gain or loss? Second rule? 312. Proof? PABnraraiR 3. Ha.l the Mm b Ex. I bit $7.">o, what part of the lose should each partner su-tain P How many dollars? Yns. A, ft ; 15, J ; C, J. 4. A, B, and G mirage in trade. A puts in $G000, B $10000, and C $8000. They gain $4000; what is each part- - -hare ? k. These rules arc equally applicable to distributing the property of a bankrupt, and many other similar problems. 5. A bankrupt whose property is worth $5000 owes A $3000. B $1500, and C $3500; to what fractional part of the property is each creditor entitled? To how many dollars? & Divide $1500 between A, B, and C so that A shall •ve $2 as often as B receives $3 and C $5, 7. A. B, and C hire a pasture, for which they pay $90; A pastures ;{ cow-, B ~>. ami C 7 ; what part of the rent shall each I low many dollars ? 8. A and li hire a pasture for $12; A's horse was in the pasture 45} weeks and B'l 7^ weeks; what rent shall each pay ? '.'. A. B. ('.and I) freight a ship to Canton; A furnishes • worth of the cargo, B $5000, C $7000, and D SI 1000. They pun $5200; what is each one's share of the gain? 10. A and B form a partnership with a joint capital of 0, of which A famishes | in cadi, and B, for his share, farntshes 160 yards of' broad cl oth. They lose $300; how shall the loss he divided? What is the price of B's cloth per yard ? 313. Problem 2. To find each partner's share of gain or loss when their capital is employed unequal times. 1. A and B trade in company; A puts in $300 for 8 months, and B $100 for 9 months. They gain $800 ; what part of the gain belongs to each ? How many dollar- ? A's $300 for 8m. = $2 1 00 for lm. B's $400 for Dm. = $3 60 for lm. 00 for lm. It is, .-., as though the joint stock were $G000 for 1 month, 31J*. What i*tlit ■ N..:. ' 313. Exy'iuii Ex. 1. 250 PERCENTAGE. of which A put in $2400, and B |9606 ; hence A is entitled to f$gg = £ of the gain, and B to gggg = £ ; i. e. A is entitled to § of $800 = $320, and B to f of $800 = $480, Ada. Hence, RULE. Multiply each man's stock by the time it is continued in trade, and, regarding the products as the respective shares of *,tock, and the sum of the products as the total stocky proceed as in Problem 1 . 2. A and B engage in trade; A furnishes $4000 for 12 months, and B $6000 for 11 months. They lose $">70 ; what is the loss of each ? Ans. A" -330. 3. A, B, and C engage in partnership; A furnishing $600 for 9m., B $800 for 8m., and C $1000 for 12m. They gain $1071 ; what is each on of the gain ? 4. A, B, and C hire a pasture for $48. A pastures 3 h< for 8 weeks, B horses for G weeks, and C 6 horses for 7 weeks; whftl part of the rent shall each pay? 5. B, T, and C enter into partner-hip, doing business in the name and ugnfttnre of B, T, and C. Jan. 1, B puts in $*5000, T May 1. 15 puts in $2000 more, ( and T take; out $1000. Sept. 1, B takes out $3000, T poll in •0, and S2O00. At the end of the year tie v Mttfe, having gained $6400; what is eaeh partner's share of the gain? Asm, Wi ttO0O,Ti £ »oo. 6. Jan. 1, 1860, B commenced business with a capital of $3000. Sept. 1, 1860, wishing to enlarge his business, he took in II as a partner, with a capital of $4000. July 1, 1861, they admit L into the partnership, with a capital of $2500. On the 1st of Jan. 1862, they dissolve partnership, having gained $75,30 ; what is each one's share of the gain ? 7. A, B, and C hire a pasture for $92. A pastures 6 horses for 8 weeks, B 12 oxen for 10 weeks, and C 50 cows for 12 weeks. Now if 5 cows are reckoned as 3 oxen, and 3 oxen as 2 horses, what part of the rent shall each pay? How many dollars ? 313. Bute for finding the shared of gain or loto when the capital id ii. lb* unoqua. timet? pARxanasHip. ■-!.,[ ft. A, B. and C hire a pa-turf for $.'500. A puts in 10 oxen i, and 96 sheep for 2G W( B puts in 7 oxen for 24 '.-for 20 weeks, and 66 ■beep for 25 weeks ; (' poti in 28 oxen for H weeks L2 cows for 12 weeks, and o."{ sheep tor 10 weeks. Now, if 11 sheep arc >ned as 1 cow, and 3 cows as 2 oxen, what i< the ooel per week tor a Bheep ? a cow ? an ox? How many dollars does man pay for sheep? cows? oxen? What part <>f (he rent each man pay? How many dollars? Ana. Coal per week for a sheep, lcV; a cow, 10c; an ox, 2le. A pay- for sheep, $:57.41; tor COWS, S.;3.f>0; for I B pays for sheep, $_> 1 ; for cows, $38.40 ; for oxen, S 10.82. (' pays for sheep, S7.20 ; for cows, $L'."».oi; for oxen; $-18. A pays Iff = $1 19.04; B, $£f = $102.72 ; C, ,gi = $ 7 !. 1 ox and S. Low enter into partnership. January 1. D $5000, bat Low puts in nothing until May 1 ; what shall he then put in that the partners may he entitled to equal shares of the profits at the close of the year? 10. Jan. 1, 1853, A, B, and C form a partnership for 1 year, and each furnishes $3000; Mar. 1, A furnishes $1000 more; June 1, B withdraws $.300, and C adds $500; Sept. 1, A with- draws S2000 and C $500, and B adds $1500. Having gained $4000, at the close of the. year the partnership is dissolved. What i> each partner's share of the pain? 1 1. A, 1>, and C traded in company. A at first put in $1000, B $12no, and C $1800; in three months A pat in $500 more and "0, and C took out $100; in 7 months from the comnienc. - Stent of beniaeSS, A withdrew all his stock but $700, B put ill as much as he at first put in, and C withdrew \ as much as A at any time had in the firm. At the end of a year they found they had pained 10 percent, on the larg est total stock at anyone time in trade. What is the total pain? What fractional part shaO eaeh have ? How many dollars? ( A's part, H? = $107.G3i?{. Ana. Total gain, $U0. ] B's part, =$ ( C's part, = $ Proof, = $ EXAMPLES IX A9AL1 EXAMPLES IN ANALYSIS. 313 a. 1. If 6 barrels of flour cost $ 12, what will 1 1 barrels cost? 2. Tf § of a cask of wine co-t $85, what will 7 casks cost ? 3. Twenty is £ of what number? 4. Fifty-one is j^g of what number ? .">. Nin.-ty-five is }J of what number? G. If £} of a ton of hay cost Do shillings, what will a ton cost ? 7. If g$ of a cask of oil is worth $7 1, what is the value of In? 8 Sixty-four is g of how many times 12? 9. Seventy-two is § of how many times 4 ? . 10. A man sold a watch for $o:j, which was f of its what was its cost? 11. A pole is f in the mud, 9 in the water, and G feet above water J what i- :h<- Length of the pole? 12. A ship's erew have provisions sufficient to last 12 men 7 months; how long would tiny la-t 2 1 men? 13. A can build 85 rods of wall in .'):; days, but B can build rials while A build- 7 | how many rods can B build in 44 d.. 14. f of 28 is ft of how many fifth 15. / T of 4 1 i- | of how many third- of 15? 16. $ of 27 is V of "ow many twelfths of 60 ? 17. A fox has 30 rods the start of a hound, but the hound runs 27 rods while the fox runs 24 ; how many rods must the hound run to overtake the fox? An-. 851. 18. A hare has 32 rods the start of a hound, but the hound rtmfl 12 rods while the hare runs 8 ; how many rod- will the hare run before the hound overtakes him ? 19. A man being aaked how many sheep he had, replied that if he had a- many more, h as many more, and I] iheep he should have 100; how many had he? 20. A detachment of 2000 soldiers was supplied with bread sufficient for 12 Ptekf, allowing each man 11 ounces a day, but Ea is. finding 1<>.*> barn 1-. containing 2<>nlh. each, wholly spoiled, how m.inv ounce- may each man eat daily, that the remainder may fast tbem IS 21. A detachment of 2000 having j of their bread spoiled, were pal apoo an allowance of 12 oc each per day for IS weeks ; what ra the whole weight of their breads good and had. and how much WSS Spoiled? 22. A detachment of 2000 soldi, r- baring lost 1 0."» barrel- of tghing 2001b. each, wen- Allowed but l2oz. each per day lor 12 weeks) hut if none had been loot, tiny mighl have had 1-1 oz. daily ; what WM the weight, including that which wa- it- t, and DOW much was left to sub-i>t on? A detachment of 2000 soldi. m, having lost j of their bread, had each 12oz. per day for 12 weeks; what Wat the weight of their bread, including the part lost, end how much per might each man have had, had none been lo-t I 21. A gentleman left bis son an estate, ] of which he ipent in 7 months, and J of the remainder in 8 months more, when he had only $5000 remaining; what was the value of the estate? •j."» The quick-step in marching being 2 paces of 28 inches each per second, what i- the rate per hour? and in what time will a detachment of soldiers reach a place GO miles distant, allowing a halt of 1^ hour-? Two men and a boy engage to reap a field of rye; one of the men can reap it in 10 days, the other in 12, and the boy in 1.") days. In how many days can the three together reap it? 27. A merchant bought a number of bales of hops, each bale containing ., at tin- rate of $3 for 111b., and sold them at the rate of $5 for 121b., and gained $248 ; how many bales did he bn Ans. 7. 28. Suppose I pay 8f{ cent- per bushel for carting my wheat to mill, the miller tUttS A ,(,r gnodingi it takes <1 .} bushels of Wheat to make a barn 1 of Hour, I pay 80 cvnN each for barrels %l\ per barrel for carrying the flour to market, where my ■ -ell- i*.' i barrel- for 'it of which he tsJo ftrrel for i - ; what do I receive per bushel for my Ans. 87 ^ cents. 22 KATI«>. RATIO. 31 1. Ratio is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind. 315. Ratio is usually indicated by two dots ; thus, 8 : 4 ( -x presses the ratio of 8 to 4. The two quantities compared are the terms of the ratio; the faK tem being the antecedent, the second the consequent, and the two terms collectively, a couplet. 310. Most math«-m:itieians consider the antecedent a divi- dend, and the consequent a divisor ; thus, 8:4=8-4 = }= 2, and 3: 12 = 3 -4- 12 = A=«i ; \ but others take the antecedent for the divisor, and the consequent for the dividend; thus B: i =4-^8=|= \, and 3 : 12 = 12 -r- 3 = V = 4. re 1. The first method w often called the English method, and the other the French; but there appears to be no good reason for such tinction. Note 2. The first is a direct ratio ; the second is an inverse or ratio. The first being considered the more simple and natural, is adopted in this work. 317. The Antecedent and consequent being a divi- dend and divisor, it follows that any change in the AB CEDENT causes a like change in the value of the ratio, and any change in the o \T causes an OPPOSITE change in the value of the ratio (Art. 84, 85, and 131). Hence, 1st. Multiplying the antecedent multiplies the ratio ; and dividing the antecedent divides the ratio (Art. 83, a and b). Hi, What is Ratio? 315. How indicated? What are the term*? The 1st? The 2d? The two collectively? 316. Which term is divisor? Is the custom uniform? Which method is here taken? Why? What is a direct ratio? An inverse ratio? 317. Explain and illustrate Art. 317 fully. RATIO. 2d. Multiply intt )/iiilti/>lics the ratio (Art. 85, <• and <1). . Malt iph/i in/ both a Mown! by the same numbtr, <>r diridiny botk by the same number, does not affect the ratio (Art. 8 I, ■ and b). 31S. The antecedent, eoMoej i eatj and ratio are so related ■h other, that, if either two of them be given, the other may md; thus, In 12:3=: 4, we have antecedent + consequent = ratio, antecedent -~ ratio = consequent, and consequent X ratio = antecedent. 310. When there is but one antecedent and one consequent the ratio is said to be simple ; thus, 15:5 = 3, is a simple ratio. 3£0. When the corresponding terms of two or more simple - ire multiplied together the resulting ratio is said to bs coinj,ouitiI ; thus, by multiplying together the corresponding terms of the simple ratios, (s=»-yli=!r we have the com- pound ratio, 4 8:4=1 2 or 48 0: 12 = 40. A compound ratio is always equal to the product of the simple ratio- of which it is compounded. Note. A compound ratio is not different in its nctture from a simple ratio, bat it is called comjwund merely to denote its origin.') 1 . Whet is the ratio of 20 to 4 ? Ans. 20 : 4 = 5. l\ What il the nitio of 2 to Ans. 2 : 9 = $. 3. What is the inverse ratio of 20 to 4 ? Ans. f a = £. 4. What is the inverse ratio of 2 to 9 ? 5. What is the ratio comjKmnded of 8 to 6 aud 9 to 2 ? 6. Which is the greater, the ratio of 9 to 7 or of 19 to 14 ? 7. Which il the greater, the ratio of 5 to 4 or of 15 to 13 ? 318. What of mtcced. . nt, and ratio? 310. What is simple ratio t ue! Its imtiii.-' Why oaJled compound? PROPORTION. PROPORTION. 3£l. Proportion is an equality of ratios. Two ratio-, and .-. 4 terms, are required to form a proportion. 322. Proportion is indicated bj neflM of doll ; A 8 : 4 : : 6 : 3, which is read, 8 is to 1 as () or, as 8 is to 4 so is 6 i or it may be Indicated t! which is mid, the ratio of 8 to 4 equals the ratio of 6 feO Any 4 numbers are in proportion, and may be written and . in like manner, if the quotient of tlie 1st divided by the :M is equal to the quotient of the 1 DJ the 4 1 1 1 . 393. Tlie [si and 1th terma are called extremes, and tl. and fljd, means. The l-t and 8d are the antecedents of the two ratios, and the 2d and 4th are the consequents. The prodnj mes is always equal to the product of tie- mean- ; thus, in the proportion 8 : 4 : : 6 : 8, we have 8 X 3 = 4 X 6. 394. Since the product of the extreaica hi opaasl bo tba product of the means, any one term may be found when the other khn en ; for the prodnd of th<- extremes dii by either mean will ghre tfa Otfcf . and the product of the means divided by either tber extreme. Fill the blank in each of the following proportions : 1. 8: 2:: : 3. Ans. ?-^-? = 12. Ans .iX8 =12> 2. C: 9::8: , 3. 4 ! : : 2 : 9, A. : 16 : : 7 : 1 1 •M« What is Proportion? 338. How indicated? Proportion, how read? When are four numbers in proportion? 343. Whut are the l«t and 4th taMM called? 2d and 3d? 1st and 3d? 2d and 4th? The product of the extremes equals what? 344. How many terms mutt be given? How can the other bt? found? 4 : : 3 1 : 4 : 3 B 1 : 8 : 6 1 : 4 : 8 4 : 6 : 8 3 : 8 : 4 8 : 3 : 4 8IMPLB 257 325. It follows from Art. 317, that if the tit and 2d, or U :>n«l 1th, or 1-t ami 3.1. or 2d ami 1th. or all tour terms of a proportion an- multiplied or divided l.v the MUM number, the rv.Miltin;: numbers will he in proportion. 3*-2<>. It' 1 numbers are proportional they will he in pro- portion in 8 different orders ; thus, (1) (liven 8 : i Alt. -matin- (1) 8 : (3) Inverting (1) 4 : i Alternating (3) 4 : (.")) Invi-rtin'j ( 1) tad transposing con pl< t- l Alternating (.".) 3 : Inverting (.".) G : (8) Alternating (7) 6 : may be written in 16 other orders, but none of t*i<>m will Ik; in proportion. Jfc\57. When the means of a proportion are alike, the term repeated i- a mean proportional between the other two, and the la-t term is a third proportional to the 1st and 2d ; thin, in 4 : G : : 6 : 9, 6 is a mean proportional between 4 and 9, and 9 is a third proportional to 1 and G. 328* A mean proportional between two numbers may be found by multiplying the two given numbers together, and then resolving the product into too equal factors ; thus, the mean pro- portional between 2 and 8 is 4, for 2 X 8 = 16 = 4 X 4; .*. 2 : 4 : : 4 : 8. 320. A third proportional to two numbers may be found by di riding the square of the 2d by the 1st. The third proportional to 5 and 10 is 20 ; for 10* -+ 5 = 20 ; .% 5 : 10 : : 10 : 20. SIMPLE PROPORTION. 330. In all examples in Simple Proportion there are three 325. What terms may be multiplied without destroying the proportion? What divided? 326. Iu how man) order* may four proportional numbers be in \ In how many not in proportion f 327. Wtmt is a mean proportional? A third proportional? 328. How is a mean proportional found? 320. A third proportional' 22 . 2.58 PROPORTION. numbers given to find a fourth ; .-. Proportion is often called the Rule of Three. Two of the three yieen numbers must be of the tame kind, and the other is of the same kind at the answer. Ex. 1. If 3 men build C rods of wall in a day, how many r will 6 men build ? This example may be :maly/ed as follows : If 3 men build rods, 1 man will build $ of 6 rods, i.e. 2 rods ; and if one man build 2 rods, 5 men will build 5 times 2 rods, i.e. 10 rod.-. Ant. ; but to solve it by proportion, we say. that ,'5 men have to 5 men th<- same ratio that the given number of rods has to the required number of rods ; thus, 3 men : 5 men : : 6 rods: required number of rod-. Now, since the means and 1-t . nn find the 2d extreme by dividing the produet of tin- Beam by the given • DM (Art. Ml' 1 ) : thin. C X 5 = 30 and 30 + 3 = 10, Ans. as before. II 331. To solve an example In Simple Proportion, Rnxn, Write that given number which is of the I as the required answer for ttie third term ; consider whether the nature of the question requires the answer to be greater or less than the third term ; if greater, write the greater of the two re- maining numbers for the second term and the less f>r the / but if less, trrite the less for tlie second and the greater for the frst; in either case, divide the product of the second and third tt mis by the frst, and the quotient will be tin t, rm $<>>njht. Notk 1. If the first and second terms are in different denominations, they should be reduced to the same before statin g ,on. \\ km ark. Everyone who intelligently solve? an example by proportion, does, in effect, solve it by analysis ; but the teacher should use much care on this point, since the scholar learns much faster when h - a question than when he merely follows 330. Of what kind must two of the three given numbers be? What th« Other! 331. Ru!e for solving an example in proportion? Note 1? Remark? PORTION. 259 a rule. Lei the following examples be solved by analysis ami by pro|M>r:i<.n. 2. If a man earn $24 in 2 months how much will he earn in [> nooi • . rm# we are seeking for dol- lars, we make ^2 1 the 3d term, o \ 9 ! I» al,, l then, M a man will earn ' ' more in 9 month- than he will in 2 months, we make 9 the 2d term and 2 the 1st. To analyze th<> above. We say. If a man earn $24 in 2 mouths then in 1 month In* will earn A of $24, i. e. $12 ; and If he earn $12 in 1 month, then in 9 months be will earn '.) times $12, i.e. $108, 3. If 15 bosh, of Wheat make 3bbl. of flour, how many bush- wheat will be required to make 7bbl. of flour? Ans. •1. It' bObosh. of wheat make 8bbL of flour, how many bar- of flour will 75 bosh, of wheat make? Ans. 15. It' a man ean walk 7o miles in 3 days, how far can he walk in h il Ans. 2(M.» miles. C. If a man travel 6 1 miles in 2 days, how long will it tak»- him to travel K><> miles? Ans. 5 days. 7 [fa locomotive run 30000 miles in 13 weeks, how far, at that rate, would it run in 52 week-? I1Y PROroHTI BT CANCELING. 1*:52 r: 39000: 4th term. 4 I x 52 = 2028000 ; 39000 x H 000-i. 18 =156000, Ana, — =lo6000, Ans. 8. If 20 men perform a piece of work in 8 days, in how many lays will 1 men perform the same? Ans. 40. [f 2 l eordi of w 1 . what will 18 cords cost. 1". If $80 pay for 5 cords of wood, how many dollars will pay for 12 cords? Ans. 72. 11. If 4 cords of wood cost $20, how many cords may be bought tor - Ans. 9. 12. If 6 bones eat -12 bu-hels of oats in 5 weeks, how many besheli will ill in the same time? coal when l t 260 PROPORTION. 14. In how many day9 can 6 men build a boott, it' 10 m< n can build it in 72 days? 15. If 7211). of cheese are worth as much u .'5<»lb. of bu- how many pounds of cheese will pay for 10 lb. of bul 1G. How many tons of coal can !• bought for $84, win tons cost $18? .11. 1 7. If 9 horses eat a ton of hay in 20 days, how many hfl will cat a ton in 30 da\ An 18. How many tons of hay will f> horses eat in 2.~> 8 hot --'" ton- in the MOM til* 19. If I pay 2s. 8d. per week f*<»r | vrhel shall I pay for pasturing 1 I 2 : 11 2s.8«L 11 2) 29s, 4d . Ans. 14s. 8d. or, 2:11 11 2)352d.: Ana 176r ? 27. llow many yard- of cloth § of a yard wide are equal to 2<» yard* 1 1 yard v. . if when Hour is worth $9 per bbl., a penny loaf weigh.* 4oz., what will it weigh when flour is worth $6 per bbl.? 29. If 10 hi 10 bmheh of oats in 3 weeks, how many BOabelfl will 12 horses eat in the same tim« ■? 30. Three men can do a piece of work in 12 days ; how many men must be added to the number to do the same in 4 d. UMFL1 m 'PORTION. 261 81. A lUp*l crew of 12 men has food for 24 days, how many mm mu-t !»<• di-chargrd that it may last 12 days tag AO for 31b. of tea ; what should I pay for 91b.? f * ship cost $8000, wh:i- i of her? \: 8M per BVt, what is the cost of 62$ lb.? It' i steeple 180 feel l'i_ r li casts a shadow 240 feet, what is the length of the shadow cast by a staff 3 feet high, at the same time? Note 2. Since each of the three terras in the above example is in /bet, the learner may be uncertain which numl>cr to place us the third term ; but he has only to notice that he is required to find the length of a shndnr, .-. die third term should be the number expressing the length of shadow in the given examplo, viz. 240ft. ; thus, 180 : 3 : : 240 : 4th term = 4 ft., Ans. 36. If a staff 3 feet long casts a shadow 4 feet, what is the high! of a steeple which, at the same time, casts a shadow 840 feet ? Ans. 180 ft. 37. It' ■ staff 3 feet long casts a shadow 4 feet, how long is the shadow of a steeple which is 180 feet high, at the 6ame time ? 38. If a steeple 180 feet high casts a shadow 240 feet, what is the liight of a staff which, at the same time, casts a shadow 4 feel 39. The interest of $300 for lyr. being $18, what is the interest of $850 for the same time? 40. The interest of $800 for 6m. being $24, what principal will gain $45 in the same time ? 41. If a man's salary amounts to $2700 in 3 years, what will it amount to in 11 years ? 12. If a man's salary amounts to $9900 in 11 years, in how many yean will it amount to $2700? If 12£ yards of silk that is | of a yard wide will make a dress, how many yards of muslin that is lg yards wide will be required to line it? 4 1. If J of an acre of land is worth $36.40, what is the value of l- r »v^ acres, at the same price? If 6 nun can mow 12a. 3r. 16rd. of grass in 2 day>, l>y working hours per day, how many days will it take thciu to do the same if they work only 4 hours per day ? iNJ PROPORTION. . If 2bbl. of flour are worth as much as 3 cords of wood, how many barrels of flour will pay for 45 cords of wood? 17. A bankrupt, owing $25000, has property worth $15000; how much will he pay on a debt of $6 48. A man, owning J of a 6hip, sells § of his share for $20000 ; what is the value of the ship ? l:». A and 15 hired a pasture for $45.90, in which A pastured 11 oxen and B 19 ; what shall each pay ? 50. If 13 men perform a piece of work in 45 days, how many in. n BMMl be added to perform the same in $ of the time ? ."»l. If tin- int. rest on $700 is $42 in one year, what will be the interest on the same sum for 3 J years? 52. How many yards of paper 2 feet in width will paper a room that i> LS| feel bog 12 feel wide, and '.♦ feel bigM Ill pay $168 for 63 gallons of wine, how much wi shall I add that I may nil it at $2 per gallon without loss? 54. A certain house was built by 30 workmen in 98 days, but, being burn< .1, it is required to rebuild it in 60 days; how many men must be employ. 55. A of 1500 men has provisions for 12 months how long will the same provi>ions last if the garri enforced by 300 m 56. If a piece of land 20 rods long and 8 rods wide contains an acre, how long nm.-t it be to contain the same when it is but 2 rod- wide. 57. If the earth revolves 366 times in 365 days, in what time does it revolve on. *, m. 58. A wall which was to be built 24 feet high was raised 8 by 6 men in 12 days; how many men must be employed to build the remainder of the wall in 12 days more ? 59. A wall was completed by 12 men in 12 days ; how many men would complete the same in 4 days ? If a man perform a journey in 6 days when the daycare 12 hours long, in how many days of 8 hours each will he form t'. 61. A cistern has a pipe that will fill it in 6 hours; how many ime use will fill it in 15 minutes? COMPOUND noteBTioH. -tern has 8 pipes; the fuvt will iill it in A hours the second in I hours, ami the third in 5 hours; in what time will they together till the cistern ? 63. Paid $8.50 for 71h. of tea ; what should I pay for 191b.? A can cut a field of gftJo fa A and B can cut it in 8 days. In w hat time can B do the saim It '2 horses can draw a load of 16 tons upon a railway, how many horses will be required to draw 72 ton A farm was sold at $25.50 per acre, amounting to $1925.25; how many acres did the farm contain ? A garrison of 1000 men have 14oz. of brood ooefa per day for 120 days; how long will the same bread last them if each man is allowed but 12oz. per day? 68. If W of a ship eool $26000, what is }). of her worth? 69. At $27 per cwt., what is the cost of 37 £lb. ? 70. The earth moves 1 9 miles per second in her orbit ; how far does she go in 3m. 27sec. COMPOUND PROPORTION. 332. Compound Proportion is an equality of two ratios, one of which is compound and the other simple ; thus, 3 • 12 ) If* o r : : ^ : 9, is a compound proportion ; and 48 : 24 : : 18 : 9, is the same reduced to a simple form. Note. Tho compound ratio may consist of any number of couplets. 333. Every compound proportion may be reduced to a simple form, and, moreover, every example in compound propor- tion may be solved by means of two or more simple proportions. Ex. 1. If 6 men in 8 hours thresh 30 bushels of wheat, in how Stay hours will 2 men thresh 5 bushels? BT 8IMPLE PROPORTION. 2 : "J 1, and 30 : 5 : : 24 : 4, Ans. 334. What is Compound Proportion? 333. May an example in compound proportion be solved by simple proportion? Analyze Ex. 1. PROPORTION. In solving this question by simple proportion, we, in the fir-t place, disregard the amount of labor, and inquire how long it will take 2 men to do as much as 6 men in 8 hours. Having found 24 hours to be the answer to this question, we next d gard the number of men* and inquire how long it will take to thresh 5 bushels of wheat if 30 bushels are threshed in 24 hours, and thus obtain 1 hours the true answer to the question. In this operation, the given number of hours, 8, is first multi- plied by 6 and the product divided by 2, then this quotic multiplied by 5 and the product divided by 30 ; but it will an- swer the lame purpose to multiply the 8 by the product of the two multipliers, 6 and . r >, Ihaa divide th«- number so obtained by die product of the two divisors, 2 and 30 ; thus, BT COMPOUND PROPORTION. 30 ! _5 \ : : 8 : 4th UnL . 2 (| multiplied by g0 3 30 for a divisor, and the g product of 6 and 5 is mul- 60)240(4, 240 tipli. •<] by 8 for a dividend. It will be seen that, of the first two couplets, i on ! e £ > ° n « «itio is less than a unit and the other greater ; but there is no impropriety in this, for one condition of the question requires the answer to be greater than the 3d term, and the other condition requires it to be less. Hence, 331. To solve questions in Compound Proportion, lvii i . Write that given number tchich is of the same kind as the required answer for the 3d term ; take any two of the re- maining terms that are alike, and, considering the quMsSm as depenpixg ox these alone, arrange them as in simple proportion; arrange each pair of like terms by the same principles ; and then multiply the continued product of the 2d terms by the 3d term, and divide this result by the continued product of the 1st terms ; the quotient will be the term sought. 334. Rule for compound proportion? COMPOUND PROPORTION. 1'ho work may often be much ■bridged by canceling factors in I ami 3d terms, with lil. K to). Ex, 2. It* 6 men in 15 days earn $135, bow many dollars will i earn in 18 - 21870 = continued product of 2d and 3d terms. 6 x 15 = 90 = continued product of Lat terms. An-. , - - . [. 2 27 : lth tena I :-.m i tt 9 x 17 = 848, Ana, 9) IS ( ** : ,M.:,An, 8. U i men, in 24 daya of hours each, build a wall 40ft. 9ft. high, and 4ft. thick, in how many days of C hour* each can 8 men build a wall 60ft. long, 12ft. high, nnd 5ft. thick ? Ans. 45. 8 men : 4 men 6 hours : '.» hours 40 ft. long : 60 ft. long 9 ft. hinh : IS ft. high 4 ft. thick : 5 ft. thick 24 days : If a family of 6 persons spend S000 in 8 months, how many dollar- will be required for a family of 10 persons in 14 asontha? Ans. 1750. 5. If a family of 6 persons spend $G00 in 8 months, how many months will $17">0 sustain a family of 1<> p If a family of 6 persona spend $000 in 8 months, how large a family may be -u-tained 1 1 months for $1750? 7. If the transportation of 12 boxes of sugar, each weighing 4cwt., 40 miles, cost $8, what must be paid for carrying 40 boxes, weighing 8|cwt each, 7-"» miles? Ans. $48.75. 8. If -1 mm dig a trench 84 feet long in 2 \ day-, how many men can dig a treni I long in 1 da;. . 10. It" 1 men dig a trench Sift, long and oft. wide in .*' da\ «, many men can dig a trench 420ft. long and 3ft. wide in 4 Ans. 9. 23 pROPORi : in. If 2 men dig ■ trench SOIL long, 5ft. wide.and 3ft. deep in S\ days, how many men can dig a trench 800ft long, 2.\ft. wide. and 4ft. deep in 7 day-? An-. L 11. If 6 DMA « I i lt a trench of 4 degrees of hardness, 30ft. loOA 5ft. Wide, end 5ft. deep in 5 days, how many men can dig ocfa of 6 degrees of hardness 105ft, long, 4ft. wideband dOOp in 2 da;. . 27. 12, It' 5 BMO, in 4 days of 10 hours < I bench of 10 Bee of hardness, 5< Wide, and (\\\\. deep, how many mm can dig a trench of 5 degree of hardn- long, l.Ut. wide. and 4 \ ft. deep, in 9 days of 8^ hours ea< I pin SO in 1 year, wliat will $900 train in 8m.? 11. It' $><><) train $1 I riths, what will $100 train in 1 1 .*». If $100 gain $G in 1 year, in what time will $300 gain 1G. If $100 gain $G in 1 year, what principal will gain $12 in 8 month- ? 17. It' a 2-penny loaf weighs 9oz. when wl . Gd. per bii-hel. how much bread may be booghi lor 3s 2d. when wheat L per wmhcl ? An*, l lib. lOoz. 18. A wall, which was to be bnilt .12 feet high, was raited I feet by B m.n in 12 dnjlj how many men must be employed to build the remainder of the wall in !• <1 | 1. 19. If f>bbl. of flour serve a family of 8 persons 12m., how many bbl. will serve a family of 12 persons 10 month- ? . If 16 horses eat 21 bushels of oats in 6 days, how many bnekeli will 2-> boreea eat in 20 d 21. A of 1600 men have bread enough to allow 24 ounces per day to each man for 3 hot, the garrison being forced by 400 men, how many ounces per day may man have in order that they mav hold out against the enemy 30 days? 22. If 3 compositors, in 2 days of 9 hours - type for 27 pagBtj each page consisting of 30 lines of 45 letters each, how may compositors will set 36 pages of 40 lines of 54 letters each, in 6 davs of 8 hours each ? COMPOIM' PROPORTION. 267 If a man, walking U boon I day for 8 days, travel 384 . in how many daya of 10 boon each would be walk 240 miles traveling at the same rate? 2 1. It" a man travel 880 miles in 7 days traveling 10 hours . bo* many miles will he go in 12 days, traveling a* the Sam<- nie, only 9 hours eaeh day } It 12 horses or 10 oxen eat 2 tons of hay in 8 wi how mueh hay will 18 horses and 2 eki ? It* it take 88 reams of paper to make 1500 copies of o book of 11 ■ h ee iBj how many nam- will be required to mako eopies of a book of 9 sheets? 27. It' 600 tile-, each 12 inches square will pave a court, how many tiles that are 1<» inches long and 8 inches wide will pave another court which is 3 times as long and half as wide? 28. How many bricks, each 8 inches long, 4 inches wide, and 2 inches thick, would occupy the same space as 600 stones, eaeh 2 feet long, 1^ feet wide, and 8 inches thick ? 29. If 7 shares in a bank yield their owner $17.50 in 3 months, how much will 12 shares yield in 2 years? If 3 men, in 16 days of 12 hours each, build a wall 30ft. long, 8ft. high, and 8ft. thick, how many men will be required to build a wall 45ft. long, 9ft. high, and 6ft. thick, in 24 days of 9 hours each? 31. If the transportation of 9hhd. of sugar, each weighing 12 cwt., 2<> leagues, cost $50, what must be paid for the transporta- tion of 50 tierces, each weighing 2^ cwt., 300 miles? |800 gain $18 in 9 months, what is the rate per cent. ? 33. If a bar of silver 2ft. lin. long, 6in. wide, and 3in. thick, 26, what is the value of a bar of gold 1ft. 9^,3 in. long, s in. wide, and 4in. thick, the specific gravity of silver to that of gold being as 10.47 to 19.26, and the value per oz. of silver Wring to that of gold as 2 to 33 ? Ans. $12> .; 1. If 196 men, in 5 days of 12h. 6m. eaeh, dig a treneh of 9 :.et long, 8] feet wide and 4$ feet deep, how many men will he required to dig a trench of 2 degfl hardness 168$ feet long, 7$ feet wide, and 2J tot deep, in 22 days of 9 hours each ? Ans. 1 5. 2G8 ALLIGATION. ALLIGATION. 33*1. Alligation treats of mixing nmple - of different 1 qualities, producing a compound of some intermediate quality. It is of two kinds, Medial and Alternate. ALLIGATION MEDIAL 336. Allh; \ i tOX IfHXLiX || tbi proe. -< liy which we find the price of the mix: R the quantities and pi of the nmfloi are giim !. A mi reliant IBtSM 5 gallons of oil worth 4s. per gal. with 1 gal. at fo, 2 gal. at 1 Is., and .'5 gal. at 12>. What is the value of a gallon of the mixt 5 gal. at At. per gal are worth 20s. 4 u 5s. « " 20s. 2 " lis. « " _3 " 1-'. u " 36s. .-. 1 I are worth 98a. and 1 gat M worth ^ of 98s=7-.. An-. All example- of tin- nature are solved on this plan. Hence, 337. To find the price of a mixture when the num- ber of articles mixed and their prices are given, Iiri.F.. Di ride the total value of the articles mixed by the sum nf the simples, and the quotient is the price of * _ A miller mixes 20 bushels of corn worth 80c. per lnidi. with 1«) hush, of rye at $1, 40bnflh. of Oil . and .''"bush. of barley at 0(>c. ; what is the price per bushel of the mixture? 3. A grocer mixes 10 pounds of sugar worth 6c. per lb. with 1211). at 8c, 41b. at 12c, and 51b. at 15c. ; what is a pound of the mixture worth? ALLIGATION ALTERNATE. 338. Alligation Altlrnatk is the process of mixing 335. What does Alligation treat of! Jt is of how many kinds? What? 336. What is Alligation Medial? 33T. Rule? AM. quantities of different prices so M to obtain ■ mixture of a lo qiiir cd intermediate prim. Then are three problems. 3tli>. Pboblhi l. The prices of Beveral kinds of ascertain how much of each kind may • impound of a proposed medium 1. A farms* wi-hes to mix oats worth 30e. per busk with \\ ( ,r:li l to make ■ mixture worth 42a ; how inaiiv bushel- of each may In- take? A\u.Mr It is evident that be must mil them in such pro- tO gain just M much M his oats as he looei 00 the barley. Now, be gains 12c on l bush, of oats and totea but 8c. oo l bash, of barley; . ., for each boahd of oats he most take 12 -r- 3=1 busheu of barley, ■BOON mi i hod. A9 (30-, 3 — 12c. X 3= — 36c., deficiency. 4 * | .[:, J 12 _|_ 3c. x 12 = +36c, surplus. Having written the prices of the oats and barley in a vertical column and the price of the mixture at the left, as above, we write the difference between the mean price (i. e. the price of the mixture) and the price of the oats against the price of the barley, and the difference between the mean price and that of the barley against the prke Of the oats, and the differences stand- jain-t the prices of the oats and barley, rcspeetively, will MO! the proportion! ffUniHHm of Oatl and barley to be taten 1 for it will be seen that the product of the deficiency in the value of a bothelof oats multiplied by the number of bushels of oats ( — 12c. X 3 = — .">•><•.), h 7y equal to the pro- dint of the surplus in the value of a bushel of barley multiplied by the number of bosheli of barley (+3c. X 12 = -f-J Cat two products are composed of the same factors ; and one representing B deficiency and the other a surplus they will balance each other. In tfai ataas manner, any number of pairs of simple- in Wlmt U Alligation Alternate? IIow many Problems? 339. 01. Problem 1? Explain the analysis of Ex. 1. Explain the 2d method. 23* 270 allr; made to balance, as in Ex. f, the price of one simple in each pair befog tar sad thai of the other greater than the mean i In performing the operation, the terms are O bj a line merelj for convenience of reference In comparing diem. 2. A mereliant has 4 kind ST, Worth several!; . and 16c. per lb. ; how mix them so as to make a mixture worth 10c. per lb. ? OPE RATIO*. —30c 6— . C —4c X 6 = — 24c 8-, 8 — 2c. X 3 = - 6c. , < 13J 2 +3cX2 = + 6c) , , 1G-J 4 +6c X 4 = +24c j "T™ * Each pair of these products, \ 1 -t and 4th, and the 2d and 3d, will necessarily balance ; for they are composed of the 8am e factors, and the one marked -|- represents a surplus and the one marked — represents a deficiency. By this method the quantities always balai. are, however many simples may il in the mixl J! 10. There evidently may be as many in: Rile. Write the prices of the several simples in a vertical column ; on the left y separated by a line, write the proposed medium price ; connect, by a line, each price that is less than the 339. Explain Ex. 2. How are the prices connected? How do they balance? 340. How many answers may there be? How proved correct? Eule? ALLIGATION. 271 hi with <>ne or more that is cu the and the price of each simple against the >< or nn 7, which the simple \ ■ I ; these differ* or their sum if two or more stand ai/ninst one price, will he the urtionai parts of the several simj>les which mat/ be to form the jnixture. 3-11. Each of the foregoing method- ii simple and correct, but, for the convenience of the merchant, there is I better mode, me the quantities of the simples, and then, by calcula- tion, correct the assumption, as follow- : 3. A merchant has kinds of wine worth . r >-., Bk„ 8fc, and 1.3s. per pal. "What quantities of each may he take to make a mixture worth 9s. per gallon ? s. pal. s. *. 7 X — 4sbs — 88 G 6X— 3=— IS % 8 6 X — 1 =— J> —52, deficiency. 13 2X +4=+ 8 15 4 X + C = + 24 _j_32, surp i us . p;al. s. — 20, deficiency. Add wine at 15s., 4 X + 6 = -f-24, surplus. -J-4, surplus. Subtract wine at 13s., — 1 X + 4= — 4, deficiency. Having assumed 7gal. at 5s., Ggal. at G^., figal. at 8s., 2gal. at and 4gaL at 1V, we find the mixture is not worth so mueh as it should be by 20s. Now this may be remedied by putting in more of the highel priced wines or tesa of the cheaper. If id -Igal. more of the l5§, wine, this will balance the d< li- cieiicy and OTIHtfl a surplus of Is., and this may be corre ct ed by taking out IgaL of the 13s. -wine. Thereare now in the mixture 7gal at 5s., 6gal. at . at 8s.. lgal. at 13s., and 8gal. at [5a, mark. The deficiencies are marked by the >ign — and the ' y -|- to aid the mind in making correctional 341. Another method, explain it. What la the remark? 9s. 878 ALLIGATION'. Notk. This mode of correcting may be indefinitely raried, hence the merchant may take the simple* in a ratio more nearly as be desires than by fither of the other modes. Let th«- poj - Pirampl b of die l th.-m ; 4. A grocer wishes to mix teas worth 2 . and o thai tin- compound may be worth pound, many pounds of each may he take? rth $IG, $20, $28, $ !<">. and $50 per 1m tad ; what unmbei of each may he sell at an average - $30 per li :i 1*2. Tuublem 2. The price of each of the bud.] the price of the compound, an. for tin g parts. Now if the 41b. at I$c 2 * will become 12 11-.. (/<>■ given owftv and if each bf tlit- proportional parts be increased in the same ratio, evidently the per lb. or the mixture will remain unaltered ; thus, 41b. at ISe. : 121b. at 18c, : : 51b. at 4c. : 151b. a 41b. at 1.) • : 121b. at 13c : : 31b. at 6c : 91b. at Cc. etc. 341. What is the Note? 34 i. Object of Problem 2? Bttlef I.xj.lanatiou? 8c. L0- AU.I';\T!"\. 2. How many gallons of win.* at % 10, and 15s. per gal. may he mixed with L5gaL of water of no exchangeable vain.-, to make a mixture worth 1--. par gal.? How many 11). of wool at 10, 10, an : CO lb. :: 71b. :281b. at 8c 15 : 601b. :: lib. : 41b. at lie. etc TVe find that the sum of the proportional parts, if linked as .is 15 lb., and if this be quadrupled, GOjb , the required comjKjund, will be obtained ; but the whole compound will be quadrupled by i each of the proportional parts in a four fold ratio. How many ounces of gold, that if 1G, 18, 20. and 21 carat* nay be taken to form a i .'1 raruts fine? 3. How many -hep worth 9, 12, 16, 18, and 2 U each, may be taken to form a Hock of 125 sheep worth 15s. each? 343. Object of rrob1«m 8? Kulc! Explanation ? 271 INVOLUTION. INVOLUTION. 344. A Power of a numl.. ■ || ih-- product obtained by u-iiiLT the number two or more timet as a factor. Involution is the process of raising a number to a p< The numlx r involved is the 1*/ power of it.-rlf. It is al.-o the root of the other powers (Art. 1 1 J ■'> and C). :M»1 The Index or Exj t ■ power is a figure placed at the right ind ■ little above the root to show how many times it is used as a factor (Art. 112, Note 1) ; thus, 4X4= 16 = 4 , ,i.e. tli<- l>,1 ji(,w,Tor^(juareof4. 4x4X4= 64=4* i.<-. the M power or cube of i. 4x 4X *X 4= 256= t 4 . Le,the 4th power of I. 4X4X4X4X 4 = 1024 = 4*, i.e. the . r ,th power of 4 1 1 nee, IM6. To involve a number to any requited power, Rule 1. Write the index of the potver or, ,- (kt root ; or, Rule 2. Multiply the number by itself ami (if a //if/her power than the second is required) multiply this product by the original number, and so on until the root has b< as a fac- tor as many times as there are units in the index of the required power. Ex. 1. What is the 3d power of 6 ? Ans. C 3 =6 X 6 X 6 = 21G. 2. What i> the 5th power of 3 ? Ans. 243. 3. What is the 4th power of 5 ? 4. What is the 8th power of 2 ? 5. What is the 2d power of 10 ? Ans. 25 G. 6. What is the 3 power of J ? 4__2 f2] s 2 2 2_2 8 ___8_ 6""3 ; UJ — 3" X 3 X 3 ~3 3 — 27* 344. What is a Tower of a number? What is Involution? Wliat is the num- ber involved? 343. What is the Iudcx or Exponent of a power? 345. Rule for involution? Second rule? iNvm.rr 7. What h the M power of ^ ? 8. What k the N i-ow.r of J ."< 1'7. *j. What h th<- hh power of .12 ? 10. What is the tqpart of I H»iJ and(S)« = U = 5A. Ans. 11. What is UM .Mil... of 3fc? l. It will be observed that ■ nixed Mother is fh-t redtteed to an ■ ■ proper. ftfccoToo, ami ■ eoonooo titanium m reduced to its lowest tcnns, uml then each term is involved separately. AleO that the number of tleci- nial j.laees in the power of a ileciinal is eipial to the nuinher of decimal In the mot multiplied bj the index of the pOWOff (Art. 171). N-ii: 2. The powers of a number greater than unity, are greater than >t, and the powers of a pro|.er fraction are less than the root ; thus, the cube of 2 is 8, which is greater than 2 ; and the square of i is J, which is greater than 3 ; but the square of Z is 4, which is loM than 3. #17. To multiply different powers of ili«- mme num- ber together : II f i .i-:. Add the i/i lumber by the index of the power to which it is to be rot Tims, the 3d power of 2* is 2 6 , for 2 2 = 2 X 2, and the 3d power of 2 X 2 is 2X2 X 2x2 X2x2 = 2x2x2X2X 2 X l > = 2 6 =64. 1 J. What is the 1th power of 3 s ? Ans. 3 12 . 15. What is iheJUh DO* 340. What is done with a mixed number? How is a common fraction in- •v many decimal places in the power of a decimal I If the root is greater than one, are its powers greater or less than the i » •« «t * If tlie root is less than one? 317. Itule for multiplying different powers of the same num- ber together? 348. Rule for involving a power? 276 !M9. To divide a power of a number by any other r of the same numl BULK. Subtract the exponent of the divisor from the exponent of the dividend. 1 G. Divide 5 r by 5 s . Ans. 5 T + 5 8 = 5«, for 5 7 -f- 5 s = - = 5X5X 5g5X5 X5><5 = 5X5X5X5 = 5 - 6 -" 1 7. Divide 8* by 8 r . 18. Divide 4 T by 4 s . B VOLUTION 3«?0. A root of a number is one of the equal factors whose continued product is that number (Art. 1 '■)). Involution or Extracting EtoOTf is the resolving of a quantity into as many equal factors as Oiere are units in of the root. tttil. There are two method- of indicating a root, one by BMiM of the radical sign, ^, and the Other by means of a tonal in--' The figure placed over the radical sign is the index of the root, and is always the same as i irfimtftff of frmtJ index ; thai, the cube root of 8 is \/8 or 8.* Tlie square root of the cube of 4, or the cube of the square root of 4, is v/4 3 or 4*. If no number is over the radical sign, 2 is understood. 35*2. Evolution m the reverse of INVOLUTION. In Involution the root is given and the power required. In Evolution the- power is given and the root required. 349. Rule for dividing one power by another power of the same number! 350. What is a Root of a number? What i* Evolution? 351. How many methods of indicating a root? What? What is the index of the root? What of the index 2? UOOT. 3*13. All numbers can be "!/ required power, but comparatively fnc can 1>< Thotc Quintan which can li.-ivr th.ir roots extracted are called perfect powers, ami their root- are rational Bombers. Nun ,:mot ta I called imperfect power.-, and their roots arc irrational or turd osnabt A Dumber may be a perfect power of one naflbe ot d ami an imperfect power ot* another; thus. 16 is a perfect aqoare, hut an imperfect cuhe. whereas i_>7 ifl I ita, hnt an Km* |H| 6 1 ifl a p re, cuhe, and power. 3*7-1. Kverv power and every root of 1 U 1. There is no other number whose powers and roots are all alike. The roots of a proper fraction are greater than the fraction, and the roots of any number greater than unity arc leal than the number ; thus, ^/J = Jj, which is greater than |; \/{jJ=j», which exceeds £J ; but */%$ = £, which is leu than y ; \/.S C3 2, which ifl toflfl than 8. . EXTRACTION OF THE SQUARE ROOT. 3*7*7. To kxtkact tiik SQUARE ROOT of a number is to resolrc it mio Iwo equal farfnrs. i. e. to find a number which, multiplied into itsrlf will produce the given num : 3*70. The tqmart of a number consists of hcice as many figures as the root, or of one less than twice as many ; thus, Ro- 1, 9, 10, 99, 100, 999. 1, 81, 100, 9801, 10000, 998001. Hence, to ascertain the number of figures in the square root of a given number, 353. Can til number* be involved' Evolved? Wbat are perfect powers! Rational ro< ' feet powers? Irrational or surd roots? May a i be a perfect power of one degree and an imperfect power of another degree? A perfect power of several degrees? Illustrate. 15*. What of 11 The roots of a proper fraction, are they greater or less than the fraction? The roots of a number greater than one? 355. To extract the square root of a number, what? 856. How many figures in the square of a number? 24 278 EVOLITI Rile. Beginning at the right, point off the number periods of two figures each, and there trill be one figure in the root for each period of two figures in Ute square, and if there is an odd figure in Ute square there will be a figure in the root for that. Ex. 1. How large a square floor can be laid frftft iare feet of boa: If we knew the length and breadth of a floor, we should find it^ area by multiplying tin- length bj the breadth (Art KM). <>r, in this example (since length and breadth an* equal), by multiplying the length by itself. But tee are now to reverse this process, and, knowing the area, to fin A ,'h of one side. Since the dumber, .">7G, consists of thro ;'6(24 U) and the square of _ (hundreds) : and, a-s 5 (hundred^) i^ i«->s | 1)17 6 than l J (liu: can be but I 17 6 -) in the root. Let na now construct 7 a square. Fjg, 1. • of which shall be S ) jn length. area of this square is 20X20 = 400 square feet, which, dedu from . will leave 176 square I seed in enlarf floor. To preserv.- the square form. tln> addition most be i uj>on the 4 sides of the floor, or, qnmlly uj>on 2 ea, a^ in 1 the nature of the oaCBj the % addi- tions, bin breadth ; and, if th«'ir length known, we could determine tin ir idth by dii iding their area, I ~ 6 feet, (An. 102). But we do know the length of bh -\- cr, viz. twice the tens of 350. To a.-certaiu tue cumber of figure* in a square root, Rule? fcxpl&in Ex. 1. r Fig. 2. n m 90 4 4 4 80 1 6 d k c _i «^ 20 20 c 4 80 20 1 > 4 ft 'J the root = 4 (tens or 40 ft), and this h s u fl cleutlj near to the whole leagtfc of the addition- tO Serve M I trial d Nuw 17»; ~ 40, or, what is .one in effect, 17-^-4, l feel fbr the breadth of tin- addition, and this Added tO the trial dieisor, 10, or annexed to tin- l (tent) will 1 1, the whole length of bm -f- CT, the // . And 41 x 4= L76] i. e. the length of the addition multiplied by it- braadtfa givei it- .■ It will be seen thai eyery foot of hoard is used, and the floor is a square, each side of which is 20 + 4 = 24ft. long, Ans. *,\r>7 . The same species of reasoning applies, however many figures there may he in the root. Hence, To Extract the Square Root of a number, Kile. 1. Separate the given number into periods of two figures each, by placing a dot over units, h/utdreds. 2. Find the greatest square in the left-hand period and set its root at the right, in the place of a quotient in long division. 3. Subtract the square of this root figure from the lrj't-ha/id period, and to the remainder annex the next period for a divi- d(/td. , 4. Double the root already found for a TRIAL divisor, ami, omitting the right-hand figure of the dividend, divide and set the II the next figure of the root. Also set it at the right of the trial divisor, and so form the tiui: divisor. 5. Multiply the, true divisor by this new root figure, arid subtract the product from the dividrnd. 6. To the remainder annex the next period fi>r a new dividend, double the part of the root already found for a trial divisor, and proceed as before until all the periods have been employed* 3.7. Kule for extracting the square root of a number? 230 Note 1 . The left-hand period may consist of bat one figure. Note 2. The trial divisor being smaller than the true divisor, the quo- tient is frequently too large, and a smaller number must be set in the root. This usually occurs when the addition to the square, a e, is wide, and, conse- quently, the square, h n, large ; or, in other words, when the trial divisor is much less than the true divisor. 358. Proof. Square the root; thus, in Ex. 1, 24 s s= 2. What is the square root of 401956? OPERATION. 4 6 1 9 5 6(6 3 4, Ans. 86 12 3)419 3 6 1 2 6 4) 5 5 6 5<' 3. What is the square root of 1918-1 1 ? Ans. 438. 4. What b the square root of 677. 4 i- 5. What is the square root of 67081 ? OPERATION. B 9, Ans. In this example, the 4 hand period consists of but 4 5)2To one fi £ ure « So» also, the 2 25 tr ' a * divisor, 4, is ftflfltfriH . tim**- : and the 2d 5 9)4581 remain d er, 4.">, equals fie . the true root re is but & What is the square root of 97 Ans. 31 7. What is the square root of 136161 ? & What hi the square root of 4201C I. W "hat is the square root of 4304C7 10. What U the square root of 22014> 11. What is the square root of 1522756 ? 12. What is the square root of 18671041 ? 13. What is the square root of 60910: 357. What is Note 1* Note 2! 358. Proof? Explain Ex. 6* SQUARE ROOT. 281 l i. What is the iqutre root of 16777116! Ol-I 167 77216(4096, figure 1 6 '. m in tin- example, 809)7772 "" x ° lu - a a j AM trial divi-or, ; 1 1 1 . 1 bring down tin- next 8186)49116 period i<> complete 49116 dow dividend. 15. What is the square root of 5701801 ? 16. What is the square root of 1048.*>7 17. What is lie- square root of 282 [TSt 19 ? Note 3. Iii tlfmiliifl tlio root of a dicimnl, put tin- first point over hurt- iredths and point toward tin- rigkt, ami if tin- bM ptriod is nut full, ■aatt 0. 18. What is the square root of .4096? .64. 19. What is tlie square root of .«'• 20. What is the square root of 39.0Cl' An-. .;.•_»:.. 21. What is the square root of 6046.6176? & What is the iqHan root of f>. Ans. 2.36+. OPERATION. SO ( 2.3 6 -j- If there is a remainder after 4 employing all the periods in 4 3)160 t ' l( " "' m ' n ^'' ,nc opera- ' , .. tion may be continued at plea- sure by annexing Miccc-ivr 466)3100 j„ Hods q£ ciphers, decimally ; 2 7 9 6 there will, however, in such 472 )3040 0. IT— pUie. cttwam be a renuiin- r the right-hand ligure of the dividend is a cipher, whereas the right*hand figure of* the subtrahend is, escesiarifo the right-hand figure of* the square of some one of the nine significant figures, the right-hand figure of the reel "/"/ of the divisor being ahocyt aKhe. Now, no one of these nine I I, will give a number ending with a cipher; .-., the last figure of the dividend and of the subtra- hend being unlike, there wmti oe a rtonainder. What is tb An-. 1.411-J1+. 358. Explain I a 1 -We 3? Explain Ex. 22. 282 EVOLl I L What fa tli 8 square root of 20? . 172-f-. What is the square root of 31 G? 20. What II thr .Mjuare root of 31. 3*19. To extract the root of a common fraction, or of a mixed number: Kile. Reduce the fraction or mixed number to its simplest form, and then tot. 360. ATin am; l Lis a figure bounded bj tfiOM straight linjBS. A right-angled triangle has one of its - a right angle, as at C The side op the rijrht angle is 1 (be hypotkmum ; tlie other two are the base and perpendicular. li Bom. 359. Rule for extracting the root of a common fraction or mixed number? *GO. What is a Triangle? A right-angled triangle? Hypothenuae? Base? 288 u The square described on the hypothecate of a right-angled triangle H im of the ret described oo the : two sides. Also the square of either of the t\ hich form the right angle Is equal to the square of tin- hv- pothc nos c diminished by qnarc of the other tide. Thi> will be seen l>y pointing the small squares in the square of the hypothenuse and those in the squares of the other two sides. Hence, 1st. To find the hypothenuse when the base and per- pendicular are given, lit i.k. Add the square of the base to the square of the per- 'pendicular, and extract the square root of the sum. 2d. To find either side about the right angle when the hypothenuse and the other side are given, BULB. From the square of the hypothenuse, subtract the stjuctre of the other given side, and extract the square root of the remainder. . l. The base of a right-angled triangle is 6 feet and the dicalar U 8 feel : what ii (be hypothem (V-' = ;io\ 8*3x64* 36 + 6-4 = 100; */100=10. Ans. 10 ft. 2. Tin* hypothennse of a right-angled triangle ia 15 and the H 12: what i< the perpendicular ? = 225, 12»= ill; 225— 111=81; a/81 = 9, km. 300. The square of the hypothenuse equals what? The square of out Other - may this appear? Rule for finding the hypothenuse? Base *r Perpendicular? Explain Ex. 1. 284 utimw 3. A ladder resting upon the ground 21 fed from a he.- just reaehei a window which k 18 feel hflghj li<>w bog ii the ladder? 4. A tree that was G4 feet liLdi U broken off I 1 feel high, part broken oil' turning iij><>n the still) as upon a hinge; at what distance from the bottom of the tree doe* the lop rtrike 1 1 1 . - ground ? 5. Two vessels nail from tin- same pott, DM due 000l 40 ■ and the Other due -<>utli 9 mfUnj be* tar apart an- they? 6. A general has 9801 men ; if he places them in a square, bOW many will there bi bO rank and file? 7. HOW many rod> of liner will 1,< n-«|uir« d to indoor CIO acres of land in a square form } 8. A farmer sets out an orchard of GOO trees so that the num- ber of rows is to the number of trees in a row as 2 to 3. trees are apart and no tree is within 12\ feet of the fence; how many square feet of land in tl Fig. 3. IUM. In li ihined ar/ . 1 •»'.»), a sqwm (Art. 1<»1 . .nil right mi fled tri Hu- ghs. Tie- line AC Kl tin .• of the square and tl>. /it/pothenuse of each of the triang The square is said to bo mierib circle and the i about the sop; The diameter of any circle i- to it- circumference in the r of 1 to &.14J .rly | benet tlie diameter multiplied }>y "lll-V.ej >v ill giro the circumference, and the circumference divided bj 3.141592 will gi\.- T 1 * « - diane The area of a circle may be found by multiplying the square of it< diameter by .785898, nearly, and if the area i> divided by .785398, the quotient will be tlie Bquare of the diameter. 301. What does Fig. 3 represent? What is the line AC! What is said of the square? Of the circle? Ratio of diameter to circumferenc. > H..w i< cir- cumference found when diameter is given? Diameter when circumference is given? Area of a circle, how found? Diameter, when area is given? SQUARE ROOT. 285 3<>£. Similar figures are figures that are of pCfinJOOlj flip form, whether large or small. The areas of all similar figures arc to etch otlicr as the squares of their co rr e sp onding li: What is tBO diameter of I circular pood which -hall contain HOB M much area as one 8 rods in diameter \ An-. lOrd 10. The area of a triangle H '1\ square inches and one si the difference in the expense of fencing a circu- lar 10-acre lot and one of the same area in a square form, the fence co-ting 7ac. per rod? 1 ">. If a pipe 3 inches in diameter will empty a cistern in 8 minutes, what is the diameter of the pipe which will empty it in 18 minute 16. The area of a rectangular piece of land (Art. 101, Note) is 50 acres, and the length of the piece is to its breadth as 5 to 1 ; what are the length and breadth? 17. A room is 16ft. long, 12ft. wide, and 9ft. high; what is the distance from one lower corner to the opposite upper cor- Ans. 21.931+ft. 18. The diameter of a circle is 10 inches; how many inches in the side of the inscribed square? Ans. \/')0 =: 7.07 1-[-. By figure 3 it is seen that the diameter of the circle i> the hvpotheiiu-e of a right-angled triangle \vhoti<-k of timber that can !>•■ hewn from a log 18 inches in diameter? 369. What are rimilar figures f The ratio of the areas of similar figures! J- 1 EXTRACTION OF THE ( DBE BOOT. 363. To extract the Cube Root of a number is to resolve it into 3 equal factors ; t. e. to find a numher w/tirh, mul- tiplied into its square, will produce the gi< : >er. 364. The cube of a number consists of three times as many ♦'S as the root, or of one or two less than ti. « as many ; ti. B^ota, 1. '\ 10, 100, (ul.... 1. 729, 1000, 97 1000000, 99700l' Hence, to ascertain the ires in the cube root of a L'iwn Hum / at the right, point off the numh,,- into periods of three figures ea>- rein the root for each period of three figures in fkk one or two figures besides full periods in the cube, there will be a figure in the root for this part of a period. 1. Suppose we have 74088 blocks of wood, each a cubic inch in Mze and form, how Ufrgt a cubical pile can be formed Un- packing these blocks tog OPERATION. Trial divisor. 1 8 , 240 4 7 4 8 8(42 Root. 64 10 8 8 Dividend. 10088 True divisor, 5 4 4 J As there are tiro periods, the root must consi-t of two figures, tens and units ; and we seek the cube of the tens in the left-hand > 1 ; the gj ube in 74 is M root is 4. We the root, I. at the rijrht of the number, and, having sub- tracted th I, from the left-hand period, we annex the next period to the remainder, 10, making 10088 for a dividend. 303. To extract the cube root of a number, what? 364. How many figures in the cube of a number? To ascertain the number of figures in a cube root, Bute? Explain Ex. 1. CUBE RO«»T. Ffe 1. 40 inches. Thus, by Ming 1000 of the blocks, a cube is formal (Fig. 1 ) wfcoaeedge ii i<> inches andwhcee ■jopteti arc 6 1000 solid be and there are 10088 blocks re- OMiniog, with which to anlai ibio pile already EbrnefL In enlarging this pile and i tng the cubic form, the nddi- must be made upon each of the 8 laces, or, more coim-ni- ently, equally opoa any 3 adja- cent d, />, and r, as in 2. What may be the thick- of the addition? By divid- iog the contents of ■ rectangular solid by the area of one ftee, we obtain the thickness (Art. 105) ; now, the remaining 10088 soli4 inches are the contents, and the sum of the area- ot the .'5 square faces, a, b, and c, is sufficiently near the area to be covered by the additions to form a *"g« •• trial divisor; for the 40 3 additions, a, 6, and c (Fig. 2), are the same as one solid 40 inches wide, 3 times 40 inches long, and of the thick- ness determined by trial. The area of these 3 is the square of 4 (tens), which is 16 (hundreds), multiplied by 3, which gives 4800; i. e., to obtain a trial di- visor, we square the root BgOffe and annex 00 cau-e the root figu; ten ) for the area of one and then multiply this area by .'!. Dividing 10088 by obtain the quotient 2, fur the ikicknm of the tuUdi ■ v the imit figure ot the root. Having made theac addirioni, a^ in i that the | > i 1 * - doOl DOfl retain the cubic form, n, hi, and/;/, being vacant Bach of tie :ig, S inc. and 2 inches thick; i.e. th* 288 EVOLUTION. area covoml to tin- depth of two ndm by filling the n Ben in Fig. 2, as seen in Fig. 3, is 2 X 40 X ,VJ = -^ n square inches ; and still there is a vacant corner n, n, n, as seen in 40 2 which is a cube of 2 inches on each edge ; Le. it is a solid 2 Fig. 1. 4<» z /~ 40 71 40 40 beta thick, (the common thickness of all the additions), covering 2X2 = 4 square inches, as seen in Fig. 4. rrr.E ROOT. 280 Now, if the several additions made in Figs. 2, 3, and 4, be spread out upon a plane, as in Fig. 5, / 7U 7\ £ 7\£ZA<£Z?[ZZA \& or, in a consolidated form, as in Fig. 6, / / / A 3 it will be readily seen that their collective solidity will be obtained by multiplying the entire area which they cover, (40 X 40 X 3 + 40 X 2 X 8 + 2 X 2 = 5044 g q» ar e inches), by their common thickness, 2, which will give 10088 solid inches; .\ a cube is formed (Fig. 4) whose edge is 40 — |— 2 = 42 inches, and no blocks remain. 36.1. If there are more than two figures in the root, the same relations subsist, and the same reasoning applies. Hence, To extract the Cube Root of a Number, Kile. 1. Separate the number into periods of three figures each by setting a dot over units, thousands, etc. 2. Find by trial the greatest cube in the left-hand period, plans its root as in square root, subtract the cube from the left-hand period and to the remainder annex the next period for a divi- dend. Z. Square the root figure, annex two ciphers and multiply this result by 3 for a tkiai. divisor ; divide the dividend by the trial divisor and set the quotient as the next figure of the root. Rale for extracting the cube root of a number ? u 290 EVOLUTION. 4. Multiply this root figure by the part of the root previously obtained^ annex one cipher and multiply this result by three ; add the last product and the square of the last root figure to the trial ■ r, and the jiM will be the TRUE DIVISOR. 5. Multiply the true divisor by the last root figure, subtract the product from the dividend, and to the remainder annex the next I for a new dividend. G. Find a new trial divisor, and proceed as before, until all the periods have been employed. Note 1. The notes in Ait. 357, with slight modifications, are equally applicable here. ; k 2. If the root consists of Tnree figures it is plain that the cube, as completed in Fig. 4, must be enlarged just as Fig. 1 has already been enlarged. Hence, the new trial divisor will consist of 3 faces of 1%. 4 ; lut the true divisor already fotuui is the sura of the significant figures in these 3 faces, except one face each of rr, xx, and », and two faces of the little cor- ner cube, nan ; moreover, the number directly above the true divisor (in the operation) represent! am face of win, and the number above that repre- sents the sum of one /ace each of the 3 long corner blocks, rr, rr, and n ; hence, tojind the next trial divisor, we have only to add the true divisor alrradj /bund to twice the number above it, and oxck the number above that, and to ike sum annex two cipher*. When there are many root figures this process is shorter than to square so much of the root as has been found, annex two ciphers, and multiply by 3, as directed in the 3d paragraph of the rule. I What is the cube root of 21- 21024576(276, Ans. ;17 j .-. we annex to the root, 00 to the 1st trial divisor for the 2d trial divisor, tnd bring down the next period :ii|.l9). What [| the value of the following expressions : 4. V**0*3H ? Ana, 1 11. 11. V ;i,; - :, - ,;,,;;7? Ans. 3.33. 5. V 317G ^3? 12. V 10077 - ( 6. V 3 82657176? 13. VW.353607? 7. V^24024008? 11. V1G6J? Ans. 5}. 8. V387420489 ? 15. V 5 61gi ? 9. V13 1217728? 16. VH* Ans. 1.65+. 10. V^ ? Ans. 1.709 +. 17. V*3f ? Application of the Cube Root. 360. Bodies which are of pr ec isel y the same form are simi- lar to each other, and the solid contents of similar bodies are to each other as the cubes of their corresponding lines, and con- v< r-< lv, the corresponding lines are to each other as the cube roots of the contents. Ex. 1. If an iron hall I inches in diameter weighs 16 pounds, what i- the weight of a ball 30 inches in diam< it*, or l 1 :6 s :: 16: Ans.; i.e. 1 : 216:: 16Uk : ;;i."m;i!,.. a.,, 365. What in Note 1' Note 2' Explain Ex. 2. Ex. 3. What it Note 8! 366. What are similar Unties? The ratio of the content* of similar bodies! 202 ARITHMETICAL PROGRESSION. 2. If a bull inches in diameter weighs 27 pounds, what is the diameter of a ball thai freight 04 pounds? V 27 : V G * : : r ' In - : Am. : i. « . 8 : 1 : : Gin. : 8in., An<=. 3. How many bullet* \ of an inch in diameter will be required to make a ball 1 inch in diameter? 1. If a globe of gold 1 inch in diameter tfl worth $100, what is the diameter of a globe worth $6400 ? 5. S ;ij i] the diameter of the earth is 7012 miles, and that it takes 1404928 bodies like the earth to make one as large as the sun, what is the diameter of the sun? 6. A bin is 8 feet long, 4 feet wide, and 2 feet deep ; what is the edge of a cubical box that will hold the same quantity of grain ? 7. If a stack of hay 2 1 foei high weighs 27 tons, what i< the bight of a similar stark which weighs 8 to: An*. 1 8. If a hell 1 inches high, 3 inches in diameter, and \ of an inch thick weighs lib., what are the dimensions of a similar bell that weight 271b.? 9. If a loaf of sugar 10 inches high weighs 81b., what is the Light of a similar loaf weighing 1 lb. ? ARITHMETICAL PROGRESSION. 367. Any series of numbers increasing or decreasing by a common difference is in Ai:i i hmf.tk ai. Progression; thus, 2, 5, 8, 11, 14, 17, etc. is an ascending series, and 35, 30, 25, 20, 15, 10, etc is a descending sen The several numbers forming a series are called Terms ; the fir»t and last terms, Extremes; the others, Means. The ditVerence between two successive terms is the Common Dif- i iiiiNCE. 367. When is a series of numbers in Arithmetical Trogreasion? How many kinds of series? What? What ara the. Terms of a series ? ARITHMETICAL 1 293 In an arithmetical alars ekrii dHoo, viz. tln> fir-t term, last term, common dUfer eoce, nomber of terms, and sum of all tho term-; and tin related to each other that If any three of them are given the other two can be found. 3GS. in an ascendin let 6 be the first term and 5 the common diflereo Then 6 = 1 -t term. C + 5 = ll=2d term, C-f-5 + 5 = 6 + 2x5 = 16 = 3d term. C + 5 + 54-5 = 6 + 3x5 = 21 = 4th term. Tin: that, in an ascending series, the second term is found by adding the common difference once to the first term ; the tliir.l term, by adding the common difference twice to the Jirst term, etc. A similar explanation may be given when the ser; descending. Hence, 369. Problem 1. To find the last term, the first term, common difference, and number of terms being given : Rule. Multiply the common difference by the number of terms less 1 / add the product to the Jirst term if the series is ascending, or subtract the product from the first term if the series is descending, and the sum or difference will be the term sought. . 1. If the first term of an ascending 5, the com- mon difference 4, and the number of terms 7, what is ths term ? 5 + 6 X 4 = 29, Ans. % The first term of a descending series is 47 and the com- mon difference 8 ; what is the Cth term ? •17 — 5X8 = 7,.' 3. What il the amount of $100, at G per cent., simple inter- .:- ': 307. What are the Extremes of a series? Means? Common Difference' 11 \r many particulars claim special attention' What ure thev ■? llow many of them must be given? 368. llow is an ascending series formed ? How a descend* ing series? 309. Object of Problem 1 ' liule? 25« 294 ARITIIMKTK AL PROGRESS I 370. Problem 2. Tu Snd the common difference, the extremes mid Dumber of terms being iri\ By inspecting the formation of the series in Art. 3C^, it will be seen that the difference ' between the extremes is equal t common difference multiplied by 1 less than the number of terms ; e.g. the difference betWW n tin- 1st and 4tli term (91 — C — is the sum of 3 equal addition- ; .-. this difference, divided (16 -$-8 = 5), gives one of these additions, i. e. the common ference. Hence, lit if. Divide the difference of the extremes by the number of terms less one, and the quotient will be the comtn Ex. 1. The extremes of an arithmetical series are 3 and 38, and the number of terms is 8 ; what is the common difference ? 38 — 3 __ 35 2. A man has 6 sons whose ages form an arithmetical set the youngest is 2 years old and the oldest 22 ; what is the dif- fen not Of their a_ Ans. 4 3. The amount of SI 00 at simple interest for 10 years is $1G0 ; what is the rate per cen: 371. Problems. To find the (lumber of terms, the extremes and common difference being gi\ By Art. MB it i- •\i.lnit that the difference of tin- extremes is the common difference multiplied by one leas than the number rms. 11 lv, Kt i i . 1 > ride the difference of the extremes by the common difference, and the quotient, increased by 1, is the number of terms. i.x. 1. The extremes of an arithmetical series are 3 and 31 and the common difference is 4; whit b the number of terms? 31— 3 , , 28 + 1= T +1=7 + 1=8, 4 2. The common difference in the ages of the children in a family is 2 petti] the youngest is 1 year old and the oldest l'J; how many children in the family ? Glomeruli' a l ii 295 372. Pbobhm t. To fM fee quo of q • extremes and number of terms l >uin of the i qoaJ to th<- tum of any two equally distant from tl .:; the •,11. i:;. we 1 ] m -f- Oth = 2d + 5th = 3d -4- 4th. 3 + 13 = 5 + H s= 7 -f 9 =16) . the ram of nil terms ii 10 x 3 = 48. 1 :• BlTLB. Multiply tkt sum "J the extremes by half the nn- of terms, ami the product is the sum of the series. Ex. 1. The e 4 a leftSS MM •"> and .')■.' and the numl»«T of terms is 10 j what i> (hi MUl of the series? 3 + 39 = 42 ; 10 -*- 2 = 5 ; 42 X 5 = 210, Ans. 2. How many strokes does a clock strike in 12 hours? GEOMETRICAL PROGRESSION. 373. Any series of numbers increasing or decreasing by a common ratio is in Geometuical Puoguession ; tlniS, 2, 0, 18. .">!, 162, etc. is an ascend in. and CI, 32, 16, 8, 1. etc. is a desoeodiii In tin- above, 8 is the ratio in the 1-t .-eric- and I in the 2d. The first tana, last term, ratio, number of terms, and sum of all the terms are so related to each other that if any three of them are given the other two can be found. 374. In a series, let 2 be the first term, and 4 the ratio ; Then 2 = 1st term. 2X4= 8 = 2d term. 2X4X4 = 2X4*= 32 = N torn 2X4X4X4 = 2X4 S =128= 4th tens. 370. Object of Problem 2? Rule? 371. Object of Problem 3' i' Kuk ' 373. What constitute* a aeries in Ceometri- Of *erie«t term 250, and the number of term* 1 ; what Ml the ratio? j 8 -a IK, -i — i = ;; ; tod N ' 2. The i Sand l*. and the nnmhnr of Ur— fl j what il the ratio? . 4 or |. mes are 3 and 243, ami the number of terms 5; what is the ra: 377. Problem 3. To find the mm of a M extreme's ami ratio being given. Having a Beriee given, e, g. 2, 10, 50, MO, 1250, C2."><>, wu/- ://)/// each trnn except the last by the rutin, 5 ; t! 2, 10, 50, M0, l M0, [6960], Produet i 10, 50, 250, U '■<) ; and we. shall evidently form a ftftp I Ties like the old, exe.-pt the lir.-t term of the pJU " "'. the remaindrr tW// ta {/«" o/ fta extremes in the old series the other terras in the two ich Other; the remainder w;'.. be 4 times the sum of all the terms except the last in the old - ; for once a leriei from 8 nfwm a series must h BfietJ ."• | of this remaindeJ plus the lust term must be the sum of all the terms in the old series ; but 4 is the ratio less 1. A similar explanation is always applicable. Hence. Kile. Divide the difference of the extremes by the ratio less one, and to the quutitnt add the greater < Ex. L The extreme, are '2 and 486, and the ratio 3 ; what is the sum of the series ? 486 — 2 = 484 ; 3—1 = 2; 484 -f- 2 = 242 ; and 212 -f 480 = 728, Ana. mes are 4 and 518 1, and the ratio ; what is the sum of the sere 3. What debt will be diecharfed by 12 monthlr payments, 1-t payment being $1, the 2d $2. and po on in i geometrical 37 7. < 'bjcet of l'roblim I I 298 ANNUITIES. ANNUITI 378. An Annuity is a sura of money payable annually, or at any regular period, either for a limited time or forever. An annuity is in arrears when the installments remain unpaid after they are due. The Amount of an annuity in arrears is the interest of the unpaid installments added to their sum. 379. Problem 1. To find the amount of an annuity in arrears, at simple interest. Ex. 1. An annuity of $100 per annum has remained unpaid 4 years; what is its amount? Ans. $436. The 4th payment is due to-day and is worth just $100; the 3d payment due 1 year ago i> worth $106 ; the 2d payment due 2 years ago ifl worth $112 j and the 1st payment due 3 years ago is worth $118. But these numbers. $100, $106, $112, and $118, are in arithmetical progression. Hence, BULB. Fmd the last term of the series by Art. 369-, and the sum of the series by Art. 372. 2. Purchased B farm for $5000, agreeing to pay for it in 5 equal annual installments ; the 5 years having elapsed without any payment being made, what is now due, allowing simple interest? Ans. $5600. 3. A salary of $600 per annum is in arrears for 8 years ; to what does it amount, allowing simple interest at 7 per cent. ? 380. Problem 2. To find the amount of an annuity in arrears at compound interest. Ex. 1. What is the amount of $1 annuity, per annum, in arrears for 3 years, at 6 per cent, compound interest ? The 3d instalment becoming due to-day, is worth just $1 ; the 2d having been due 1 year, is worth $1.06; and the 1st having 378. What is an Annuity ? When is an annuity in arrears ? What is the Amoant of an annuity? 3T9. Object of Problem 1 7 Rule? 3S0. rrot>:eui2? . 299 due 2 years, is worth $1 $1 -f$1.06 + $l r=$3.133t*>, the >um sought. Hut these uumbON are in geomet- rical progression. Hence, Ki i | 1. Find the hist term of the series by Ar < compound interest, from 1 to 20 yean. V.arv 4 p«r Cent. 6 per Cent. 7 per 1 Ye»n. 1 1.000000 1 000000 1.000000 1.000000 1 2 2040000 2 050000 2.060000 2.070000 2 3 3.121600 3.152500 a. ii 3.214900 3 4 4.3101 2 j 4.374616 4.4 : 4 5 5.41' 15581 <»739 5 6 6.632975 6.801913 6975319 7 18 6 7 7.89- 8.142008 •5838 :o21 7 8 921 i 9.549109 •7468 8 9 10.562795 11.0* 11.491316 11. '.i7 9 10 .,107 12.577 18.151 15 61 10 11 13.486351 14.206787 14.971645 18 788589 11 12 15.085806 15.9171-7 16 869941 17.888451 12 13 16.626838 17 712983 18.882138 80.140548 13 14 18.291') 11 19.598632 81.01 0488 14 15 20.023588 21.578564 25.129022 15 16 21.824531 88.853 27.888054 16 17 23.697 :.12 25.840366 2880 10817 17 18 85.545419 28.13 <85 19039 18 L9 27.671229 »992 37 :i7S965 19 20 S079 33.065 4 88.788 40.995492 20 2. What is the amount of an annual salary of $1000, iu ■mm l\>r it 6 per cent.? m, $5637.093. 3. What is the amount of an annual rent of 100£, in arrears for 15 years, at 5 per cent. ? Ans. 2157.85G4£ = 2157£ 17s. Id. 2qr. 1. What i> the amount of an annual pension of $500, in arrears for 12 years, at 6 per cent.? lit EuU? Id Bul.r PERMUTATIONS 381. Pi-KMii ation it tin- arranging of a given number of things in every possible order of miOOfionloil 382. Problem. To find the number of permutations of a given number of thin The single letter, a, can have but 1 position, i.e. it cannot ftand either before or after itself; the 2 letters, a and b, furnish the *J permutation-, ■] ? >■ , the number of which is expressed by the product of 1 X 2 = 2; and if a 3d letter, c, be introduced, we have (c a by c b a) 1. Two parallel line- ere equally distant from each other. When two lines meet so as to form * angles, the lims an- perpendicular to Other and the angh a an right angles. A right angle contains 90°. An acute angle is an angle of less than 90°. An obtuse angle is an angle of more than 90°. Two lines are oblique to each other whoa the j meet so as to form acute or obtuse angles, and the a are oblique angles. 385. A Triangle is a plane figure which is bounded by three Hoes. Tlie base of a triangle (or any other figun ) i- the side on which it i< Mp pOOed to >tand. altitude of a triangle is tlie ncfpnndinnhir d'wtance from the ungle opposite the 1. to the led. 380. PbOjUM 1. To find the area of a tri lit ii;. Multiply the base by half the altitude. Bat 1. Tlie base ol a triangle is 7 inches and the altitude 8 inches ; what i< its ari ;. in. J. The hase is 8ft. and the hight lift.; what i- the area? 883. What is Mensuration? 384. What of two parallel line** What it a right angle? An acute angle? Obtuse angle? What are oblique line*? angle*? 3*3. What is a Triangle? IU baae? lu alUtude? 380. Bule for ^ it* area* 2G 302 MENSURATION. 387. A Quadrilateral or Quadrangle is a plan^ figure, having four sides and four angles. There are three kinds of quadrilaterals, viz. : 1st. Trapeziums, none of whose sidej Are parallel ; 2d. Trapezoids, as A B C D, only one pair of whose sides are parallel ; and, A E 3d. Parallelograms, each pair of whose opposite sides are parallel, as A B C D, or F E C D. A F BE The diagonal of a figure is a line which joins two opposite angles, as A C in the above trapezoid, and B D in the parallelo- gram. The altitude of a trapezoid or parallelogram is the per- pendicular between two parallel lidet. 388. Problem 2. To find the area of a trapezium : Rule. Draw a diagonal dividing the trapezium into tiro triangles, and find the area of each triangle by Problem 1. The sum of these triangles will be the area of the trapezium. Ex. What jl the area of a trapezium, one of who*e diagonals is 20 inches, and the length of the perpendiculars let fall upon it, from the other angles of the trapezium, G and 8 inches? Ans. 140sq. in. 389. Problem 3. To find the area of a trapezoid : Rule. Multiply the half sum of the parallel sides by the alti- tude, and the product will be the area. 387. What in a Quadrilateral? How many kinds? What is a trapezium? Trapezoid? Parallelogram? What ia the diagonal of a figure? Altitude of a trapezoid? Of a parallelogram? 388. Rule for finding the area of a trapezium? 3S9. Rule for finding the area of a trapezoid? MENSURATI' M Ex. 1. The parallel sides of a trapezoid are 10 and II and its altitude its area? j.ft. 2. What Kl Um ar« « of a board, whose length is 1 its area ? 391. A Polygon is a plain figure bounded by straight lines. ml. Three straight lines, at least, are required to bound a polygon. The lines which bound a polygon, taken together, are called the perimeter of the polygon. A polygon of 5 sides is called a pentagon ; of 6, a hexagon ; 7, a heptagon ; 8, an octagon ; 9, a nonagon ; 10, a decagon ; 11, an undecagon ; 1 2, a dodecagon ; etc. Note 2. A polygon may he divided into triangles by drawing diagonals, and then its area may be found by Problem 1. 399. Problem 5. To find the area of a circle when tho radius and circumference are given (Art. 109 and 361): Rile 1. Multiply the circumference by half the radius; or, Rule 2. Multiply the square of the radius by 3.141592, and the product is the area. 1. What is the area of a circle, whose radius is C and cir- cumference 37.699104? Ans. 113.097 2. What is the area of a circle whose radius is 10? 390. Rufe for finding the are* of a parallelogram? 391. What U a Polygon? miter of a polygon? Name the different polygons? 393. Rale for finding the area of a circle? Second Rale? 30-4 MENSURATION. 393. A Prism is a solid that has two similar, equal, parallel faces, called bases, and all its other faces parallelograms. Note. A prism is triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrangles, peutagons, etc. A Cylinder is a round body whose diameter is the same throughout its entire length, and whose ends or bases are equal, parallel circles. 394. Problem G. To find the surface of a prism or cylinder : RULE. Multiply the perimeter or circumference of the base by the length of the solid, and to the product add the area of the two ends. Ex. 1. What is the surface of a prism, whose length is 10 inchi's and l>a~<- \ inches squan ? Ans. L92#q.in. 2. What ifl the surface of a cylinder, whose length is 80 and diam at? 395. Problem 7. To find the solid contents of a prism or cylinder : K ii. Multiply the area of the base by the altitude. V.\. 1. What are the contents of a cylinder, whose length is 20imli< % and whose diameter is 10 inches? An<. l.-»7<>.7'JGc. in. 2. What are the contents of a quadrangular prism, whoso length is 2b feel and whose base is 3 feet square ? 396. A Pyramid is a solid, having a polygonal face, called the base, and all its other faces are trian- . hich meet at a common point, called the vertex of the pyramid. The slant hight is the distance from the vertex to the middle of one side of the base. 393. What is a Priam! A Cylinder? 394. Rule for finding the surface of a prism or cylinder? 395. Rule for finding the contents of a prism or cylin- der? 390. What is a Tyramid? Its vertex? Slant hight? now. 805 p.. A pyramid u triangular, quadrangular, vt _; a* iU base ii a triangle, quadrangle, etc. A I ■ solid, like a pyramid, except that ita base is a circle. The altitude of the pyramid or cone . crpemlicular hiyht. 397. Problem 8. To find the contents of a pyra- mid or of a cone : Kile. Multiply the area of the base by one third of the altitude. Ex. 1. What are the contents of a cone, whose base is 10 feet in diameter and whose altitude is 24 fe. Ans. G28.3184cu.n. 2. Whftl are the contents of a pyramid, whose altitude ii 12 Inches and whose base is a triangle, having its base G inches and its altitude 8 inches ? 398. The Frustum of a pyramid or cone is the part remaining after a portion next the vertex has been cut off by a plane parallel to the base. The two ends are called the upper and lower bases. 399. Problem 9. To find the contents of the frus- tum of a pyramid or cone : Krir. Multiply the sum of the ttco bases, added to the mean proportional between the two bases, by one third of the altitude of the frustum. Ex. 1. What are the contents of the frustum of a quadrangu- 390. What is a Cone? Altitude of a pyramid or cone? 397. Rule for find- ing the solid contents? 399. What la the Frustum of a pyramid or cone? 899. Content* of a frustum, how found? 30G ■■■■»■ I lf«> lar pyramid, whose altitude is 21 feet and whose bases are 5 feel ami .'} feet square? Ans. 3-i«'k*u, ft, 2. What are the contents of the frustum of a cone, whose hijHit is 12 feet and whose bases are feet and 4 feel in diam- eter ? 100. A Spiiere or Globe is a solid bounded by a curved surface, all peril of the surface bq. in. 2. What is the surface of the earth, supposing it to be a sphere 8000 miles in diameter? 3. What is the surface of the sun, supposing it a sphere diameter is 885680 mi 403. Problem 11. To find the contents of a sphere : Rule 1. Multiply the surface of the sphere by one third of the radius. Role 2. Multiply the cube of the diameter by the decimal ?99; i.e.byl of 3.141592. Ex. 1. What are the contents of a sphere, whose diameter is 100 inches? . 523598§c.iu. 2. What ia the volume or solidity of the earth, supposing it a sphere whose diameter is 8000 miles ? 3. What is the volume or solidity of the sun, supposing it a sphere whose diameter is 885680 mil. 400. What is a sphere? Its diameter? 401. Rule for finding the surface of a sphere? 403. Rule for finding the volume or solid contents of a sphere? Second rule? KXAAIl'LES. Ml MIm I LLANEOUS EXAMPLES. 1 What number In cre as ed t _'0? 2. What Dumber diminished by 1$ gives 21 t of two numb. ; d 0116 of the numb. I) times the other; what an- the numb. 1 Odi and tttfl fOdi are what part of an acre? 5. The difference between two Demberi i- 87| smaller nember if 12*; what is the lai a hat number multiplied by 86] givm 1000? 7. What Bember divided by • 8. What is the greatest common divisor of 84 and 1 '.'. What is the least common multiple of 72 and 364? 10. Wbal k the interest of 9766.64 fcr *m. 17 11. The difle reooe b et w e en two oamben k 2$, and me smaller number is 10; what is the larger? Whet the sum of th< numbers ? 1 2. The difference of two numbers is 563492, and the larger number is .'3042.338 ; what is the smaller? What the mm of the two numbers ? let Ana 8079046. 13. How many bricks 8 inches long, 1 inches wide, and 2 inches thick, will be required to build I wall 20 feet long, 16 feet hi«rh,and 2\ feet thick? 1 1. How many brick> whole dimensions are 8', 4', and 2', will it take* to build the walls of a DOOM 40ft, long, 28ft. wid. 22ft. high, the walls to be 1ft. 6' thick, and no allowance made for doors and windows ? 1."). The salary of the President of the T'nitcd State* is ") per annum; what sum may he expend daily, and yet 560 in one term of office, viz. 4 years? Ans. ^ 1 6. What number, multiplied by h of itself, will pro 17. What number, multiplied by •} of itself, will pfodnet 18. How mai will it take to lay a floor 20ft. long and 16ft. wide? How large a square floor can be laid with 676 6quar« i 308 MISCELLANEOUS EXAMPLES. 20. The fore wheel of a carriage ia 9 feet, and the hind wheel 10^ feet in circumference j how many times will each turn round in running from Boston to Andover, 20^ mile- ? 21. A rectangular piece of land, containing 00 acres, has its length to its breadth as 3 to 2, what are its length and breadth? 22. Bought a cask of molasses, containing 84 gallons, for J but 9 gallons having leaked out, at what price per gallon must I sell the remainder to gain $1.2.;? Ans. 48 cents. 23. If a pipe G inches in diameter will discharge a certain quantity of water in 1 boors, in what time will a 4-inch pipe discharge the lantity? . 1) liours. 2 1. In l2gaL .""jt. ipt 2gi., how many pills ? 2j. In 1846542 seconds how many week etc? 26. Resolve 207 1 ( » into its prime factors. Ana. 2, 2,3,3, 5, 11, 13. 27. Reduce ft, ^ 3 , ft, and *V to equivalent fractions having the least common denominator. 28. Reduce 3s. 4d. 2qr. to the fraction of a pound. 29. Reduce ft of a pound to shillings and pence. 30. Ajdd g 11). joz. Jdwt. 3g r . together. 31. From 9 lb take JJ. -. A colonel, arranging his men in a square battalion, found that he had ;;i nun remaining; but, increasing the rank and file by 1 soldier, he wanted 20 men to make up the square. Of how many men did his regiment consist? Ans. 656. 83. How shall I mark gloves that cost me 80c. per pair so that I may discount 33^ per cent, from the marked price and yet gain 2'> per cent, on the cost? Ans. $1.50. 31. Suppose that in a shower the water falls to the depth of 2 inches, how many gallons will fall upon a township that is 6 miles square, each gallon containing 231 cubic inches? 35. How many bricks 8' long, 4' wide, and 2' thick, will be required to build a house 32ft. long, 24§ft. wide, and 20ft. high, the walls being 1ft. 4' thick, the house having 2 doors, each 4ft. wide and 8ft. high, and 21 windows, each 3ft. wide and Cfu high, no allowance being made for the space occupied by the mortar? 36. What is the square root of the square root of 16 times 81 ? MISCELLANEOUS EXAMPLES. 809 37. If ■ horse travels C>\ vaSkt pet hour, how many hours will it take bim t<> travel m tar m i rail car will run in G boos car running 82] miles pet hour? 38. Light moves about 192000 miles per second and about 11 12 second; what is the ratio of tin* v.l-.city of light to that of nood? Ans. 887705fff. 30. What is the square root of 4 times the square of 8 ? What i- the cube of tin- square root of j 41. What is the cube root of the square of 8? 42. What ia the square of the culx' root of 8? 43. Two ships sail from the same port, one due north and the other due west, one at the rate of G miles and the other 8 per hour. Suppose the surface of the ocean to be plane, how far Apart are the ships in 10 hours ? 44. An army consists of 59049 men; how many shall be placed in rank and file to form them into a square? 1">. "What is the diameter of a circular pond which shall con- tain 36 times as much area as one 20 rods in diam 46. What is the mean proportional between 16 and 6 47. What is the third proportional to 3 and 30? 48. A ladder 41 feet long, will reach a window 40 feet Ugh on one side of a street, and, without moving the foot, it will reach a window 9 feet high on the other side; how wide is the street? Ans. 49ft. 49. What is the difference in the expense of fencing a circular 40-acre lot and one of the same area in a square form, the fence costing 50c. per rod ? 50. Sold to J. P. F. goods as follows : Jan. 18, 1862, on 6m., 75yd. of cloth, at $4, $800, Mar. 12, " " 3m., 600gal. of molasses, " 33&c, 200. June 15, " "4m., 50bbl. of flour, " $8. 400. Also bought of him : Feb. 18, 1862, on 4m., 30c of wood, at $ 6, $180. Mav 24, * 6m., I0L of hav, 120. July 6, " " 5m., 10 o u 30, 300. ■ 24, " " 4m., 1 ho ■ 100. Wh^n shall he pay me the balance of the debt ? 310 MISCELLANI. MI'LES. 51. What is the side of a square equivalent in area to a rec- tangular field, which is 81 rods long and 49 rods wide! 52. Sent an invoice of goods to my agent in Liverpool which he sold for $2.3000 ; what sura can he invest for me, his commis- sion for selling being 2 per cent, and for investing 1 per cent. ? 53. A house worth $8000 is insured for I its value ; what b the premium at g of 1 per cent. ? 54. Wli.it is the amount of $325, at 6 per cent, compound st, for3yr. 8m. 12d.? 55. $1200. Boston, Mmj 12, lft 60. :• value received of A. B. I promix- t<» pay him, or his order, one thousand two hundred dollars, on demand, with inter Chaki.i.s D\\r. Ini.mi:- .;0, $300 : [fee. 1*, 1860, $10; May 6, 1861, $16.50; Jure Jl. 1861, S100; Dee. 21, 1861, $100 ; what was due Apr. 12, 1# 56. A bushel mra-ure is 18^ inch«s in diameter and ft inches d« •.•]> ; what are the dimensions of a nmilar measure that holds halt" a peek ? An-. 9}in. diameter; 4in. deep. 57. Sold a lot of goods for $100 and thereby gained 25 per cent. ; what per emt. >hould I have gained, had I sold them for $12< 58. A garden whose breadth is 5 rod*, and whose length is 1$ times its I»n a-lth, has a wall Z\ feet thick and 4 feet high, around it, outride of the line; what was the cost of this wall at 3£c. per cubic foot ? 59. What will be the cost of digging a ditch around the above- mentioned garden, within and adjacent feo the wall 3£ feet wide and 2^ feet deep, at $ of a cent per cubic foot? 60. What would be the cost of walling the above-mentioned garden, the central line of the wall to be on the bounding line, Ike wall to be 3^ feet thick and 3J feet high and to cost G£c. per cubic foot ? 1 1 . A hare has 45 rods the start of a hound, but the hound runs 12 rods while the hare runs 9 ; how many rods will the hare run before the hound overtakes him ? 62. A hare has 32 rods the start of a hound, but the hare runs only : iiile the hound runs 20 ; how far will the hound run before be overtake! the 1 the inter. JO from Aug. 8, 18G1, to July A. K and (' engage to do a piece of work] A DM do it •i 2 1, and C in 30. In what time ean the time together do the work ? \ .nth-man left his Bon an i f which be .-|»ent in UP and {\ 2 of the remainder in •) months nioiv, when bl only $! Ion remaining ; what was tin- value of lL 66. The I'lHiiiiKiiiilcr of a he>ieged fbftCMI bai 2lh. of* hnad pOf day tor each -oldier for 46 day-, hut wi-hrs to prolofl 60 days ; what DDOSt he the allowance per day? G7. A man Bold a wateh for >»)<», which w;u> j of its cost; what was lost hy the transaction? It' a btt of silver 1ft. f>in. long, tin. wideband 2in. thick, is worth 11240, what is the value of a har of gold 1ft. 3in. long 8in. wide, and lin. thick, the weight of a euhic im h of .-ilvrr ; to the weight of a euhic inch of gold as 10 to 10, and tho value per ounce of silver being to that of gold ■ 2 to 3.'3 ? 69. dan. 1, 1861, A, I>, and C form a partnership for 1 and each furnishes $20<» 1, A furnishes $1000 more; June 1, B furnishes SloOO and C withdraws $500; Oct. 1. A withdraws $500, and IS and Q furnish $1000 each. II gained $3000, at the close of the year the partnership i- solved. What is *-aeh partner's share of the gain ? 7<>. How many gallons of wine at 6, 10, l.">, and 20s. per gal. may he taken to form a mixture of ( J5 gallons worth 12 gallon ? 71. Find the difference in time due to a difference of 17° 20' 40" in longitude. 72. The difference in the time of two places is 3h. 18m. 15eec; what i- the difference in longitude? A merchant hought a numher of ha the rate of $7 and sold them out at the for 7\.l , and gained $200 by the bargains; An-. 0. miscellan: 71. The trans-Atlantic telegraph laid in 1S">7 from St. John's, foondland, to Valentia, Ireland, 1640 miles En a straight line, consisted of 7 copper wires, twisted together, imbedded in prima percha, and surrounded by 18 bundles of iron wire Bach bun- of iron wire consisted of 7 wires which were twisted together, and the bandies ran spirally round the cable. Now, to allow lor deviations ftw ;hfl coarse, ineqnaUttea of die sea-bottom, 1 1 1 tunes as l«>iig as would be required for a straight coarse, ami that it was necessary to i ncr e as e the wiVfl 1 Bails in every 20 in con-eqnence of twisting the wires, and 1 milr in eve: •_. uise of the bundles running spirally, what length of wire was required faff the cable? 0905) ni: 7'). By the census of 1860, the number of inhabitants of Ala- bam.. J96; ofArka: Vtt | of California, 880016] of ( 1; of Delaware, 112218; of Florida, 140439] of G I of Illinois 1711753] Of Indi- ana. 1350941 . 1 <>71 10; of Ken- tucky, 11557131 of Louisiana, 700290; of Maine, 528276; of Man lane ! ; of Ma of Mich 749111; of Minnesota, 172022; of Mississippi, ; of JOOri, 1182317; of New Hampshire, .120)72; of New Jeff- 672031 ; of New York, 3880735 ; of North Carolina, 992667 ; of Ohio, 2339599 ; of Or- of Pennsylva- nia, 2906370; of Rhode Island, 174621; of South Carolina, 703812; of Tennessee, 1109847; of T -132; of \ monr. B) of Virginia, 1596079; of Wisconsin, 775873; of the District of Columbia, 75076; and of the Territories, 220143 ; what was the population of the United States in 1860 ? Ans. 31443790. STANDARD ARITHMETICS. EATON'S COMPLETE SERIES, VVTEI) TO TUT BEST u ;»'H: OF INSTKHTIOV I. TIT IV. THE PRIMATT TIC THE IJS THE NEW TREATISE- ON WRITTEN ARITHMETIC ;i£,^H the ■ THOMPSON, i>ul Ushers, 29 CoraMll, B6ston.