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 IN MEMORIAM 
 FLORIAN CAJORl 
 
 ^K>^ 
 
 V 
 
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PREFACE. 
 
 
 n->n'h'rn^lci 
 
 However forcibly an author may be impressed with 
 the conviction that his work constitutes an important 
 improvement on all similar efforts which have preceded 
 it, he is still aware that its favorable reception by the 
 public depends upon the recognition of its merits by 
 other minds than his own. 
 
 In regard to this book, if the improvements which 
 I have attempted to incorporate into it are not readily 
 recognized by the experienced teacher as he peruses it, 
 they are certainly of so little value as not to be worth 
 pointing out. I have, then, only to suggest to the 
 reader to turn to the subjects, say of Ratio and Pro- 
 portion, and critically peruse the articles as they occur, 
 including the examples in the application of principles 
 and the references. If I mistake not, a few pages will 
 reveal to him many of the iinportant features which 
 distinguish this work; and, if an experience of nea.rly 
 thirty years in the school-room justifies me in express- 
 ing the opinion, he will be surprised that they have not 
 been hitherto developed. 
 
q-v 
 
 PREFACE. 
 
 There is only one point to which I will expressly 
 refer. It will be seen that decimal fractions are the 
 orflfepring of decimal notation, and not of ^^vidgar" 
 fractions, and that their notation is early introduced for 
 the sake of scientific accuracy, as well as the early in- 
 sertion of problems involving United States currency; 
 for if there is any concrete quantity which the Amer- 
 ican child readily understands, it is that involving 
 
 dollars and cents. 
 
 P. A. TOWNE. 
 Mobile, Ala., January^ 1866. 
 
r^h), 
 
 
 CONTENTS 
 
 a 
 
 PAGE 
 
 Definitions 9 
 
 Notation 11 
 
 Arabic Notation 11 
 
 Numeration 17 
 
 Decimals 20 
 
 Notation and Numeration 20 
 
 Principles of Arabic Notation 25 
 
 United States Money — Notation and Numeration 26 
 
 Roman Notation and Numeration 30 
 
 Addition 31 
 
 Subtraction 45 
 
 Multiplication 55 
 
 Division 69 
 
 Short Division 71 
 
 Long Division 78 
 
 Properties of Integral Numbers 87 
 
 Definitions 87 
 
 Factoring 91 
 
 Greatest Common Divisor 93 
 
 Least Common Multiple 99 
 
 Fractions 108 
 
 Nature of Fractions 108 
 
 Notation of Fractions 109 
 
 Classification of Fractions 112 
 
 Value of a Fraction 113 
 
 Propositions in Fractions 113 
 
 Pteduction of Fractions 113 
 
 (•5) 
 
% 
 
 ♦ 
 
 ^■yt t-ti'^ 
 
 6 ' " *■-** ■ COSTHNIS."" 
 
 D 
 
 Fractions — [Continued.) pagk 
 
 Addition of Fractions 119 
 
 Subtraction of Fractions 121 
 
 Multiplication of Fractions 124 
 
 Division of Fractions 130 
 
 Reduction of Common Fractions to Decimal Fractions 136 
 
 Compound Numbers 151 
 
 Definitions 151 
 
 English Money 152 
 
 French Money ; 152 
 
 Troy Weight 153 
 
 Avoirdupois Weight 154 
 
 Apothecaries Weight 155 
 
 Long Measure 155 
 
 Cloth Measure 156 
 
 Superficial or Square Measure 158 
 
 Solid Measure 159 
 
 Wine Measure 160 
 
 Ale or Beer Measure 161 
 
 Dry Measure 161 
 
 Time 162 
 
 Circular Measure 164 
 
 Reduction of Compound Numbers 166 
 
 Compound to Concrete 166 
 
 Concrete to Compound 166 
 
 Denominate Fractions to Compound Numbers 169 
 
 Compound Numbers to Denominate Fractions 169 
 
 Compound Numbers to Decimal Fractions 173 
 
 Denominate Decimal Fractions to Compound Numbers 173 
 
 Addition of Compound Numbers 175 
 
 Subtraction of Compound Numbers 177 
 
 Time between Dates 179 
 
 Multiplication of Compound Numbers 181 
 
 Division of Compound Numbers 184 
 
 Longitude in Time 186 
 
 Analysis by Aliquot parts 187 
 
 Review in Addition 195 
 
 Review in Subtraction 198 
 
 Review in IMultiplicatiou and I'ivision 200 
 
CONTENTS. -^ 7 
 
 PAGE 
 
 Percentage 202 
 
 Applications of Percentage 210 
 
 Insurance 210 
 
 Commission 212 
 
 Stock 213 
 
 Brokerage 215 
 
 Profit and Loss 215 
 
 Duties or Customs 220 
 
 Interest 222 
 
 Problems in Interest 232 
 
 Present Worth *.... 235 
 
 Bank Discount 236 
 
 Promissory Notes 237 
 
 Compound Interest 241 
 
 Ratio 244 
 
 Proportion 246 
 
 Rule of Three 250 
 
 Partnership 261 
 
 Equation of Payments 265 
 
 Alligation Medial 266 
 
 Alligation Alternate 268 
 
 Position 273 
 
 Single Position 273 
 
 Double Position 276 
 
 Involution 281 
 
 Evolution 285 
 
 Square Root 286 
 
 Cube Root 295 
 
 Problems 303 
 
 Arithmetical Progression 305 
 
 Geometrical Progression 311 
 
 Permutations, Arrangements, and Combinations 317 
 
 Practical Geometry 319 
 
 Definitions , 319 
 
 Pythagorean Proposition 323 
 
 Proposition on the Triangle 326 
 
 Mensuration 327 
 
 Area of Triangle, I...^ 327 
 
8 CONTENTS. 
 
 a 
 
 Practical Geometry — [Continued.) page 
 
 Area of Triangle, II 328 
 
 Area of Quadrilateral with Parallel Sides 328 
 
 Area of Trapezium 329 
 
 Proposition on Similar Figures 330 
 
 The Grindstone Problem 331 
 
 Mensuration of Solids 332 
 
 Definition 332 
 
 Contents of a Cylinder or Prism 333 
 
 Contents of a Cone or Pyramid 334 
 
 Contents of a Frustrum of a Cone or Pyramid 335 
 
 ConCents of a Cistern 335 
 
 Proposition on Spheres 335 
 
 Surface and Contents of a Sphere 336 
 
 Miscellaneous Examples 336 
 
 Appendix 351 
 
 Table of Multiplication 351 
 
 Table of Square Roots 362 
 
 Table of Cube Roots 353 
 
 Strength of Building Materials 354 
 
 Annuities •. 355 
 
 French Weight 357 
 
 French Linear Measure 357 
 
 French Superficial Measure 358 
 
 French Solid Measure 358 
 
 French Measure of Capacity 358 
 
 Table of Foreign Money (fixed by law) 358 
 
 Books and Paper 360 
 
ELEMENTARY ARITHMETIC. 
 
 DEFINITIONS 
 
 1. Science is knowledge reduced to order. 
 
 2. Art is the practical application of the principles 
 of a science. 
 
 3. Quantity is a term that is applied to any thing 
 that can be measured. 
 
 4. A Unit is a quantity to which the term one may 
 be applied. Thus: one horse, one ten, one half. — 
 (Vide 128.) 
 
 5. A Number is a unit, or a collection of units. 
 
 6. Figures are characters used to represent any given 
 number. They include the cipher, naught, or zero, and 
 nine digits. Thus : 
 
 naught 
 
 one 
 
 two 
 
 three 
 
 four 
 
 five 
 
 six 
 
 seven 
 
 eight 
 
 nine 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 The ni7ie digits are called significant figures, to distin- 
 guish them from the cipher, which has, when written 
 alone, no value whatever. Its effect when joined to 
 other figures is explained under Notation. 
 
 7. Integral numbers or integers are tvhole numbers. 
 Thus: seven, twenty, one hundred and six, etc., are 
 
 integral numbers. 
 
 (i>) 
 
10 DEFINITIONS. 
 
 8. A Fractional number or fraction represents one, 
 or more than one, of the equal parts of a unit. Thus : 
 one seventh, five twentieths, two thirds, four ninths, etc., 
 <ire fractional numbers or fractions. — (Vide 127.) 
 
 O. A Decimal fraction represents 07ie, or more than 
 one, of the parts of a unit which is divided into ten, one 
 hundred, one thousand, ten thousand, etc., equal parts. 
 Thus : one tenth, four tenths, thirty-seven hundredths, 
 forty-five thousandths, etc., are decimal fractions. 
 
 10. Mathematics is the science of quaiitity. 
 
 11. Arithmetic is that branch of mathematics which 
 treats of numbers. It is a science and an art. 
 
 13. A Problem is a question proposed which requires 
 a solution. 
 
 13. An Operation is the method of solving a problem. 
 
 14. An Analysis is an investigation of the several 
 parts of a problem. An a^ialysis leads to the operation 
 which obtains the answer to a problem. 
 
 15. A Rule is a direction for performing an opera- 
 tion. A rule is usually derived from an analysis of one 
 or more problems. 
 
NOTATION. 11 
 
 NOTATION. 
 
 IG. Notation is the method of representing num- 
 bers by figures or letters. Arabic notation represents 
 numbers by figures. Roman notation represents num- 
 bers by letters. 
 
 ARABIC NOTATION. 
 
 PROBIiEM I. 
 
 17. To represent an integral number between naught 
 and ten, 
 
 Write the necessary figure as in Definition 6. 
 
 EXAMPLE. 
 
 1. Write in figures, three, seven, one, five, four, eight, 
 six, nine, two, and naught. 
 
 2. How many units does the figure 5 represent ? 
 
 3. How many marbles does the figure 9 represent? 
 
 4. How many apples have you when they can be 
 represented by 0? 3? 4? 6? 7? 8? 2? 
 
 PROBIiEM II. 
 
 18. To represent any integral number between te7i 
 and nineteen, inclusive. 
 
 Place the several figures to the right of the figure 1. 
 
 EXAMPLE. 
 
 1. Write, by means of figures, 
 
 ten eleven twelve thirt'n fonrt'n fift'n sixt'n sevont'ii eiglil'n nitict'n 
 
 Ans. 10 11 12 13 14 15 16 17 18 19 
 
12 NOTATION. 
 
 The figures at the right hand, or first place, in these 
 numbers represent units of the first order. 
 
 The figure in the second place represents a unit of 
 THE SECOND ORDER. — (Vide Def. 4.) 
 
 PROBI.EM III. 
 
 lO. To represent the units of the second order, 
 between ten and ninety inclusive, 
 
 Place the cipher at the right of the several digits. 
 
 example. 
 1. Write, by the aid of figures, 
 
 ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. 
 
 Ans. 10 20 30 40 50 60 70 80 90 
 
 PROBI.EM IV. 
 
 SO. To represent any integral number between 
 twenty and one hundred, 
 
 Unite the required units of the first order to those of 
 the second. 
 
 examples. 
 
 1. Write, by the aid of figures, the number twenty- 
 five. Here five units of the first order are required, and 
 two of the second. Ans. 25. 
 
 2. Write on your slate all the integral numbers be- 
 tween naught and one hundred, by the aid of figures. 
 
 PROBLEM V. 
 
 21. To represent the units of the third order, 
 called hundreds. 
 
 Place hvo ciphers to the right of the several digits, 
 thus : 
 
NOTATION. 13 
 
 One hundred... 100 Four hundred, ..400 Seven hundred. ..700 
 Two hundred. ..200 Five hundred. ..500 Eight hundred. ..800 
 Three hundred. 300 Six hundred 600 Nine hundred. ...900 
 
 PROBLEM TI. 
 
 22. To represent any integral number between one 
 hundred and one thousand, 
 
 Write down the figure representing the required units 
 of the third order, and annex to it the figures repre- 
 senting the required units of the second and first orders. 
 
 EXAMPLES. 
 
 1. Write, by the aid of figures, one hundred and 
 twenty-three. Here one unit of the third order, two of 
 the second, and three of the first, are required. 
 
 Ans. 123, 
 
 2. Write, by the aid of figures, seven hundred and 
 eight. Here seven units of the third order, none of the 
 second, and eight of the first, are required. 
 
 Ans, 708. 
 
 3. Write on your slate five hundred and forty-nine, 
 three hundred and forty, six hundred and one, three 
 hundred and seven, nine hundred and nine, seven 
 hundred and ten. 
 
 4. Write all the integral numbers between three 
 hundred sixty-five and four hundred. 
 
 5. Write all the integral numbers between seven 
 hundred three and seven hundred fifty. 
 
 PROBLEM Til. 
 
 23. To represent units of the fourth order, 
 
 called THOUSANDS, 
 
J 4 NOTATION. 
 
 Place three ciphers to the right of the several digits^ 
 thus: 
 
 One thousand 1000 Four thousand. .4000 Seven thousand. .7000 
 
 Two thousand '2000 Five thousand..5000 Eight thousand.. 8000 
 
 Three thousand. ..3000 Six thousand. ..6000 Nine thousand. ..9000 
 
 Units of the fifth order are called tens of thou- 
 sands. 
 
 Units of the sixth order are called hundreds of 
 'Thousands. 
 
 probxem viii. 
 
 24. To represent any number of thousands by the 
 aid of figures, 
 
 Write the figures as if they zvere to represent a number 
 less than one thousand, and then annex three ciphers. 
 
 EXAMPLES . 
 
 1. Write, by the aid of figures, one hundred and 
 twenty-three thousand. Ans. 123000. 
 
 2. Write seven hundred and eight thousand. 
 
 A71S. 708000. 
 
 3. Write five hundred and forty-nine thousand, six 
 hundred and one thousand, nine hundred and ten thou- 
 sand, five hundred and fifty-five thousand, twenty-one 
 thousand. 
 
 PROBIiEM IX. 
 
 25. To represent any integral number less than one 
 million. 
 
 Write the number as if it were required to write the 
 thousands only, but instead of the three ciphers place that 
 part of the number less than a thousand. 
 
NOTATION. 15 
 
 EXAMPLES. 
 
 1. Write one hundred and twenty-three thousand one 
 hundred and ninety-one. Ans. 123191. 
 
 2. Write seven hundred and eight thousand two hun- 
 dred and five. Ans. 708205. 
 
 3. Write nine hundred and nine thousand nine hun- 
 dred and nine. Ans. 909909. 
 
 4. Write four hundred and forty-four thousand four 
 hundred and forty-four. Ans. 444444. 
 
 5. Write twenty-seven thousand and one. 
 
 Ans. 27001. 
 
 Remark. — Two ciphers are retained because no mention is 
 made of units of the second or third order. 
 
 6. Represent in figures twenty-seven thousand three 
 hundred and twenty-one, twenty-seven, three hundred 
 and one, twenty-seven thousand and twenty-one, tw^enty- 
 seven thousand and five, sixty-seven. 
 
 Units of the seventh order are called millions. 
 
 Units of the eighth order are called tens of mill- 
 ions. 
 
 Units op the ninth order are called hundreds of 
 millions. 
 
 PROBI.£i!»I X. 
 
 26. To represent in figures any number of millions, 
 Write the number as if no mention were made of mill- 
 ions and then annex six ciphers. 
 
 examples. 
 
 1. Write one hundred and twenty-three millions. 
 
 Ans. 123 000 000. 
 
 2. Write seven hundred and eight millions. 
 
 Ans. 708 000 000. 
 
10 NOTATION. 
 
 27. In the previous section, the three figures on the 
 right of the answer to each example form the period op 
 
 UNITS. 
 
 The three next figures form the period of thousands. 
 The three next figures form the period of millions. 
 A few succeeding periods are named as follows : 
 
 4. Billions. 7. Quintillions. 10. Octillions. 
 
 5. Trillions. 8. Sextillions. 11. Nonillions. 
 
 6. Quadrillions. 9. Septillions. 12. Decillions. 
 
 Remark. — These names might be continued to any extent. 
 PROBIiEM XI. 
 
 28. To represent in figures any number whatever, 
 Write each period in the order named, as if it were the 
 
 period of units; hut if intermediate periods are not 
 named, fill their places luifh ciphers. 
 
 exajiples. 
 
 1. Write four hundred and twenty-three trillions two 
 hundred and seven billions five hundred and six thou- 
 sand and one. Ans. 423,207,000,506,001. 
 
 Remark. — For convenience the periods may be separated by a 
 comma. 
 
 2. Write in figures twenty-seven; forty-three; one 
 hundred and fifty-four ; five thousand and ten ; twenty- 
 six thousand and forty -five; three hundred thousand 
 and seven ; two millions and five ; forty-seven millions 
 and thirty-three ; three decillions one nonillion twenty- 
 seven octillions three quintillions one hundred and 
 twenty billions and thirty-four. 
 
 Last Ans. 3,001,027,000,000,003,000,000,120,000,000,034. 
 
NUiMEllATION. 17 
 
 NUMERATION 
 
 29. Numeration exhibits the method of reading num- 
 bers represented by figures. 
 
 EXA]\IPLES. 
 
 1. Read, in common language, the number 631230- 
 405078901. 
 
 We begin by separating the number into periods of 
 three figures each, commencing at the right. Thus : 
 631, 230, 405, 078, 901. 
 
 Next, apply, either mentally or in writing, the riames 
 of each of the jjeriods as given in Notation, beginning at 
 the right. Thus : 
 
 Trill. Bill. Mill. Tlions. Units. 
 
 631, 230, 405, 078, 901. 
 
 Next, hegin at the left^ and give to the third figure 
 the name of the unit of the third order, (vide 21,) and 
 to the second figure the name of the unit of the second 
 order, (vide 19,) and to the first figure the name of the 
 digit which it represents, (vide 6.) Finally, apply the 
 name of the period itself. 
 
 Do the same thing to the successive periods toward 
 the right, omitting the names of places filled by ciphers. 
 Thus: 
 
 Six hundred thirty-one trillions two hundred thirty 
 billions four hundred and five millions seventy-eight 
 thousand nine hundred and one. 
 
18 NUMERATIOX. 
 
 Remark 1. — The word and is used when required by custom. 
 
 Rkmark 2. — It is not customary to apply the name of the period 
 on the right. Nine hundred and one is the same as nine hundred 
 and one units. 
 
 2. Read the number 63021457823675301742601930. 
 
 OPERATION. 
 Scptil. Sextil. Quin. Quad. Trill. Bill. Mill. Thoiis. 
 
 63, 021, 457, 823, 675, 301, 742 601, 930. 
 Sixty-three septillions twenty-one sextillions, etc. 
 
 Remark. — It is observed that the period on the left need not 
 be full, for a cipher in the vacant place would be of no service. — 
 (Vide 6.) 
 
 3. Read the number 319415012. 
 Arts. Three hundred nineteen million four hundred 
 fifteen thousand and twelve. — (Vide 18.) 
 
 30. From these examples we have the following 
 
 RULE FOR READING NUMBERS. 
 
 1. Separate the number into periods of three figures 
 each, beginning at the right. 
 
 2. Apply to the third, second, and first figures of the 
 period on the left, if it is fidl, the names of the units of 
 the third, second, and first orders, and afterivard apply 
 the name of the period, omitting names* in all cases where 
 ciphers occur. 
 
 3. Read each period in the same way, passing to the 
 right. 
 
 4. If the middle figure of a p)eriod is 1, apply the name 
 which it, in connection ivith the figure on its right, rep- 
 resents. 
 
NUMEllATION. 
 
 19 
 
 EXAMPLES. 
 
 Read the following numbers 
 
 1. 
 
 2. 
 3* 
 4. 
 5. 
 6. 
 
 12. 
 
 184. 
 
 3261. 
 
 81765. 
 
 987123. 
 
 7. 6345555. 14. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 
 3. 
 
 10. 
 
 530. 
 
 4021. 
 
 56304. 
 
 740357. 
 
 15. 
 16. 
 17. 
 
 18. 
 19. 
 20. 
 
 1312547. 
 
 24340586. 
 
 382003125. 
 
 4960606544. 
 
 51730004163. 
 
 664800000532. 
 
 6401836. 21. 7562345940051. 
 
 22. 30014820016. 
 
 23. 372536370001. 
 
 24. 8134000500010. 
 
 25. 370010050020. 
 
 26. 1010101010101. 
 
 27. 4040506070809. 
 
 28. 3004005006001. 
 
 29. 7000600050004. 
 
 30. 9000040000700. 
 
 31. 4000000130000. 
 
 32. Read the number 3014056000451. 
 
 33. Read the number 40700369997823. 
 
 34. Read the number 100370059431001. 
 
 35. Read the number 6001478900462357. 
 
 36. Read the number 81705430267891456. 
 
 37. Read the number 123456789098765432. 
 
 38. Read the number 9876543210123456789. 
 
 39. Read the number 10203040506070809011. 
 
 40. Read the number 908070605040302010999. 
 
 41. Read the number 1002003004005006007008. 
 
 42. Read the number 199:^991^91^Q1^g]^9^1. 
 
 f 
 
20 DECIMALS. 
 
 DECIMALS 
 
 NOTATION AND NUMERATION. 
 SI. The several units in decimals are named TENTH, 
 
 HUNDREDTH, THOUSANDTH, TEN-THOUSANDTH, HUNDRED- 
 THOUSANDTH, MILLIONTH, etc. 
 
 32. The relation between integral and decimal units, 
 and the manner of representing them by figures, are 
 given below. 
 
 One million 1000000. 
 
 One hundred thousand 100000. 
 
 One ten thousand 10000. 
 
 One thousand 1000. 
 
 One hundred 100. 
 
 One ten 10. 
 
 One t One. 
 
 .1 One tenth. 
 
 .01 One hujidredth. 
 
 .001 One thousandth. 
 
 .0001 One ten-ihotisandth. 
 
 .00001 One hundred-thoiisandlh. 
 
 .000001 One millionth. 
 
 (1.) The Decimal Point distinguishes the decimal from 
 the in!9jfa0m^Sy^tsJpJa/e0>emg at the left of all 
 decimals. The period (.) is commonly employed for this 
 purpose. 
 
 (2.) By one tenth is meant one of the ten equal parts 
 into which the nnit one is divided. 
 
DECIMALS. 21 
 
 (3.) Bj cue hundredth is meant one of the hundred 
 equal parts into which the unit one is divided. 
 (4). By one thousandth is meant, etc. 
 
 l>ROBL,£M I. 
 
 33. To represent by the aid of figures any number 
 of tenths, 
 
 Place the decimal point to the left of the proper digit. 
 
 EXAMPLES. 
 
 1. Represent five tenths with a figure. Ans. .5 
 
 2. Represent two tenths, three tenths, four tenths, 
 six tenths, seven tenths, eight tenths, nine tenths, each 
 by the proper digit. 
 
 Hemakk. — By two tenths we mean two of the ten equal parts into 
 which the unit one is divided. 
 
 PROBIi£M II. 
 
 34. To represent by the aid of figures any number of 
 hundredths. 
 
 (1.) If the number of hundredths is less than ten, 
 
 Place a cipher between the decimal point and the proper 
 digit. 
 
 (2.) If the number of hundredths is ten or more than 
 ten, 
 
 Place the decimal point to the left of the given number. 
 
 examples'. 
 
 1. Represent two hundredths by figures. 
 
 Ans. .02 
 
 2. Represent three hundredths, four hundredths, etc., 
 to nine hundredths by figures. Ans. .03 etc. 
 
22 DECIMALS. 
 
 3. Represent ten hundredths by figures. 
 
 A71S, .10 
 
 4. Represent eleven hundredths, twelve hundredths, 
 etc., to ninety-nine hundredths. Ans. .11 etc. 
 
 Remark. — By tivo hundredths we mean two of the hundred equal 
 parts into which the unit one is divided. 
 
 PROBIiEM III. 
 
 35. To represent by the aid of figures any number of 
 thousandths. 
 
 (1.) If the number of thousandths is less than ten, 
 
 Place two ciphers between the decimal point and the 
 proper digit. 
 
 (2.) If the number of thousandths is ten or more, and 
 less than a hundred. 
 
 Place a cipher between the decimal point and the given 
 number. 
 
 (3.) If the number of thousandths is one hundred, or 
 more than one hundred, 
 
 Place the decimal point to the left of the given number. 
 
 EXAMPLES. 
 
 1. Represent three thousandths by figures. 
 
 Ans. .003 
 
 2. Represent one* thousandth, two thousandths, etc., 
 to nine thousandths. Ans. to last, .009 
 
 3. Represent ten thousandths by figures. 
 
 Ans. .010 
 
 4. Represent eleven thousandths, twelve thousandths, 
 etc., to ninety-nine thousandths. Ans. to last, .099 
 
DECIMALS. 23 
 
 5. Represent one hundred thousandths. 
 
 Ans, .100 
 
 6. Represent six hundred and twenty-one thou- 
 sandths. Ans. .621 
 
 7. Represent three hundred and six thousandths. 
 
 Ans. .306 
 
 8. Represent one hundred and one thousandths, one 
 hundred and two thousandths, etc. 
 
 Ans. .101 .102 etc. 
 
 Remark. — By /our thousandlhs we mean /our of the thousand equal 
 parts into which the icnit one is divided. 
 
 PROBLEM IV. 
 
 36. To represent by figures any decimal whatever, 
 
 (1.) Write the given number, as in sections 17 to 28. 
 
 (2.) WJien necessary, prefix ciphers enough to make the 
 right-hand figure of the number occupy the place of 1 
 when representing the given decimal unit. — (Vide 32.) 
 
 (3.) To the left place the decimal point. 
 
 Remark. — Observe that one figure only on the right of the point 
 is required to represent tenths^ two figures to represent hundredths, 
 three figures to represent thousandths, etc. (Vide 32.) 
 
 EXAMPLES. 
 
 1. Represent by figures one ten-thousandth. 
 
 Ans. .0001 
 
 2. Represent by figures two ten-thousandtJis. 
 
 Ans. .0002 
 
 3. Represent twenty-one ten-thousandths. 
 
 Ans. .0021 
 
 4. Represent three hundred and six ten-thousandths. 
 
 Ans. .0306 
 
24 DECIMALS. 
 
 5. Represent three thousand and five ten-thousandths. 
 
 Ans. .3005 
 
 6. Represent one hundred-thousandth. 
 
 Ans. .00001 
 
 7. Represent six hundred and one hundred-thou- 
 sandths. Ans. .00601 
 
 8. Represent one millionth. Ans. .000001 
 
 9. Represent one thousand and five millionths. 
 
 Ans. .001005 
 
 10. Represent bj figures, four tenths^ twelve hun- 
 dredths, seven hundredths, five thousandths, thirty-seven 
 thousandths, one hundred and eleven thousandths, forty- 
 six ten-thousandths, nine hundred and one ten-thou- 
 sandths, three hundred and sixty-one hundred-thou- 
 sandths, ten thousand four hundred and fifty-six 
 millionths, one ten-millio7ith, twenty-seven ten-millionths, 
 one hundred-millionth, sixty-five hundred-millionths, one 
 billionth, three thousand and fifty-seven hillionths. 
 
 Ans. to last, .000003057 
 
 PR OBI. EM V. 
 
 •17. To read any decimal represented by figures, 
 (1.) Read the figures as if represe7iting an integral 
 
 number. — (Vide 29.) 
 
 (2.) Apply the name of the decimal unit indicated by 
 
 the right-hand figure. — (Vide 36, Remark.) 
 
 EXAMPLES. 
 
 1. Read .5; .05; .005; .0005; .00005; .000005 in 
 words. 
 
 2. Read .3; .13; .213; .1111; .22222; .999999 in 
 words. 
 
DECIMALS. 25 
 
 3. Read .9; .24; .031; .0461; .00231; .009999 in 
 
 words. 
 
 Remark. — An integral number and a decimal may be written 
 together. Thus : 3.7 are three and seven tenths. 
 
 4. Read 2.5; 4.05; 8.005; 21.0005 in words. 
 
 5. Read 8.1; 2.12; 9.224; 27.1234 in words. 
 
 6. Read the following expressions : 
 
 3.004 23.005 67.431 5.6789 
 
 17.115 48.673 12.6001 27.3004 
 
 126.432 12.6432 1.26432 .126432 
 
 .1345 1.345 13.45 134.5 
 
 38. Principles of Arabic Notation. 
 I. All numbers are derived from the unit one. 
 II. Removing any figure one place toward the left 
 increases its value .ten times. Thus, in the expressions 
 
 .001 .01 .1 1. 10. 100. 
 
 the value of the figure 1 is increased ten times in each 
 step of its passage from right to left past the decimal 
 point. 
 
 Remark 1. — Any digit, then, may have a simple or it may have 
 a local value; it has a simple value when written alone, and a local 
 value in all other cases. 
 
 m, ^ r The simple value of three is 3. 
 
 ' \ Some local values of three are 30. ; .3 ; .03 
 
 Remark 2. — The principal use of the cipher is to give a local 
 value to the digits. — (Vide 6.) 
 
 Remark 3. — It is evident that one tenth of any quantity, as for 
 instance a dollar, is the same as ten hundredths of the same quan- 
 tity; that is, .1 is the same as .10; hence. 
 
 Placing a cipher to the right of a decimal, does not 
 change the value of the decimal. 
 3 
 
26 DECIMALS. 
 
 UNITED STATES MONEY. 
 
 NOTATION AND NUMERATION. 
 
 39. The several units of the currency of the United 
 States are named the Eagle, Dollar, Dime, Cent, and 
 Mill. Of these only the dollar, cent, and mill are con- 
 sidered in arithmetical Notation. 
 
 40. The dollar is the primary unit, and figures rep- 
 resenting dollars are considered as integral numbers. 
 
 41. The cent is the one hundredth part of one dollar. 
 
 42. The mill is the one thousandth part of one dollar. 
 
 43. The sign $, when placed before figures, denotes 
 that United States money is meant. 
 
 Remark. — The gold coins of the United States are the double- 
 eagle, eagle, half-eagle, quarter-eagle, and dollar. 
 
 The silver coins are the dollar, half-dollar, quarter-dollar, dime, 
 and half-dime. The nickel coin is the three-cent piece. 
 
 The copper coins are the two-cent and the one-cent pieces. ' 
 
 The value of the eagle is ten dollars, and of the dime ten cents. 
 
 The eagle weighs 10 pennyweights 18 grains. 
 
 PROBIiEM I. 
 
 44. To represent by figures any number of dollars, 
 cents, and mills, 
 
 (1.) Place the figures indicating the dollars on the left 
 of the decimal point. 
 
 (2.) Consider the figures indicating the cents as so 
 many hu7idredths, arid write them as directed by 34. 
 
 (3.) Write the figure indicating the mills in the third 
 place on the right of the decimal point. 
 
 (4.) To the whole prefix the sign |. 
 
 Remark. — If mills only are to be represented, the places of the 
 dollars and cents must be filled with ciphers. — (Vide 35, 1.) 
 
DECIMALS. 27 
 
 EXAMPLES. 
 
 1. Represent one mill by figures. Arts, f 0.001. 
 
 2. Represent two mills, three mills, four mills, etc., 
 to nine mills. Ans. to last, |0.009. 
 
 3. Represent one cent by figures. Ans. §0.01. 
 
 4. Represent 2 cents, 3 cents, 4 cents, etc., to 9 cents. 
 
 Ans. to last, $0.09. 
 
 5. Represent twenty-five cents by figures. 
 
 Ans. 10.25. 
 
 6. Represent 6 dollars 27 cents. An^. |6.27. 
 
 7. Represent 8 dollars 10 cents 4 mills. * 
 
 Ans. $8,104. 
 
 8. Represent 24 dollars 25 cents 1 mill. 
 
 Ans. $24,251. 
 
 9. Represent 103 dollars 6 mills. Ans. $103,006. 
 
 10. Represent 904 dollars 37 cents. 
 
 Ans. $904.37. 
 
 11. Represent 1 dollar 20 cents 5 mills. 
 
 Ans. $1,205. 
 
 12. Represent 4 cents; 5 mills; 13 cents; 65 cents; 
 5 dollars 14 cents 2 mills; 167 dollars 55 cents 7 mills; 
 1 dollar 1 mill; 65 dollars 6 mills; 125 dollars. 
 
 Ans. to last, $125.00. 
 
 13. Represent 4 dollars 3 cents 1 mill; 6000 dollars 
 1 cent; 1245 dollars 3 mills; 45 dollars; 7 mills; 5 cents; 
 222 dollars 22 cents 2 mills; 9167 dollars 54 cents 9 
 mills. 
 
 14. What is meant by the expression $1,251 ? 
 
 Alls. 1 dollar 25 cents 1 mill. 
 
 15. Read in words the following sums of money : 
 
28 
 
 
 DECI3: 
 
 [ALS. 
 
 
 13.265 
 
 §3.043 
 
 $20,072 
 
 $0,714 
 
 19.118 
 
 §6.259 
 
 $18,013 
 
 $7.14 
 
 $0,001 
 
 $0,141 
 
 $00,162 
 
 $71.40 
 
 $1,111 
 
 $5,001 
 
 $11,001 
 
 $714. 
 
 45. Exercises ix Review. 
 
 1. Express by figures twenty-six; one hundred and 
 one ; six hundred and forty ; seven hundred and fifty- 
 three ; five hundred and sixty-seven ; three hundred and 
 eleven; six thousand and four; eight thousand and 
 ninety. 
 
 2. Expre'ss by figures two thousand and twenty-two ; 
 three million and twenty-two; forty -five million and 
 twenty-two ; three hundred and eight thousand. 
 
 3. Express by figures six billion and five million; 
 eight trillion and seven thousand; six quadrillion and 
 one ; three quintillion and sixty-seven. 
 
 4. Express by figures four hundred and forty-three 
 thousand five hundred and twenty-five. 
 
 5. Express by figures sixty-eight billion two hundred 
 and three million five hundred and five thousand six 
 hundred and forty-five. 
 
 6. Express by figures twenty-six Jiundredths; one 
 hundred and one thousandths; six hundred and forty 
 millionths; sixty-four hundred-thousandths. 
 
 7. Express by figures seven tenths ; seven hundredths ; 
 seven thousandths ; sixty-five thousandths; one hillmith; 
 one trillionth; one quadrillionth; one quintilliontli ; one 
 sextillionth; one septilUonth; three octilllonths; seven 
 nonillionths; one hundred and eleven deeilliontlis. 
 
 8. The number of inches from the Equator to the 
 
DECIMALS. 29 
 
 North Pole is three hundred ninety-three million seven 
 hundred seven thousand nine hundred. What figures 
 express them? 
 
 9. The numher of seconds in a year is thirty-one 
 million five hundred fifty-six thousand nine hundred 
 twenty-seven and fifty-seven hundredths. What figures 
 express them ? 
 
 10. The distance from the earth to the moon is two 
 hundred thirty-eight thousand six hundred and fifty 
 miles. What figures express this distance ? 
 
 11. The distance from the earth to the sun is about 
 ninety-five million miles. What figures express this 
 distance ? 
 
 12. It is about twenty trillion miles to the nearest 
 star. Express the distance in figures. 
 
 13. Express by the aid of figures two dollars sixteen 
 cents four mills ; six dollars seven cents ; eight dollars 
 and one mill; one hundred twenty-five dollars sixty 
 cents. Ans. to last, §125.60 
 
 14. Express by the aid of figures one hundred 
 twenty-five ; one hundred twenty -five thousandths ; one 
 dollar twenty -five cents ; twelve dollars fifty cents ; 
 twelve cents five mills ; one and twenty-five hundredths ; 
 twelve and five tenths; twelve and fifty hundredths. 
 
 15. Read the following expressions: 3071; $3071; 
 3.071; §3.071; 30.71; §30.71; 307.1; §307.10; 
 307.10; 307.100. 
 
30 ROMAN NOTATION AND NUMERATION. 
 
 ROMAN NOTATION AND NUMERATION 
 
 46. The Roman Notation makes use of seven Capi- 
 tal Letters to represent numbers. They are 
 
 I, V, X, L, C, D, M; 
 and their values are, respectively, 
 
 1, 5, 10, 50, 100, 500, 1000. 
 
 47. Principles of Roman Notation. 
 
 I. The repetition of a letter repeats the value of the 
 letter. Thus : II are 2, III are 3, XX are 20, XXX are 
 30, CCC are 300. 
 
 II. If a letter is placed hefore another of greater value 
 than itself, the value of the less is taken from that of the 
 greater. Thus : IV represent 4, XL represent 40, XC 
 represent 90. 
 
 III. If a letter is placed after another of greater value 
 than itself, and a letter of greater value does not follow 
 both of them, the value of the less is added to that of 
 the greater. Thus : XI represent 11, XIV represent 14, 
 OX represent 110, CXL represent 140. 
 
 48. Examples. 
 
 1. Represent by the aid of letters the numbers 1, 2, 
 3, 4, 5. Ans. I, II, III, IV, V. 
 
 2. Represent by the aid of letters the numbers, 6, 7, 
 8, 9, 10. A71S. VI, VII, VIII, IX, X. 
 
ADDITIOX. 31 
 
 3. Represent by the aid of letters the numbers 11, 
 12, 13, etc., to 50. Ans. XI, XII, XIII, etc., L. 
 
 4. Represent 54, 80, 90, 100, 150, 199, 500, 1099, 
 1865. Ans. to List, MDCCCLXV. 
 
 5. Represent 60, 63, 71, 94, 83, 101, 565, 1741. 
 
 Ans. to last, MDCCXLI. 
 
 6. Represent 1001, 1005, 1008, 1010, 1499. 
 
 Ans. to last, MCDXCIX. 
 
 7. Represent 1410, 1951, 1673, 1467, 1866. 
 
 Ans. to last, MDCCCLXVI. 
 
 ADDITION 
 
 49. Addition is the operation of finding the sum of 
 two or more numbers. 
 
 50. The sum is a number which contains as many 
 units as all the numbers taken together. Thus : the sum 
 of 5 and 3 is 8. 
 
 • SIGNS. 
 
 51. The sign + is called plus, and signifies that the 
 numbers between which it is placed are to be added to- 
 gether. Plus is a Latin word, signifying more. 
 
 52. The sign = is called the sign of equality, and 
 signifies that the lohole expression placed before it is 
 equal to that placed after it. Thus: 5-f 3=8, is read 
 five plus three equals eight, and the meaning is that 
 there are the same number of units in 8 as in 5 and 
 3 taken together. 
 
32 
 
 ADDITION. 
 
 53. In the following table the sign + may be read 
 by the word and, the sign = by the word are. Thus : 
 5 and 3 are 8. 
 
 
 ADDITIO 
 
 N TABLE. 
 
 
 2 + 0=2 
 
 3 + 0=3 
 
 4 + 0=4 
 
 5 + 0=5 
 
 2 + 1=3 
 
 3 + 1=4 
 
 4 + 1=5 
 
 5 + 1 = 6 
 
 2 + 2=4 
 
 3 + 2=5 
 
 4 + 2 = 6 
 
 5 + 2=7 
 
 2 + 3=5 
 
 3 + 3=6 
 
 4 + 3=7 
 
 5 + 3=8 
 
 2 + 4=6 
 
 3 + 4=7 
 
 4+4=8 
 
 5 + 4=9 
 
 2 + 5=7 
 
 3 + 5=8 
 
 4 + 5=9 
 
 5 + 5 = 10 
 
 2 + 6=8 
 
 3 + 6=9 
 
 4 + 6 = 10 
 
 5 + 6 = 11 
 
 2 + 7=9 
 
 3 + 7 = 10 
 
 4 + 7 = 11 
 
 5 + 7 = 12 
 
 2 + 8 = 10 
 
 3 + 8 = 11 
 
 4 + 8 = 12 
 
 5 + 8 = 13 
 
 2 + 9 = 11 
 
 3 + 9 = 12 
 
 4 + 9 = 13 
 
 5 + 9 = 14 
 
 6 + = 6 
 
 7 + 0=7 
 
 8 + 0=8 
 
 9 + 0=9 
 
 6 + 1=7 
 
 7 + 1=8 
 
 8 + 1 = 9 
 
 9 + 1 = 10 
 
 6 + 2=8 
 
 7 + 2=9 
 
 8 + 2 = 10 
 
 9 + 2 = 11 
 
 6 + 3 = 9 
 
 7 + 3 = 10 
 
 8 + 3 = 11 
 
 9 + 3 = 12 
 
 6 + 4 = 10 
 
 7 + 4.= 11 
 
 8 + 4 = 12 
 
 9 + 4 = 13 
 
 6 + 5 = 11 
 
 7 + 5 = 12 
 
 8 + 5 = 13 
 
 9 + 5 = 14 
 
 6 + 6 = 12 
 
 7 + 6 = 13 
 
 8 + 6 = 14 
 
 9 + 6 = 15 
 
 6 + 7 = 13 
 
 7 + 7 = 14 
 
 8 + 7 = 15 
 
 9 + 7 = 16 
 
 6 + 8 = 14 
 
 7 + 8 = 15 
 
 8 + 8 = 16 
 
 9 + 8 = 17 
 
 6 + 9 = 15 
 
 7 + 9 = 16 
 
 8 + 9 = 17 
 
 9 + 9 = 18 
 
 PROBLEM I. 
 
 54. To add any number of figures representing units 
 of the^rs^ order, (vide 18,) 
 
 (1.) Set the figures under each other, and add from the 
 bottom upward or from the top downward. 
 
 (2.) Place the sum under the column, so that the figure 
 representing units of the first order shall fall exactly/ 
 underneath the figures above. 
 

 
 AJJi 
 
 UllUiN. 
 
 
 i5 
 
 
 
 EXAMPLES. 
 
 
 
 (1.) (2.) 
 
 (3.) 
 
 (4.) (5.) 
 
 (6.) (7.) 
 
 (8.) (9.) 
 
 (10.) 
 
 4 5 
 
 1 
 
 6 8 
 
 7 5 
 
 2 4 
 
 5 
 
 6 8 
 
 
 
 7 2 
 
 2 9 
 
 3 
 
 9 
 
 3 2 
 
 7 
 
 2 9 
 
 4 8 
 
 2 
 
 6 
 
 7 9 
 
 3 
 
 5 4 
 
 3 1 
 
 9 5 
 
 7 
 
 20 24 
 
 11 
 
 20 23 
 
 16 23 
 
 11 14 
 
 27 
 
 33 
 
 11. Add 6, 3, 5, 9, 4, 0, 2. 15. Add 9, 8, 7, 6, 5, 4, 3, 2, 1. 
 
 12. Add 8, 3, 9, 0, 9, 9, 9. 16. Add 1, 4, 7, 2, 5, 8, 3, 6, 9. 
 
 13. Add 4, 1, 3, 2, 5, 7, 9. 17. Add 3, 2, 5, 7, 9, 1, 4, 6, 8. 
 
 14. Add 8, 2, 6, 4, 1, 5, 0. 18. Add 1, 3, 5, 7, 9, 2, 4, 6, 8. 
 
 PR OB I. EM II. 
 
 55. To add any number of figures representing units 
 of the second order, (vide 19,) 
 
 (1.) aS'^^ the figures under each other, and add as in 
 54, (1.) 
 
 (2.) Place the sum under columns, so that the figure 
 represeyiting units of the second order shall fall exactly 
 underneath the digits above. 
 
 
 
 
 EXAMPLES 
 
 . 
 
 
 
 (1.) 
 
 (2.) 
 
 (3.) 
 
 (4.) (5.) 
 
 (6.) 
 
 (7.) 
 
 (8.) 
 
 40 
 
 50 
 
 10 
 
 90 50 
 
 50 
 
 80 
 
 70 
 
 60 
 
 80 
 
 00 
 
 80 40 
 
 90 
 
 50 
 
 30 
 
 30 
 
 20 
 
 70 
 
 70 70 
 
 80 
 
 30 
 
 20 
 
 70 
 
 90 
 
 30 
 
 60 30 
 
 70 
 
 40 
 
 40 
 
 200 240 110 300 190 290 200 160 
 9. Add 60, 30, 50, 90, 40. 12. Add 10, 20, 90, 80. 
 
 10. Add 10, 20, 40, 70, 80. 13. Add 30, 50, 70, 90. 
 
 11. Add 60, 50, 40, 30, 20. 14. Add 20, 40, 60, 80. 
 
34 
 
 ADDITION. 
 
 Remark. — It is evident that figures, all of which represent any 
 one given order of units, may be added in the same way. 
 
 (15.) 
 400 
 
 600 
 300 
 700 
 
 (16.) 
 5000 
 8000 
 2000 
 9000 
 
 (17.) 
 10000 
 00000 
 70000 
 30000 
 
 (18.) (19.) 
 
 900000 6000000 
 
 800000 7000000 
 
 700000 1000000 
 
 600000 9000000 
 
 2000 24000 110000 3000000 23000000 
 
 In the same manner add — 
 
 (20.) (21.) (22.) (23.) 
 
 8 70 500 4000 
 
 6 20 400 3000 
 
 5 70 800 9000 
 
 19 160 1700 
 
 24. Add 4, 9, 3, 7, 6. 
 
 25. Add 5, 8, 4, 3, 0, 7, 1. 
 
 26. Add 30, 40, 50, 60. 
 
 27. Add 400, 300, 500, 100. 
 
 28. Add 9000, 4000, 3000. 
 
 29. Add 40000, 30000, 10000. 
 
 16000 
 
 Ans. 29. 
 
 Ans. 28. 
 
 Ans. 180. 
 
 Ans. 1300. 
 
 Ans. 16000. 
 
 Ans. 80000. 
 
 l»ROBI.EM III. 
 
 56. To add any numbers together where the sum of 
 the corresponding orders of units in all the numbers is 
 9 or less than 9, 
 
 (1.) Write the numbers so that the corresponding orders 
 of units may stand mider each other. 
 
 (2.) Begin at the right, and add each column separately, 
 placing the smn exactly under the column added. 
 
ADDITION. 
 
 35 
 
 EXAMPLES. 
 
 1. Add together 19, 160, 1700, and 16000. 
 
 OPERATION. 
 
 19 
 160 
 
 1700 
 16000 
 
 Ans. 9999. 
 Ans. 8778 
 
 17879 Ans. 
 
 2. Add 1025, 6712, 1111, 1151. 
 
 3. Add 1234, 4321, 2222, 1001. 
 
 4. Add 31004, 13121, 22102, 21101, 11210. 
 
 Ans. 98538 
 
 5. Add 9, 10, 300, 130, 4110, 71100. Ans. 75659 
 
 PROBIiEMIV. 
 
 57. To add any numbers whatever together. 
 
 EXAMPLES. 
 
 1. Add together 4578, 3426, and 9875. 
 
 
 
 OPERATION. 
 
 
 
 4578 
 
 
 
 • 3426 
 
 
 
 9875 
 
 Vide 55, Ex. 
 
 20 . 
 
 . . 19 
 
 Vide 55, Ex. 
 
 21 . 
 
 . . 160 
 
 Vide 55, Ex. 
 
 22 . 
 
 . . 1700 
 
 Vide 55, Ex. 
 
 23 . 
 1 . 
 
 . 16000 
 
 Vide 56, Ex. 
 
 . nS79 Ans 
 
36 ADDITION. 
 
 A moment's attention shows how the above operation 
 may be contracted. 
 
 The sum of the first column is 19, which is composed 
 of 9 units of the first order and 1 of the second. Set 
 down the 9 units under the units of the first order, j^r ^o 
 and add the 1 unit of the second order to the 342(3 
 column of units of the same order, making 17, 9875 
 which is composed of 7 units of the second order, TI^Z^ 
 and 1 of the third. Set down the 7 units of the 
 second order under that column, and add the 1 unit of 
 the third order to the column of units of that order, 
 making 18, which is composed of 8 units of the third 
 order and 1 of the fourth. Set down the 8 units of the 
 third order under that column, and add the 1 unit of the 
 fourth order to the column of units of that order, making 
 17, which is written down as in 55. Hence, 
 
 RULE. 
 
 (1.) W^Hte the numbers so as to j^l^toe the figures in the 
 corresponding orders of units directly under each other, 
 and draw a line underneath. 
 
 (2.) Begin at the right hand, and add each column 
 separately, setting doivn the right-hand figure of the result 
 under the column added, and add- the left-hand figure or 
 figures to the next column on the left. 
 
 (3.) Set dozvn the tuhole amount of the last column. 
 
 2. Add 234, 589, 613. Ans. 1436. 
 
 3. Add 7123, 6054, and 9123. Ans. 22300. 
 
 4. Add 70561, 23564, and 34625. Ans. 128750. 
 
 5. Add 123456, 654321, 456123. Ans. 1233900. 
 
 6. Add 123, 240, 85, 36, and 7. Ans. 491. 
 
ADDITION. 
 
 7. Add 1, 7, 43, 76, 65, 15, and 100. Ans. 307. 
 
 8. Add 13, 165, 48, 6251, and 19. A71S. 6496. 
 
 9. Add 108, 5012, 4103, 60450, and 6. 
 
 Ans. 69679. 
 
 10. Add 3456, 6543, 4563, 3645, and 5634. 
 
 A71S. 23841. 
 
 11. Add 31236, 415, 621437, 90053, and 34. 
 
 Ans. 743175. 
 
 12. Add 31, 280, 4560, 78930, and 672140. 
 
 Ans. 755941. 
 
 (13.) 
 
 (14.) 
 
 (15.) 
 
 12343247 
 
 213673 
 
 13021654 
 
 6015400 
 
 13021654 
 
 12343247 
 
 13021654 
 
 6015400 
 
 6015400 
 
 213673 
 
 12343247 
 
 213673 
 
 31593974 
 
 
 
 (16.) 
 
 
 (17.) 
 
 12346721305 
 
 
 3126754 
 
 8917259679 
 
 
 25678960 
 
 763421893 
 
 
 763421893 
 
 25678960 
 
 
 8917259679 
 
 3126754 
 
 
 12346721305 
 
 22056208591 
 
 
 58. To add several decimals together, proceed exactly 
 as in 57, and then place the decimal p>oint in the sum 
 directly under the decimal points above. — (Yide 38, Re- 
 mark 3.) 
 
8 
 
 
 ADDITION. 
 
 
 
 EXAMPLES. 
 
 
 (1.) 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 ^34 
 
 71.23 
 
 705.61 
 
 314.5 
 
 .589 
 
 60.54 
 
 235.64 
 
 21.346 
 
 .613 
 
 91.23 
 
 * 346.25 
 
 5.17 
 
 1.436 
 
 223.00 
 
 1287.50 
 
 341.016 
 
 (5.) 
 
 
 (6.) 
 
 (7.) 
 
 345.012 
 
 
 785.432 • 
 
 987.65 
 
 45.78 
 
 1234.6 
 
 12.1453 
 
 121.3 
 
 
 257.87 
 
 1.67 
 
 87.125 
 
 2 
 
 12.431 
 
 1436.123 
 
 599.217 
 
 1290.333 
 
 2437.5883 
 
 8. Add 12.4, 3.47, 27.67, and 86. Ans. 129.54. 
 
 9. Add 1.24, 34.7, 2.767, and .86. Ans. 39.567. 
 
 10. Add .124, .347, .2767, and 8.6. Ans. 9.3477. 
 
 11. Add 57.76, 98.54, 38.72, and 43.65. 
 
 Ans. 238.67. 
 
 12. Add 5.776, 985.4, 38.72, and 4365. 
 
 Ans. 5394.896. 
 
 13. Add 577.6, 9.854, 3.872, and .4365. 
 
 Ans. 591.7625. 
 
 14. Add 4.8, 43.31, 74.019, and 11.204. 
 
 Ans. 133.333. 
 
 15. Add 29.0029, 3.4476, and 58.123. 
 
 Ans. 90.5735. 
 
 16. Add twelve and four tenths, three and forty-seven 
 hundredths, twenty-seven and sixty-seven hundredths, 
 and eighty-six. Ans. 129.54. 
 
 17. Add twenty-nine and twenty-nine ten-thou- 
 
» 
 
 ADDITIOX. 39 
 
 sandths, three and four thousand four hundred and 
 seventy-six ten-thousandths, fifty-eight and one hun- 
 dred and twenty-three thousandths. Ans. 90.5735. 
 
 18. Add three and seven tenths, four and five hun- 
 dredths, one hundred, five thousandths, sixty-seven 
 millionths, five hundred and three, eight and six ten- 
 thousandths. Ans. 618.755667. 
 
 59. To add United States Money, consider the several 
 items as decimals, adding as in 58 ; then prefix the sign 
 % to the sum.— (Vide 44.) 
 
 EXAMPLES. 
 
 1. Add two dolhirs sixteen cents four mills, six dol- 
 lars seven cents, eight dollars one mill, one hundred 
 twenty-five dollars and sixty cents. 
 
 OPERATION. 
 
 _ $2,164 
 6.07 
 8.001 . 
 125.60 
 
 §141.835 Ans. 
 
 2. Add §241.075, $45.06, |37.05, §1216.131. 
 
 Ans. §1539.316. 
 
 3. Add §3124.162, §812.95, §67.12, §2145.75. 
 
 A71S. §6149.982. 
 
 4. Add §1.132, §56.075, §931.87, §4621.953. 
 
 Alls. §5611.03. 
 
 5. Add §27.413, §45.084, §607.219, §205.03, §25.25, 
 and §405.006. Ans. §1315.002. 
 
40 ADDITION. 
 
 6. Add $136,255, §10.30, $248.50, $100,125, and 
 $65.38. Ans. $560.56. 
 
 7. Add $2600, $1927.404, $1603.40, $3304.17, 
 $165.47, and $2600.08. Ans, $12200.524. 
 
 8. Add $170, $400.02, $130, $250.10, and $845.22. 
 
 Ans. $1795.34. 
 9 Add $17.15, $23.43, $7.19, $8.37, and $12,315. 
 
 Ans. $68,455. 
 
 10. Add $6.75, $2.30, $0.92, $0,125, and $0.06. 
 
 Ans. $10,155. 
 
 11. Add $56.18, $7,375, $280.00, $0,287, $17.00, 
 and $90,413. A71S. $451,255. 
 
 12. Add 241 dollars 7 cents 5 mills, 45 dollars 6 cents, 
 37 dollars 5 cents, and 1216 dollars 13 cents 1 mill. 
 
 Ans. $1539.316. 
 
 Remark. — In adding a long column of figures, it is of much 
 assistance to divide it into several parts at pleasure, add each of 
 the parts separately, and finally the several partial sums for the 
 sum total. 
 
 (13.) 
 
 (14.) 
 
 (15.) 
 
 45678 
 
 76.345 
 
 $27,251 
 
 12345 
 
 18.237 
 
 43.026 
 
 37425 
 
 5.404 
 
 126.007 
 
 3128- 98576 
 
 12.36 -112.346 
 
 185.214 
 
 8462 
 
 1.1 
 
 243.671 
 
 71351 
 
 33.33 
 
 453.172-1078.341 
 
 81250 
 
 45.54 
 
 999.999 
 
 11111-172174 
 
 8.8 - 88.77 
 
 471.862 
 
 3333 
 
 75.464 
 
 125.281 
 
 7812 
 
 21.853 
 
 931.452 
 
 4512 
 
 27.306 
 
 813.161 
 
 76251- 91908 
 
 31.452-156.075 
 
 13.20 -3354.955 
 
 362658 
 
 357.191 
 
 $4433.296 
 

 
 ADDITION. 
 
 
 41 
 
 (16.) 
 
 
 (17.) 
 
 
 (18.) 
 
 
 43267 
 
 
 143.01 
 
 
 $25.04 
 
 
 14567 
 
 
 26.435 
 
 
 87.05 
 
 
 76543 
 
 
 506.146 
 
 
 125.113 
 
 
 81234 
 
 
 81.237 
 
 
 37.40 
 
 
 30506 
 
 
 67.21 - 
 
 - 824.038 
 
 103.046 
 
 
 4736- 
 
 -250853 
 
 1.004 
 
 
 95.062 
 
 
 154 
 
 
 65.042 
 
 
 127.111 
 
 
 58463 
 
 
 121.251 
 
 
 1237.086 
 
 
 81460 
 
 
 67.132 
 
 
 906.07 - 
 
 -2742.978 
 
 70120 
 
 
 9.25 - 
 
 - 263.679 
 
 81.023 
 
 
 93126 
 
 
 14.062 
 
 
 3410.192 
 
 
 47615- 
 
 -350938 
 
 87.643 
 
 
 1.20 
 
 
 82361 
 
 
 100.916 
 
 
 19.02 
 
 ' 
 
 95864 
 
 
 2147.05 - 
 
 -2349.671 
 
 127.45 
 
 
 3729 
 
 
 432.876 
 
 
 87.40 - 
 
 -3726.285 
 
 26- 
 
 -181980 
 
 91.91 
 
 
 487.103 
 
 
 9428 
 
 
 125.125 
 
 
 45.073 
 
 
 32193 
 
 
 37.126 
 
 
 110.029 
 
 
 86159- 
 
 -127780 
 
 85.437- 
 
 - 772.474 
 
 $ 
 
 3145.671- 
 
 -3787.876 
 
 911551 
 
 4209.862 
 
 10257.139 
 
 
 PRACTICAL EXAMPLES. 
 
 19. A gentleman purchased 234 bushels of corn at 
 one time, 589 at another, and 613 at another. How 
 many bushels did he buy in all? — (Vide 57, Ex. 2.) 
 
 20. During one year my crop of cotton was sold for 
 $7123.00 ; the next year it brought $6054, and the year 
 after I received |9123.00. How much did I receive for 
 cotton during the three years? — (Vide 57, Ex. 3.) 
 
 21. January has 31 days, February 28, March 31, 
 
 4 
 
42 ADDITION. 
 
 April 30, May 31, June 30, July 31, August 31, Sep- 
 tember 30, October 31, Noveraber 30, December 31. 
 How many days in the year? Ans. 365. 
 
 22. Washington was born in 1732 and lived 67 years. 
 In what year did he die? Ans. 1799. 
 
 23. From the creation of the world to the flood, there 
 were 1656 years ; from the flood to the siege of Troy, 
 1164 years; from the siege of Troy to the building of 
 Solomon's Temple, 180 years; from the building of the 
 Temple to the birth of Christ, 1004 years. In what 
 year of the world did the Christian Era commence ? 
 
 Ans. 4004. 
 
 24. How many years have intervened from the 
 creation of the world to the year 1865? Ans. 5869. 
 
 25. Homer was born 733 years before the Christian 
 Era. How many years from the birth of Homer to the 
 year 1865 ? Ans. 2598. 
 
 26. I bought a barrel of flour for |6.78; ten pounds 
 of raisins for |2.30; seven pounds of sugar for |0.92; 
 one pound of coWeQ for $0,125, and two oranges for 
 10.10. What was the whole amount? Ans. |10.225. 
 
 27. A collector has bills in his possession of the fol- 
 lowing amounts: one of §43.75; another of $29.18; 
 another of $17.63; another of $268.95, and anotlicr of 
 $718.07. What amount has he to collect? 
 
 Ans. $1077.58 
 
 28. A man has the following sums of money due him, 
 viz: $420,197, $105.50, $304,005, $888,455. What is 
 the amount due him? Ans. $1718.157. . 
 
 29. What is the. sum of 429, 21.37, 355.003, 1.07, 
 and 1.7? Ai^s. 808.143. 
 
ADDITIOX. 43 
 
 30. What is the sum of .2, .80, .089, .006, .9000, and 
 .005? Ans.2. 
 
 31. A gentleman bought at one thne 13.25 bushels of 
 corn; at another, 8.4 bushels; at another, 23.051 
 bushels ; at another, 6.75 bushels. How many bushels 
 did he buy in all? Ans. 51.451 bushels. 
 
 32. A gentleman owns five farms ; the first is worth 
 §11500; the second, §3057; the third, §2468; the 
 fourth, §9462; and the fifth is worth as much as the 
 four together. What is the value of the five farms ? 
 
 Ans. §52974. 
 
 33. By the census of 1850, the population of the ten 
 largest cities of the United States was as follows : New 
 York, 515547 ; Philadelphia, 340045 ; Baltimore, 169054 ; 
 Boston, 136881; New Orleans, 116375; Cincinnati, 
 115436; Brooklyn, 96838; St. Louis, 77860; Albany, 
 50763; Pittsburg, 46601. What was the population of 
 all combined? Ans. 1665400. 
 
 34. By the census of 1860, the population of the 
 following cities was ascertained to be — of New York, 
 805651; Philadelphia, 562529; Brooklyn, 266661; 
 Baltimore, 212418; Boston, • 177812; New Orleans, 
 168675; St. Louis, 160773; Cincinnati, 161044; Chi- 
 cago, 109260; Bufi-alo, 81129; Louisville, 68033 ; New- 
 ark, 71914; San Francisco, 56802; Washington, 61122; 
 Providence, 50666; Rochester, 48204; Detroit, 45619; 
 Milwaukee, 45246; Cleveland, 43417; Charleston, 
 40578; Troy, 39232 ; New Haven, 39267; Richmond, 
 37910; Lowell, 36827; Mobile, 29258; Jersey City, 
 29226; Portland, 26341 ; Cambridge, 26060; Roxbury, 
 25137; Charlestown, 25063; Worcester, 24960; Utica, 
 
44 
 
 ADDITION. 
 
 22529; Reading, 23161; Salem, 22252; New Bedford, 
 22309; Dayton, 20081; Nashville, 16988. How many 
 inhabitants in all these cities combined? Ans. 
 
 35. By the census of 1860, the population of the 
 several States and Territories was as follows : 
 
 Alabama 964201 
 
 Arkansas 435450 
 
 California 379994 
 
 Connecticut 460147 
 
 Delaware 112216 
 
 Florida 140425 
 
 Georgia 1057286 
 
 Illinois 1711951 
 
 Indiana 1350428 
 
 Iowa 674948 
 
 Kentucky 1155684 
 
 Louisiana 708002 
 
 Maine 628279 
 
 Maryland 687049 
 
 Massachusetts 1231066 
 
 Michigan 749113 
 
 Minnesota 173855 
 
 Mississippi 791305 
 
 Missouri 1182012 
 
 New Hampshire 326073 
 
 New Jersey 672035 
 
 New York 3880785 
 
 North Carolina 992G22 
 
 Ohio 2339502 
 
 Oregon 52465 
 
 Pennsylvania 2906115 
 
 Rhode Island 174620 
 
 South Carolina 703708 
 
 Tennessee 1109801 
 
 Texas 604215 
 
 Vermont 315098 
 
 Virginia 1596318 
 
 Wisconsin 775881 
 
 Colorado 34277 
 
 Dakotah ,. 4837 
 
 District of Columbia... 75080 
 
 Kansas 107206 
 
 Nebraska 28841 
 
 New Mexico 93516 
 
 Utah.." 40273 
 
 Washington 11594 
 
 Nevada 6857 
 
 How many inhabitants in the United States in 1860? 
 
 Ans. 31445080. 
 
 36. How many inhabitants in the six New England 
 States taken together? Ans. 3135283. 
 
 37. How many inhabitants in the States bordering on 
 the Gulf of Mexico? Ans. 3208148. 
 
 38. How many inhabitants in the States watered bj 
 the Tennessee River? Ans. 4020991. 
 
SUBTRACTIOX.' 45 
 
 39. How many inhabitants in the States bounded in 
 part by the Ohio River? Ans. 8153883. 
 
 40. How many inhabitants in the States watered by 
 the Mississippi River? Ans. 8718889. 
 
 41. How many inhabitants in the States and Terri- 
 tories lying wholly west of the Mississippi River? 
 
 Ans. 3830340. 
 
 SUBTRACTION 
 
 60. Subtraction is the operation of finding the dif- 
 ference between two numbers. 
 
 61. The difference is such a number as added to the 
 
 less will give the greater. 
 
 SIG-NS. 
 
 63. The sign — is called minus, and when placed 
 between two numbers it signifies that the one on the 
 right is to be subtracted from that on the left. 3Iinus 
 is a Latin word, signifying less. 
 
 63. The expression 8 — 5=3, is read eight minus five 
 EQUALS three, and the meaning is, that three is the dif- 
 ference between eight and five. The expression may 
 also be read, five from eight are three. 
 
 64. The greater of the two numbers is called the 
 minuend, and the smaller is called the subtrahend. The 
 result of the subtraction is called the difference, and 
 oftentimes the reniainde''\ 
 
46 
 
 SUBTIl ACTION. 
 SUBTRACTION TABLE. 
 
 2 — 2 = 
 
 3-3 = 
 
 4 — 4 = 
 
 5-5 = 
 
 3-2 = 1 
 
 4-3 = 1 
 
 5-4=1 
 
 6-5 = 1 
 
 4 — 2 = 2 
 
 5-3 = 2 
 
 6—4 = 2 
 
 7 — 5 = 2 
 
 5-2 = 3 
 
 6-3 = 3 
 
 7-4 = 3 
 
 8-5 = 3 
 
 6-2 = 4 
 
 7-3 = 4 
 
 8 — 4 = 4 
 
 9 — 5 = 4 
 
 7-2 = 5 
 
 8-3 = 5 
 
 9-4 = 5 
 
 10 - 5 = 5 
 
 8 — 2 = 6 
 
 9-3 = 6 
 
 10 - 4 = 6 
 
 11 — 5 = 6 
 
 9-2 = 7 
 
 10 — 3 = 7 
 
 11 _ 4 = 7 
 
 12 — 5 = 7 
 
 10 — 2 = 8 
 
 11 - 3 = 8 
 
 12 - 4 = 8 
 
 13 - 5 = 8 
 
 11 - 2 = 9 
 
 12 — 3 = 9 
 
 13 — 4 = 9 
 
 14 — 5 = 9 
 
 6-6 = 
 
 7-7 = 
 
 8 — 8 = 
 
 9-9 = 
 
 7-6 = 1 
 
 8-7 = 1 
 
 9-8=1 
 
 10 — 9 = 1 
 
 8-6 = 2 
 
 9 — 7 = 2 
 
 10 — 8 = 2 
 
 11 - 9 = 2 
 
 9-6 = 3 
 
 10 - 7 = 3 
 
 11 -^ 8 = 3 
 
 12 - 9 = 3 
 
 10 — 6 = 4 
 
 11 - 7 = 4 
 
 12 - 8 = 4 
 
 13 — 9 = 4 
 
 11 - 6 = 5 
 
 12 - 7 = 5 
 
 13 - 8 = 5 
 
 14 - 9 = 5 
 
 12 - 6 = 6 
 
 13 - 7 = 6 
 
 14 - 8 = 6 
 
 15 - 9 = 6 
 
 13 - 6 = 7 
 
 14 - 7 = 7 
 
 15 - 8 = 7 
 
 16 - 9 = 7 
 
 14 - 6 = 8 
 
 15 - 7 = 8 
 
 16 - 8 = 8 
 
 17 - 9 = 8 
 
 15 - 6 = 9 
 
 16 - 7 = 9 
 
 17 - 8 = 9 
 
 18 — 9 = 9 
 
 65. If tlie same number is added to any hvo niim- 
 bers, the difference bettveen the resulting numbers is the 
 same as that between the given numbers. Thus : 
 
 The diiFerence between 7 and 2 is 5. If now 6 be 
 added to both 7 and 2, the difference between the re- 
 sulting numbers, 13 and 8, is still 5. 
 
 PROBI.KM I. 
 
 00. To subtract one number from another, Avhen each 
 figure of the subtrahend is equal to, or less than, the cor- 
 responding figure of the minuend, counting from the right, 
 
 1. Write the less number under the greater, so as to 
 
SUBTRACTION. 
 
 47 
 
 place the figures representing the corresponding orders of 
 units directly under each other. 
 
 2. Begin at the right, and subtract each figure of the 
 subtrahend from the figure of the minuend above it; the 
 results placed under the figures from which they were 
 obtained Avill express the difference between the two 
 numbers. 
 
 (1.) 
 From 5346 
 Take 2145 
 
 EXAMPLES. 
 
 (2.) (3.) 
 
 7890 4567 
 
 3450 1023 
 
 (4.) 
 67814 
 6514 
 
 Ans. 
 
 3201 4440 
 
 3544 
 
 
 61300 
 
 5. 
 
 From 76503 take 65402. 
 
 
 Ans. 
 
 11101. 
 
 6. 
 
 From 84321 take 62100. 
 
 
 Aiis. 
 
 22221. 
 
 7. 
 
 From 54360 take 21030. 
 
 
 Ans. 
 
 33330. 
 
 8. 
 
 From 74215 take 3115. 
 
 
 Alls. 
 
 71100. 
 
 9. 
 
 From 21036 take 24. 
 
 
 Ajis. 
 
 21012. 
 
 10, 
 
 From 762137 take 1025. 
 
 
 Ans. ' 
 
 761112. 
 
 11. 
 
 From 12345 take 2345. 
 
 
 Ans. 
 
 10000. 
 
 12. 
 
 From 54321 take 4321, 
 
 
 Alls. 
 
 50000. 
 
 13. 
 
 From 20037 take 10036. 
 
 
 Ans. 
 
 10001. 
 
 
 (14.) (15,) 
 
 (16.) 
 
 
 (17.) 
 
 From 
 
 45.13 78.64 
 
 §3.105 
 
 
 §41.043 
 
 Take 
 
 34.02 67.64 
 
 2.104 
 
 
 21.032 
 
 Ans. 11.11 11.00 
 
 18. From 87.36 take 43.15. 
 
 19. From 96.125 take 5.01. 
 
 20. From 128.41 take 127.2 
 
 21. From |45.16 t:ike |3Lr 
 
 §1.001 
 
 §20.011 
 
 Ans. 44.21 
 
 A71S. 91.115. 
 
 Ans. 1.21 
 
 Ans. §11.01 
 
48 SUBTIIACTION. 
 
 22. From $3426.45 take §113.23. Ans. 3313.22. 
 
 23. From §4327.871 take §3216.461. 
 
 Ans. §1111.410. 
 
 24. From §945.375 take §334.275. 
 
 Ans. §611.100. 
 
 25. From §12.032 take §1.021. Ans. §11.011. 
 
 26. From §119.457 take §8.236. Ans. §111.221. 
 
 PROBLiEM II. 
 
 G7. To find the difference between any two numbers 
 whatever. 
 
 EXAMPLES. 
 
 1. From 493 take 287. 
 
 OPERATION. 
 493 
 
 287 
 
 206 Ans. 
 
 Here 7 can not be taken from 3, because 7 is larger 
 than 3. Mentally add 10 units to the 3 units, and from 
 the sum 13 units, take 7 units, placing the difference, 
 6 units, under the figures representing units of the first 
 order. 
 
 Since now the minuend has 'been increased by 10 
 units, we must increase the subtrahend by the same 
 number of units to preserve the true difference. — (Vide 
 65.) 
 
 Mentally add 1 unit of the second order (which is the 
 very same thing as 10 units of the first order) to the 8 
 units of the second order, and we have 9 units of the 
 second order. Proceed now as in 66. Hence, 
 
SUBTRACTION. 49 
 
 Tt U L E . 
 
 1. Write the numhers as in G6. 
 
 2. If the unit figure of the minuend is equal to or 
 greater than the unit figure of the subtrahend, subtract as 
 in (j6; but if the unit figure of the minuend be less than 
 that of the subtrahetid, mentally add 10 ^o the upper 
 figure and subtract the loiuer figure from the sum, placing 
 the result under the unit column. 
 
 3. Add 1 to the next figure of the subtrahend, and 
 proceed tvith the result precisely as with the figures repre- 
 senting units of the first order. 
 
 4. Do the same thing to the figures representing each 
 
 order of U7iits, and the results will express the difference 
 
 between the two numbers. 
 
 Remark 1, — The work may be verified by adding the remainder to 
 the subtrahend. 
 
 2. From 7804 take 5936. 
 
 OPERATION. 
 
 7804 min. 
 5936 sub. 
 
 1868 
 
 7804 proof. 
 
 Remark 2. — The pupil should say at once, 6 fi'om 14 leave 8; 4 
 from 10 leave 6; 10 from 18 leave 8; 6 from 7 leave 1. Do not say 
 ^from 4 you can^t, 1 to carry, etc. 
 
 (3.) 
 From 5432 
 Take 1685 
 
 (4.) 
 * 34.57 
 23.19 
 
 (5.) 
 56.19 
 24.396 
 
 (6.) 
 125.4 
 37.236 
 
 Alls. 3747 
 
 11.38 
 34.57 
 
 31.794 
 
 56.19 
 
 88.164 
 
 Proof 5432 
 5 
 
 125.4 
 
50 
 
 SUBTRACTION. 
 
 
 
 (7.) (8.) 
 
 (9.) 
 
 From 
 
 $125,456 $1243.18 
 
 $7256.372 
 
 Take 
 
 87.25 125.914 
 
 199.20 
 
 Ans. 
 
 $38,206 $1117.266 
 
 $7057.172 
 
 
 (10.) 
 
 (11.) 
 
 
 From 645.00037 
 
 10000.0000 
 
 
 Take .00198 
 
 .0111 
 
 
 Ans. 644.99839 
 
 9999.9889 
 
 12. 
 
 From 40000 take 9. 
 
 Ans. 39991. 
 
 13. 
 
 From 123456789 take 87654321. 
 
 
 
 Ans. 35802468. 
 
 14. 
 
 From 101010101 take 90909090. 
 
 
 
 Ans. 10101011. 
 
 15. 
 
 From 303030303 take 40404040. 
 
 
 
 A71S. 262626263. 
 
 16. 
 
 From 234702358 take 54321987. 
 
 
 
 Ans. 180380371. 
 
 17. 
 
 From 1000000 take 1. 
 
 Ans. 999999. 
 
 18. 
 
 From 1000000 take .1. 
 
 Ans. 999999.9. 
 
 19. 
 
 From 1000000 take .01. 
 
 Ans. 999999.99. 
 
 20. 
 
 From 34567 take .003. 
 
 Ans. 34566.997. 
 
 21. 
 
 From 1 take .000001. 
 
 Ans. .999999. 
 
 22. 
 
 From 5 take 4.000001. 
 
 Ans. .999999. 
 
 23. 
 
 From 12.4 take 3.756. 
 
 Ans. 8.644. 
 
 24. 
 
 From 100.25 take 75.12. 
 
 Ans. 25.13. 
 
 25. 
 
 From $100.25 take $75.12. 
 
 Ans. $25.13. 
 
 26. 
 
 From $20.05 take $5.50. 
 
 Ans. $14.55. 
 
 27. 
 
 From $90.00 take $70,045. 
 
 Ans. $19,955. 
 
 28. 
 
 From $1000 take $1,111. 
 
 Ans. $998,889. 
 
SUBTRACTION. 51 
 
 29. From 6 dollars take 5 mills. Ans. $5,995. 
 
 30. From 8 dollars take 7 cents. Ans. $7.93. 
 
 31. America was discovered in A. D. 1492 by Chris- 
 topher Columbus. How many years from that event to 
 A. D. 1865 ? A71S. 373 years. 
 
 32. George Washington was born in A. D. 1732, and 
 died in A. D. 1799. To what age did he live? 
 
 Ans. 67. 
 
 33. The Declaration of Independence was published 
 July 4, 1776. How many years have intervened up to 
 July 4, 1865? A^is. 89. 
 
 34. Henry Hudson sailed up the river of his name, 
 A. D. 1609. How many years from that time to A. D. 
 1865? Ans.2b6. 
 
 35. The Mariners' Compass was invented in A. D. 
 1302. How many years to A. D. 1865 ? Aiis. 563. 
 
 36. What length of time from the birth of Francis 
 Bacon, A. D. 1561, to A. D. 1865? Ans. 304 years. 
 
 37. What length of time from the birth of Shake- 
 speare, A. D. 1564, to A. D. 1865 ? Ans. 301 years. 
 
 38. What length of time from the birth of John 
 Milton, A. D. 1608, to A. D. 1865? Ans. 257 years. 
 
 39. Pliny died in A. D. 17. How many years to A. 
 D. 1865 ? A71S. 1848. 
 
 40. Sir William Herschel was born in A. D. 1738,^ 
 Galileo, A. D. 1564. How many years elapsed from the 
 birth of the one to that of the other? Ajis. 174. 
 
 41. Oliver Cromwell was born A. D. 1599. How 
 many years from that time to the death of Washington ? 
 
 A71S. 200. 
 
 42. Patrick Henry was born A. D. 1 736. How many 
 
52 SUBTRACTION. 
 
 years from that; time to the publication of the Decla- 
 ration of Independence ? Aiis. 40. 
 
 43. The Revolutionary War began A. D. 1775 ; the 
 last war with Great Britain, A. D. 1812. How many 
 years from the beginning of the one to the beginning 
 of the other? Ans. 37. 
 
 44. What was the increase in the population of New 
 York from A. D. 1850 to A. D. I860?— (Vide 59, Ex. 
 33 and 34.) Ans. 290104. 
 
 45. How many more inhabitants in New York in 
 1850 than in Philadelphia? Ans. 175502. 
 
 46. How many more in 1860 ? A^is. 243122. 
 
 47. How many more in Boston than in New Orleans 
 in 1850? A71S. 20506. 
 
 48. How many more in New Orleans than in Cincin- 
 nati? Ans. 939. 
 
 49. How many more in Cincinnati than in Brooklyn ? 
 
 Ans. 18598. 
 
 50. How many more in Brooklyn than in St. Louis ? 
 
 Ans. 18978. 
 
 51. How many more in St. Louis than in Albany? 
 
 A71S. 27097. 
 
 52. How many more in Albany than in Pittsburg ? 
 
 A71S. 4162. 
 53. . How many more in New York than in Pittsburg ? 
 
 A71S. 468946. 
 
 54. The polar diameter of the earth is 7898.973 
 miles; the equatorial diameter, 7925.249 miles. How 
 much greater is the equatorial than the polar diameter ? 
 
 Ans. 26.276 miles. 
 
 55. The length of a degree of longitude at the equator 
 
SUBTRACTION. 58 
 
 is 69.161 miles ; at New York it is 52.536 miles. What 
 is the difference ? A^is. 16.625 miles. 
 
 56. A is worth §6542.37; B is worth $9341.95 ; C 
 is worth §18425.63. How much are all three together 
 worth? Ans. §34309.95. 
 
 How much is B worth more than A ? 
 
 Ans. §2799.58. 
 How much more is C worth than B? 
 
 Ans. §9083.68. 
 How much more is C worth than A? 
 
 Ans. §11883.26. 
 How much more is C worth than A and B together? 
 
 Ans. §2541.31. 
 
 57. To stock a farm, the land of which was worth 
 §22475.96, I bought two horses for §327.80 ; two yoke 
 of oxen at §175.47 per yoke ; five cows at §27.36 each ; 
 a pair of mules for §275; and sixty-seven sheep for 
 §201.45. How much more is the land worth than the 
 stock? Ans. §21183.97. 
 
 How much more were the oxen worth than the horses ? 
 
 Ans. §23.14. 
 How much more were the oxen worth than the cows ? 
 
 A71S. §214.14. 
 Which were worth most, the oxen and horses together, 
 or the cows, mules, and sheep together, and by how 
 much? Ans. Oxen and horses, by §65.49. 
 
 58. A merchant bought at one time 3476 yards of 
 cloth; at another, 5426 yards; at another, 4221 yards. 
 He sells 3210 yards to one person, and 4345 to another. 
 How many yards has he left? Ans. 5568 yards. 
 
 59. A farmer bought of a merchant broadcloth to the 
 
54 SUBTRACTION. 
 
 value of $137.50; cotton cloth, §93.45; sugar, |37.63; 
 molasses, 114.37; coffee, §11.45; flour, §28.13. He pays 
 the merchant, in corn, §123.65 ; in hay, §47.24; and the 
 balance in cash. What was the amount of cash paid? 
 
 Ans. §151.64. 
 
 60. Bought a yoke of oxen for §150 ; a horse for 
 §237; three cows for §87.45; and sold the whole for 
 §500. What was my gain? Ans. §25.55. 
 
 61. The length of a pendulum which vibrates once a 
 second, at London, is 39.1393 inches. One ten-millionth 
 of the meridian distance from the Equator to the North 
 Pole is 39.37079 inches. What is the difference? 
 
 Ans. .23149 inches. 
 
 62. The equatorial circumference of the earth is 
 24897.883 miles, and the circumference on a meridian 
 is 24855.296 miles. What is the difference ? 
 
 Ans. 42.587 miles. 
 
 63. What is the difference between 25 dollars 1 cent 
 4 mills and 6 dollars 17 cents 9 mills ? 
 
 A71S. §18.835. 
 
 64. What is the difference between 181 dollars 7 
 cents 9 mills and 140 dollars 9 cents 7 mills? 
 
 A71S. §40.982. 
 
 65. What is the difference between 9 dollars 5 cents 
 3 mills and 10 dollars 3 cents 5 mills? 
 
MULTIPLICATION. 65 
 
 MULTIPLICATION. 
 
 68. Multiplication is the operation of increasing 
 one number as many times as there are units in another. 
 
 69. The number to be increased is called the multi- 
 plicand. 
 
 70. The number indicating how many times the 
 multiplicand is to be increased is called the multi- 
 plier. 
 
 Tl. The result of the operation is called the product. 
 
 SIGNS. 
 
 72, The sign X is called sign of multiplication, and 
 when placed between two numbers, signifies that they 
 are to be multiplied together. Thus, the expression, 
 
 8x5-=40, 
 is read, eight multiplied hy five equal forty, or eight 
 times five equal forty, or eight times five are forty, 
 
 73. A bar or imrenthesis is used to indicate that 
 several numbers are to be taken as a single number, thus : 
 
 2+6X5--40, or (2-f 6)X5=-40; but 2+6x5=32. 
 
 •74. When two or more numbers are multiplied to- 
 gether to produce a single number, each of the numbers 
 involved is called a factor of the product. Thus, in 
 the expression 3X2X5=30, each of the numbers, 3, 2, 
 5, is a factor of 30. 
 
 The product of the factors is a composite number. 
 
56 
 
 MULTIPLICATION. 
 
 75. It is evident that 3X5 is the same as 5X3. For 
 
 3+3+3+3+3 is the same as 5+5+5. 
 
 76. Any number multiplied by produces 0; and 
 any number multiplied by 1 produces the number itself. 
 
 Thus, 8X0=0, and 8X1=8. 
 
 TABLE OF MULTIPLICATION. 
 
 2x0 = 
 
 
 
 3x0 
 
 = 
 
 4x0 
 
 = 
 
 5x0=0 
 
 2x1 = 
 
 2 
 
 3 X 1 
 
 = 3 
 
 4 X 1 
 
 = 4 
 
 5x1=5 
 
 2x2 = 
 
 4 
 
 3x2 
 
 = 6 
 
 4x2 
 
 = 8 
 
 5 X 2 = 10 
 
 2x3 = 
 
 6 
 
 3x3 
 
 = 9 
 
 4x3 
 
 = 12 
 
 5 X 3 = 15 
 
 2x4 = 
 
 8 
 
 3x4 
 
 = 12 
 
 4x4 
 
 = 16 
 
 5 X 4 = 20 
 
 2x5 = 
 
 10 
 
 3x5 
 
 = 15 
 
 4x5 
 
 = 20 
 
 5 X 5 = 25 
 
 2x6 = 
 
 12 
 
 3x6 
 
 = 18 
 
 4 X 6 
 
 = 24 
 
 5 X 6 = 30 
 
 2x7 = 
 
 14 
 
 3X7 
 
 = 21 
 
 4x7 
 
 = 28 
 
 5 X 7 = 35 
 
 2x8 = 
 
 16 
 
 3x8 
 
 = 24 
 
 4x8 
 
 = 32 
 
 5 X 8 = 40 
 
 2x9 = 
 
 18 
 
 3x9 
 
 = 27 
 
 4x9 
 
 = 36 
 
 5 X 9 = 45 
 
 6x0 = 
 
 
 
 7x0 
 
 = 
 
 8x0 
 
 = 
 
 9x0=0 
 
 6x1 = 
 
 6 
 
 7 X 1 
 
 = 7 
 
 8 X 1 
 
 = 8 
 
 9x1=9 
 
 6x2 = 
 
 12 
 
 7x2 
 
 = 14 
 
 8x2 
 
 = 16 
 
 9 X 2 = 18 
 
 6x3 = 
 
 18 
 
 7x3 
 
 = 21 
 
 8x3 
 
 = 24 
 
 9 X 3 = 27 
 
 6x4 = 
 
 24 
 
 7x4 
 
 = 28 
 
 8x4 
 
 = 32 
 
 9 X 4 = 36 
 
 6x5 = 
 
 30 
 
 7x5 
 
 = 35 
 
 8x5 
 
 = 40 
 
 9 X 5 = 45 
 
 6 X G = 
 
 36 
 
 7x6 
 
 = 42 
 
 8x6 
 
 = 48 
 
 9 X 6 = 54 
 
 6x7 = 
 
 42 
 
 7x7 
 
 = 49 
 
 8x7 
 
 = 56 
 
 9 X 7 = 63 
 
 6x8 = 
 
 48 
 
 7x8 
 
 = 56 
 
 8x8 
 
 = 04 
 
 9 X 8 = 72 
 
 6x9 = 
 
 54 
 
 7x9 
 
 = 63 
 
 8x9 
 
 = 72 
 
 9 X 9 = 81 
 
 PROBIiEM I. 
 
 77. To multiply any number by any other number 
 less than 10. 
 
 EXAINIPLES. 
 
 1. Multiply 357 by 7. 
 
MULTIPLICATIOX. 57 
 
 OPERATION. VERIFICATION. 
 
 357 357 
 
 49 
 
 350 
 2100 
 
 7 357 
 
 357 
 
 357 vide 68 and 75. 
 
 357 
 
 357 
 
 2490 357 
 
 2499 
 
 7 times 7 units of the first order are 49 units of tlie 
 first order; that is, 9 units of i\iQ first order and 4 of the 
 second. 
 
 7 times 5 units of the second order are 35' units of the 
 second order; that is, 5 units of the second order, and 3 
 of the third. 
 
 7 times 3 units of the third order are 21 units of the 
 tliird order; that is, 1 unit of the third order and 2 of 
 the fourth. 
 
 The sum of these partial products is the product 
 required. 
 
 This operation may be contracted thus : 
 
 7 times 7 are 49. Write down the 9 units operation 
 of the first order and mentally add the four oerr 
 units of the second order to the 35 units of 7 
 
 the same order, making 39. Write down 
 
 the 9 units of the second order, and men- 
 tally add the 3 units of the third order to 
 the 21 units of the same order, making 24. 
 Hence, 
 
58 
 
 MULTIPLICATION. 
 
 RULE, 
 
 1. Write the midtiplier under the midtipliccmd, so that 
 units of the same order may stand under each other. 
 
 2. Ifultijyly the right-hand figure of the midtiplicand 
 hy the multiplier^ and set doivn the figure of tlie product 
 representing units of the first order under the column of 
 units of that order, and add the figure representing units 
 of the second order to the product of the second figure of 
 the multiplicand hy the multii^lier. Set down the figure 
 representing units of the second order under the units of 
 the multiplicand of that order, and add the figure repre- 
 senting units of the third order to the next product, and so 
 on till all the figures of the midtiplicand have been mul- 
 tij)lied. The result is the product required. 
 
 (2.) (3.) (4.) (5.) 
 
 Multiply 1736 4530 7106 2400 
 
 By 3 4 5 6 
 
 Ans. 5208 
 
 18120 
 
 35530 14400 
 
 Multiply 303479 
 By 2 
 
 (7.) 
 9854321 
 
 : 7 
 
 (8.) 
 123456789 
 9 
 
 Ans. 606958 
 
 68980247 
 
 1111111101 
 
 9. Multiply 
 
 10. Multiply 
 
 11. Multiply 
 
 12. Multiply 
 
 13. Multiply 
 
 14. Multiply 
 
 456031 by 3. 
 32467 by 8. 
 10054 by 5. 
 999999 by 9. 
 5432 by 7. 
 142857 by 7. 
 
 Ans. 1368093. 
 
 Ans. 259736. 
 
 Ans. 50270. 
 
 A71S. 8999991. 
 
 Ans. 38024. 
 
 Ans. 999999. 
 
MULTIPLICATION. 
 
 59 
 
 15. Multiply 101010 by 9. A71S. 909090. 
 
 16. Multiply 3421 by 4. Ans. 13684. 
 
 17. Multiply 123456789 by 2, 3, 4, 5, 6, 7, 8, 9. 
 
 18. Multiply 987654321 by 2, 3, 4, 5, 6, 7, 8, 9. ^ 
 
 PROB1.EM11. 
 
 78. To multiply any number by a unit of any order, 
 that is, by 10, 100, 1000, etc. 
 
 RULE. 
 
 An7iex as many ciphers to the right of the multiplicand 
 as there are ciphers in the multiplier. — (Vide 38, II.) 
 
 EXAMPLES. 
 
 1. 
 
 Multiply 357 by 1.— (Vide 76 
 
 >') 
 
 Ans. 357, 
 
 2. 
 
 Multiply 357 by 10. 
 
 
 Am. 3570, 
 
 3. 
 
 Multiply 357 by 100. 
 
 
 Ans. 35700, 
 
 4. 
 
 Multiply 358 by 1000. 
 
 
 Ans. 358000, 
 
 5. 
 
 Multiply 3476 by 100. 
 
 
 Ans. 347600, 
 
 6. 
 
 Multiply 35760 by 10. 
 
 
 Ans. 357600, 
 
 7. 
 
 Multiply 473 by lOOOOOO. 
 
 A: 
 
 ns. 473000000, 
 
 8. 
 
 Multiply 473000 by 1000. 
 
 Ans. 473000000, 
 
 9. 
 
 Multiply 473000000 by 1. 
 
 A: 
 
 ns. 473000000, 
 
 PROBIiEM III. 
 
 79. To multiply any number by a figure represent- 
 ing units of any order, 
 
 (1.) Consider the figure as representing units of the 
 first order, and zvrite the numbers as in 77. 
 
 (2.) After multiplying annex the ciphers, and the result 
 will he the j^roduct required. 
 
60 
 
 MULTIPLICATION. 
 
 Multiply 357 
 By 40 
 
 Ans. 14280 
 
 (5.) 
 
 Multiply 4530 
 
 By 
 
 E X A JI P L E S . 
 
 (2.) (3.) 
 357 1736 
 
 200 70 
 
 30 
 
 71400 121520 
 (6.) 
 4530 
 900 
 
 (4.) 
 1736 
 400 
 
 694400 
 
 (7.) 
 3456 
 7000 
 
 Ans. 135900 
 
 24192000 
 A71S. 6321000. 
 A71S. 355300. 
 
 4077000 
 
 8. Multiply 9030 by 700. 
 
 9. Multiply 7106 by 50. 
 
 10. Multiply 303479 by 20000. 
 
 Ans. 6069580000. 
 
 11. Multiply 987654321 by 900000. 
 
 Ans. 888888888900000. 
 
 PROBIiEM IV. 
 
 80. To multiply one number by another. 
 
 EXAMPLES. 
 
 (1.) (2.) 
 
 Multiply ... 357 Multiply . . 
 
 By . . ; . . ^ By ... . 
 
 Vide 77, Ex. 3, 
 
 4530 
 934 
 
 Vide 77, Ex. 1, 2499 
 Vide 79, Ex. 1, 1428 
 Vide 79, Ex. 2, 714 
 
 1812 
 
 Vide 79, Ex. 5, 1359 
 Vide 79,, Ex. 6, 4077 
 
 Ans. 
 
 88179 
 
 Ans. 
 
 4231020 
 
 Remark. — The cipher of 79, Ex. 1, need not appear in (he opera- 
 tion, and so of the ciphers in the other examples referred to. Hence, 
 
 I 
 
MULTIPLICATIOX. 
 
 61 
 
 1. Write iJie niimhers so that the riglit-ltand significant 
 figures of the multiplicand and multiplier may stand 
 under each other. 
 
 2. Multiply the multiplicand hy each figure of the mul- 
 tiplier, placing the first figure of each product directly 
 tinder the figure used in multiplying. 
 
 3. Add the several products together and annex to their 
 sum all the ciphers on the right of both factors. The 
 result is the product required. 
 
 Remark 1. — The second point of the rule does not apply to the 
 ciphers on the right of either factor. 
 
 Multiply 
 
 By . 
 
 Ans. 
 
 (3.) 
 
 (4.) 
 
 1476 
 470 
 
 Multiply . 
 By . . 
 
 . . 34300 
 . . 4310 
 
 10332 
 5904 
 
 343 
 
 1029 
 1372 
 
 fip.q790 
 
 Ans. . . 147833000 
 
 Remark 2. — When ciphers occur between the significant figures 
 of the multiplier, their product into the multiplicand need not ap- 
 pear in the operation. 
 
 (5.) 
 
 (6.) 
 
 Multiply 
 By. . 
 
 . 459 
 . 307 
 
 3213 
 1377 
 
 Multiply 
 By . , 
 
 2134 
 5004 
 
 8536 
 10670 
 
 Ans 140913 Ans. 
 
 7. Multiply 46834 by 4060. 
 
 8. Multiply 47042 by 47042. 
 
 . . . 10678536 
 Ans. 190146040. 
 Ans. 2212949764. 
 
62 
 
 MULTIPLICATION. 
 
 9. Multiply 123 by 125. 
 
 10. Multiply 328 by 67. 
 
 11. Multiply 75432 by 47. 
 
 12. Multiply 678954 by 24. 
 
 13. Multiply 789563 by 570. 
 
 14. Multiply 1579126 by 1710. 
 
 15. Multiply 67853 by 8765. 
 
 16. Multiply 3678543 by 4567. 
 
 17. Multiply 492 by 625. 
 
 18. Multiply 1312 by 335. 
 
 19. Multiply 603456 by 94. 
 
 20. Multiply 1357908 by 144. 
 
 21. Multiply 2368689 by 190. 
 
 22. Multiply 8432 by 6350. 
 
 23. Multiply 27496 by 1658. 
 
 24. Multiply 82488 by 555. 
 
 81. When one or both factors 
 
 Ans. 15375. 
 
 Ans. 21976. 
 
 Ans. 3545304. 
 
 Ans. 16294896. 
 
 Ans. 450050910. 
 
 ^Tis. 2700305460. 
 
 Ans. 594731545. 
 
 Ans. 16799905881. 
 
 A71S. 307500. 
 
 Ans. 439520. 
 
 Ans. 56724864. 
 
 Ans. 195538752. 
 
 Ans. 450050910. 
 
 Ans. 53543200. 
 
 A71S. 45588368. 
 
 Ans. 45780840. 
 
 are decimals. 
 
 EXAMPLES. 
 
 (1-) 
 
 Multiply 357 
 
 By ..... . 24.7 
 
 2499 
 1428 
 714 
 
 Ans 8817.9 
 
 In this example, since the mul- 
 tiplier is decreased ten times, 
 (vide 38, II,) the product ought 
 to be decreased ten times, which 
 is done by placing one figure 
 to the right of the decimal 
 point. 
 
 (2.) 
 
 Multiply 85.7 
 
 By 24.7 
 
 2499 
 1428 
 714 
 
 Ans 881.79 
 
 In this example, since the mul- 
 tijDlicand is also decreased ten 
 times, (vide 80, Ex. 1,) the pro- 
 duct ought to be decreased ten 
 times more than in Ex. 1, which 
 is done by putting the decimal 
 point one place further to the left 
 
MULTIPLICATION. 
 
 63 
 
 Hence : 
 
 (1.) Proceed 'precisely as in 80. 
 
 (2.) Place the decimal ijoint in the lyroduct so as to cause 
 as many figures to stand on its right as there are figures 
 on the right of the point in both the factors, supplying 
 any deficiency by prefixing ciphers. 
 
 (3.) (4.) (5.) 
 
 Multiply 45.9 4.59 .459 
 
 By 3.07 3.07 .307 
 
 3213 3213 
 
 3213 
 
 1377 1377 
 
 1377 
 
 Ans. 140.913 14.0913 
 
 .140913 
 
 (6.) (7.) 
 
 (8.) 
 
 Multiply .0008 .00008 
 
 .000008 
 
 By .0007 .007 
 
 .07 
 
 Ans. .00000056 .00000056 
 
 .00000056 
 
 (9.) 
 
 (10.) 
 
 Multiply .0716 
 
 .1234 
 
 By 1.326 
 
 1234 
 
 Ans. .0949416 152.2756 
 
 11. Multiply $3.57 by 7. 
 
 Ans. $24.99. 
 
 12. Multiply ^3.57 by 40. 
 
 Ans. $142.80. 
 
 13. Multiply $3.57 by 200. 
 
 Ans. $714.00. 
 
 14. Multiply ?3.57 by .7 
 
 Ans. $2,499. 
 
 15. Multiply 171.61 by 365. Ans. $26137.65. 
 
 16. Multiply $0.93 by 63. 
 
 Ans. $58.59. 
 
 17. Multiply $13.75 by 43. 
 
 Ans. $591.25. 
 
 18. Multiply $4.68 by 169. 
 
 Ans. $790.92. 
 
64 
 
 MULTIPLICATION. 
 
 19. Mult 
 
 20. Mult 
 
 21. Mult 
 
 22. Mult: 
 
 23. Mult 
 
 24. Mult 
 
 25. Mult 
 
 26. Mult 
 
 27. Mult 
 
 28. Mult 
 
 29. Mult 
 
 30. Mult 
 
 31. Mult 
 
 32. Mult 
 
 ply 
 ply 
 
 ply 
 
 ply 
 ply 
 ply 
 ply 
 ply 
 ply 
 ply 
 ply 
 ply 
 ply 
 
 ,^0.057 by 84C). 
 1132.55 by 369"! 
 ^0.299 by 69. 
 §69.748 by 144. 
 §3.75 by 47. 
 $6.79 by 163. 
 §1.375 by 19. 
 §4.57 by 18. 
 §15.89 by 9. 
 §0.75 by 125. 
 §58.90 by 45. 
 §0.058 by .07 
 §0.904 by .025 
 §1.287 by .9 
 
 Ans, §48.222. 
 
 Ans. §48910.95. 
 
 Ans. §20.631. 
 
 Ans. §10043.712. 
 
 Ans. §176.25. 
 
 Ans. §1106.77. 
 
 Ans. §26.125. 
 
 Ans. §82.26 
 
 Ans. §143.01. 
 
 A71S. §93.75. 
 
 Ans. §2650.50. 
 
 Ans. §0.00406. 
 
 Ans. §0.022600. 
 
 Ans. §1.1583. 
 
 82. In Multiplication the j-^'^'oduct is always of the 
 same name as the multiplicand^ and the multiplier in the 
 operation must be considered as simply a number without 
 name. Thus: §75X47=-§3525.— (Vide 166.) 
 
 PKACTICAL APPLICATION. 
 
 1. What will 47 oxen cost, at §75 each? ' 
 
 Ans. $3525. 
 
 2. If a man walk 23 miles a day, how far will he 
 walk in 17 days? Ans. 391 miles. 
 
 3. If a vessel sail 451 miles a day, how far will it 
 sail in 9 days ? Ans. 4059 miles. 
 
 4. What will 495 yards of cloth cost, at 11 dollars a 
 yard?— (Vide 75.) Am. §5445. 
 
 5. What will 569 hogsheads of molasses cost at $37 
 each? Ans. §21053. 
 
MULTir LIGATION. '65 
 
 6. What will 451 bales of cotton cost at |53 per bale? 
 
 Alls. §23903. 
 
 7. There arrived in market 18 wagons during one 
 ■week, each wagon containing 6 bales of cotton, worth 
 |56 per bale. How much was the cotton worth ? 
 
 Ans. §6048. 
 
 8. Bought 8 bales of cotton, each bale containing 
 530 pounds, worth §0.13 per pound. How much did I 
 pay for it? . A^is. §551.20. 
 
 9. I purchased at one time 3 bales of cotton, each 
 bale weighing 554 pounds, at §0.11 per pound; at 
 another time, 5 bales, each weighing 535 pounds, at 
 §0.12 per pound. What was the whole worth ? 
 
 Ans. §503.82. 
 
 10. I sold the cotton in the preceding example at 
 §0.115 per pound. Did I make or lose, and how much? 
 
 Ans. lost §5.065. 
 , 11. I bought 27 hogsheads of molasses, each contain- 
 ing 63 gallons, at §0.53 per gallon. How^ much did 1 
 pay for it? Ans. §901.53. 
 
 12. I sold a sack of hops, weighing 396 pounds, at 
 §0.113 per pound. How much did I get for it? 
 
 Ans. §44.748. 
 
 13. What cost 5342 barrels of flour at §8.50 per 
 barrel ? A71S. §45407. 
 
 14. One chest of tea contains 69 pounds, and costs 
 §0.299 per pound; another chest contains 74 pounds, 
 and costs §0.274 per pound. How much is lost by sell- 
 ing the whole at §0.28 per pound? Ans. §0.867 
 
 15. What cost 169 boxes of oranges at §6.71 per 
 box? Ans. §1133.99. 
 
66 MULTIPLICATIOX. 
 
 16. What cost 357 barrels of potatoes at §2.47 per 
 barrel? Ans. ^SS1.19. 
 
 17. I purchase 243 casks of butter, each containing 
 57 pounds, at §0.34 per pound, and sell the same at 
 S0.40 per pound. How much do I gain ? 
 
 Ans, §831.06. 
 
 18. My vineyard produces 5342 bottles of wine, at a 
 cost of §0.37 per bottle, and I retail the wine at §1.25 
 per bottle. How much do I clear? Ans. §4700.96. 
 
 19. Two passenger trains of cars meet at a station, 
 and each train runs, on an average, 37 miles an hour. 
 Suppose them to start at the same time, how far apart 
 will they be at the end of 17 hours ? 
 
 Ans. 1258 miles. 
 
 20. A passenger train and a freight train of cars 
 start from a given station and run in the same direction. 
 The passenger train moves at the rate of 37 miles an 
 hour, and the freight train 19 miles an hour. How far 
 apart will they be at the end of 13 hours? 
 
 A71S. 234 miles. 
 How far apart had they run in different directions? 
 
 Ans. 728 miles. 
 
 21. A drover bought 357 oxen, at the rate of §47 a 
 head. In going to market, 12 oxen fell through a bridge 
 and were killed. The cost of driving the remainder to 
 market was §5 a head. The oxen were then sold at an 
 average of §49 each. What was the loss ? 
 
 A71S. §1599. 
 
 22. How much would have been gained by selling the 
 oxen in the preceding example at §55 a head? 
 
 Ans.Un. 
 
MULTIPLICATION. 
 
 67 
 
 MERCHANTS' BILLS. 
 
 83. A Bill is a written statement of articles bought 
 or sold, their prices, entire cost, etc. 
 
 Buffalo, April 26, 1865. 
 Messrs. J. H. Reed & Co., Chicago, 111., 
 
 Bought of D. Ransom & Co., 
 Terms cash. 121 Main Street. 
 
 10 
 
 5 
 
 Gross Trask's Magnetic Ointmeut, @ $21 
 
 " Ransom's Hive Syrup Tolu, @ $24 
 
 . $210 
 . 120 
 . 160 
 . 160 
 . 95 
 
 00 
 
 00 
 
 5 
 
 5 
 
 " Mrs. Winslow's Soothing Syrup, @ $32 
 
 " Brown's Bronchial Troches, @ $32 
 
 00 
 00 
 
 5 
 
 " Judson's Mt. Herb Pills, @ $19 
 
 00 
 
 
 
 
 
 $745 
 
 00 
 
 Received payment. , 
 
 D. Ransom & Co., 
 By John M. Sabin, Cl'k. 
 10 Cases per Lake Shore Railroad. 
 
 (2.) 
 New York, Jan. 1, 1865. 
 Mr. James Bryant, Mobile, Ala., 
 
 Bought of A. Smith & Co. 
 
 24 
 
 Pounds of Tea, @ $0.63 
 
 $15jl2 
 
 55 
 
 Barrels of Potatoes, @ $5 
 
 275 00 
 
 98 
 
 Pounds of Raising @ $0.45 
 
 41 
 
 807 
 
 85 
 
 85 
 
 Barrels of Shad, @ $9.50 
 
 50 
 
 
 
 
 
 ^1139 
 
 47 
 
 6'p^i-'i 
 
 ^ 
 
68 MULTIPLICATION. 
 
 (8.) 
 
 New York, Jan. 1, 1865. 
 Mr. Lewis Rollings, 
 
 • To Otis Howe & Co., Dr. 
 
 1864. 
 
 Sept. 13. To 30 doz. Wool Hose @ $1.50 
 
 " 13. " 6 prs. Gloves @ $0.75 
 
 " 13. " 4 doz. Napkins @ $1.20 
 
 Oct. 12. " 4 " Shirt Bosoms @ $2.40 
 
 " 12. " 22 yds. Drilling : .@ $0.10 
 
 " 12. " 6 " Broadciotli @ $4.00 
 
 $90.10 
 Received payment. 
 
 S. Billings, 
 For Otis Howe & Co. 
 
 - • (4.) 
 
 ^ St. Louis, July 15, 1865 
 Mr. S. H. DeCamp. 
 
 To A. T. Campbell, Dr. 
 
 1865. 
 
 Jan. 5. To 12 doz. Scythes @ $10.00 
 
 " 9. " 6J " Hoes @ $7.00 
 
 Feb. 1. " 10 " Rakes @ $1.80 
 
 March 4. " 3 Plows @ $11.00 
 
 " 4. " 7 doz. Pitchforks @ $9.50 
 
 " 1. " 9 « Padlocks @ $24.00 
 
 « 12. " 1 Coffee-mill @ $5.00 
 
 July 15. Settled by due bill $504.00 
 
 A. T. Campbell. 
 
DIVISION. 69 
 
 DIVISION. 
 
 84. Division is the operation of finding how many 
 times one number is contained in another. 
 
 85. The number to be divided is called the dividend. 
 80. The member by which to divide is called the di- 
 visor. 
 
 8T. The number of times the divisor is contained in 
 the dividend is called the quotient. 
 
 88. IVlien there is a number left after dividing, it is 
 called the remainder. 
 
 SIGN. 
 
 89. The sign -^- is known as the sigri of division, and 
 when placed between two numbers, it signifies that the 
 former is to be divided by the latter. Thus, the ex- 
 pression, 
 
 40-f-5=8, 
 
 is read, forty divided by five equal eight, or five in forty 
 eight times. 
 
 OO. Operations in division are carried on under two 
 forms. Thus : 
 
 (1.) (2.) (3.) 
 
 5)13 5)13(2 13 1 5 
 
 -:. o 10 10 T 
 
 2-3 — — 
 
 3 3 
 
70 
 
 DIVISION. 
 
 The first is known as the method of Short Division. 
 
 The second is known as the method of Long Di- 
 vision. 
 
 In either of these forms, 13 is the dividend, 5 is the 
 divisor, 2 is the quotient, and 3 the remainder. 
 
 In the operations of Long Division, the divisor is some- 
 times written as in (3.) 
 
 Ol. The remainder added to the product of iJie divisor 
 and quotient produces the dividend. Thus : 
 
 5x2+3=13. 
 
 (1.) Any number divided by itself produces 1 ; and 
 (2.) Any number divided by 1 produces the number 
 itself. Thus, 7^7=1, and 7-^1=7. 
 
 
 TABLE OF 
 
 division. 
 
 
 2-4-2=1 
 
 3 -f-3 = 1 
 
 4 -r-4= 1 
 
 5-4-5 = 1 
 
 4-^2 = 2 
 
 6 -r-3 = 2 
 
 8-^4 = 2 
 
 10 -^ 5 = 2 
 
 6h-2 = 3 
 
 9^3 = 3 
 
 12 -T-4 = 3 
 
 15 -=- 5 = 3 
 
 8-^2 = 4 
 
 12 -H 3 = 4 
 
 16 --4 = 4 
 
 20 H- 5 = 4 
 
 10 -i- 2 = 5 
 
 15 -V- 3 = 5 
 
 20 ^ 4 = 5 
 
 25 -4- 5 = 5 
 
 12 -V- 2 = 6 
 
 18 -V- 3 == 6 
 
 24 -H 4 = 6 
 
 30 -4- 5 = 6 
 
 14 ~ 2 = 7 
 
 21 -4- 3 = 7 
 
 28 ~ 4 = 7 
 
 35 -4- 5 = 7 
 
 16 -^ 2 = 8 
 
 24 -^ 3 = 8 
 
 32 -T- 4 = 8 
 
 40 -4- 5 = 8 
 
 18 -r- 2 = 9 
 
 27 -r- 3 = 9 
 
 36 ~ 4 = 9 
 
 45 -4- 5 = 9 
 
 6 -=-6 = 1 
 
 7 -T-7 = 1 
 
 8-4-8 = 1 
 
 9-4-9 = 1 
 
 12 -- 6 = 2 
 
 14 -^ 7 = 2 
 
 16 ~ 8 = 2 
 
 18 -4- 9 = 2 
 
 18 -^ 6 = 3 
 
 21 -4- 7 = 3 
 
 24 H- 8 = 3 
 
 27 -T- 9 = 3 
 
 24 ^ 6 = 4 
 
 28 -^ 7 = 4 
 
 32 -4- 8 = 4 
 
 36 -4- 9 = 4 
 
 30 ^ 6 = 5 
 
 35 -i- 7 = 5 
 
 40 -4- 8 = 5 
 
 45 -4- 9 = 5 
 
 36 -T- 6 = G 
 
 42 -v- 7 = 6 
 
 48 -T- 8 = 6 
 
 54 -4- 9 = 6 
 
 42 ~ 6 = 7 
 
 49 -V- 7 = 7 
 
 56 -4- 8 = 7 
 
 63 -4- 9 = 7 
 
 48 -f- 6 = 8 
 
 56 -^ 7 = 8 
 
 64 -^ 8 = 8 
 
 72 -4- 9 = 8 
 
 54 -r- 6 = 9 
 
 63 -^ 7 = 9 
 
 72 -:- 8 = 9 
 
 81 -^ 9 = 9 
 
DIVISION. 71 
 
 SHORT DIVISION. 
 02. To divide a number by a number less than 10. 
 
 
 EXAMPLES. 
 
 
 (1.) 
 
 (2.) 
 
 (3.) 
 
 7)2499 
 
 5)35530 
 
 3)3698 
 
 357 7106 1232-2 
 
 7 is contained in 24 units of the tldrd order, 3 units 
 of the same order and 3 over. Write down the 3 units 
 of the third order, and prefix the 3 over to the 9 units 
 of the second order, making 39. 
 
 7 is contained in 39 units of the second order 5 units 
 of the same order and 4 over. Write down the 5 units 
 of the second order, and prefix the 4 over to the 9 units 
 of the first order, making 49. 
 
 7 is contained in 49 units of the first order , 7 units of 
 the same order. Then 357 is the quotient. — (Vide 77, 
 Ex. 1.) 
 
 In the second example, we may then say at once, 5 in 
 35 give 7; 5 in 5 give 1 ; 5 in 3 give 0; 5 in 30 give 6. 
 —(Vide 77, Ex. 4.) 
 
 In the third example, 3 in 3 give 1 ; 3 in 6 give 2 ; 3 
 in 9 give 3 ; 3 in 8 give 2, and 2 remainder. Hence-, 
 
 B u L E . 
 
 1. Write the divisor on the left of the dividend, as in 
 90,(1.) 
 
 2. Find hoiv many times the divisor is contained in 
 the left-hand figure, or, if that is smaller than the divisor^ 
 
72 
 
 DIVISION. 
 
 in two of the left-hand figures of the dividend, mid ivrite 
 the quotient diredhj under the figure of the lowest order 
 used. 
 
 3. Prefix the remainder to the figure of the next lower 
 order, and divide as before, continuing the tvorJc till all 
 the figures have been used. The result is the quotient re- 
 quired. 
 
 Remark 1. — The final remainder, if tliere is any, is to be written 
 as in 90, (1.) 
 
 (4.) (5.) (6.) 
 
 2)606958 7)68980247 9)1111111101 
 
 303479 Vide 77, Ex. 7. 
 
 Vide 77, Ex. 
 
 7. 
 
 Divide 1368093 by 3. 
 
 Ex.9. 
 
 .8. 
 
 Divide 259736 by 8. 
 
 Ex. 10. 
 
 9. 
 
 Divide 8999991 by 9. 
 
 Ex. 12. 
 
 10. 
 
 Divide 5208 by 3. 
 
 Ex.2. 
 
 11. 
 
 Divide 38024 by 7. 
 
 Ex. 13. 
 
 12. 
 
 Divide 999999 by 7. 
 
 Ex. 14. 
 
 13. 
 
 Divide 13684 by 4. 
 
 Ex. 16. 
 
 14. 
 
 Divide 340974 by 9. • 
 
 Ans. 37886. 
 
 15. 
 
 Divide 10101 by 3. 
 
 Ans. 3367. 
 
 16. 
 
 Divide 270192 by 6. 
 
 'Ans. 45032. 
 
 17. 
 
 Divide 3530380 by 5. 
 
 Ans. 706076. 
 
 18. 
 
 Divide 4236456 by 6. 
 
 Ans. 706076. 
 
 19. 
 
 Divide 4942532 by 6. 
 
 Ans. 
 
 20. 
 
 Divide 818181 by 9. 
 
 Ans. 90909. 
 
 21. 
 
 Divide 103701 by 3. 
 
 Ans. 34567. 
 
 22. 
 
 Divide 2751840 by 1, 2, 3, 4, 
 
 etc. 
 
 23. 
 
 Divide $4725 by 4. 
 
 Jlem. §1. 
 
 24. 
 
 Divide 13257 by 8. 
 
 Pern. |1. 
 

 DIVISION. 
 
 
 25. 
 
 Divide $4321 by 2. 
 
 Rem. $1. 
 
 26. 
 
 Divide 16721 by 3. 
 
 Rem. $1. 
 
 27. 
 
 Divide $3454 by 5. 
 
 Rem. $4. 
 
 28. 
 
 Divide $2348 by 7. 
 
 Rem. $3. 
 
 29. 
 
 Divide 4532 by 9. 
 
 Rem. 5. 
 
 30. 
 
 Divide 17623 by 3. 
 
 Rem. 1. 
 
 31. 
 
 Divide 30407 by 5. 
 
 Rem. 2. 
 
 32. 
 
 Divide 40321 by 2. 
 
 Rem. 1. 
 
 83. 
 
 Divide 76541 by 9. 
 
 'Rem. 5. 
 
 34. 
 
 Divide $451 by 5. 
 
 Rem. $1. 
 
 73 
 
 * Remark 2. — The divisor may be written under the remainder 
 with a line between them, the whole being considered as forming 
 a part of the quotient. Thus, (vide 134,) 
 
 (35.) (36.) (37.) 
 
 2)7 3)16 8)245 
 
 3i 5i 81f 
 
 The answer of (35) is read three and one half. — 
 (Vide 8.) 
 
 The answer of (36) is read five and one third. 
 
 The answer of (37) is read eighty-one and two thirds. 
 
 88. 
 
 Divide 4725 by 4. 
 
 Ans. 1181 J. 
 
 39. 
 
 Divide 8890 by 4. 
 
 Ans. 972f . 
 
 40. 
 
 Divide 5341 by 5. 
 
 Ans. 1068J. 
 
 41. 
 
 Divide 8459 by 5. 
 
 Ans. 691f. 
 
 42. 
 
 Divide 3005 by 6. 
 
 Ans. 5001. 
 
 43. 
 
 Divide $451 by 2. 
 
 Ans. $225J. 
 
 44. 
 
 Divide $650 by 3. 
 
 Ans. $216f . 
 
 45. 
 
 Divide $121 by 4. 
 
 Ans. $301. 
 
 46. 
 
 Divide $154 by 4. 
 
 Ans. $38f . 
 
 47. 
 
 Divide $3459 by 7. 
 
 7 
 
 Ans. $4944. 
 
74 DIVISION. 
 
 48. Divide P258 by 8. ^^s. *407f . 
 
 49. Divide $1111 by 9. Ans. |123|. 
 
 50. Divide $2222 by 5. Ans. §444f . 
 
 51. Divide $3333 by 6. Ans. $555|. 
 
 52. Divide $4444 by 7. Ans. $634f . 
 
 53. Divide $5005 by 8. Ans. $625|. 
 
 54. Divide $3003 by 9. Ans. $333f . 
 55.- Divide $221 by 2. Ans. $110|. 
 
 Remark 8;— r-Ciphers may be annexed to tlie dividend, and the 
 division continued till there is no remainder, or it may terminate 
 at any convenient point. The figures of the quotient, after annexing 
 ciphers, are decimals. Thus, 
 
 (56.) (57.) (58.) 
 
 4)375.00 3)$5678.000 9)$4573.000 
 
 93.75 $1892.6661 $508,111^ 
 
 The answer of (57) is 1892 dollars 66 cents 6 mills 
 and two-thirds of a mill. 
 
 59. Divide 4725 by 4. ^ns. 1181.25. 
 
 60. Divide 3890 by 4. Ans. 972.5. 
 
 61. Divide 5341 by 5. ^ws. 1068.2. 
 
 62. Divide 3459 by 5. Ans. 691.8. 
 
 63. Divide 3005 by 6. Ans. 500.83|. 
 
 64. Divide $451 by 2. Ans. $225.50. 
 
 65. Divide $4357 by 6. ^m. $726.166|. 
 
 66. Divide $1 by 2, 3, 4, 5, 6, 7, 8, 9. 
 
 Ans. $0.50, $0,331, $0.25, $0.20, $0.166|, $0.142f, 
 
 $0,125, $0.111f 
 
 03.' When the divisor, dividend, or both contain 
 decimals, 
 
 1. If (he numher of decimal figures in the divisor 
 
DIVISION. 75 
 
 exceeds that in the dividend^ make it equal in both hy 
 annexing ciphers as decimals to the dividend. 
 
 2. Divide as in 92, and place the decimal point in the 
 quotient, so as to cause as many figures to stand on its 
 right as all the decimal figures used in the dividend exceed 
 those in the divisor, supplying any deficiency of quotient 
 figures hy prefixing ciphers. — (Vide 81.) 
 
 EXAMPLES. 
 
 Divide 249.9 by 7. 
 
 OPERATION. 
 
 7)249.9 
 
 35.7 
 
 Here the dividend has one decimal figure, the divisor 
 none. 
 
 2. Divide 2499 by .7. 
 
 OPERATION. 
 
 .7)2499.0 (Vide 93, 1.) 
 3570 
 
 Here the dividend has one decimal figure, and the 
 divisor one. The quotient is an integral number. 
 
 3. Divide 24.99 by .07. 
 
 OPERATION. 
 
 .07)24.99 
 
 357 
 Here the dividend has two decimal figures, and 
 
76 DIVISION. 
 
 the divisor two, and the quotient is then an integral 
 number. 
 
 4. Divide 249.9 by .0007. 
 
 OPERATION. 
 
 .0007)249.9000 (Vide 93, 1.) 
 
 357000 
 Here the dividend has four decimal figures, and"^ .3 
 divisor four. „ 
 
 5. Divide .2499 by 7. 
 
 OPERATION. 
 
 7).2499 
 
 .0357 (Vide 93, 2, last clause.) 
 
 Here the dividend has four decimal figures, the divisor 
 none. One cipher had to be prefixed to the three quo- 
 tient figures. 
 
 6. Divide 52.08 by .03. 
 
 
 OPERATION. 
 
 
 
 .03)52.08 
 
 
 
 1736 Ans. 
 .03 
 
 
 
 52.08 Proof 
 
 '. (Vide 91.) 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 
 Divide 375 by A 
 
 Divide 430 by .008 
 Divide 4725 by .4 
 Divide 32570 by .08 
 Divide 43.21 by .002 
 
 Ans. 937.5 
 
 Ans. 53750, 
 
 Ans. 11812.5 
 
 Ans. 407125. 
 
 Ans. 21605. 
 
DIVISION. 77 
 
 12. Divide 7.621 by .0003. Am. 25403.331. 
 
 13. Divide .00214 by .002. Am. 1.07, 
 
 14. Divide .03017 by .007. Am. 4.31. 
 
 15. Divide 93.276 by .007. Am. 13325.142f . 
 
 16. Divide 48.35 by .005. Am. 9670. 
 
 17. Divide 14535 by 9, .9, .09, .009, .0009, .00009, 
 and .000009. Am. 1615, 16150, 161500, etc. 
 
 B4:. To divide an integral number or decimal by a 
 UMt of any order, that is, by 10, 100, 1000, etc., 
 
 T Remove the decimal point of the dividend as many 
 figures to the left as are indicated hy the ciphers in the 
 divisor.— {VidiQ 38, II.) 
 
 EXAMPLES. 
 
 1. Divide 3570 by 10, (vide 78, Ex. 2.) 
 
 Am. 357. 
 
 2. Divide 35700 by 100. Am. 357. 
 
 3. Divide 357 by 10. Am. 35.7. 
 
 4. Divide 357 by 100. Am. 3.57. 
 
 5. Divide 35.7 by 10. Am. 3.57. 
 
 6. Divide 31.4 by 100. Am. .314. 
 
 7. Divide §451.30 by 10. Am. $45.13. 
 
 8. Divide §4321 by 100. Am. §43.21. 
 
 9. Divide §23456 by 1000. Am. §23.456. 
 
 10. Divide §1.00 among 10 men, 100 men, 1000 
 men. Am. §0.10, §0.01, §0.001. 
 
 11. Divide §45.20 equally among 10 men; 100 men; 
 1000 men. -Aws. §4.52; §0.452; §0.0452. 
 
 12. Divide §3120 equally among 10000 men. 
 
 A71S. §0.312. 
 
78 
 
 DIVISION. 
 
 LONa DIVISION. 
 
 05. To divide a number by any number greater 
 than 10. 
 
 EXAMPLES. 
 
 1. Divide 1716 by 11. 
 
 (1.) We find how many times 
 11 is contained in 17, and write 
 the quotient figure 1 to the right 
 of the dividend. 
 
 (2.) Multiply the divisor 11 
 by the quotient figure 1, and 
 place the product 11 under 17. 
 
 (3.) Subtract 11 from 17, and 
 
 OPERATION. 
 
 11)1716(156 
 
 11 
 
 61 
 55 
 
 66 
 
 Am. 
 
 place the remainder 6 under 11. 
 
 (4.) Bring down the next q 
 
 figure 1 of the dividend to the 
 
 right of the 6, and proceed with the result 61 exactly as 
 with 17. That is, 
 
 (1.) 11 in 61 are 5, which is placed in the quotient. 
 
 (2.) Then 5 times 11 are 55, which is placed under 
 61. - 
 
 (3.) 55 from 61 are 6. 
 
 (4.) Bring down the next figure 6 of the dividend. 
 In the resulting number 66 the divisor 11 is contained 
 
 Hence, 
 
 6 times, with no remainder. 
 
 RULE, 
 
 (1.) Find how many times the whole divisor is con- 
 tained in the same niuiiher of figures on (he left of the 
 
DIVISION. 79 
 
 dividend considered as representing so many units ; or, 
 if this number is smaller than the divisor, find how 
 many times the divisor is contained in a number of figures 
 of the dividend^ greater by one than the number of figures 
 in itself, and place the quotient on the right of the divi- 
 dend. 
 
 (2.) Multiply the whole divisor by this quotient figure, 
 and place the p)roduct under the figures of the dividend 
 just compared ivith the divisor. 
 
 (3.) Subtract the product from the figures above it. 
 
 (4.) To the right of the difference bring doion the next 
 figure of the dividend. 
 
 (5.) If the new dividend is noiv smaller titan the divi- 
 sor, place a cipher in the quotient, and bring down the next 
 figure of the dividend; but if the dividend is equal to or 
 larger than the divisor, proceed in obtaining the second 
 figure of the quotient as ivith the first, and so on till all 
 the figures are brought down. 
 
 Remark 1. — Notice that no product can be greater than the 
 figures of the dividend above it. 
 
 Remark 2. — Notice that no difference can be equal to or greater 
 than the divisor. 
 
 2. Divide 1716 by 12. Ans. 143. 
 
 3. Divide 1716 by 13. Ans. 132. 
 
 4. Divide 3360 by 14. Ajis. 240. 
 
 5. Divide 3360 by 15. Ans. 224. 
 
 6. Divide 3360 by 16. Ans. 210. 
 
 7. Divide 7429 by 17. Ans. 4S7. 
 
 8. Divide 7429 by 19. Ans. 391. 
 
 9. Divide 7429 by 23. Ans. 323. 
 10. Divide 26691 by 21. Ans. 1271. 
 
80 DIVISION. 
 
 11. Divide 7371 bj 91. Arts. SI. 
 
 (12.) (13.) 
 
 91)7371(81 184)56488(307 
 
 728 552 
 
 91 1288 (Vide Rule, 5.) 
 
 91 1288 
 
 
 
 14. Divide 56488 by 92. Ajis. 614. 
 
 15. Divide 84487 by 97. Ans. 871. 
 
 16. Divide 24108 by 98. Ans, 246. 
 
 17. Divide 24108 by 246. Ans. 98. 
 
 18. Divide 5768 by 56. A^is. 103. 
 
 19. Divide 48071 by 53. . Ans. 907. 
 
 20. Divide 2448 by 16, 17, and 18. 
 
 Ans. 153, 144, and 136. 
 
 21. Divide 6072 by 22, 23, and 24. 
 
 Ans. 276, 264, and 253. 
 
 22. Divide 5544 by 36, 44, and 56. 
 
 Ans. 154, 126, and 99. 
 
 23. Divide 12259 by 13, 23, and 41. 
 
 Ans. 943, 533, and 299. 
 
 (24.) (25.) 
 
 509)461663(907 999)708291(709 
 
 4581 6993 
 
 3563 8991 
 
 3563 8991 
 
 26. Divide 73647 by 147 and 167. 
 
 Ans. 501 and 441. 
 
DIVISION. 81 
 
 27. Divide 36942 by 131 and 141. 
 
 Ans. 282 and 262. 
 
 28. Divide 526587 by 191 and 919. 
 
 Ans. 2757 and 573. 
 
 29. Divide 3203055 by 711 and 901. 
 
 Ans, 4505 and 3555. 
 
 30. Divide 193581 by 137 and 157. 
 
 Ans. 1413 and 1233. 
 
 31. Divide 404278 by 431 and 134 
 
 Ans. 938 and 3017. 
 
 32. Divide 670592745 by 54321. 
 
 OPERATION. 
 
 670592745154321 
 54321 12345 
 
 127382 
 
 108642 (Yide 90, (3.) 
 
 187407 
 162963 
 
 244444 
 
 217284 
 
 271605 
 271605 
 
 33. Divide 45780840 by 82488. Ans. 555- 
 
 34. Divide 1234549380 by 12345. Ans. 100004. 
 
 35. Divide 497477808387 by 987. 
 
 Ans. 504030201. 
 
 36. Divide 49419533647761876 by 9876. 
 
 Ans. 5004003002001. 
 
82 DIVISION. 
 
 Remark 3. — If there are cipliers on the right of the divisor, 
 they may be neglected in the operation, together with a like num- 
 ber of figures on the right of the dividend. 
 
 37. Divide 34705 by 
 
 700. 
 
 
 38. Divide 34705 by 
 
 760. 
 
 
 OPERATIONS 
 
 3. 
 
 (37.) 
 
 
 (38.) 
 
 7'00)347'05 
 
 
 76^0)3470'5(45 
 
 49—405 
 
 
 304 
 
 430 
 380 
 
 505 
 
 The remainder of (37) is 405, and that of (38) is 505. 
 
 39. Divide 3521 by 200. Rem, 121. 
 
 40. Divide 3521 by 30. Bern. 11. 
 
 41. Divide 4561 by 40. Rem. 1. 
 
 42. Divide 3457 by 50. Rem. 7. 
 
 43. Divide 7654 by 70. Rem 24. 
 
 44. Divide 3420 by 90. Arts. 38. 
 
 45. Divide 1716000 by 12000. Ans. 143. 
 
 46. Divide 1716000 by 130000. . Rem. 26000. 
 
 47. Divide 3360000 by 1400000. Rem. 560000. 
 
 48. Divide 73647000 by 1470000. Rem. 147000. 
 
 49. Divide 123456789 by 2000000. 
 
 Rem. 1456789. 
 
 Remark 4. — In decimals observe the Rule in 93 and 92, Rem. 3. 
 
 50. Divide 143 by 25. 
 
 51. Divide 143 by 125. 
 

 DIVISION. 1 
 
 (50.) 
 25)143.00(5.72 
 125 
 
 OPERATIONS. 
 
 (51.) 
 125)143.000(1.144 
 125 
 
 180 
 175 
 
 50 
 50 
 
 180 
 125 
 
 550 
 
 500 
 
 
 
 500 
 500 
 
 83 
 
 52. Divide 206.166492 by 4.123. 
 
 53. Divide .102048 by 3189. 
 
 OPERATIONS. 
 
 (52) (53) 
 
 206.166492| 4.123 .102048| 3189 
 
 206.15 50.004 9567 .000032 
 
 
 16492 (Yide 90, (3.) 6378 
 
 
 16492 6378 
 
 54. 
 
 Divide 4567 by 25, 125, and 625. 
 
 - 
 
 Ans. 182. 68, etc. 
 
 55. 
 
 Divide 1460 by 16, 32, and 64. 
 
 
 Ans. 91. 25, etc. 
 
 56. 
 
 Divide 7623 by 40, 80, and 1600. 
 
 
 Ans. 190. 575, etc. 
 
 57. 
 
 Divide 143 by 15. 
 
 58. 
 
 Divide 100 by 24. 
 

 84 I DIVISION. 
 
 ' OPERATIONS. 
 
 (57.) (58.) 
 
 15)143.00(9.53 X 24)100.00(4.16|| 
 
 135 \ 96 
 
 80 (Vide 92, Rem. 2.) 40 
 
 75 24 
 
 50 , 160 
 
 45 ^ 144 
 
 5 16 
 
 59. Divide 140.913 by 3.07 and 45.9. 
 
 60. Divide 281.826 hj 30.7 and 4.59. 
 
 Ans. 9.18 and 61.4. 
 
 61. Divi^ .0949416 by 1.326. Ans. .0716. 
 
 62. Divide 152.^756 by .1234. Ans. 1234. 
 
 63. Divide $14|.80 equally among 40 men. 
 
 Ans. 13.57 each. 
 
 64. Divide §714.00 equally among 200 men. 
 
 Ans. 13.57 each. 
 
 65. Divide §2.499 equally among 7 men. 
 
 Ans. §0.357 each. 
 
 66. Divide $26137.65 equally among 365 men. 
 
 Ans. $71.61 each. 
 
 67. Divide $58.59 equally among 63 men. 
 
 Ans. $0.93 each. 
 
 68. Divide $10043.712 equally among 144 men. 
 
 Ans. $69,748 each. 
 
 69. Divide $176.25 equally among 47 men. 
 
 Ans. $3.75 each. 
 
DIVISION. 85 
 
 70. Divide $48910.95 equally among 369 men. 
 
 Ans. $132.55 each. 
 
 71. Divide |20.631 equally among 69 men. 
 
 Ans. fO.299 each. 
 96. In Division the quotient is always of the same 
 name as the dividend; and the divisor in the operation 
 must be considered as simply a number without name. 
 Thus, $3525^47-?75.— (Vide 167.) 
 
 PRACTICAL APPLICATIONS. 
 
 1. If 47 oxen cost $3525, what will one ox cost? 
 
 Ans. $75. 
 
 2. If a man can walk 391 miles in 17 days, how far 
 can he walk in one day ? Ans. 23 miles. 
 
 3. A vessel has sailed 4059 miles in 9 days. What 
 was the average rate per day? Ans. 451 miles. 
 
 4. I sell cloth to the amount of $5445, at the rate of 
 $11 per yard. How many yards are sold ? 
 
 Ans. 495 yards. 
 
 5. Eleven men divide 5445 boards equally among 
 them. How many does each have ? 
 
 Ans. 495 boards. 
 
 6. If 5445 marbles are equally divided among 495 
 boys, how many does each have ? Ans. 11 marbles. 
 
 7. If 495 yards of cloth are sold for $5445, what is 
 the average price per yard? Ans. $11. 
 
 8. If 569 hogsheads of molasses cost $21053, what is 
 the price per hogshead ? Ans. $37. 
 
 9. I buy molasses to the amount of $21053, at the 
 rate of $37 per hogshead. How many hogsheads are 
 bought ? 
 
86 DIVISION. 
 
 10. If $21053 are equally divided among 37 men, 
 how many dollars does each man receive? 
 
 11. There arrived in market a number of wagons, 
 each loaded with 12 bales of cotton, worth |56 per bale. 
 The cotton was sold for $12096. How many wagons 
 came to market? 
 
 12. A train of 18 cars brought down the Mobile and 
 Ohio Railroad cotton to the amount of $12096, at the 
 rate of $56 per bale. How many bales did each car 
 carry, supposing the cotton equally distributed? 
 
 13. Eighteen cars took into Memphis $12096 worth 
 of cotton, each car being loaded with 12 bales. What 
 was the cotton valued at per bale? 
 
 14. The exact distance round the earth at the equa- 
 tor, is 24897.883 miles, which is divided into 360 
 degrees. What is the length of each degree ? 
 
 Ans. 69.161 miles. 
 
 15. How long would it take a man to travel round 
 the earth at the equator, traveling at the rate of 60 
 miles per day ? A718. 414.9647 J days. 
 
 16. Since the earth turns on its axis once in 24 hours, 
 how far does any point on the equator move per hour ? 
 Per minute? Per second? 
 
 Last Ans. .2881699 mHeS. 
 
 17. When the moon is 240,000 miles from the earth, 
 how long would it take a car to make the distance, at 
 the rate of 45 miles per hour ? Ans. 5333 Jf hours. 
 
 18. When the sun is 95,000,000 miles from the earth, 
 how long would a car be occupied in making the dis- 
 tance, running at the rate of 57 miles per hour ? 
 
 Ans. 16666661 f hours. 
 
PROPERTIES OP INTEGRAL NUMBERS. 87 
 
 t 
 
 PROPERTIES OF INTEGRAL NUMBERS, 
 
 DEFINITIONS. 
 
 97. Any number is exactly divisible by another when 
 the q^tient is an integral number. Thus, 51 is exactly 
 divisible by 17, because the quotient 3 is an integral 
 number. — (Vide 7.) But 51 is not exactly divisible by 
 25, because the quotient, 2j^^ contains a fractional 
 number. — (Yide 8.) 
 
 98. An EVEN NUMBER is one that is exactly divisible 
 by 2. The even numbers, then, are 2, 4, 6, 8, 10, etc. 
 
 99. An ODD NUMBER is one that is not exactly divisible 
 by 2. The odd numbers are, then, 1, 3, 5, 7, 9, 11, etc. 
 
 100. A PRIME NUMBER is One that is exactly divisible 
 by no number except itself and 1. 
 
 Remark 1. — No even number^ except 2, can he a prime number. 
 
 Remark 2. — The prime numbers less than 100, are 
 
 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 
 
 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. 
 
 101. A COMPOSITE NUMBER is the product of two or 
 more prime numbers. — (Vide 74.) Thus, 
 
 24=2X2X3X2. 
 
 Remark. — Any composite number is exactly divisible by either of its 
 prime factors, or by the product of any two or more of them. Thus, 
 30 is divisible by 2, 3, or 5, or by 2X3=::6, 2X5=10, 3X5=15. 
 
88 PROPERTIES OP INTEGRAL NUMBERS. 
 
 10!S. Any two numbers are said to be prime with 
 respect to each other when no factor of the one is a factor 
 of the other. Thus, 26=2X13, and 63-=3x3x7, are 
 prime with respect to each other. 
 
 103. A COMMON DIVISOR of two or more numbers is 
 a number which exactly divides each of them. Thus, 2 is 
 a common divisor of 12 and 20. 
 
 Remark. — Numbers which are prime with respect to each pther, have 
 no common divisor other than 1. 
 
 104. The GREATEST COMMON DIVISOR of two or more 
 numbers is the greatest number which exactly divides 
 each of them. Thus, 4 is the greatest common divisor of 
 12 and 20. 
 
 105. A MULTIPLE of a number is the product of that 
 number by some other number. Thus, the multiples of 
 2 are 4, 6, 8, 10, etc. The multiples of 5 are 10, 15, 
 20, etc. The multiples of 36 are 72, 108, 144, etc. 
 
 106. A COMMON MULTIPLE of two or more numbers 
 is a number that is exactly divisible by each of them. 
 Thus, 84 is a common multiple of 2, 3, 6, and 7, because 
 it is exactly divisible by each of them. 
 
 107. The LEAST COMMON MULTIPLE, of two or more 
 numbers is the least number that is exactly divisible by 
 them. Thus, 84 is the least common multiple of 7, 12, 
 and 14, because it is the least number exactly divisible 
 by each of them. ^ 
 
 Remark. — Every number is the least common multiple of all its prime 
 factors, or of all its divisors. Thus, 42 is the least common multiple 
 of 2, 3, and 7, its prime factors, or of 2, 3, 6, 7, 14, and 21, all its 
 divisors, except 1 and 42, which may be included. 
 
plloperties of integral numbers. 89 
 
 Properties of the Integral Numbers, from 2 to 13, 
 inclusive. 
 
 108. Every number, and no other, is exactly divisible 
 by 2, wJiose rigid-hand figure is a 0, or an even number. 
 Thus, 
 
 120, 374, 336, 95678, are exactly divisible by 2. 
 
 341, 753, 255, 24683, are not exactly divisible by 2. 
 
 101]>.. Every number, and no other, is exactly divisible 
 by 3, the sum of whose digits is divisible by 3. Thus, 
 
 1035, 1305, 1350, 5031, are exactly divisible by 3, 
 because 1+0+3+5=9 is exactly divisible by 3. 
 
 liO. Every number, arid no other, is exactly divisible 
 by 4, tvhose two right-hand figures, considered as repre- 
 senting a number less than 100, are exactly divisible 
 by 4. Thus, 
 
 120, 324, 428336, are exactly divisible by 4, because 
 
 20, 24, 36, are exactly divisible by 4. But 
 
 121, 463, 782354, are not exactly divisible by 4, because 
 
 21, 63, 54, are not exactly divisible by 4. 
 
 111. Every number, and no other, is exactly divisible 
 by 5, whose right-hand figure is a 0, or a S. Thus, 
 
 3240, 4535, etc., are exactly divisible by 5. 
 
 112. Every numher, and no other, is exactly divisible 
 by 6, which is even, and exactly divisible by 3. Thus, 
 534, 2136, 6348, etc., are exactly divisible by 6. 
 
 113. Every number, and no other, is exactly divisi- 
 ble by 7, when the difference, of the sum of the products 
 produced by multiplying the figures standing in the odd 
 PERIODS, by 1, 3, 2, respectively, counting from right to 
 
90 PROPERTIES OF INTEGRAL NUMBERS. 
 
 leftj and the sum of the products of the even periods, 
 found in the same way, is exactly divisible by 7. Thus, 
 
 6, 883, 905, 
 is exactly divisible by 7, because the difference between 
 the sum of 5+18+6=29=5Xl+9x2+6Xl=products 
 in odd periods; and of 3+24+16=43=3x14-8X3+ 
 8x2=products in even periods, which is 14, is exactly 
 divisible by 7. 
 
 Remark 1. — In practice, the multiplication and addition should 
 be done mentally, and as soon as the sum amounts to 7 or more 
 omit the 7, and use the surplus. Thus, 
 
 5 and 4 are 2 and 6 are 1. 
 3 and 3 are 6 and 2 are 1. 
 Remark 2. — If the figure 7 or a is found in the number it need 
 not be multiplied. Thus, 34, 707, 205. 
 
 5 and 4 are 2 and 4 are 6 and 2 are 1; 
 and the number is not exactly divisible by 7, but will have a re- 
 mainder of 1. 
 
 114. Every number, and no other, is exactly divisi- 
 ble by 8, whose three right-hand figures, considered as 
 representing a number less than 1000, are exactly di- 
 visible by 8. Thus, 
 
 387528; 135752, etc., are exactly divisible by 8, because 
 528; 752, are exactly divisible by 8. 
 
 115. Every number, and no other, is exactly divisi- 
 ble by 9, the sum of whose digits is divisible by 9. 
 Thus, 
 
 14454, 44451, 54414, etc., are exactly divisible by 9, 
 because 4+4+4+1+5=18, is exactly divisible by 9. 
 
 IIG. Every number, and no other, is exactly divisi- 
 ble by 11, when the difference of the sujn of the figures 
 
PROPERTIES OF INTEGRAL NUMBERS. 91 
 
 standing in the odd places^ and the sum of the figures 
 standing in the even places, counting either way, is 
 exactly divisible by 11. Thus, 64532146723 is exactly 
 divisible by 11, because the difference between 27=6+5 
 +2+4+7+3, sum in odd places, and 16=4+3+1+6 
 +2, sum in even places, which is 11, is exactly divisible 
 by 11. 
 
 117. Every number, and no other, is exactly divisible 
 by 12, which is exactly divisible by 3 and 4. 
 
 118. The proposition in reference to 7 is true also of 
 13, if the numbers 1, 10, 9, are substituted respectively 
 for 1, .3, 2. 
 
 Remark 1. — Any number composed of two full periods^ with the same 
 figures in each, and the same arrangement, is exactly divisible by 7 or 13. 
 
 144144, 305305, 101101, 352352, 111111, are eacli exactly divisi- 
 ble by 7 or 13. 
 
 Remark 2. — The number 1001, or any multiple of it, is exactly di- 
 visible by 7, 11, or 13. 
 
 FACTORING. 
 
 119. To resolve a number into its prime factors, 
 
 1. Divide the number by any prime number that will 
 exactly divide it. 
 
 2. Divide the quotient by any prime number that will 
 exactly divide it. 
 
 3. Continue thus till a quotient of 1 is obtained. 
 The several divisors are the prime factors sought, 
 
 EXAMPLES. 
 
 1. What are the prime factors of 13860 . 
 
92 
 
 PROPERTIES OF INTEGRAL NUMBERS. 
 
 Vide 108. 
 
 Vide 109. 
 Vide 111. 
 Vide 113. 
 Vide 116. 
 
 OPERATION. 
 
 2 13860 
 
 2 6930 
 , 3 3465 
 
 5 1155 
 
 7 
 11 
 
 231 
 
 33 
 
 The prime factors of 
 13860 are 2, 2, 3, 3, 
 5, 7, 11. 
 
 2. What are the prime factors of 4, 6, S, 12, 14, and 
 16? A)is. 2 and 2, etc. 
 
 3. What are the prime factors of 18, 20, 22, 26, 28, 
 and 30 ? A71S. 2, 3 and 3, etc. 
 
 4. What are the prime factors of 32, 34, 36, 38, 40, 
 and 42? Ans. 2,2, 2, 2, and 2. 
 
 5. What are the prime factors of 44, 45, 46, 48, 49, 
 and 50 ? Ans. 2, 2, and 11, etc. 
 
 6. What are the prime factors of all the composite 
 numbers between 50 and 100 ?— (Vide 100, Rem. 2.) 
 
 A71S. 51=3X17, 52=2X2X13, etc. 
 
 7. What are the prime factors of 121, 144, 169, 196, 
 225, 256, 289, 324, 361, 400, 441, and 484. 
 
 Ans. 121=11X11, etc. 
 
 8. What are the prime factors of 143, 187, 209, 253, 
 319, 341, 451, 561, 737, 913, and 957. 
 
 Ans. 143=11X13, etc. 
 
PROPERTIES OF INTEGRAL NUMBERS. 93 
 
 9. What are the prime factors of 371, 413, 469, 497, 
 623, 1001, 3003, 1309, 1463, and 1771? 
 
 Last Ans. 7, 11 and 23. 
 
 10. What are the prime factors of 2940, 4620, 5460, 
 7140, 7690, 930, 1330, 1610, 6020, and 4350? 
 
 Last Ans. 2, 3,5, 5 and 29. 
 
 11. Wliat are the prime factors of 744 ? 
 
 Ans. 2, 2, 2, 3 and 31. 
 
 12. What are the prime factors of 1680 ? 
 
 Ans. 2, 2, 2, 2, 3, 5 and 7. 
 
 13. What are the prime factors of 636 ? 
 
 Ans. 2, 2, 3 and 53. 
 
 14. What are the prime factors of 1080? 
 
 A71S. 2, 2, 2, 3, 3, 3 and 5. 
 
 15. What are the prime factors of 5000? 
 
 Ans. 2, 2, 2, 5, 5, 5, and 5. 
 
 16. What are the prime factors of 221, 299, and 387? 
 
 Ans. 13 and 17, etc. 
 
 17. What are the prime factors of 2431? A^is. 
 
 GREATEST COMMON DIVISOR. 
 
 ISO. To find the greatest common divisor of two or 
 more numbers, 
 
 Resolve the numbers into their prime factors, and then 
 find the 2?roduct of such as are common to all the num- 
 bers. The result will be the greatest common divisor 
 sought. 
 
 EXAMPLES. 
 
 1. What is the greatest common divisor of 168, 420, 
 and 6006? 
 
94 
 
 PROPERTIES OF INTEGRAL NUMBERS. 
 
 
 OPERATION 
 
 168 
 
 2 
 
 420 
 
 84 
 
 2 
 
 210 
 
 42 
 
 3 
 
 105 
 
 21 
 
 5 
 
 35 
 
 7 
 
 7 
 
 7 
 
 1 
 
 
 1 
 
 2 
 
 6006 
 
 3 
 
 3003 
 
 7 
 
 1001 
 
 11 
 
 143 
 
 13 
 
 13 
 
 The factors common to all the numbers are 2, 3, and 
 7. Hence, 2X3X7 = 42, is their greatest common 
 divisor. 
 
 2. What is the greatest common divisor of 2, 4, and 
 6 ? Ans. 2. 
 
 3. What is the greatest common divisor of 4, 6, and 
 8? ' Ans. 2. 
 
 4. What is the greatest common divisor of 6, 8, and 
 10 ? Ans. 2. 
 
 5. What is the greatest common divisor of 2, 4, and 
 8? ^7is. 2. 
 
 6. What is the greatest common divisor of 3, 6, and 
 9? Ans. 3. 
 
 7. What is the greatest common divisor of 4, 8, and 
 12? Ans. 4. 
 
 8. What is the greatest common divisor of 8, 12, 
 and 20? A7is.4. . 
 
 9. What is the greatest common divisor of 12, 18, 
 and 24? Am. 6. 
 
L 
 
 PROPERTIES OF INTEGRAL TSTUMBERS. 95 
 
 10. What is the greatest common divisor of 24, 36, 
 and 48 ? Ans. 12. 
 
 11. AVhat is the greatest common divisor of 14, 21, 
 and 35 ? Ans. 7. 
 
 12. What is the greatest common divisor of 26, 39, 
 and 52? Aiis. IS. 
 
 13. What is the greatest common divisor of 16, 24, 
 and 48? ^ Ans.S. 
 
 14. What is the greatest common divisor of 252, 180, 
 and 288 ? Ans. 36. 
 
 • 15. What is the greatest common divisor of 144, 
 196, and 256 ? Ans. 4. 
 
 16. What is the greatest common divisor of 744, 
 1680, 636, and 1080 ?— (Vide 119, Examples 11, 12, 
 13, and 14.) Ans. 2X2X3=12. 
 
 17. What is the greatest common divisor of 375, 
 1100, 120, and 1440? Ans. 5. 
 
 18. What is the greatest common divisor of 780, 
 1560, 720, and 1008? Ans. 12. 
 
 19. What is the greatest common divisor of 144, 
 196, 256, and 324? Ans. 4. 
 
 20. What is the greatest ^common divisor of 143, 
 187, 209, and 253? Ans. 11. 
 
 21. W^hat is the greatest common divisor of 216, 
 408, 740, and 810 ? Ans. 2. 
 
 22. What is the greatest common divisor of 187, 
 209, 52, and 161 ?— (Vide 103, Rem.) Ans. 1 . 
 
 121. The greatest common divisor of two numbers is 
 the greatest common divisor of the less of the two num- 
 bers, and the difference found by taking the largest 
 multiple of the less number, which is smaller than the 
 
96 PROPERTIES OF INTEGRAL NUMBERS. 
 
 greater, from the greater number. Thus, the greatest 
 common divisor of 
 
 36 and 136, 
 is also the greatest common divisor of 
 36 and 136— 108-=28, 
 108 being 3X36, the largest multiple of 36 less than 
 136. (Vide 105.) This must be so. For any divisor 
 of 36 also divides 36x3, and since it divides 36X3, if 
 it divides 36x3+28, it must divide 28. There can, 
 then, be no common divisor of 36, and 36x3+28, -which 
 is not a common divisor of 36 and 28. Therefore the 
 greatest common divisor of 36 and 28 is the greatest 
 common divisor of 36 and 136. Hence, 
 
 \'22. To find the greatest common divisor of two 
 numbers, 
 
 1. Write the numbers on a line far enough from each 
 other to draw one or two perpendicular lines between 
 them. 
 
 2. Find the greatest multiple of the smaller, that is less 
 than the larger, and subtract it from the larger number. 
 ,i.--.3. Find the greatest multiple of the remainder, less 
 than the smaller number, and subtract it from the smaller 
 number. 
 
 4. Find the greatest multiple of this remainder, less 
 than the previous remainder, and subtract as before, and 
 continue the work till there is no remainder. 
 
 The last integral remainder will be the greatest common 
 divisor sought. 
 
 EXAMPLES. 
 
 1. What is the greatest common divisor of 36 and 
 136 ? 
 
PROPERTIES OF INTEGRAL NUMBERS. 
 
 97 
 
 OPERATIONS. 
 
 (1.) 
 
 (2.) 
 
 36 
 
 28 
 
 8 
 8 
 
 136 
 
 108 
 
 28 
 24 
 
 36 
 
 28 
 
 8 
 8 
 
 136 
 
 108 
 
 28 
 24 
 
 4 Am. 4 Ans. 
 
 Remark. — The multiples are found precisely as in division, and 
 the quotient figures may be retained as in (1), or neglected as in (2). 
 
 2. What is the greatest common divisor of 285 and 
 465 ? Ans. 15. 
 
 3. What is the greatest common divisor of 532 and 
 1274 ? Ans. 14. 
 
 4. What is the greatest common divisor of 693 and 
 1815 ? Ans. 33. 
 
 OPERATIONS. 
 
 285 
 
 465 
 
 532$ 
 
 21274 
 
 693 
 
 1815 
 
 180 
 
 285 
 
 420$ 
 
 U064 
 
 429 
 
 1386 
 
 105 
 
 180 
 
 112] 
 
 L 210 
 
 264 
 
 429 
 
 75 
 
 105 
 
 981 
 
 L 112 
 
 165 
 
 264 
 
 30 
 
 75 
 
 14^ 
 
 r 98 
 
 99 
 
 165 
 
 30 
 
 60 
 
 
 98 
 
 66 
 
 99 
 
 
 
 15 
 
 
 
 33 
 
 66 
 
 
 
 
 
 
 66 
 
 
 
 5. What is the greatest common divisor of 1054 and 
 1426? Ans. 62. 
 
 9 
 
98 PROPERTIES OF INTEGRAL NUMBERS. 
 
 6. What is the greatest common divisor of 725 and 
 3175? Ans,2^. 
 
 7. What is the greatest common divisor of 4585 and 
 6685 ? Am. 35. 
 
 8. What is the greatest common divisor of 4605 and 
 5505? Ans.U. 
 
 9. What is the greatest common divisor of 636 and 
 1080? of 744 and 1680? of 972 and 1260? of 3471 
 and 1869 ? Ans. 12, 24, 36, and 267. 
 
 10. What is the greatest common divisor of 1137 and 
 9475? of 3447 and 9575? of 2359 and 8425? of 1903 
 and 4325 ? Ans. 379, 383, 337, and 173. 
 
 11. What is the greatest common divisor of 117869 
 and 137773? ^ns. 311. 
 
 Remark, — If there are inore than two numbers, ,find the greatest 
 common divisor of any two of them, and then of this divisor and a third 
 number, and so on till all the numbers have been used. The last integral 
 remainder will be the greatest common divisor of all the members. 
 
 12. What is the greatest common divisor of 285, 465, 
 and 35?— (Vide Ex. 2.) Ans. 5. 
 
 13. What is the greatest common divisor of 532, 
 1274, and 21?— (Vide Ex. 3.) Ans. 7. 
 
 14. What is the greatest common divisor of 1815, 
 693, 66, and 88?— (Vide Ex. 4.) Ans. 11. 
 
 15. What is the greatest common divisor of 620, 
 1116, and 1488 ? Ans. 124. 
 
 16. What is the greatest common divisor of 1054, 
 1426, and 2263? Ans. 31. 
 
 17. What is the greatest common divisor of 233, 587, 
 and 653?— (Vide 103, Rem.) Ans. 1. 
 
 18. What is the greatest common divisor of 739, 
 503, and 439? * Ans. 1, 
 
 I 
 
E^ 
 
 PROPERTIES' OF INTEGRAL NUMBERS. 99 
 
 19. What is the greatest common divisor of 97343, 
 139639, and 206193? Ans. 311. 
 
 LEAST COMMON MULTIPLE. 
 
 123. To find the least common multiple of two or 
 more numbers, 
 
 1. Write the numbers in a line, with convenient in- 
 tervals. 
 
 2. Divide by any prime number that ivill exactly divide 
 any two or more of them, and write the quotients and 
 numbers not exactly divisible in a line below the given 
 numbers. 
 
 3. Divide this line by any prime number as before, and 
 continue to divide till no prime number, except 1, will 
 exactly divide any two numbers in the line. 
 
 4. Multiply the divisors and numbers left in the last 
 line together. 
 
 The product will be the least common multiple 
 sought. 
 
 Remark, — If any of the given numbers will exactly divide any other 
 of them, it may be neglected in the operation, and not affect the result. 
 
 EXAMPLES. 
 
 1. What is the least common multiple of 4, 8, 9, 
 and 21. 
 
 OPERATION. 
 
 8, 9, 21 
 
 8, 3, 7 
 Hence, 3X8x3X7=504, is the least common mul- 
 tiple required. ' 
 
100 
 
 I'KOPEIITIES OF INTEGRAL NUMBERS. 
 
 The figure 4 is neglected in the operationj because it 
 will exactly divide 8. 
 
 2. What is the least common multiple of 8, 64, and 
 72? ^^s. 576. 
 
 3. What is the least common multiple of 26, 39, and 
 52 ? A71S. 156. 
 
 4. What is the least common multiple of 14, 56, and 
 
 196? 
 
 (2.) 
 
 2 
 
 64, 
 
 72 
 
 2 
 
 32, 
 
 36 
 
 2 
 
 16, 
 
 18 
 
 
 8, 
 
 9 
 
 OPE 
 
 13 
 
 RATIONS. 
 
 (3.) 
 39, 52 
 
 
 3, 4 
 
 A71S. 392. 
 
 (4.) 
 2 
 
 2 
 
 56, 
 
 196 
 
 28, 
 
 98 
 
 14, 
 
 49 
 
 2, 
 
 7 
 
 (2.) • 
 2X2X2X8X9=576. 
 
 (3.) 
 13X3X4=156. 
 
 (4.) 
 
 2X2X2X7X7=392. 
 
 5. What is the least common multiple of 2, 3, 4, 5, 
 6,7,8,9,12? 
 
 OPERATION. 
 
 2 
 
 5, 7, 8, 9, 12 
 
 2 
 
 5, 7, 4, 9, 6 
 
 8 
 
 5, 7, 2, 9, 3 
 
 X 
 
 5, 7, 2, 3, 1 
 6X7X2X3=2520 Ans 
 
PROPERTIES OF INTEGRAL NUMBERS. 101 
 
 6. What is the least common multiple of 7, 8, and 
 14? of 5, 25, and 50? of 2, 4, and 6? 
 
 Ans. 56, 50, and 12. 
 
 7. What is the least common multiple of 4, 6, and 8 ? 
 of 6, 8, and 10 ? of 2, 4, and 8 ? Ans. 24, 120, and 8. 
 
 8. What is the least common multiple of 3, 6, and 9 ? 
 of 4, 8, and 12 ? of 8, 12, and 20 ? 
 
 Ans. 18, 24, and 120. 
 
 9. What is the least common multiple of 12, 18, and 
 24 ? of 24, 36, and 48 ? of 8, 32, and 16 ? 
 
 Ans. 72, 144, and 32. 
 
 10. What is 'the least common multiple of 8, 18, and 
 36 ? of 21, 42, and 14 ? of 7, 14, and 70 ? 
 
 Ans. 72, 42, and 70. 
 
 11. What is the least common multiple of 9, 18, and 
 27 ? of 12, 16, and 20 ? of 5, 10, and 15 ? 
 
 Ans. 54, 240, and 30. 
 
 12. What is the least common multiple of 2, 6, and 
 8 ? of 7, 14, and 21 ? of 3, 4, and 5 ? 
 
 Ans. 24, 42, and 60 ? 
 
 13. What is the least common multiple of 2, 3, and 
 4 ? of 2, 5, and 7 ? of 3, 7, and 11. 
 
 _ Ans. 12, 70, and 231. 
 
 14. What is the least common multiple of 14, 28, 
 and 98? of 8, 14, and 35 ? of 24, 25 and 32? 
 
 Ans. 196, 280, and 2400. 
 
 15. What is the least common multiple of 63, 12, 84? 
 of 54, 63, 81 ? of 21, 35, 84 ? Ans. 252, 1134, 420. 
 
 16. "\^niat is the least common multiple of Q6, 143, 
 55 ? of 144, 196, 128 ? of 18, 45, 63 ? 
 
 Ans. 4290, 56448, 630. 
 
102 PROPERTIES OF INTEGRAL NUMBERS. 
 
 17. What is the least common multiple of 20, 35, 80 ? 
 of 16, 24, 56 ? of 26, 39, 65 ? Ans. 560, 336, 390. 
 
 18. What is the least common multiple of 34, 51, 85 ? 
 of 57, 95, 133? of 69,115,161? 
 
 Ans. 510, 1995, 2415. 
 
 19. What is the least common multiple of 8, 7, 10, 
 14 ? of 2, 6, 7, 29 ? of 14, 21, 35, 49 ? 
 
 Ans. 280, 1218, 1470. 
 
 20. What is the least common multiple of 272, 238, 
 204, 170 ? Ans. 28560. 
 
 21. What is the least common multiple of 11, 12, 13, 
 14, 15, 16, 17, 18, 19, 20 ? Aiis. 232792560. 
 
 134. To find the least common multiple of two 
 numbers, 
 
 (1.) Find the greatest common divisor of the numhei^s. 
 
 (2.) Divide one of the members by this divisor, and 
 multiply the quotient by the other. 
 
 The product will be the least common multiple of the 
 numbers. 
 
 EXAMPLES. 
 
 1. What is the least common multiple of 1903 and 
 4325 ? 
 
 OPERATION. 
 
 1903 
 
 1557 
 
 346 
 346 
 
 
 
 4325 
 
 3806 
 
 519 
 346 
 
 173 
 
 Then, 1903-v-173x4325-:47575 Ans, 
 Or, 4325-^173Xl903--47575 Aris. 
 
PROPERTIES OF INTEGRAL NUMBERS. 103 
 
 2. What is the least common multiple of 3471 and 
 1869 ?— (Vide 122, Ex. 9.) Ans. 24297. 
 
 3. What is the least common multiple of 1137 and 
 9475 ?— (Vide 122, Ex. 10.) Ans. 28425. 
 
 4. What is the least common multiple of 3447 and 
 9575? J.7i.§. 86175. 
 
 5. What is the least common multiple of 2359 and 
 8425 ? Ans. 58975. 
 
 6. What is the least common multiple of 117869 and 
 137773? * ^ris. 52215967. 
 
 125. Practical Applications. 
 
 1. The diJBference between two numbers is 7, and the 
 less number is 25. What is the greater? Ans. 32. 
 
 2. The difference between two numbers is 19, and the 
 less number is 43. What is the greater? Ans. 62. 
 
 3. The difference between two numbers is 184, and 
 the less is 325. What is the greater? Ans. 509. 
 
 4. The difference between two numbers is 7, and the 
 greater is 32. What is the less? Ans. 25. 
 
 5. The difference between two numbers is 19, and the 
 greater is 62. What is the less? Ans. 43. 
 
 6. The difference between two numbers is 184, and 
 the greater is 509. What is the less? Ans. 325. 
 
 7. The difference between two numbers is 7, and the 
 less is 25. What is their sura? Ans. 57. 
 
 8. The difference between two numbers is 19, and 
 the less is 43. What is their sum ? Ans. 105. 
 
 9. The difference between two numbers is 184, and 
 the greater is 509. What is their sum ? Aiis. 834. 
 
104 PROPERTIES or INTEGRAL NUMBERS. 
 
 10. The sum of two numbers is 105, and tlieir differ- 
 ence 19. What are the numbers ? Ans. 62 and 43. 
 
 (1.) Add the difference to the sum, and divide by 2. 
 
 (2.) Subtract the difference from the sum, and divide 
 by 2. 
 
 The result will be the numbers. Thus, 
 
 (1.) (2.) 
 
 105 105 
 
 19 . 19 
 
 2)124 2) 86 
 
 62=Greater. 43:^Lcss. 
 
 11. The sum of tAvo numbers is 5, and their difference 
 1.4. What are the numbers ? Ans. 3.2 and 1.8. 
 
 12. The sum of two numbers is 783, and their difference 
 141. What are the numbers ? A71S. 462 and 321. 
 
 13. The sum of two numbers is 46.4, and the greater 
 is 29.3. What is the less? Ans. 17.1. 
 
 14. There are ^47.32 in two boxes. One of them 
 contains §24.85. How much money in the other? 
 
 Ans. §22.47. 
 
 15. I have §323.67 in two purses. One of them 
 contains §125.63. How much in the other? 
 
 Ans. §198.04. 
 
 16. I have two purses. One of them contains 
 §198.04, which is more money than is in the other by 
 §72.41. How much does it contain ? Ans. 125.63. 
 
 17. My money is in two purses, both of which con- 
 tain §323.67, and there are §72.41 more in the one tlian 
 in the other. How many dollars in each purse ? 
 
PROPERTIES • OF INTEGRAL NUMBERS. 105 
 
 18. Two men together own 3521.25 acres of land, 
 but one of them owns 45.75 acres more than the other. 
 How many acres does each man own ? 
 
 Ans. 1783.5 and 1737.75. 
 
 19. If to a certain number I add 45, the sum will be 
 223. What is the number? Ans. 178. 
 
 20. If from a certain number I take 34, the remain- 
 der will be 213. What is the number ? Ans. 247. 
 
 21. If to a certain number I add 14, and then sub- 
 tract 75 the result will be 268. What is the number? 
 
 Ans. 329. 
 
 22. If from a certain number I subtract 123, and 
 then add 329, the result will be 930. What is the num- 
 ber ? Ans. 724. 
 
 23. The divisor of a number is 45, and the quotient 
 is 73. What is the dividend? Ans. 3285. 
 
 24. The divisor of a number is 73, and the quotient 
 is 45. What is the dividend ? 
 
 25. The dividend is 3285, and the quotiei\t is. 73. 
 What is the divisor ? 
 
 26. The dividend is 3285, and the quotient is 45. 
 What is the divisor? 
 
 27. The multiplier is 45, and the \product 3285. 
 What is the multiplicand? 
 
 28. The multiplicand is 45, and the product 3285. 
 What is the multiplier ? • .^ 
 
 29. If a certain number is divided by 321^ the quo- 
 tient will be 23. What is the number ? , . ^ 
 
 30. If a cej^tain number is multiplied by 321, the 
 product will be 7383. What is the number? 
 
 31. If a certain number is multiplied by 4, and the 
 
106 PllOPERTIES OF INTEGRAL NUMBERS. 
 
 product is then divided by 7, the quotient will be 16. 
 What is the number ? A^is. 28. 
 
 32. If a certain number is divided by 7, and the quo- 
 tient is then multiplied by 4, and the product increased 
 by 46, the sum diminished by 37, the result will be 25. 
 What is the number? 
 
 33. If to a number you add 65, and from the sum 
 subtract 38, divide the diiference by 2, multiply the 
 quotient by 3, the result will be 141. What is the 
 number? Ans. 67. 
 
 34. The dividend is 251, the divisor 13. What is the 
 remainder ? 
 
 35. The dividend is 251, the quotient 19, and the 
 remainder 4. What is the divisor ? 
 
 36. The divisor is 13, the quotient 19, the remainder 
 4. What is the dividend ? 
 
 37. ^he divisor is 13, the dividen4 251, the remain- 
 der 4. What is the quotient ? 
 
 3^ Wliat is the least number of marbles that can be 
 divided eqiially among 2, 3, 4, or 6 boys? Ans. 12. 
 
 39. What is the least number of dollars that can be 
 divided equally among 8, 14, or 21 men? Ans. 168. 
 
 40. A can dig 7 rods of ditch per day; B caia dig 13 
 rods, and C 14 rods in the same time. Wjbat isAe least 
 number of rods that will make a number^of JBl day's 
 
 ^ork for each of the three men? '* f-^^* ■^^•2* 
 
 41. A/gentleman has 145 gallons, of /ftatawbst', 203 
 gallons ^ Madeira, and 319 gallons of Sci^ppjernong, 
 and he desires, to fill a number of casks of equal size 
 without mixing or wp^ting the wine. How many gallons 
 must each cask hold (—(Vide 121.) Ans. 29 or 1. 
 
PROPERTIES OE INTEGRAL NUMBERS. 107 
 
 42. A farmer has 482 bushels of corn, 622 bushels of 
 wheat, and 758 bushels of barley. He wishes to fill a 
 number of sacks of equal capacity, not mixing the grain, 
 or leaving any out. IIo^y many bushels must each sack 
 hold? Ans. 1 or 2 bushels. 
 
 43. A merchant has 64 silver dollars, 72 half dollars, 
 and 144 quarters. He wishes to place an equal number 
 of each in several drawers, not mixing them, or leaving 
 any out. What is the least number of drawers that will 
 answer the purpose ? Ans. 35. 
 
 44. A certain number, on being divided by 11, 12, 
 13, 14, 15, 16, 17, 18, 19, and 20, respectively, has a 
 remainder on each division just one less than the divisor. 
 What is the number ?— (Vide 123, Ex. 21.) 
 
 Ans. 232792559. 
 
 45. A, B, C, and D start together, and travel the 
 same way, round an island 500 miles in circumference. 
 A goes 8 miles an hour, B 12, C 16, and D 20. What 
 is the least number of hours that will bring them to- 
 o;ether aoiain ? Ans. 125. 
 
 How many times round the island will each have 
 traveled? Ans. A 2, B 3, C 4, and D 5 times. 
 
 46. Suppose three railroad trains to start at the same 
 time from Quito, and to run round the earth on the 
 equator at the rate of 3G^ 60, and 75 miles an hour re- 
 spectively. What is the least number of days in which 
 all will arrive at Quito at the same time ? — (Vide 96, 
 Ex. 14.) Ans. 69.1608 days. 
 
 How many times round the earth will each train have 
 gone? Ans. First 2, second 4, and third 5 times. 
 
108 FRACTIONS. 
 
 FRACTIONS 
 
 NATURE OF FRACTIONS. 
 
 126. If an apple is divided into two equal parts, each 
 part is said to be a half of the whole apple. 
 
 If an orange is divided into tJwee equal parts, each 
 part is said to be a third of the whole orange. 
 
 If a line is divided into four equal iJarts, each part is 
 said to be di, fourth of the whole line. 
 
 127. If any quantity whatever is divided into a given 
 nuraher of equal parts, each pai^t takes a name lohich 
 indicates the nurdher of parts into tuhich the quantity is 
 divided. Thus, 
 
 A half indicates a division into two equal parts. 
 
 A third indicates a division into three equal p>arts. 
 
 A fourth indicates a division into four equal p)arts. 
 
 A fifth, sixth, seventh, eighth, ninth, etc., indicate, 
 when applied to any quantity, that it has been divided 
 into five, six, seven, eight, nine, etc., equal parts. 
 
 12S. If an apple is divided into any number of equal 
 parts, each part is a whole part or unit, and the word 
 one may therefore be applied to it. (Vide 4.) Thus, 
 one half, one third, one fourth, one fifth, one tenth, one 
 twentieth, etc. 
 
 More than one part may be indicated, just as more 
 than one of any other quantity is indicated. Thus, two 
 thirds, two fourths, three fourths, two halves, four 
 fourths, seven tenths, nineteeii twentieths, etc., any of 
 which expressions is called a fraction. Hence, 
 
FRACTIONS. 109 
 
 (1.) A FRACTION represents one or more than one of 
 the equal parts of a unit. — (Vide 8.) 
 
 (2.) The UNIT OF A FRACTION is the whole quantity 
 from which the fraction is derived. 
 
 (3.) A FRACTIONAL UNIT is ONE of the equal parts of 
 the whole quantity that is divided. — (Yide 38, I.) 
 
 NOTATION OF FRACTIONS. 
 
 120. The unit of a fraction may always be repre- 
 sented by the figure 1. 
 
 130. A fractional unit is represented by drawing a 
 horizontal line under the figure 1, and placing under- 
 neath it the figure denoting the number of parts into 
 which the unit of the fraction is divided. Thus, 
 
 One half is represented by J. 
 
 One third is " " -J. 
 
 One fourth is " " J. 
 
 One fifth is " " i- 
 
 One sixth is " " %. 
 
 One seventh is " " 4- 
 
 One twentieth is " " . j*^. 
 
 One eighty-fifth is represented by -i^. 
 
 Remark. — The reciprocal of a number is represented by placing 
 the number under the figure 1 in the manner of a fractional unit. 
 Thus, the reciprocal of 2 is ^; of 3 is \] of 100 is j-^^, etc. 
 
 131. Any given number of fractional units is repre- 
 sented by writing the given number above the line in 
 place of the figure 1. Thus, 
 
 Two thirds is represented by |. 
 
 Two fourths is " " f . 
 
 Three fo\irths is " " f . 
 
110 
 
 FRACTIONS 
 
 23 
 
 S5' 
 
 Two fifths is represented by 
 
 Three fifths is " " 
 
 Four fifths is " " 
 
 Seven twentieths is " " 
 
 Twenty-three eighty-fifths is represented by 
 
 XS2. Properly a fraction represents a less number of 
 fractional units than is contained in the unit of the 
 fraction. The unit of the fraction may, however, be 
 represented in the form of a fraction, by writing that 
 figure above the line which denotes the entire number 
 of fractional units contained in it. Thus, two halves 
 is represented by §, three thirds by |, etc. 
 
 133. A greater number of fractional units than is 
 contained in the unit of the fraction, may also be repre- 
 sented in the form of a fraction. Thus, 
 
 Three halves is represented by §. 
 
 Four halves is " "...... 4 
 
 Five halves is 
 Four thirds is 
 Five thirds is 
 Six thirds is 
 Seven thirds is 
 Seventeen sixths is 
 
 134. An integral number is jo 
 thus, (vide 92, Rem. 2,) 
 One and one half is represented by 
 One and one third is represented by 
 Two and one half is " " 
 Three and one fifth is " " 
 Five and three fifths is " " 
 Eight and six thirteenths is represented by 
 
 ned to a fraction 
 
 
 21. 
 
 5|. 
 8A- 
 
FRACTIONS. Ill 
 
 135. When one quantity is to be divided by another, 
 the dividend and divisor may be written in the form of 
 a fraction. Thus, 
 
 Seven divided by five is represented by .... J 
 
 One half divided by three is represented by ... | 
 
 One divided by one half is represented by • • • i 
 One third divided by one fifth is represented by . . f 
 
 3" 
 
 2 
 
 Two divided by five and one third is represented by gy 
 Two and one half divided by three and one third is 
 
 represented by rf 
 
 136. Since any fractional unit is a whole part of the 
 unit of the fraction, it may itself he divided into any 
 number of equal parts, and each part into any number of 
 other equal parts, and so on to any extent whatever. 
 Thus, a half dollar may be divided into two equal parts, 
 and each of these into five other equal parts, etc. 
 
 This division is indicated by figures, thus : 
 One half of one half is indicated by ... J of J. 
 One fifth of one half is " "... i of J. 
 
 One fifth of one seventh is " " ... i of 4. 
 One fifth of one third of one fourth is indicated 
 
 i>y J of J- of 1. 
 
 NUMERATOR AND DENOMINATOR. 
 
 137. In every fraction, the figure above the line 
 indicates the number of fractional units taken. It is 
 thence called the numerator of the fraction. 
 
112 FK ACTIONS. 
 
 138. In every fraction the figure below tlie line in- 
 dicates the number of equal parts into which the unit of 
 the fraction is divided. It therefore determines the 
 name to be applied to the fractional unit, and is thence 
 called the denominator of the fraction. 
 
 Remark 1. — The numerator and denominator taken together are 
 called the terms of a fraction. 
 
 Remark 2. — The terms of a fraction are said to be inverted when 
 the numerator takes the place of the denominator, and the reverse. 
 Thus, the fraction f inverted becomes |. 
 
 Remark 3. — Every fraction is said to be in its lowest terms when 
 they are prime with respect to each other. — (Vide 102.) 
 
 Remark 4. — When a quantity is written in the form of a fraction^ 
 
 the part above the principal line is still called the numerator^ and 
 
 1 
 that below, the denominator. Thus, in the expression | , one half is 
 
 the numerator, and one third the denominator. 
 
 CLASSIFICATION OF FRACTIONS. 
 
 139. A Simple Fraction is one whose terms are each 
 single integral numbers. Thus, J, |, f, f, y, |J, etc.,. 
 are simple fractions. 
 
 140. A Proper Fraction is a simple fraction whose 
 numerator is less than the denominator. Thus, J, |, f , 
 ?> A? IJj ^*^-? ^^® proper fractions. — (Vide 132.) 
 
 141. An Improper Fraction is a simple fraction whose 
 numerator is equal to, or greater than, the denominator. 
 Thus, I, I, I, f, f, V? if J ^tc., are improper fractions. — 
 (Vide 133.) 
 
 142. A Compound Fraction is an expression consist- 
 ing of two or more simple fractions, united by the word 
 OF. thus, I of i, I of J, \ of I of i of I of f, are 
 compound fractions. — (Vide 136.) 
 

 FRACTIONS. ' 113 
 
 143. A Mixed Fraction is an integral number united 
 to a proper fraction. Thus, 1^, Ij, 2^, 31, 8^^, 9^1, 
 etc., arc mixed fractions. — (Vide 134.) 
 
 144. A Complex Fraction is a quantity written in 
 the form of a fraction, in which one of its terms is a 
 fraction, or both. Thus, tt^ v? vjvt,^,^^ are complex 
 fractions. — (Vide 135.) 
 
 VALUE OF A FRACTION. 
 
 145. The value of a fraction is the quotient of the 
 numerator divided by the denominator. 
 
 Remark 1. — The value of a. proper fraction is less than 1. 
 
 Remark 2. — The value of an improper fraction is equal to or 
 greater than 1. Thus, f=l; |=1 ; f=l^; V=2j etc. 
 
 Remark 3. — The value of a fraction is such a part of the nu- 
 merator as is denoted by the reciprocal of the denominator. Thus, 
 1=1 of 5; -|=i of 4; -|=i- of 4; |=i of 5, etc.— (Vide 127 and 
 130, Rem.) " 
 
 PROPOSITIONS. 
 
 I. The value of a fraction is not changed by multi- 
 plying or dividing both terms by the same number. 
 
 II. The value of a fraction is multiplied when the 
 numerator is multiplied or denominator divided. 
 
 III. The value of a fraction is divided when the nu- 
 merator is divided or denominator multiplied. 
 
 REDUCTION OF FRACTIONS. 
 
 140. The reduction of a fraction consists in changing 
 its form, or the value of its terms, without altering the 
 value of the fraction. 
 10 
 
114 
 
 FRACTIONS 
 
 147. To reduce a simple fraction to its lowest terms, 
 Divide the numerator and denominator hy their greatest 
 common divisor , or cancel the factors common to the nu- 
 merator and denominator. 
 
 EXAMPLES. 
 
 1. Reduce f to its lowest terms. Ans. J. 
 
 2. Reduce §, f , ^%, |, /^-, Jg, /^, and >f , each to its 
 lowest terms. Ans -f , f, |, |, |, f, J, and f . 
 
 3. Reduce f J, f§, if J, |||, if^ ||, and |f, each to 
 its lowest terms. Arts. J, f , |, g, ^, f , and y%. 
 
 OPERATION. 
 
 (Vide 115.) 9|i-||=ig==-Mn5. 
 
 4. Reduce H, -j3^<V, ^^(j. JS. IS. ??, and ^%%, each to 
 its lowest terms. Ans. |, f, |, |, f, |, and f. 
 
 ^ "Rprlnpp 18 105 84 9_9 78 117 o^lfl 19^ paph 
 
 o. xveuute -3 q, 175? T40' les? Tsn? Tirs? '^^^^ 325? ^^^^ 
 to its lowest terms. Ans. f. 
 
 6"RprInf»P IT 3 8 6 9 116 155 J2 5 9 104 nQoli+nif^ 
 . xieauce -34, ^t^, ^^j T45j ths? 296? 1 it? ^^^^^ ^^ ^^^ 
 
 lowest terms. Ans. J, |, |, etc. 
 
 7 PprlnPA 39 51 69 87 169 183 213 popVl tn 
 
 *• xieauce g^, g-g, yyg, y45, 25^? sTJSJ 355> ^^^^i ^^ 
 its lowest terms. Ans. |. 
 
 8. Reduce f ||g to its lowest terms. 
 
 OPERATION. 
 
 (Viae iiy, j^x. iv.; 4^2 0—2x2x3X5X7X11 ' ' 
 
 Remark. — In practice a line may be drawn across the common 
 factors, and the factors in each term not crossed must be multi- 
 plied together. Thus, 
 
 9. Reduce -f f g to its lowest terms. 
 
FRACTIONS. 115 
 
 OPERATION. 
 ,4 6_^X3>aXl3_3 9 A^ 
 
 10. Reduce J-JiJ to its lowest terms. Ans. {J. 
 
 11. Reduce ?J|J to its lowest terms. Aiis ^|f. 
 
 12. Reduce jii J to its lowest terms. A')is, -J|. 
 
 13. Reduce Iff § to its lowest terms. Ans. [f. 
 
 14. Reduce Jitlfl to its lowest terms. 
 
 OPERATION. 
 
 (Vide 122, Ex. 11.) 3ii|J.J.|B..|^ 
 
 15. Reduce Iff ? to its lowest terms. 
 
 16. Reduce -JJ|| to its lowest terms. 
 
 17. Reduce J|§f to its lowest terms. 
 
 18. Reduce |f-|J to its lowest terms. 
 
 19. Reduce |||| to its lowest terms. 
 
 20. Reduce } J|| and j%\% to their lowest terms. 
 
 Ans. hi and ||. 
 
 148. To reduce a simple fraction to another fraction 
 having a given denominator, 
 
 (1.) If necessary, reduce the fraction to its lowest terms. 
 
 (2.) Divide the proposed denominator by the denomi- 
 nator of the reduced fraction. 
 
 (3.) Midtiply both terms of the reduced fraction by the 
 quotient. 
 
 EXAMPLES. 
 
 1. Reduce \l\% to a fraction whose denominator 
 shall be 69. 
 
 OPERATIONS. 
 (1.) (2.) (3.) 
 
 — 23X3 ^^ ^* 
 
 379 J^n 
 
 
 44 3 -^^ 
 
 * ■ 
 
 Ans. 
 
 t\- 
 
 Ans. 
 
 s^- 
 
 Ans. 
 
 M- 
 
 Ans. 
 
 A- 
 
 Ans. 
 
 2\- 
 
116 
 
 FRACTIONS. 
 
 2. Reduce y°o to fractions whose denominators shall 
 be 9, 15, 18, 21, and 27. Ans. f , J g, jf, if, and Jf. 
 
 3. Reduce !§ to fractions whose denominators shall 
 be 15, 25, 30, 35, 40, 45, etc. 
 
 477.9 -9- J S 18 2 1 pf« 
 
 4. Reduce f, ;|, g, and /q, to fractions whose denomi- 
 nators shall each be 60. Ans. |J, |g, |g, and |f. 
 
 5. Reduce |, Z^, 2§5 ^'^i^d i\? t<^ fractions whose de- 
 nominators shall each be 30. A71S. fg, §J, -Jf, and -Ig. 
 
 6. Reduce |J, J|, y'/g, and J|, to fractions whose 
 denominators shall all be 105. Ans. /q\, -^^-g, etc. 
 
 7. Reduce f i, 4§, Jf I' ^^^^ iii' *^ fractions whose 
 denominators shall be 280. Ans. Jl^, etc. 
 
 8. Reduce -J|, ||, ||, and [if, to fractions whose 
 denominators shall be 60. Ans. |J, etc. 
 
 9. Reduce f |, iff, -|f }, and ||f, to fractions whose 
 denominators shall be 504. Ans. |§J, etc. 
 
 10. Reduce Jf |, o^, f |, and /g^, to fractions whose 
 denominators shall be 126. Ans. j^^^, etc. 
 
 a fraction whose denominator 
 
 Ans. If. 
 a fraction whose denominator 
 Ans. \2. 
 to fractions whose denominators shall 
 Ans. f, y, y, etc. 
 
 14. Reduce 13, 14, 15, 16, 17, 18, 19,^ and 20 to 
 fractions whose denominators shall all be 17. 
 
 Ans. ^iV> ¥-7% etc. 
 
 15. Reduce 11, 12, 21, 22, 23, 24, and 25 to fr^- 
 tions whose denominators shall be 19. 
 
 Ans. Yg% t/> etc. 
 
 11. Reduce Jff J 
 
 to 
 
 shall be 52. 
 
 
 12. Reduce 3=f 
 
 to ; 
 
 shall be 4. 
 
 
 13. Reduce 5 to 
 
 fraci 
 
 be 1, 2, 3, 4, 5, 6, 7, 
 
 etc. 
 
FRACTIONS. 117 
 
 16. Reduce 19 to fractions.Tvhose denominators shall 
 be 11, 12, 13, 14, etc., to 20. Ans. \\9, ^^^^ etc. 
 
 17. How many half dollars in 23 dollars ? 
 
 Ans. 46 half dollars. 
 
 18. How many quarters in 23 dollars ? 
 
 Ans. 92 quarters. 
 
 149. To reduce a mixed number to an improper 
 fraction which shall be in its lowest terms, 
 
 (1.) Reduce the fractional part to its lowest terms, if it 
 is not so given. 
 
 (2.) Multiply the whole number by the denominator of 
 the reduced fraction, and add to the product the nu- 
 merator. 
 
 (3.) Write the sum over the reduced denomiiiator. 
 
 EXAMPLES. 
 
 1. Reduce 23j|fg, 15ff§g, and 7|4§, to improper 
 fractions in their lowest terms. — (Vide 147, Ex. 9 and 
 10; also, 148, Ex. 1.) 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) 
 
 23i§!S-23H 154fiS-15H 7f4g=r|f 
 
 SgV Ans. Y_6 J^^jg. 4_2_4 ^^^s. 
 
 2. Reduce 13f, 15|, 17|, and I^^q, to improper 
 fractions in their lowest terms. 
 
 Ans. y, V? V^ and V- 
 S. Reduce 21f , 23/o, 27i§, and 31|f, to improper 
 fractions in their lowest terms. 
 
 J/*,<? 6 5 7 1 13 8 ori,l 1 27 
 
 JLns. 3,3, 5 , aiiu ^ . 
 
118 FRACTIONS. 
 
 4. Reduce 17j J, 16||, 12f |, and 19; J |, to improper 
 fractions in their lowest terms. 
 
 Ans. %^, y^, V, and %K 
 
 5. Reduce 14||, 15|i, IGyVs? and ITyVs, to improper 
 fractions in their lowest terms. Aiis. \^, ''/, etc. 
 
 6. Reduce lSj%, ITy^, 19tV, and 2d.^\, to improper 
 fractions in their lowest terms. A7is. \y, etc. 
 
 7. Reduce 124jf|, 256/r\^ 211|ff-g, and 112j-||i, 
 to improper fractions in their lowest terms. 
 
 J^a 1120 1795 3600 and^^^^ 
 
 150. To reduce an improper fraction to an integral 
 or mixed number, 
 
 Divide the numerator hy the denominator, and place 
 the excess of fractional units to the right of the quotient. 
 
 Remark. — Let it be understood that, unless special direction is 
 given to the contrary, all answers are to be given in their lowest 
 terms. 
 
 ^-dB-l^A^FL E S . 
 
 1. Reduce '4'* to an improper fraction. A71S. 3^. 
 
 2. Reduce |, f, |, y , y, y, ^/, to integral or mixed 
 numbers. Ans. 4, 4, 4.^, 2^, 2i, 3, 3]. 
 
 3. Reduce \4, V? 'P? 'i¥' ^V^ ^^^d \8^, to mixed 
 numbers. Ans. 13i, 15|, 17|, 18|, 21|, 23f . 
 
 4. Reduce ^^f, YsS W? ¥? Wj to mixed numbers. 
 
 J.WS. 23^1, etc. 
 
 5. Reduce 1//, ^^{, 4_y, \¥? and 1/^2^ to integral 
 numbers. Ans. 13, 14, etc. 
 
 6. Reduce W> W? ¥eS \V> ¥2S to mixed num- 
 bers. Ans. ISJ^, I7J5, 20/g, 20 Jj, 20i. 
 
 7. Reduce W? \¥j W? \V? ^^^ W? to mixed 
 numbers. j-ws. 19, \, etc. 
 
FRACTIONS. 119 
 
 8. Reduce -%^^, '"'^S^^, ^V/^ and 5f|f^ to mixed 
 numbers. Ans. 3574, 1228^^, etc. 
 
 ADDITION OF FKACTIONS. 
 
 151. To add two or more proper fractions together, 
 
 (1.) Reduce the fractions to their lowest terms ^ if they 
 are not so given. — (Vide 147.) 
 
 (2.) Find the least common multiple of the reduced 
 denominators, — (Vide 123.) 
 
 (3.) Reduce each fraction to one which shall have a 
 denominator denoted hy the least common multiple. — (Vide 
 148.) 
 
 (4.) Add the numerators of the resulting fractions, and 
 if the sum placed over the denominator is an improper 
 fraction, reduce it to an integral or mixed number. — 
 (Vide 150.) 
 
 Remark. — If the given fractions all have the same denominator, 
 their numerators should, of course, be added at once by (4.) 
 
 EXAMPLES. 
 
 1. Add together f , |, |, and f^. 
 
 OPERxVTION. 
 
 f+i+i+A =Given fractions. 
 
 i+l+f+l (Vide 147, Ex. 1 and 2.) 
 (Vide 148, Ex. 4.) iHiHiS+ig= W^Sfi Ans, 
 
 2. Add together f , f^, J §, and ^-^.—(yide 148, Ex. 
 5.) ^ Ans. 2t3. 
 
 3. Add together | J, Jf . iVa. and J]-. Ans. 2-f^%. 
 
 4. Add together if, fj-, -,\%, and ^.. Ans. 2f. 
 
 5. Add together §, f i, |-f, and |i. Ans. IJ. 
 
120 FRACTIONS. 
 
 6. Add together §, y\, -^^^ and ^V- ^'^s. Ijoo- 
 
 7. Add together A 8, 1.0 5;_8_4_^ and /gV -^^s. 2|. 
 
 8. Add together 4, f , f , and 4- -^^s. If. 
 
 9. Add together -J, f, -|, and |. ^ws. 1-J. 
 
 10. Add together I, J, i, and J. Ans. l^J. 
 
 11. Add together J, |, f , and |. ^ns. 2jf . 
 
 12. Add together |, -/g, |, and j4_^. JlWS. 2/^. 
 
 13. Add together 4, Z^, ^\, and 3%-. ^tis. Jf g. 
 
 14. Add together f |, i?-f , ifi, and -|if. 
 
 ^/^§^ 3229^ 
 
 15. Add together h^, /,%, ff, and ^%\. 
 
 Ans. 1{^. 
 
 16. Add together |, -j^, 4, f , and ^^g. J.?zs. 2/^. 
 
 17. Add together y/^g and ^Z^^.— (Vide 124, Ex. 
 1.) Ans. 44J45. 
 
 18. Add together -^^^j and yj§^. ^tis. of oIt-. 
 19 Add together y 1^377 and 9 yV?- ^ns. 5/4^0 5. 
 
 Kemark. — When mixed numbers are required to be added, add 
 the sum of the fractional parts to the sum of the integral numbers. 
 
 20. Add together 3j and 4j; also, 7j and 6| ; also, 
 14| atid 135. 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) 
 
 3i= 
 
 =3^ 
 
 n=n 
 
 14i=14|f 
 
 H= 
 
 =4A 
 
 ^=^ 
 
 13f=13|§ 
 
 7A 
 
 Ans. 
 
 14i Ans. 
 
 28|f Ans. 
 
 21. 
 
 Add together 13f and 14|. 
 
 Ans. 28J. 
 
 22. 
 
 Add together 
 
 171 and 18 fV 
 
 Ans. 36iJ. 
 
 23. 
 
 Add together 
 
 21f and 23,8^-. 
 
 Ans. 45 J. 
 
 24. 
 
 Add together 
 
 17||andl6|?. 
 
 Aws. 34^. 
 
FRACTIONS. 121 
 
 25. Add together 4j, 3j, 4i, and 6|. Ans. 18io §. 
 
 26. Add together 7^, ISj^^-, 16|, and 5j. 
 
 Ans. 42f 3.. 
 
 27. Add together 4i, 2i, 28|f, and 29j-J. 
 
 28. Add together 9i, 8^^, 7o^5, and 65. 
 
 ^7^s. 64|. 
 
 Ans. 89f f. 
 
 29. Add together 1 J and 2i ; also, Sj and 4J ; also, 
 2| and Ij; also, Ij and 2j. 
 
 ^"*' ^15? '"30? "^Bf ^40* 
 
 30. Add together 4Jq and 3^^; also, 2 J^ and 5-J; 
 also, 5J3 and 7^^. Ans. 7/- 7/^; 12/^. 
 
 31. Add together 2^ and 1 J^ 5 ^^so, 3Jj and 4J- ; 
 also, 51 and 3J. ^ns. 3|f ; 7if ; 8/0- 
 
 32. Add together 3f and 4f ; also. If and 5| ; also, 
 4|and6|. Am.7U; 6fi; llji. 
 
 SUBTRACTION OF FRACTIONS. 
 
 XS2. To subtract one proper fraction from another, 
 
 (1.) Reduce the fractions to their lowest terms, if they 
 are not so given. 
 
 (2.) Find the least common multiple of the reduced de- 
 nominators, and reduce each fraction to the denominator 
 denoted by it. 
 
 (3.) Take the numerator of the subtrahend from that 
 of the minuend, and place the difference over the denomi- 
 nator. — (Vide 150, Rem.) 
 
 
 
 EXAMPLES. 
 
 
 
 
 
 
 . From 1 
 
 take 
 
 2\- 
 
 2. 
 
 From 
 
 1% 
 
 take 
 
 1% 
 
 
 11 
 
 3. 
 
 From 
 
 Jl take 
 
 hh 
 
 
 
 ^ 
 
 >•"' 
 
 - 
 
12-2 FRACTIONS. 
 
 OPERATIONS. 
 
 (1.) (2.) (8.) 
 
 8 — /f 1% — 1\ if — ii Given fractions. 
 
 1. .3-_i 13. 
 
 3 8 4 1^ 
 
 (Vide 147.) 
 i-f-^ A718. l-i=^iAns. ||-i|-3^^^s. (Vide 148.1 
 
 4. What is the value of |-A? of |-|? of jl.—^^'i 
 
 of A-^r' ^^^^- 8%; T6; §§; f- 
 
 5. What is the value of i— A? of J— J^V of ^—^J 
 
 6. What is the value of y%— §'? of Ji — J? of Ji— |? 
 
 7. What is the value of /f— 1\? of |— ^? of 4— _i-? 
 
 of 6 189 J ^,9 1. 7. 37. 1 
 
 8. From {J.j take «|. ^?«8. J,. 
 
 9. From f || take f|9, j^^s^ _i^^ 
 
 10. From Hf take f|J. ^^s. 1. 
 
 11. From ^^%% take 5%. J.^s. ^. 
 
 12. From 34^^ take y^V?- ^^s- ^eVf 5- 
 
 13. From ^g^^^ take ^f^^. A7is. ^gV^^. 
 14.' From j^%^ take 34^^,-. J.ws. ^ J/g^. 
 
 15. From ^/g^ take ^4^-5. ^ws. olfl^. 
 
 Remark 1. — A proper fraction is taken from 1 by writing the dif- 
 ference between the numerator and denominator over the denominator. 
 Thus, 1—3^1 since f — | = 1 (Vide 132.) 
 
 And 1—^^=3^ since \l—^^=3^ 
 
 16. From 1 take §, |, J, J, a, |, f , |, |, and f 
 
 yd 790 1 1 4 1 2 p + p 
 
 yiAts. 25 -3, 5, 2, 3, eio. 
 
 17. From 1 take J§, /„ J|, ,*t, /„ Jg, and |J. 
 
 ^- 
 
 18. From 3 take §, |, 
 
 
 
 Ans 
 
 •5^0 
 
 1 5 
 ? 2 2"? 
 
 iJ, 
 
 etc, 
 
 1, 
 
 4,i, 
 
 5 
 
 A, 
 
 and 
 
 sV 
 
 
 
 4??s. 
 
 Ol, 
 
 01 
 
 04 
 
 25, 
 
 etc, 
 
 
FRACTIONS. 
 
 19. Findthe value of 9— J§; of 24— ^^. ; of37— '«; 
 of 86-/^. Ans. 8/^; 23||; 36 Ji; 85/^. 
 
 20. Find the value of 4— f ; of 121— J§ ; of 187—^^ ; 
 of 2145-IJ-. Ans. 3|; 120,^^; 186,3^; 2144f. 
 
 21. From 8 take 24.. 22. From 4j take 3J-. 
 
 23. From 5i take 3 
 
 OPERATIONS. 
 
 (21.) (22.) (23.) 
 
 8 4j-4i 5i=5§ 
 
 51 Ans. 1| Ans. 1| Ans. 
 
 The first two examples need no explanation. In the 
 last, having reduced the fractional parts to the same 
 denominator, the numerator of the upper fraction is 
 added to the difference hetiveen the terms of the lower 
 fraction for the numerator of the ansiver. 
 
 This is the same as adding a fractional unit, in terms 
 of the reduced fraction, to the upper fraction, and then 
 subtracting the lower fraction. Thus, |+§ — i=i. 
 
 The true difference is preserved by adding 1 to the 
 lower integral number, before taking i.t from that above 
 it. Hence, 
 
 Remark 2. — Always proceed in this way when the fractional 
 part of a mixed subtrahend is greater than that of the minuend. 
 
 24. From 28i take l^. ■ Ayis. 14j 
 
 25. From 36ji take 17f . Ans. 18f 
 
 26. From 45| take 23/^. Ans. 21|. 
 
 27. From 34i take 17J J. Ans. 16f . 
 
 28. From 123| take 45/^. Ans. 1^^^, 
 
4 
 
 FRACTIONS. 
 
 
 29. 
 
 From 106| take 103g. 
 
 Ans. 3 J. 
 
 30. 
 
 From 165/j- take 63^^. 
 
 Ans. 102//^-. 
 
 31. 
 
 From 71Ji take 3f J. 
 
 Ans.QlUi. 
 
 32. 
 
 From 19^- take lOff. 
 
 Alls. 8|||. 
 
 33. 
 
 From 55j% take 37i-gi. 
 
 Ans. 17lil. 
 
 34. 
 
 From 25jV5 take 18|4|. 
 
 Ans. 6i|. 
 
 35. 
 
 From 122 J take 16 j|. 
 
 Ans. 105f f . 
 
 36. 
 
 From 1345f take 237|. 
 
 Ans. 1107f i. 
 
 37. 
 
 From 1000 take 555 j\. 
 
 A71S. 4:4:4: ^^j. 
 
 Remark 3. — Improper fractions may be subtracted precisely like 
 proper fractions, but it is generally much better to reduce them to 
 mixed numbers before subtracting. Thus, 
 
 .8 1 — 8 8—1 44—1 42 (Vide Ex. 24.) 
 
 MULTIPLICATION OF FRACTIONS. 
 
 153. To multiply a simple fraction by an integral 
 number, 
 
 (1.) Multiply the numerator by the integral number, 
 and plaee the product over the denominator; or, (vide 
 145, II,) _ . . 
 
 (2.) Divide the denominator by the integral number, if 
 it is exactly divisible, and place the quotient under the 
 numerator. 
 
 Remark 1; — All answers should be integral numbers, mixed 
 numbers, or proper fractions. — (Vide 150, Rem.) 
 
 EXAMPLES 
 
 1. Multiply I by 10. Ans. 6. 
 
 3. Multiply f by 6. 
 
 4. Multiply T^y by 2. 
 
 Ans. 1\. 
 Ans. fo-. 
 
 6. Multiply Jy by 5. Ans. 2\. 
 
 6. Multiply ^jy by 4. Ans. 1^. 
 
 7. Multiply \l by 15. Ans. ^. 
 
 8. Multiply V by 14. Ans. 24. 
 
FRACTIONS. 125 
 
 9. Multiply I by 2, 3, 4, 5, 6, 7, 8, 9, and 10. 
 
 Ans. If, 2|, 3i, etc. 
 
 10. Multiply 1^ by 11, 12, 13, 14, 15, 16, 17, 18, 
 and 19. Ans. 7f^g, 8i, etc. 
 
 11. Multiply JJ by 20, 25, 30, 35, 40, 45, 50, and 
 100. Ans. 13f , 17, etc. 
 
 Remark 2. — Any simple fraction multiplied by its denominator 
 produces the numerator for a product. Thus, 
 
 VX5-17; f|fX163=2o9. 
 154. To multiply a mixed number by an integral 
 number, 
 
 (1.) Multiply the fractional part as in 153. 
 (2.) Multiply the integral part as in 80. 
 (3.) Add the products together. 
 
 EXAMPLES. 
 
 
 1. Multiply 4| by 10. 2. Multiply 7| by 6. 
 
 3. Multiply 9/5 by 5 
 
 • 
 
 OPERATIONS. 
 
 
 (1.) (2.) 
 
 (3.) 
 
 4f (vide Ex. 1, 153.) 7i (vide Ex. 3.) 
 10 6 
 
 9/g (vide Ex. 5.) 
 5 
 
 46 Ans. 44 i Ans. 
 
 47| A71S. 
 
 4. Multiply 27/3 by 2. 
 
 Am. 54} J. 
 
 i 5. Multiply 29/0 by 4. 
 
 Ans. 117|. 
 
 6. Multiply 121 § by 15. 
 
 Ans. 1831. 
 
 7. Multiply 15f by 5. 
 
 Ans. 78. 
 
 8. Multiply 13| by 18. 
 
 Ans. 250. 
 
 9. Multiply 14/^ by 34. 
 
 Ans. 486. 
 
126 FRACTIONS. 
 
 10. Multiply 1311 by 11, 12, 13, 14, 15, 16, 17, 18, 
 and 19. Last Ans. 260^^^. 
 
 Remark, — A complex fraction may be reduced to a simple frac- 
 tion bj/ multiplying both terms of the fraction by the least common mul- 
 tiple of the denominators of the fractional part s^. 
 
 11. Reduce |, -f, and ~^ to simple fractions. 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) 
 
 3X4 T^^^^- 6ix6-3^^^'^* 3^-3^X10-31-11 ^''^- 
 
 • 1 1 1 1 1 -2_ 33 ' < 
 
 12. Reduce rf, rf, r|, and rf, to simple fractions. 
 
 ^3 ^S ^1 ^¥ 
 
 ^^s- hh A. h and |. 
 
 13. Reduce J, r? "?5 and -_% to simple fractions. 
 
 J_?2S. -/o5 14, I, and ^%. 
 
 14. Reduce |, ?^, | |, ^, and ^^, to simple 
 
 f^^^ti^^l- ^ ^ ^^.. 4?, il, Y, etc. 
 
 155. To multiply an integral number by a simple 
 fraction, ' 
 
 (1.) Divide the integral number hy the denominator 
 of the fraction, and multiply the quotient hy the nu- 
 merator ; or, 
 
 (2.) Multiply the integral number by the numerator of 
 the fraction, and divide the p)roduct by the denominator. 
 
 EXAMPLES. 
 
 
 1. Multiply 20 by f . 
 
 2. Multiply 60 by ^\. 
 
 3. Multiply 76 by J |. 
 
 Ans. 15, 
 Ans. 16, 
 Ans. 26, 
 
FRACTIONS. 127 
 
 4. Multiply 169 by J|. 
 
 5. Multiply 5 by y\. 
 
 6. Multiply 320 by /g. 
 
 7. Multiply 480 by j|. 
 
 8. Multiply 75 by f . 
 
 9. Multiply 87 by ^f . 
 
 10. Multiply 147 by /g. 
 
 11. Multiply 1728 by J, 
 
 5 7 ^5 9_ 1 ; 
 
 ^ns. 
 
 156. 
 
 J.?25 
 
 ^2f 
 
 J-TiS. 
 
 180. 
 
 Ans. 
 
 440. 
 
 Ans. 
 
 64f. 
 
 Ans 
 
 ■.39. 
 
 Ans. 
 
 lOi. 
 
 , and 
 
 U' 
 
 Last Ans. 1692. 
 
 156. To multiply two or more simple fractions to- 
 gether, 
 
 Place the product of the numerators over that of the 
 denominators. 
 
 Remark 1. — The rule applies also to tlie reduction of compound 
 fractions to simple ones, 
 
 EXAMPLES. 
 
 1. Multiply I by | ; that is, find the value of g of |. 
 
 
 OPERATION. 
 
 
 
 fX?-/5^^^^- 
 
 
 2. 
 
 Multiply i by |. 
 
 Ans. |, 
 
 3. 
 
 Multiply 1 by f . 
 
 , Ans. IJ, 
 
 4. 
 
 Multiply f by ji. 
 
 ^ns. if, 
 
 5. 
 
 Multiply ^ by II 
 
 Ans. If?, 
 
 6. 
 
 Multiply f by |. 
 
 ^^s. If. 
 
 7. 
 
 Multiply ,^3 by y. 
 
 Ans. .v.. 
 
 8. 
 
 • 
 
 Multiply 1 by f 
 
 J.WS. if. 
 
 9. 
 
 Multiply II by If. 
 
 ^r^s. f if 
 
 10. 
 
 Multiply 1 by | of f. 
 
 Ans. 3^0. 
 
 11. 
 
 Multiply ,3, of 1 by f . 
 
 ^?js. -igl, 
 
 12. 
 
 Multiply i of 1 by ^' of |>. 
 
 Ans. 4||. 
 
128 FRACTIONS. 
 
 13. Multiply V of i ^J A- ^^^' ioi- 
 
 14. Multiply f of I of /j by V._(yi(ie 147, Ex. 9.) 
 
 Remark 2. — All the factors common to the numerators and de- 
 nominators should be canceled before multiplying. 
 
 15. Multiply f of V of V ^J t\- 
 
 OPERATION. 
 
 f X V X V X i\. Given fractions. 
 
 ^ ^^ ^X^ ^ ^X2X2 ^ ^X^~^' 
 
 16. Multiply f I of fl of J by /g. ^tzs. f. 
 . 17. Slultiply f § of 1/ of V by J^. J.m. 8. 
 
 ■ _18. Find tlje value of fX VX^j^. A71S. 511 1. 
 
 19. Findthe^valueof JfXfJXfXlXV- 
 ' A71S. 58f §. 
 
 V » 20. Find the value of ;|gXVXifgX|X-|Xf. 
 . \ /. ' ^^«- 24j3y. 
 
 # '21, Find the value of ^X V-X|fgX4Xi>^f 
 
 * % >-^ ■' . A71S. s^di. 
 
 ♦ , ^2. -^Firid the value of %^ X 3^- >^10. Ans^20. ' 
 
 GENERAL RULE. V '^ 5f 
 
 157. To multiply fractional numbers, ^. *^ • ' 
 
 (1.) In compound fractions consider the word of as a 
 sign of multiplication. 
 
 (2.) Reduce complex fractions, integral and mixed 
 numbers, to simple fractions. 
 
 (3.) Cancel all the factors common to the numerators 
 and denominators. 
 
 (4.) Multiply the remaining factors of the numerators 
 
FRACTIOXS. 129 
 
 iog ether, and also those of the deiiominators, forming a 
 simple fraction of the products. — (Vide 153, Rem. 1.) 
 
 Remark. — The best mode of canceling common factors will be 
 learned by practice. It is hardly ever necessary to resolve each 
 term into lis prime factors. 
 
 EXAMPLES. 
 
 1. Multiply 2^X61x31X1^3X2x1, forming a simple 
 inber. 
 
 OPERATION. 
 
 fraction or integral nuinber, 
 
 ^ ^ ^ 73 1 // 1 
 Draw a line across 4 and 32, writing 8 in place of 32. 
 
 2. Multiply 2 J by 2h # Ans. 61. 
 
 3. Multiply 61 by 61. ^ ^ ^ -Ans. Z^-^^. 
 
 4. Multiply 31 by H. r4Nl Ans. l^. 
 
 5. Multiply 2i by ^. ^ ^ ^ Ai^. 1^\. 
 
 6. Jiultiply 5^ by:5i/' Ans. 30i. 
 "7.' fli^tiply XQV^y 16^. Ans. 272i. 
 
 8. Multiply 4i; byv4i.^i% Ans. 20i. 
 
 9. Multiply 7i by 1'^ig > Ans. h^. 
 
 10. Multiply ^ by^3j t ^ .Aws. 8f . 
 
 11. Multiply i\ hj!%. Ans. 1. 
 ,12. Multiply 8^ by 3 J^. Ans. 2B. 
 
 13. Multiply 91 by 2^\. Ans. 18J. 
 
 JL4. Multiply 3} by 3^^. ' Ans. 9 J. 
 
 15. Multiply 21 by 4^6. Ans. 8f. 
 
 16. Find the value of I4x||-X^. 
 
 OPERATION. 
 
 fifXTViXlg. (Vide 154, Rem.) 
 
 mX4 5X10 5X3 _ ^ 3 . 
 
 113X5^3X47 ^5X2~-^^- ^'''' 
 
X) niACTIONS. 
 
 J 
 
 17. Find the value of ?X^x4 Am. |. 
 
 18. Find the value of ^X^fX^f • Am. 1/A. 
 
 19. Find the value of ^X#,X^X207. 
 
 Am. 16|. 
 
 20. Find the value of 4 X 8 ^^ X f X V X 24 J. 
 
 J.72S. 614^. 
 
 21. Find the value of |^x||xgx^. 
 
 Am. If. 
 
 22. Find the value of f X^^X^xp'. Am. ^%^^%. 
 
 23. Find the value of V/XyX— ^xSx^ of ^ 
 
 " 465 3^ 2^ ' 
 
 X5i. ^^s. 132. 
 
 24. Find the value of 1 § x 3^0,0 x § X | X ^s"" X 15. 
 
 Ans. 120. 
 
 25. Find the value of T\X|X|Xy. Am. 1^. 
 
 26.. Find the value of IxO/^Xf X4|X8|. 
 
 Am. 240|. 
 
 27. Find the value of f gX }iX V0VXI8. 
 
 Ans. 9. 
 
 28. Find the value of 5iX5i XS^. Ans. 166|. 
 
 29. Find the value of 3^ x^lxS^ X3|. 
 
 A71S. 150 Jg 
 
 DIVISION OF FRACTIONS. 
 
 158. To divide a simple fraction by an integral 
 number, 
 
 (1.) Divide the numerator by the integral number, if it 
 is exactly divisible, and place the quotient over the de- 
 nominator; otherwise, 
 
FRACTIONS. 131 
 
 (2.) Multiply the denominator hy the integral nitmher, 
 and place the product under the numerator. — (Vide 
 145, III.) 
 
 EXERCISES. 
 
 5. Divide | by 5. Am. -^. 
 
 6. Divide y by 4. Ans. 1^^. 
 
 7. Divide %^ by 13. Ans. If. 
 
 8. Divide f| by 19. Ans. ^. 
 
 2. Divide ^% by 18. Ans. ^^ 
 
 3. Divide y by 36. Ans. ^\. 
 
 4. Divide ^ by 51. Ans. ji-^. 
 
 9. Divide JiJ§ by 11, 12, 13, 14, 15, 16, 17, and 18. 
 
 A.71S. 2 2 0' 2 4 05 ^^^' 
 
 10. Divide ^^g^ ]by 17; s^o^e by 18; ^^^^ by 19, and 
 36ihv1Q ' An9 §' 1^- g and J 9 
 
 150. To divide a mixed number by an integral 
 number. 
 
 Reduce the mixed number to an improper fraction, and 
 then divide as in 158. 
 
 EXERCISES. 
 
 1. Divide 2J by 5.— (Vide Ex. 5, 158.) 
 
 2. Divide 7 J by 8. Ans.'\l. 
 
 3. Divide 9| by 12. Ans. f . 
 
 4. Divide Ij by 2. Jltis. f. 
 
 5. Divide 5f by 4. J.?is. l/^- 
 
 6. Divide 2-i by 5. A71S. /g. 
 
 7. Divide If by 36. Ans. ^^. 
 
 8. Divide 41 by 6. ' Ans. ^^. 
 
 9. Divide ^ by 11. J.7is. f . 
 10. Divide 5j by 13. A71S. §. 
 
 Remark 1. — If the dividend is larger than the divisor, its inte- 
 gral part may be first divided for the integral part of the answer; 
 then divide the remainder united ^ the fractional part of the divi- 
 dend for the fractional part of the answer. 
 
132 FRACTIONS. 
 
 11. Divide 27-1 by 5; 71^ by 8; and 153f by 12. 
 
 OPERATIONS. 
 
 
 (1.) (2.) 
 
 
 (3.) 
 
 5)271 8)711 
 
 
 12)153f 
 
 5/5(yideEx.l) 8li(yideEx.i 
 
 2.) 12|(VideEx.3.) 
 
 12. Divide 54i| by 2. 
 
 
 Ans. 27/^. 
 
 13. Divide 117i by 4. 
 
 
 Ans. Ex. 5, 154. 
 
 14. Divide 1831 by 15. 
 
 
 Ans. Ex. 6, 154. 
 
 15. Divide 150/g by 11. 
 
 
 . Ans.lS}^. 
 
 16. Divide 1641 by 12. 
 
 
 Ans. im. 
 
 17. Divide 471 by 5. 
 
 
 A71S. 9/5. 
 
 18. Divide 441 by 6. 
 
 
 Ans. 7|. 
 
 19. Divide 100 by 3. 
 
 
 Ans. 331. 
 
 20. Divide 1274J by 11, 
 
 12,1^ 
 
 5, 14, 15, 16, 17, 18, 
 
 and 19. 
 
 Ans. 
 
 115||, 106^4, etc. 
 
 Remark 2. — If the divisor is a composite number, it is usually 
 best to divide first by one of the component parts, and the resulting 
 quotient by another part, and so on till all the component parts are 
 used. 
 
 21. Divide 4567f by 25, 32 and 51. 
 
 
 
 OPEKATIONS. 
 
 
 (1-) 
 
 
 (2.) 
 
 
 (3.)- 
 
 5)4567| 
 
 
 4)4567| 
 
 
 3)4567f 
 
 5) 913>-J 
 
 Ans. 
 
 8)1141}i 
 142 i»A 
 
 Ans. 
 
 17)1522/^ 
 
 182tV0 
 
 89i-Jf Ans. 
 
 22. Divide 34027f by 36, 42, 65, 56, 72, and 49. 
 Ans. 945,^6, 810, Vg, ^23,^6, etc. 
 
FRACTIONS. 133 
 
 23. Divide 72431 i by 15, 21, 28, and 35. 
 
 Ans, 4828f 3, 3449 Z^, etc. 
 
 160. To divide an integral number by a simple 
 fraction, 
 
 Multiply the whole niimher hy the denominator of the 
 fraction, and divide the product by the numerator ; or, 
 
 Invert the divisor, and then multiply t?s m 155, (2.) — 
 (\^ide 138, Rem. 2.) 
 
 
 EXERCISES. 
 
 
 
 1. 
 
 Divide 15 by f. 
 
 
 
 
 
 Ans. 20. 
 
 2. 
 
 Divide 16 by j\. 
 
 
 
 
 Am 
 
 f. Ex. 2, 155. 
 
 3. 
 
 Divide 26 by ^f. 
 
 
 
 
 
 Ans. Ex. 3. 
 
 4. 
 
 Divide 156 by a|. 
 
 
 
 
 
 Ans. Ex. 4. 
 
 5. 
 
 Divide 3 by 5. 
 
 
 
 
 
 ^ns. 2i. 
 
 6. 
 
 Divide 49 by J. 
 
 
 
 
 
 Ans. 28. 
 
 7. 
 
 Divide 57 by '/. 
 
 
 
 
 
 J.71S. 24. 
 
 8. 
 
 Divide 32 by V- 
 
 
 
 
 
 J.??s. 114. 
 
 9. 
 
 Divide 45 by J, |, 
 
 f. 
 
 h 
 
 h\h 
 
 1 6 
 16? 
 
 V, and ^%. 
 
 Ans. 90, 67J, 60, etc. 
 
 10. Divide 4290 by f f , VoS ^^^l f |. 
 
 ^ns. 2015, etc., (vide 123, Ex. 16.) 
 
 11. Divide 28560 by ^p, ^f ^ YrS and V/- 
 
 ^ns. 525> 840, 1540, 2184. 
 161. To divide one simple fraction by another, 
 Invert the divisor, and then place the product of the 
 numerators over that of the denominators. 
 
 EXAMPLES. 
 
 
 1. Divide f by |. 
 2 Divide | by |. 
 
 Ans. \ 
 Ans. 
 
134 
 
 14 
 
 FRACTIONS. 
 
 
 
 3. 
 
 Divide | J by |. 
 
 
 Ans. |. 
 
 4. 
 
 Divide §1 by J J. 
 
 
 Ans. If. 
 
 5. 
 
 Divide f^_o.^^ II . 
 
 
 JLtzs. ^^. 
 
 6. 
 
 Divide 3y, by i|. 
 
 
 A71S. |. 
 
 7. 
 
 Divide Jgf by |f. 
 
 
 ^Tis. /r- 
 
 8. 
 
 Divide | by f . 
 
 
 Ans. Jf . 
 
 9. 
 
 Divide -/ by |. 
 
 
 ^/^s. 10. 
 
 10. 
 
 Divide J| by V- 
 
 
 J.n.s. 1^\. 
 
 11. 
 
 Divide If? by ]J. 
 
 
 Ans. ;|. 
 
 12. 
 
 Divide |f | by If. 
 
 
 ^7ZS. if. 
 
 13. 
 
 Divide J by f. 
 
 Ans. 
 
 Ex. 14, 154. 
 
 14., 
 
 Divide ^% by j%. 
 
 Ans. 
 
 Ex. 13, 154. 
 
 
 GENERAL RULE. 
 
 
 162. To divide fractional numbers, 
 
 (1.) In compound fractions consider the tvord of as a 
 sign of multiplication. 
 
 (2.) Reduce complex fractions^ integral and mixed 
 numbers, to simple fractions. 
 
 (3.) Invert each of the reduced or given simple frac- 
 tions considered as divisors, 
 
 (4.) Cancel all the factors common to the numerators 
 and denominators of the simple fractions. 
 
 (5.) Multiply the remaining factors of the numerators 
 together, and also those of the denominators, forming a 
 simple fraction of the products. — (Vide 153, Rem. 1.) 
 
 EXAMPLES. 
 
 1. Divide 4 of § of 5i by if of 48. 
 
 OPERATION. 
 
 4X|XVXifX^xV5=/6 ^ns. 
 
FRACTIONS. 135 
 
 2. Divide f of | by J of 2^. Ans. Jf. 
 
 3. Divide 39-jig by 6\. Ans. Gj. 
 
 4. Divide 272} by 16\, Am. 16j. 
 
 5. Divide 3061 by 17 J. Ans. 17^. 
 
 6. Divide f of J/ by ^\ Ans. 1. 
 • 7. Divide 1419^X11X^5 by ^. ^?is. 2f. 
 
 8. Divide i|x^ by ^,Xl8f . Ans. f . 
 
 9. Divide 4iX^" by %^X~. Ans. I. 
 
 10. Find the value of ^X-f divided by ^XlOj. 
 
 Ans. ii^. 
 
 11. Find the value. of |X^ divided by ?X^X^| 
 X^-lf . ^^^ ^ • ^.;. i-. ' 
 
 12. Find the value of ilX^Xyi divided by ^X 
 
 167 ^ 6f 
 
 13. Find the value of 2i- divided by 5i. 
 
 Ans. Ex. 14, 154. 
 
 14. Find the value of ?±-^. Ans. Ex. 14, 154. 
 
 15. Find the value of ^4- -^^s- lA- 
 
 16. Reduce | to a fraction whose denominator shall 
 be 4.— (Vide 148.) Ans. ^. 
 
 17. Reduce J| to a fraction whose denominator shall 
 be 2J.— O^ide 154, Ex. 12.) Ans. ^. 
 
 3 
 
 18. Reduce J J to a fraction whose denominator shall 
 be 5i. Ans. rr. 
 
 X^. ^ns. 5if. 
 
 H 
 
136 FRACTIONS. 
 
 REDUCTION OF COMMON FRACTIONS TO DECIMAL 
 FRACTIONS. 
 
 163. To reduce a simple fraction to a decimal 
 fraction, 
 
 Divide the numerator hy the denominator. — (Vide 92, 
 Rem. 3, and 145.) 
 
 EXAMPLES. 
 
 1. Reduce J, |, |, and ]|, to decimal fractions. 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) (4.) 
 
 2)1.0 . 4)3.00 8)5.000 16)13.0000 
 
 .5 Ans. .75 Ans. .625 Ans. .8125 Ans. 
 
 2. Reduce ^-q, ^l^, iJif? ^^^ IS *^ decimal frac- 
 tions. — (Vide 94 and 95, Rem. 3.) 
 
 Ans. .1, .03, 1.728, and .925. 
 
 3. Reduce -||, H, ff, ^^-3%, ^i^, i^%, and if to deci- 
 mal fractions. 
 
 Ans. .4, .85, .5, .025, .128, .096, and .36. 
 
 4. Reduce ^y^, J/^, ,-%, /„ hi, |f, and ,-Ji^ to 
 decimal fractions. 
 
 Ans. .06640625, .08, .4, .25, .48, .0765625, .006875. 
 
 5. Reduce ^fS n^, f§, i§, 0^, ||, and W to 
 decimal fractions. 
 
 Ans. 30.25, 35.2, 2.0625, 1.2, 4.09375,1.171875, 5.08. 
 
 6Rp(lnPP 3 26 7 210 80 24 nrifl 5 1 f^ /Ippi'mnl 
 
 fractions. Ans. 1.5, 2.5 .28, 8.75, .128, 1.6, and .75. 
 
 7. Reduce 1^, 3j, 81, 6/^, 30f, 35f, and 7S to 
 mixed decimals. 
 
 ^?zs 1.5, 3.25, 8.2, 6.7, 30.75, 35.6, and 7.625. 
 
FRACTIONS. 137 
 
 8. What is the value of ^3^, $4|, $7f , $Sl, and ^6\ |? 
 
 Ans. $3.50, §4.625, |7.75, §8.20, and §6.81i. 
 
 Remark. — If the denominator of a fraction in its lowest terms 
 contains a prime factor other than 2 or 5, the value of the fraction 
 can not be exactly expressed by a decimal. The exact value may, 
 however, be preserved by placing the excess of fractional units to 
 the right of the quotient, stopping the division at pleasure. 
 
 9. Reduce ^, |, f.j, f, |, and y\ ^^ mixed decimals. 
 
 Ans. .331, .16§, .416f , .66|, .83J, and .583i. 
 
 10. Reduce |, J^, fj, |, f J, |f, and || to mixed 
 decimals. 
 
 Ans. .4281, .06|, .846/3, .33^, 1.16f , .66|, and 1.66f . 
 
 11. What is the value of $7^, |4|, |6|, $8J, and 
 §12 J^? 
 
 A71S. §7.33J, |4.83i, |6.66|, ?8.773, and |12.06|. 
 
 164. To reduce a decimal fraction to a common 
 fraction in its lowest terms, 
 
 (1.) For the numerator of the fraction, write the 
 figures composing the given number. 
 
 (2.) For the denominator, write 1, with as many 
 cij)he7's annexed as there are decimal places in the given 
 number. 
 
 (3.) Reduce the resulting fraction to its loivcst terms. 
 
 EXAMPLES. 
 
 1. Reduce .5, .75, .625, and .8125 to simple fractions. 
 
 OPEKATIONS. 
 
 (1.) (2.) (3.) (4.) 
 
 ,»,=jAns. j\%=\Am. {ii-,=lAm. -^^yii,^\lArw. 
 12 
 
138 FRACTIONS. 
 
 2. Reduce .06640625 and 30.25 to simple fractions. 
 
 OPERATIONS. 
 
 (1.) (2.) or (2.) 
 
 3. Reduce .25, .85, .55, .125, .135, and .325 to simple 
 fractions. Aiis. i, JJ, J i, -J, i^\, and i-§. 
 
 4. Reduce .025, .0085, .9375, .0008, and .16 to 
 simple fractions. Ans. J^, ^ij^, i|, ^ J^o' and 5*^. 
 
 5. Reduce .34375, .1328125, and .203125 to simple 
 fractions. Ans. J J, y^g, and ||. 
 
 6. Reduce $3.50, $4,625, $7.75, §8.20, and $3.40 to 
 dollars. Ans. $3j, $4|, $7f , $8|, and $3f. 
 
 Remark. — "When there is an irreducible fraction at the end of 
 the decimal, 
 
 (1.) Consider the given number as integral^ and reduce it to an im- 
 proper fraction. 
 
 (2.) Annex to the denominator as many ciphers as there are decimal 
 p)laces in the given number. 
 
 (3.) Reduce the resulting fraction to its lowest terms. 
 
 7. Reduce .428|, .066|, and .83j to simple fractions. 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) 
 
 .428f .066| .83i 
 
 3000 S-d/Mo 200 1 J*iQ 2 50 5 A'i'}^ 
 
 8. Reduce .47J, 4.7|, 47.33i, .047^, and .43J to 
 simple or mixed fractions. 
 
 ^ns. ^5^5, 4j-i, 47^, tJJ^, and J^. 
 
 9. Reduce .473j, .33^, .45|, .125^, and .25| to simple 
 fractions. Ans. -jVo> irs^ ft? /^^j ^^^ «o- 
 
FRACTIONS. 189 
 
 10. Reduce §7.33|, |4.83|, |7.25J, |0.754f, and 
 |4.50f to dollars. 
 
 Ans. $7J, Uh ^n, Psih and |4f|. 
 
 11. Add together $240,172, §120.75f , and §255.136f . 
 
 Ans. $616,066^. 
 
 12. Add together $5.87|, $3,187^, and |2|. 
 
 ^ ^718. $11,687^. 
 
 13. Add together |, .066f , and .8]. 
 
 OPERATION. 
 
 1= .4281 
 
 .066| 
 
 M= .833A 
 
 1.328f=:lf g Ans. 
 
 14. Add together f , .18|, and 7-|. 
 
 Ans. 8.7652|=8f |i. 
 
 15. From $25.41 take $17|. Ans. $7.66. 
 
 16. From $28,026 take $19.15|. Ans. $8.872f . 
 
 17. From $12.25 take $8i. J^ns. $3.75. 
 
 18. From $5| take $4.25. Ans. $1,08 «. 
 19.* From $6i take $5.25. Aiis. $1.00. 
 20. From $7f take $1|. " ^?^s. $6.58'. 
 
 165. Problems Involving Preceding Principles. 
 
 1. If a horse consume ^ a bushel of oats in one day, 
 I of a bushel in another, | of a bushel in another, and 
 f of a bushel in another, how many bushels are con- 
 sumed in the four days? — (Vide 151, Ex. 1.) 
 
 Ans. 2|g bushels- 
 
140 FKACTIONS. 
 
 2. If I buy 4 of ^ yard of ribbon at one store, y\ at 
 another, o^- ^^ another, and -f^ at another, hoAV many 
 yards have been purchased? — (Vide 151, Ex. 13.) 
 
 Ans. if J yards. 
 
 3. A man bought two pieces of cloth, one containing 
 13f yards, and the other 14| yards. How many yards 
 in both pieces ?— (Vide 151, Ex. 21.) 
 
 Ans. 28^ yards. 
 
 4. In one pile of wood I have 4 J cords; in another 
 3J cords; in another 4i; and in another 6^ cords. 
 How many cords of wood in the four piles? — (Ex. 25) 
 
 Ans. 18j2§ cords. 
 
 5. If I make purchases to the amount of 'fl7|-| at 
 one time, and §16|| at another, how many dollars have 
 I expended in all?— (Ex. 24.) Ans. $34i. 
 
 6. If I travel 47 f miles in one day, 33 ig in another, 
 and 19 in another, how many miles have I traveled in 
 all? Ans. 100 miles. 
 
 7. Add together §4.25i, §3.37 J, §6.753, and §7.52 j^,. 
 
 Ans. §21.91. ' 
 
 8. Add together §240.17^, §120.75|, and §255.13f . 
 
 A9is. §616.06J2. 
 
 9. Add together §16.254, §40.20|, §13.60 J, and 
 §24.035. Ans. §94.094 J. 
 
 10. What is the difference between § of a dollar and 
 2'j- of a dollar?— (Vide 152, Ex. 1.) A7is. §0.16|. 
 
 11. What is the difference between §3.00 and -| of a 
 dollar ? Ans. §2.25. 
 
 12. From a cask of wine containing 8 gallons, 2J 
 gallons were drawn. What quantity remained in the 
 cask?— (Vide 152, Ex. 21.) "^Ans. 5j gallons. 
 
FRACTIONS. 141 
 
 13. If I purchase flour at 3 J dollars per barrel, and 
 sell it for 5| dollars, ^'hat doJ^ gain? Ayis. |1.83j. 
 
 14. A farmer sold 55 y^j- ajp^ from a farm of 100 acres. 
 IIow many acre^did lie s^ own? ^ A71S. 4:4: f^ acres. 
 
 15. If I buy a'p'iece of land for |1 03.33 J, and sell 
 the same for $106.66 f, Avhat do I gain ? Ans. $3.33 J. 
 
 16. A merchant bought a piece of cloth containing 
 123| yards, and from it sold 45j'^(j yards. How many 
 yards remained? Ans. 78 /^ yards. 
 
 17. A railroad train has 13 1 hours in which to run 
 550 miles. Having run lOj hours, the conductor finds 
 that only 41 6| miles have been made. What distance 
 is yet to be run, and in what time? 
 
 Ans. 133 J miles in 2| hours. 
 
 18. From 1000 yards of cloth I sold at one time 
 479j'^g yards, and at another 275 1. How many yards 
 have I still on hand? A71S. 245 /g. 
 
 19. A merchant bought at one time 234| yards of 
 cloth, at another time 753y'^Q yards, and sold of the two 
 lots 843^^ yards. How much cloth has he yet on hand? 
 
 Ans. 145 i yards. 
 
 20. I bought 30 cords of wood for $105 J-, and sold 
 17 cords of the wood for $65,25. If I sell the remain- 
 ing 13 cords for $45^, how much do I gain in the trans- 
 actions? Ans. $5,585. 
 
 21. The sum of two numbers is 4b^ ; one of them is 
 23/2-. What is the other ?— (Vide 151, Ex. 23.) 
 
 Ans. 21|. 
 
 22. The difference between two numbers is J, and 
 the less is 4|. What is the greater? Ans. 4|. 
 
 23. The difference between two numbers is 13?, and 
 
142 FRACTIONS. 
 
 the greater is 28 f|. What is the less? — (Vide 151, 
 Ex. 20, (3.) ^ .^ A71S. 14f . 
 
 24. The difference betA^kn two nuiflbers is 184i, and 
 the greater is 509^. What il the sumVf^the two num- 
 bers?— (Vide 125"rte*x. 9.) . " • ' Ans. 8343^. 
 
 25. The sum of two numbers is 34^3, and one of 
 them is 16||. What is the other? Ans. ITJ. 
 
 26. The sum of four numbers is 89ff. Three of the 
 numbers are 65, 7^^, and SjL. What is the fourth? 
 
 Ans. 9^. 
 
 27. A farmer, who had wheat worth |4325.75, sold at 
 one time 120 bushels for $135 J ; at another time 45J 
 
 "imshels for §70 1 ; and at another time 87| bushels for 
 /, »; ii60. What is the value of the 4000 bushels he still 
 . finds he has on hand? Ans. $4019.58 J. 
 
 '^^ *^ 4 lOO, To find the cost of a number of things, when 
 jy^ 1!||e cost of one i^ given, 
 
 **^^ iKidtipm the cost of one hy the .numher of tilings. The 
 t* ^ plbduct ^vi'il be the cost of the w^hole. — (Vide 82.) 
 ^ •*. ^*28f^If*lt yard of cloth cost | of a dollar, what will 
 *'*-4^;^ards cost? (Vide 153, Ex. 1.) 15 yards? 20? 
 40? J.ns. $6.00, etc. 
 
 29. If a pound of butter cost | of a dollar, what 
 will 6 pounds cost? (Vide 153, Ex. 3.) 8 pounds? 
 12? 20? Alls. $2.25, etc. 
 
 30. If a pound of raisins cost ^^ of a dollar, what 
 will 5 pounds cost? (Vide 153, Ex. 5.) 10 pounds? 
 12? 15? 20? Ans. $2.33 J, etc. 
 
 31. At J§ of a dollar a pound, what will 15 pounds 
 of beef cost? (Vide Ex. 7.) 20 pounds? 30? 45? 60? 
 
 Ans. $3.25, etc. 
 
FRACTIONS. 143 
 
 32. At J I of a dollar a bushel, what will 20 bushels 
 of apples cost? (Vide Ex. 11.) 25 bushels? 30? 35? 
 40? Ans. §13.60, etc. 
 
 33. If a ton of hay cost |27i^3, what will be the 
 cost of 2 tons? (Vide 154, Ex. 4.) 3 tons? 4? 5? 
 12? Ans. 154.76; I, etc. 
 
 34. If one coat cost 13 J | dollars, what will be the 
 cost of 11 coats? 12? 13? 14? 15? 
 
 Ans. $150,561, etc. 
 
 35. If a ton of hay cost $20, what will | of a ton 
 cost? (Vide 155, Ex.'l.) | of a ton? ^^'i -/^? 
 
 ' '\ ^ ^ ^^ Ans. $15.00, etc. 
 
 36. If a bale of cott<)^ cost ,^6p, what will /g of a 
 bale cost ? (Vide Ex. 2.) " ^\'? .^% ? 'i § ? |J ? 
 
 Ans. $16.00, etc. 
 
 37. When flour is woYtJi $5 a barrel, what is the 
 value of j^5 of a barrel? o%? -/^ ? J§? 
 
 Ans. $2,331, etc. 
 
 38. If a flock of sheep is wbrth $1728, what are | 
 of it worth"? ^'^ l'^ A? -9 ? 13? 4 7 7' 
 
 Ans. $1382.40, etc. 
 
 39. If an acre of land is worth $128, what are -| of 
 it worth? ^'^ -%.*? -K'^ -5- *? Si? is? 
 
 Ans. $112.00, etc. 
 
 40. If a yard of cloth cost § of a dollar, what are | 
 of a yard worth?— (Vide 156, Ex. 1.) Ans. $0.174 . 
 
 41. If a yard of cloth cost f of a dollar, what are f 
 of a yard worth? Ans. $0,174. 
 
 42. If a pound of tea cost \ of a dollar, what cost 
 f of a pound? (Vide 156, Ex.~2.) | of a pound? 4? 
 i ? ^ ? A71S. 37i cents, etc. 
 
144 FRACTIONS. 
 
 43. If silk is worth || of a dollar a yard, what are ^\ 
 of a yard worth? Ji? ^\ ? ff ? Ans. ?0.112|0f. 
 
 44. What will -^\ of a pound of tea cost at J of 
 a dollar a pound? f? |? |? J of ^ of a dollar a 
 pound? • Last Ans. $0.07 o\. 
 
 45. At 2 J cents each, what will be the cost of 2h 
 apples? (Vide 157, Ex. 2.) Sj? 4j? 4i? 12i ? 12i? 
 
 Ans. 6\ cents, etc. 
 
 46. At 16^ dollars an acre, what will 16^ acres of 
 land cost? 25|? SO-J? 401? 75|? JLtis. |272.25. 
 
 167. To find the cost of one thing when the number 
 of things and the cost of the whole are given, 
 
 Divide the cost of the tvhole hy the number of things. 
 The quotient will be the cost of one. — (Vide 96.) 
 
 47. If 2 yards of cloth cost f of a dollar, what will 
 one yard cost?— (Vide 158.) Ayis. $0.16|. 
 
 48. If 18 yards of ribbon cost ^^ of a dollar, what 
 will one yard cost? Ans. 2\ cents. 
 
 49. If 5 sheep cost 27j dollars, what will be the 
 price of one sheep? — (Vide 159, Ex. 11, (1.) 
 
 Ans. |5.46f . 
 
 50. If 8 yards of broadcloth are worth 71 J dollars, 
 what is one yard worth ? Ans. -$8.9 If 
 
 51. If 12 acres of land cost 153| dollars, what will 
 one acre cost? Ans. §12.80. 
 
 52. If 6 pounds of butter cost 2 J dollars, what will 
 one pound cost? Ans. 37^ cents. 
 
 53. If 9 bushels of wheat cost 10^- dollars, what will 
 one bushel cost? Ans. $1.16|. 
 
 54. If 4 acres of land cost fll7|. Avliat will one acre 
 of the same land cost? Ans. §29.45. 
 
FRACTIOXS. l-i5 
 
 55. I have four small farms containing, respectively, 
 11, 12, 13, and 14 acres, and I value each farm at 
 |1274j. What is the value per acre of each farm ? 
 What is the average price per acre ? 
 
 Ans. $115.8G^*j-, $10G.208i-, $98.038yfij, $91,035^, $101.96. 
 
 56. If I of a ton of hay cost §15, what is the price 
 per ton ?— (Vide 160, Ex. 1.) Ans. §20. 
 
 57. If y'^W of a man's salary per month amounts to 
 §84, what is his salary per ^ear? Ans. §1728. 
 
 58. A gentleman divided j\ of a lot of marbles 
 among 4 boys, giving each boy 21 marbles. How many 
 marbles would each boy have received if the whole lot 
 had been divided ? Ans. 36 marbles. 
 
 59. If J of a bale of cotton are worth §48, what is 
 one bale worth? Aois. §54.85|. 
 
 167^. To find the number of things when the cost .• 
 of one is given, 
 
 Divide the cost of all hy the cost of one thing. The 
 quotient will be the number of things? — (Vide 96.) 
 
 60. At 12f dollars an acre, how many acres of land 
 can be bought for 153 1 dollars? Ans. 12 acres. 
 
 61. At I of a dollar a pound, how many pounds of 
 butter can be bought for 2 J- dollars. Ans. 6 pounds. 
 
 62. If cherries are worth 7| cents a quart, how many 
 quarts can be bought for 1^^^ dollars? 
 
 Ans. 16 quarts. 
 
 63. How many sheep can be bought for 27-| dollars, 
 if the average price is §5.46f . Ans. 5 sheep. 
 
 64. At 2 J- dollars a yard, how many yards of cloth 
 can be bought for ^^-^ dollars ? Ans. 3 J yards. 
 
 13 
 
146 FKACTIONS. 
 
 65. If a man boards at 3| dollars a week, how long, 
 can he board for 188. i dollars? Ans. 52 Avecks. 
 
 66. At 2 dollars a yard, how many yards can be 
 bought for 5i dollars? ^ws. 2| yards. 
 
 67. If I pay 656J dollars for 75 tons of coal, what 
 is the price per ton ? 
 
 68. If a ton of coal is worth 8| dollars, what will bo 
 the cost of 80 tons ? 
 
 69. If I pay §656.25 for 75 tons of coal, what will 
 be the cost of 80 tons ? 
 
 70. If 75 tons of coal cost §656.25, how many tons 
 can be bought for §700 ? 
 
 71. Ii 80 tons of coal cost §700, how many tons can 
 be bought for §656.25 ? 
 
 72. I^ 75| yards of cloth cost §643.87|, how much 
 ^nust I pay for 36 yards? 
 
 73. When §306 are paid for 36 yards of cloth, how 
 many yards can be purchased for §643 J ? 
 
 74. For 36 yards of cloth 306 dollars were paid. 
 How many yards of the same cloth can be purchased 
 for §643.875 ? 
 
 75. For 75|.jards of cloth' I pay §643.87^. How 
 many yards can I get for §306 ? 
 
 76. How many coats, each containing 1| yards of 
 cloth, can be made of 18| yards? 
 
 77. What will ten barrels of apples cost at 1 J dollars 
 per barrel ? 
 
 78. If 10 boxes of oranges cost 18j dollars, what is 
 the price per box ? 
 
 79. If 10 pounds of copper cost §18.75, what num- 
 ber of pounds can be had for §171.25. 
 
FiiACTlONS. 147 
 
 80. If ^p3|P^ifcds of copper can be liad for 171 J 
 dollars, howma^y pounds will 18| dollars buy? 
 
 81. A merchant sells | of his ship. What part of it 
 does he still own^ Ans. J. 
 
 82. A merchant owning | of a ship sells J of his 
 interest. What part of the ship does he still own ? — 
 (Vide 156, Ex. 2.) Ans. |. 
 
 83. Having | of an apple, I give away half of it. 
 What part of the apple is now gone? Ans. |. 
 
 84. Having | of a ship, I sell half my interest for 
 9420 dollars. What is the whole ship worth at the same 
 rate? 
 
 85. If a §hip is worth |25120, what are | of her 
 worth ? What are | worth ? ' g- ? Ans. -|=|21980. 
 
 86. If I of a ship are worth $15700, what is the 
 value of I ? ^ ' 
 
 87. If J of a bale of cotton are worth 48 dollars, 
 what is the value of f of a bale? Ans. $41^. 
 
 88. What fraction is that to which if i be added, the 
 sum will be 1 ? 2 ? 3 ? 4 ? 5 ? Ans. f , |, etc. 
 
 89. What fraction is that to which if J^be added, 
 the sum will be 1 ? 20 ? 45 T 100 ? Ans. ~'^, %%\ etc. 
 
 90. What number is tllaj which, if it be taken from 
 57, will leave a remainder of | ? | ? ^ J ? J| ? 
 
 A71S. 56|, 56|, etc. 
 
 91. What number is that which, on being added to 
 357/5, will make the sum 455?? ^72^5? 9570^*5? 
 
 Ans. 983 J, etc. 
 
 92. What fraction is that to which if |- of f be 
 added, the sum will be 1 ? 15 ? 8| ? 4,^o ^ 
 
 hsist Ans. 4i§. 
 
 \ 
 
148 FIIACTIOXS. 
 
 93. The sum of two numbers is 47-J-, an{>tlie differ- 
 ence 7l. What are the numbers ? — (Vide 125, Ex. 10.) 
 
 A71S. 21^^ and 19iJ. 
 
 94. If a certain number be divided by 2 J, and the 
 quotient be multiplied by 8i, the product diminished by 
 5 4, the difference increased by 7^, the sum will be 62 f. 
 What is the number? Ans. 18-|. 
 
 95. I have a fortieth interest in an oil well, and am 
 willing to sell half of it for $1340. What is the value 
 of my interest, and of the whole well, at the same rate ? 
 
 Ans. 12680 and $107200. 
 
 3 T 
 
 96. 
 
 What 
 
 is 
 
 i 
 
 of 360 ? 
 
 1 of 720? 
 
 ii 
 
 of 378? - 
 
 of 1643? 
 
 
 
 
 Am 
 
 f. 270, 600, 
 
 351 
 
 , and 636. 
 
 97. 
 
 What 
 
 is 
 
 H 
 
 of 
 
 38? 1 
 
 3 of 331? 
 Ans. ^\8^, 
 
 ifo 
 29f 
 
 f 32|? 
 , and 31 1. 
 
 > 98. 270 is I of what number ? 600 is § of what num- 
 ber? 351 is H of what number? 
 
 99. 636 is Jf of what number? j\%is i| of what 
 number? 29| is J J of what number? 31j is || of 
 what number? 
 
 100. J^of 68 is If of what number? Ans. 58. 
 
 101. 11 of 341 is J J of how many times 5 ? 
 
 Ans. 781 times 5. 
 
 102. i| of 49| is If of 'how many thirds of 18? 
 
 Alts. '9 thirds of 18. 
 
 103. On a trip from New Orleans to New York, I 
 expend i of my money, and still have »^270. What did 
 the trip cost me ? Ans. §90. 
 
 104. During a storm a captain threw overboard ^ of 
 his cargo of cotton, and still lias 600 bales on board. 
 How many bales were thrown overboard? Ans. 120. 
 
FRACTIONS. 149 
 
 105. If, after reserving ^^ of my wine, I sell 351 
 gallons, how many gallons do I reserve ? Ans. 27. 
 
 106. Multiply the fractions J and | by the least 
 common multiple of their denominators. 
 
 Ans. 3 and 4. 
 
 107. Multiply the fractions | and J by the least 
 common multiple of their denominators. 
 
 Ans. 8 and 9. 
 
 108. Multiply the fractions y\ and ^^ Ja^he least 
 common multiple of their denominators. ^^B 
 
 Ans. 12 and 5. 
 
 109. Multiply 35 1 and 15 1 by the least common 
 multiple of their denominators. Ans. 214 and 91. 
 
 110. Multiply ^, -i, and | by the least common mul- 
 tiple of their denominators. Ans. 6, 4, and 9. 
 
 111. Multiply ^, I, y^Q, and §, by the least common 
 multiple of their denominators. 
 
 Ans. 15, 20, 9, and 12. 
 
 112. Multiply j J-, JJ, /g, and | by the least common 
 multiple of their denominators. 
 
 Ans. 44, 33, 32, and 24. 
 
 113. Which fraction is the greater, /^ or -f^ ? 
 
 Ans. ^ 
 
 Remark. — Of two fractions, that wliich gives the gi'eater product 
 on multiplying both by the least common multiple of the denomi- 
 nators is the greater. 
 
 114. Which fraction is the greater, J J or J J ? -J-f or 
 i J ? and by how much ? A71S. ] ^ by ,1 3 ; J J by -,- j^. 
 
 115. Which fraction is greatest, J, /j, or f^ 
 
 Alls. /_. 
 
 116. If A and B together can do /^ of a piece of 
 
150 FRACTIONS. 
 
 work in one day, and A alone can do J of it in one day, 
 what part can B do in one day? Ans. J. 
 
 117. If A and B together can do a piece of work in 
 2| days, and A alone can do it in 4 days, in what time 
 can B alone do the work ? Ans. 5 days. 
 
 118. A and B can do ^^ of a piece of work in one 
 day; A and C can do j% pf'the same work in one day; 
 B and C can do J J in one day. What part of the work 
 could al^M^ether^do in one day ? What part could A 
 alone d^^Bn'e day ? What part could B alone do in 
 one day 'f^ What part could C alone do in one day ? 
 
 A7is.A\\, ij; Ai; Bi; Ci. 
 
 119. A can do a piece of work in 4 days, B in 5 
 days, and C in 6 days. What part of the work can A 
 and B together do in one day ? What part B and C 
 together ? What part A and C together ? What part 
 can all together do in one day ? In how many days can 
 all together do the work? Last Ans. Iff days. 
 
 120. A cistern has three pipes. The first will fill it 
 in 2 hours, the second in 3 hours, the third in 4 hours. 
 In what time will the cistern be filled when the three 
 pipes are running together ? 
 
 Ans. In ]| of an hour. 
 
 121. A cistern has 3 pipes, two at the top and one 
 at the bottom. One of the top pipes would fill it in 5 
 hours, the other in 6; but the pipe at the bottom 
 empties it in 8| hours. In what time will the cistern 
 be filled when the pipes are running together ? 
 
 Ans. In 4 hours. 
 
 122. A man and his wife could drink a cask of beer 
 in 10 iays. In the absence of the man it lasted his 
 
COMPOUND NUMBERS. 151 
 
 -wife 30 days. How long Avould the man be occupied in 
 drinking it? A7is. 15 days. 
 
 123. A, B, and C could do a piece of work in .f'^ 
 days; A, B, and D in | days; A, C, and D in J§ days; 
 B, C, and D in j § days. In what time could they all 
 do the work, and in what time could each man do it 
 alone ? 
 
 Ans. All in Jf days; A in 1; B in 2; C in 3; and D in 4 days. 
 
 COMPOUND NUMBERS. 
 
 DEFINITIONS. 
 
 168. An ABSTRACT NUMBER is a number whose unit 
 has no name other than that given it as a mere number. 
 Thus, 5, 29, 3i, are abstract numbers. 
 
 169. A CONCRETE or DENOMINATE NUMBER is a num- 
 ber ivkose unit has a name other than that given it as a 
 mere number. Thus, 5 dollars, 29 feet, 3 J apples, are 
 concrete numbers. 
 
 ITO. A SIMPLE NUMBER IS a unit or collection of units 
 of the same kind. A simple number is either abstract 
 or co7icrete. Thus, 5, 5 dollars; 3^, 3^ apples, are 
 simple numbers. 
 
 171. A COMPOUND NUMBER is a number consisting of 
 two or more concrete numbers of different unit values, but 
 reducible to a simfle number. Thus, 3 feet 4 inches is 
 a compound number, and equal to 40 inches, which is a 
 simple number; 8 dollars 5 cents = 805 cents. 
 
 1 
 
f 
 
 152 COMPOUND NUMBEilS. 
 
 TABLES OF COMPOUND NUMBERS. 
 
 M N E Y. 
 
 172. United States Money is the national currency 
 of the United States. The relative value of its differ- 
 ent units or denominations has already been given. — 
 (Vide 39-43.) 
 
 173. English Money is the national currency of 
 Great Britain. The units or denominations are named 
 Guinea, Pound, Crown, Shilling, Penny, and Farthing. 
 
 TABLE. 
 
 4 farthings (far.) make 1 penny, abbreviated d. (denarius.) 
 12 pence " 1 shilUng, " s. (solidus.) 
 
 20 shillings " 1 pound, " £. (hbra.) 
 
 21 shillings " 1 guinea, " G. 
 
 5 shillings " 1 crown, " Cr. 
 
 Remark 1. — The Pound, also called Sovereign, is the Primary 
 Unit of English Money, and is Avortli $4.84. X Crown is worth 
 §1.21; a Shilling, $0,242; a Penny, $0.0201; a Farthing, G^Jj mills. 
 
 Remark 2. — The Sovereign is 22 carats fine, the other 2 parts 
 being copper; it weighs 5 dwts. 3-J-|J- gx*s. 
 
 174. French Money is the currency of the Empire 
 of France. The Franc is the unit, and is worth 18 1 
 cents. 
 
 EXERCISES. 
 
 1. In 2 dollars how many cents? 3? 4? 5? 15? IJ? 
 
 2. In 200 cents how many dollars? 300? 400? 500: 
 1500? 150? 
 
 3. In U eagles how many mills? 2? 2^? 3}? 
 
 4. In 15000 mills how many eagles? 20000? 25000? 
 32500? 
 
COMPOUND NUMBERS. 153 
 
 5. In 2 pounds how many shillings ? 3? 4? 5? 7? 
 13? 15? 25? 
 
 6. In 40 shillings how many pence? 60? 80? 100? 
 140? 260? 300? 500? 
 
 7. In 480 pence how many farthings? 720? 960? 
 1200? 1680? 3120? 3600? 6000? 
 
 8. In 1920 farthings how many pence? 2880? 3840? 
 4800? 6720? 12480? 14400? 24000? 
 
 9. In 480 pence how many 'shillings? 720? 960? 
 12.00? 1880? 3120? 3600? 6000? 
 
 10. In 40 shillings how many crowns? 60? 80? 100? 
 140? 260? 300'? 500? 
 
 WEiaHT. 
 
 175. Troy Weight is used in weighing gold, silver, 
 and precious stones. The units are named Grain, 
 Pennyweight, Ounce, and Pound. 
 
 TABLE. 
 
 24 grains (gr.) make 1 pennyweight, abbreviated dwt. 
 
 20 pennyweights " 1 ounce, " oz. 
 
 12 ounces " 1 pound, " lb. 
 
 Remark 1. — The Pound is the Primary Unit of Troy Weight, and 
 is determined by the weiglxt of 22.794422 cubic inches of distilled 
 water. 
 
 JIemark 2. — The carat by which the diamond is weighed and 
 valued is equal to 4 grains, 
 
 EXERCISES. 
 
 1. In 2 lbs. how many grs.? 3? 
 4? 5? 6? 13? 15? 27? 
 
 3. In 23 oz. how many grs.? 
 25? 26? 27? 28? 29? 
 
 2. In 11520 grs. how many lbs.? 
 17280? 23040? 28800? 74880? 
 
 4. In 11040 grs. how many oz.? 
 12000? 12480? 12960? 
 
ail 
 
 COMPOUND NUMBERS. 
 
 5.|[n 1 lb. how many|oz.? how 
 Vmany dwt.? how many gr. ? 
 7. In 3 J lb. how many gr. ? 
 
 6. In 5760 gr. how many dwt.? 
 how many oz.? how many lb.? 
 8. In 20160 gr. how many lb.? 
 
 /170) Avoirdupois Weight is used in weighing gro- 
 ceries, and cheap commodities of every description. 
 fThc units are named Dram, Ounce, Pound, Quarter, 
 Hundredweight, and Ton. 
 
 abb] 
 
 ^eviatc 
 
 )d oz. 
 
 
 a 
 
 lb. 
 
 
 u 
 
 qr. 
 
 ht, 
 
 n 
 
 cwt. 
 
 
 u 
 
 T. 
 
 TABLE. 
 
 16 drams" .) n^^ke 1 ounce, 
 
 16 ounces.'. -^ 1 pound, 
 
 25 pounds ^ 1 quarter, 
 
 4 quarters ^ 1 hundredweight, 
 
 20 hundredweight " 1 ton. 
 
 Remark 1. — The Pound is the Primary Unit of Avoirdupois 
 Weight, and is determined by the weight of 27.701554 cubic inches 
 of distilled water. 
 
 Remark 2. — The Troy Pound is the same as \ji lbs. Avoirdu- 
 pois, and the Troy Ounce is the same as ^|| oz. Avoirdupois. 
 
 EXERCISES. 
 
 1. In 1 T. how many dr.? 
 3. In 1 cwt. how many oz.? 
 
 5. In 17^. how many dr.? 
 
 7. In 37 lb. how many dr.? 
 
 9. In 47 dr. how many/oz.? 
 
 11. In 13 cwt. how many qr.? 
 
 13. In 113 T. how many dr.? 
 
 33? 45? 127? 254? 
 
 15.' In 21 lb. how many oz.? 23? 
 17? 13? 15? 19? 
 
 2. In 512000 dr. how many T.? 
 4. In 1600 oz. how many cwt.? 
 6. In 272 dr. how many oz.? 
 
 8. In 9472 dr. how many lb.? 
 
 10. In 18800 oz. how many qr.? 
 
 12. In 52 qr. how many cwt.? 
 
 14. In 57856000 dr. how many T.? 
 16896000? 23040000? G5024000? 
 
 16. In 336 oz. how many lb.? 
 368? 272? 208? 240? 304? 
 
 ^7^ At 1 cent per ounce, what v/ill a ton of raisins 
 cost? Jns. P20.00. 
 
 18. At 50 cents per pound, what will 7 tons of butter 
 cost? Ans. $7000.00. 
 
 /V^ 
 
COMPOUND NUxMBERS. 
 
 155 
 
 19. At 5^0 cents per pound, how many tons of lard 
 can be bought for $35000 ? ' ^ Ans. 85. 
 
 177. Apothecaries Weight is used in mixing medi- 
 cines. The units arc named Grain, Scruple, Dram, 
 Ounce, and Pound. 
 
 TABLE. 
 
 20 grains (gr.) make 1 scruple, marked . . 9 
 
 3 scruples " 1 dram, " • . 5 
 
 8 drams " 1 ounce, " • • ^ 
 
 12 ounces " 1 pound, " . . lb 
 
 Remark. — The Pound is the Primary Unit, and is the same as 
 the Pound Troy. 
 
 EXERCISES. 
 
 I. In 2 ft) how many §? 5? 7? 
 9? 11? 13? 15? 
 
 3. In 2 ft) how many 5? 6? 
 8? 10? 12? 14? 
 
 5. In 15 ft) how many 9? 17' 
 19? 21? 23? 25? 29? 
 
 7. In 16 ft) how many gr.? 18 
 20? 22? 24? 2G? 28? 
 
 9. In ^ lb how many 5? -J? | 
 
 II. In f of an § how many 9 
 
 t? F I? t\? 
 
 13. In y'j of a 5 ^^^^ many 
 
 2. In 24 § how many ft)? 60? 
 84? 108? 132? 156? 180? 
 
 4. In 192 3 how many ft)? 576? 
 768? 960? 1152? 
 
 6. In 4320 g how many ft)? 
 4890? 5472? 6048? 6G24? 7200? 
 
 8. In 23040 gr. how many ft)? 
 34560? 74880? 104440? 
 
 10. In 48 5 how many ft)? 32? 
 24? IG? 13f? 
 
 12. In 18 9 how many § ? 16? 
 20? 21? 10? 
 
 14. In 35 gr. how many 5? IG? 
 54? 45? 21? 
 
 LINEAR MEASURE. 
 
 178. Long Measure is used in measuring the length 
 of all quantities, except cloth. The units are named 
 Inch, Foot, Yard, Rod, Furlong, Mile, and League. 
 
156 COMPOUND NUMBERS. 
 
 TABLE. 
 12 inches (in.) make 1 foot, abbreviated ft. 
 " 1 yard, " yd. 
 
 " 1 rod, " r. 
 
 " 1 furlong, " fur. 
 
 " 1 mile, " m. 
 
 " 1 league, " 1. 
 
 IvEMARK 1. — The Imperial Yard is the standard of English linear 
 measure, and is determined by the length of the pendulum vibra- 
 ting once a second at London, temperature 62} degrees Fah. This 
 length is 39.1393 inches. 
 
 Remark 2. — Gunter's chain, used in surveying land, is 4 rods 
 long, and consists of 100 links. 
 
 EXERCISES. 
 
 3 
 
 feet 
 
 H 
 
 yards 
 
 :0 
 
 rods 
 
 8 
 
 furlongs 
 
 3 
 
 miles 
 
 1. In 1 m. how many in.? 
 
 3. In 7 fur. how many in.? 
 
 5. In 2 r. how many yd.? 3? 
 4? 5? 6? 75? 
 
 7, In 3 r. how many in.? 12? 
 13? 15? 17? 19? 
 
 9. In 1 fur. how many in.? 
 11. In 1 1. how many ft.? 
 
 2. In 63360 in. how many m.? 
 4. In 55440 in. how many fur.? 
 6. In 11 yd. how many r.'i 
 16^? 22? 27J? 33? 412}? 
 
 8. In 594 in. hov/ many r.^ 
 2376? 2574? 2970? 3366? 3762? 
 10. In 7920 in. how many fur.? 
 12. In 15840 ft. how many 1.? 
 
 13. How many inches through the earth from pole to 
 pole?— (Vide 67, Ex. 54.) Ans. 500478929.28. 
 
 14. How many inches through the earth at the equa- 
 tor? Ans. 502143776.64. 
 
 15. In 1 link of Gunter's chain how many inches? 
 
 Ans. 7 If inches. 
 
 16. In 1 mile how many chains? Ans. 80 chains. 
 179. Cloth Measure is used in measuring goods 
 
 bought or sold by the yard. The units are named Inch, 
 Nail, Quarter, Yard, Ell Flemish, Ell English, and Ell 
 French. 
 
COMPOU^^'D NUMBERS. 
 
 157 
 
 2:^ inches 
 4 
 
 ike 
 
 nails 
 
 quarters 
 
 quarters 
 
 quarters 
 
 quarters 
 
 TAELE. 
 
 
 
 
 1 nail, 
 
 abbrev. 
 
 na. 
 
 1 quarter, 
 
 
 
 qr. 
 
 1 yard, 
 
 
 
 yd. 
 
 1 ell Flemish, 
 
 
 
 E. Fl. 
 
 1 ell English, 
 
 
 
 E. E. ^ 
 
 1 • ell French, 
 
 
 
 E. F. 
 
 of this measure is 
 
 that of Long 
 
 Remark 1. — The stau 
 Measure. 
 
 Remark 2. — In mercantile practice only the yard and quarter 
 are in general use. 
 
 EXERCISES. 
 
 1. In 1 yd. how many in.? 
 
 3. In 3 qr. how many in.? 
 
 5. In 5 E. Fl. how many in.? 
 
 7. In 7 E. E. how many in.? 
 
 9. In 8 E. Fl. how many in.? 
 
 11. In 24 E. Fl. how many yd.? 
 
 13. In 70 E. E. how many yd.? 
 
 15. In 120 yd. how many E. FL? 
 
 17. In 1 E. F. how many in.? 
 
 2. In 30 in. how many yd.? 
 
 4. In 27 in. how many qr.? 
 
 6. In 135 in. how many E. FL? 
 
 8. In 315 in. how many E. E.? 
 10. In 216 in. how many E. FL? 
 12. In 18 yd. how many E. FL? 
 14. In 87^ yd. how many E. E. ? 
 16. In 160 E. FL how many yd.? 
 18. In 54 in. hoAV many E. F.? 
 
 19. In 18360 inches how many quarters? yards? 
 ells Flemish ? ells English? ells French? 
 
 Ans. 2040 ; 510 ; 680 ; 408 ; 340. 
 
 20. What will -| of a yard of cloth cost at 7 cts. per 
 nail ? Atis. 98 cts. 
 
 21. What will ^^ of a yard of calico cost at 15 cts. 
 per nail ? Aiis. 12 cts. 
 
 22. In 2.5 feet how many inches? Aiis. 30 in. 
 
 23. In 3.75 furlongs how many rods? An§. 150 r. 
 
 24. What is the number of miles from the Equator to 
 the North Pole?— (Vide 67, Ex. 61.) 
 
 Ans. 6213.824 m. 
 
158 
 
 COMPOUND IsUMBERS. 
 
 SUPERFICIAL OR SQUARE MEASURE. 
 
 ISO. Square Measure is used in measuring sur- 
 faces; as land, plastering, etc. The units are named 
 Square Inch, Square Foot, Square Yard, Square Rod, 
 Rood, Acre, and Square Mile. 
 
 9 square feet 
 30J square yards 
 40 square rods 
 4 roods 
 G40 acres 
 
 1 Inch. 
 
 sq. yd. 
 sq. r. 
 R. 
 A. 
 M. 
 
 TABLE. 
 
 144 square inches (sq. in.) make 1 square foot, abb. sq. ft. 
 
 " 1 square yard, 
 " 1 square rod, 
 " 1 rood, 
 " 1. acre, 
 " 1 square mile, 
 
 Remark 1. — The standard is the same 
 as that of Long Measure. 
 
 Remark 2. — 16 square rods make 1 
 i_i square chain, and 10 square chains make 
 ^ 1 acre. 
 
 g Remark 8, — The figure in the margin 
 is exactly 1 square inch; that is, it is 
 1 linear inch on each side. 144 such 
 squares are equivalent to a square foot, 
 however the arrangement may be. They 
 are equal to a square foot when arranged so as to make another 
 square. 
 
 EXERCISES. 
 
 1 SQUARE INCH. 
 
 1 L\Gir. 
 
 1. In 1 A. how many sq. in.? 
 
 3. In 1 A. how many sq. r.? 
 2? 3? 4? 5? 
 
 5. In 1 sq. r. how many sq. ft.? 
 2? 5? 8? 
 
 7. In 1 R. liow many sq. yd.? 
 3? 13? 17? 
 
 2. In 6272640 sq. in. how many 
 A.? 
 
 4. In IGO sq. r. how many 
 A.? 320? 480? 640? 800? 
 
 6. In 272J sq. ft. how many 
 sq. r.? 544^? 1361^^? 2178? 
 
 8. In 1210 sq. yd. how many 
 R.? 3030'' ir)7:50? 20570? 
 
COMPOUND NUMBER! 
 
 159 
 
 0. la 1 sq. r. how many sq. 
 in.? 
 
 11. In 1 A. lioAv many sq. 
 ft.? 
 
 13. In 1 sq. yd. how many sq. 
 in.? 121? 242? 
 
 10. In 39204 sq. in. how many 
 sq. r.? 
 
 12. In 130680 sq. ft. how many 
 A.? 
 
 14. In 1296 sq. in, how many 
 sq. yd.? 156816? 313632? 
 15. In 252| A. how many sq. I 16. In 2524 sq. ch. how many 
 ch.? A.? 
 
 17. In -^^ m. how many fur.? 18. In ^j fur. how many r.? 
 
 Ans. 5^j. I Ans. Z^j. 
 
 SOLID MEASURE. 
 181. Cubic Measure is used in measuring solids, as 
 timber, earth, and such other things as have length, 
 breadth, and thickness. The units are named Cubic 
 Inch, Cubic Foot, Cubic Yard, Ton, Cord Foot, and 
 Cord. - 
 
 TABLE. 
 1728 cubic inches (cu. in.) make 1 cubic foot, abb. cu. ft. 
 
 ton, " 
 
 ton of shipping, " 
 cord foot, " 
 
 cord, " 
 
 cu. yd. 
 
 T. 
 
 T. 
 
 T.ofS. 
 
 CO. ft. 
 CO. 
 
 27 cubic feet " 1 cubic yard, 
 
 40 feet of round timber " 1 ton, 
 
 50 feet of hewn timber " 1 
 
 42 cubic feet " 1 
 
 16 cubic feet " 1 
 
 8 cord feet " 1 
 
 Remark 1. — The standard of this meas- 
 ure is that of Long Measure. 
 
 Remark 2. — A cube is a solid bounded 
 by 6 equal squares. 
 
 Remark 3. — The figure in the margin 
 represents an exact cubic inch. Its 
 squares are called faces, and the bounda- 
 ries of the faces are called edr/es. Each 
 edge represents 1 linear inch. Each edge 
 of a cubic foot contains 12 linear inches, so that there are 
 12x12x12: that is, 1728 cubic inches in a cubic foot. 
 
 / 
 
 
 /' 
 
 
 1 Cubic Inch. 
 
 
 / 
 
 Y 1 Linear Inch. 
 
 / 
 
160 
 
 COMPOU]S'D NUMBERS. 
 
 EXERCISES, 
 
 1. In 1 CO. liow many cu. ft.? 
 
 2? 3? 4? 10? 
 
 1-V? 
 
 3. In 1 CO. liow many cu. in.? 
 6? 6? 7? ^? -I? f? 
 
 5. In 3.125 CO. liow many cu. ft. ? 
 7. In 1 cu. yd. how many cu. 
 
 2. In 128 cu. ft. how many co.? 
 256? 384? 512? 1280? 96? 192? 
 
 4. In 221184 cu. in. how many 
 CO.? 27648? 41472? 
 
 6. In 400 cu. ft. how many co.? 
 
 8. In 46656<»cu. in. how many 
 cu. yd.? 3888? 324? 36? 
 
 182. Wine Measure is used for measuring alh 
 liquors, except ale, beer, and milk. The units are 
 named Gill, Pint, Quart, Gallon, Tierce, Barrel, Hogs- 
 head, Pipe, and Tun. 
 
 TABLE. 
 4 gills (gi.) make 1 pint, abbreviated pt. 
 
 2 pints 
 
 4 quarts 
 
 31J- gallons 
 
 42 gallons 
 
 63 
 
 2 
 
 gallons 
 hogsheads 
 
 pipe 
 
 1 quart, " 
 
 1 gallon, " 
 
 1 barrel, " 
 
 1 tierce, ^" 
 
 1 hogshead, " 
 
 1 pipe, " 
 
 1 tun. ^ 
 
 qt. 
 
 gal. 
 
 bbl. 
 
 ti. 
 
 hhd. 
 
 py 
 
 Remark. — The Wine- Gallon contains 231 cubic inched 
 
 1. In 1 tun how many hhd.? 
 gal.? qt.? pt.? gi.? 
 
 3, In 13 gal. how many gi.? 
 15? 17? 19? 21? 23? 
 
 6. In 5 tuns how many gi.? 7? 
 9? 11? 13? 17? 
 
 7. In 3 bbl. how many gal.? 
 4? 7? 10? 13? 16? 
 
 9. In 1 hhd. how many bbl.? 
 11. In 126 ti. how many hlid.? 
 13. In 2.5 qt. how many gal.? 
 15. In 23.625 gal. how many hhd ? 
 
 EXERCISES. 
 
 2. In 8064 ^\A\( 
 pi 
 
 (ii€ , -1 i 
 
 hhd.? pi.? tuns? 
 
 4. In 416 gi. how many gal.? 
 480? 544? 608? 072? ^? 
 
 6. In 40320 gi. how m^y= tuns/ 
 56448? 72-576? 88704? 
 
 8. In 94J.,gal. how many bbl.? 
 126? 220i'j^l5? 409 J? 
 10. In 1«"0 bbl. how many hhd.? 
 12. In 168 bbl. how many ti.? 
 14. In .625 gal. how many qt.? 
 16. In .375 hhd. how many gnl.? 
 
3^ 
 
 COMrOUXD XUxMBERS. 161 
 
 183. Ale or Beer Measure is used for measuring 
 ale, beer and milk. The units are named Pint, Quart, 
 Gallon, Barrel, and Hogshead. 
 
 table. 
 
 2 pints (pt.) make 1 quart, abbreviated qt. 
 
 4 quarts " 1 gallon, " ' gal. 
 
 36 gallons " 1 barrel, " bbl. 
 
 \\ barrels . " 1 hogshea'd, " hhd. 
 
 Remark. — The Beer Gallon contains 282 cubic inches. 
 EXERCISES. 
 
 1. In I hhd. how many gal.? 
 qt..? pt.? / 
 
 3. In 693 Beer Gal. how many- 
 Wine Gal.? 2079? 
 
 6. In Jg of a gal. how many pt. ? 
 
 7. In -^-^ of a hhd. how mamy 
 pt.? ' '"^ 
 
 9. In \ of a bbl. how many 
 qt.? 
 
 2. In 432 pt. how many qt.? 
 gal.? bbl.? 
 
 4. In 846 Wine Gal. how many 
 Beer Gal.? 2538? 
 
 6. In I a pt. how many gal.? 
 
 8. In 1 pt. what part of a 
 hhd.? 
 
 10. In 36 qt. what part of a 
 bbl.? 
 
 184. Dry Measure is used in m^easuring such arti- 
 cles as grain, fruit, etc. The units are named Pint, 
 Quart, Peck, Bushel, and Quarter. 
 
 TABLE. 
 
 2 pints (pt.) make 1 quart, abbreviated qt. 
 
 8 quarts " 1 peek, " pk. 
 
 4 pecks " 1 bushel, " bu. 
 
 8 bushels " 1 quarter, " qr. 
 
 Remark. — The Winchester Bushel is a cylinder, 18^- inches 
 internal diameter, and 8 inches deep. It contains 2150.4 cubic 
 inches. 
 
 14 
 
162 COMPOUJS^D XUMBEilS. 
 
 EXERCISES. 
 
 1. What cost 25 quarters of wheat at 90 cents per 
 bushel? A71S. $180. 
 
 2. At 90 cents per bushel how many quarters of 
 wheat can be bought for $360 ? Ans. 50 quarters. 
 
 3. What cost 17 bushels of apples at 27 cts. a peck? 
 
 4. At 27 cents a peck, how many bushels of apples 
 can be bought for $18.36 ? 
 
 5. What must be paid for 7 bushels of chestnuts at 
 3 cents a pint? Ans. $13.44. 
 
 6. What cost I of a pint of blackberries at $3.20 per 
 bushel? Ans. 3 cents. 
 
 7. W^hat cost 25-J bushels of potatoes at 20 cents a 
 peck? ^ns. $20.70. 
 
 8. How many bushels of potatoes can I buy for 
 $41.40, at 2-J cents a quart? Ans. 51|- bushels. 
 
 TIME. 
 
 185. The units of Time are named Second, Minute, 
 Hour, Day, Week, Month, Year, Century. 
 
 TABLE. 
 
 GO seconds (sec.) make 1 minute, abbreviated m. 
 
 60 minutes " 1 hour, " h. 
 
 24 hours " 1 civil day, " d. 
 
 % days " 1 week, " w. 
 
 12 months " 1 year, " y. 
 
 Remark 1. — The standard unit of time is the period occupied by 
 
 the eurth in making one revolution on its axis, wliich period is 
 
 called a Sidereal Day and consists of 23 h. 5G m, 4 sec. 
 
 Remark 2.— The Tropical Year consistR of 305 d. 5 h. 4S m. 47.57 
 
 S3C. 
 
COM IHJ V .\ D N U M 13 ERS . 
 
 163 
 
 REMAini: 3.— The Civil, Legal, or Julian Year consists of 365 
 days, except Leap Year, "tvhich consists of 300 days. 
 
 Remark 4. — Every year which is exactly divisible by 4 is a 
 Leap Year, excepting those centennial years not exactly divisible 
 by 400. Thus, 1868 will be a Leap Year; 1900 will not be a Leap 
 Year; but the year 2000 will be a Leap Year. 
 
 TABLE OF THE MGNTIH 
 
 Mouth. 
 
 Abb. 
 
 Order. 
 
 Xo. D. 
 
 Mouth. 
 
 Abb. 
 
 Order. 
 
 Xo. D. 
 
 January. 
 
 Jan. 
 
 1st. 
 
 31. 
 
 July. 
 
 
 7th. 
 
 31. 
 
 February. 
 
 Feb. 
 
 2d. 
 
 28. 
 
 August. 
 
 Aug. 
 
 8th. 
 
 31. 
 
 March. 
 
 Mar. 
 
 3d. 
 
 31. 
 
 September, 
 
 Sept. 
 
 9th. 
 
 30. 
 
 April. 
 
 Apr. 
 
 4th. 
 
 30. 
 
 October. 
 
 Oct. 
 
 10th. 
 
 31. 
 
 May. 
 
 
 5th. 
 
 31. 
 
 November. 
 
 Nov. 
 
 11th. 
 
 30. 
 
 June. 
 
 
 6th. 
 
 30. 
 
 December. 
 
 Dec. 
 
 r2th. 
 
 31. 
 
 Remark 5. — In Leap Year, February has 29 days. 
 Remark 6. — In finding the interval between two dates, it is 
 customary to consider the months as having 30 days each. 
 
 TABLE 
 
 Showing the time in days from any day in one month to the corref^pond- 
 ing day in another month. 
 
 January... 
 February... 
 
 March 
 
 April 
 
 Mav 
 
 June 
 
 •'^ny 
 
 AugXTSt 
 
 Sepr'ember. 
 
 October 
 
 November . 
 December.. 
 
 365 
 1 334 
 
 ■306! 
 
 !275| 
 
 i245 
 
 l214! 
 
 184 
 
 153 
 
 122 
 
 92 
 
 61 
 
 31 
 
 31 
 365 
 337 
 306 
 
 59 
 
 28 
 
 365 
 
 334 365 
 276|304l335 
 245I273!304 
 21o;2!3274 
 184 212 243 
 
 89 
 
 61 
 
 30 
 
 365 
 
 90;120il51181 
 
 59 
 
 31 
 
 120 150 
 
 1334 365 
 304 335 
 
 122 
 91 
 61 
 30 
 
 365 
 
 212243 
 1811212 
 153;184 
 122 153 
 
 t\^ 
 
 153 
 
 123 
 
 92 
 
 62 
 
 181J212 
 
 151 i 182 
 
 120;i51 
 
 90:i21 
 
 242!273!303 
 2121243:273 
 18l!212;242 
 151:182212 
 
 92 
 
 61 
 
 31 
 
 365 
 
 334 
 
 123 
 92 
 62 
 31 
 
 365 
 
 2731304 334 
 242i273 303 
 214|245j275 
 183l214|244 
 !153 1841214 
 
 153{1 
 
 122 
 
 92 
 
 61 
 
 30 
 [365 
 13341365 
 
 3041335 
 273|304 
 243:274:3041 3351 365 
 
 1231153 
 
 92J122 
 
 61 91 
 
 3l| 61 
 
 30 
 
164 COMPOUND NUMBERS. 
 
 EXERCISES. 
 
 1. How many days from January 10th to June 
 10th? Ans, 151. 
 
 Find January in the left-hand column, and follow the line to 
 the right till you come to June, 
 
 2. How many days from February 6th to May 
 6th? Ans. 89. 
 
 3. How many days from January 1st to July 
 4th? ■ Ans. 184. 
 
 Here add 3 to the tabular number, which is 181. 
 
 4. How many days from December 25th to July 
 10th? Ans. 197. 
 
 Here subtract 15 from the tabular number, 212, 
 
 5. How many days from January 17, 1868, to July 
 17, 1868? Ans. 182. 
 
 Here add 1 to the tabular number for Leap Year, as the dates 
 include the month of February, 
 
 CIRCULAR MEASURE. 
 
 186. Circular Measure is used in estimating Lati- 
 tude and Longitude, and in measuring the relative dis- 
 tances of the Planets and other heavenly bodies. The 
 units are named Second, Minute, Degree, Sign, and 
 Circumference of Circle. 
 
 TABLE. 
 
 60 seconds (") make 1 minute, marked ' 
 60 minutes " 1 degree, " ° 
 
 30 degrees " 1 bign, abbreviated S. 
 
 12 signs make 1 circumference of circlcj abb. circ. 
 
COMPOUND NU3IBE11S. 
 
 166 
 
 Remark 1. — The length of a degree measured on the equatoi- is 
 69.161 m. The length of a degree measured on a meridian is 
 69.042 m. 
 
 Remark 2. — A degree contains 60 geo- 
 graphic miles. 
 
 Remark 3. — Since every circumference of 
 a circle contains 360 degrees, the length of 
 the degree varies as the diameter of the cir- 
 cle varies. The circumference of a circle 
 is always about 3.1416 times its diameter. 
 
 Remark 4. — The Celestial Equator is di- 
 vided into 12 Signs; their names are Aries, 
 Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Saggitarius, 
 Capricornus, Aquarius, and Pisces. 
 
 EXERCISES. 
 
 1. In 1 circle how many degrees? minutes? seconds? 
 
 2. What is the length of 1' along the equator? 
 
 AiiB. 1.15268 m. 
 
 3. What is the length of V along a meridian ? 
 
 Am. 1.1507 m. 
 
 4. What is the length of V along the equator? 
 
 An%. .01921. 
 
 5. What is the length of V^ along a meridian ? 
 
 Am, .01918 nearly. 
 
 6. If the semi-circumference of a circle is 3.1415926- 
 535897932, etc., inches in length, as it really is when 
 the diameter is 2 inches, what is the length of 1' ? 
 
 Am. .000290888298665721 in. 
 
 7. What is the circumference of a circle whose diam- 
 eter is 10 inches? Am. 31.4159 in. 
 
 8. What is the circumference of a circle whose diam- 
 eter is 10,000,000 miles? 
 
 Am. 31415926.535 m. 
 
166 
 
 COMPOUND ^'UMBERS. 
 
 KEDUCTION OF COMPOUND NUMBERS. 
 
 187. Reduction in Aritlimetic consists in making 
 some change in the method of i^epresenting a quantity. 
 Hence, reduction makes no change upon the value of a 
 quantity. 
 
 188. It has been seen that many simple concrete 
 numbers may be reduced to other concrete numbers of 
 a lower unit value By multijMcation, Thus, 
 
 2 £=40 s., because 20x2=40. (Vide 174, Ex. 5.) 
 
 189. *lt has also been seen that simple concrete 
 numbers may be reduced to other simple concrete num- 
 bers of a higher unit value by division. Thus, 
 
 72 in.= 6 ft., because 72-f-12=6. (Vide 178.) 
 
 190- To reduce a compound 
 number to a simple concrete 
 number, 
 
 Arrange the different units com- 
 posing the compound number in a 
 horizontal line^ and over each place 
 the number connecting it with the 
 next higher unit. 
 
 Multiply the units of the highest 
 value hy that number which stands 
 over the next lower units, and to the 
 ])roduct add the same loiver units; 
 then multiply the sum by the num- 
 ber standing over the next lower 
 units, etc., continuing the work till 
 the lowest units given have been 
 added. 
 
 EXAMPLES. 
 
 1. Reduce 17 £ 68. Od. 8 far. to 
 fartliinss. 
 
 191. To reduce a simple con- 
 crete number to a compound 
 number. 
 
 Divide the given number by that 
 number which connects it ivith the 
 unit of the next higher value, plac- 
 ing the remainder, if there be any, 
 to the right. 
 
 Continue the work till the unit 
 of the highest value is reached, or, 
 till the next divisor would be greater 
 than the dividend. 
 
 Ulc last quotient and the several 
 remainders, written in the order of 
 their tmit values, will be the com' 
 2^ound number. 
 
 E X A ai P L E S . 
 
 2. Reduce 1G599 farthings to 
 pounds, etc. 
 
CO M PO L X D >: U:.I BERS . 
 
 167 
 
 b. 
 
 7. 
 
 9. 
 11. 
 grai 
 13. 
 15. 
 17. 
 19. 
 21. 
 23. 
 25. 
 
 OPERATION. 
 20 12 4 
 
 17 5 9 3 
 20 
 
 345 
 
 12 
 
 4149 
 4 
 
 16599 far. Ans. 
 
 Reduce 20 <£ 15 s. 9d. 3 far. 
 Reduce 240£ Os. 7d. 2 far. 
 Reduce 17s. Ifar. to far. 
 Reduce lib. loz.ldwt, Igr. 
 Reduce 17 lb. 5 dwt. to 
 ns. 
 
 Reduce Soz, 3 dwt. 3gr. 
 Reduce 7 T. 15 oz. to oz. 
 Reduce 3T. 7cwt. 3qr. 
 Reduce 3qr. 11 oz. 13 dr. 
 Reduce 1 lb 3 § 55. 
 Reduce lib 2 3 to grains. 
 Reduce 5r. 4 yd. 2 ft. 7 iu. 
 
 OPERATION. 
 5K 3 12 
 
 5 4 2 7 
 54 
 
 3U 
 3^ 
 
 96^ 
 12 
 
 OPERATION. 
 
 1G599 
 
 12 
 
 20 
 
 4149—3 
 
 345—9 
 
 4. 
 
 6. 
 
 8. 
 10. 
 12. 
 etc. 
 14. 
 16. 
 18. 
 20. 
 22. 
 24. 
 26. 
 
 17—5 
 17-£ 5s. 9d. 3far. Ai 
 
 Reduce 19959 far. to £, etc. 
 Reduce 230430 far. to £. 
 Reduce 817 far. to s., etc. 
 Reduce 6265 gr. to pounds. 
 Reduce 98040 gr. to pounds, 
 
 Reduce 1515gr. to oz., etc. 
 Reduce 224015 oz. to T., etc. 
 Reduce 27;igr. to T., etc. 
 Reduce 19389 dr. to qr., etc. 
 Reduce 7500 gr. to To, etc. 
 Reduce 5800 gr. to lb, etc. 
 Reduce 1165 in. to r., etc. 
 
 OPERATION. 
 
 1165 
 97—1 
 32—1 
 
 64 =lialf yds. 
 5— 4 J 
 
 1165 iry.AriS. ^/Js. 5r. 4)/^yd.lft.lin.=5r. 4j'd.2ft.7m. 
 
 27. Reduce 401. 6 fur. 2 in. to I 28. Reduce 7650722 in. to 
 inches. | leagues, etc. 
 
 29. Reduce 22 fur. IGr. 3yd. | 30. Reduce 1479-4 ft. to fur- 
 
 1ft. to ft-et. longs, etc. 
 
168 
 
 31. Reduce 4 m. 7 fur. 20 r 16 
 ft, to inches. 
 
 33. Reducel yd. Iqr, 2na. 7}in. 
 
 35. Reduce 4 tuns 5hhd. 3qt. 
 to quarts. 
 
 37. Reduce Itun Igal. 3qt. to 
 gills. 
 
 39. Reduce 47bbl. 18 gal. of 
 ale to pints. 
 
 41. Reduce 15 bu. 2pk. 7qt. to 
 quarts. 
 
 43. Reduce 9bu. 5qt. Ipt. to 
 pints. 
 
 45. Reduce 14 A. IR. 17 r. to 
 rods. 
 
 47. Reduce 17 A. 3R. 12 r. to 
 square feet. 
 
 49. Reduce 3 da. 55 m. to min- 
 utes. 
 
 51. Reduce 9S. 13° 25^ to sec- 
 onds. 
 
 63. Reduce 25° 14^ V^ to sec- 
 onds. 
 
 55. Reduce 5 fur. 3 r. 10 ft. 6 in. 
 to inches. 
 
 57. Reduce 21$ 3 m. to mills. 
 
 COMPOUND NUMBERS. 
 
 32. Reduce 313032 in. to miles, 
 
 etc. 
 
 34. Reduce 50 in. to yards, etc. 
 36. Reduce 5295 qt. to tuns, 
 
 etc. 
 
 38. Reduce 8120 gi. to tuns, 
 etc. 
 
 40. Reduce 13680pt. to barrels, 
 etc. 
 
 42. Reduce 503 qt. to bushels, 
 etc. 
 
 44. Reduce 587 pt. to bushels, 
 etc. 
 
 46. Reduce 2297 r. to acres, 
 etc. 
 
 48. Reduce 776457 sq. ft. to 
 acres, etc. 
 
 50. Reduce 4375m. to days, 
 etc. 
 
 52. Reduce 1020300^^ to signs, 
 etc. 
 
 54. Reduce 90847^'' to degrees, 
 etc. 
 
 56. Reduce 40320 in. to fur- 
 longs. 
 
 68. Reduce 21003 m. to dollars. 
 
 59. What cost 1 lb. 1 oz. 1 dwt. 1 gr. of gold, at 3^ 
 cts. per grain? Ans. 208.83 J. 
 
 60. If gold is worth 3 J cts. per. grain, hoy>^ many 
 pounds can be bought for $626.50 ? 
 
 Ans. 3 lb. 3 oz. 3 dwt. 3 gr. 
 
 61. What cost 17 lb. 5 dwt. of silver, at 31^ cts. 
 per dwt. ? 
 
 62. What weight of silver can be bought for |1260.- 
 93|-, at the rate of 31 J cts. per dwt? 
 
COMPOUXD NUMBERS. 1G9 
 
 63. What will 3 T. 7 cwt. 3 qr. of rice cost at 3d. 
 English money per- pound ? 
 
 64. How much rice at 3d. per pound can be bought 
 for £84 13s. 9d.? 
 
 65. At the rate of |0.060J per pound, how many tons 
 of hay can be had for $4D9.887i ? 
 
 66. What cost 4 m. 7 fur. 20 r. 16 ft. of railroad, at 
 an expense of |3.78f| per ft.? 
 
 67. How many miles of railroad can be built for 
 ^98810.60f §, at an expense of 3f | dollars per ft. ? 
 
 68. What will 4 tuns 5 hhd. 3 qt. of molasses cost, 
 at 15 cts. per qt. ? 
 
 69. What number of tuns of claret can be had for 
 $794.25, at the rate of 7J cts. per pt,? 
 
 70. What will 15 bu. 2 pk. 7 qt. of chestnuts amount 
 to in dollars, at 3 farthings English money per pt. ? 
 
 71. How many bushels of chestnuts can be had for 
 §15.215-1 at the rate of IJd. per qt.? 
 
 72. If a rod of land produce 3i pounds of cotton, 
 how many bales can be taken from 14 A. 1 R. 17 r. of 
 land, at 500 pounds to the bale ? 
 
 73. If a rod of land produces 3| pounds of cotton, 
 what number of acres will produce 14yYo bales ? 
 
 74. A gentleman having 17 A. 3 R. 10 r. of land, laid 
 it off in lots, each containing 25 rods, and sold them at 
 |450 per lot. How much did he get for his land? 
 
 75. If I sell a quantity of land at the rate of $450 
 for 25 sq. r., and obtain $51300, how many acres do I 
 sell? 
 
 192. To reduce a denominate 103. To rediioe a compound 
 
 fraction to a compound number, number to a denominate fraction, 
 
 Multiply the fraction by thenum- Reduce the unit of the proposed 
 
 15 
 
170 
 
 COMPOUND NUMBERS. 
 
 her zvhich connects it with the next 
 loicer imitj and the fractional part 
 of the product by the number which 
 connects it with the next lower unit, 
 and so on till the lowest unit of the 
 table is reached. 
 
 The several integral parts of the 
 product will form the compound 
 number, retaining the fractional 
 part, if there be any, of the last 
 quotient. 
 
 EXAMPLES. 
 
 compound number. 
 
 OPERATION. 
 
 11)56(5 (Vide 180, Ex. 17.) 
 65 
 
 1 
 
 40 
 
 11)40(3 (Vide 180, Ex. 18.) 
 
 7 
 
 m 
 
 ll)115i(10 
 
 no 
 
 5J 
 12 
 
 11)66(6 
 66 
 
 Ans. 5 fur. 3r. lOft. Gin. 
 
 fraction to the lowest units men- 
 tioned in the compound number for 
 the denominator of the required 
 fraction. 
 
 Reduce the compound number to 
 the same units for the numerator of 
 ths fraction. 
 
 Reduce the fraction to its lowest 
 terms. 
 
 Remark. — If there is a fraction 
 connected loith the lowest units, mul- 
 tiply both parts by the denominator 
 of the fraction before reducing. 
 
 
 examples. 
 
 
 2. Reduce 5 fur. 3r. 10 ft. 6 in. 
 
 to a denominate fraction. 
 
 
 
 operation. 
 
 
 
 40 
 
 16>^ 12 
 
 1 
 
 8 
 
 5 3 
 40 
 
 10 6 
 
 8 
 40 
 
 203 
 16J 
 
 
 320 
 16^ 
 
 3359* 
 12 
 
 
 5280 
 12 
 
 40320 
 
 
 63360 
 
 (Vide 178, Ex. 1, and 190, Ex.55.) 
 
COMPOUND NrMBERS. 
 
 171 
 
 3. Reduce |ra. to a compound 
 number. 
 
 ^ 5. Reduce y'gCwt. to a com- 
 pound number. 
 
 7. Reduce f T. to a compound 
 number. 
 
 9. Reduce ^£ to a compound 
 number. 
 
 11. Reduce |s. to a compound 
 number. 
 
 13. Reduce ^d. to a compound 
 number. 
 
 15. Reduce iituns to a com- 
 pound number. 
 
 17. Reduce flilid. to a com- 
 pound number. 
 
 19. Reduce f bbl. wine to a 
 compound number. 
 
 21. Reduce If A. to a com- 
 pound number. 
 
 23. Reduce 17|fA. to a com- 
 pound number. 
 
 4. Reduce 3 fur. 22 r. 3 ft. Sin. 
 to fraction of mile. 
 
 6. Reduce Iqr. 181b. 12 oz. to 
 fraction of hundredweight. 
 
 8. Reduce 8cwt. 2qr. 71b. 2 oz. 
 4idr., etc. 
 
 10. Reduce 8s. 6 d. 3ffar., etc. 
 
 6 3f 
 
 OPERATION. 
 
 
 OPEEATION. 
 
 3 
 
 20 
 
 1 
 20 
 
 8 
 
 12 
 
 7)60(8 
 66 
 
 20 
 12 
 
 102 
 4 
 
 4 
 
 240 
 
 411f 
 
 12 
 
 4 
 
 2880 
 
 7)48(6 
 42 
 
 960 
 
 7 
 
 
 6 
 
 6720 
 
 (191 Rem.) 
 
 4 
 
 
 |f|^=3£ Ans. 
 
 7)74(3^ 
 
 
 ' 
 
 8s. 6d. 33 far. Ans. 
 
 
 
 12. Reduce 4d. 2 far. to frac- 
 tion of a shilling. 
 
 14. Reduce If- far. to fraction 
 of a penny. 
 
 16. Reduce Ipi. Ihhd. 42 gal. 
 to fraction of a tun. 
 
 18. Reduce 23gal. 2qt. Ipt., 
 etc. 
 
 20. Reduce 18 gal. 3qt. Ipt. 
 f gi., etc. 
 
 22. Reduce 1 A. IR. 28fr. to 
 fraction of acre. 
 
 24. Reduce 17 A. 3R. 12 r. to 
 fraction of acre. 
 
172 
 
 COMPOUND XL'MEERS. 
 
 25. Reduce ^-^ yr. to a com- 
 pound number. 
 
 27. Reduce |- w. to a compound 
 number. 
 
 29. Reduce | h. to a compound 
 number. 
 
 31. Reduce ^bu. to a compound 
 number. 
 
 33. Reduce || bu. to a com- 
 pound number. 
 
 35. Reduce 3| lb. Troy to a 
 compound number. 
 
 2(i. Reduce TOO 
 fraction of year. 
 
 da. 12 h. to 
 
 28. Reduce 2 da. 19 h. 12 m. to 
 fraction of week. 
 
 30. Reduce 22 m. 30 sec. to 
 fraction of hour. 
 
 32. Reduce 3 pk. 2qt. IJpt. to 
 fraction of bushel. 
 
 34. Reduce 1 pk. 6 qt. § pt. to 
 fraction of bushel. 
 
 36. Reduce 3 lb. 10 oz. to frac- 
 tion of pound, 
 
 38. Reduce 9 dwt. 9 gr. to frac- 
 pound number. tion of pennyweight. 
 
 39. If I ride 5 fur. 3 r. 10 ft. 6 in. in a railroad car, 
 what ought to be my exact fare at the rate of 11 cts. a 
 mile ? Ans. 7 cts. 
 
 40. What cost the iron on a track measuring 3 fur. 
 22 r. 3 ft. 8 in., at the rate of §4500 per mile ? 
 
 Ans. §2000. 
 
 41. Sold 1 pi. 1 hhd. 42 gal. of molasses at the rate 
 of 175.60 per tun. What did I get ? Ans. $69.30. 
 
 42. A grocer bought 8 cwt. 2 qr. 7 lb. 2 oz. 44 dr. of 
 coffee at §9.50 per cwt., and sold it at a retail price of 16 
 cts. per lb. How much did he make? A7is. §55.71 f. 
 
 43. Bought 1 pk. 6 qt. f pt. of chestnuts at the rate 
 of §2.88 per bushel, and retailed them at 9 cts. per 
 quart. Do I make or lose ? 
 
 44. If a boy could count 6000 marbles in an hour, 
 how many could he count in 22 m. 30 sec. ? 
 
 Ans. 2250. 
 
 45. If I buy iron at §45 per ton, and sell it at 2J cts. 
 per pound, what do I gain by selling 13 cwt. 2 qr. 15 
 lb.? Ans.^S.41h 
 
COMPOUND NUMBERS. 
 
 173 
 
 194- To reduce a compound 
 number to a decimal fraction, 
 
 Divide the lowest units by that 
 number which connects them with 
 the next higher^ and annex the quo- 
 tient as a decimal to the given num- 
 ber of those higher units. 
 
 Continue to divide till the units 
 required are reached. 
 
 The last quotient will be the re- 
 quired decimal. 
 
 EXAMPLES. 
 
 1. Reduce 23 gal. 2 qt. 1 pt. to 
 a decimal in hogshead. 
 
 OPERATION. 
 
 2.5. (Vide 182, Ex. 13. 
 
 63123.625 (Vide 182, Ex. 15.) 
 .375 hhd. Ans. 
 (Vide 192, Ex. 17.) 
 
 3, Reduce 3 fur. 22 r. 3 ft. 8 in. 
 to decimal fraction of mile. 
 
 5. Reduce 1 qr. 18 lb. 12 oz. to 
 fraction of hundredweight. 
 
 7. Reduce 8 s. 6d. 3^ far. to 
 fraction of pound sterling, 
 
 9. Reduce 15 £ 10s. 9d. to 
 fraction of pound sterling. 
 
 11. Reduce 18 h. 9 m. to frac- 
 tion of day. 
 
 13. Reduce 5 cwt. 2 qr, 15 lb. to 
 fraction of ton. 
 
 195. To reduce a denominate 
 decimal fraction to a compound 
 number. 
 
 Multiply the given decimal frac- 
 tion by the number which connects 
 it with the next lower units, and the 
 decimal part of the product by the 
 number connecting it tvith the next 
 lower units, and so on till the lowest 
 unit is reached. The integral parts 
 of the products will form the num- 
 ber required. 
 
 EXAMPLES. 
 
 2. Reduce .375 hhd. to a com- 
 pound number. 
 
 OPERATION. 
 
 .375 hhd. 
 63 
 
 23.625 gal. 
 4 
 
 2,500 qt. 
 2 
 
 1.000 pt. 
 Ans. 23 gal. 2qt. Ipt. 
 
 4. Reduce .44| m to a com- 
 pound number. 
 
 6. Reduce .4375 cwt. to a com- 
 pound number. 
 
 8. Reduce .42857^ £ to a com- 
 pound number. 
 
 10. Reduce 15.5375 <£ to a com- 
 pound number. 
 
 12, Reduce .75625 da. to a com- 
 pound number. 
 
 14. Reduce .2825 T. to a com- 
 pound uumbci'. 
 
174: COMPOUND N^UMBERS. 
 
 15. Reduce 3 ft. 9 in. to frac- 
 tion of yard. 
 
 17. Reduce 3 pk. 2 qt. lipt. to 
 fraction of bushel. 
 
 19. Reduce 22 m. 30 sec. to 
 fraction of hour. 
 
 21. Reduce 2 da. 19 h. 12 m. to 
 fraction of week. 
 
 23. Reduce 3° 30^ 36^^ to frac- 
 tion of degree. 
 
 16. Reduce 1,25 yd. to a com- 
 pound number. 
 
 18. Reduce .83^ bu. to a com- 
 pound number. 
 
 20. Reduce .375 h. to a com- 
 pound number. 
 
 22. Reduce .4 w. to a compound 
 number. 
 
 24. Reduce 3.51° to a compound 
 number. 
 
 25. What will 23 gal. 2 qt. 1 pt. of wine cost at $60 
 per hhd. ? A71S. $22.50. 
 
 26. What will the above wine bring at $1 per gal. ? 
 
 Alls. §23.625. 
 
 27. What will 3 pk. 2 qt. IJ pt. of corn cost at 90 
 cts. per bushel ? Ans. 75 cts. 
 
 28. If I buy 13 A. 2 R. 35 r. of land at $17.28 per 
 acre, and sell it in lots of 1 rood each at 12 dollars a 
 lot, how much do I make? Ans. $421.44. 
 
 29. What will .2825 tons of rice bring at 7 cts. per 
 lb.? at $1.50 per qr.? at $5 per cwt. ? 
 
 Ans.$S9.bD; $33.90; $28.25. 
 
 30. At $25 per acre, how much land can be bought 
 for $648.75 ? Aits. 25.95 A.-=25 A. 3 R. 32 r. 
 
 31. Bought 18 cwt. 1 qr. 18 !b. of tea at $65 per cwt., 
 and sell the same at 75 cts. per lb. How much do I 
 I make? Ans. $184.30. 
 
 32. Bought 56 hhd. 16 gal. 3 qt. of molasses at 
 $46 per hhd. What did it amount to? 
 
 Alls. $2588.23. 
 
 33. I sell the above molasses at 87J cts. per gal. 
 What do I gain ? Ans. $513.43. 
 
 34. A planter sold 270 bales of cotton at $60 per 
 
COMPOUND NUMBERS. 175 
 
 bale, and invested the proceeds in land at $28 per acre. , 
 How much land did he purchase ? 
 
 Ans. 578 A. 2 R. llf r. 
 
 35. A merchant imported 325 yards of silk, at an 
 expense of £1 4s. 6d. per yard, and desires to clear 
 $1200 in retailing it. What must be the price per yard ? 
 (Vide 173, Rem. 1.) A7is. $9,621. 
 
 36. I import 275 yards of French broadcloth, at an 
 expense of 24 francs per yard, and clear, in retailing it, 
 $500. What do I charge per yard? (Vide 174.) 
 
 Ans, $6.23. 
 
 ADDITION OF COMPOUND NUMBERS. 
 
 190. To add compound numbers, 
 
 (1.) Wrile the units of the same name under each other ^ 
 and place over each column that number which connects 
 its unit with the next higher unit. 
 
 (2.) Add the coluynn of units of the least value, and 
 divide the sum hy that number which stands over it, 
 placing the remainder under the column. 
 
 (3.) Add the column of units of the next higher value, 
 including the quotient of . the preceding division, and 
 divide by the number which stands over it, placing the re- 
 mainder under the column. 
 
 (4.) Add all the colmnns of units in the same ivay, 
 writing down, hoivever, the entire sum of the column 
 ivhich can have no number over it. 
 
 ' EXAMPLES. 
 
 1. Add together 20£ 15s. 9d. 3 far. ; 240c£ 7d. 2 far.; 
 and 17s. 1 far. 
 
176 COMPOUND NUMBERS. 
 
 2. Add together 40 1. 6 fur. 2 in.; 2 m. 6 fur. 16 r. 
 3 yd. 1ft.; 4 m. 7 fur. 20 r. 16 ft.; 11. 2 m. 7 fur. 13 r. 
 1 ft. 11 in. 
 
 OPERATIONS. 
 
 (1-) (2.) 
 
 
 20 
 
 12 
 
 4 
 
 20 
 
 15 
 
 9 
 
 3 
 
 240 
 
 
 
 7 
 
 2 
 
 
 17 
 
 
 
 1 
 
 40 5% 3 12 
 
 40 
 
 6 
 
 
 
 
 
 
 
 2 
 
 2 
 
 6 
 
 16 
 
 3 
 
 1 
 
 
 4 
 
 7 
 
 20 
 
 
 
 16 
 
 
 1 2 
 
 7 
 
 13 
 
 
 
 1 
 
 11 
 
 Ans. 261 £ 13 s. 5d. 2 far. 
 
 441. 2 m. 3 fur. 10 r. 3 ^yd. 1ft. 1 in. 
 
 Ans. 441. 2 m. 3 fur. 10 r. 3 yd. 2 ft. 7 in., since ^ yd.=l ft. 6 in. 
 
 3. Add together 1 hhd. 25 gal. 3 qt. 1 pt. ; 8 hhd. 2 qt. ; 
 3 hhd. 27 gal. 1 pt. ; and 21 hhd. and 1 pt. 
 
 Ans. 8 T. 1 hhd. 53 gal. 2 qt. 1 pt. 
 
 4. Add together 13 bu. 2 pk. 7 qt. 1 pt. ; 150 bu. 1 pk. 
 5 qt.; 200 bu. 3 pk. 5 qt. 1 pt. Ans. 365 bu. 2 qt. 
 
 5. Add together J J of a tun, § of a hhd., and | of a 
 bbl. 
 
 OPERATION. 
 
 2 2 f)3 4 2 
 
 \i tun = 
 1 hhd.- 
 
 f bbl.=: 
 
 1 1 42 
 23 
 
 18 
 
 
 2 
 3 
 
 (Vide 192, Ex. 15.) 
 
 1 ( " 192, " 17.) 
 li ( " 192, " 19.) 
 
 ItunO 21 gal. 2 qt. J pt. 
 
 6. Add together | £ g s. and f d. 
 
 Ans. 8s. lid. 3|far. 
 
 7. Add together 1| A. and 17f § A. 
 
 Ans.^lO A. 1 R. 4 r. 
 
 8. Add together /jj yr. § w. and | h. 
 
 Ans. 112 da. 7 h. 34 m. 30 sec. 
 
COMPOUND NUMBERS. 177 
 
 9. Add together § bu. and || bu. 
 
 Ans. 1 bu. 1 pk. 1 qt. 
 
 10. Add together j i m. and f fur. 
 
 Ans. 7 fur. 26 r. 5 ft. 4,% in. 
 
 11. Add together 15.5375£ and .4285|£. (Vide 195, 
 Ex. 8 and 10.) Ans. 15£ 19s. 3d. 3|far. 
 
 12. Add together .4 bu. and .7 pk. 
 
 Ans. 2 i)k. 2 1 qt. 
 
 13. Add together 5.88125 A. and4 A. 2 R. 35 r. 
 
 Ans. 10 A. 2 R. 16 r. 
 
 14. Add together J lb. Troj and .583J oz. 
 
 Ans. 6 oz. 11 dwt. 16 gr. 
 
 15. Add together .875£ and .75s. Ans. 18s. 3d. 
 
 SUBTRACTION OF COMPOUND NUMBERS. 
 
 197. To subtract compound numbers, 
 
 (1.) Write the nitmhers, as in 196. 
 
 (2.) Subtract the units of the loioest value in the sub- 
 irahend from the corresponding units of the 7ninuend, 
 and iJ^lace the difference under the same column; but if 
 the number in the subtrahend is larger than that in the 
 minuend, add the number standing over the column to the 
 number in the minuend, and subtract from the sum the 
 number in the subtrahend. 
 
 (3.) If tfte jurmher standii^g above any column has 
 been employed in the subtraction, add 1 to the units of 
 next higher value in the subtrahend ; after which proceed 
 exactly as with the preceding column, and so on till all 
 the columns have been subtracted. 
 
178 COMPOUND NUMBERS. 
 
 EXAMPLES. 
 
 1. From 21 r. 3 ft. 5 in. take 17 r. 16 ft. 9 in. 
 
 (1.) 
 
 12 
 
 OPERATIONS. 
 
 A7IS. 
 
 
 (2.) 
 
 12 
 
 21 3 
 17 16 
 
 5 
 9 . . 
 
 21 
 
 18 
 
 3 
 
 
 
 5 
 
 3 
 
 
 
 
 
 3r. 24ft. 
 
 Sin. 
 
 3r. 
 
 3ft. 
 
 2. 
 
 Remark. — The first result is easily reduced to tlie second by ob- 
 serving that 6 in.= ^ ft. 
 
 2. From 3 fur. 29 r. 2 yd. 1 ft. take 1 fur. 39 r. 3 yd. 
 2 ft. A71S. 1 fur. 29 r. 4 yd. 6 in. 
 
 3. From 63 T. 1 hhd. 15 gal. take 19 T. 3 lihd. 17 
 gal. A71S. 43 T. 1 lilid. 61 gal. 
 
 4. From 8 bu. 3 pk. 1 qt. take 3 bu. 2 pk. 7 qt. 
 
 Ans. 5 bu. 2 qt. 
 
 5. From 25° 4' 27^^ take 17° 20^ 40^^ 
 
 A71S. 7° 43' 47'^ 
 
 6. JV^hat is the difference between 40 m. and 39 m. 
 7 fur. 39 r. 16 ft. 7 in. ? Ans. 1 inch. 
 
 7. From 77° 0' 15'^ take 71° 3' 30^'. 
 
 Ans. 5° 56' 45''. 
 
 8. From 85° 30' take 77° 0' 15". Ans. ^ 29' 45". 
 
 9. From 86° 49' 3" take 79° 55' 38". 
 
 Ans. 6° 53' 25". 
 
 10. Washington is 77° 0' 15" West Longitude, and 
 Boston 71° 3' 30" West Longitude. What is the differ- 
 ence in the Longitude of these places ? 
 
 Ans. 5° 56' 45". 
 
COMPOUND NUMBERS. 179 
 
 11. Louisville is 85° 30' W. L., and Mobile 88° 1' 29" 
 W. L. What is the difference in the Longitude of these 
 places ? Ans. 2° 31' 29''. 
 
 12. The City of Mexico is in North Latitude 19° 25' 
 45", and Cincinnati is in N. L. 39° 5' 54". What is the 
 difference ? Ans. 19° 40' 9". 
 
 13. The difference of time between Greenwich and 
 Milledgeville, Ga., is 5 h. 33 m. 19 sec. When it is 
 noon at Greenwich, what is the time at Milledgeville ? 
 
 Ans. 26 m. 41 sec. past 6 A. M. 
 
 Remark. — The time of a place being given, the time of all places 
 east of it is later, and of all places zcest^ earlier. 
 
 198. To find the time between two dates, 
 Write the year, the order of the month, (Vide 185, 
 Rem. 4,) and the day of the month of each date, respect- 
 ively, under each oilier. 
 Then proceed as in 197. 
 
 ■ ' 
 
 
 EXAMPLES. 
 
 
 
 
 
 1. Find the 
 
 :, 1867. 
 
 time 
 
 from 
 
 January 
 
 27, 
 
 1865, 
 
 to 
 
 July 
 
 
 
 OPERATION. 
 
 
 
 
 
 
 
 
 12 30 
 
 
 
 
 
 
 
 1867 
 1865 
 
 7 4 
 1 27 
 
 
 
 
 
 Ans. 2yr. 5 m. 7 da. 
 
 2. Find the time from July 4, 1865, to August 1, 
 1866. Ans. 1 yr. 27 da. 
 
 3. Find the time from August 1, 1869, to September 
 9, 1871. Ans. 2 yr. 1 m. 8 da. 
 
180 COMPOUND NUMBERS. 
 
 4. Find the time from November 15, 1866, to Decem- 
 ber 8, 1879. Ans. 13 yr. m. 23 da. 
 
 5. Find the time from January 27, 1866, to Septem- 
 ber 9, 1871. Ans. 5 yr. 7 m. 12 da. 
 
 6. The Independence of the United States was de- 
 clared July 4, 1776. What interval has passed on Jan- 
 uary 1, 1867? A71S. 90 yr. 5 m. 27 da. 
 
 Remark.— The true interval found by the table, (185, Rem. 6,) 
 is 90 yr. 181 da.) 
 
 7. A merchant bought at one time 8120 gills of wine, 
 at another J J of a tun, at another | of a hhd., and at 
 another time .375 of a hhd. What did the whole cost 
 at 11.00 per gallon ? Ans. $532.00. 
 
 8. Bought at one time 17 yd. 3 qr. 2 na. of broad- 
 cloth; at another time lo^% yd.; at another 87.8125 
 yd.; at another 27 yd. Ij qr. ; and at another time 
 29.375 yd. What did the whole cost at 5 J dollars per 
 yard? Ans. ^$965.93 1 . 
 
 9. From a lot of land containing 10 A. 3 R. ip r., I 
 sell at one time 1 A. 2 R. 13 r.; at another time 2 A. 2 R. 
 5r. I gave §600.53 J- for the land; sold the first lot 
 at $70 per acre, and the second lot at $75 per acre. 
 For how much per acre could I sell what remains, and 
 lose nothing ? Ans. $44.78. 
 
 10. Sold corn in three lots, viz:. 13 bu. 2 pk. 7 qt. 
 1 pt., at 60 cts. per bu.; 150 bu. 1 pk. 5 qt., at 50 cts. 
 per bu. ; 200 bu. 3 pk. 5 qt. 1 pt., at 55 cts. per bu. 
 What should I have gained by selling all the corn at 56 
 cts. per bu. Ans. $10.48|J. 
 
 11. A load of hay weighs 43 cwt. 2 qr. 18 lb., includ- 
 ing the wagon which weighs 9 cwt. 3 qr. 23 lb. What 
 is the hay worth at $7.50 per ton? 
 
COMPOUND NUMBERS. 181 
 
 12. While in London, I paid for a vest 1£ 13s. 4d.; a 
 coat, 7£ 12s. 9d. ; pants, 2£ 3s. 9d. ; boots and hat, 
 9£ 8s. How many dollars did the whole come to ? 
 
 Am. $101.115f . 
 
 13. If from a cask of molasses containing 118 gal., 
 20 gal. 1 pt. leak out, for how much must the remainder 
 be sold per gallon to lose nothing, the whole cost having 
 been $75? Ajis. |0.766. 
 
 MULTIPLICATION OF COMPOUND NUMBERS. 
 
 199. To multiply a compound number. 
 Multiply each of the simple numhei^s composing the 
 compound number hy the multiplier^ reducing lower to 
 higher units, as in Addition of Compound Numbers. 
 
 EXAMPLES. 
 
 1. Multiply 3 m. 2 fur. 4 r. 2 ft. 5 in. by 13. 
 
 OPERATION. 
 
 8 40 16)^ 12 
 
 5 
 13 
 
 42m. 3fur. 13r. 14A-ft. 5in. 
 Ans. 42 m. 3 fur. 13 r. 14 ft. 11 in., since \ ft.=6 in. 
 
 2. Multiply 12£ 4s. 6d. 2far. by 13. 
 
 Ans. 158£ 19s. 2far. 
 
 3. Multiply 1 lb. 9 oz. 13 dwt. by 15. 
 
 Ans. 27 lb. 15 dwt. 
 
 4. Multiply 19 cwt. 3 qr. 23 lb. by 18. 
 
 Ans. 359 cwt. 2 qr. 14 lb. 
 
182 COMPOUND NUMBERS. 
 
 5. Multiply 18 gal. 3 qt. 1 pt. by 27. 
 
 Ans. 509 gal. 2 qt. 1 pt. 
 
 6. Multiply 365 da. 5 h. 48 m. 47.57 sec. by 7. 
 
 Alls. 2556 da. 16 h. 41 m. 32.99 sec. 
 
 7. Multiply 4 A. 3 R. 20 r. 4 yd. 7 ft. 47 in. by 84. 
 
 OPERATION. 
 
 4 40 30)^ 9 144 
 
 4 
 
 3 
 
 20 
 
 4 
 
 7 
 
 47 
 
 7 
 
 34 
 
 
 
 21 
 
 2f 
 
 6 
 
 il 
 12 
 
 409A. 2R. 13r. lOfyd. 3ft. 60 in. 
 
 Ans. 409A. 2R. 13r. lly^. 1ft. 24in., since 36in.=^ft. and 2| ft.=Jyd. 
 
 8. Multiply 4 cwt. 1 qr. 7 lb. 6 oz. 6.5 dr. by 44. 
 
 Ans. 9 T. 10 cwt. 1 qr. 9 oz. 14 dr. 
 
 9. Multiply 4 cwt. 18| lb. by 476. 
 
 Ans. 99 T. 12 cwt. 6 lb. 
 
 10. Multiply 21 m. 65 r. 13 ft. by 5. 
 
 Ans. 106 m. 8 r. 15 ft. 6 in. 
 
 11. Multiply 1 sq. r. 57 sq. ft. 55 sq. in. by 7. 
 
 Ans. 8 sq. r. 129 sq. ft. 61 sq. in. 
 
 12. Multiply 13£ 5s. 4.75d. by 24. 
 
 Ans. 31 8£ 9s. 6d. 
 
 13. Multiply 81£ 14s 9d. by 80. 
 
 Ans. 6539£. 
 
 14. Multiply 13s. 6d. Ifar. by 519. 
 
 Ans. 350£ 17s. 3d. 3far. 
 
 15. Multiply 17 cwt. 3 qr. 10 lb. by 60. 
 
 Ans. 1071 cwt, 
 
 I 
 
COMPOUND NUMBERS. l«o 
 
 16. If a bale of cotton Is worth 12£ 14s. 6d. 2far., 
 what will be the cost of 13 bales at the same rate? 
 
 Ans. 1800.79. 
 
 17. I own 16 lots of land, each containing 5 A. 3 R. 
 20 r. What is the land worth at $40 per acre ? 
 
 Ans. §3760. 
 
 18. If a man travel 24 m. 4 fur. 4 r. per day, what 
 will it cost to travel 5 days at the rate of 12 J cts. per 
 mile ? Ans. |15.32. 
 
 19. What will 9 casks of sugar cost at $12 per cwt., 
 each cask weighing 8 cwt. 2 qr. 12 lb. ? Ans. $930.96. 
 
 20. If one silver cup weighs 8 oz. 4 dwt. 10 gr., what 
 will 6 cups, each of the same weight, be worth at the 
 rate of $1.25 per ounce ? Ans. $61.66. 
 
 21. What will 9 pieces of broadcloth cost at 5 dollars 
 a yard, each piece containing 29 yd. 2 qr. 3 na. ? 
 
 Ans. $1335.93|. 
 
 22. A steamship in crossing the Atlantic makes an 
 average distance of 250 m. 3 fur. per day. How far 
 will she sail in 9 days? Ans. 2253 m* 3 fur. 
 
 23. I send to market 5 casks of wine, each containing 
 123 gal. 2 qt. 1 pt. What is the wine worth at 1 dollar 
 per gallon ? Ans. $618.12 J. 
 
 24. A ship sails on the line of the Equator 5 days, 
 at the rate of 2° 15' 20^' per day. How many miles 
 does the ship make? (Vide 186, Rem. 1.) 
 
 Ans. 779.98 miles. 
 
 25. Suppose the ship in the preceding problem had 
 sailed the same number of days on a meridian, what 
 would have been the distance made? 
 
 Ans. 778.64 m. 
 
184 COMPOUND NUMBERS. 
 
 DIVISION OF COMPOUND NUMBERS. 
 
 200. To divide a compound number, 
 
 (1.) Divide that term of the compound number which 
 has the highest U7iit value by the divisor^ and write the 
 quotient as the highest term of the answer. 
 
 (2.) Reduce the remainder, if there he any, to the next 
 loiver unit, adding that term of the dividend which has 
 the same unit value. 
 
 (3.) Divide the sum as before, and 2:)roceed in the same 
 manner till the lowest units are reached. 
 
 EXAMPLES. 
 
 1. Divide 165£ 9s. 2 far. by 13. 
 
 OPERATIONS. 
 
 20 12 4 20 12 4 
 
 13)165 9 2 13)165 9 2 (12 £ 14s. Gd. 2 far. .4ns. 
 
 150 
 
 A)is. 12 £ 14s. 6d. 2 far. 
 
 9 
 
 Analysis: 165£-f-13=12£ 20 Remark. — The processes by 
 
 and 9£ Rem. 9£x20-j-9s. ~~ Short and Long Division are 
 
 =]89s. (More exact, 20s. -.qo essentially the same. The 
 
 X9-{-9s.= 189s.) 189s.-r- operation by Long Division 
 
 13 = 148. and 7s. Rem. 7s. 7 is preferable till it has become 
 
 Xl2=84d. _2^ quite familiar, since no figures 
 
 84d.-i-13— 6d. and 6d. 34 requiring attention are sup- 
 
 Rem. 6d.X4-f2far.= 26 78 pressed. 
 
 far. Finally, 26 far.-^13 
 
 =2 far. 5 
 
 26 
 
 2. Divide 42 m. 3 fur. 13 r. 14 ft. 11 in. by 13. 
 
 Ans. 199, Ex. 1. 
 
COMPOUND NUMBERS. 185 
 
 3. Divide 27 lb. 15 clwt. by 15. Ans. 199, Ex. 3. 
 
 4. Divide 359 cwt. 2 qr. 14 lb. by 18. 
 
 Ans. 199, Ex. 4. 
 
 5. Divide 509 gal. 2 qt. 1 pt. by 27. 
 
 Ans. 199, Ex. 5. 
 
 6. Divide 2556 da. 16 li. 41 m. 32.99 sec. by 7. 
 (Vide 185, Rem. 2.) Ans. 199, Ex. 6. 
 
 7. Divide 84° 18^ by 15. Aiis. 5° 37' 12^- 
 
 8. Divide 83° 19' 45'' by 15. Aiis. 5° 33' 19". 
 
 9. Divide 88° 1' 29" by*'l5. Ans. 5° 52' 5.9". 
 
 10. Divide 86° 49' 3" by 15. Ans. 5° 47' 16.2". 
 
 11. If 4 of a ship be worth 235£ 16s. lid., what is 
 the whole ship worth in United States money? 
 
 A71S. $7990.46. 
 
 12. If I of a ship be worth 943£ 7s. 8d.-, what is the 
 whole ship worth in francs ? Ans. 43485.48 francs. 
 
 13. If 8 bbl. of flour cost 2£ 12s., \vhat will 29 bbl. 
 cost? Ans. $45,617. 
 
 14. I bought a tract of land containing 486 A. 2 R. 
 30 r. for §1000; subsequently I divide the land into 12 
 farms, and sell 11 of these farms for |6 an acre, reserv- 
 ing the twelfth as a homestead. How much do I make 
 over and above the homestead? A7is. |1676.78-|. 
 
 15. From 7 acres of land I harvest 299 bu. 1 pk. 7 qt. 
 of wheat. The tillage of the land cost me $2 an acre, 
 and I sell the wheat at |1.25 per bushel. What do I 
 clear from each acre? A71S. $51.47||. 
 
 16. If 15T. 7 cwt. 2qr. 181b. of cotton cost $3384.48, 
 what will 1 lb. cost ? What 1 qr. ? What 1 cwt. ? What 
 1 T. ? What would 100 bales be worth, each bale 
 weighing 511 lb.? Ans. 1 lb. is worth 11 cts. 
 
 16 
 
186 COMPOUND NUMBERS. 
 
 LONGITUDE IN TIME. 
 
 201. To cliange °, \ ", of Longitude, into h. m. sec. 
 of Time, 
 
 (1.) Every point of the Earth's surface, except the 
 poles, moves through 360° in 24 hours. Hence, 
 15° of Lon.=l hour of Time. 
 (2.) Now 15°=900^ and lhour==60 minutes. Hence, 
 
 15' of Lon.=l m. of Time. 
 (3.) But 15^:^900^ and 1 m.=60 sec. Hence, 
 
 15'' of Lon.=l sec. of Time. Therefore, 
 Divide the °, ', ", hy 15, and consider tlie terms' of the 
 quotient as h. m. sec. 
 
 EXAMPLES. 
 
 1. The city of Lexington is 84° 18' W. L. When it 
 is noon at Lexthgton, Avhat time is it at Greenwich? 
 (Vide 200, Ex. 7, and 197, Ex. 13, Rem.) 
 
 Ans. 37 m. 12 sec. past 5 P. M. 
 
 Remark. — The Meridian from wliicli Longitude is reckoned • 
 passes through Greenwich near London. 
 
 2. Milledgeville is 83° 19' 45" W. L. When it is 
 noon at Greenwich, what is the time at Milledgeville ? 
 (Vide 197, Ex. 13.) Ans, 26 m. 41 sec. past 6 A. M. 
 
 3. When it is noon at Lexington, what time is it at 
 Milledgeville? Ans. 3 m. 53 sec. P. M. 
 
 4. When it is noon at Milledgeville, what is the time 
 at Lexington? Ans. 11 o'clock 56 m. 7 sec. A. M. 
 
 5. The city of Mobile is 88° 1' 29" W. L. When it 
 is noon at Mobile, what is the time at Greenwich? 
 
 Ans. 5 o'clock 52 m. 5.9 sec. P. M. 
 
C03IP0UND NUMBERS. 187 
 
 6. Louisville is 85° 30' W. L. What is the difference 
 in time between Lexington and Louisville ? 
 
 7. Washington is 77° 0' 15'' W. L. When it is noon 
 at Washington, what is the time at Louisville ? 
 
 8. Boston is 71° 3' 30'' W. L. When it is noon at 
 Washington, what is the time at Boston ? 
 
 Ans. 2S m. 47 sec. P. M. 
 
 9. The Cape of Good Hope is 18° 29' E. L. When 
 it is noon at Washington, what is the time at Good 
 Hope ? Ans. 6 o'clock 21 m. 57 sec. P. M. 
 
 10. San Francisco is 122° 26' 48" W. L. When it is 
 8 o'clock 9 m. 47.2 sec. P. M. at London, what is the 
 time' at San Francisco ? Ans. Noon. 
 
 11. New York is 74° 0' 3" W. L. When it is noon 
 at New York, what is the time at all the places men- 
 tioned in the preceding problems ? 
 
 12. On the morning of November 15, 1859, a re- 
 markable meteor was seen at New York, Albany, Wash- 
 ington, and Fredericksburg. At Washington the time 
 was 9 o'clock and 30 m. What w^as the time at the other 
 places, Albany being 73° 44' 39" W. L., and Fredericks- 
 burg 77° 38' W^ L. ? 
 
 ANALYSIS BY ALIQUOT PARTS. 
 
 202. An aliquot part of any number is an exact 
 half, third, fourth, etc., of the number. Thus, 
 12J- cts. is an aliquot part of ^1, because 12J is J- of 100. 
 
 2*^ is J of 4. 
 
 6 is J of 24. 
 
 5 is 1 of 20. 
 
 121 is 1 of 25. 
 
 2 R. 
 
 u 
 
 a 
 
 u 
 
 lA, 
 
 u 
 
 6 gr. 
 
 u 
 
 (( 
 
 u 
 
 1 dwt., 
 
 ii 
 
 5 s. 
 
 ii 
 
 u 
 
 u 
 
 1£, 
 
 'i. 
 
 21 lbs. 
 
 u 
 
 u 
 
 « 
 
 1 qr., 
 
 a 
 
188 
 
 COMPOUND NUMBERS. 
 
 EXAMPLES. 
 
 - 1. What cost 39 A. 2 R. 15 r. of land, at §87.375 
 per acre. 
 
 OPERATION. 
 
 §87.375 
 39 
 
 Therefore, 
 
 Price of 1 A. is . . . 
 
 " 39 A. is . . 
 
 " 2 R. is . . . 
 
 " 10 r. is . . . 
 
 " 5 r. is . . . 
 
 Price of 39 A. 2 R. 15 r. is |3459.503f ^ Ans. 
 
 2. What cost 39 A. 2 R. 15 r. of land, at |139.80 
 per acre? ylii-s. 15535.206 J-. 
 
 3. What cost 176 A. 3 R. 25 r. of land, at §75.375 
 
 3407.625 
 
 
 43.687.1 
 
 . J of 1 A. 
 
 5.460 }i 
 
 . I of 2 R. 
 
 2.73011 
 
 . i of 10 r. 
 
 per acre 
 
 Ans. §13334.308^9. 
 
 4. What cost 20 A. 2 R. 24 r. of land, at §30 per 
 acre? Ans. §619.50. 
 
 5. What cost 10 yd. 3 qr. 2 na. of silk, at §1.80 per 
 yard ? 
 
 OPERATION. 
 
 Price of 1 yd. is 
 
 .80 
 10 
 
 Therefore, 
 
 a 
 
 10 yd. is 
 
 . 18.00 
 
 
 <£ 
 
 2 qr. is . 
 
 . .90 . 
 
 . i of 1 yd. 
 
 u 
 
 1 qr. is . 
 
 . .45 . 
 
 . i of 2 qr. 
 
 a 
 
 2 na. is . 
 
 . .225 . 
 
 . J of 1 qr. 
 
 Price of 10 yd. 3 qr. 2 na. is §19.575 Ans. 
 
 6. What cost 15 yd. 2 qr. 3 na. of cloth at 25 cents 
 per yard? Ans. ^S.02^^^. 
 
COMPOUND NUMBERS. 
 
 189 
 
 7. What cost 25 yd. 1 qr. 3 na. of broadcloth, at $5.50 
 per yard? A/i5. $139.90 1. 
 
 8. What cost 67 bu. 3 pk. 7 qt. of cranberries, at §2 
 per bushel? 
 
 OPERATION. 
 
 Price of 1 bu. . . §2.00 Therefore, 
 
 67 
 
 " 67 bu. . 
 
 . 134.00 
 
 
 ^' 2pk. . 
 
 1.00 
 
 1 bu. 
 
 " Ipk. . 
 
 .50 
 
 i of 2 pk 
 
 " 4 qt. . 
 
 .25 
 
 i of 1 pk 
 
 " 2 qt. . 
 
 .125 
 
 I of 4 qt. 
 
 " 1 qt. . 
 
 .0625 . 
 
 \ of 2 qt. 
 
 Price of 67 bu. 3 pk. 7 qt. $135.9375 Ans. 
 
 9. What cost 125 bu. 3 pk. 1 qt. of wheat, at 87^ cts. 
 per bushel? Ans. $110,058. 
 
 10. What cost 25 bu. 1 pk. 3 qt. of clover seed, at 
 $5.00 per bushel? Ans. $126.72. 
 
 11. What cost 503 bu. 4 qt. of corn, at 43j- cts. per 
 bushel?. ^7?s. $220,117. 
 
 12. What cost 76 bu. 1 qt. of peas, at $1.66| per 
 bushel? J.ns. $126.72. 
 
 13. What cost 10 bu. 3 pk. of apples, at 50 cts. per 
 bushel? Ans. $5.37^. 
 
 14. What cost 25 bu. 1 pk. of potatoes, at 35 cts. per 
 bushel? A71S. $8.83f. 
 
 15. What cost Ibu. 1 pk. 1 qt. of chestnuts, at $1.00 
 per bushel? Ans. $1.28 J. 
 
 16. What cost 17 cwt. 3 qr. 23 lb. of hay, at $13 
 per ton? 
 
190 
 
 COMPOUND NUMBERS. 
 
 
 
 
 OPERATION. 
 
 
 Price of 
 
 IT.... 
 
 10 cwt. . . 
 
 . $13.00 
 
 . therefore, 
 
 iC 
 
 6.50 
 
 . J of 1 T. 
 
 u 
 
 5 cwt. 
 
 
 
 3.25 
 
 . i of 10 cwt 
 
 a 
 
 2 cwt. 
 
 
 
 1.30 
 
 J of 10 cwt. 
 
 a 
 
 2 qr. . 
 
 
 
 .325 
 
 1 of 2 cwt. 
 
 u 
 
 1 qr. . . 
 
 
 
 .1625 . 
 
 i of 2 qr. 
 
 a 
 
 5 1b. . 
 
 
 
 .0325 
 
 I of 1 qr. 
 
 a 
 
 15 1b. . . 
 
 
 
 .0975 . 
 
 3 times 5 lb. 
 
 a 
 
 lib. . . 
 
 
 
 .0065 . 
 
 I of 5 lb. 
 
 Ci 
 
 2 1b. . 
 
 
 
 .013 
 
 2 times 1 lb 
 
 Price of 17 cwt. 3 qr. 231b. $11,687 Ans. 
 
 17. What cost 3 T. 10 cwt. 3 qr. of iron, at $30,375 
 per ton? Ans. $107.45. 
 
 18. What cost 16 boxes of sugar, each box contain- 
 ing 4 cwt. 3 qr. 18 lb., at $6.65 per hundredweight? 
 
 Ans. $524,552. 
 
 19. What cost 9 casks of sugar, each cask weighing 8 
 cwt. 2 qr. 12 lb., at $12 per cwt. ? A7is. 199, Ex. 19. 
 
 20. What cost 16 cwt. 3 qr. 21 lb. 6 oz. of rice, at 
 $7.00 per hundredweight? Ans. $118.75. 
 
 21. What cost 5 cwt. 2 qr. of hay, at $27 per ton? 
 
 Ans. $7.42f 
 
 22. What cost 2 T. 3 cwt. 3 qr. of hay, at $30 per 
 ton? ^ws. $65,625. 
 
 23. What cost 7 T. 15 cwt. 1 qr. of hay, at $40 per 
 ton? Ans. $S10.bO. 
 
 24. What cost the iron on a track measuring 3 fur. 
 22 r. 3ft. 8 in., at the rate of $4500 per mile? 
 
COMPOUND NUMBERS. 
 
 191 
 
 OPERATION. 
 
 Price of 1 m. 
 
 §4500.00 
 
 Therefore, 
 
 
 2 fur. . 
 
 . 1125.00 
 
 . J of a mile 
 
 
 1 fur. . 
 
 562.50 
 
 . J of 2 fur. 
 
 
 20 r. 
 
 281.25 
 
 . J of 1 fur. 
 
 
 2r. 
 
 28.125 
 
 . /^ of 20 r. 
 
 
 3 ft. . 
 
 2.556 f9- 
 
 . Jf of 2 r. 
 
 
 6 in. . 
 
 0.426/, 
 
 1 of 3 ft. 
 
 
 2 in. . 
 
 0.142 J, . 
 
 I of 6 in. 
 
 Price ofS fur. 22 r. 3 ft. 8 in. 12000,000 (192-3; Ex. 40.) 
 
 25. If I ride 5 fur. 3 r. 10 ft. 6 in. in a railroad car, 
 what ought to be my exact fare, at the rate of 11 cts. 
 per mile? Ans. 192-3; Ex. 39. 
 
 26. What will 23 gal. 2 qt. 1 pt. of wine cost, at §60 
 per hogshead? Ans. 194-5; Ex. 25. 
 
 27. What cost 11 hhd. 17 gal. 2 qt. of wine, at |49.77 
 per hogshead? Ans. §561.29 J. 
 
 28. If one silver cup weighs 8 oz. 4 dwt. 10 gr., what 
 will 6 cups, each of the same weight, be worth, at the 
 rate of §1.25 per ounce? Ans. 199; Ex. 20. 
 
 29. What will 537 'bushels of wheat cost, at §1.374 
 per bushel? 
 
 OPERATION. 
 
 Price at §1.00 per bu. 
 " 25 cts. " 
 
 §537.00 
 134.25 
 
 1 of §1.00. 
 
 121 cts. " 
 
 67.125 . h of 25 cts. 
 
 Price at §1.37. 
 
 §738.375 A71S. 
 
192 
 
 COMPOUND NUMBERS. 
 
 80. What cost 327 bushels of potatoes, at 62^ cts. 
 per bushel? 50 cts.= i of §1; 121 = i of 50 cts. 
 
 Ans. 1204.375. 
 
 31. What cost 453 bushels of corn, at 87 i cts. per 
 bushel? 87i'cts.= 50 cts.-{-25 cts.+ 12i cts. 
 
 Ans. 1396.375. 
 
 32. What cost 1999 gal. of wine, at |1.62i per 
 gallon? JLws. 13248.375. 
 
 33. What cost 5794 yd. of cloth, at |3.16| per yard? 
 16| cts.= i- of $1.00. Ans. $18347. 66f. 
 
 34. What cost 3579 yards of cloth, at $1.12i- per 
 yard? $1.18|? |2.26? $3.37J? 18| cts.=3 times 6| cts. 
 =j\ of $1.00. First Ans. $4026.375. 
 
 35. What cost 2468 gal. of wine, at $1.43} per 
 gallon? $2.50? $3.62 J? $4.56i? 43|- cts.=:25 cts.+l2^ 
 ctB.+6^ cts. Last Ans. $11260.25. 
 
 36. What cost 3 T. 10 cwt. 3 qr. of iron, at 6£ 4s. 
 6d. per ton? 
 
 OPERATION. 
 
 Price of 1 T. . . 6£ 4s. 6d. Therefore, 
 
 3 
 
 a 
 
 3T- . 
 
 . 18 13 
 
 6 
 
 
 u 
 
 10 cwt. . 
 
 . 3 2 
 
 3 . 
 
 . J of IT. 
 
 (.6 
 
 2qr. . 
 
 3 
 
 l/o 
 
 J-^ of 10 cwt 
 
 a 
 
 Iqr. . 
 
 1 
 
 m 
 
 . ^i of 2 qr. 
 
 Priceof3T. lOcwt. 3qr.22£ Os. 5 J^d-, or $106.58. 
 
 37. What cost 40 yd. 3 qr. 1 na. of broadcloth, at 1 £ 
 per yard? Ans. 40 £. 16s. 3d.=-$197.53i. 
 
 38. What cost 25 bu. 8 pk. 5 qt. of wheat, at 5 s. 6 d. 
 per bushel ? Ans. 7£ 2 s. 5 J ^ d---$34.48l3^.. 
 

 OPERATION. 
 
 3 for 1 jr. is 
 
 . . §325.00 
 
 2 
 
 " 2 yr. is 
 
 . . 650.00 
 
 " 3 m. is 
 
 . . 81.25 
 
 " 10 da. is 
 
 . . 9.02J- 
 
 COMPOLXi) XUMBEllS. 193 
 
 39 If I pay $325 for the use of a certain sum of 
 money 1 year, what ought I to pay for the use of the 
 same money 2 yr. 3 m. 10 da.? 
 
 Therefore, 
 
 . 1 of 1 yr. 
 . i of 3 m. 
 
 Use for 2 yr. 3 m. 10 da. is $740.27-J Ans. (185 ; Rem. 6.) 
 
 40. At the rate of $125 per year, what ought I to 
 pay for a sum of money 1 yr. 3. m. 15 da. ? 
 
 A71S, $161.45f. 
 
 41. At the rate of SI 75 per year, what ought I to 
 pay for a sum of money 1 yr. 4 m. 21 da. ? 
 
 A71S. $243.54^. 
 
 42. At the rate of $400 per year, what ought I to 
 pay for a sum of money 1 yr. 5 m. 7 da. ? 
 
 A71S. $574.44^, 
 
 43. At the rate of $525 per year, what ought I to 
 pay for a sum of money 2 yr. 3 m. 12 da. ? 
 
 Ans. $1198.75. 
 
 44. At the rate of $1000 per year, what ought I to 
 pay for a sum of money 5 yr. 5 m. 5 da.? 
 
 A71S. $5430.55|. 
 
 45. At the rate of $2000 per year, what ought I to 
 pay for a sum of money 10 yr. 10 m. 10 da. ? 
 
 Ans. $21722.22|. 
 
 46. At the rate of $750 per year, what ought I to 
 pay for a sum of money 3 yr. 7 m. 19 da.? 
 
 17 
 
194 
 
 COMPOUND NUMBERS. 
 
 
 OrERATION. 
 
 
 J for 1 yr. is . 
 
 . .?750.00 
 3 
 
 Therefore, 
 
 " 3 yr. is . 
 
 . 2250.00 
 
 
 " 6 m. is . 
 
 . 375.00 . 
 
 • J of 1 .yr- 
 
 " 1 m. is . 
 
 62.50 . 
 
 1 of 6 m. 
 
 " 15 da. is . 
 
 31.25 . 
 
 . I of 1 m. 
 
 " 3 da. is . 
 
 6.25 . 
 
 . 1 of 15 da 
 
 " 1 da. is . 
 
 2.08J . 
 
 . J of 3 da. 
 
 Use for 3 yr. 7 m. 19 da. |2727.08J Ans. 
 
 47. At the rate of §1000 per year, what ought I to 
 pay for a sum of money 5 yr. 9 m. 27 da.? 
 
 Ans. S5825.00. 
 
 48. At the rate of |2500 per year, what ought I to 
 pay for a sum of money 4 yr. 11 m. 25 da. ? 
 
 Am. §12465.27|. 
 
 49. At the rate of §825 per year, what ought I to 
 pay for a sum of money 6 yr. 2 m. 18 da.? 
 
 Ans. §5128.75. 
 
 50. The rent of a farm is §1150 per year. What 
 ought to be paid during an occupancy of 7 yr. 7 m.? 
 
 Ans. §8720.831 . 
 
 51. What ought I to pay for the use of §5000 during 
 4 yr. 3 m. 20 da., at the rate of §400 per year? 
 
 J.7is..§1722.22|. 
 
 52. What ought I to pay for §1850 during 3 m. 3 da., 
 at the rate of §148 per year? Ans. §38.23 1. 
 
 53. What ought I to pay for §9215 during 3 m. 3 da., 
 at the rate of §737.20 per year? Ans. §190.44 J. 
 
 54. What ought I to pay for the use of §8000 from 
 
COMPOUND NUMBERS. 195 
 
 January 27, 1866, to July 4, 1868, at the rate of $640 
 per year?— (198; Ex. 1.) Ans. $1559.111. 
 
 55. What ought I to pay for the use of $7250 from 
 July 4, 1865, to August 1, 1866, at the rate of $580 
 per year? Ans. $623.50. 
 
 56. What ought I to pay for the use of $5280 from 
 August 1, 1869, to September 9, 1871, at the rate of 
 $422.40 per year? Ans. 889.38|. 
 
 57. If I pay 12 cts. per year for the use of $1, "what 
 ought I to pay for its use 3 yr. 5 m. 15 da. ? 
 
 Ans. $0,415. 
 
 58. If I pay $99.75 per year for the use of $950, 
 what ought I to pay for the use of the same sum 2 yr. 
 4m. 20 da.? Ans. $238.29^. 
 
 59. What ought I to pay for the use of $421.40 for 
 3 yr. 5 m. 15 da., at the rate of $25,284 per year? 
 
 OPERATION. 
 
 Use for 1 yr. is . . $25,284 
 
 3 
 
 a 
 
 3 yr. is . 
 
 75.852 
 
 
 a 
 
 4 m. is 
 
 8.428 
 
 . 1 of 1 yr. 
 
 a 
 
 1 m. is 
 
 2.107 
 
 . 1 of 4 m. 
 
 u 
 
 15 da. is 
 
 . , 1.0535 
 
 . 1 of 1 m. 
 
 Use for 3 yr. 5 m. 15 da. $87.4405 
 
 203. Review in Addition. 
 
 1. Find the value of 123+45+2004. Ans. 2172. 
 
 2. Find the value of 21+105+710. Ans. 836. 
 
 3. Find the value of 7+90+1041. Aiis. 1138. 
 
196 COMPOUND NUMBERS. 
 
 4. Find the value of 50+75+432ia ^m 43335. 
 
 5. Find the value of ?1 2 3+^45+12 004. 
 
 Ans. $2172. 
 
 6. Find the value of 21 £+105 £+710 £. 
 
 Ans. 836 £. 
 
 7. Find the value of 7 m.+90 m.+1041 m. 
 
 Ans. 1138 ra. 
 
 8. Find the value of 50 pk.+75 qt.+43210 pt. 
 
 Ans. 44160 pt. 
 
 9. Find the value of .123+.45+.2004. Ans. .7734. 
 
 10. Find the value of .21+.105+.71. J.ns. 1.025. 
 
 11. Find the value of .7+.9+.1041. Ans. 1.7041. 
 
 12. Find the value of .5+.75+.4321. Ans. 1.6821. 
 
 13. Find the value of $0.123+S0.45+S0.2004. 
 
 Ans. $0.7734. 
 
 14. Find the value of .21 £+.105 £+.71 £. 
 
 Ans. 1.025 £. 
 
 15. Find the value of .7 1.+.9 m.+ .1041 fur. 
 
 Ans. 24.1041 fur. 
 
 16. Find the value of .5 pk.+.75 qt.+.4321 pt. 
 
 A71S. 9.9321 pt. 
 
 17. Find the value of 1.23+4.5+20.04. 
 
 Ans. 25.77. 
 
 18. Find the value of 2.1+1.05+7.1. Ans. 10.25. 
 
 19. Find the value of 7+90+10.41. Ans. 107.41. 
 
 20. Find the value of 50+7.5+.4321. 
 
 Ans. 57.9321. 
 
 21. Find the value of |1.23+$4.50+|20.04. 
 
 A71S. $25.77. 
 
 22. Find the value of 2.1 £+1.05 £+7.1 £. 
 
 Ans. 10.25 £. 
 
COMPOUND NUMBERS. 197 
 
 23. Find the value of 7 l.-f9 m.+10.41 fur. 
 
 Ans. 250.41 fur. 
 
 24. Find the value of 50 pk.+7.5 qt.+.4321 pt. 
 
 A71S. 815.4321 pt. 
 
 25. Find the value of |+i+|. ^ns. IJ. 
 
 26. Find the value of J + | + |. Ans. If §. 
 
 27. Find the value of ^^-{-f^-\-^^. Ans. l^gf . 
 
 28. Find the value of ii-hyVs + Z/s- -^ns. f i§. ' 
 
 29. Find the value of ||+li+li. Ans. |1.12i. 
 
 30. Find the value of i £+| s.-[-| d. 
 
 Ans. 128J d. 
 
 31. Find the value of y^^ 1.+/^- m.+ ^g fur. 
 
 Ans. lOif fur. 
 
 32. Find the value of ^| pk.+ fVs ^t-+i¥3 P^- 
 
 Ans. 4|| pt. 
 
 33. Find the value of 123|+45|+2004|. 
 
 Ans. 2173|. 
 
 34. Find the value of 21i+105H-710|. 
 
 Ans. 837|g. 
 
 35. Find the value of 7i\+90/o+1041/5. 
 
 A71S. 1139^i§. 
 
 36. Find the value of 50^J+75/5V+43210//3. 
 
 Ans. 43335|i§. 
 
 37. Find the value of 1.231+4.5-1+20.041. 
 
 Ans. 25.791. 
 
 38. Find the value of 2.1i+1.05|+7.10|. 
 
 Ans. 10.31/5. 
 
 39. Find the value of .7f,-+9.0/3+10.41/5. 
 
 ^ Ans. 20.18H§. 
 
 40. Find the value of 5.0iJ+7.5jV5+4321.0//3. 
 
 Ans. 4333.56.\. 
 
198 COMPOUND NUMEERi. 
 
 41. Find the value of $1.23|+§4.51i+§20.04|. 
 
 A71S. $25,791. 
 
 42. Find the value of 2.15X+1.05|£+T.108£. 
 
 A71S. 10.3141 £. 
 
 43. Find the value of .7i\ r.+9.0/^ ft.H- J in. 
 
 Ans. 1 r. 4 ft. 7 in. 
 
 44. Find the value of 5.0^| A.+7.5jV5 R.+l r. 
 
 Ans. 6 A. 3 R. 21.48 r. 
 
 45. Add together l£ 2s. 6d.; 9£ 8s. 9d.; 12£ 13s. 
 2d.; and 4£ 7s. 3d. Ans. 27£ lis. 8d. 
 
 46. Add together 15 r. 16 ft. 6 in.; 33 r. 14 ft. 7 in.; 
 and 19 r. 8 ft. 9 in. Ans. 69 r. 6 ft. 10 in. 
 
 47. Add together 5 A. 2 R. 7 r. ; 8 A. 3 R. 32 r. ; and 
 9 A. 1 R. 27 r. Ans. 23 A. 3 R. 26 r. 
 
 48. Add together 3 hhd. 62 gal. 3 qt.; 2 hhd. 16 gal. 
 1 qt.; 3 hhd. 57 gal. 2 qt.; 3 hhd. 45 gal. 3 qt.; 2 hhd. 
 59 gal. 3 qt. ; and 3 hhd. 39 gal. 2 qt. 
 
 Ans. 20 hhd. 29 gal. 2 qt. 
 
 204. Review in Subtraction. 
 
 1. From 1001 take 763. Ans. 238. 
 
 2. From 3999 take 455. Ans. 3544. 
 
 3. From 31015 take 9999. Ans. 21016. 
 
 4. From 11111 take 7778. Ans. 3333. 
 
 5. From $245 take $78. Ans. $167. 
 
 6. From 24 pears take 19 pears. Ans. 5 pears. 
 
 7. From 91 apples take 74 apples. Ans. 17 apples. 
 
 8. From 375 da. take 109 da. Ans. 266 days. 
 
 9. From .763 take .1001. Ans. .6629. 
 
 10. From .455 take .3999. Ans. .0551. 
 
 11. From .45 take .0073. Ans. .4427. 
 
COMPOUND NUMBERS. 199 
 
 12. From .39 take .039. Ans. .351. 
 
 13. From §0.37 take |0.227. Ans. |0.143. 
 
 14. From .5 pears take .25 pears. Ans. .25 pears. 
 
 15. From .91 apples take .74 apples. 
 
 Ans. .17 apples. 
 
 16. From .67 lb. take .4937 lb. Ans. .1763 lb. 
 
 17. From 10.01 take 7.63. Ans. 2.38. 
 
 18. From 39.99 take .455. Ans. 39.535. 
 
 19. From 3.8 take 1.005. Ans. 2.795. 
 
 20. From 37 take 19.04. Ans. 17.96. 
 
 21. From |3.25 take $2.12. Ans. |1.13. 
 
 22. From 3.5 da, take 2.85 da. Ans. .65 days. 
 
 23. From 43.7 m. take 27.43 m. Ans. 16.27 m. ' 
 
 24. From 9.5 sec. take 4.67 sec. Ans. 4.83 sec. 
 
 25. Find the value of 3— ^ ; of 4— |. Ans. 2^ ; 3}. 
 
 26. Find the value of |— | ; of |— ^. Ans. | ; |. 
 
 27. Find the value of 3^—2 J ; of 4i— 3l. 
 
 Ans. 1|; IgL. 
 
 28. Find the value of 8— 3^ ; of 12—41. 
 
 Ans. 4i ; 7|. 
 
 29. Find the value of 3.8— 2j ; of 37— 19-^^ 
 
 Ans. 1.3 ; 17.96. 
 
 30. Find the value of 4.3^—2.331; of 84— 3.14f. 
 
 Ans. 2.01f ; 5. 
 
 31. Findthe value of 42— .OOOJ; 37—4.000'". 
 
 Ans. 41.9991; 32.999 J. 
 
 32. Find the value of 860.4581— 25.1 J. 
 
 Ans. 835.3248f . 
 
 33. From | h. take i m. Ans. 29 m. 45 sec. 
 
 34. From i da. take J h. Ans. 5. h. 45 m. 
 
 35. From | m. take * in. Ans. 191 r. 16 ft. 5J in. 
 
200 
 
 COMPOUND NUMBERS. 
 
 36. From 
 
 J A. take J 
 
 sq. m. 
 
 ».75i. 
 
 A71S. 19r. 272 ft. 35f in. 
 Ans. $0,495/5. 
 Ans. 7 ft. 8|g in. 
 Alts. 2 gal. -i qt. 
 
 A718. |1.00/g. 
 
 Ans. $12.48. 
 
 37. From §1.25| take 
 
 38. From 8.3 J ft. take 7.0| in. 
 
 39. From 2J gal. take J qt. 
 
 40. From 7i francs take 371 cts. 
 
 41. From 3|£ take 3} dollars. 
 
 42. From 4i bu. take 3j pk. J.ws. 3 bu. 2 pk. 4 qt. 
 
 43. From 9 m. take 8 m. 7 fur. 39 r. 16 ft. 5 in. 
 
 Ans. 1 inch. 
 
 44. From 8 m. 7 fur. 16 ft. 5 in. take 5 m. 6 yd. 11 in. 
 
 A71S. 3 m. 6 fur. 39 r. 14 ft. 
 
 45. From 1£ 21s. 13d. 5 far. take 5s. 3d. 7 far. 
 
 Ans. l£16s. 9d. 2 far. 
 
 46. From 9 bu. 3 pk. 5 qt. take 4 bu. 7 pk. 15 qt. 
 
 Ans. 3 bu. 2 pk. 6 qt. 
 
 I305. Review in Multiplication and Division. 
 
 1. Multiply 45 by 45. 
 
 3. Multiply 101 by 1001. 
 
 5. Multiply .45 by .45. 
 
 7. Multiply .101 by .1001. 
 
 9. Multiply 4.5 by 45. 
 11. Multiply .0009 by .0009. 
 13. Multiply $45 by 75. 
 15. Multiply 365 da. by 17. 
 17. Multiply 375.61b. by .125. 
 19. Multiply 03.5 bu. by .78. 
 21. Multiply 74 by ^f-y. 
 23. Multiply vjVj by 370. 
 25. Multiply H by ff. 
 27. Multiply 4 by zll. 
 29. Multiply 6^| by 24. 
 31. Multiply 4^ by 3f 
 
 Am. 145. 
 
 2. Divide 2025 by 45. 
 
 4. Divide 101101 by 1001. 
 
 6. Divide .2025 by .45. 
 
 8. Divide .0101101 by .1001. 
 10. Divide 202.5 by 4.5. 
 12. Divide .00000081 by .0009. 
 14. Divide $3375 by 75. 
 10. Divide 6205 da. by 17. 
 18. Divide 4G.95 by .125. 
 20. Divide 49.53 bu. by .78. 
 22. Divide 6 by ^i^^. 
 24. Divide i||o by -870. 
 20. Divide j% by j^^. 
 28. Divide 14 by 3i|. , 
 30. Divide 7| by 3. 
 32. Divide 14^ by S^V 
 
 Ans. 415. 
 
{J03il'0UXD ^' UMBERS. 201 
 
 33. Multiply .OOOi by 4.8. 
 85. Multiply S.Of by 2.00i. 
 37. Multiply 375f r. by |. 
 39. Multiply 63^ A. by |f 
 41. Multiply 3.75ir. by 51. 
 43. Multiply 8.61 bu. by 129. 
 
 31. Divide .0016 by .0001. 
 
 36. Divide 6.031 1| by 2.00^. 
 
 38. Divide 46 |i r. by i. 
 
 40. Divide 49.53 A. by ff. 
 
 42. Divide 191f i by 51. 
 
 44. Divide 1112f bu. by 8|. 
 
 45. Multiply 26 £ 14 s. 8d. 3 far. by 11. 
 
 Ans. 294: £ 2 s. Od. Ifar. 
 
 46. Multiply 1 m. 1 fur. 1 r. 1 ft. 1 in. by 27. 
 
 Ans. 30 m. 3 fur. 28 r. 12 ft. 9 in. 
 
 47. Multiply 8 T. 1 cwt. 3 qr. 7 lb. 5 oz. 6 dr. by 3^ 
 
 Ans. 28 T. 6 cwt. 1 qr. 13 lb. 2 oz. 13 dr. 
 
 48. Multiply 365 da. 5 li. 48 m. 47.57 sec. by 6. 
 
 Ans. 2191 da. 10 h. 52 m. 45!'42 sec. 
 
 49. Add together 10 apples, 4 pears, and 6 peaches 
 (Vide 171.) 
 
 50. Add together 3 ft. and 4 in. Ans. 40 inches. 
 
 51. From §100 take 4 bu. 2 pk. 1 qt The problem 
 is absurd. Why? 
 
 52. I buy 4 bu. 2 pk. 1 qt, of cherries at 50 cts. per 
 qt., and pay for them out of a §100 bill. How much 
 money do I receive in change? Aiis. §27.50. 
 
 53. Multiply 25 cts. by 25 cts.— (Vide 82.) 
 
 54. Divide 25 rails by 25 rails.— (Vide 96.) 
 
 55. Reduce each of the following expressions to a 
 compound number: 3.23125 £; S^Vo^; 3.23|£; 64.- 
 625s.; 64gs; 64.6is.; 775 id.; 775.5 d.; 3102 far. 
 
 Ans. 3£4s. 7d. 2 far. 
 
 56. Reduce 1 yr. 2 mo. 6 da. to months and decimals 
 of a month. 1 yr. 3 mo. 9 da.; 2 yr. 4 mo. 12 da.; and 
 3 yr. 6 mo. 15 da. 
 
 J.ns. 14.2 mo. ; 15.3 mo.; 28.4mo.; 42.5 mo. 
 
202 PERCENTAGE. 
 
 PERCENTAGE. 
 
 206. Percentage embraces those operations in 
 which numbers are compared with 100 as a unit. 
 
 SOT. Per centum, in Latin, signifies by the hundred, 
 but the contraction per cent, is used, for the most part, 
 as a synonym of the word hundredth or hundredths. 
 Thus, 
 
 1 per cent, of 25 is the same as 1 hundredth of 25. 
 
 2 per cent, of 50 is the same as 2 hundredths of 50. 
 25 per cent, of a number is the same as 25 hundredths 
 of it. 
 
 The sign ^ is the same as the contraction per cent. 
 
 The RATE PER CENT, is the number of hundredths. 
 Thus, 
 
 ■ 5 ^ of 400 is read 5 per cent, of 400, and has the 
 same meaning as 5 hundredths of 400, that is, yj^ of 
 400, or .05 of 400. 
 
 7 % of a number is 7 hundredths of it. 
 
 1% of a number is f hundredths of it, that is ^f^ 
 of it. 
 
 ^iOS. To represent decimally any given rate per 
 cent., 
 
 Write the given rate as so many hundredths. 
 
 EXAMPLES. 
 
 1. Represent the following rates per cent, decimally, 
 viz: 1%; 2%; 12%; 10%; 25%; 47%; 100%; 
 125%; 200%; 1000%; 1275%. 
 
 Ans..O\] .02; .12; .10; .25; .47; 1.00; 1.25; 2.00; 10.00; 12.75. 
 
TERCENTAGE. 203 
 
 2. Represent decimally h%;i%; ~l%; Jo%> i o % ; 
 
 A71S. .005; .0025; .001; .0005; .001; .0002; .0075; .004. 
 
 3. Represent decimally 1^^; 2\%; 3j%; lj^%; 
 ^i\fo; 11%; H%; H%. 
 
 Ans. .015; .0225; .032; .0105; .061; .075; .051; .041. 
 
 4. Represent decimally ^^-^ ; ^%%; |%; 125|%; 
 4Jg%. ^7is. .00025; .00075; .00375 ; 1.25|; .040625. 
 
 209. To find what rate per cent, a given decimal 
 represents, 
 
 Multiply the decimal hy 100 and reduce the result^ if 
 necessary^ as in 164. 
 
 EXAMPLES. . 
 
 1. What rate per cent, does .01 represent? .04? .05? 
 .07? .1? .2? Ans,l%',4.%', 5%; 1 %; 10%; 20%. 
 
 2. What rate per cent, does .00428f represent? .000|? 
 .008 1 ? (Vide 164, Ex. 7.) Ans, | %; -^^%', %%. 
 
 3. What rate per cent, does .001 represent? .0075? 
 .041^? 1.00? Ans, ^\%; f %; 4i%; 100%. 
 
 4. What rate ^er cent, does .007 represent? .0007? 
 1.07? Ans. /^%; tJo%; 107%. 
 
 !S10. To find the per cent, (or percentage) of a num- 
 ber at a given rate, 
 
 Multiply the given number hy the rate per cent, ivrit- 
 ten as a decimal. The product will be the per cent. 
 required. 
 
 EXAMPLES. 
 
 1. Find 3% of 25; -J% of |500 ; and 1.^^% of 
 7000 lbs. of coff'ee. 
 
204 PERCENTAGE. 
 
 OPERATIONS. 
 
 25 $500 7000 lb. 
 
 .03 .005 .0105 
 
 .75=1 Ans. 12.500 Ans. 73.5000=-73i lb. Ans. 
 
 2. Find 1 % of $120 ; 2 % of $410 ; 7 % of 140 bu. ; 
 9% of 555 da. (211, Ex. 2.) 
 
 3. Find 6% of 333 fur.; 5% of 400 m.; 1% of 
 $1000; J % of $3333. 
 
 4. Find 1% of $7000; n% of $9000; 2^% of $700; 
 J^% of $10000. 
 
 5. Find 2% of$125;7^;ll%;25%;75%;100%; 
 150%. 
 
 6. Find 11% of Gets.; 7cts.; $1.20; $1.75; $3.50; 
 $7.62i; $9.45. 
 
 7. Find 25% of $25; 33i % of $500; 16f % of 
 $8000. 
 
 8. Find 101 J % of $505; 202 J % of $404; 1000% 
 of $23.10. 
 
 9. Received at Mobile from New Orleans 200 lihds. 
 of sugar ; but in discharging the cargo 2 % of the sugar 
 was lost. How much remains to be sold? 
 
 Ans. 196 hhds. 
 
 10. The steamer Indiana started from Vicksburg with 
 500 bales of cotton on board. On the trip to New Or- 
 leans 14% of the cotton was transferred to another boat. 
 How much remained on board? Ans. 430 bales. 
 
 11. Bought at New Orleans 75 hhds. of molasses, but 
 on its arrival at Cincinnati 4% is missing. How much 
 remains to be sold? Ans. 72. hhds. 
 
 12. Shipped at Havana for New York 1250 boxes of 
 
PERCENTAGE. 205 
 
 oranges; on the passage 14 fo of the oranges Avere 
 thrown overboard. How many boxes arrived in New 
 York? Ans. 1075 boxes. 
 
 13. On a trip from New Orleans to New York I ex- 
 pend 25% of my money. I started with. §360. How 
 much did I have on arriving at New York? (Vide 167, 
 Ex. 103.) Ans. §270. 
 
 14. During a storm a captain threw overboard 16| ^ 
 x)f a cargo of cotton. He left New Orleans with 720 
 bales; with how many did he arrive at Liverpool^ (167, 
 Ex. 104.) Ans. 600 bales. 
 
 15. My wine made during the year 1865 was 378 
 gallons. Reserving 7^%, I sell the remainder at §4.50 
 per gallon. What did the crop bring me? (167, Ex. 
 105.) Ans. §1579.50. 
 
 16. My wine made during the year 1866 w^as 450 
 gallons. Reserving 10 ^ for private use, I sell the re- 
 mainder at §3.75 per gallon. Which year was most 
 profitable, 1865 or 1866, and by how much? 
 
 Ans. 1865, by §60.75. 
 
 17. I buy sugar for §1700, and sell so as to clear 5 ^ 
 on the cost. How much do I get? Ans. §1785. 
 
 18. A merchant failing, pays his creditors 30^. He 
 owes A §2500; B §4000; and C §4500. What will each 
 receive? Ans. A §750; B §1200; and C §1350. 
 
 19. I pay 10% of my salary for board; i% for 
 washing ; l2 % for clothes, and 8 % for other expenses. 
 How much do I clear from a salary of §2000 ? 
 
 Ans. §1395. 
 Sll. To find the rate per cent, of a number at a 
 given percentage, 
 
' c 
 
 } 
 
 206 PEllCEXTAGE. 
 
 Divide the given percentage hy the number of which 
 the rate per cent, is required. The quotient will repre- 
 sent the rate per cent, decimally, and may be changed 
 as in 209. 
 
 EXAMPLES. 
 
 1. What rate % of 25 is .75? of $500 is $2.50? of 
 7000 lb. of coffee is 73. \ lb. of coffee? 
 
 OPERATIONS. 
 
 (1.) " (2.) (3.) 
 
 25).75 500)2.50 7000)73.5 
 
 .03=% Ans. .005= J % Aiis. .0105=1 ^.^^^ -^ns. 
 
 2. What rate % of $120 is $1.20? of $410 is $8.20? 
 of 140 bu. is 9| bu. ? of 555 da. is 49 ^ § da. ? 
 
 3. What rate % of 333 fur. is 19|~§fur.? of 400 m. 
 is 20 m.? of $1000 is $5.? of $3333 is $11.11? 
 
 4. What rate % of $7000 is $17.50? of $9000 is 
 $135? of $700 is $15f ? of $10000 is $5? 
 
 5. What rate % of $125 is $2.50? $8.75? $13.75? 
 $31.25? $93.75? $125? $187.50? 
 
 6. What rate % of 6 cts. is j% m. ? of 7 cts. is 1 j^ m.? 
 of $1.20 is If cts.? of $1.75 is 2| cts.? of $3.50 Is 5J- 
 cts.? of $7.62i is 11/5 cts.? of $9.45 is 14/^ cts.? 
 
 7. What rate % of $25 is $6 J? of $500 is $166|? 
 of $8000 is $1333| ? 
 
 8. What rate % of $505 is $511.06? of $404 is 
 $818.10? of $23.10 is $231? 
 
 9. From a ship having on board 200 bales of cotton, 
 4 bales fell into the sea, and wer^ lost. What rate % 
 were these four bnles of the whole number on board? 
 
rEllCEXTAGE. 207 
 
 10. A boy commenced play with 200 marbles, and 
 ended with 196. What was his rate ^ of loss? 
 
 11. A ship sailed from New Orleans for Liverpool 
 with a cargo of 500 bales of cotton. The ship reached 
 her port with only 430 bales, the remainder "having been 
 thrown overboard. With what rate % of her cargo did 
 she reach Liverpool, and what rate % had been thrown 
 into the sea? 
 
 12. Out of 1250 boxes of oranges shipped at Havana, 
 1075 arrived in New York. What rate % of the cargo 
 had been lost ? 
 
 13. I started from Charleston with $360, and arrived 
 in Quebec with §270. What rate % of my money had 
 been expended in the trip? 
 
 14. Out of my wine made in 1865, which was 378 
 gallons, I sold a quantity amounting, at the rate of ?4j 
 per gallon, to $1579|, having reserved the remainder 
 for private usfe. What rate ^ on the whole wine was 
 reserved ? 
 
 15.. My wine made in 1866 had increased at the rate 
 of 19 2^- ^/o on that made in 1865, but tHe market being 
 dull I sold a portion of it for §3.75 per gallon, clearing 
 §60.75 less than on the previous year. What rate ^ of 
 the wine of 1866 remained unsold? 
 
 !S1!^. To find a number on which, at a given rate ^, 
 a given percentage may be obtained. 
 
 Divide the given 'percentage hy the given rate per cent, 
 expressed decimally. .The quotient Avill be the required 
 number. 
 
208 ^ PERCENTAGE. 
 
 EXAMPLES. 
 
 1. What number is that of which 3% is .75? 1% of 
 how many dollars is $2.50 ? l^^ % of how many pounds 
 is 73i pounds? 
 
 OPERATIONS. 
 
 (1.) (2.) (3.) 
 
 .03).75 .005)12.500 .0105)73.5000 lb. 
 
 25 Ans. 1500 Ans. 7000 lb. Ans. 
 
 2. 8^ of what number will produce 125? 
 
 Am. 1562.50. 
 
 3. 8^ of how many dollars will produce |175? 
 
 A71S. 12187.50. 
 
 4. 5 ^ of how many dollars will produce |400 ? 
 
 Ans. 18000. 
 
 5. 4^ of how many dollars will produce §250? 
 
 Ans. 16250. 
 
 6. On a pleasure excursion I spend $90, which I find 
 is 25 fo of the money with which I started. How much 
 money have I still on hand? Ans. (211, Ex. 13.) 
 
 7. If I pay 8 ^ of a sum of money for its use during 
 a year, and thereby pay |525, what amount of money 
 do I have the use of? Ans. $6562.50. 
 
 8. If I pay 6^ of Sb sum of money for its use during 
 a year, and thereby pay $750, what amount of money 
 do I have the use of? Ans. $12500. 
 
 9. I borrow $350 for 1 year, and agree to pay 7% 
 of the sum for its use. How much do I pay ? 
 
 Ans. $24.50. 
 
 10. If I pay $24.50 for the use of $350 for 1 year, 
 what rate % on the money do I pay? * Ans. 7%, 
 
PERCENTAGE. 209 
 
 11. What amount of money can I get the use of for 
 1 year by paying $24.50, that being 7 ^ on the money 
 borrowed? Ans. $350. 
 
 S13. A number being given which is a given rate 
 per cent, more than another number, to find that other 
 number, 
 
 Divide the given number by \ -\- the rate per cent, 
 written as a decimal. 
 
 EXAMPLES. 
 
 1. 560 is 12 fo more than a certain number. What 
 is that number? 
 
 Remark. — Thp question is precisely the operation. 
 
 same as this; 560 is iif of what number? 
 and (560-112) X 100=560-1.12. 1.12 )560.00 
 
 500 Ans. 
 
 2. 1000 is 33| fo more than a certain number. What 
 is that number? 1000-^1.33|=| of 1000. 
 
 Ans. 750. 
 
 3. $150 is 20% more than what sum? Ans. $125. 
 
 4. $140 is 16? fo more than what sum? 
 
 Ans. $120. 
 
 214. A number being given which is a given rate p'er 
 cent, less than another number, to find that other number. 
 
 Divide the given number by X — the rate per cent, ivr it- 
 ten as a decimal. 
 
 EXAMPLES. 
 
 1. 270 is 25 % less than what operation. 
 
 number? 1-^9 
 
 Remark. — The question is the same as 
 
 this: 270 is ^^-^ of what number? and (270 •'^5 )270.00 
 -f-75) X 100=270— 75. (Vide 167. Ex. 98.) 36O im 
 
 18 
 
210 PERCENTAGE. 
 
 2. $1.40 is 30% less than what sum? 
 
 Ans. $2.00. 
 
 3. $4.50 is 25% less than what sum? 
 
 A71S. $6.00 
 
 4. $8.75 is 33J% less than what sum? (| of 8.75.) 
 
 Ans. $13.12J. 
 
 APPLICATIONS OF PERCENTAGE. 
 
 215. Insurance is a contract made between parties, 
 by which the one binds itself, for a consideration, to re- 
 imburse the other for losses of property occasioned by 
 fires, or other casualties. 
 
 (1.) The party taking the risk is called the Under- 
 writer. 
 
 (2.) The party protected is called the Insured. 
 
 (3.) The Policy is the written contract of insurance. 
 
 (4.) The Premium is the sum paid for insurance. 
 
 (5.) The premium is usually a percentage on the value 
 of the property insured, and is paid at the time the pol- 
 icy is drawn. 
 
 ' 310. To find the Premium, when the amount insured 
 and the rate per cent, of insurance are given. 
 
 Multiply the amount insured hy the rate %, wiHtten 
 decimally. — (Vide 210.) 
 
 EXAMPLES. 
 
 1. What premium must be paid annually for insuring 
 a house worth $4500, at -J % ? Ans. $11.25. 
 
 2. What is the premium on a cargo of cotton valued 
 at $2500, at 1 % ? Ans. $3,125. 
 
PEllCE^'TAGE. 211 
 
 3. What is the premium on a cargo of goods carried 
 from New York to Mobile, the goods being valued at 
 112500, and insured at Ij % ? Ans. $187.50. 
 
 4. I have a house worth §4500, and insure it for | of 
 its value at 1|% per annum. What is the expense of 
 insurance, including $2.00 for the policy? 
 
 Ans. §50.00. 
 
 5. I lose by fire four houses, valued at §25000. On 
 this property I had paid annually, during five years, a 
 premium of lj% on the entire value. The policy be- 
 ing good, what have I saved by insuring the property? 
 
 Ans. §28437.50. 
 
 217. To find for what sum a policy must be taken 
 out, at a given rate per cent., to cover both property and 
 premium, 
 
 Divide the sum for wJiich the property is to be insured 
 hy 1 — the rate ^, written as a decimal. — (Yide 214.) 
 
 EXAMPLES. 
 
 1. Shipped a cargo of flour from New York to Mata- 
 moras, valued at §23940. For what sum must it be in- 
 sured to cover the value of the flour and the premium, 
 the rate of insurance being 5 ^ ? Ans. §25200. 
 
 2. For what sum must I insure §45000 worth of cot- 
 ton, shipped from Yicksburg to New Orleans, so as to 
 cover the cotton and premium, the rate of insurance 
 being 2 J % ? Ans. §46153 |f 
 
 3. The premium for insuring a house is ^50, including 
 §2.00 for the policy. The rate of insurance is If %. 
 What is the value of the house ?— (Yide 212.) 
 
 Ans. §3000. 
 
212 PERCENTAGE. 
 
 318. Commission is a sum allowed to a Commission 
 Merchant, Agent, or Factor, by a Principal, for his 
 services in buying or selling goods. The Agent, if re- 
 siding in a different part of the country, or in a foreign 
 country, is called a Consignee; the goods shipped, a 
 Consignment; the Principal, a Consignor. 
 
 The commission is usually a given percentage of the 
 money involved in the purchase or sale. 
 
 210. To find the commission on a given sum, at a 
 given rate per cent.. 
 
 Multiply the given sum hy the given rate ^, written as 
 a decimal. — (Vide 210.) 
 
 EXAMPLES. 
 
 1. I receive an order in Mobile, from Liverpool, for 
 the purchase of 300 bales of cotton. The cotton I buy 
 weighs on an average 500 pounds to the bale, and costs 
 in Mobile 11 J cts. per lb. What is my commission, at 
 the rate of 1^% on the cost of the cotton? 
 
 Ans. §253.12^. 
 
 2. A commission merchant sells goods to the amount 
 of §4375, on which he receives a rate of 2 ^. 'To what 
 does his commission amount? Ans. |87.50. 
 
 3. What is the commission on a purchase of 50 bales 
 of cotton, at 500 pounds to the bale, and costing 10| cts. 
 per pound, the commission being IJ % ? 
 
 Ans. §45.39Jg. 
 
 330. To find the commission when the amount in- 
 cludes the sum to be invested, and also the commission, 
 
 Divide the given amount by l-\- the rate ^, ivritten as 
 a dscimal, and subtract the result from the given amount. 
 
<^ ■^^ 
 
 EXAMPLES. 
 
 •L3 
 
 1. I receive in Mobile |7500, with which to purchase 
 cotton. My commission is to be 2 ^ on the purchase, 
 which is to be deducted from the money. What is my 
 commission? Ans. $147.05] |. 
 
 2. Suppose I had received $22400, with the under- 
 standing that my commission should be 2j ^ on the 
 amount purchased. AVhat amount should I have to ex- 
 pend for cotton? Ans. $21853.658. 
 
 3. I send a commission merchant $1000, with which 
 to buy cotton, after deducting his commission of 5 J^ on 
 the money invested. What amount is invested in cotton, 
 and how much is retained as commission? 
 
 Ans. $952.38 and $47.62. 
 
 221. Stock is money belonging to a collection of in- 
 dividuals, called a Corporation, authorized by law to do 
 business together. 
 
 (1.) The owners of the stock are called Stockholders. 
 
 (2.) A Share is one of the equal parts into which the 
 stock is divided. Such a part is usually $100. 
 
 (3.) A Certificate is a written evidence of ownership 
 of stock. 
 
 (4.) The par value of stock is the number of dollars 
 mentioned in each share. 
 
 (5.) The market value of stock is the number of dol- 
 lars a share w^ll bring when sold for cash. 
 
 (6.) Stock is above par, or below par, according as the 
 market value is above or below the par value. 
 
 (7.) If above par, stock is at a premium; if below 
 par, it is at a discount. 
 
sft 
 
 PERCENTAGE. 
 
 22S. To find the value of stock at a given rate per 
 cent, premium, 
 
 Multiply the par value hy l-\- the rate ^ ivritten as a 
 decimal. 
 
 EXAMPLES. 
 
 1. What is the value of 17 shares of stock, at 5 ^ , 
 
 ? 
 
 premium 
 
 OPERATION. 
 
 11700 
 1.05 
 
 85.00 ^"^^ ■/ 
 1700 ^-^ '^^ 
 
 ?^ 
 
 $1785.00 Ans. (Vide 210, Ex. 17.) 
 
 2. What is the value of 14 shares of railroad stock, 
 at a premium of 7 % ? Ans. $1498. 
 
 3. What is the value of $32000 in State bonds, at a 
 premium of J ^ ? ' Ans. $32040. 
 
 223. To find the value of stock at a given rate per 
 cent, discount. 
 
 Multiply the par value hy 1 — the rate % written as a 
 decimal. 
 
 EXAMPLES. 
 
 1. What is the value of 17 shares of stock at 5 % 
 discount ? 
 
 OPERATION. 
 $1700 
 
 .95 
 
 85.00 
 1530.0 
 
 $1615.00 
 
PERCENTAGE. 215 
 
 2. What is the value of 14 shares of railroad stock, at 
 a discount of 7 % ? Ans. §1302. 
 
 3. What is the value of 12 J shares of stock, at a dis- 
 count of 14 % ?— (210, Ex. 12.) Ans. |1075. 
 
 4. What is the value of |13000 in State bonds, at a 
 discount of 8% ? Ans. $11960. 
 
 5. I invest $10000 in railroad stock at the par value. 
 In a year the stock depreciates 3 fo, and, fearing a 
 further decline, I sell all my certificates. What is my 
 loss? Ans. $300. 
 
 6. If I buy 15 shares of stock at a premium oi Sfo, 
 and sell at a discount of 3%, what do I lose? 
 
 Ans. $90. 
 
 7. If I buy 18 shares of railroad stock at 5 ^ below 
 par, and sell at 7 % above par, what do I gain ? 
 
 !S34. Brokerage is the percentage charged by money 
 dealers, called Brokers, for negotiating Bills of Exchange. 
 Brokers all deal in stocks and other monetary matters. 
 
 225. To find the brokerage on a given sum. 
 Multiply the sum hy the rate fo written as a decimal. 
 
 EXAMPLES. 
 
 1. Wishing to rais,e an amount of money, I sell to a 
 broker 100 shares of railroad stock at a discount of i %. 
 What is the amount of brokerage? Ans. $25.00. 
 
 2. What must I pay a New Orleans broker for cashing 
 bills on New York to the amount of $5000, brokerage 
 at the rate of > % ? Ans. $25.00. 
 
 22%. Profit and Loss are terms used by merchants 
 and other business men in reference to the purchase and 
 sale of goods. 
 
2r6 PERCENTAGE. 
 
 (1.) The cost is the price paid for an article. 
 
 (2.) The selling price is the amount received for an 
 article. 
 
 (3.) The profit is the amount received less the cost. 
 
 (4.) The loss is the cost less the amount received. 
 
 2217. To find the profit or loss when the cost price and 
 the rate per cent, of profit or loss are given, 
 
 Multiply the cost price by the rate fo written as a 
 decimal. The result will be the profit or loss. 
 
 EXAMPLES. 
 
 1. A merchant bought goods for ^500, and sold them 
 at a profit of 12^. What does he clear? Ans. $60. 
 
 2. If I buy goods for |750, and sell them at a profit 
 of 33A %, what do I clear ? Ans. ,^250. 
 
 3. What is the profit on oil, valued at $175, retailed 
 at 25% above the cost? Aiis. $43.75. 
 
 4. Buy sugar for $700, $800, and $1000; clear 25% 
 on the first lot, 33^ % on the second, and lose 50 % on 
 the third. How did I come out of the trade? 
 
 Ans. Lost $58 J. 
 
 228. To find the rate per cent, of profit or loss when 
 the cost and selling prices are givQn, 
 
 Divide the difference hetiveen the cost and selling prices 
 hy the cost price. Change the quotient by 209. 
 
 EXAMPLES. 
 
 1. If I buy goods for $500, and sell the same for 
 $560, what is the rate % of profit? 
 
 2. If I buy a quantity of flour for $750, and sell it 
 for $1000, what is the rate % of profit? 
 
PERCENTAGE. £17 
 
 f 
 
 3. If I buy flour at |4 per barrel, and sell it at §5.50 
 per barrel, what is the rat^ fo of profit? 
 
 OPERAftoNS. 
 
 (1.) (^.) (3.) 
 
 §560 11000 $5.50 
 
 500 750 4.00 
 
 500) 60.00(.12 750) 250.00(.33J 4.)1.50(.37i 
 
 60.00 250.00 1.50 
 
 Arts. 12 fc- Ans. 33|%. Ans.S1l%> 
 
 4. If I purchase tea at 60 cts. per pound, and sell it 
 at 90 cts., what is the rate ^ of profit? Ajis. 50^. 
 
 5. If I buy 40 yards of broadcloth at §2.50 per yard, 
 and sell the whole for $120, what is the rate ^ of profit? 
 
 Ans. 20%. 
 
 6. I have in my storehouse 300 barrels of damaged 
 flour, which cost me §1450. I am willing to sell the lot 
 at §4 per barrel. What w^ould be the rate fo of loss ? 
 
 Ans, nj^fo. 
 
 7. Cost price §1.20, selling price §1.50. Rate % of 
 profit? Ans, 229, Ex. 5. 
 
 8. Cost price §1.25, selling price §1.75. Bate % of 
 profit ? 
 
 9. Cost price §1.40, selling, price §2.00. Rate % of 
 profit ? 
 
 10. Cost price §4.50, selling price §6.00 Rate % of 
 profit ? 
 
 11. Cost price §6.00, selling price §4.50. Rate.^ of 
 loss? 
 
 12. Cost price §2.00, selling price §1.40. Rate % of 
 loss? 
 
 19 
 
PERCENTAGE. 
 
 229. To find the selling price, when the cost price is 
 known, so that a given per cent, may be made or lost, 
 
 (1.) If profit is to be made, multiply tlie cost^rice hy 
 l-\- the rate ^, ivritten as u decimal. (Vide 222.) . 
 
 (2.) If loss is to be susfained, multiply the cost price 
 hy 1 — the rate fo, ivritten as a decimal. (Vide 223.) 
 
 EXAMPLES. 
 
 1. I buy goods for ^500, and propose to clear 12 fo- 
 What must be the selling price? |500xl.l2. 
 
 Ans. $560. 
 
 2. If I buy flour for §750, and in the sale of it clear 
 33| ^, what is my selling price? 
 
 $750X1.33J=§750X-|. Ans. |1000. 
 
 3. I buy cloth at §4 per yard, and wish to make a 
 profit of 37| % on the cost. What must be my selling 
 price per yard? §4Xl.37i-=PX V- 
 
 Ans. $5.50 
 
 4. I have in store 300 barrels of flour, which cost 
 $1450. It being damaged, I am willing to lose VI J^fc 
 What must I charge per barrel? Ans. $4. 
 
 5. Cost price $1.20, rate % of profit 25. What is 
 the selling price ? 
 
 6. Cost price $1.25, rate % of profit 40. What is 
 the selling price ? 
 
 7. Cost price $1.40, rate % of profit, 42f . What is 
 the selling price? 
 
 8. Cost price $4.50, rate % of profit 33 J. What is 
 the selling price? 
 
 9. Cost price $6.00, rate % of loss 25. What is the 
 selling price? 
 
PERCENTAGE. 219 
 
 10. Cost price 12.00, rate % of loss 30. What is the 
 selling price? 
 
 11. To make 12 » % profit, for how much must I sell 
 cloth that cost 16 cts. per yard? 24 cts.? 32 cts.? 64 cts.? 
 72 cts.? 88 cts.? Am. % of 16 cts.=18 cts., etc. 
 
 12. To make 16f % profit, what must a merchant 
 mark calico that cost 36 cts. per yard? 42 cts.? 54 cts.? 
 72 cts.? 11.26? $1.50? $1.80? . 
 
 13. To make 33^^, what must I mark books which 
 cost 24 cts.? 27 cts.? 30 cts.? 42 cts.? §1.02? $1.05? 
 $1.08? $1.24? $1.44? 
 
 230. To find the cost price, when the selling price 
 and the rate per cent, of profit or loss are given. 
 
 (1.) If a profit has been made, divide the Belling 'price 
 hy\-\- the rate % written as a deciftiaL (Vide 213.) 
 
 (2.) If a loss has been sustained, divide the selling 
 pnce hy 1 — the rate % written as a decimal, (Vide 214.) 
 
 EXAMPLES. 
 
 1. Selling price $560, rate % of profit 12. What is 
 the cost price? 
 
 2. Selling price $1000, rate % of profit 33^. What 
 is the cost? 
 
 3. Selling price $5.50, rate ^ of profit 37|. What 
 is the cost? 
 
 4. Selling price $1.50, rate % of profit 25. What 
 is the cost? 
 
 5. Selling price $1.75, rate % of profit 40. What 
 is the cost? 
 
 6. Selling price $2.00, rate % of profit 42f . What 
 is the cost? 
 
220 PERCENTAGE. 
 
 7. Selling price $4:.50, rate % of loss 25. What is 
 the cost? 
 
 231. Duties or Customs are sums of money assessed 
 by government upon imported goods. 
 
 (1.) Specific duties are assessed upon goods at a cer- 
 tain rate per hogshead, gallon, bale, etc., with no refer- 
 ence to their value. 
 
 (2.) Ad valorem duties are a certain percentage of the 
 cost of goods. 
 
 (3.) An Invoice is a written account of the goods of a 
 cargo containing a statement of the cost of each article 
 in the currency of the country whence imported. 
 
 EXAMPLES. 
 
 1. The invoice of 'a cargo of goods which arrived in 
 Mobile from Liverpool, contained the following among 
 other items : 
 
 325 yd. Broadcloth cost 26 s. sterling per yd. 
 623 yd. Muslin " 4 s. " " 
 
 600 yd. Lace " Is. lOd. " " 
 
 975 yd. Carpeting " 6 s. " " 
 
 1280 yd. " " 4s. 8d. " " 
 
 The duty on the broadcloth was 15 ^ ; on the muslin 
 12| fo ; on lace 12 J ^ ; carpeting 15 %. What was 
 the amount of duty in United States money? 
 
 Ans. 1844.58. 
 
 232. Miscellaneous Examples. 
 1. A and B invest $550 in a speculation of which A fur- 
 nishes $330, and B the balance. They gain $70. What 
 is the rate % of profit on the money invested ? What is 
 the share of each ? Ans. 12 f\ %-, A $42; B $28. 
 
PERCENTAGE. 221 
 
 2. A man failing owes A §175; B $500; C $600; D 
 1210; E $42.50; F $20; and G $10. His property is 
 sold for $934.50. What is the rate % of loss? What 
 is the share of each creditor? 
 
 Ans. Loss 40 per cent. A loses $70; B $200; C $240; D $84.00; 
 E $17; F $8; G $4. 
 
 3. A bankrupt owes A $500; B $1200; and C S4300. 
 The net cash proceeds of his estate amount to only 
 $1500. What rate ^ does he pay on his debts? What 
 does each creditor receive? 
 
 Am. 2h%, A $125; B $300; C $1075. 
 
 4. If the money and effects of a bankrupt amount to 
 $3361.74, and he is indebted to A $1782.24, to B 
 $1540.76, and to C $2371.17, how much will each re- 
 ceive? Ans. A $1052.20; B $909.64; C $1399.90. 
 
 5. I send a commission merchant $1000, with which 
 to buy cotton. If I allow him 5 % commission on the 
 money sent, how much will he have to expend in cot- 
 ton? ^Tis. $950. 
 
 6. How much railroad stock can be obtained for 
 $3860, when the stock is at a discount of ^l%t 
 
 Ans. $4000. 
 
 7. What rate % of $700000 is $700? Ans. J^%. . 
 
 8. What rate % of $450000 is $2250? Ans. \%. 
 
 9. A certain town, whose property is valued at $750- 
 000, proposes to raise a tax of $1875. What will be 
 the rate % ? Ans. J % , or 2^ mills on the dollar. 
 
 10. AVhat will be the tax of a man whose property 
 is valued at $12000, at the rate of i % ? Ans. $30. 
 
 11. A city agrees to loan a railroad company $1000- 
 000, which amount is to be raised on a property valued 
 
222 PERCENTAGE. 
 
 at $175000000. What is the rate % of taxation, and 
 what does A pay, whose property is taxed for |35000 ? 
 
 Ans. 4%. A pays $200. 
 
 12. Sold tea at 90 cts. per pound, and gained 20 %. 
 What % should I have gained had I sold it for $1.00 
 per pound? Am. 33i %, 
 
 13. I sell tea at $1.28 per pound, and thereby lose 
 20 % . What would be gained or lost ^ by selling the 
 same tea at $1.68 per pound? Am. Profit of 5%. 
 
 14. iBy selling coffee at 67 J cts. per pound I make a 
 profit of 12J%, but I desire to make 30%. What must 
 be my selling price per pound? An%. 78 cts. 
 
 15. I bought a quantity of broadcloth for $2.59 per 
 yard, but on measuring it I find it falls short 12|% in 
 length. What must be my selling price per yard in 
 order to clear 12 1% on the real cpst? Am. $3.33. 
 
 16. I bought a quantity of calico at 40 J cts. per yd.; 
 but 10% of the calico proved to be damaged, and 10% 
 of the balance was lost by bad debts, and yet I cleared 
 10 % on the cost. What was the selling price per 
 yard? An%. 55 cts. 
 
 INTEREST. 
 
 1333. Interest is a percentage paid for the use of 
 
 money. 
 
 (1.) The Principal is the money for which interest is 
 paid. 
 
 (2.) The Amount is the sum of the principal and 
 interest. 
 
 (3.) The Rate per cent, per annum is the number of 
 cents paid fur the use of 1 tlollar for a year. 
 
PERCENTAGE. 223 
 
 (4.) The Time is the period for which interest is paid. 
 Thus, 
 
 July 4, 1865, A borrowed of B $7250, agreeing to 
 pay at the rate of 8^ per annum. August 1, 1866, 
 he paid $623.50. 
 
 The principal is $7250; the interest is $623.50; the 
 amount is $7873.50; the rate per cent, per annum is 
 8; the time, 1 yr. 27 da. (Vide 198, Ex. 2; and 202, 
 Ex. 55.) 
 
 Resiark 1. — The rate per cent, in tlie various states is estab- 
 lished by law, and is thence called the legal rate; a higher than 
 legal rate is usury. The legal rate in Maine, New Hampshire, 
 Vermont, Massachusetts, Rhode Island, Connecticut, New Jersey, 
 Pennsylvania, Delaware, Maryland, Virginia, North Carolina, 
 Tennessee, Kentucky, •••Ohio, ^Indiana, •••Illinois, *Iowa, "^Nebraska, 
 ^Missouri, *Kansas, ^Arkansas, -^Mississippi, Florida, District of 
 Columbia, and debts in favor of the United States, is 6 per cent.; 
 ^Michigan, New York, Minnesota, Georgia, and South Carolina, 7 
 per cent.; Alabama and Texas, 8 per cent.; California, 10 per cent.; 
 Louisiana, 5 per cent. By special contract, parties, in the states 
 marked * can take interest as high as 10 per cent. 
 
 Remark 2. — The month, in computing interest, is regarded as 
 having 30 days. Custom has made it lawful. (Vide 185, Rem. 6.) 
 
 !334. To find the interest of any principal for 1 year, 
 at any given rate per cent, per annum. 
 
 Multiply the principal hy the rate fo per annum^ writ- 
 ten as a decimal. 
 
 EXAMPLES. 
 
 1. What is the interest of $6500 for 1 year, at 1 % 
 per annum? 2%? 3%? 4%? etc., to 12%? 
 
 Ans. $65; $130; $195; $260, etc. . . $780. 
 
 2. At 12 % per annum, what is the interest of $1041| 
 forlyear? $1458i? $3333i? $4375? $6250? $8333i? 
 $20833^? $6875? ^ws. $125; $175, etc. . $825. 
 
224 PERCENTAGE. 
 
 3. At Sfo per annum, what is the interest of §5000 
 for 1 year? $1850? |9215? $8000? $7250? $5280? 
 
 Ans. 202, Ex. 51. . . 56. 
 
 4. What is the amount of $500, at 2^ per annum, 
 for 1 year? $250 at 5%? $375 at 6%? $475 at 6%? 
 $450 at 7 % ? $1250 at 8 % ? (Vide 233, (2.) 
 
 Ans. $^10; $262.50. . . $1350. 
 
 5. What is the amount of $1.00, at 5^ per annum, 
 fori year? 6%? 7%? 8%? 10%? 
 
 Ans. $1.05; $1.06; $1.07; $1.08; $1.10. 
 
 6. What is the amount of $1.05, at 5% per annum, 
 for 1 year? $1.06 at 6% ? $1.07 at 7% ? $1.08 at 8% ? 
 $1.10 at 10%? 
 
 Ans. $1.10i; $1.12/^; $1.1449; $1.1664; $1.21. 
 ' 235. To find the interest of $1.00, at 12 % per 
 annum, for any given time. 
 
 EXAMPLES. 
 
 1. Find the interest of $1.00, at 12% per annum, for 
 3 yr. 5 mo. 15 da. 
 
 ANALYSIS. 
 
 (1.) 12 per cent, per annum, means 12 cents for the use of $1 
 one year. (Vide 233, (3.) 
 
 (2.) 12 cents per year gives a rate of 1 cent per month. 
 
 (3.) 10 mills (1 cent) for 30 days (1 month) gives a rate of 1 
 mill for 3 days. 
 
 The interest of $1.00 for 3 yr. 5 mo. is therefore 3Xl2-f-5 $0.41 
 The interest of $1.00 for 15 da. is . . 15-5-3 . . 0.005 
 
 The interest of $1.00 for 3 yr. 5 mo. 15 da. is (202, Ex. 57.) $0,415 
 
 Hence, 
 
 (1.) Call the months in the given time so many cents. 
 
 (2.) Call one third of the days so many mills. 
 
PERCENTAGE. 
 
 225 
 
 Tlie sum of tliese results will be the interest required. 
 
 Find the interest of |1.00, at 12 ^ per annum, for 
 the following times. The work should be done mentally. 
 (Vide, also, 205, Ex. 56.) 
 
 2. 1 yr. 2 mo. 6 da. 
 
 Ans. §0.142. 
 
 3. 1 yr. 3 mo. 9 da. 
 
 Ans. $0,153. 
 
 4. 2 yr. 4 mo. 12 da. 
 
 .4ns. $0,284. 
 
 5. 3 yr. 6 mo. 15 da. 
 
 Ans. $0,425. 
 
 6. 4 yr. 1 mo. 18 da. 
 
 Ans. $0,496. 
 
 7. 5 yr. 3 mo. 21 da. 
 
 Ans. $0,637. 
 
 8. 6 yr. 7 mo. 24 da. 
 
 Ans. $0,798. 
 
 9. 1 yr. 3 mo. 29 da. 
 
 Ans. $0.159f. 
 
 10. Oyr. 4 mo. 12 da. 
 
 Ans. $0,044. 
 
 11. Oyr. 8 mo. 3 da. 
 
 Ans. $0,081. 
 
 12. yr. mo. 24 da. 
 
 Ans. $0,008. 
 
 13. 20 yr. 1 mo. 3 da. 
 
 Ans. $2,411. 
 
 14. 1 yr. 1 mo. 27 da. 
 
 A71S. $0,139. 
 
 15. 8yr. 4 mo. da. 
 
 Ans. $1.00. 
 
 16. 2 yr. 1 mo. 1 da. 
 
 Ans. $0.25O|. 
 
 17. 3 yr. 2 mo. 2 da. 
 
 Ans. $0.380f. 
 
 18. 4 yr. 3 mo. 4 da. 
 
 Ans. $0.511|. 
 
 19. 5 yr. 7 mo. 11 da. 
 
 A71S. $0.673f. 
 
 20. 3 yr. 4 mo. 25 da. 
 
 A71S. $0.4081-. 
 
 21. 5 yr. 3 mo. 9 da. 
 
 .4ns. $0,633. 
 
 22. yr. 1 mo. 1 da. 
 
 A71S. $0.0101 
 
 23. yr. 8 mo. 2 da. 
 
 Ans. $0.080f. 
 
 24. 12 yr. mo. da. 
 
 Ans. $1.44. 
 
 25. 25 yr. 5 mo. da. 
 
 Ans. $3,052. 
 
 S30. To find the interest on any principal, for any 
 time, at any rate per cent, per annum, 
 
 Multiply the principal by the interest^of §1 at 12% per 
 annum for the given time. (Vide 235.) 
 
 (1.) The interest at 1% is one twelfth that at 12%. 
 
 (2.) The interests at 2%, 3%, 4%, and 6% are, re- 
 spectively, one sixth, one fourth, one third, and one half 
 that at 12%. 
 
226 ' PERCENTAGE. 
 
 (3.) The interests at S% and 9 % are, respectively, two 
 thirds and three fourths that at 12 %. 
 
 (4.) The interest at any rate whatever may he found hy 
 multiplying that at Ifo hy the numher representing the 
 rate % per annum. 
 
 EXAMPLES. 
 
 1. What is the interest of $421.40 for 3 yr. 5 mo. 
 15 da., at the rate of 6^ per annum? 
 
 OPERATION. 
 
 1421.40 
 (Vide 235, Ex. 1.) .415 
 
 210700 
 42140 
 
 168560 
 
 (Vide 235, (2.) 2)174.88100 
 
 (Vide 202, Ex. 59.) |87.4405 Ans. 
 
 2. What is the interest of $19.35 for lyr. 2 mo. 6 da., 
 at the rate of 6^ per annum? Ans. $1,374. 
 
 3. What is the interest of $17.21 for 1 yr. 3 mo. 9 da., 
 at the rate of 6^ per annum? Ans. $1.316565. 
 
 4. What is the interest of $140.10 for 2 yr. 4 mo. 
 12 da., at the rate of 6 ^ per annum ? 
 
 Ans. $19.8942. 
 
 5. What is the interest of $75.15 for 3 yr. 6 mo. 
 15 da., at the rate of 6^ per annum? 
 
 Ans. $15.96||. 
 
 6. What is the interest of $1000 for 4 yr. 1 mo. 
 18 da., at the rate of 6% per annum? 
 
 Ans. $248. 
 
PERCENTAGE. 227 
 
 7. What is the interest of |175 for 5 jr. 3 mo. 21 da., 
 at the rate of 6% per annum? Ans. |55.73|. 
 
 8. What is the interest of $2141 for 6 yr. 7 mo. 24 
 da., at the rate of 6% per annum? Aiis. §854.259. 
 
 9. What is the interest of $1041| for 1 yr. 3 mo. 15 
 da., at the rate of 6^ per annum? Ans. |80.72iJ. 
 
 10. What is the interest of |1458| for 1 yr. 4 mo. 21 
 da., at the rate of 6^ per annum? Ans. $121.77y'2. 
 
 11. What is the interest of ?3333| for 1 yr. 5 mo. 
 7 da., at the rate of 6 ^ per annum ? 
 
 Ans. $287.22f . 
 
 12. What is the interest of $4375 for 2 yr. 3 mo. 12 
 da., at the rate of 6% per annum? Ans. $599. 37^. 
 
 13. What is the interest of $421.40, from January 1, 
 
 1865, to June 16, 1868, at 6% per annum? 3^? 4%? 
 
 Last Ans. $58.293|.. 
 
 14. What is the interest of $19.35, from April 4, 
 
 1866, to June 10, 1867, at 2 % ? 3 % ? 4 % ? 
 
 Last Ans. $0,916. 
 
 15. What is the interest of $25.14, from February 6, 
 1866, to April 3, 1867, at 6% ? 3% ? 4^ ? 
 
 First Ans. $1,747. 
 
 16. What is the interest of $200, from March 14, 
 1865, to July 14, 1873, at 6 % ? 3 % ? 4 % ? 
 
 First Ans. $100.00. • 
 
 17. What is the interest of $525, from May 17, 1868, 
 to June 18, 1870, at 6%? at 3%? at 4%? 
 
 First Ans. $65.71i. 
 
 18. What is the interest of $700, from August 14, 
 1865, to October 16, 1868, at 8%? at 2%? at 4%? 
 
 • Ans. $177,644 
 
228 PEIICENTAGE. 
 
 19. What is the interest of §925.16, from October 15, 
 
 1865, to January 19, 1870, at 8%? at 2%? at 4%? 
 
 Ans. $315,377. 
 
 20. What is the interest of $375, from December 8, 
 
 1869, to Julj 19, 1875, at '8%? at 2%? at 4^? 
 
 Ans. $168,416. 
 
 21. What is the interest of $1400, from November 4, 
 1868, to March 29, 1872, at.8%? at 2%? at 4%? 
 
 Ans. $381,111. 
 
 22. What is the interest of $2000, from September 
 10, 1869, to February 5, 1873, at 8%? at 2%? at 4%? 
 
 Ans. $544,444. 
 
 23. What is the interest of $4062.50, from July 5, 
 
 1866, to October 15, 1868, at 8^ per annum? 
 
 Ans. 202, Ex. 39. 
 
 24. What is the interest of $1562.50, from June 6, 
 
 1870, to September 21, 1871, at 8%? 2%? 4%? 
 
 Ans. 202, Ex. 40. 
 
 25. What is the interest of $2187.50, from August 
 10, 1868, to January 1, 1870, at 8 % ? 2 % ? 4 % ? 
 
 Ans. 202, Ex. 41. 
 
 26. What is the interest of $8000, from January 1, 
 1866, to June 8, 1867, at 5 % ? 10 % ? 
 
 Ans. 202, Ex. 42. 
 
 27. What is the interest of $10500, from February 
 1, 1867, to May 13, 1869, at 5 % ? 10 % ? 
 
 Ans. 202, Ex. 43. 
 
 28. What is the interest of $15000, from March 4, 
 1868, to October 23, 1871, at 5 % ? 10 % ? 
 
 Ans. 202, Ex. 46. 
 
PERCENTAGE. 229 
 
 29. What is the interest of ^20000, from April 9, 
 1867, to February 6, 1873, at 5%? at 10%? 
 
 Ans. 202, Ex. 47. 
 
 30. What is the interest of |50000, from May 16, 
 
 1865, to May 11, 1870, at 5%? at 10%? 
 
 Am. $12465.27?. 
 
 31. What is the interest of $425.30, from March 4, 
 
 1866, to May 19, 1868, at 7%? at 3j%? 
 
 Ans. $65,744. 
 
 32. What is the interest of $510.83, from March 21, 
 
 1867, to December 30, 1867, at 7%? at 3^-%? 
 
 Ans. $27,713. 
 
 33. What is the interest of $170, from June 19, 1865, 
 to July 1, 1866, at 7%? at 3^%? Ans. $12,296. 
 
 34. What is the interest of $966, from January 1, 
 
 1867, to March 20, 1869, at 7%? at 3j%? 
 
 Ans. $150,078. 
 
 35. What is the interest of $213.27, from August 15, 
 1872, to March 13, 1875, at 7%? at 3J%? 
 
 Ans. $38,483. 
 
 36. What is the interest of $426.50, from Sept. 4, 
 
 1868, to May 4, 1874, at 9 %? 4i- %? Ans. $217,515. 
 
 37. What is the amount of $164.06, from July 4, 
 1864, to February 15, 1870, at 3%? 1^%? 6%? 9%? 
 4J %? (Vide 233, (2.) Ans. $191.69 ; $177.87, etc. 
 
 38. What is the amount of $120.10, for 8 yr. 4 mo., 
 at 12% per annum? 100 yr., at 1%? 50 yr., at 2%? 
 
 Ans. $240.20. 
 
 39. What is the amount of $120.10, for 12 yr. 6 mo., 
 at 8 % per annum? 25 yr., at 4 % ? 33 yr. 4 mo., at 
 3 % ? Ans. $240.20. 
 
230 PERCENTAGE. 
 
 40. What is the amount of $120.10, for 16 yr. 8 mo., 
 at 6% per annum? for 20 yr., at 5%? 12 yr., at 8|%? 
 
 Ans. $240.20. 
 
 41. What is the amount of $450, from January 1, 
 1865, to March 16, 1865, at 8^ per annum? 
 
 Ans. $457.50. 
 
 42. What is the amount of $382.50, from March 16, 
 
 1865, to January 1, 1866, at 8^ per annum? 
 
 An§. $406,725. 
 
 43. What is the amount of $306,725, from January 
 1, 1866, to April 4, 1866, at 8% per annum? 
 
 Ans. $313,064. 
 
 44. What is the amount of $113,064, from April 4, 
 
 1866, to January 1, 1867, at 8^ per annum? 
 
 - Ans. $119,772. 
 
 45. What is the amount of $700, from January 1, 
 
 1865, to July 28, 1865, at 6% per annum? 
 
 Ans. $724.15. 
 
 46. What is the amount of $624.1 5, from July 28,1865, 
 to April 4, 1866, at 6^ per annum? Ans. $649.74. 
 
 47. What is the amount of $149.74, from April 4, 
 
 1866, to January 1, 1867, at 6^ per annum? 
 
 Ans. $156.40. 
 
 48. What is the amount of $3000, from Jan. 1, 1865, 
 to April 1, 1-865, at 10% per annum? Ans. $3075. 
 
 49. What is the amount of $2075, from April 1, 
 1865, to January 1, 1866, at 10% per annum? 
 
 Ans. $2230.625. 
 
 50. What is the amount of $1230.625, from January 
 1, 1866, to January 1, 1867, at 10% per annum? 
 
 Ans. $1353.68f. 
 
PERCENTAGE. 231 
 
 51. What is tlie amount of $620.53, for 4 mo. 6 da., 
 at 5% per annum? Arts. |631.39. 
 
 52. What is the amount of |1123.60, for 8 mo. 15 da., 
 at 6% per annum? Ans. |1171.353. 
 
 53. What is the amount of $1531.301, for 3 mo. 
 24 da., at 7% per annum? Ans. $1565.24. 
 
 54. What is the amount of $4709.25, for 5 mo. 27 
 da., at 10% per annum? Ans. $4940.788. 
 
 2S*7. To find interest for days, counting 365 to the year, 
 Multiply one year's interest (vide 234) by the number 
 of days in the time, and divide the product by 365. 
 
 EXAMPLES. 
 
 1. What is the interest of $6500, from April 15 to 
 December 15, 1866, at 6 %? (Vide 185, Rem. 6 ; Table.) 
 
 $390X244--365. J^ns. $260.71. 
 
 2. At 6fo per annum, what is the interest of $375 
 from January 1 to March 15? $22.50 x 73 -4-365 = i of 
 $22.50. Ans. $4.50. 
 
 3. What is the interest of $1000, for 365 days, at the 
 rate of 6% for 360 days? $60Xftf=$60Xf|. 
 
 Ans". $60,831. 
 
 4. What is the interest of $1000, for 360 days, at the 
 rate of 6 ^ for 365 days ? $60Xf |f=$60X-ff • 
 
 Ans. $59.18. 
 
 5. What is the interest of $500000, for 365 days, at 
 the rate of 6% for 360 days? Ans. $30416f. 
 
 6. What is the interest of $500000, for 360 days, at 
 the rate of 6% for 365 days? Ans. $29589^^5. 
 
 7. What is the interest of $500000, for 90 days, at 
 the rate of 6% per annum of 360 days? Ans. $7500. 
 
232 PERCENTAGE." 
 
 8. What is the interest of $500000, for 90 days, at 
 the rate of 6% per annum of 365 days? 
 
 Ans. $7397.26. 
 
 9. What is the interest of a §1000 bond, for 75 days, 
 at the rate of 7.30 fo per annum of 365 days ? 20 cts. 
 
 per day. Ans. $15.00. 
 
 10. What is the interest of a $100 bond, for 67 days, 
 at the rate of 7.30 fo per annum of 365 days ? 2 cts. 
 per day. Ans. $1.34. 
 
 11. What is the interest of a $10000 bond, for 93 
 days, at the rate of 7.30 % per annum of 365 days? 
 
 $2 per day. AnS. $186. 
 
 Remark. — In New York, interest for years and months is com- 
 puted by the rule under sec. 236, but for the odd days by the rule 
 under this section, 237. 
 
 12. What is the interest on a note of $1000, in New 
 York, having run from March 4, 1865, to March 25, 
 1866? • ^ns. $74.03. 
 
 13. What wbuld be the interest of the above note 
 computed in Kentucky? Ans. $63.50. 
 
 PROBLEMS IN INTEREST. 
 
 238. To find the principal, when the time, rate per 
 cent., and interest are given. 
 
 Divide the given interest hy the interest o/i $1.00 at 
 the given rate and time. 
 
 EXAMPLES. 
 
 1. The interest of a certain sum, at 6 % per annum, for 
 Syr. 5 mo. 15 da. is $87.4405. What is the principal? 
 (Vide 236, Ex. 1.) $87.4405-.2076. Ans. $421.40. 
 
PERCENTAGE. 233 
 
 2. The interest of a certain sum at 6 ^ per annum, 
 for 1 yr. 2 mo. 6 da. is ^1.374. What is the principal? 
 
 Ans. 236, Ex. 2. 
 
 239. To find the principal, when the time, rate per 
 cent., and amount are given, 
 
 Divide the given amount bij the amount of §1.00 at 
 the given rate and time. 
 
 EXAMPLES. 
 
 1. The amount of a certain sum at 6% per annum, 
 for 3yr. 5 mo. 15 da. is §508.8405. What is the prin- 
 cipal? $508.8405--1.2075. Ans. $421.40. 
 
 2. The amount of a certain sum, at 8^ per annum, 
 from January 1, 1865, to March 16, 1865, is $457.50. 
 What is the principal ? Ans. 236, Ex. 41. 
 
 3. The amount of a certain sum, at 10% per annum, 
 from January 1, 1866, to January 1, 1867, is $1353. 68|. 
 What is the principal? Ans. 236, Ex. 50. 
 
 240. To find the time, when the principal, interest, 
 and rate per cent, are given. 
 
 Divide the given interest by the interest on the principal 
 at the given rate for 1 year. 
 
 EXAMPLES. 
 
 1. The interest of $421.40, at 6 % per annum, is 
 $87.4405. What is the time? 87.4405 --25.284 =3.458^ 
 
 yr.=3yr. 5mo. loda. (Vide 195.) 
 
 2. The interest of $75.15, at 6 % per annum, is 
 $15.96|-§. What is the time? 
 
 Ans. 236, Ex. 5. 
 20 
 
234 PERCENTAGE. 
 
 3. The interest of §525 to June 18, 1870, at 6 % 
 per annum, is $65.71 1. What is the date from which 
 interest is computed? Ans. 236, Ex. 17. 
 
 4. The interest on §100 is also §100. What has the 
 time been at the rate of 1 % per annum? 2%? 3%? 
 4%? 5%? 6%? 
 
 5. The interest on §120.10 is also §120.10. What 
 has the time been at the rate of 6 ^ per annum? 
 8%? 12%? Ans. 236, Ex. 38, 39, 40. 
 
 6. In what time will a^iy sum double itself at a rate of 
 5% per annum? 7%? 9%? 4i%? 
 
 Ans. 20 yr., etc. 
 
 7. In what time will any sum triple itself at the rate 
 of 6% per annum? Aiis. 33 yr. 4 mo. 
 
 S41. To find the rate per cent., when the principal, 
 interest, and time are given, 
 
 Divide the given interest hy the intei^est on the principal 
 at Ifo p^T annum for the given time. 
 
 EXAMPLES. 
 
 1. The interest of §421.4t) for 3yr. 5 mo. 15 da. is 
 §87.4405. What is the rate % per annum? 87.4405-- 
 14.5734^ Ans. 6 % . 
 
 2. The interest of §700 from August 14, 1865, to 
 October 16, 1868, is §177.64|. What is the rate % 
 per annum? Ans. 236, Ex. 18. 
 
 3. The interest of §750 for 3 yr. 4 mo. is §162.50. 
 What is the rate % per annum? Ans. 6j%. 
 
 4. The interest of §950 for 2 yr. 4 mo. 20 da. is 
 §238.29 J. What is the rate % per annum? (Vide 202, 
 Ex. 58.) ' Ans. 10J%. 
 
PERCEXTAGE. 235 
 
 PRESENT Yv^OKTH. 
 
 242. The Present Worth of a sum of money due 
 at some future time, is a principal ^vliicli, being put at 
 interest at the time \)f payment, Avill amount to the sum 
 at the time it is due. 
 
 The Discount is the diiference between the present 
 worth and the sum of money due. 
 
 243. To find the present worth of a sum of money 
 for a given time and rate per cent, per annum, 
 
 Divide the given sum by the amount of fl.OO at the 
 given rate and time. (Vide 239.) The quotient is the 
 present worth. 
 
 EXAMPLES. 
 
 1. What is the present worth of |508.8405, due in Syr. 
 5 mo. 15 da., at the rate of G ^ per annum? §508.8405-^- 
 1.2075. Alls. S421.40. 
 
 2. What is the discount of §457.50, due March 16, 
 1865, but paid January 1, 1865, at 8% per annum? 
 
 Ans. ^7.50. 
 
 3. What is the discount on §1353.68 1, due January 
 1, 1867, but paid January 1, 1866, at 10 J/^ per annum? 
 
 A71S. $123,061. 
 
 4. What is the discount of $1800, due 1 yr. 3 mo. 
 hence, at the rate of 6 ^ per annum ? 
 
 Ans. $125,582. 
 
 5. What is the discount of $475, due 1 yr. hence, at 
 the rate of 7% per annum? Ans. $31,075. 
 
 6. What is the discount of $2500, due 3 mo. hence, 
 when money is worth 4J%? A71S. $27.81. 
 
236 PERCENTAGE. 
 
 7. Four notes are due as follows : §900 in 6 mo. ; 
 $2700 in 1 yr.; |3900 in 1 yr. 6mo.; and |4200 in 2yr. 
 If the notes are all paid at the present time, what will 
 be the entire discount, at 6$^ ? Ans. ?951.06. 
 
 8. What is the discount of §400, due three months 
 hence, but paid now, at the rate of 12^ per annum? 
 S%2 r^/ot 6%? 5^,? 
 
 >lns. §11.65; §7.84; §6.88; §5.91; §4.94. 
 
 BANK DISCOUNT. 
 
 244. Bank Discount is a deduction made by a bank 
 upon a sum of money borrowed of it, at the time the 
 money is taken. 
 
 (1.) By custom, this discount is the interest on the 
 face of the note for the time it is drawn, increased by 
 three days, called Days of Grace, 
 
 (2.) The Proceeds of a bank note is what remains 
 after the discount has been deducted. 
 
 245. To find the bank discount on a note or draft, 
 Find the interest on the face of the note for three days 
 
 more than the time for ivhich it is draivn. 
 
 EXAMPLES. 
 
 1. What is the bank discount on a note of §400, dis- 
 counted for 90 days, at 8^; ? 6 % ? 2 of $400X.031; | of 
 $.loox.03i. Ans. §8.26f ; §6.20. 
 
 2. Find the proceeds of a note of §150, discounted 
 forGOdays, at 6%? 7%? 8%? 10%? 1 of $i50x.02i; 
 
 -i^jj of $150X.021, etc. 
 
 ^715. §148.42i; §148.16.1; §147.90; §147.37i. 
 
TEJICENTAGE. 237 
 
 3. Find the proceeds of a note of §177.75, discounted 
 for 90 days, at 6^ per annum. Ans. §175.00. 
 
 4. Find the proceeds of a note of §505.305, discounted 
 for 60 days, at 6^ per annum. A7is. §500. 
 
 5. Find the proceeds of a note of §375.00, discounted 
 for 30 days, at 6^ per annum. Ans. §372. 93| 
 
 246. To make a note of which, when discounted, the 
 proceeds shall be a given sum. 
 
 Divide the given sum by the proceeds on §1.00 at the 
 given time and rate per cent, per annum. 
 
 EXAMPLES. 
 
 1. I wish to obtain §175 for 90 days. For what sum 
 must my note be drawn, at 6 ^^ per annum ? 
 1.00-§0.0155=§0.9845, then §175-^.9845--§177.755. 
 
 2. I buy produce worth §500, but it will be 60 days 
 before I shall have money with which to pay for it. 
 For what sum must I draw, at 6 ^ per annum ? 
 
 Ans. §505.305. 
 
 3. I buy cotton to the amount of §372.93 J, and bor- 
 row money at the bank with which to pay for it. For 
 what sum do I draw, payable in 30 days, at 6^ per 
 annum? Ans. §375. 
 
 PROMISSORY NOTES. 
 
 247. A Promissory Note is a promise in writing to 
 pay a certain sum of money to a person, named in the 
 note, or order, or to the bearer. 
 
 (1.) The Drawer of a note is the person signing it. 
 (2.) The Payee of a note is the person to whom the 
 money is to be paid. 
 
238 PERCENTAGE. 
 
 (3.) The Indorser of a note is a person who guaran- 
 tees the payment of it. He does this by writing his 
 name on the back of the paper on which the note is 
 written. 
 
 (4.) An Indorsement is an acknowledgment on the 
 back of the note that, at a given date, a part of the 
 money was paid. 
 
 (5.) The Face of a note is the sum promised to be 
 paid. 
 
 S48. To find the amount due at the maturity of a 
 note upon which one or more indorsements have been 
 made, 
 
 I. When the time of the note is one year or 
 less, 
 
 (1.) Find the amount of the face of the note from its 
 date to the time of maturity/. 
 
 (2.) Find the amount of each payment from the time it 
 was made till the time the note matures. 
 
 (3.) Subtract the sum of the amounts of all the pay- 
 ments from the amount of the face of the note. (Vide 
 233, (2.) 
 
 EXAMPLES. 
 
 $500. Richmond, Jan. 1, 1865. 
 
 (1.) Ninety days after date, I promise to pay to the 
 order of Frank H. Ransom Five Hundred Dollars, 
 with interest, value received. 
 
 John M. Sabin. 
 
 Indorsements : January 20, $100; February 10, $50; 
 February 25, $100; March 1, $150. 
 
 What was due at maturity ? 
 
PERCENTAGE. 
 
 239 
 
 OPERATION. 
 
 
 A^mottnt of JJ500 for 93 days is 
 
 §507.75 
 
 " §100 " 74 " 
 
 §101.231 
 
 " §50 '' 53 " 
 
 50.44^ 
 
 §100 " 38 " 
 
 100.631 
 
 " §150 " 34 " 
 
 150.85 
 
 Sum of the amounts of payments, 
 
 .§403.15§ 
 
 Sum due at maturity, April 4, 1865, §104.59^ 
 
 II. When the time of the note is more than 1 year, 
 (1.) Find the amount of the face of the note to the date 
 
 of the first payment, and deduct the payment. 
 
 (2.) Find the amount of the remainder to the date 
 
 of the next payment, and, after deducting the payment, 
 
 find the amount of the remainder to the date of the 
 
 next indorsement, and so on till the last payment is 
 
 reached. 
 
 (3.) Find the amount of the remainder, on deducting 
 
 the last payment, from the date of that payment till the 
 
 time the note matures. 
 
 Remark. — Unless a payment or payments are eqnal to or exceed 
 
 the interest due at the date of the last, they must be added to the 
 
 succeeding payment, and the sum considered as a single payment. 
 
 In business, therefore, no payment should be made on a note unless 
 
 it exceeds th« interest then due. 
 
 §450. Mobile, Jan. 1, 1865. 
 
 (2.) Two years after date, I promise to pay to the order 
 of James Boone Four Hundred and Fifty Dollars, with 
 interest, value received. David Miller. 
 
 Indorsements: March 16, 1865, §75; January 1, 
 1866, SlOO; April 4, 1866, §200. 
 
 What was due at maturity? 
 
240 PERCENTAGE. 
 
 OPERATION. 
 
 Amount of $450 to March 16, is (236, Ex. 41.) |457.50 
 
 75.00 
 
 Payment deducted is 382.50 
 
 24.225 
 
 Amount of $382.50 to Jan. 1, is (236, Ex. 42,) 406.725 
 
 100.000 
 
 Pa^^ment deducted is 306.725 
 
 6.339 
 
 Amount of $306,725 to April 4, is (236, Ex. 43,) 313.064 
 
 200.000 
 
 Payment deducted is 113.064 
 
 6.708 
 
 Amount of $113,064 to Jan. 1, '62, (236, Ex. 44) $119,772 
 $700. Louisville, Jan. 1, 1865. 
 
 (3.) Two years after date, for value received, I 
 promise to pay A. B., or order, Seven Hundred Dollars, 
 with interest. M. Greene. 
 
 Indorsements: July 28, 1865, $100; April 4, 1866, 
 $500. 
 
 What was due at maturity? (Vide 236, Ex. 45, 46, 
 47.) Ans. $156.40. 
 
 $3000. San Francisco, Jan. 1, 1865. 
 
 (4.) Two years after date, for value received, I promise 
 to pay James Monroe, or order, Three Thousand Dol- 
 lars, .with interest. 
 
 Indorsements: April 1, 1865, $1000; January 1, 
 1866, $1000. 
 
 What was due at maturity? Ans. $1353.68|. 
 
PEKCEXTAGE. 241 
 
 S.SOO. Mobile, June 10, 1865. 
 
 (5.) June 2, 1866, for value received, I promise to 
 pay S. S. Bryant, or order, Three Hundred Dollars, with 
 interest. P. Hamilton. 
 
 Indorsements : January 20, 1866, §116 ; March 2, 
 1866, §49.50; April 26, 1866, §85. 
 
 What was due at maturity by both rules ? 
 
 Ans. I. $67.89; II. §68.17. 
 §1000. Galveston, Jan. 1, 1869. 
 
 (6.) July 1, 1870, for value received, I promise to pay 
 C. Q. M., or order. One Thousand Dollars, with interest. 
 
 Wm. Daniel. 
 
 Indorsements: July 1, 1869, §30; Jan. 1, 1870, §470 
 
 What was due at maturity by both rules? 
 
 Ans. 11. §603.20 ; I. §598.80. 
 §400. Buffalo, Jan. 1, 1870. 
 
 (7.) One year after date, for value received, I promise 
 to pay N. Stacy, or order, Four Hundred Dollars, with 
 interest. M. M. DeYoung. 
 
 Indorsements: March 16, 1870, §200; July 1, §100. 
 
 What was due at maturity ? 
 
 Ans. II. §113.88 ; I. §112.25. 
 
 COMPOUND INTEREST. 
 
 249. Compound Interest is interest on the principal, 
 and then, after the interest becomes due, on the amount. 
 (Vide 233, (2.) 
 
 examples. 
 1. What is the amount of §1.00 at compound interest, 
 for3yr., atS^perannum? 6%? 7%? 8%? 10%? 
 21 
 
242 
 
 PEllCENTAGE. 
 
 OPERATIONS. 
 
 $1.05X1.05X1.05=11.157625 Ans. 
 $1.06Xl.06Xl.06-=:|1.191016 Ans. 
 $1.07Xl.07Xl.07=$1.225043 Ans. 
 $1.08Xl.08Xl.08==$1.259712 Ans. 
 $1.10X1.10X1.10=$1.331 (234, Ex. 5, G.) 
 
 by suljtracting 
 
 Remark. — The compound interest is obtained 
 $1 from the amounts. 
 
 TABLE, 
 
 Showing the amount of §1.00 at compound interest, for any number 
 of years from 1 to 25, at 5, 6, 7, 8, and 10 per cent. 
 
 Yeaks 
 
 5 Per Cent. 
 
 G Per Cent. 
 
 7 Per Cent. 
 
 8 Per Cent. 
 
 10 Per Cent. 
 
 "ijMooo" 
 
 1 
 
 1.050000 
 
 1.060000 
 
 1.0700U0 
 
 1.080000 
 
 2 
 
 1.102500 
 
 1.123600 
 
 1.144900 
 
 1.166400 
 
 1.210000 
 
 3 
 
 1.157625 
 
 1.191016 
 
 1.225043 
 
 1.259712 
 
 1.331000 
 
 4 
 
 1.215506 
 
 1.262477 
 
 1.310796 
 
 1.360488 
 
 1.464100 
 
 5 
 
 1.276282 
 
 1.338226 
 
 1.402551 
 
 1.469328 
 
 1.610510 
 
 6 
 
 1.340096 
 
 1.418519 
 
 1.500730 
 
 1.586874 
 
 1.771561 
 
 7 
 
 1.407100 
 
 1.503630 
 
 1.605781 
 
 1.713824 
 
 1.948717 
 
 8 
 
 1.477455 
 
 1.593848 
 
 1.718186 
 
 1.850930 
 
 2.143589 
 
 9 
 
 1.551328 
 
 1.689479 
 
 1.838459 
 
 1.999004 
 
 2.357948 
 
 10 
 
 1.628895 
 
 1.790848 
 
 1.967151 
 
 2.158924 
 
 2.593742 
 
 11 
 
 1.710339 
 
 1.898299 
 
 2.104851 
 
 2.331638 
 
 2.853117 
 
 12 
 
 1.795856 
 
 2.012196 
 
 2.252191 
 
 2.518170 
 
 3.138428 
 
 13 
 
 1.885649 
 
 2.132928 
 
 2.409845 
 
 2.719623 
 
 3.452271 
 
 14 
 
 1.979932 
 
 2.260904 
 
 2.578534 
 
 2.937193 
 
 3.797498 
 
 15 
 
 2.078928 
 
 2.396558 
 
 2.759031 
 
 3.172169 
 
 4.177248 
 
 16 
 
 2.182875 
 
 2.540352 
 
 2.952163 
 
 3.425942 
 
 4.594973 
 
 17 
 
 2.292018 
 
 2.692773 
 
 3.158815 
 
 3.700018 
 
 5.054470 
 
 18 
 
 2.406619 
 
 2.854339 
 
 3.379932 
 
 3.996019 
 
 5.559917 
 
 19 
 
 2.526950 
 
 3.025600 
 
 3.616527 
 
 4.315701 
 
 6.115909 
 
 20 
 
 2.653298 
 
 3.207135 
 
 3.869084 
 
 4.660957 
 
 6.727500 
 
 21 ■ 
 
 2.785963 
 
 3.399564 
 
 4.140562 
 
 5.033834 
 
 7.400250 
 
 22 
 
 2.925261 
 
 3.603537 
 
 4.430402 
 
 5.436540 
 
 8.140275 
 
 23 
 
 8.071524 
 
 3.819750 
 
 4.740530 
 
 5.871404 
 
 8.954302 
 
 24 
 
 8.225100 
 
 4.048935 
 
 5.072367 
 
 6.341181 
 
 9.849733 
 
 25 
 
 3.386354 
 
 4.291871 
 
 5.427433 
 
 6.848475 
 
 10.834700 
 
PERCENTAGE. 243 
 
 S50. To find the amount at compound interest of 
 any principal, at any rate ^ per annum, and for any 
 given time, 
 
 (1.) For the given integral number of years, multiply 
 the principal hy the amount of |1.00, for the same time 
 and rate %. 
 
 (2.) 0)1 this amount find the amount for the months 
 and days, as in §236. 
 
 Remark. — The compound interest will be the diflference between 
 the amount and the given principal. 
 
 EXAMPLES. 
 
 1. Find the compound interest of §400 for 9 yr., at 
 bfo per annum. §400Xl.551328=f620.5312. 
 
 Ans. $220.53. 
 
 2. Find the compound interest of $400 for 9 yr. 4 mo. 
 6 da., at 5% per annum. (Vide 236, Ex. 51.) 
 
 Ans. $231.39. 
 
 3. Find the compound interest for $1000 for 2 yr. 
 8 mo. 15 da., at 6 % per annum. (Vide 236, Ex. 52.) 
 
 Ans. $171,353. 
 
 4. Find the compound interest of $1250 for 3yr. 3 mo. 
 24 da., at 7% per annum. (Vide 236, Ex. 53.) 
 
 Ans. $315.24. 
 
 5. Find the compound interest of $700 for 20 yr. 
 5 mo. 27 da., at 10 % per annum. (Vide 236, Ex. 54.) 
 
 Ans. $4240.788. 
 
244 n.vTio. 
 
 RATIO. 
 
 251. Ratio is the quotient obtained by dividing one 
 number by another of the same kind. Thus, 
 
 The ratio of 5 to 15 is '^f=^, commonly expressed 
 by 5 : 15. 
 
 (1.) The two numbers forming a ratio are together 
 called terms. 
 
 (2.) The first term is called the antecedent. 
 
 (3.) The second term is called the consequent. 
 
 (4.) The antecedent and consequent form, a couplet. 
 
 (5.) The value of a ratio is the quotient of the conse- 
 quent divided hy the antecedent. 
 
 (G.) A ratio is in its simjjlest or loivcst terms -when the 
 terms are integral and prhyie tvith resjject to each other. 
 (Vide 102.) 
 
 252. To reduce a ratio to its lowest terms, 
 
 (1.) If fractions are involved, multijjlt/ the terms hy the 
 least common multiple of the denominators of the frac- 
 tions. (Yide 107.) 
 
 (2.) Divide the resulting terms hy their greatest common 
 divisor^ (vide 104,) or cancel such factors as are common 
 to hoth terms, 
 
 EXAMPLES. 
 
 1. Reduce 15:20; 14:21; 16:24 to their lowest 
 terms, and find their values. Values !{ ; li ; IJ. 
 
 2. Reduce 9:63; 26:169; 34:187 to their lowest 
 terms, and find their values. Values 1\ 6^; 5^. 
 
RATIO. 
 
 245 
 
 3. Reduce 5| : 4f ; 2i : If ; and Jf : 3i to their lowest 
 terms, and find their values. 
 
 OPERATIONS. 
 
 (2.) 
 21 : If 
 
 5|:4f 
 
 119 : 102 68 : 48 
 
 7 : 6 Value f 
 
 4'. Reduce | : I ; j 
 and find their values. 
 
 (3.) 
 
 21 : 16 Value J f , 
 
 M:3i 
 52 : 351 
 4 
 
 27 Value 6^, 
 
 1 . 
 
 4 ' 
 
 2 . 7 
 6 • 1 Oi 
 
 to their lowest terms, 
 Values i; f; If. 
 
 5. Reduce 4i:5i; 6j:7i; and 4^ : 2|f to their 
 lowest terms, and find their values. 
 
 
 
 
 VahcesV^; Ig^; and |. 
 
 6. 
 
 Reduce 3i 
 
 .92 . 4.1 
 
 :f; and J: 
 
 : 7^ to their lowest 
 
 term 
 
 s, etc. 
 
 
 Values 
 
 3 ? T6 ? ^"^ ^g- 
 
 7. 
 
 Reduce 5 : 
 
 2.5; .3:21; and 1/^ 
 
 ^ : 3.4 to their low- 
 
 est terms, etc. 
 
 
 
 
 
 
 OPERATIONS. 
 
 
 (1.) 
 
 (2.) 
 
 
 (3.) 
 
 5; 
 
 :2.5 
 
 .3:21 
 
 l/,:3.4 
 
 50: 
 
 :25 
 
 3 : 210 
 
 
 17:34 
 
 2 ; 
 
 : 1 Value J. 
 
 1:70 
 
 Value 70. ' 
 
 1:2 Vahie2. 
 
 8. 
 
 Reduce .5 : 
 
 .2; Mi: 
 
 .25 ; and .4 
 
 : .7 to their lowest 
 
 terms, etc. 
 
 
 
 Values, Ex. 4. 
 
 9. 
 
 What is the ratio of 50 cts. : 20 cts.? 33 1 cts. : 25 
 
 cts.? 
 
 40 : 70 cts. 
 
 ? 
 
 
 Ans. f; I; If. 
 
 10. What is the ratio of 14 hu. : 35 bu.? 2 qt. : 8 qt.? 
 30 sec. : 50 sec? Ajis. 2^-; 4; f. 
 
 11. What is the ratio of 2 qt. : 3 pk.? 30 sec. :7m.? 
 lpt.:l gal.?— (Vide 251.) 
 
246 RATIO. 
 
 OPERATIONS. 
 (1.) (2.) (3.) 
 
 2 qt. : 3 pk, 30 sec. :7 m. 1 pt. : 1 gal. 
 
 2 qt. : 24 qt. 30 sec. : 420 sec. 1 pt. : 4 qt. 
 
 1 qt. : 12 qt. Value 12. 1 sec. : 14 sec. Value 14. 1 pt. : 8 pt. Value 8. 
 
 12. What is the ratio of 1 mile to 5 fur. 3 r. 10 ft 
 6 in.?— (193, Ex. 2.) Ans. j\. 
 
 13. What is the ratio of 1 mile to 3 fur. 22 r. 3 ft. 
 8in.?— (193, Ex. 4.) Ans.^^. 
 
 PROPORTION. 
 
 353. A Proportion is an equality/ of ratios. 
 
 The equality is indicated by four dots or a double 
 colon written between the couplets. Thus, 
 
 5: 15:: 6: 18 
 is a proportion, and is read 5 is to 15 as 6 is to 18, the 
 meaning of which is that 15-^-5 is the same as 18-^-6; 
 that is, M=:ig^ 
 
 (1.) The first and last terms of a proportion are called 
 extremes. 
 
 (2.) The second and third terms are called means. 
 
 (3.) The first and second terms form the first couplet. 
 
 (4.) The third ' and fourth terms form the second 
 couplet. Thus, in the proportion above, 
 
 5 and 18 are the extremes; 15 and 6 the means; 5 
 and 15 the first couplet; 6 and 18 the second couplet; 
 5 and 6 the two antecedents, and 15 and 18 the tAvo 
 consequents. 
 
 !354. Proposition. — If four numbers are in propor- 
 tion, the product of the extremes' is equal to the product 
 of the means. Thus, 
 
RATIO. 247 
 
 From any proportion as 3 : 9 : : 7 : 21, 
 we have by 251 and 253, |=V) ^^^ ^J multiply- 
 
 ing both of these fractions by the least common multiple 
 of their denominators we have 9x7=3x21, and it is 
 evident that any proportion may be treated in a similar 
 way. 
 
 Remark. — In any proportion, if the second term is less than the 
 first, the fourth will be less than the third, and if the second term is 
 greater than the first, the fourth will be greater than the third. 
 
 S55. Proposition. — If in any j^'^oportion the terms 
 of either couplet he multiplied or divided hy the same 
 number, the proportion luill not he destroyed. Thus, from 
 the proportion, 
 
 7 : 21 : : 8 : 24 we have 
 
 1 : 3 : : 8 : 24 or 7 : 21 : : 1 : 3 
 
 1^256. Proposition. — If, in any proportion, the two 
 
 antecedents or the two consequents he midtiplied or divided 
 
 hy the same numher, the proportion will not he destroyed. 
 
 Thus : From the proportion 
 
 7 : 8 : : 21 : 24 we have 
 
 1 : 8 : : 3 : 24 or 7 : 1 : : 21 : 3 
 
 257. Problem. — The two extremes of a proportion 
 and one mean being given, to find the other mean, 
 
 (1.) Reduce the given terms as'loiv as possible hy 255 
 and 256. 
 
 (2.) Divide the product of the resulting extremes hy the 
 mean. The quotient will be the other mean. 
 
 EXAMPLES. 
 
 1. Given the terms 7 : 21 : : : 24, to find the other 
 mean. 
 
248 
 
 RATIO. 
 OPERATION. 
 
 Divide first couplet by 7. 7 : 21 
 
 Divide consequents by 3. 1:3 
 
 1: 1 
 
 2. Given the terms 7 : 
 
 mean. 
 
 :24 
 
 : 24 (Vide 255.) 
 
 : 8 Ans. (Vide 256.) 
 
 8 : 24, to find the other 
 Ans. 21. 
 
 3. Given, 10 : 14 : : 
 
 : 35, to find the other mean. 
 
 Ans. 25. 
 
 4. Given, 10 : : : 25 : 35, to find the other mean. 
 
 Ans. 
 
 5. Given, 85 : 102 : : : 306, to find the other mean. 
 
 A71S. 255. 
 
 6. Given, J : | : : : f , to find the other mean. 
 
 OPERATION. 
 
 Multiply first couplet by 6. A : | 
 
 Divide consequents by 2. 3:4 
 
 3:2 
 
 -I (Vide 1671 Ex. 106.) 
 
 f (Vide 255.) 
 
 I (Vide 256.) Ans. j^^. 
 
 7. Given, J : : : ^^^ : §, to find the other mean. 
 
 8. Given, -/g : /,- 
 
 9. Given, f : 24 : 
 
 Ans. f . 
 : 100, to find the other mean. 
 Ans. 120. 
 : 20, to find the other mean. 
 Ans. j\. 
 10. Given, | : ^5_ . . . 20, to find the other mean. 
 
 258. The two means of a proportion and one extreme 
 being given to find the other extreme, 
 
 (1.) Reduce the terms as low as possible by 255 and 
 256. 
 
 (2.) Divide the product of the resulting means by the 
 extreme. The quotient will be the other extreme. 
 
RATIO. 249 
 
 EXAMPLES. 
 
 1. Given of : 4f : : 21, to find the fourth term. 
 
 OPERATION. 
 
 Multiply couplet by 21. 5| : 4f : : 21 (Vide 252, Ex. 3.) 
 Divide couplet by 17. 119 : 102 : : 21 (Vide 255.) 
 Divide antecedents by 7. 7 : 6 : : 21 (Vide 256.) 
 
 1 : 6 : : 3 : 18 Ans. 
 
 2. Given, 1 : 2 : : 3, to find the fourth term. A7is. 6. 
 
 3. Given, 3:9:: 12, to find the fourth term. 
 
 Ans. 36. 
 
 4. Given, 4 : 16 r : 15, to find the fourth term. 
 
 Ans, 60. 
 
 5. Given, ^% : /o : : 120, to find the fourth term. 
 
 Ans. 100. 
 
 6. Given, 1 : | : : J, to find the fourth term. 
 
 Ans. I. 
 
 7. Given, 9 : 18 : : ^, to find the fourth term. 
 
 A71S. 1. 
 
 8. Given, 35| : 15^ : : 4, to find the fourth term. 
 
 Ans. IjV^. 
 
 9. Given, 4^ : 22^ : : 9, to find the fourth term. 
 
 Ans. 45. 
 
 10. Given, ^^ : 28? : : 3, to find the fourth term. 
 
 Ans. 27. 
 
 11. Given, 2.5 : 45 : : 63, to find the fourth term. 
 
 A71S. 1134. 
 
 12. Given, 4.5 : 22.5 : : 9, to find the fourth term. 
 
 A71S. 45. 
 
 13. Given, J-i : ^J- : : ^%, to find the fourth term. 
 
 Ans. §. 
 
250 RATIO. 
 
 14. Given, 5 : f : : |, to find the fourth term. 
 
 Ans. /j. 
 
 15. Given 7| : 6 : : 5|, to find the fourth term. 
 
 Ans. 4j|-f. 
 
 16. Given, 6 : 7| : : 5f , to find the fourth term. 
 
 Ans. 6§§. 
 
 17. Given, 1 : 2| : : 2|, to find the fourth term. 
 
 Ans. 6i. 
 
 18. Given, 75| : 36 : : 643|, to find the fourth term. 
 
 Ans. 306. 
 
 19. Given, 1.50 : 1.00 : : .30, to find the fourth term. 
 
 Ans. .20. 
 
 20. Given, 3X32 : 5X16 :: 120, to find the fourth 
 term. Ans. 100. 
 
 21. Given, 8X21 : 56X6 :: 10, to find the fourth 
 term. Ans. 20. 
 
 22. Given, 2\ : If : ; |f , to find the fourth term. 
 
 Ans. ||. 
 
 23. Given, • :^%:'. 120 : 100, to find the first term. 
 
 A71S. /g. 
 
 RULE OF THREE. 
 
 259. Every problem in proportion involves at least 
 three quantities, so related to each other that a fourth 
 may be found from them. 
 
 260. The statement of a problem consists in properly 
 arranging the three quantities mentioned in it, so as to 
 form the first, second, 'dud third terms of a proportion. 
 
 261. The Rule of Three consists of directions by 
 which a problem in proportion may be stated. They 
 are as follows : 
 
HATIO. 251 
 
 (1.) Of the three quantities mentioned, make that the 
 
 THIRD TERM wMcJl JiaS the SAME NAME aS the ANSWER 
 
 required. 
 
 (2.) Of the two remaining quantities, make the 
 GREATER the SECOND TERM if the ANSWER ought to be 
 GREATER than the third term; make the less the sec- 
 ond term if the answer ought to be less tha7i the third 
 TERM. — (Vide 254, Rem.) 
 
 (3.) Place the remaining quantity for the first term. 
 
 The fourth term, found by 258, will be the answer, 
 
 examples. 
 
 1. If 5f lb. of sugar cost 21 cts„ what will 4!^ lb. cost? 
 (1.) 21 cts. has the same name which the answer 
 
 should have. 
 
 (2.) 4| lb. will evidently cost less than 21 cts., the 
 jorice of 5f lb. 
 
 (5.) 5| should then be the first term, 4? the second, 
 and 21 cts. the third. 
 
 Thus 5§ : 4^ : : 21 (vide 258, Ex. 1.) Ans. 18 cts. 
 
 2. If j\ of a quantity of sugar cost S120, what will 
 /j of the same quantity cost?— (See 167 J, Ex. 113, 
 Rem.; and 258, Ex. 5.) Ans. |100. 
 
 3. If 3 yd. of cloth cost $12, what will 9 yd. cost? 
 
 A71S. §36. 
 
 4. If 4 lb. of rice cost 15 cts., what will 16 lb. cost? 
 
 Ans. 60 cts. 
 
 5. If 1 lb. of tea cost J of a dollar, what will J lb. 
 cost? Ans. 16| cts. 
 
 6. If a staff 9 feet high cast a shadow \ of A foot 
 
-OJ RATIO. 
 
 LONG, what will be the length of the shadow of a post 
 which is l^feet high f Ans. 258, Ex. 7. 
 
 7. If a tree 35| ft. high casts a shadow 4 ft. long, 
 what will be the length of the shadow of a tree 15 1 ft. 
 Mgh 1 Am. 1 ^V? ^.=1 ft. .8 ^V^ in. 
 
 8. If a staff 10 ft. high casts a shadow 12 ft. in lengthy 
 what will be the hight of a tree whose shadow measures 
 70 ft. ? Am. 58 J ft.=58 ft. 4 in. 
 
 9. If 34 yd. of cloth cost ^3, what will 28| yd. cost? 
 
 Am. |27. 
 
 10. If 2.5 A. of land cost |63, what will 45 A. cost? 
 
 Am. §1134. 
 
 11. If 45 A. of land cost' |1134, what will 2.5 A. 
 cost ? Am. 
 
 12. If 45 A. of land cost §1134, what quantity can 
 be bought for §63 ? 1134 : 63 : : 45 Am. 
 
 13. If 4.5 yd. of cloth cost §9, what will 22.5 yd. 
 cost ? 
 
 14. If 25.5 yd. of cloth cost §45, what will 4.5 yd. 
 cost? 
 
 15. If J J of a pound of butter cost -^^ of a dollar, 
 how much will ij of a pound cost? 
 
 16. If W of a lb. of butter cost 40 cts., what will 
 jj lb. cost? 
 
 17. If \\ of a lb. of butter cost 53 1 cts., what quan- 
 tity can be bought for 40 cts.? An8. 8| oz. 
 
 18. If 8 J oz. of butter can be bought for 40 cts., what 
 quantity can be purchased for 53 J cts. 
 
 An%. 11J4 oz. 
 
 19. If 6 yd. of cloth cost §4.55||, what will 71 yd. 
 cost ? 
 
RATIO. 253 
 
 20. If 75 yd. of cloth cost ^5.60, Avhat will G yd. 
 cost? 
 
 21. If 1 yd. of carpeting cost |2-J, what will 2 J yd. 
 cost? 
 
 22. If 2 J yd. carpeting cost ^6.25, what will .one yd. 
 cost? 
 
 23. If 75f yd. of cloth cost |643|, what will 3G yd. 
 cost?— (Vide 167i, Ex. 72; and 258, Ex. 18.) 
 
 24. If 1 coat require If yd. of cloth, how many coats 
 can be made of 18 J yd.? Ans. 10 coats. 
 
 25. If 10 lb. of copper cost ^18.75, what number of 
 pounds can be had for ^171.25?— (Vide 167J, Ex. 79 
 and 80.) 
 
 26. If f of a ship are worth $15700,^ what is the 
 value of i of the ship? • Ans. 167}, Ex. 85. 
 
 27. If 1 lb. of butter cost 3 pence, how many tons 
 can be bought for 84 £ 13 s. 9d.? 
 
 Ans. 191, Ex. 63. 
 
 28. If I buy cloth at §1.50 and sell it for §1.20, 
 what shoiild I lose on §1.00?— (Vide 258, Ex. 19.) 
 
 Ans. §0.20. 
 
 29. If I buy cloth at §1.25 and sell it for §1.75, 
 what should I make on §1.00?— (Vide 228, Ex. 8.) 
 
 Ans. 40 cts. 
 
 30. If I pay at the rate of 11 cts. per mile, how far 
 can I ride for 7 cts.? Ans. 193, Ex. 39. 
 
 31. If I sell a quantity of land at the rate of §450 
 for 25 sq. r., and obtain §51300, how many acres do I 
 sell? Ans. 191, Ex. 74:. 
 
 32. If 2.5 lb. of tobacco cost 75 cts., how much will 
 185 pounds cost? Ans. §55.50. 
 
254 RATIO. 
 
 33. If 2 oz. of silver cost ^2M, wliat will f oz. cost? 
 
 Ans. $0.84. 
 
 34. If 7 lb. of sugar cost 75 cts., what will 12 pounds 
 cost? Ans. $1,284. 
 
 35. If 141 tons of coal cost 85 £, what will 94 tons 
 cost? Ans. $274.26. 
 
 36. If the interest of $19.35 is $2.7477, what will be 
 the interest of $17.21 for the same time and rate per 
 cent.? Ar^s. $2.44382. 
 
 37. If the interest of $17.21 is $2.63313, what will 
 be the interest of $140.10 for the same time and rate 
 per cent.? Ans. $21.4353. 
 
 38. If 4 men can do a piece of work in 100 days, in 
 how many days would 5 men do the same work? 
 
 Ans. 80 days. 
 
 39. If 20 men can do a piece of work in J of a day, 
 how^ long would it take 2 men to do the same w^ork? 
 
 Ans. 5 days. 
 
 40. If 60 men can do a piece of work in 8 days, how 
 many men would perform the same w^ork in 20 ^ays? 
 
 Ans. 24 men. 
 
 41. If 2 men cai> dig a ditch in 40 days, in how many 
 days would 10 men dig the same ditch? 
 
 A71S. 8 days. 
 
 42. Having read 120 pages of a book, I find that I 
 have still to read | of the book. How many pages does 
 the book contain? 
 
 § : I : : 120 Ans. 300 pages. 
 
 43. A person owning | of a coal mine sells | of his 
 share for $400. What is the mine worth at this rate? 
 
 Ans. $1000. 
 
iiATio. 255 
 
 44. If 10 covfs eat 8 tons of hay in a given time, 
 how many cows would eat 56 tons in the same time? 
 
 Ans. 70 cows. 
 
 45. If 70 cows eat a certain quantity of hay in 6 
 weeks, how many coavs would eat the same hay in 21 
 weeks? Ans. 20 cows. 
 
 46. If 20 men have been at work 18 days to con- 
 struct a given length of railroad, how many days will 
 76 men require to construct the same length of road ? 
 
 Ans. 4i| days. 
 
 47. If, by making 10 hours a day, a certain number 
 of men complete a work in 18 days, how many days 
 would be required to complete a similar work, at 12 
 hours a day? Ans. 15 days. 
 
 48. If 18 days are required to construct 500 feet of 
 railroad, how many days will be occupied on 1140 feet 
 of the same road? Aiis. 41^^ days. 
 
 49. If a passenger train of cars gain on a freight 
 train at the rate of 8 miles in 3 hours, how many hours 
 will it take to gain 60 miles? Ans. 22^ hours. 
 
 50. A passenger train of cars moves at the rate of 
 45 miles in 3 hours, and a freight train at the rate of 
 37 miles in the same time. If the freight train has 60 
 miles the start, in what time will it be overtaken ? 
 
 51. A hand car, running at the rate of only 1 mile 
 an hour, has 12 miles the start of a passenger train, 
 which runs at the rate of 12 miles an hour. In what 
 time will the hand car be overtaken? 
 
 Ans. 1 h. 5y\ m. 
 
 52. If the long hand of a clock move at the rate of 
 12 spaces an hour, and the short hand 1 sjmce an hour, 
 
256 RATIO. 
 
 in what time will the long hand gain 12 spaces upon the 
 short hand? 
 
 53. At what time between 9 and 10 o'clock will the 
 hands of a clock be together? 
 
 Ans. 9 o'clock, 49 Jj- m. 
 
 S62. To state a j)roblem when it involves more than 
 three terms, 
 
 (1.) Of the quantities mentioned, maJce that the third 
 term which has the same name as the answer required. 
 
 (2.) Select tivo terms of the same name, and arrange 
 the couplet as though it were entirely disconnected with all 
 other conditions of the i^rohlem. (Vide 261, (2.) 
 
 (3.) Select two other terms of the same name, and ar- 
 range as before, and so on, till all the teyms are arranged. 
 
 Having canceled all the factors common to any ante- 
 cedent and consequent, (vide 255,) or common to either 
 one of the first terms and the third, (vide 256,) the co7i- 
 tinued product of the means divided hy that of the first 
 terms ivill he the ansiuer. 
 
 EXAMPLES. 
 
 1. If 10 cows eat 8 tons of hay in 6 weeks, how many 
 cows will eat 56 tons in 21 weeks? 
 
 It would take more cows to eat 56 tons than 8 tons, 
 and less cows to eat it in 21 weeks than 6 weeks. 
 
 STATEMENT. 
 
 (Vide 261, Ex. 44 and 45.) 8 : 66 .^ (vide 258, Ex. 21.) 
 
 21: 6"^" 
 
 SOLUTION. 
 Divide upper couplet by 8. 1-7..;^q (vide also 156, Ex. 22.) 
 
 Divide lower couplet by 3 and cancel the 7s. Ans. 20 cows. 
 
RATIO. 257 
 
 2. If the wages of 6 men for 14 days be §84, what 
 will be the wages of 9 men for 16 days? Ans. ^144. 
 
 3. If 12 oz. of wool make 21 yd. of cloth which is 
 1^ yd. zvide, how many pounds will it take to make 
 150 yd. only 1 yd. wide? Ans. 30 lb. 
 
 4. If the interest of §19.35 for 1 yr. 2 mo. 6 c!a. is 
 §2.7477, what is the interest of §17.21 for 1 yr. 3 mo. 
 9 da., the rate per cent, being the same in each case ? 
 (Vide 205, Ex. 56; also 236, Ex. 3.) Ans. §2.63313. 
 
 5. If the interest of §140.10 for 2 yr. 4 mo. 12 da. 
 is §39.7884, what is the interest of §75.15 for 3 yr. 
 6 mo. 15 da., the rate per cent, being 12 in each case? 
 (Vide 205, Ex. 56 ; .236, Ex. 5.) Ans. §31.93|. 
 
 6. How much hay will 32 horses eat in 120 days, if 
 96 horses eat 3| tons in 7 J weeks? A^is. 2f tons. 
 
 7. If 6 laborers dig a ditch 34 yd. long in 10 days, 
 how many yards will 20 laborers dig in 15 days? 
 
 8. If 20 laborers dig a ditch 17-0 yd. long in 15 days, 
 how many days will 6 laborers require to dig a ditch 
 34 yd. long ? Ans. 
 
 9. If 14 men can reap 84 acres in 6 days, how many 
 men must be employed to reap 44 acres in 4 days? 
 
 10. If 20 men, by Avorking 10 hours a day, have been 
 employed 18 days in constructing 500 ft. of railroad, 
 how many days of 12 hours each must 76 men bo em^ 
 ployed to construct 1140 feet of the same road. 
 
 STATEMENT. 
 
 (Vide 261, Ex. 46, 47, 48.) 76 : 20 
 (Vide 157, Ex. 27) 12 
 
 500 
 
 99 
 
 10:: 18 
 1140 A71S. 9 davs. 
 
258 RATIO. 
 
 11. If 50 men, by working 5 hours a day, can «^'> 9A 
 cellars in 54 days, each cellar being 36 ft. lo- 
 
 wide, and 10 ft. deep, how many men can dig lb ctii.. s 
 in 27 days, each cellar being 48 ft. long, 28 ft. wide, and 
 9 ft. deep, by working 3 hours a day? 
 
 Ans. 200 men. 
 
 12. If 496 men, in 5^ days of 11 hours each, dig a 
 trench of 7 degrees of hardness, 465 ft. long, 3| ft. wide, 
 2| ft. deep, in how many days, of 9 hours long, will 24 
 men dig a trench, of 4 degrees of hardness, 337 i ft. 
 long, 5f ft. wide, and 3 J ft. deep? 
 
 Ans. 157, Ex. 23. 
 
 13. If 12 men can build a wall 30 ft. long, 6 ft. high, 
 and 3 ft. thick in 15 days, when the days are 12 hours 
 long, in what time will 60 men build a wall 300 ft. long, 
 8 ft. high, and 6 ft. thick, when they work only 8 hours 
 a day? ^?zs. 157, Ex. 24. 
 
 14. If 25 pears can be bought for 10 lemons, and 28 
 lemons for 18 pomegranates, and 1 pomegranate for 48 
 almonds, and 50 almonds for 70 chestnuts, and 108 
 chestnuts for 2 J cts., how many pears can I buy for 
 $1.35? Ans. 375 pears. 
 
 15. If the interest of J2187.50, from Aug. 10, 1868, 
 to Jan. 1, 1870, at 8 per cent, per annum, is |243.54^, 
 what is the interest of |10500, from Feb. 1, 1867, to 
 May 13, 1869, at the rate of 5 per cent, per annum? 
 
 STATEMENT. 
 
 218750:1050000 
 
 167:274 : : 243.54J 
 8:5 
 
 Ans. 236, Ex. 27. 
 
KATIO. 250 
 
 2$IS. To divide a given number into parts "which 
 shall be proportional to given numbers, 
 
 (1.) If the given numbers are fractions, midfij^Ii/ them 
 all hy the least common multiple of the denominators. 
 
 (2.) 3Iake the sum of the results the first term of a 
 proportion, any one of the residts the second term, and 
 the given number the third term. 
 
 The fourth term, found by 258, will be one of the 
 required parts, and the others may be found in like 
 manner. 
 
 EXAMPLES. 
 
 1. Divide 49 into two parts which shall have the ratio 
 of I • ^ 
 
 2 
 
 OPERATION. 
 
 iX6=3. (Vide 167J, Ex. 106.) 
 PROOF. fx6=4. 
 
 21+28=49. 7 : 3 : : 49 : 21 1st part. 
 
 21 : 28 : : J : I 7 : 4 : : 49 : 28 2d part.^ 
 
 2. Divide 136 into two parts which shall be propor- 
 tional to the numbers | and f . (Vide 167J, Ex. 107.) 
 
 Ans. 64 and 72. 
 
 3. Divide 544 into two parts which shall be propor- 
 tional to the numbers /^ and ^^. (Vide 167J, Ex. 108.) 
 
 Ans. 384 and 160. 
 
 4. Divide 2135 into two parts which shall be pro- 
 portional to the numbers 35 § and 15^. (Vide 167 J, 
 Ex. 109.) Ans. 1498 and 637. 
 
 5. Divide 209 into three parts which shall be propor- 
 tional to the numbers J, |-, and |. 
 
260 
 
 PvATIO. 
 
 OPEEATION. 
 
 ^.X12=6. (Vide 167J, Ex. 110.) 
 J X 12-4. 
 
 fXl2-=9. 
 
 19: 6:: 209:66 1st part. 
 
 PROOF. 
 
 66+44+99=209. 
 
 66 : 44 :: J : J (vide 254.) 19 : 4 : : 209 : 44 2d part. 
 
 66 : 99 :: i : I 19 : 9 : : 209 : 99 3d part. 
 
 6. Divide 504 into four parts which shall be propor- 
 tional to the numbers J, f , j%, and f . (167J, Ex. 111.) 
 
 Ans. 135; 180; 81;lindl08. 
 
 7. Divide 30; 45; 75; 135; 180; and 750, each into 
 five parts which shall be proportional to the numbers 1, 
 2, 3, 4, and 5. Ans. 2; 4; 6; 8; 10, etc. 
 
 8. Divide 36 into three parts, so that J the first, | the 
 second, and i the third, shall all be equal to each other. 
 
 Remark, — The parts will evidently be as the numbers 2, 3, and 4. 
 
 Ans. 8; 12; and 16. 
 
 9. Divide 136 into two parts, so that -f of the first 
 and I of the second shall be equal. 
 
 Remabk. — The numbers will be as ii : i, or as 9 : 8. 
 
 A71S. 72 and 64. 
 
 10. A gold and a silver watch together cost §132, and 
 the gold watch costs 10 tijnes as much as the silver 
 watch. What did each cost? Ans. §120 and $12. 
 
 11. A's age is double B's, and B's is triple C's. The 
 sum of all their ages is 140. What is the age of each? 
 
 Remark. — Their ages are as the numbers, 1, 3, and G. 
 
 An8.A8i; B42; C 14. 
 
 12. A gentleman bought a certain number of oxen, 
 and double the number of cows; and also three tiiiics 
 
RATIO. 261 
 
 as many sheep as cows. He gave ^50 each for oxen, 
 $25 each for coavs, and |3 each for sheep; the whole 
 costing §354. What number of each did he purchase ? 
 Ans. 3 oxen; 6 cows; 18 sheep. 
 
 13. A man paid |74 for a sheep, a cow, and an ox. 
 The cow was valued at 12 sheep, and the ox two cows. 
 What was the price of each? 
 
 Ans. sheep §2; cow §24; ox §48. 
 
 14. I wish to make a mixture of 360 pounds of tea, 
 using at the rate of 30 lb. worth 30 cts. a pound ; 11 lb. 
 worth 33 cts. a pound ; 23 lb. worth 67 cts. a pound ; and 
 26 lb. worth 86 cts. per pound. What quantity of each 
 kind must be used? 
 
 Ans. 120, 44, 92, and 104 lbs., respectively. 
 
 204. Partnership is an association of two or more 
 individuals for the transaction of business. 
 
 (1.) The partners constitute the company, firm, or 
 house. 
 
 (2.) The capital is the money invested by the company 
 in business. 
 
 (3.) The profit or loss to be shared is called a divi- 
 dend. 
 
 S05. To ascertain the dividend of the partners, 
 when the money of each has been invested the same 
 length of time. 
 
 Make the capital of the company the first term of a 
 projwrtion, the money of any partner the second term, 
 and the profit or loss the third term. 
 
 The fourth term, found by 258, will be the divi- 
 dend of the partner whose money forms the second 
 term. 
 
262 RATIO. 
 
 EXAMPLES 
 
 1. A and B invest $550 in a speculation, of which 
 A furnishes $330 and B $220. They gain §70. What 
 is the dividend of each? 
 
 OPERATION. 
 
 550 : 330 : : $70 : A's share. 
 5: 3 ::$70:$42 
 Divide antecedents by 110. 
 
 550 : 220 : : $70 : B's share. 
 5: 2 ::$70:$28 
 (Vide 232, Ex. 1.) Ans. A $42; B $28. 
 
 2. A invests $300- in ti speculation, and B $400. 
 They gain $49. What is the share of each? (Vide 
 263, Ex. 1.) Ans. A $21 ; B $28. 
 
 3. A, B, and C form a partnership. A furnishes 
 $1200, B $1600, and C 2000. What is each partner's 
 share in a gain of $960? 
 
 Alls. A $240; B $320; C $400. 
 
 4. A, B, and C form a partnership; A furnishing 
 $800, B $1500, and C $3000. They gain $500. What 
 is the dividend of each? 
 
 Ans. A 75.471; B $141,509; C $283.02. 
 
 5. A and B form a partnership. A furnishes $1200, 
 and B $500. They gain $544. What is the share of 
 each? Ans. 263, Ex.- 3. 
 
 6. A, B, C, and D make up a purse to buy lottery 
 tickets. A furnishes $15, B $20, C $9, and D $12. 
 What is each one's share in a prize of $504? 
 
 Ans. 263, Ex. 6. 
 
 7. A, B, C, D, E, F, and G engage in an oil specula- 
 
RATiO. . 263 
 
 tion. A furnislies §175 of the capital, B S500, C $600, 
 D ,^210, E ^42.50, F §20, and G §10. They expend 
 §623 in prospecting, and then give the matter up as a 
 total failure. What does each lose? ^ns. 232, Ex. 2. 
 
 8. Three gentlemen engage in a gold speculation. A 
 furnishes §500, B §1200, and C §4300. They clear 
 §1500. What is each man's share? A71S. 232, Ex. 3. 
 
 !S66. To ascertain the dividend of the partners, when 
 the money of each has been invested different lengths 
 of time, 
 
 (1.) Multiply each jjartner^s money hy the time it is 
 invested. 
 
 (2.) Make the swn of the ijroducts the first term of a 
 'proportion^ any one j^rodiict the second term, and the gain 
 or loss the third term. 
 
 The fourth term, found by 258, will be the share of 
 the partner whose product forms the second term. 
 
 EXAMPLES. 
 
 1. A and B are associated in trade. A has furnished 
 of the joi7it stock §330 for 5 months, and B §220 for 8 
 months. They gain §170.50. What is the share of 
 each? 
 
 OPERATION. 
 
 330X5=1650. 
 
 220X8=1760. 
 
 Divide the coup- 3410 : 1650 : : §170.50 
 let by 10, and the 2 : 165 : : §1 
 
 two antecedents by 3410 : 1760 : : §170.50 
 170.1 2: 176:: SI 
 
 A's share. 
 §82.50 
 B's share. 
 
264 TvATIO. 
 
 2. A and B enter into partnership. At first A, with 
 a capital of §3G0, does business alone. At the expira- 
 tion of two months B comes in with a capital of §520, 
 and the partners do business together 5 months. The 
 profits of the concern, from the time A commenced 
 business, were $128. What should be the share of each? 
 
 Ans. A $63; B $65. 
 
 3. A, B, and C form a partnership. " A's part of the 
 capital is $4300, B's $2000, and C's $1500. At the end 
 of 2 months A withdrew with his stock. At the end 
 of another 2 months B withdrew with his stock. C con- 
 tinued the business alone for another 2 months, when 
 the entire profits were found to have been $1280. What 
 is the share of each? 
 
 A71S. A $430; B $400; C $450. 
 
 4. On the 1st of January, 1866, A commenced busi- 
 ness wdth a capital of $8000. On the 1st of July B 
 joins him with a capital of $16000. On the 1st of 
 July, 1867, it is found that $4000 have been cleared since 
 A began business. What is the share of each? 
 
 Ans. A $171^; B $2285|. 
 
 5. A, B, and C, at the end of a partnership, have 
 jointly $1000 in trade. A's stock has been in the busi- 
 ness 7 months, B's stock 8 months, and C's 12 months. 
 A's dividend is $21 ; B's $40; C's 24. What amount of 
 money did each invest? 
 
 A71S. A $300; B $500; C $200. 
 
 6. Two cousins, George and Frank Ransom, com- 
 menced business on the 1st of January, each partner 
 putting in $10000. On the 1st of June George in- 
 creased his stock by $2000, Frank withdravring the 
 
RATIO. 265 
 
 same sura. On the 1st of September George withdrew 
 §4000, but Frank increased his by §3000. At the end 
 of the year they had made §6720. What was the divi- 
 dend of each? Ans, George §3360; Frank §3360. 
 
 EQUATION OF PAYMENTS. 
 
 267. To ascertain the mean time for the payment of 
 
 several sums due at different times, 
 
 (1.) Multipli/ each ijayment by its lime of credit. 
 
 (2.) Divide the sum of the products by the sum of the 
 payments. The quotient will be the mean time. 
 
 EXAMPLES. 
 
 1. I owe a merchant §30, due in 4 months; §40, due 
 in 5 months, and §50, due in 6 months. What is the 
 mean time for the payment of all the bills ? 
 
 OPERATION'. 
 
 30X4=120 
 40X5=200 
 50X6=300 
 
 120 ) 620(5J 
 
 Ans. 5^ months. 
 
 Remark. — The interests of the several sums, at any rate per cent, 
 per annum, for the given times, when added together, must be the 
 same as the interest of the sum of the payments for the mean time. 
 Hence the following rule: 
 
 (1.) Find the interest of each payment for its time of credit. 
 
 (2.) Divide the sum of the interests hy the interest of the sum of the 
 payments for 1 month. The quotient will be the mean time. 
 
 Remark. — If the rate of 12 per cent, per annum is selected, the 
 operation will be identical with that above. 
 
 23 
 
266 RATIO. 
 
 2. A merchant owes §200, payable in 4 months ; $400, 
 in 5 months; |500, in 6 months; and §600, in 8 months. 
 What is the mean time of payment? 
 
 Ans. 6j\ months. 
 
 3. A merchant has given three notes to the same 
 creditor : §200, due in 2 months ; §200, in 4 months ; 
 and §200, in 6 months. What is the mean time of 
 payment ? Ans. 4 months. 
 
 4. I have several bills due at a store; §40 due in 20 
 days from January 1, 1866; §30 due in 40 days from 
 the same time; amd §50 due in 45 days. What is the 
 mean time of payment ? 
 
 A71S. 35/^ days; i. e. Feb. 4, 1866. 
 
 5. I buy a house and lot for §1600, with the under- 
 standing that I am to pay one fourth cloiV7i, one third of 
 the balance in 3 months, and the remainder in 6 months. 
 What would be the mean time for the payment of the 
 whole sum ? Ans. 3| months. 
 
 ALLIGATION MEDIAL. 
 
 268. To ascertain the mean price of a compound 
 consisting of ingredients of which the quantity and 
 value of each are given, 
 
 Divide the entire cost of the compound hy the sum of 
 the ingredients. The quotient will be the mean price. 
 
 EXAMPLES. 
 
 1. A merchant bought 160 gallons of wine at 40 cts. 
 per gallon ; 75 gallons at 60 cts. per gallon ; 225 gal- 
 lons at 48 cts. per gallon ; 40 gallons at 87 J cts. per 
 gallon, What wns the mean cost of the wine? 
 
RATIO. 267 
 
 OPERATION. 
 
 160X.40 = 64.00 
 75X.60 = 45.00 
 
 225X.48 =-108.00 
 40X.87J= 35.00 
 
 500 ) 252.00(.505 
 252.00 
 Ans. 10.504 per gallon. 
 
 2. A farmer mixed wheat, viz : 5 bushels worth |1.10 
 per bushel; 10 bushels worth 60 cts. per bushel; 5 
 bushels worth 70 cts. per bushel. What is the mean 
 price of the mixture ? Ans. 75 cts. per bushel. 
 
 3. A wine merchant mixes wine, viz: 88 gallons of 
 Canary, worth 50 cts.' per gallon ; 88 gallons of Sherry, 
 worth 76 cts. per gallon ; and 48 gallons of Claret, worth 
 ^1.75 per gallon. What is a gallon of the mixture 
 w^orth ? A71S. 87 cents. 
 
 4. A goldsmith mixes 7 ounces of gold 23 carats fine 
 with 3 ounces 16 carats fine, 3 oz. of 18, and 3 of 19 
 carats fine. What is the fineness of the mixture ? 
 
 Ans. 20 carats. 
 
 5. If I mix 30 lb. of tea at 30 cts. per lb.; 11 lb. at 
 33 cts. per lb. ; 23 lb. at 67 cts. per lb. ; and 26 lb. at 
 86 cts. per lb., what is one pound of the mixture worth? 
 
 Ans. 56 cents. 
 
 6. If I mix 11 lb. of tea at 30 cts.; 30 lb. at 33 cts.; 
 26 lb. at 67 cts. ; and 23 lb. at 86 cts., what is one pound 
 of the mixture worth? Ans. 56 cts. 
 
 7. If I mix 30 lb. of tea at 30 cts.; 41 lb. at 33 cts.; 
 23 lb. at 67 cts. ; and 49 lb. at 86 cts., what is one pound 
 of the mixture worth? Ans. 56 cts. 
 
268 
 
 RATIO. 
 
 8. If I mix 41 lb. of tea at 30 cts. ; 30 lb. at 33 cts.; 
 26 lb. at 67 cts. ; and 49 lb. at 86 cts., what is one pound 
 of the mixture worth ? Aiis. 56 cts. 
 
 9. If I mix 41 lb. of tea at 30 cts.; 11 lb. at 33 cts. ; 
 49 lb. at 67 cts. ; and 26 lb. at 86 cts., Avhat is one pound 
 of the mixture worth ? Ans. 56 cts. • 
 
 10. If I mix 11 lb. of tea at 30 cts.; 41 lb. at 33 cts.; 
 49 lb. at 67 cts.; and 23 lb. at 86 cts., what is the mean 
 price? Ans. 56 cts. 
 
 11. During 14 hours on a certain day the mercury of 
 a thermometer stood as follows : 2 hours at 60 degrees ; 
 3 hours at 62 degrees ; 4 hours at 64 degrees ; 3 hours 
 at 67 degrees ; 1 hour at 72 degrees ; and 1 hour at 75 
 degrees. What was the mean temperature during the 
 time? Ans. 65 degrees. 
 
 ALLIGATION ALTERNATE. 
 260. To find the quantity that may be used of each 
 ingredient of a proposed compound, the price of each 
 being given, and the mean price of the compound. 
 
 EXAMPLES. 
 
 1. A farmer has wheat worth $1.10 per bu. ; wheat 
 worth 60 cts. per bu. ; and wheat worth 70 cts. per bu. 
 He desires to mix it so that a bushel may be worth 75 
 cents. What quantity of each may be used ? 
 
 75 
 
 110 
 
 60 
 
 70- 
 
 OPERATION. 
 
 (1) (2) (3) (4) (5) 
 
 15 
 
 5 
 
 3 
 
 1 
 
 4 
 
 35, 
 
 
 7 
 
 
 7 
 
 
 35 
 
 
 7 
 
 7 
 
 Am. 4 bu. at $1.10; 7 bu. at 60 cts.; and 7 bu. at 70 cts. 
 
KATIO. 269 
 
 Explanation. — 110, which is greater than the mean rate, is 
 connected by a dotted line with 60, which is less than the mean 
 rate; also, 110 is connected with 70, forming another couplet, 
 one term of which is greater, and the other less than the mean 
 rate, 75. 
 
 In column (1), 15 is the difference between 60 and 75, and 35 is 
 the difference between 110 and 75 — each difference being opposite 
 the other term of the couplet. 
 
 In column (2), 5 is the difference between 75 and 70, and 35 is 
 the difference between 110 and 75 — each difference opposite the 
 other term of the couplet. 
 
 Column (3) is formed by dividing the terms of (1) by their 
 greatest common divisor. \ . 
 
 Column (4) is formed by dividing the terms of (2) b;y their 
 greatest common divisor. 
 
 Column (5) is formed by adding the corresponding terms of (3) 
 and (4). 
 
 Any number of answers may now be found. 
 
 OPERATION CONTINUED. 
 
 (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) 
 
 7 
 
 10 
 
 13 
 
 5 
 
 6 
 
 7 
 
 9 
 
 1 
 
 6 
 
 1 
 
 14 
 
 21 
 
 28 
 
 7 
 
 7 
 
 7 
 
 14 
 
 2 
 
 10 
 
 1 
 
 7 
 
 7 
 
 7 
 
 14 
 
 21 
 
 28 
 
 21 
 
 1 
 
 ■ 5 
 
 4 
 
 Columns (6), (7), and (8) are found by multiplying the terms 
 of (3) by 2, 3, and 4, respectively, and adding the corresponding 
 terms of (4). 
 
 Columns (9), (10), and (11) are found by multiplying the terms 
 of (4) by 2, 3, and 4, respectively, and adding the corresponding 
 terms of (3). 
 
 Column (12) is found by multiplying the terms of (3) by 2, and 
 those of (4) by 3, and then adding the corresponding products. 
 
 Column (13) is that, of (6) divided by 7. 
 
 Column (14) is (13) multiplied by 5. (Vide 268, Ex. 2.) 
 
 Column (15) is (11) divided by 7. All the answers may be ver- 
 ified by 268, and as many others as one is curious to find. 
 
 2. I wish to mix tea at 30 cts. ; tea at 33 cts. ; tea at 
 67 cts. ; and tea at 86 cts. per pound, so as to make the 
 
270 
 
 llATIO. 
 
 What quantity of 
 
 mixture worth 56 cts. per pound, 
 each can be used ? 
 
 OPERATIONS. 
 
 56 
 
 Multiply or divide the terms in columns (1) or (2) 
 by any numbers whatever (merely preserving their 
 ratio) for other answers. Thus, 
 
 ( 
 
 ;i.) 
 
 
 
 
 (2.) 
 
 
 
 (1) (2) (3) (1) (2) (3) 
 
 30 30 
 
 
 30 
 
 
 30 , 
 
 11 
 
 
 11 
 
 33 1 
 
 
 11 
 
 11 
 
 56 
 
 33~4n 
 
 
 30 
 
 30 
 
 67 •• 
 
 
 23 
 
 23 
 
 67 -^ 
 
 26 
 
 
 26 
 
 86 — 26 
 
 
 26 
 
 
 86 — ' 
 
 
 23 
 
 23 
 
 (Vide 
 
 268, 
 
 Ex. 
 
 5.) 
 
 
 (Vide 
 
 268, 
 
 Ex. ( 
 
 >.) 
 
 15 
 
 15 
 
 45 
 
 22 
 
 11 
 
 ^ 
 
 11 
 
 22 
 
 44 
 
 30 
 
 60 
 
 30 
 
 23 
 
 46 
 
 92 
 
 52 
 
 26 
 
 13 
 
 13 
 
 13 
 
 39 
 
 23 
 
 46 
 
 23 
 
 ,30 
 
 56 
 
 130- 
 
 !33=1 I 
 |67— li 
 186 -^'26 
 
 (3.) 
 
 (1) (2) (3) (4) 
 
 30 
 
 23 
 
 56 
 
 30^ 
 33 
 
 67- 
 
 (4.) 
 
 (1)(2)(3 
 
 30 
 30 
 
 (Vide 268, Ex. 7.) 
 
 (Vide 268, Ex. 8. 
 
 Change the numbers in columns (1), (2), and (3) at 
 pleasure, merely preserving their ratio, and add the cor- 
 responding terms, for other answers. 
 (5.) 
 
 (1) (2) (3) (4) 
 
 zw 
 
 30111 
 
 261 
 
 26 
 
 56 
 
 30. 
 33 
 
 67 
 86 
 
 (6.) 
 
 (1) (2) (3) (4) (5) 
 
 11 
 
 26 
 
 30 
 30 
 
 23126 
 
 (Vide 268, Ex. U.; 
 
RATIO. 271 
 
 Having arranged tlie several prices and the mean price 
 as in the examples, 
 
 (1.) Connect in couplets each price that is less than the 
 mean price with one that is greater, hy lines which may 
 he readily distinguished from each other. 
 
 (2.) Select any couplet, and write the difference between 
 each of its terms and the mean price opposite the other 
 term, in a column to the right. 
 
 (3.) Treat each couplet in the same manner, forming 
 as many columns as there are couplets. 
 
 (4.) Add together the numbers standing opposite each 
 'price, and place the results in an additional column. 
 
 These sums will be the quantities that may be used 
 of the price opposite which they stand. 
 
 Remark 1. — To find other answers, divide or multiply the terms 
 of any column by any number whatever, and then add as in (4), 
 observing that it is only necessary that the two numbers in any 
 column preserve their ratio. 
 
 Remark 2. — The corresponding terms of any two or more col- 
 umns, containing answers, may be added together for other^ 
 answers. 
 
 Remark 3. — If the prices are so connected that each price is 
 linked with only one other price, only one column to the right is 
 necessary. (See above, operations (1) and (2), where only one 
 column was necessary. 
 
 Remark 4. — When the mean price is the same as one of the 
 given prices, the latter need not be included in the operation. 
 
 3. A merchant would mix wines, worth 16 shillings, 
 18 shillings, and 22 shillings per gallon, so that the 
 mixture may be worth 20 shillings per gallon. What 
 quantity of each may be used ? 
 
272 
 
 RATIO. 
 
 OPERATION. 
 
 20 
 
 The steps are precisely liko 
 those in Ex. 1. 
 
 (1) (2) (3) (4) (5) 
 
 16- 
 
 18- 
 22- 
 
 Ans. 1 gal. of IG; 1 of 18; and 3 of 22 shillings. 
 
 The following are among other answers easily obtained : 
 
 (6) (7) (8) (9) (10) (11) (13) (14) (15) (16) 
 
 2 
 
 
 1 
 
 
 1 
 
 
 2 
 
 
 1 
 
 1 
 
 4 
 
 2 
 
 2 
 
 1 
 
 3 
 
 2 
 
 3 
 
 4 
 
 1 
 
 1 
 
 1 
 
 3 
 
 5 
 
 * 
 
 9 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 2 
 
 4 
 
 9 
 
 10 
 
 5 
 
 7 
 
 9 
 
 4 
 
 5 
 
 6 
 
 8 
 
 14 
 
 10 
 
 28 
 
 (15) is half of (3) added to 9 times (4). 
 
 (16) is the sum of (11), (13), and (14). (See above, 
 Rem. 2.) 
 
 4. A merchant has wine at 40 cts. ; 75 cts. ; 60 cts. ; 
 
 and 48 cts., and he wishes to make a mixture of all 
 
 worth 50 cts. How many gallons can he use of each? 
 
 Remark 5. — Since the answers are limitless in number and va- 
 riety, it would be difficult to obtain those which might be given. 
 Let the pupil verify those which he may find by 268. 
 
 5. What quantity could be used of each to make the 
 mixture worth 70 cts. per gallon ? 45 cts. ? 48 ? 60 ? 
 (See Rem. 3.) 
 
 6. A merchant has three kinds of sugar, worth 6, 8, 
 and 15 cts. per pound. He wishes to make a mixture 
 worth 11 cts. a pound. What quantity of each may be 
 used ? 
 
 7. What quantity of each kind may he use to make 
 the mixture worth 12 cts.? 13 cts.? 14 cts.? 10 cts.?' 
 Sets.? 
 
 Remark 6. — If the quantity to be used of a given price is fixed, 
 multiply the terms of any answer, obtained as above, by the quo- 
 tient of the fixed quantity, divided by the number opposite the 
 
RATIO. 273 
 
 price mentioned. Thus, if, in Ex. 1, it is required to use 21 bushels 
 at $1.10, multiply the terms of column (5) by 2_i__5i^ qj. column 
 (6) by V=3, or(7)by2TV, etc. 
 
 Remark 7, — If the entire quantity to be used is fixed, apply the 
 principle of 263 to the fixed quantity, using any answer as the pro- 
 portional numbers. (Vide Ex. 2, Operation (1); and 263, Ex. 14.) 
 
 8. A grocer has currants at 4 cts. ; 6 cts. ; 9 cts. ; and 
 11 cts. ; and he wishes to make a mixture of 240 lb., 
 worth 8 cts. per pound. What quantity of each may 
 be used ? 
 
 Am, 72; 24; 48; 96, or 48; 48; 72; 72, etc. 
 
 POSITION. 
 
 270. Position is the operation of finding the true 
 answer to a problem through the aid of one or two as- 
 sumed answers. 
 
 (1.) Single Position assumes one ansiver. 
 (2.) Double Position assumes two answers. 
 
 271. Single position is applicable to those problems 
 in which, if any number is assumed as the answer, and 
 the steps, indicated by the problem, performed upon it, 
 the ratio of the result, and the result in the question., is 
 the same as that between the assumed ansiver and the true 
 answer. Hence, the 
 
 RULE. 
 
 (1.) Assume any convenient number to he the true answer, 
 
 (2.) Perform the operations indicated hy the pi^oblem 
 upon the assumed number. 
 
 (3.) Make the result the first term of a proportion, the 
 RESULT in the problem the secoimTterm, and the assumed 
 answer the third term. 
 
 The fourth term, found by 258, will be the true answer. 
 
274 RATIO. 
 
 EXAMPLES. 
 
 1. The sum of | and -J- of a certain number is 87 J. 
 What is the number ? 
 
 OPERATION. 
 
 Suppose 24 to be the answer. 
 
 1 of 24 is 8 
 
 1 of 24 is 6 
 
 Then 14 : 87J : : 24 
 
 Multiply couplet by 2. (255.) 28 : 175 : : 24 
 
 Divide couplet by 7. (255.) 4 : 25 : : 24 
 
 Divide antecedents by 4.' (256.) 1 : 25 : : 6 : 150 Ans. 
 
 Proof.— I'of 150+i of 150=87^. 
 
 2. What number is that which being increased by A, 
 I, aiid i of itself, the result will be 75 ? Ans. 36. 
 
 Remark. — In order to avoid fractions, assume tlie least common 
 multiple of the denominators of the fractions, or any multiple 
 of it. 
 
 3. What number is that, the sum of | and J of which 
 is 376 ? Assume 45 or 90. A71S. 360. 
 
 4. What number is that which being increased by |, 
 I, and f of itself, the result will be 48i? Ans. 15. 
 
 5. The difference between the fifth and tenth of a 
 certain number is 17. What is the number? 
 
 Ans. 1^0. 
 
 6. The difference between | and | of a certain num- 
 ber is 15. What is the number? Ans. 180. 
 
 7. The rent of a farm is 20 per cent, greater this 
 year than last year. This year it is $1800. What was 
 the rent last year? Ans. $1500. 
 
RATIO. 275 
 
 ^ OPERATION. 
 
 Suppose $100 to be the rent last year. 
 
 20 per cent. (J) of §100 is 20 
 
 Then $120 : $1800 : : $100 : $1500 
 
 8. If I of a number be multiplied by 7, and | of the 
 number itself be added to the product, the result will be 
 219. What is the number? Ans. 45. 
 
 9. A gold watch is worth ten times as much as a 
 silver watch, and both together are worth $132. What 
 is each watch worth? Ans, 263, Ex. 10. 
 
 10. The rent of a farm is 10 J^ less this year than 
 last year. This year it is $1000. What was it last 
 year? Aiis. $1111 1. 
 
 11. The difference between | and | of a number is 
 38f. What is the number? Ans. 100. 
 
 12. The sum of | and § of a number is 920. What 
 is the number? Ans. 720. 
 
 13. One fifth of all the sheep I have die in 1866, and 
 three fourths of the remainder in 1867, when I have 
 only 20 sheep left. What number had I at first. 
 
 Ans. 100. 
 
 14. A certain gentleman, at the time of his marriage, 
 agreed to give his wife | of his estate, if at the time of 
 his death he left only a daughter, and if he left only a 
 son she should have I of his property ; but5 as it hap- 
 pened, he left a son and daughter, in consequence of 
 which the widow received in equity $2400 less than she 
 would have received if there had been only a daughter. 
 What would have been the wife's dowry if he had left 
 only a son? A71S. $2100. 
 
276 KATIO. 
 
 S72. Double Position is applicable to those prob- 
 lems in which the difference between the true and first as- 
 sumed number is to the difference between the true and 
 second assumed number as the first error is to the second. 
 
 (1.) If the assumed numbers are both too large, or both 
 too small, the errors arising from them are said to be 
 
 ALIKE. 
 
 (2.) If one of the assumed numbers happens to be too 
 large, and the other too small, the errors are said to be 
 
 UNLIKE. 
 
 RULE, 
 
 (1.) Assume any convenient number, and perform the 
 ojjerations upon it indicated by the problem. 
 
 (2.) Take the differ eyice between the result and that 
 pointed out by the problem. This difference is the first 
 
 ERROR. 
 
 (3.) Assume a second number, and in like manner find 
 
 the SECOND ERROR. 
 
 (4.) If the errors, as shown by the results, are aljke, 
 multiply the first error by the second assumed num- 
 ber, and the second error i»^ the first assumed' nui^- 
 BER, and divide the difference of^ the products by the 
 difference of the errors. - The quotient is the^true an- 
 swer. But, 
 
 (5.) If the errors, as shown by the results, are unlike, 
 multijyly as before, and divide the sum of the products by 
 the SUM of the errors. The quotient will be the true 
 answer. 
 
 EXAMPLES. 
 
 1. A merchant expended ^1500 of his capital for the 
 support of his family a year. At the end of the year, 
 
RATIO. 
 
 277 
 
 however, lie has added to the capital not expended a sum 
 equal to 3 times that part, thereby tripling his original 
 capital. With what sum did he begin the year? 
 
 OPEKATION. 
 
 Pirst assumed No. $2000 $3000 
 1500 1500 
 
 (2.) 
 
 Second assumed No. 
 
 $500X3= 
 
 $500 
 $1500 
 
 $1500 
 $4500= 
 
 =$1500X3 
 
 Result too small, 
 Capital X3= 
 
 $2000 
 $6000 
 
 $6000 
 $9000 
 
 Result too small. 
 Capital X3. 
 
 $6000-$2000=First 
 
 error. $4000 
 
 3000 
 
 $3000=Second error. 
 2000 
 
 
 12000000 
 6000000 
 
 60000 
 
 00 
 
 
 1000)6000000 
 
 
 
 $6000 A 
 
 ns. 
 
 See Rule (4.) 
 
 (1.) 
 
 First assumed No. 
 
 SECOND OPERATION. 
 
 $2400 $9000 
 1500 1500 
 
 (2.) 
 
 Second assumed No. 
 
 
 $900 
 $2700 
 
 $7500 
 $22500 
 
 
 Result too small, 
 Capital X3= 
 
 $3600 
 $7200 
 
 $30000 
 $27000 
 
 Result too large. 
 Capital X3. 
 
 $7200— $3600=First 
 
 error. $3600 
 
 9000 
 
 $3000 Second error. 
 2400 
 
 
 32400000 
 7200000 
 
 720000 
 
 
 
 
 6600)39600000 
 
 See Rule (5.) 
 
 
 $6000 Ans. 
 
 
278 RATIO. 
 
 2. Divide tlic number 13 into two parts, so tliat 9 
 times one of the parts may be equal to 17 times the 
 other. 
 
 
 
 
 OPERATION. 
 
 
 
 
 
 (1-) 
 
 
 
 (2.) 
 
 
 
 Assume 10 
 9 
 
 and 
 
 3 
 17 
 
 
 Assume 7 
 9 
 
 and 
 
 6 
 
 17 
 
 90 
 51 
 
 
 51 
 
 
 63 
 
 
 102 
 63 
 
 .39 
 
 7 
 
 
 39 
 6 
 
 errors un 
 
 dike. 39 
 3 
 
 
 39 
 10 
 
 273 
 390 
 
 
 234 
 
 
 117 
 
 234 
 
 
 390 
 
 78)663 78)351 
 
 8^ one part. 4i other part. 
 
 3. Divide 100 into two parts, so that if \ of one of the 
 parts be subtracted from \ of the other, the remainder 
 may be 11. Ans. 24 and 76. 
 
 4. A and B had §85 between them, and f of A's 
 added to f of B's money make §60. How much had 
 each? Ans. A$50;B §35. 
 
 5. One half a certain number is the same as one third 
 another number. But if 5 is added to the first, and 10 
 to the second, then one fifth of the first is the same as 
 one eighth of the second. What are the numbers? 
 
 Ans. 20 and 30. 
 
 6. Triple a certain number is the same as double an- 
 other number. But if 10 is added to the first, and 5 
 subtracted from the second, then 5 times the sum will 
 be equal to 6 times the difi'erence. What are the num- 
 bers ? Ans. 20 and 30. 
 
■^'^l'^s>\^^j>-4..^. 
 
 281 
 
 7. In a mixture of wine and Avater, J the whole +25^ 
 gallons was wine; J the whole — 5 gallons was water. 
 What was the quantity of each in the mixture? 
 
 Ans. Wine 85; water 35 gallons. 
 
 8. A gentleman supported himself 3* years for 50 £ a 
 year, and at the end of each year added to that part of 
 his capital not thus expended a sum equal to | of this 
 part. At the end of 3 years his original capital was 
 doubled. With what sum did he begin business? 
 
 Ans. 74:0 £. 
 
 ""Eemark. — 200 £ and 290 £ are convenient suppositions. 
 
 9. A boy had a number of marbles. He laid aside 2, 
 and then Avon in play as many as he had left. He then 
 laid aside 3, and Avon in play as many as were left. He 
 noAv lays aside 4, and, on winning as many as he had 
 left, finds he has 13 in all. How many did he begin 
 with? Ans. 5. 
 
 10. A gentleman has tAVO horses, and a saddle worth 
 $50. If the saddle be placed on the first horse it makes 
 his value double that of the second ; but if the saddle 
 be placed on the second horse, it makes his \'alue triple 
 that of the first. . What was the value of each? 
 
 Ans. First $30 ; second $40. 
 
 Remark. — If the first horse is assumed to be worth §70, then by 
 the frst condition of the problem the second horse must be worth 
 ($504-§70)-r-2, or $60, and the second condition of the problem is 
 not filled by $100. 
 
 If the first horse is assumed to be worth $90, then the second 
 must be worth ($90-f$50)^2, or $70, and the second error will 
 be $150. 
 
 11. Divide 55 into tAvo parts, so that the less part di- 
 vided by the difference of the parts shall be 2. 
 
 Ans. 33 and 22. 
 
278 ,^ 
 
 RATIO. 
 
 12. A gentleman was asked the time of day, and re- 
 plied tliat I of the time past noon was equal to /g of 
 the time to midnight. What was the time? 
 
 An8. 12 minutes past 3 o'clock. 
 
 Remark.— In tlie first place f : -/j : : i : ^3 : : 11 : 4. (Vide 252.) 
 So that the reply was the same as if he had said that 11 times 
 the number of minutes past noon were equal to 4 times the number 
 of minutes to midnight. 
 
 13. In a mixture of corn and wheat, \ the whole -|-5 
 bushels was corn, and \ the whole +10 .bushels was 
 wheat. What was the quantity of each? 
 
 14. The sum of the first and second of three numbers 
 is 13; of the first and third 19; of the second and third 
 24. What are the numbers? An%. 4; 9; and 15. 
 
 15. A and B have the same income. A saves^ J of 
 his annually; but B, by spending |50 per annum more 
 than A, at the end of 6 years finds himself §150 in 
 debt. What is the income of each? 
 
 Am. $125 per year. 
 
 16. A commenced trade, and at the end of the third 
 year found his original stock tripled. Had his gains 
 been $1000 per year more than they actually were, he 
 would have doubled his stock each year. What was his 
 original stock ? An%. |1400. 
 
 17. Three persons. A, B, and C, were seen traveling 
 in the same direction. At first A and B were together, 
 and C 12 miles in advance of them. A goes 7, B 10, 
 and C 5 miles per hour. In what time will B be half- 
 way between A and C? How long before C will be mid- 
 way between A and B? How long since A was midway 
 between B and C? 
 
 Ann. Respectively, 1 h. 30 m., 3 h. 25 f m., and 12 h. 
 
INVOLUTION. 281 
 
 INVOLUTION. 
 
 2^72. (1,) The first poiver of a number is the num- 
 be'r itself. 
 
 (2.) The second power or square of a number is the 
 number multiplied by itself. Thus, the square of 1 is 
 1X1=1; the square of 2 is 2x2^4; the square of 3 
 is 3X3=9; the square of 25 is 25x25=625. 
 
 (3.) The tliird power or cube of a number is the pro- 
 duct of a number taken as a factor three times. Thus, 
 the cube of 1 is 1X1X1=1; the cube of 2 is 2x2x2 
 =8; the cube of 3 is 3x3x3=27; the cube of 4 is 
 4X4X4=64. 
 
 (4.) Any given power of a number is indicated by a 
 small figure placed at the right and a little above the 
 number. Thus, 2* denotes that the fourth power of 2 is 
 to be taken, and that 2 is to be taken as a factor 4 times. 
 
 The small figure is called the exponent of the number 
 or index of the poiver. 
 
 The expression 2^=16, is read fourth poiver of 2 
 equals 16. 
 
 5573. Involution is the operation of finding any 
 givcn_ pow^j ' Uf i" "^^1^^^ er. 
 
 GENERAL EULE. 
 
 3Iultiply the number by itself till it is used as a factor 
 as many times as there are units in the exponent. 
 
 Remark. — Fractions should always be in their lowest terms be- 
 fore multiplying. 
 
 24 
 
282 INVOLUTION. 
 
 EXAMPLES. 
 
 1. Find the values of 7^ 8^; 25^; and 125^ 
 
 (1.) 
 
 1st power, 7 
 7 
 
 OPERATIONS 
 
 (2.) 
 8 
 8 
 
 64 
 8 
 
 512 
 
 (3.) 
 25 
 25 
 
 625 
 
 25 
 
 (4.) 
 125 
 125 
 
 Square, 49 
 
 7 
 
 15625 
 125 
 
 Cube, 343 
 
 15625 
 
 1953125 
 
 2. Find the values of 11^; 12^; 13^; lOP; 111 2; 
 and 1111^ 
 
 3. Find the values of 14^; 15'; 16^; 17"^; 21*^; 24*; 
 and 27^ 
 
 4. Find the values of 15^; 25^; 35^; 45^; 55'^; 65^; 
 75^; 852; 952^ 
 
 Remark. — The value of these expressions may be found men- 
 tally. Thus, 
 
 152=10X20+25; 252=20X30+25; 352=30X40+25, etc. 
 
 5. Find the values of 15^; 25^; 35^; 45^; 55^; 65^; 
 753 ; 85^; 95^ 
 
 6. Find the values of 2*; 3*; 4*; 5*; 6^; 7^ 8^ 
 
 Ans. In Table, page 283. 
 
 7. Find the squares of f ; | ; £f^ ; and y'^g^^-. 
 
 ^'^'5. J; i; -,^g; and -J^, 
 
 8. Find the squares of |; |J; ||; and ^|. 
 
 Ans. 4; /g; ^1; and ^4^. 
 
 9. Find the squares of 2^; 6J ; 31; 16}/, and 51. 
 
 A71S. 157, Ex. 2, etc. 
 10. Find the cube of 5^ and the 4th power of S^. 
 
 Ans. 157, Ex. 28 and 29. 
 
INV'OLUTIOX. 
 
 283 
 
 11. Find the 4tli power of .025. 
 
 Ans. .000000390625. 
 
 12. Find the 5th power of .029. 
 
 Ans. .000000020511149. 
 
 13. Find the first nine powers of all the numbers 
 below 10. 
 
 TABLE. 
 
 
 02 
 
 o 
 
 
 o 
 
 n 
 
 I? 
 s 
 
 rji 
 
 a 
 
 H 
 
 
 
 2 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 1 
 
 4 
 
 9 
 16 
 25 
 36 
 49 
 64 
 81 
 
 1 
 
 8 
 27 
 G4 
 125 
 216 
 343 
 512 
 729 
 
 1 
 
 16 
 
 81 
 
 256 
 
 625 
 
 1296 
 
 2401 
 
 4096 
 
 6561 
 
 1 
 
 32 
 
 243 
 
 1024 
 
 3125 
 
 7776 
 16807 
 32768 
 59049 
 
 1 
 
 64 
 
 729 
 
 4096 
 
 15625 
 
 46656 
 
 117649 
 
 262144 
 
 531441 
 
 1 
 
 128 
 
 2187 
 
 16384 
 
 78125 
 
 279936 
 
 823543 
 
 2097152 
 
 4782969 
 
 1 
 
 256 
 
 6561 
 
 6553G 
 
 390625 
 
 1679616 
 
 5764801 
 
 16777216 
 
 43046721 
 
 1 
 
 612 
 
 19683 
 
 262144 
 
 1953125 
 
 10077696 
 
 40353607 
 
 134217728 
 
 387420489 
 
 274. Several useful facts may be gathered from the 
 table. 
 
 (1.) Any power of 1 is 1. 
 
 (2.) If the square of a number ends with 1, the num- 
 ber itself ends with 1 or 9. 
 
 (3.) If the square of a number ends w^ith 4, the num- 
 ber itself ends with 2 or 8. 
 
 (4.) If the square of a number ends with 6, the num- 
 ber itself ends with 4 or 6. 
 
 (5.) If the square of a number ends with 9, the num- 
 ber itself ends with 3 or 7. 
 
 (6.) If the cube of a number ends with either 
 
 1; 2; 3; 4; 5.; 6; 7; 8; 9, the number itself 
 ends with 1 ; 8; 7; 4; 5; 6; 3; 2; 9. 
 
284 INVOLUTION. 
 
 (7.) The 5th power, 9th power, 13th power, etc., of a, 
 number ends Avith the same figure as the number itself. 
 
 (8.) It is evident that if any power of a number ends 
 with a 0, the number itself ends with a 0. 
 
 575. Any powder of a number, multiplied by any 
 other power of the same number, produces a power 
 whose index is the ^ sum of the exponents of the given 
 number. Thus 3"^X3*^3^ and 7^X7'=7^ which may 
 be verified by the table. Hence, 
 
 576. To find a given power of a number, when all 
 the low^er powers are not wanted, 
 
 3Iultiply any tivo or more powers iogetlier^ the sum of 
 whose indices equals the index of the required p>oicer. 
 
 EXAMPLES. 
 
 1. Find the 9th power of 3. 
 
 Ans. 3-^X3*-=243x81-=19683 Ans. 
 
 2. Find the 15th power of 2. 
 
 Ans. 2^X2'--256X 128-32768 Ans. 
 
 3. Find the 13th power of 3. Ans. 3^ X 3*^=1594323. 
 
 4. Find the 20th power of 2. 
 
 Ans. 2^X29X22=1048576-=512X512X4. 
 2T7. Any number squared produces a number con- 
 sisting of exactly double the figures in the number itself, 
 or of one less Hum double the figures. For, 1 is the 
 smallest number consisting of one figure, and 9 is the 
 largest; 10 is tlie smallest number consisting of two 
 figures, and 99 is the largest; 100 is the smallest of 
 tliree figures, and 999 the largest. Now we have 
 
 V^l ; 10---100 ; 100- -10000 ; 
 and 9---81 ; 99^=9801 ; 999^-998001 ; etc. 
 
\ 
 
 EVOLUTION. 285 
 
 ST8. Any number cubed produces a number consist- 
 ing of exactly triple the figures in the number itself, or of 
 one or two less than triple the figures. For, 
 
 P-= 1 ; 10^=1000 ; lOO^^-^lOOOOOO ; 
 
 d'=^n9 ; 99^--970299 ; 999'^=997002999. 
 
 EVOLUTION 
 
 S79. Any given root of a number is the number 
 whicli, on being taken as a factor as many times as is 
 indicated by the name of the root, will produce the 
 number itself. Thus : 
 
 (1.) The^Vs^ root of a number is the number itself. 
 
 (2.) Th'e second or square root of a number is one of 
 its two equal factors. Q^hus : the square root of 1 is 1, 
 because 1X1=1; the square root of 4 is 2, because 
 2X2=4; the square root of 9 is 3, because 3x3=9. 
 
 (3.) The cube or third root of a number is one of its 
 three equal factors. Thus : the cube root of 1 is 1, be- 
 cause 1X1X1=1; the cube I'oot of 8 is 2, because 
 2X2X2=8; the cube root of 27 is 3, because 3X3X 
 3=27; and the cube root of 64 is 4, because 4X4X 
 4=64. 
 
 (4.) The 4th; 5th; 6th, etc., root of a number is one 
 of its 4; 5; 6, etc., equal factors. Thus: the 4th root 
 of 1 is 1, because 1x1X1X1=1 ; the 4th root of 625 
 is 5, because 5x5X5X5=625; the 5th root of 7776 is 
 6, because 6x6x6x6x6=7776. 
 
 (5.) A perfect square is a number which has an exact 
 
286 EVOLUTION. 
 
 SQUARE BOOT. Tlius, 16 ; 64; 81; and 144 are perfect 
 squares. 
 
 (6.) A perfect cube is a number wliicli has an exact 
 cube root. Thus, 8 ; 27 ; 64, etc., are perfect cubes. 
 
 (7.) Any given root of a number is indicated by a 
 fractional exponent, or by the aid of the character ]/, 
 called the radical sign. Thus : 
 
 The square root of 16 is indicated by 16^, or by 
 
 l/l6; the cube root of 8 is indicated by 8% or by V'8; 
 
 the 7th root of 2187 is indicated by 2187^ or by f 2187. 
 The figure at the left of the radical sign is called the 
 index of the root. 
 
 (8.) A surd is a number which requires a radical sign 
 
 or exponent to exactly express it. Thus, l/2; 1^4; 5^ 
 are surds. 
 
 280. Evolution is the operation of finding any 
 given root of a number. 
 
 SQUARE ROOT. 
 
 281. The extraction of the square root of a number is 
 the operation of finding a number which multiplied by 
 itself will produce the given number. 
 
 282. The square root of any integral number which 
 is a perfect square and less than 400, may be found from 
 memory, or by inspecting Table I of the Appendix. 
 
 c 
 
 EXAMPLES. 
 
 1. What is the square root of 1? 4? 9? 16? 25? 
 36? (Vide 273, Ex. 13.) 
 
EVOLUTION. 
 
 287 
 
 2. What is the square root of 49? 64? 81? 100? 
 121? 
 
 3. What is the square root of 196? 324? 361? 169? 
 
 4. What is the square root of 225? 289? 256? 400? 
 
 IIemark, — All the integral numbers less than 400, other than 
 those in the last four examples, have only approximate square roots; 
 that is, their square roots are surds. The same is true of all 
 numbers not perfect squares. 
 
 S83. The general method of extracting the square 
 root of a number may be readily understood from the 
 following operations. We will first square some num- 
 ber, as 37. 
 
 OPERATION FIRST. 
 
 (1.) (2.) 
 
 37 30+7 
 
 37 30+7 
 
 49 
 210 
 210 
 900 
 
 1369 
 
 30x7+7^ 
 302+30X7 
 
 30^+2x30x7+7^ 
 
 In (2) it is evident that 30x7 added to 30X7 is twice 
 30X7, or 2X30X7. To reverse these steps, that is, to 
 extract the square root of 1369, we proceed as follows: 
 
 OPERATION SECOND. 
 
 1369 1 37 30 
 9 ^ 
 
 469 2x30+7 
 
 469 
 
 
 
 30^+2x30x7+7^ 30+7 
 302 
 
 2x30X7+7' 
 2x30x7+7' 
 
 67 
 
 Explanation, — By putting a point or dot over every alternate 
 figure of the given number, commencing with the right-hand 
 
288 EVOLUTION. 
 
 figure, we not only can determine the number of figures of wliicli 
 the root will consist, (vide 277,) but we can also determine the 
 left-hand figure of the root; for the square root of the largest per- 
 fect square, less than 13, must be that figure. Now 9 is the largest 
 perfect square less than 13, and its square root is 3. This 3 is 
 written at the right and also at the left of the given number. 
 
 Having taken 9 from 13, the remainder, 469, is the same as 2X 
 30X7-j-7^. Now if we double the 3 just obtained, it in eifect pro- 
 duces 60, which is 2X30. If we now divide 469 by 60, and annex 
 the quotient to 6, and also to. the right of the root, as Operation (2) 
 points out, and then multiply 67 by 7, the exact remainder, 469, 
 will be produced whenever the given number is a perfect square. 
 Hence, 
 
 284. To extract the square root of any integral 
 number, 
 
 (1.) Divide the number into periods of ttvo figures each, 
 counting from right to left. 
 
 (2.) Fbid the greatest perfect square in tJie left-hand 
 period, and 'place its square root at the left and also at 
 the right of the given number. Subtract this greatest 
 perfect square from the left-hand p)eriod, and to the re- 
 mainder annex the next period to the right; the result is 
 the First Dividend. 
 
 (3.) Double the figure of the root just found and place 
 it to the left of the first dividend. See how many times 
 tills double figure, with, a cipher mentally annexed, is con- 
 tained in the first dividend. Place the quotient figure as 
 the second figure of the root, and also annex it to double 
 tJt^ first figure as the First Divisor. 
 
 (4.) Multiply the first divisor by the second figure of 
 the root, and subtract the ijroduct from the first dividend, 
 and to the remainder annex the next period to the right 
 for the Second Dividend. 
 
EVOLUTION. 
 
 289 
 
 (5.) Double iJie part of the root now found and place 
 it to the left of the second dividend ; see how many ti7nes 
 this double root, luith a cipher mentally annexed, is con- 
 tained in the second dividend, and place the quotient at 
 the third figure of the root, and also to the right of double 
 the previous pa7i of the root, as the Second Divisor. 
 
 Proceed with the second divisor as with the first, and 
 continue the operation till all the periods have been used. 
 
 Remark 1. — Double the root already found is sometimes called 
 the trial divisor. 
 
 Remark 2. — It sometimes happens, as in Operation (2) below, 
 that in making the division for the next figure of the root the 
 quotient is too large. In such a case make the quotient figure so 
 small that the product will be less than the dividend. 
 
 Remark 3. — It sometimes happens, as in Ex. 15, Operations (1) 
 and (2) below, that a dividend is too small to contain the trial 
 divisor. In such a case place a cipher in the root and to the right 
 of the trial divisor, and then annex the 7iezt period to the dividend. 
 
 Remark 4. — In the case of decimals, the first two figures on the 
 right of the decimal point constitute one period, the next two 
 figures another, and so on. 
 
 EXAMPLES. 
 
 1. Find the square roots of 6084; 1521; and 98.01. 
 
 148 
 
 6084178 Ans. 
 49 
 
 1184 
 1184 
 
 OPERATIONS. 
 
 (2.) 
 
 152i|39 Ans. 
 9 
 
 69 
 
 621 
 621 
 
 18.9 
 
 (3.) 
 
 98.0i|9.9 Ans. 
 81 
 
 17.01 
 17.01 
 
 2. Find the square roots of 2025; 3969; 6241; and 
 6561. Ans. 45; 63; 79; and 81. 
 
 25 
 
290 EVOLUTION. 
 
 3. Find the square roots of 1936; 2401; 8281; and 
 8836. Ans. 44; 49; 91; and 94. 
 
 4. Find the square roots of 10.24; 10.89; 11.56; 
 12.25. Ans. 3.2 ; 3.3 ; 3.4 ; 3.5. 
 
 5. Find the square roots of 6400 ; 2500 ; 841 ; and 
 729. Ans. 80 ; 50; 29; an d 27. 
 
 6. What is the value of i/8281+t/6561+i/50.41. 
 
 Ans. 179.1. 
 
 7. What is the value of 1/10:89+^1444+1/18.49. 
 
 Ans. 45.6. 
 
 8. What is the value of t/5776— 1/40.96. 
 
 Ans. 69.6. 
 
 9. What is the value of l/5625— 1/3025. 
 
 Ans. 20. 
 
 10. What is the value of V 9025— l/l225. 
 
 Ans.-60. 
 
 11. What are the square roots of 21904 and 3564.09. 
 
 OPERATIONS. 
 
 1 
 
 21904 1 
 1 
 
 .48 Ans. 
 the square 
 
 5 
 
 109 
 
 118.7 
 
 (2.) 
 
 3564.09 59.7 ^ws. 
 25 
 
 24 
 
 119 
 96 
 
 1064 
 981 
 
 288 
 
 2304 
 2304 
 
 83.09 
 83.09 
 
 12. What are 
 
 roots of 
 
 332929 and 467856. 
 ins. 577 and 684. 
 
 13. Find the value of t/473344+i/48.8601. 
 
 Ans. 694.99. 
 
EVOLUTION. 291 
 
 14. Find the value of 1-/44.0896+1/3080.25. 
 
 Ans. 62.14. 
 
 15. Find the square roots of 366025 and 49126081. 
 
 OPERATIONS. 
 
 (1.) (2.) 
 
 6 
 1205 
 
 366025 I 605 Ans. 7 
 
 36 
 
 6025 14009 
 6025 
 
 49126081 1 7009 Ans. 
 49 
 
 126081 
 126081 
 
 16. Find the square roots of 259081 and 826281. 
 
 A71S. 509 and 909. 
 
 17. Find the square roots of 49.4209 ; 404.01 ; and 
 .822649. Ans. 7.03 ; 20.1 ; and .907. 
 
 18. Find the square roots of 12321 ; 1234321 ; and 
 123454321. 
 
 19. Find the square roots of 49284 and 4937284. 
 
 20. What is the value of l/ 110889— l/ 40376081. 
 
 Ans. 312.91. 
 
 21. Find the square root of .0011943936. 
 
 Ans. .03456. 
 
 22. Find the square roots of 99980001 and 9999- 
 800001. 
 
 23. Find the square root of 152399025. 
 
 Ans. 12345! 
 
 24. Find the square root of 2950771041. 
 
 Ans. 54321. 
 
 25. Find the square root of 8264446281. 
 
 26. Find the square root of 6529932864. 
 
 27. Find the square root of 4999479849. 
 
292 
 
 EVOLUTION. 
 
 28. Find the square root of 2, and also of 3, to 7 
 decimal places. 
 
 OPERATIONS. 
 
 (1-) (2.) 
 
 1 
 
 2.00000011.4142136 
 
 1 
 
 : 1 3.000000|1.7320508 
 
 2.4 
 
 1.00 
 96 
 
 400 
 
 281 
 
 2.7 
 
 3.43 
 
 3.462 
 
 2.00 
 
 1.89 
 
 2.81 
 
 1100 
 1029 
 
 2824 
 
 11900 
 11296 
 
 7100 
 6924 
 
 2.8282 
 
 ) 60400 
 56564 
 
 3.4640 
 
 ) 17600 
 17320 
 
 3836 
 2828 
 
 280 
 
 272 
 
 1008 
 848 
 
 
 
 160 
 168 
 
 
 
 Remark. — After finding four figures of the root as usual, the re- 
 maining figures are found by simply dividing the last dividend by 
 the last divisor, except that, instead of annexing ciphers to the 
 dividend, we take away a figure at the right of the divisor at each 
 new figure of the root. Attention is of course paid to the rules for 
 division of decimals. In general, 
 
 Having found one more than half the required figures of the root in 
 the ordinary way, the remaining figures may he found by dividing the last 
 dividend hy the last divisor, carefully observing the rules in division of 
 decimals. 
 
 In this manner Table II of the Appendix may be verified. 
 
EVOLUTION. 293 
 
 285. The square root of the product of two numbers 
 is the same as the product of the square roots of the 
 numbers. Thus : 
 
 l/4x9 is the same as l/4 Xv'9 ; also V2 xS=V2 xVB, 
 as may be easily verified. 
 
 S8G. To find the square root of a composite number, 
 Find the product of the roots of its factors, mahing 
 one of the factors a perfect square if possible. 
 
 EXAMPLES. 
 
 '1. What is the square root of 8? 
 ■■ We have t/8= 1/4x^ = 1/4 X"l/2 = 2Xl/2. 
 ' N'ow t/2=i1.4142136. Hence, l/8=2X1.4142136. 
 
 Ans. 2.8284272. 
 
 2. Find the values of l^FS; VM; VM', and 1/72. 
 
 Ans. 4.2426408, etc. 
 
 3. Find the values of l/'98 ; l/l28 ; VlQ2 ; and 1/200. 
 
 Ans. 9.899495, etc. 
 
 4. Find the values of Vl2', VVJ; t/48; and VJb. 
 
 Ans. i/4x3=2Xt/3=3.464102. 
 
 5. Find the values of 1/IO8; l/l47; T^IM; and 
 1/243. Ans. 10.392305, etc. 
 
 6. Find the values of l/20 ; 1^28; 1^99; and 1/8O. 
 
 t/20=2Xi/5; t/28=2X>^7; T/99=3X"/li; VW):^4:XV^. 
 
 7. Find the values of 1/120 ; l/270 ; 1/8OO ; and 
 1/450. Ans. 2X1/30=10.954451, etc. 
 
 Remark.— In practice reject as many figures on the right of the 
 product of the tabulated decimal as are found in the root of the 
 factor which is a perfect square. 
 
294 EVOLUTION. 
 
 287. The square root of a fraction is the square root 
 of the numerator divided bj the square root of the de- 
 nominator. Thus : 
 
 The square root of | is -7^ = 1, because |X|=|. 
 
 288. To find the square root of a fraction, 
 
 (1.) If necessary, multiply both terms of the fraction 
 hy the smallest number that will make the denominator a 
 perfect square. 
 
 (2.) Divide the square root of the resulting numerator 
 by that of the denominator^ for the required root. 
 
 EXAMPLES. _^ 
 
 1. What is the square root of J ? ^ / ' ^ 
 
 „, , ^ /_ ._ 1/2 1/2 1.4142L36 
 We have J=f ; then V | = l/f -:^ — ^= -— 
 
 =.7071068. 
 
 2. What are the square roots of J; J ; 4; ^^1 
 
 -^Tis. 1x1/3^.5773502. 
 
 3. What are the square roots of f ; | ; f ; i ? (Vide 
 
 286, Ex. 6. Ans. ^^^^ =.8944272, etc. 
 
 4. What are the square roots of f ; | ; 4 > I '• j 
 
 ^Tis. ]^=.9258201, etc. 
 
 5. What are the square roots of 2| ; 3 J ; 4§ ; 2 J ? 
 
 ^ns. 2^=1.5811388^ etc. 
 
 6. What are the square roots of 30/(j; 11/^; Hi?; 
 and 6|f ? Jtzs. 5'; sf; 3|.; and 2|. 
 
 7. What are the square roots of 2| ; 18 J^; 272^; 
 51 A? ^ns.li; 41; 5^; 74. 
 
EVOLUTION. 205 
 
 CUBE II GOT. 
 
 289. The extraction of the cube root of a number is 
 the operation of finding one of itfe three equal factors. 
 
 290. The cube root of any integral number which is 
 a perfect cube and less tlian 1000, may be found from 
 memory, or by inspecting the third column of the Table 
 under 273, Ex. 13. 
 
 EI^AMPLES. 
 
 1. What are the cube roots of 1 ; 8; 27; 64; 12^? 
 
 2. What are the cube roots of 216 ; 343 ; 512 ; and 
 
 729? •^ 
 
 Remark. — All the integral numbers less than 1000, other than 
 those in the above two examples, have only approximate cube roots, 
 that is, their cube roots are surds. (Vide 279, (8.) The same is 
 true of all numbers not perfect cubes. 
 
 291. The general method of extracting the cube root 
 of a number may be drawn from the following opera- 
 tions. We will cube the number 37 by continuing 
 Operation (2) of 283. 
 
 
 
 OPERATION FIRST. 
 
 37^= 
 
 1369= 
 
 30^+2x30x7+72 
 
 37- 
 
 .37= 
 
 30+7 
 
 30^+2X30^x7+30x7^ 
 
 302x7+2x30x7-f7^ 
 
 37^= 50653= 303+3x30^X7+3x30X7^+7^ 
 
 In the operation, the quantity 30^+2X30X7+7^ is 
 first multiplied by 30 and then by 7, and the two indi- 
 cated products are added together. It is evident that 
 
296 
 
 EVOLUTION. 
 
 twice 30^X7 added to once 30^X7 is the same as three 
 times 30^X7, that is, the sum is 3x30'X7; also, that 
 once 30X7^ added to twice 2>^Y.1^ is three times 30 X7^ 
 that is, the sum is 3X30x7^. 
 We will now reverse these steps. 
 
 1st Col. 
 30 
 
 2X30 
 3X30+7 
 
 OPERATION SECOND. 
 
 2d Col. 
 302 
 
 3X30= 
 3X30^-1-3X30X7+72 
 
 Quantity. Root. [30+7 
 
 30^+3X302X7+3X30X72+78 
 30^ 
 
 3X302X7+3X30X72+73 
 3X302x7+3X30X72+73 
 
 The above operation contracted is as follows : 
 
 OPERATION THIRD. 
 
 Number, Root, 
 
 50653 I 37. 
 
 27 
 
 1st Col. 
 
 2d Col. 
 
 3 
 
 6 
 97 
 
 9 
 
 2700 
 
 3379 
 
 . 
 
 
 23653 
 23653 
 
 The steps in Operation Second are as folloAvs: 
 Having arranged the quantity so as to form two 
 columns on the left, with 30 in the 1st col., its square 
 in the 2d col., and its cube under the quantity itself. 
 We now indicate double the 30 of the 1st col., and write 
 it under 30 of the same column. Next multiply this 
 2X30 by 30, which gives 2X30^, and add the product to 
 30^ of the second column, and write the indicated sum 
 or 3x30^ under 30^ in the same column. Next indicate 
 the triple of the 30 in the 1st col., and write the indi- 
 
EVOLUTION. 297 
 
 oated product, viz., 3x30, under 2X30 of the 1st col. 
 If we now divide the first term of the remainder, viz., 
 3X30^X7, by 3x30-, the second term of the 2d col., 
 the quotient will of course be 7, the second term of the 
 root. This 7 is added to the right of 3x30 of the 1st 
 col., making 3x30+7, which is then multiplied by 7, 
 and added to 3x30^ of the 2d col., giving 3x30^+3 
 X30x7+7^ which is now multiplied by 7, and the pro- 
 duct is the same as the remainder, after subtracting 30'^ 
 from the quantity itself. 
 
 The indicated products and additions of Operation 
 Second are actually made in Operation Third, omitting 
 ciphers, which would be of no service if written down. 
 Hence, 
 
 !S9S. To extract the cube root of any integral 
 number, 
 
 (1.) Divide the number into periods of three figures 
 each, counting from right to left. The left-hand period 
 may consist of one, two, or three figures. (Vide 278.) 
 
 (2.) Find bi/ 290 the largest perfect cube less than the 
 left-hand period ; place its cube root to the right of the 
 number J and also to the left, as the first term of the 1st 
 col. ; square the first term of the 1st col., and write it as 
 the first term of the 2d col.; cube the first term of the 
 1st col., and subtract it from the left-hand period of the 
 number, and annex the next period to the right. The 
 result is the first dividend. Add the figure in the 1st 
 col. to itself, and lurite the sum as the second term of the 
 1st col.; midtiply the second term of the 1st col. by the 
 figure in the root, and add the product to the first term 
 of the 2d col.; the sum with two ciphers annexed is the 
 
298 EVOLUTION. 
 
 second term of the 2d col. Add the figure in the root 
 to the second terin of the 1st col., and place the sum under 
 the second term. 
 
 (3.) See hoiv many times the second term of the 2d 
 col. is contained in the fikst dividend, and write the 
 quotient as the second figure of the root, and also to the 
 right of the last number in the 1st col., forming its third 
 term; multiply the third term of the 1st col. by this 
 second figure of the root, and add the product to the second 
 term of the 2d col. as its third ter7n; multiply this third 
 term by the second figure of the root, and subtract the p)ro- 
 ducf from the first dividend, and annex the next period of 
 the number for the second dividend. 
 
 Proceed with the second figure of the root precisely 
 as with the first, and continue the operation till all the 
 periods have been used. 
 
 Remark 1. — It often happens that in making the division for 
 the figure of a root the quotient will be found too large. In such 
 a case erase the work as far back as that point, and make the root 
 figure so small that the subtrahend will be less than the dividend. 
 (Vide Ex. 13.) 
 
 Remauk 2. — If a dividend is too small to contain the divisor, 
 place a cipher in the root, and also to the right of the last term 
 of the first column, and two ciphers to the last term of the second 
 column. (Vide Ex. 19.) 
 
 Remark 3. — In the case of decimals, the first three figures on the 
 right of the decimal point constitute one period, the next three 
 figures another, and so on. 
 
 Remark 4. — This method of taking the cube root is so superior, 
 it is not a little surprising that it has not been adopted, to the ex- 
 clusion of all others. Aside from its value in this connection, it 
 prepares the student for an easy understanding of the best mode 
 of finding the numerical values of the roots of equations above 
 the second degree in Algebra. 
 
EVOLUTIOX. 
 
 299 
 
 EXAMPLES. 
 
 1. Find the cube root of 753571. 
 
 OPERATION. 
 
 9 
 
 18 
 
 271 
 
 81 
 
 24300 
 
 24571 
 
 753571 I 91 Ans. 
 
 729 
 
 24571 
 24571 
 
 2. Find the cube roots of 531441 and 357911. 
 
 Ans. 81 and 71. 
 
 3. Find the cube roots of 132651 and 72507. 
 
 A71S. ^r^ 43. 
 
 4. Find the cube roots of 970299 and 681472. 
 
 Ans. 99 and 88. 
 
 5. Find the cube roots of 456533 and 287496. 
 
 Ans. 77 and 66. 
 
 6. Find the cube roots of 6.859 and 24.389. 
 
 Ans. 1.9 and 2.9. 
 
 7. Find the value of 1^704969+1^421875. 
 
 Ans. 164. 
 
 8. Find the value of 1^884736+1^314432. 
 
 Ans. 164. 
 
 9. Find the value of 1^110592—1^^103823. 
 
 Ans. 1. 
 
 10. Find the value of f" 91125 Xl^4287o. 
 
 An.s. 1575. 
 
 11. Find the value of 1^456533—1^421875. 
 
 Ans. 2. 
 
 12. Find the value of 1^4096X1^15625. Am. 400. 
 
300 EVOLUTION. 
 
 13. Find the cube root of 2863288. 
 
 OPERATION. 
 
 1 
 
 2 
 
 34 
 38 
 422 
 
 1 
 
 300 
 436 
 58800 
 59644 
 
 2863288 [142 
 1 
 
 1863 
 1744 
 
 119288 
 119288 
 
 14. Find the cube roots of 2299968 and 2352637. 
 
 -^ Am. 132 and 133. 
 
 15. Find the cube roots of 10793861 and 14526784. 
 
 Am. 221 and 244. 
 
 16. Find the cube roots of 36926037 and 63521199. 
 
 Am. 333 and 399. 
 
 17. Find the cube root of 212776173. Am. 597. 
 
 18. Find the cube root of 997002999. Am. 999. 
 
 19. Find the cube root of 743677416. 
 
 OPERATION. 
 
 9 
 
 18 
 2706 
 
 81 
 
 2430000 
 2446236 
 
 743677416 I 906 
 
 729 
 
 14677416 
 14677416 
 
 20. What is the cube root of ip30301? Am. 101. 
 
 21. What is the cube root of 128787625? 
 
 22. What is the cube root of 225866,529 ? 
 
 Am. 609. 
 
EVOLUTION. 
 
 101 
 
 23. Find the cube root of 2 to seven places of 
 decimals. 
 
 OPERATION. 
 
 2 |1.2599211 An90:, 
 
 1 
 
 1 
 
 2 
 
 300 
 
 32 
 
 364 
 
 34 
 
 43200 
 
 365 
 
 45025 
 
 370 
 
 4687500 
 
 3759 
 
 4721331 
 
 3768 
 
 475524300 
 
 37779 
 
 4758)64311 
 
 1.000 
 
 .728 
 
 272000 
 225125 
 
 46875000 
 42491979 
 
 4383021000 
 
 4282778799 
 
 10024)2201 
 9517 
 
 507 
 475 
 
 32 
 47 
 
 We proceed in the usual way, and find 1.2599, at 
 which point divide the dividend 10024 by 4758, reject- 
 ing other figures on the right as not afi'ecting the result. 
 In general, having found one more than half the figures 
 of the decimal in the ordinary way, divide the last divi- 
 dend hy the last term of the 2d col., rejecting from the 
 right of the dividend one figure less than from the right 
 of the divisor. In this manner Table III of the Ap- 
 pendix may be verified. 
 
 24. Find the cube roots of 3, 4, etc., to 20. 
 
302 EVOLUTION. 
 
 29S. The cube root of the product of two numbers 
 is the same as the product of the cube roots of the num- 
 bers. Thus : 
 
 1^8x27-1^8 X1^27, and f'2xE=f2xf'S, 
 as may be verified from the Table. Hence, 
 
 294. To find the cube root of a composite number, 
 Find the product of the cube roots of its factors, makiiig 
 
 one of the factors a perfect cube, if possible. 
 
 EXAMPLES. 
 
 1. What is the cube root of 16? We have #"^16= 
 #"8X2=1^8 X#'2 =2X1^2. Now, if 2 -=1.2599211. 
 Hence, 1^16=2X1.259921. Ans. 2.519842. 
 
 2. Find the values of #"24^ 1^321 1^40^ and 1^48. 
 fSxf'^; 1^8X1^4; 1^8X1^5; 1^8X1^6. 
 
 Ans. 2.8845, etc. 
 
 3. Find the values of 1^54; 1^128; #'250. 
 
 Ans. #^27 X^'g -3.779763. 
 
 4. Find the values of ^88^ ^297"; and ^704. 
 (Vide 286, Rem.) Ans. 2X^11=4.44796. 
 
 295. The cube root of q> fraction is the cube root of 
 the 7iumerator divided by the cube root of the denomi- 
 
 nator. Thus, the cube root of /^ is -— ^ = |, because 
 
 F 27 
 
 fX|X|=3V Hence, 
 
 296. To find the cube root of a fraction, 
 
 (1.) If necessary, multiply both terms of the fraction 
 by the smallest number that ivill make the denominator a, 
 perfffct cube. 
 
EVOLUTION. 303 
 
 (2.) Divide the cube root of the resulting numerator hy 
 that of the denominator for the required root. 
 
 EXAMPLES. 
 
 1. What is the cube root of i? We have 
 
 1=^ Then irT_iri=^4_f4 _]L587401_ 
 
 .7937005 Ans. 
 
 2. What are the cube roots of | ; J ; J ; and I ? 
 
 Ans. 1X1^9 =.693361, etc. 
 
 3. What are the cube roots of f ; |; |; and li? 
 
 Jxrg^ Jxiri2i fxif 9] ixif To: 
 
 ^ns. .908561, etc. 
 
 4. What are the cube roots of | ; | ; y^g ; | ; -J| ? 
 
 ^?is. -I X 1^18 =.87358. 
 
 5. What are the cube roots of ff ; iff; ||f ' 
 
 Remark. — When the root of a perfect square, cube, fourth power, 
 etc., contains no more than three figures, it may be taken mentally, 
 after a little practice, and on observance of the points in sec. 274. 
 All roots of integral numbers containing not more than six figures, 
 decimals and fractions, are easily found by a common Table of 
 Logarithms. (Algebra, 152.) 
 
 PROBIiEMS. 
 
 29T. The sum of two numbers and their product 
 being given, to find the numbers. 
 
 Find the square root of four times the product 
 SUBTRACTED FROM the Square of the sum. This root 
 will be the diiference of the numbers. (Vide 125, 
 Ex. 10.) 
 
 V 
 
304 EVOLUTION. 
 
 EXAMPLES. 
 
 1. The sum of two numbers is 105 and their product 
 2666. What are the numbers? l/l052— 4X2666=19. 
 
 Ans. 62 and 43. 
 
 2. The sum of two numbers is 10 and their product 
 24. What are the numbers? Ans. 6 and 4. 
 
 3. The sum of two numbers is 8j and their product 
 17 J. What are the numbers? A71S. 5 and 3|. 
 
 4. The sum of two numbers is 28 and their product 
 196. Wliat are the numbers? Ans. 14 and 14. 
 
 S98. The difference of two numbers and their pro- 
 duct being given, to find the numbers, 
 
 JFind the square root of four times the product added 
 TO the square of the difference. This root is their sum. 
 
 EXAMPLES. 
 
 1. The difference of two numbers is 19 and their pro- 
 duct 2666. What are the numbers ? 
 
 y 192+4x2666=105. Ans. 62 and 43. 
 
 2. The difference of two numbers is 3 and their pro- 
 duct 40. What are the numbers? Ans. 8 and 5. 
 
 3. The difference of two numbers is 7 and their pro- 
 duct 294. What are the numbers ? Ans. 21 and 14. 
 
 4. The difference of two numbers is 3^ and their pro- 
 duct 73 J. What are the numbers? Ans. 10 J and 7. 
 
 5. The sum of two numbers is 23j and their product 
 135. What are the numbers? Ans. 11} and 12. 
 
 6. The difference of two numbers is J and their pro- 
 duct 135. What are the numbers ? 
 
ARITHMIiTICAL PROGRESSION. 305 
 
 ARITHMETICAL PROGRESSION. 
 
 299. An Arithmetical Progression is a series of 
 numbers in which any term is found by adding a given 
 number to the preceding term, or by subtracting a given 
 number from tJie precediyig term. 
 
 (1.) The common difference is the number to be added 
 or subtracted. 
 
 (2.) The progression is increasing when the common 
 difference is added. 
 
 (3.) The progression is decreasing when the common 
 difference is subtracted. 
 
 300. In every progression these five points may be 
 considered, viz., the first term, the last term, the common 
 difference, the number of terms, and the sum of all the 
 terms. Thus, in the arithmetical progression, 
 
 1, 3, 5, 7, 9, 11, 13, 
 the first t'erm is 1, the last term 13, the common differ- 
 ence 2, the number of terms 7, the sum of all the terms 
 49, and the progression is increasing. 
 In the decreasing progression, 
 
 22, 19, 16, 13, 10, 7, 4, 1, 
 the first term is 22, the last term 1, the common differ- 
 ence 3, the number of terms 8, and the sum of all the 
 terms 92. 
 
 (1.) The first and last terms are sometimes called 
 extremes, and the intermediate terms the means. The 
 means must of course be less by 2 than the number of 
 terms. 
 
 20 
 
306 AEITHMETICAL PROGRESSION. 
 
 (2.) In a progression of three terms, the middle term 
 is called the arithmetical mean, and is half the extremes. 
 
 301. To find the last term, when the first term, 7mm- 
 her of terms, and common difference are known. 
 
 EXAMPLES. 
 
 1. The first term of a progression is 4, common dif- 
 ference 3, number of terms 50. What is the last term? 
 
 ANALYSIS. 
 
 If we take a few terms of the progression ; thus, 
 1st. 2d. 3d. 4th. 5th. 
 
 4, 4+3, 4+2x3, 4+3x3, 4+4x3; 
 that is, 4 7 10 13 16; 
 
 we see that any term, as, for instance, the 5th, is found 
 by adding to the 1st the product of the common differ- 
 ence into a number less by 1 than that denoting the 
 term. If we should continue the progression to the 
 50th term, the quantity standing under 50th Avould 
 evidently be 4+49x3, and the 50th term would there- 
 fore be 151. Hence, 
 
 When the progres'sion is increasing, 
 
 (1.) To the first term add the product obtained by mul- 
 tiplying the common difference by the number of terms 
 less 1. 
 
 2. In the progression 2, 4, 6, 8, etc., what is the 20th 
 term? 40th term? 60th? 71st? 100th? 
 
 Ans. 40, 80, etc. 
 
 3. In the progression 3, 11, 19^ 27, etc., what is the 
 50th term? 80th? 150th? 200th? 500th? 
 
 Ans. 395, 635, etc. 
 
ARITHMETICAL PROGRESSIOX. 307 
 
 4. In the progression 7, 11, 15, etc., what is the 10th 
 term? 45th? 75th? 101st? Ans. 43, 183, etc. 
 
 5. In the progression 8, 11, 14, etc., what is the 12th 
 term? 24th? 47th? 81st? 1000th? 
 
 Last Ans. 3005. 
 
 6. In the progression 1, 1^, 2, 2|, etc., what is the 
 900th term? 1200th term? 1500th term? 
 
 Ans. 450 1. 
 
 7. In the progression 5, 5-|, 5f , 6, etc., w4iat is the 
 45th term? 90th? 750th? 100th? Ans. 19f, etc. 
 
 When the progression is decreasing, 
 
 (2.) From the first term subtract the product obtained 
 by 7nultiplying the common difference by the number of 
 terms less 1. 
 
 8. In the progression 500, 496, 492, etc., what is the 
 20th term? 25th? 30th? 50th? ^^is. 424, etc. 
 
 9. In the series 320, 318, 316, etc., what is the 10th 
 term? 21st? 31st? 41st? Ans. 302, etc. 
 
 10. In the series 412, 409, 406, etc., what is the 15th 
 term? 85th? 45th? Ans. 370, etc. 
 
 11. In the series 100, 95, 90, etc., what is the 12th 
 term? 15th? 20th? 21st? Last J[?is. 0. 
 
 302. To find the common difference^ when the ex- 
 tremes and the numbers of terms are known. 
 
 Divide the difference of the extremes by the number of 
 terms less 1. 
 
 EXAMPLES. 
 
 1. The extremes of a progression are 4 and 151; 
 number of terms 50. What is the common diiference? 
 What is the series? (151— 4)--(50— 1)=3, the common 
 difference. The series is then 4, 7, 10, etc. 
 
308 AlHTHMETICAL PROGRESSION. 
 
 2. The extremes are 2 and 40 ; number of terms 20. 
 What is the series? Common difference is 2. 
 
 Ans. 2, 4, 6, 8, etc. 
 
 3. The extremes are 8 and 395 ; number of terms 50. 
 What is the series? Ans. 3, 11, 19, etc. 
 
 4. Insert 8 means between the extremes 7 and 43. 
 (Vide 300, (1.) 
 
 Ans. 7, 11, 15, 19, 23, 27, 31, 35, 39, 43. 
 
 5. Insert 10 means between 8 and 41. 
 
 A71S. 8, 11, 14, etc. 
 
 6. Insert 18 means between 500 and 424, 
 
 A71S. 500, 496, etc. 
 
 7. Insert 8 means between 320 and 302. 
 
 8. Insert 2 means between 4 and 14. 
 
 Ans. 4, 7J, lOf, 14. 
 
 9. Insert 3 means between 1 and 2. 
 
 Ans. 1, 1J-, 11, If, 2. 
 
 10. Insert 5 means between 3 and 4. 
 
 Ans. 3, 31, 3|, etc. 
 
 11. Insert 7 means between 3 and 6. 
 
 Ans. 3, 3|, 3-|, etc. 
 
 12. Insert 5 means between 1 and 13. 
 
 Ans. 1, 3, 5, 7, 9, 11, 13. 
 
 14. What is the arithmetical mean between 4 and 10? 
 (Vide 300, (2.) Ans. 7. 
 
 15. What is the arithmetical mean between 1 and 2? 
 
 Ans, li=1.5. 
 
 16. What is the arithmetical mean between 7 and 15? 
 
 Ans. 11. 
 
 17. What is the arithmetical mean between 1.5 and 1 ? 
 
 Ans. 1.25 
 
ARITHMETICAL PROGRESSION. 309 
 
 303. To find the sum of all the terms, when the 
 first term, the last term, and the number of terms are 
 known. 
 
 EXAMPLES. 
 
 1. The extremes are 1 and 13, number of terms 7. 
 What is the sum of all the terms ? 
 
 ANALYSIS. 
 
 We have just found the series to be (Ex. 12 above) 
 
 1, 3, 5, 7, 9, 11, 13. This series 
 
 reversed is 13, 11, 9, 7, 5, 3, 1. The sum of double 
 
 th« series is then 1-1+144-14+14-f 14+14+14:=7Xl^=98, and 
 
 7V14 
 the sum of the series -itself is -^o — =49, as may be verified by 
 
 adding the terms. (Vide 300.) ilence. 
 
 Take half the product obtained by multiplying the sum 
 of the extremes by the number of terms. 
 
 2. What is the sum of 50 terms of the series 4, 7, 
 10, etc.? Last term is 151, by 301. 
 
 An,. 5^X^^=3875. 
 
 3. What is the sum of 20 terms of the series 2, 4, 6, 
 etc.? 40 terms? 60 terms? 71 terms? Ans. 420, etc. 
 
 4. What is the sum of 50 terms of the series 3, 11, 
 19, etc.? 80 terms? 150 terms? 200 terms? 
 
 ^?2s. 9950, etc. 
 
 5. What is the sum* of IQ terms of the series 7, 11, 
 15, etc.? 35 terms? 45 terms? 75 terms? 
 
 6. What is the sum of 12 t^rms of the series 1, 2, 3, 4, 
 etc.? 24 terms? 48 terms? 96 terms? Ans. 78, etc. 
 
 7. What is the sum of 2 terms of the series 1, 3, 5, 
 7, etc.? 3 terms? 4 terms? 5 terms? 6 terms? 7 terms? 
 25 terms? Last Ans. 625. 
 
310 ARITHMETICAL niOGRESSION. 
 
 8. What is the sum of 40 terms of the series 1, 4, 7, 
 etc.? 50 terms? 60 terms? 70 terms? 
 
 Ans. 2380, etc. 
 
 9. A gentleman started on a journey, traveling 5 
 miles the first day, 7 miles Jie second day, and so on, 
 gaining 2 miles each day. Another gentleman, starting 
 from the same place, travels over the same road at a 
 uniform rate of 34 miles per day. How far apart were 
 they at the end of 10 days? 20? 30? 
 
 Last Ans. They are together. 
 
 10. A gentleman started on a journey, traveling 55 
 miles the first day, 51 the second day, and so on, losing 
 4 miles each day. Another gentleman travels uniformly 
 40 miles per day over the same road and from the same 
 place. How far apart are they at the end of 10 days? 
 
 Ans. 30 miles. 
 
 11. A boy buys 12 marbles, giving 1 cent for the first, 
 2 cts. for the second, 3 cts. for the third, and so on. 
 How many cents did he give for all? Ans. 78 cts. 
 
 12. Suppose the city of Quito to be precisely on the 
 equator, and that this circle is exactly 25000 miles in 
 circumference; suppose, furthermore, that the earth 
 were entirely of land, and that stakes have been set up 
 at intervals of 1 mile, tho enti|;e distance round the 
 equator. Now allowing a bag of gold dollars to be at 
 the bottom of each stake, what distance would a person, 
 setting out from Quito, h^v^e to travel in order to carry 
 the bags, one at a time, to that city? 
 
 A71S. 312500000 miles. 
 
GEOMETRICAL PROGRESSION. 811 
 
 GEOMETRICAL PROGRESSION. 
 
 304. A Geometrical Progression is a series of num- 
 bers, in which any term is found by multiplying the pre- 
 ceding term by a given number. 
 
 (1.) The ratio is the number used as a multiplier. 
 
 (2.) The progression is increasing when the ratio is 
 greater than 1. 
 
 (3.) The progression is decreasing when the ratio is 
 less tha7i 1. 
 
 305. In every geometrical progression these five 
 points may be considered, viz., the first term, the last 
 term, the ratio, the number of terms, and the sum of all 
 the terms. Thus, in the series, 
 
 1, 3, 9, 27, 81, 243, 
 the j^rs^ term is 1, last term ^4:^^^atio 3, number of terms 
 6, sum of all the terms 364, and the series is increasing. 
 In the decreasing progression, 
 
 1 1 j_ _L ^ ^ _ _ 1 _ 
 
 the f.rst term is 1, last term Y-0^24' '^^^^^ h 'f^umber of 
 terms 6, and the sum of all the terms i§||. 
 
 (1.) The first and last terms are called the extremes, 
 and the intermediate terms the means. 
 
 (2.) In a series of three terms the middle term is 
 called the geometrical mean, and is the square root of the 
 product of the extremes. 
 
 (3.) The ratio is the quotient of any term divided by 
 the preceding term. 
 
312 GEOMETRICAL PROGRESSION. 
 
 306. To find the last term, when the first term, num- 
 ber of terms, and ratio are known. 
 
 EXAMPLES. 
 
 1. If the fi7'st term of a series is 2, ratio 3, and num- 
 her of terms 10, what is the last term? 
 
 ANALYSIS. 
 
 If we take a few terms of the progression thus, 
 1st. • 2d. 3d. 4th. 5tli. 
 
 2, 2X3, 2X3^ 2X3\ 2X3^ 
 that is, 2 6 18 54 162 
 
 we see that any term, as, for instance, the 5th, is found 
 by multiplying the first term by that power of the ratio 
 denoted by the number of terms less 1. If we should 
 continue the series to the 10th term, the quantity repre- 
 senting ii would eviden{;|y be 2X3^, and the 10th term 
 is therefore 39366. (Vj^je ?76, Ex. 1.) Hence, 
 
 Multiply that poiver of the ratio denoted by the number 
 of terms less 1 by the first term. 
 
 2. In the series 3, 21, 147, etc., what is the 4th term? 
 6th term? 9th term? (Vid^e 273, Ex. 13, Table.) 
 
 Last Ans. 17294403. ' 
 
 3. In the series 4, 8, 16, etc., what is the 8th term? 
 9th term? 16th terra? (Vide 276, Ex. 2.) 
 
 Last Ans. 131072. 
 
 4. In the series 2, 4, 8, etc., what is the 5th term ? 
 9th terra? 13th term? Last Aiis, 8192. 
 
 5. In the series 1, J, -}, etc., what is the 10th term ? 
 6th term? 4th term? Ans. ^{.t, etc. 
 
GEOMETRICAL PROGRESSION. 313 
 
 SOT. To find the ratio^ when the extremes and the 
 number of terms are known, 
 
 (1.) Divide the last term by the first term. 
 
 (2.) Take the root of the quotient denoted by the number 
 of terms less 1. 
 
 EXAMPLES. 
 
 1. The extremes of a series are 2 and 39366 ; number 
 of terms, 10. What is the ratio and the consequent 
 series ? 39366 -- 2 =-- 19683. Then 1^19683 = 3, the 
 ratio. (Vide 273, 13, Table ; 274, (7.) 
 
 • The series is 2, 6, 18, 54, 162, 486, 1458, 4374, 
 13122, 39366. 
 
 2. Insert 2 means between 3 and 1029. The number 
 of terms will of course be 4. 1029 -r- 3 --=343; then 
 #"343=7, the ratio. Ans. 3, 21, 147, 1029. 
 
 3. Insert 3 means between 7 and 1792. 
 
 Ans. 7, 28, 112, 448, 1792. 
 
 4. Insert three means between J and g^^- 
 
 Ans. i, J^, Jq, Jq, g*^. 
 
 5. What is the geometrical mean between 4 and 16 ? 
 
 Ans. 8. (Vide 305, (2.) 
 
 6. What is the geometrical mean between 5 and 16 J? 
 
 Ans. 9. 
 
 7. What is the arithmetical mean between 1 and ? 
 (Vide 300, (2.) Ans. .5 
 
 8. What is the geometrical mean between 10 and 1 ? 
 
 Ans. 3.162278. 
 
 9. What is the arithmetical mean between 1 and .5 ? 
 
 Ans. .75 
 
 10. What is the geometrical mean between 10 and 
 3.162278? Ans. 5.623413. 
 
 27 
 
314 GEOMETRICAL PROaRESSION. 
 
 11. What is the arithmetical mean between .75 and 1 ? 
 
 Ans. .875. 
 . 12. What is the geometrical mean between 10 and 
 5.623413 ? Ans. 7.498942. 
 
 13. What is the arithmetical mean between 1 and 
 .875? Ans. .9375. 
 
 14. What is the geometrical mean between 10 and 
 7.498942 ? Ans. 8.659643. 
 
 15. Find the values of 1±^ and l/lOX 8.659643. 
 
 16. Find the values of 
 
 •''"'+•'''' and 1/9.305720x8.659643. 
 
 17. Find the values of 
 
 .96875+.953125 ^^^ i/9.305720x 8.976871. 
 
 18. Find the values of 
 
 ■960988+.9a3i25 ^^^ ^ 9.139817x8.976871. 
 
 19. Find the values of 
 
 .957031+.953125 ^^^ ^- 9.057978x8.976871. 
 
 20. Find the values of 
 
 .955078+ .953125 ^^^ t/9.017333x8.976871. 
 
 21. Find the values of 
 
 .954102+ . 955078 ^^^ i/8.997079x9.017333. 
 
 22. Find the values of 
 
 .954590+.954102 ^^^ ^^ 9.007200X8.9970 7-9. 
 
 28. Find the values of 
 
 .954346+.954102 
 
 --- and 1^9.002138X8.997079. 
 
and 1/9.000873x8.999608. 
 
 GEOMETRICAL PROGRESSION. 3^15 
 
 24. Find the values of 
 
 ■?^^^^!^ and • 8.999608X.9002138 . 
 
 25. Find the values of 
 
 .954285+.954224 
 2 
 
 26. Find the values of 
 
 ^'^^'Y'"^' -d 1^ 9-000241X8.999608. 
 
 27. Find the values of 
 ■954239+.904254 ^^^ ^ 8.999924x9.000241. 
 
 28. Find the values of 
 
 .9542474-.954235 
 
 and 1^9.000082X8.999924. 
 Ans. .954243 and 9.000003. 
 
 Remark. — Each arithmetical mean in the above examples is the 
 Logarithm of the corresponding geometrical mean, beginning with 
 example 7. The logarithm of 9 is .954243. (Vide Algebra, 153.) 
 
 308. To find the sum of all the terms, when the ex- 
 tremes and ratio are given. 
 
 EXAMPLES. 
 
 1. The extremes are 2 and 162, the ratio 3. What is 
 the sum of the series? 
 
 ANALYSIS. 
 
 The series is 2, 2x3, 2X3^, 2x3^ 2x3^ Multiplied 
 by 3 it is 2X3, 2X3^, 2x3^ 2x3S 2X3^ 
 
 The first series added would give the sum of all the 
 terms ; the second series added would give three times 
 the sum of all the terms. 
 
 If, then, the first series be subtracted from the second, 
 
316 GEOMETRICAL PROGRESSION. 
 
 the remainder will be twice the sum of all the terms. 
 The actual subtraction gives 2X3^ — 2, the other terms 
 canceling each other. Hence, the sum of the series is 
 
 2 — 2"^^ -^^^- Hence, 
 
 Multiply the last term hy the ratio, and from the pro- 
 duct subtract the first term; divide the remainder by the 
 ratio, less 1. 
 
 2. In the series 3, 21 . . . 1029, what is the sum of 
 all the terms? (Yide 305, (3.) 
 
 Ans, ^^^^f-^ :^1200. 
 
 3. In the series 4, 12 . . . 78732, what is the sum of 
 all the terms? Ans. 118096. 
 
 4. In the series 5, 20 . . . 327680, what is the sum 
 of all the terms? Ans. 436905. 
 
 5. In the series 4, 8, 16, etc., what is the sum of 16 
 terms? (Yide 306, Ex. 3.) J.ws. 262140. 
 
 6. In the series 2, 4, 8, etc., what is the sum of 13 
 terms? Ans. 16382. 
 
 7. In the series 5, 50, 500, etc., what is the sum of 8 
 terms? Ans. 55555555. 
 
 309. If the series is decreasing, and is carried on 
 infinitely, the last term will become 0. Hence, to find 
 the sum of an infinitely decreasing series. 
 
 Divide the first term by the ratio subtracted from 1. 
 
 EXAMPLES. 
 
 1. What is the sum of the series J, J, ^, etc., to in- 
 finity? Ans. »^(1— i)=l. 
 
 2. What is the sum of J, y^^, ^^5, etc.? Ans. |. 
 
 3. What is the sum of ^, i, ^, etc.? Am. \. 
 
PERMUTATIONS. 817 
 
 4. What is the sum of 4, 1, J, y^^, etc.? Ans. 5-|. 
 
 5. What is the value of .l-f-.Ol+.OOl, etc.? 
 
 Ans, ^. 
 
 6. What is the value of .1+.05+.025, etc.? 
 
 7. 
 8. 
 
 What 
 What 
 
 A 
 is the value of l+f+g^, etc.? 
 is the value of |+/u+i3, etc.? 
 
 Vns. i- 
 Ans. 
 
 Ans. 
 
 =.2. 
 If. 
 
 2f. 
 
 9. 
 
 What 
 
 is the value of 
 
 UA+t¥8, etc. 
 
 ? 
 Ans* 
 
 If 
 
 PERMUTATIONS 
 
 310. To find the number oi permutations that can be 
 made with any given number of things, each one differ- 
 ent from the other, 
 
 Multiply all the consecutive integral numbers^ from 1 
 up to the given number of things, continually together. 
 The product will be the permutations that can be made. 
 
 EXAMPLES. 
 
 1. How many permutations can be made of the first 
 four letters of the alphabet? Ans. 1X2X3X4=24. 
 
 2. How many days can 7 persons be placed in differ- 
 ent positions at table? Ans. 5040. 
 
 3. A captain of 26 men told his company that he 
 should not consider them perfectly drilled till each man 
 had occupied all possible positions in the arrangements 
 that might be made of them. Suppose his men drill 12 
 
318 PERMUTATIONS. 
 
 hours a day, and make a change every hour, how many 
 years must they drill before becoming perfect, reckoning 
 321 days to the year? 
 
 Ans. 107716786412020736000000 years. 
 
 311. To find how many arrangements may be made 
 by taking each time a given number of different things 
 less than all. 
 
 Take a series of nwnhers beginning with the number 
 of things given and decreasing by 1 until the number of 
 terms equals the number of things to be taken at a time; 
 then find the product of all the terms. 
 
 EXAMPLES. 
 
 1. How many sets of 3 letters each may be made out 
 of 4 letters ? Ans. 4 X 3 X 2==24. 
 
 proof: 
 
 abc, acb, adb, bac, bca, bda, cab, cba, cda, dab, dba, dca, 
 
 abd, acd, adc, bad, bed, bdc, cad, cbd, cdb, dac, dbc, deb. 
 
 2. How many integral numbers can be expressed, 
 each composed of any 5 of the 9 digits ? 
 
 Ans. 9X8X7X6X5=15120. 
 
 3. How many arrangements can be made out of the 
 26 letters of the alphabet, 6 being taken at once? 
 
 Ans. 165765600. 
 
 312. To find the number of combinations that can be 
 made of any number of things in sets of 2 and 2; 3 
 and 3, etc., 
 
 (1.) Form a series of numbers^ as in 311, /or a dividend. 
 
 (2.) Form a series of numbers^ as in 310, up to the 
 number of things to be combined at a time, for a divisor. 
 
 The quotient will be the number of combinations 
 sought. 
 
PRACTICAL GEOMETRY. 319 
 
 EXAMPLES. 
 
 1. How many combinations can be made out of 4 
 letters, having 3 different letters in each set? 
 
 4X3X2 
 ^ns. -^^2x3~'*- 
 Proof. — abc, abd, acd, bed. 
 
 2. How many combinations can be made out of 10 
 letters, each combination having in it 7 letters, but no 
 two of them to have all their letters alike ? 
 
 Ans. 120. 
 
 3. How many different combinations of 3 colors can 
 be made of the 7 prismatic colors? Ans. 35. 
 
 4. How many different combinations of 4 colors can 
 be made of the 7 prismatic colors? Ans. 35. 
 
 5. How many different combinations of 3 may be 
 made of 10 different things? Ans. 120. 
 
 6. How many different combinations of 7 may be 
 made of 10 different things? Ans. 120. 
 
 PRACTICAL GEOMETRY. 
 
 DEFINITIONS. 
 
 313. Geometry is that branch of Mathematics which 
 treats of the' relations of extension. Extension has three 
 dimensions — length, breadth, and thichiess. 
 
 (1.) A point is mere position, with no length, breadth, 
 or thickness. 
 
 (2.) A line is length, without breadth or thickness. 
 
320 PRACTICAL GEOMETRY. 
 
 (3.) A surface is a figure having length and breadth, 
 but no thickness. 
 
 (4.) A solid is a figure having length, breadth, and 
 thickness. 
 
 LINES. 
 
 (5.) A straight line is one, all points of which lie in 
 the same direction. It is designated bj letters placed 
 near its extreme points. Thus, the line ab is repre- 
 sented by A B 
 
 (6.) A curved line is one, ail points of which lie in 
 different directions. Thus, the curved line 
 A B is represented by 
 
 (7.) Parallel straight lines are two or 
 
 more straight lines lying in the same c d 
 
 direction. Thus, ab, cd, ef, are par- ^ ^ 
 
 allel straight lines. 
 
 (8.) Oblique lines are those which do not lie in the 
 same direction. Thus, A b and c D are . __ — _— — -^ 
 
 oblique lines. c d 
 
 Remark. — When two oblique lines meet each other, they form an 
 angle. 
 
 (9.) An angle is the divergence of two b 
 
 straight lines proceeding from the same ^^ o 
 
 'point. Thus, a, or b A c, or c A b is aa 
 
 angle. In reading, the letter at the vertex . » 
 
 is placed in the middle, or the letter at / ^ 
 
 the vertex may be used alone to desig- ^ 
 nate the angle when other angles arc not ^^ 
 adjacent. ^^ o 
 
 A B 
 
PRACTICAL GEOMETRY. 321 
 
 Eemark. — An angle is greater or less^ according as the lines di- 
 verge more or less. 
 
 (10.) A right angle is one of the equal angles which 
 two straight lines may make 
 in meeting or intersecting 
 
 each other. Thus, A o c, B o c, ^ 
 
 A D, or B D is a right angle 
 when each is equal to any other. 
 
 An acute angle is one which is less than a right angle. 
 
 An obtuse angle is one which is greater • „ 
 
 than a rio;ht angle. Thus, B o c is acute. ^^ 
 
 and A c is obtuse. ° 
 
 (11.) Perpendicular lines are those which form a right 
 angle with each other. Thus, A o is per- 
 pendicular to c 0, and c o is perpendicular 
 to A 0, when A o c is a right angle. 
 
 SURFACES. 
 
 (12.) A plane is a surface in which if any two points 
 be assumed and connected by a straight line, that line 
 ivill lie wholly in the surface. Other surfaces are called 
 curved surfaces. Thus, the surface of a common slate 
 represents a plane, and the surface of a slate globe 
 represents a curved surface. 
 
 (13.) A plane figure is a figure bounded by straight 
 or curved lines. 
 
 (14.) A polygon is a plane bounded by straight lines. 
 
 (15.) A triangle is a polygon of three sides, and con- 
 sequently of three angles. 
 
 (a.) An equilateral triangle has its three sides equal; 
 
322 
 
 PRACTICAL GEOMETRY. 
 
 an isosceles triangle has tivo of its sides equal; a scalene 
 triangle lias its three sides unequal, 
 
 (b.) A right-angled triangle has one of its angles a 
 right angle. 
 
 (c.) ArT acute-angled triangle has all of its angles acute. 
 
 (e.) An obtuse-angled triangle has one of its angles 
 an obtuse angle. Thus : 
 
 A B c is equilateral and acute-angled ; D E F is isosceles 
 and right-angled; mno is scalene and obtuse-angled, 
 p Q R is scalene and acute-angled. 
 
 (/.) Any side of a triangle may be assumed as the 
 base, and its altitude is the perpendicular line extending 
 from the vertex of the opposite angle to the base. 
 
 (g.) In a right-angled triangle the side opposite the 
 right angle is called the hypothenuse. The other sides 
 are then designated as the base and perpendicular. 
 
 (16.) A quadrilateral is a polygon of four sides. 
 
 (17.) A parallelogram is a quadrilateral with its oppo- 
 site sides parallel. 
 
 (18.) A rectangle is a parallelogram 
 having a right angle. 
 
 (19.) A rhomboid is a parallelogram 
 having an oblique angle. 
 
 (20.) A square is a rectangle hav- 
 ing its sides all equal. 
 
 (21.) A rhombus is a rhomboid having 
 its sides all equal. 
 
 S, ,Q 
 
 a) |r 
 
 Vi IB 
 
PRACTICAL GEOMETRY. 323 
 
 (22.) A trapezium is a quadrilateral 
 with its opposite sides 7iot parallel. 
 
 (23.) A trapezoid is a quadrilateral 
 with two sides parallel. Thus: 
 
 A diagonal is a line joining the vertices of two opposite 
 
 angles. Thus T E is a diagonal. 
 
 Remakk. — A diagonal divides a rectangle into two right-angled 
 triangles. 
 
 (24.) A pentagon is a polygon of five sides; a hexagon, 
 of six sides; a heptagon, of seven; an octagon, of eight; 
 a nonagon, of nine ; and a decagon, of ten sides. 
 
 (25.) A circle is a plane figure 
 bounded by a curved line called the 
 circu7nference, every point of which 
 is equally distant from a point within 
 called the center. The diameter is 
 a straight line drawn through the 
 center and terminating in the cir- 
 cumference. A radius is any line draivn from the center 
 to the circumference. An arc is a part of the circumfer- 
 ence. Thus, A B is a diameter, o B, o A, and o M are 
 radii ; A M is an arc. 
 
 (26.) Similar figures are such as .are mutually equi- 
 angular, and have the sides containing the equal angles, 
 taken in the same order, proportional. 
 
 314. Proposition. — The square of the hypothenuse 
 of a right-angled triangle is equivalent to the sum of the 
 squares on the other two sides. Hence, 
 
 315. To find the hypothenuse, the ttvo sides being known, 
 (1.) Square the two sides, and add together the results. 
 (2.) Find the square root of the sum. 
 
324 PEACTICAL GEOMETRY. 
 
 EXAMPLES. 
 
 1. The sides of a right-angled triangle are 3 and 4 
 feet. What is the length of the hypothenuse ? 
 
 Ans. -1/32 + 42 = 1/25 = 5 feet. 
 
 2. The sides of a right-angled triangle are each 1 
 foot. What is the length of the hypothenuse ? 
 
 ^Tis. 1/2 = 1.4142136 feet. 
 
 3. The sides of a right-angled triangle are each 2 feet 
 in length ; 3 feet ; 4 feet ; 5 feet in length. What are 
 the corresponding lengths of the hypothenuse? 
 
 Ans. 1^8 = 2.828427, (Vide 286, Ex. 1.) 
 
 4. The bottom of a window is 40 feet from the ground, 
 and I wish to place the foot of a ladder 30 feet from the 
 bottom of the wall in which the window is situated. 
 What is the length of a ladder that will reach the win- 
 dow? Ans. 50 feet. 
 
 5. An acre of land is laid out in the form of a square. 
 What is its distance between opposite corners ? 
 
 Ans. 1/320 = 8X1/5 = 17.888544 rods. 
 
 6. Suppose a floor to measure 16 feet by 20. What 
 is the distance between opposite corners ? 
 
 Ans. 1/656 =4X l/41 = 25.612497. 
 
 7. What is the length of the lower edge of a brace 
 which touches a post 3^- feet from the corner, and a 
 beam 4^ feet from the same point ? 
 
 Ans. i X 1/130 = 5.7008771 feet. 
 
 316. To find the side of a right-angled triangle, when 
 the hypothenuse and the other side are known. 
 
 From the square of the hypothenuse subtract the square 
 of the given side. The square root of the remainder will 
 be the other side. 
 
PRACTICAL GEOMETRY. 325 
 
 EXAMPLES. 
 
 1. The hypothenuse is 5 and a side 4. Wliat is the 
 other side of the triangle? 
 
 2. A ladder 50 feet long is placed at a distance of 30 
 feet from the bottom of a house, and just reaches the 
 sill of a window. What is the hight of the sill? 
 
 Ans. 40 feet. 
 
 3. If a line 144 feet long will reach from the top of a 
 fort to the opposite side of a river 64 feet wide, on 
 whose brink it stands, what is the hight of the fort ? 
 
 A71S. 129 feet, nearly. 
 
 4. In the ruins of Persepolis are left two columns 
 standing upright ; one is 70 feet high and the other 50 
 feet ; in a line between them stands a small statue 5 feet 
 high, the top of which is 100 feet from the summit of 
 the higher, and 80 feet from that of the lower column. 
 What is the distance between the tops of the two col- 
 umns ? Ans. 143 J feet. 
 
 317. To find the side of a square when its diagonal 
 is known. 
 
 Multiply half the diagonal by the square root of 2. 
 
 EXAMPLES. 
 
 1. The diagonal of a square is 1 foot. What is the 
 side ? Ans. h X V2 =.7071068 feet. 
 
 2. The diagonal of a square is 2 feet ; 3 feet ; 4 feet ; 
 5 feet, etc. What is a side? Ans. 1.4142136, etc. 
 
 3. The distance from a corner to an opposite corner 
 of a square room is 20 feet. What is a side ? 
 
 Ans. 14.142136 feet. 
 
326 PRACTICAL GEOMETRY. 
 
 SIS. Proposition. — In any triangle, if a perpen- 
 dicular be drawn from its vertex to its base, then the 
 whole base is to the sum of the other two sides as the dif- 
 ference of those sides is to the difference of the parts of 
 the base made by the perpendicular. c 
 
 Thus, suppose c d to be a perpendicular 
 drawn from the vertex c to the base A B 
 of the triangle ABC, then, 
 
 A B : A c+B c : : A c— b c : A d— b d. 
 
 Remark. — The perpendicular must always fall within the tri- 
 angle. 
 
 310. To find the perpendicular, when the th'ee sides 
 of a triangle are known, 
 
 (1.) Make the base the first terin of a proportion; the 
 sum of the other tivo sides the second term; the difference 
 of the two sides the third term. The fourth term found 
 by 258 will be the difference of the parts of the base. 
 
 (2.) Find the parts by 125, example 10, (1) and (2). 
 
 (3.) Find the perpendicular by 316. 
 
 EXAMPLES. 
 
 1. In a triangle ABC we have ab=5 rods; AC=4 
 rods, and B C— 3 rods. What is the length of the per- 
 pendicular CD? 
 
 OPERATIONS. 
 
 (1.) 5 : 4+3 : : 4-3 : 1.4. 
 
 (2.) (125,Ex.ll.) -^±i:^=3.2-:ADand^::^=1.8=DB. 
 
 (3.) 1/42-3.2^=2.4 or 1/32— 1.82=2.4=:D c. 
 
 Ans. 2.4 rods. 
 Why is this triangle right-angled at c'i 
 
PRACTICAL GEOMETET. 327 
 
 2. In an isosceles triangle the sides are 5, 5, and 8 
 rods. What is the length of the perpendicular ? 
 
 Ans, 3 rods. 
 
 Remark. — Assume the side which is unequal to the other two to 
 be the base ; then the parts of the base made by the perpendicular 
 will be each 4 rods. 
 
 3. In an equilateral triangle the sides are each 1 rod. 
 What is the length of the perpendicular ? 
 
 Ans. |/r=i = Vi = hxVS = .8660254. 
 
 4. The sides of a triangle are 4, 5, and 6 rods. What 
 is the length of the perpendicular? 
 
 Ans. 3.307187 rods. 
 
 5. The sides of a triangle are 10, 10, and 16 rods. 
 What is the length of the perpendicular? Ans. 6 rods. 
 
 6. The sides of a triangle are each 25 rods. What is 
 the length of the perpendicular? Ans. 21.6506 rods. 
 
 7. The sides of a triano-le are each 2 rods in leno-th ; 
 3 rods; 4 rods; 5 rods; 6 rods, etc. What is the 
 length of the perpendicular? 
 
 Ans.l/S; iXV^S; 2X1^3, etc. 
 
 MENSURA.TION. 
 
 320. Mensuration is the method of finding the 
 area of surfaces in square units from their known or 
 implied linear dimensions. By mensuration we also find 
 the cubical contents of solids. 
 
 321. To find the area of a triangle, when its sides 
 are known, 
 
 (1.) Find the perpendicular by 319. 
 (2.) Multiply the base by the perpendicular, and take 
 half the product. 
 
328 PRACTICAL GEOMETRY. 
 
 EXAMPLES. 
 
 1. What is the area of a triangle whose sides are 3, 
 
 4, and 5 rods? (Vide 319, Ex. 1.) ?^, or, because 
 the triangle is right-angled, -^ . Ans. 6 rods. 
 
 2. What is the area of a triangle whose sides are 5, 
 
 5, and 8 rods? Ans. 12 rods. 
 
 3. What is the area of a triangle whose sides are each 
 1 rod? Ans. .4330127 rods. 
 
 4. What is the area of a triangle whose sides are 4, 
 5, and 6 rods? Ans. 9.9215 rods. 
 
 5. What is the area of a triangle whose sides are 10, 
 10, and 16 rods? Ans. 48 rods. 
 
 6. What is the area of a triangle whose sides are each 
 25 rods? 13 rods? 40 rods? Ans. 270.633 rods, etc. 
 
 7. How many square feet of boards will cover the 
 gable end of a house whose rafters measure 23 feet, the 
 length of the beam at the ends of the building being 
 34.8712 feet? Ans. 261.534 feet. 
 
 Rule II. — Add together the three sides, and taJce half 
 their sum. From this half sum subtract each side sepa- 
 rately; mtdtiply together the half sum and the three re- 
 mainder. The square root of the product will be the area. 
 
 8. What is the area of a triangle whose sides are 10, 
 15, and 20 rods? 15+10+20--45 ; 45-^2=22.5; 
 V'22.5X2.5X12.5X7.5==72.62. Ans. 72.62 rods. 
 
 333. To find the area of any quadrilateral, two of 
 whose sides are parallel, 
 
 Multiply the sum of the parallel sides hy the perpendic- 
 ular distance between them, and take half their product. 
 
PRACTICAL GEOMETRY. 329 
 
 EXAMPLES. 
 
 1. How many yards of carpeting will cover a floor 20 
 feet long and 16 feet Avide, carpet 1 yard wide? 20 X 
 16-^-9. Ans. 35 f yards. 
 
 2. How many acres in a rectangular piece of land 
 17 chains long and 5 ch. and 41. wide? (Vide 180, 
 Rem. 2.) 17x5.04^10 Ans. 8 A. 2 R. 10/o'(j r- 
 
 3. How many acres in a piece of land in the form of 
 a rhombus, each side measuring 70 rods and the width 
 being 8 rods ? Ans, 3 A. 2 R. 
 
 4. How many acres in a square piece of land whose- 
 sides are each 35 ch. 25 1. ' Ans. 124 A. 1 R. 1 r. 
 
 5. A board measures 25 feet in length and 1 ft. 6 in. 
 in width. What is the area in feet? Ans. 37 J feet. 
 
 6. In the trapezoid, 313 (23,) I D is 20 rods; z o, 
 27 rods; P R, 12 rods. How many acres in the lot? 
 m^}><}l Ans. 1 A. 3 R. 2 r. 
 
 7. In a trapezoid the parallel sides are 25 ch. 13 1. 
 and 30 ch. 1 1. ; the perpendicular distance between them 
 is 40 ch. How many acres ? Ans. 110 A. 1 R. 4.8 r. 
 
 S23. To find the area of a trapezium, when all its 
 sides and a diagonal are known, 
 
 Find the area of each triangle hy 321, and their sum 
 will be the area of the trapezium. 
 
 EXAMPLES. 
 
 1. In a trapezium represented by the figure ZIUM, 
 813, (22,) z I is 25 rods; mi, 23 rods; zu, 17 rods; 
 U M, 21 rods ; and z m, 30 rods. How many acres ? 
 
 Ans. 2A. 3R. 13.7 r. 
 
 28 
 
330 PRACTICAL GEOMETRY. 
 
 324. Any iivo similar figures have their areas in pro- 
 portion to the squares of any ttvo lines similarly situated 
 in each. Hence, all circles are to each other as the squares 
 of their radii, or the squares of their diameters. 
 
 325. To find the area of a circle, 
 
 Multiply half of the diameter by half the circumfer- 
 ence. 
 
 Remark. — The circumference of a circle whose diameter is 1 is 
 3.141592053589793238462643383279502884197169399375105820974- 
 94459230781640628620899862803482534211706798214808651327230- 
 66470938446, etc. 
 
 EXAMPLES. 
 
 1. What is the area of a circle whose diameter is 1 ? 
 
 Ans. .7853981633974483, etc. 
 Hence, to find the area of a circle, midtiply the square 
 of the diameter hy enough of the figures composing the 
 decimal .785398163397, etc., to make the area sufficiently 
 exact. 
 
 2. What is the area of a circle whose diameter is 2 
 rods? 3 rods? 4 rods? 5. rods? etc. 
 
 Ans. 3.1416, etc., rods; 7.06858, etc., rods. 
 
 3. What is the area of a circle whose diameter is 20 
 rods? 30 rods? 40 rods? etc. 
 
 4. What is the area of a circle whose radius is 10 
 rods? 15 rods? 20 rods? etc. 
 
 5. What is the area of a circle whose radius is 100 
 rods? 150 rods? etc. 
 
 6. What is the area of a circle whose radius is 5000 
 rods? Ans. 490873 A. 3 R. 16 r. 
 
 7. What is the area of a circle whose radius is 1 
 mile? 2 miles? Smiles? etc. J.?is. 3.1416 miles, etc. 
 
rRACTICAL GEOMETRY. 331 
 
 8. What is the area of a circle whose radius is 5 
 yards? 6 yards? etc. Ans. 78 J yards, etc. 
 
 9. Three men purchase a grind- 
 stone with a radius of 30 inches. '' / \''~''"-^/\\ 
 How much of the radius must each ,' 
 grind oiF to secure J of the stone, \ 
 making no allowance for loss at '' ^ 
 the center? We have ""-" 
 
 
 Circ. A : 
 
 : circ. c o : 
 
 : AO^ 
 
 : C 0^ ; that is, 
 
 1 ; 
 
 : 1 : 
 
 : 30- ; 
 
 :30^Xf. 
 
 Circ. A ; 
 
 : circ. P : 
 
 : AO^ 
 
 : P 0- : that is, 
 
 1 
 
 -• i -• 
 
 : 30-^ : 
 
 ; 30^X^ 
 
 Since c 0^=30^X1, c o=30Xl/|-=10Xl/6--24.4949. 
 Since p o2=30-Xl, p o=30Xi/1=10XV^3=17.3205. 
 
 Hence, the fii?st takes 30—24.4949=5.5051 inches; 
 the second takes 24.4949—17.3205=7.1744 inches; the 
 third takes 17.3205 inches. 
 
 Remark. — It is only necessary to use the ratio of the .areas in 
 the proportions. 
 
 10. Five men purchased a grindstone, and each man 
 paid ^ of the price. The radius was 40 inches. How 
 much of it must each man grind off to secure his 
 share? Ans, First man 40— 16x/5=4.2229 in. 
 
 11. I have a circular garden 25 rods in diameter, 
 and wish to make a walk around it that shall take up 
 I of the entire area. What must be the width of the 
 walk? Ans. 1.0891 rods. 
 
 12. I have a circular garden 50 rods in diameter, and 
 wish to make a walk around the outside of it whose area 
 shall be J^ the area of the garden. What must be the 
 width of the walk? Ans. 1.22 rods. 
 
332 
 
 PRACTICAL GEOMETRY. 
 
 MENSURATION OF SOLIDS. 
 
 326. Demnition. — (1.) A cylinder \^ a solid described 
 by the revolution of a rectangle about one of its sides, 
 which remains fixed. 
 
 (2.) A cone is a solid described by the revolution of 
 a right-angled triangle about one of its sides^ which re- 
 mains fixed. 
 
 (3.) A sphere is a solid described by the revolution 
 of a semicircle about its diameter, which remains fixed. 
 Thus : 
 
 P B D revolved about o P generates a cylinder ; c B 
 revolved about c o generates a cone ; and A M B revolved 
 about A B generates a sphere, o p is the altitude of the 
 cylinder, o c of the cone, and A B is the diameter of the 
 sphere. 
 
 (4.) A prism is a solid whose bases are j[?<2raZ?eZ and 
 its sides parallelograms. A right prism has its edges 
 perpendicular to its bases, 
 
 (5.) A parallelopiped is a prism whose bases as well 
 as sides are parallelograms. A cube has six equal square 
 faces. 
 
 (6.) A pyramid is a solid, having a polygon for a 
 base and three or more triangles for its sides, whose 
 
PRACTICAL GEOMETllY. 
 
 333 
 
 vertices meet in a common point called the vertex of 
 the pyramid. The frustrum of a pyramid is the part 
 left after cutting off a portion of the top by a plane 
 parallel to the base. Thus : 
 
 /^. \ 
 
 \ 
 
 
 \o 
 
 \ 
 
 F 
 
 D 
 1 
 
 li 
 
 A B c-D E F is a right triangular prism ; A B c D-E is a 
 right quadrangular prism, or parallelopiped; A B c-D is 
 a triangular pyramid ; and A B c-s represents the frus- 
 trum of a pyramid. 
 
 3S7. To find the contents of a cylinder or prism, 
 Multiply the area of the base by its altitude. 
 
 EXAMPLES. 
 
 1. Each side of the base of a triangular prism is 1 
 foot, and the altitude is 3 ft. 2 in. What are the con- 
 tents in feet? in inches? (Vide 321, Ex. 3.) 
 .4330127X31. Ans. 1.3712 feet. 
 
 2. The sides of a triangular prism are 7, 8, and 9 
 inches. Its altitude is Ij feet. Find the contents in 
 inches. Ajis. 482.99 inches. 
 
 3. The diameter of each end of a cylinder is 8 feet, 
 and the hight is 5 J feet. Find the contents in feet. 
 
 Alts. 276.46 feet. 
 
334 PRACTICAL GEOMETRl. 
 
 4. The diameter of a cylindrical water-pail is 10 
 inches, and the hight is 1 foot. How many wine gal- 
 lons does the pail hold? (Vide 182, Rem. 1.) 
 
 A71S. 4.08 gallons. 
 
 5. How many bushels in a box 15 feet long, 5 feet 
 wide, and 8 feet deep? (Vide 184, Rem.) 
 
 Ans. 482.142 bushels. 
 
 6. How many bushels in a box 6 feet long, 1^ feet 
 wide, and 2 J feet deep? Ans. IS. OS. 
 
 7. What are the contents in cubic feet of a wall 24 
 feet 3 inches long, 10 feet 9 inches high, and 2 feet 
 thick? Ans. 521|. 
 
 3S8. To find the contents of a cone or pyramid, 
 Multiply the area of the base by the altitude^ and take 
 I of the product. 
 
 EXAMPLES. 
 
 1. Each side of the base of a triangular pyramid is 1 
 foot, and the hight is 14 inches. What are the con- 
 tents? Ans. 290.9844 in. 
 
 2. The sides of the base of a triangular pyramid are 
 10, 11, and 12 feet, and the hight is 12 feet. What are 
 the contents? Ans. 206.085 feet. 
 
 3. The base of a cone is 10 feet in diameter and the 
 hight is 5 feet. Find the contents. 
 
 Ans. 130.899 feet. 
 
 4. A square pyramid, 477 feet high, has each side of 
 its base 720 feet in length. Find the contents. 
 
 Ans. 3052800 cu. yd. 
 
 5. The sides of the base of a triangular pyramid, 
 which is 14J feet high, are 5, 6, and 7 feet. Find the 
 contents. Ans. 71.0352 feet. 
 
PRACTICAL GEOMETRY. 335 
 
 3S9. To find the contents of the frustrum of a cone 
 or pyramid, 
 
 Find the sum of the areas of the two ends and the geo- 
 metrical mean between them. Multiply this sum hy the 
 altitude, and take one third of the product 
 
 EXAMPLES. 
 
 1. A stick of timber is 15 in. square at one end and 
 6 in. square at the other and 24 feet long. What arQ 
 the contents? (^^5+B«+«0)X8 ^„,.- 19, feet. 
 
 2. A conic frustrum is 18 feet high, 8 feet in diam- 
 eter at one end and 4 feet at the other. Find the 
 contents. Ans. 527.7888 feet. 
 
 3. A cask, in the form of two equal conic frustrums, 
 has a bung diameter of 28 inches, a head diameter of 20 
 inches, and a length of 40 inches. How many gallons 
 of wine will it hold? Ans. 79.0613 gallons. 
 
 7. A cistern is 12 feet in diameter at the top, 10 feet 
 in diameter at the bottom, and 14 feet deep. What 
 number of gallons will it hold? 
 
 OPERATION. 
 Area of top = 144 X -7854 = 113.09 
 Area of bottom .-= 100 X .7854= 78.54 
 Geomet. mean =t/113.09X78.54 = 94.25 
 
 Sum = 285.88 feet. 
 Then 285.88XV*X1728--231 =9980 gallons, nearly, ^hs. 
 
 8. How many Winchester bushels will the above cis- 
 tern hold? A71S. 1072 bushels. 
 
 330. Proposition. — All spheres are to each other as 
 the cubes of their radii, or diameters. 
 
336 MISCELLANEOUS EXAMPLES. 
 
 331. To find the surface and also the solid contents 
 of a sphere, 
 
 (1.) Multiply the square of the diameter by 3.1415^, 
 eic.^ for the surface. 
 
 (2.) ^lultiply the cube of the diameter hy .523598, etc.^ 
 for the solid contents. 
 
 Remark. — The latter decimal is i of the former. 
 
 EXAMPLES. 
 
 1. An artificial globe is 24 inches in diameter; what 
 
 is its area, and what the solid contents ? 
 
 Arts. Area 1809.556992 sq. in.; solidity 2738.218752 cu. in. 
 
 2. A slate globe is 6 feet in diameter; what is its 
 area, and what its contents ? 
 
 Ans. Area 113.0973 sq.ft.; contents 113.0973 cu. ft. 
 
 3. A sphere is 40 feet in diameter; what will be the 
 diameter of one containing | as many cubic feet? J- as 
 many? i as many? (Vide 296, Ex. 3, 2, and 1.) 
 
 Ans. 40X^^1=36.34 ft.; 40X'^ 1=31.75 ft.; 40X#'i=25.19 ft. 
 
 332. MISCELLANEOUS EXAMPLES. 
 
 1. If a certain number be multiplied by 5, and the 
 product divided by J, and 3 be added to the quotient, 
 and 7 taken from the sum, the remainder will be 76. 
 What is the number ? Ans. 8. 
 
 2. If to a certain number 12 be added, and the square 
 root of the sum taken, the cube of that root will be 64. 
 What is the number ? Ans, 4. 
 
 ^ 
 
APPENDIX. 
 
 353 
 
 
 
 III. 
 
 TABLE OP CUBE ROOTS. 
 
 
 No. 
 
 1 
 
 Cube Root. 
 
 No. 
 
 Cube Root. 
 
 No. 
 
 Cube Root. 
 
 No. 
 
 Cube Root. 
 
 1.000000 
 
 Ig" 
 
 3.583048 
 
 91 
 
 4.497941 
 
 136 
 
 5.142563 
 
 2 
 
 1.259921 
 
 47 
 
 3.608826 
 
 92 
 
 4.514357 
 
 137 
 
 6.155137 
 
 3 
 
 1.442250 
 
 48 
 
 3.634241 
 
 93 
 
 4.530655 
 
 138 
 
 5.167649 
 
 4 
 
 1.587401 
 
 49 
 
 3.659306 
 
 94 
 
 4.546836 
 
 139 
 
 5.180102 
 
 5 
 
 1.709976 
 
 50 
 
 3.684031 
 
 95 
 
 4.562903 
 
 140 
 
 5.192494 
 
 6 
 
 1.817121 
 
 51 
 
 3.708430 
 
 96 
 
 4.5J8857 
 
 141 
 
 5.204828 
 
 7 
 
 1.912931 
 
 52 
 
 3.732511 
 
 97 
 
 4.594701 
 
 142 
 
 5.217103 
 
 8 
 
 2.000000- 
 
 53 
 
 3.756286 
 
 98 
 
 4.610436 
 
 143 
 
 5.229322 
 
 9 
 
 2.080084 
 
 54 
 
 3.779763 
 
 99 
 
 4.626065 
 
 144 
 
 5.241483 
 
 10 
 
 2.154435 
 
 55 
 
 3.802953 
 
 100 
 
 4.641589 
 
 145 
 
 5.253588 
 
 11 
 
 2.223980 
 
 56 
 
 3.825862 
 
 101 
 
 4.657009 
 
 146 
 
 5.265637 
 
 12 
 
 2.289429 
 
 57 
 
 3.848501 
 
 102 
 
 4.672329 
 
 147 
 
 5.277632 
 
 13 
 
 2.351335 
 
 58 
 
 3.870877 
 
 103 
 
 4.687548 
 
 148 
 
 5.289573 
 
 14 
 
 2.410142 
 
 59 
 
 3.892996 
 
 104 
 
 4.702669 
 
 149 
 
 5.301459 
 
 15 
 
 2.466212 
 
 60 
 
 3.914868 
 
 105 
 
 4.717694 
 
 150 
 
 5.313293 
 
 16 
 
 2.519842 
 
 61 
 
 3.936497 
 
 106 
 
 4.732624 
 
 151 
 
 5.325074 
 
 17 
 
 2.571282 
 
 62 
 
 3.957892 
 
 107 
 
 4.747459 
 
 152 
 
 5.336803 
 
 18 
 
 2.620742 
 
 63 
 
 3.979057 
 
 108 
 
 4.762203 
 
 153 
 
 5.348481 
 
 19 
 
 2.668402 
 
 64 
 
 4.000000 
 
 109 
 
 4.776856 
 
 154 
 
 5.360108 
 
 ■ 20 
 
 2.714418 
 
 65 
 
 4.020726 
 
 110 
 
 4.791420 
 
 155 
 
 5.371685 
 
 21 
 
 2.758924 
 
 66 
 
 4.041240 
 
 111 
 
 4.805896 
 
 156 
 
 5.383213 
 
 22 
 
 2.802039 
 
 67 
 
 4.061548 
 
 112 
 
 4.820285 
 
 157 
 
 5.394691 
 
 23 
 
 2.843867 
 
 68 
 
 4.081655 
 
 113 
 
 4.834588 
 
 158 
 
 5.406120 
 
 24 
 
 2.884499 
 
 69 
 
 4.101566 
 
 114 
 
 4.848808 
 
 159 
 
 6.417502 
 
 25 
 
 2.924018 
 
 70 
 
 4.121285 
 
 115 
 
 4.862944 
 
 160 
 
 6.428835 
 
 2G 
 
 2.962496 
 
 71 
 
 4.140818 
 
 116 
 
 4.876999 
 
 161 
 
 5.440122 
 
 27 
 
 3.000000 
 
 72 
 
 4.160168 
 
 117 
 
 4.890973 
 
 162 
 
 5.451362 
 
 28 
 
 3.036589 
 
 73 
 
 4.179339 
 
 118 
 
 4.904868 
 
 163 
 
 5.462556 
 
 •29 
 
 3.072317 
 
 74 
 
 4.198336 
 
 119 
 
 4.918685 
 
 164 
 
 5.473704 
 
 30 
 
 3.107233 
 
 75 
 
 4.217163 
 
 120 
 
 4.932424 
 
 165 
 
 5.484807 
 
 31 
 
 3.141381 
 
 76 
 
 4.235824 
 
 121 
 
 4.946087 
 
 166 
 
 5.495865 
 
 32 
 
 3.174802 
 
 77 
 
 4.254321 
 
 122 
 
 4.959676 
 
 167 
 
 5.506878 
 
 33 
 
 3.207534 
 
 78 
 
 4.272659 
 
 123 
 
 4.973190 
 
 168 
 
 5.517848 
 
 34 
 
 3.239612 
 
 79 
 
 4.290840 
 
 124 
 
 4.986631 
 
 169 
 
 5.528775 
 
 35 
 
 3.271066 
 
 80 
 
 4.30^70 
 
 125 
 
 5.000000 
 
 170 
 
 6.539658 
 
 36 
 
 3.301927 
 
 81 
 
 4.326749 
 
 126 
 
 5.013298 
 
 171 
 
 5.550499 
 
 37 
 
 3.332222 
 
 82 
 
 4.344482 
 
 127 
 
 5.026526 
 
 172 
 
 5.561298 
 
 38 
 
 3.361975 
 
 83 
 
 4.362071 
 
 128 
 
 5.039684 
 
 173 
 
 5.672055 
 
 39 
 
 3.391211 
 
 84 
 
 4.379519 
 
 129 
 
 5.052774 
 
 174 
 
 5.582770 
 
 40 
 
 3.419952 
 
 85 
 
 4.396830 
 
 130 
 
 5.065797 
 
 175 
 
 5.593445 
 
 41 
 
 3.448217 
 
 86 
 
 4.414005 
 
 131 
 
 5.078753 
 
 176 
 
 6.604079 
 
 42 
 
 3.476027 
 
 87 
 
 4.431048 
 
 132 
 
 5.091643 
 
 177 
 
 6.614672 
 
 43 
 
 3.503398 
 
 88 
 
 4.447960 
 
 133 
 
 5.104469 
 
 178 
 
 6.625226 
 
 44 
 
 3.530348 
 
 89 
 
 4.464745 
 
 134 
 
 5.117230 
 
 179 
 
 6.635741 
 
 i 45 
 
 3.556893 
 
 90 
 
 4.481405 
 
 135 
 
 5.129928 
 
 180 
 
 5.646216 
 
354 
 
 APPENDIX. 
 
 IV. 
 
 TABLE: 
 
 Shoiving the ultimate fratisveise strength of a bar 1 foci long and 1 inch 
 
 1 inch in diameter, made of either of the materials mentioned. The bar is loaded 
 in the middle, and lies loose at both ends. 
 
 Matebials. 
 
 Square Bar 
 
 Onk Third. 
 
 Round Bar. 
 
 One Third. 
 
 Oak 
 
 800 
 
 1137 
 
 o69 
 
 916 
 
 600 
 
 2580 
 
 4013 
 
 269 
 379 
 189 
 305 
 200 
 860 
 1338 
 
 628 
 893 
 447 
 719 
 471 
 2026 
 3152 
 
 209 
 298 
 149 
 239 
 157 
 675 
 1050 
 
 Ash 
 
 Elm 
 
 Pitch-pine 
 
 Pine 
 
 
 Wroui'lit-iron 
 
 
 To find the ultimate transverse strength of any rectangular 
 beam, supported at both ends and loaded in the middle, 
 
 Multiply the strength of an inch square bar 1 foot long, as in the 
 Table, by the breadth, and by the square of the depth, in inches, and 
 divide the product by the length, in feet. 
 
 The quotient will be the weight in avoirdupois pounds. 
 
 Remark 1. — When a beam is supported in the middle and loaded at each end, 
 it will bear the same weiglit as when supported at both ends and loaded in the 
 middle ; that is, each end will boar half the weight. 
 
 Remark 2. — When the Aveight is applied somewliero between the middle and 
 the end of the beam, multiply twice the length of the long end by twice the length of 
 the short end, and divide the product by the whole length of the beam. The quotient^, 
 is the effective length of the beam. 
 
 Remark 3. — If the beam is round, multiply the ultimate strength of the round bar, 
 in the Table, by the cube of the diameter, in inches, and divide the product by the 
 length, IN FKET. 
 
 Remark 4. — When a beam is fixed at both ends and loaded in the middle, it 
 will bear one half more than when loose at both ends. If the beam is loose at 
 both ends, and tho weight is applied uniformfy along its length, it will bear 
 double; but if fixed at both ends, and tho weight applied uniformly along its 
 length, it will bear triple tho weight. 
 
 EXAMPLES. 
 
 1. What weight will break a beam of asli 5 inches broad, 7 inches 
 deep, and 26 feet deep between the supports? 
 
 1137X^X7^ 
 
 Ans. 
 
 :111431b., nearly. 
 
APrENDix. 355 
 
 2. What is the ultimate strength of an oak Lcam 20 feet long, 
 4 inches broad, 8 inches deep, and the weight placed G feet from 
 
 the end? ??^ =10.8 feet. (Vide Rem. 2.) Then, 
 
 800X4X8^^^,,,^,,^ ^,,. 
 lb,8 
 
 3. What is the ultimate transverse strength of a wrought-iron 
 solid cylinder, 10 feet long and 5 inches in diameter ? 
 
 Ans. "-^ =39400 lb. (Vide Rem. 3.) 
 
 ANNUITIES. 
 
 An Annuity is an estate which entitles its owner to the pay- 
 ment of a fixed sum, at regular intervals of time. 
 
 The annuity, time, and rate of interest being given, to find the 
 amount, 
 
 Raise the ratio to a power denoted by the time, from which subtract 1; 
 divide the remainder by the ratio less 1, and multiply the quotient by the 
 annuity. The product will be the amount. 
 Remabk 1. — For the powers of the ratio, see Table under 249. 
 
 EXAMPLES. 
 
 1. What is the amount of a pension of $100 per annum, which 
 has remained unpaid for 5 years, interest 6 per cent.? 
 
 1.06^=1.338226. Then .338226--.06Xl00=$563.71. Ans. 
 
 2. What is the amount of an annual rent of $150, in arrears 
 for 12 years, at 6 per cent, compound interest? 
 
 Ans. $2530.489999. 
 
 3. Wliat is the amount of a pension of $900, in arrears 17 years, 
 at 7 per cent, compound interest? Ans. 27756.193. 
 
 4. What is the amount of an annual salary of $6000, in arrears 
 for 8 years, at 5 per cent, compound interest? 
 
 Ans. $57294.60. 
 
 5. What is the amount of a $100 pension, in arrears 20 years, 
 at 5 per cent.? 6 per cent.? 7 per cent.? 
 
 Ans. $3306.596 ; $3678.558 ; $4099.55. 
 G. What is the amount of a pension of $1000, in arrears for 12 
 years, at 7 per cent, compound interest? Ans. $17888.45. 
 
356 
 
 APPENDIX. 
 
 The annuity, time, and rate of interest being given, to find the 
 present worth, 
 
 Divide the amount^ as found ahove^ hy the ratio raised to a power 
 denoted hy the time. 
 
 EXAMPLES. 
 
 1. What is the present worth of a pension of $100, to continue 
 5 years, at 6 per cent, per annum? 
 
 ^563.71-f-1.0G'^=$421.24. Ans. 
 Remauk 2. — In this way the following Table may be constructed : 
 
 v. — TABLE: 
 
 Shoioing the presetit value of $1.00 for any number of years, from 1 to 25, at 5, G, 
 7, 8, and 10 per cent. 
 
 Yeaes 
 
 5 Per Cent. 
 
 6 Per Cent. 
 
 7 Per Cent. 
 
 8 Per Cent. 
 
 10 Per Cent. 
 
 1 
 
 0.952381 
 
 0.943396 
 
 0.934579 
 
 0.925926 
 
 0.909091 
 
 2 
 
 1.859410 
 
 1.833393 
 
 1.808018 
 
 1.783265 
 
 1.735537 
 
 3 
 
 2.723248 
 
 2.673012 
 
 2.624316 
 
 2.577097 
 
 2.486852 
 
 4 
 
 3.545951 
 
 3.465106 
 
 3.387211 
 
 3.312127 
 
 3.169865 
 
 5 
 
 4.329477 
 
 4.212364 
 
 4.100197 
 
 3.992710 
 
 3.790787 
 
 6 
 
 5.075692 
 
 4.917324 
 
 4.766540 
 
 4.622880 
 
 4.455261 
 
 7 
 
 5.786373 
 
 5.582381 
 
 5.389289 
 
 5.206370 
 
 4.868419 
 
 8 
 
 6.463213 
 
 6.209794 
 
 5.971299 
 
 5.746639 
 
 5.334926 
 
 9 
 
 7.107822 
 
 6.801692 
 
 6.515232 
 
 6.246888 
 
 5.759024 
 
 10 
 
 7.721735 
 
 7.360087 
 
 7.023582 
 
 6.710081 
 
 6.144557 
 
 11 
 
 8.306414 
 
 7.886875 
 
 7.498674 
 
 7.138964 
 
 6.495061 
 
 12 
 
 8.863252 
 
 8.383844 
 
 7.942686 
 
 7.536078 
 
 6.813692 
 
 13 
 
 9.393573 
 
 8.852683 
 
 8.357651 
 
 7.903776 
 
 7.103356 
 
 14 
 
 9.898641 
 
 9.294984 
 
 8.745468 
 
 8.244237 
 
 7.366687 
 
 15 
 
 10.379658 
 
 9.712249 
 
 9.107914 
 
 8.559479 
 
 7.606080 
 
 16 
 
 10.837770 
 
 10.105895 
 
 9.446649 
 
 8.851369 
 
 7.823701 
 
 17 
 
 11.274066 
 
 10.477260 
 
 9.763223 
 
 9.121638 
 
 8.021553 
 
 18 
 
 11.689587 
 
 10.827603 
 
 10.059087 
 
 9.371887 
 
 8.201412 
 
 19 
 
 12.085321 
 
 11.158116 
 
 10.335595 
 
 9.603599 
 
 8.364920 
 
 20 
 
 12.462210 
 
 11.469921 
 
 10.594014 
 
 9.818147 
 
 8.513564 
 
 21 
 
 12.821153 
 
 11.764077 
 
 10.835527 
 
 10.016803 
 
 8.648694 
 
 22 
 
 13.W3003 
 
 12.041582 
 
 11.061241 
 
 10.200744 
 
 8.771540 
 
 23 
 
 13.488574 
 
 12.303379 
 
 11.272187 
 
 10.371059 
 
 8.883218 
 
 24 
 
 13.798642 
 
 12.550358 
 
 11.469334 
 
 10.528758 
 
 8.984744 
 
 25 
 
 14.093945 
 
 12.783356 
 
 11.653583 
 
 10.074776 
 
 9.077040 
 
 2. What is the present worth of a pension of $800, to continue 
 25 years, at 10 per cent.? Ans. $2723.11. 
 
APPENDIX. 
 
 357 
 
 MISCELLANEOUS TABLE. 
 
 12 units make 1 dozen. 
 
 12 dozen " 1 gross. 
 
 12 gross " 1 great gross. 
 
 20 units " 1 score. 
 
 100 years " 1 century. 
 
 10 centuries " 1 chiliad. 
 
 100 pounds " 1 quintal of fisli. 
 
 196 pounds " 1 barrel of flour. 
 
 200 pounds " 1 barrel of pork. 
 
 14 pounds " 1 stone, 
 
 21J stones " 1 pig- 
 
 8 pigs " 1 fother. 
 
 18 inches " 1 cubit. 
 
 6 feet " 1 fathom. 
 
 24 sheets " 1 quire. 
 
 20 quires r... " • 1 ream. 
 
 2 reams " 1 bundle. 
 
 5 bundles " 1 bale. 
 
 FRENCH WEIGHT. 
 
 Frexch weight is that used in the empire of France. Its units 
 are named milligramme, centigramme, decigramme, gramme, decagramme, 
 hectogrmme, kilogramme, myriagramme, quintal, millier or bar. The 
 milligramme is the unit of lowest value, and the bar the highest. 
 It takes 10 of any order to make one of the next higher order of 
 units, except that 100 quintals make 1 bar. 
 
 The GRAMME is the fundamental unit, and is the weight of a 
 centimetre of pure water, at the temperature of melting ice, which 
 is 15.43402 grains Troy. 
 
 1 quintal = 1 cwt. 3 gr. 25 lb.; 1 millier or bar = 9 T. 16 cwt. 
 3 gr. 12 lb. 
 
 1 pound Avoirdupois = 453| 
 
 1 pound Troy 
 grammes. 
 
 FRENCH LINEAR MEASURE. 
 
 The units of this measure are named millimetre, centimetre, deci- 
 metre, METRES, decametre, hectometre, kilometre, and myriametre; and 
 it takes 10 units of any lower order to make 1 of the next higher. 
 
358 APPENDIX. 
 
 The METRE is the fundamental unit, and one of the ten million 
 equal parts into which the meridian distance from the equator 
 to the north pole is divided. This distance is 39.37079 English 
 inches. It is thence easy to find the value of any French unit in 
 English measure. 
 
 FRENCH SUPERFICIAL MEASURE. 
 
 The units of this measure are named milliare, centiare^ deciare, 
 ARE, decare, hectare, kilare, and myrlare; and it takes 10 units of a 
 lower order to make 1 of the next higher. 
 
 The ARE is the fundamental unit, and is a square decametre^ 
 which is equivalent to 1076.4298 square feet. 
 
 FRENCH SOLID MEASURE. 
 
 The units of this measure are named centistere, decistere, stere, 
 and decastere; and it takes 10 units of a lower order to make 1 of 
 the next higher. 
 
 The stere is the fundamental unit, and is a cubic metre, which 
 is equivalent to 35.3174 cubic feet. 
 
 FRENCH MEASURE OF CAPACITY. 
 
 The units of this measure are named millitre, centilitre, decilitre, 
 LITRE, decalitre, hectolitre, kilolitre, and myrialitre. 
 
 The LITRE is the fundamental unit, and is a cubic decimetre, 
 which is 61.027051 cubic inches. 
 
 1 litre = 1| English pints; 1 hectolitre = 22 English gallons. 
 
 1 decalitre = 2.6414308 wine gallons; 1 hectolitre =2.834 Win- 
 chester bushels. 
 
 RATES OF FOREIGN MONEY OR CURRENCY. 
 
 (FIXED BY LAW.) 
 
 Ducat of Naples $0 80 
 
 Florin of the Netherlands 40 
 
 Florin of the Southern States of Germany 40 
 
 Florin of Austria and Trieste 48J 
 
 Florin of Nuremburg and Frankfort 40 
 
 Florin of Bohemia 48^ 
 
 Guilder of the Netherlands 40 
 
 Lira of Lombardo and the Vcuetiuu Kingdom 16 
 
ArpENDix. 3-59 
 
 Livre of Leghorn $0 16 
 
 Lira of Tuscany 16 
 
 Lira of Sardinia 18f 
 
 Livre of Geneva 18f 
 
 Milrea of Portugal 1 12 
 
 Milrea of ]\Ladeira 1 00 
 
 Milrea of Azores 83i- 
 
 Marc Banco of Hamburg 35 
 
 Ounce of Sicily 2 40 
 
 Pound Sterling of Jamaica 4 84 
 
 Pound Sterling of the British Provinces 4 00 
 
 Pagoda of India 1 84 
 
 Real Vellon of Spain 05 
 
 Real Plate of Spain 10 
 
 Rupee of British India 44| 
 
 Rix Dollar (or Thaler) of Prussia and North Germany.. 69 
 
 Thaler (or Rix Dollar) of Bremen 78| 
 
 Thaler (or Rix Dollar; of Berlin, Saxony, and Leipsic... 69 
 
 Rouble (Silver) of Russia 75 
 
 Specie Dollar of Denmark 1 05 
 
 Specie Dollar of Norway 1'06 
 
 Specie Dollar of Svreden 1 06 
 
 Tale of China 1 48 
 
 Banco Rix Dollar of Sweden and Norway 39| 
 
 Banco Rix Dollar of Denmark 53 
 
 Crown of Tuscany 1 05 
 
 Curacoa Guilder 40 
 
 Leghorn Dollar or Pezzo ^Oj^iny 
 
 Livre of Catalonia 53| 
 
 Livre of N\ifchatel 26| 
 
 Swiss Livre 27 
 
 Scudi of Malta 40 
 
 Roman Scudi 99 
 
 St. Gall Guilder 403-^^^ 
 
 Rix Dollar of Batavia 75 
 
 Roman Dollar 1 05 
 
 Turkish Piastre 05 
 
 Current Mark 28 
 
 Florin of Prussia..! , 22f 
 
360 APPENDIX. 
 
 Florin of Basle $0 41 
 
 Genoa Livre 21 
 
 Livre Tournois of France. 18| 
 
 Rouble (paper) of Russia, varies from 4j^^^ to i^^^jj to the dollar, 
 
 BOOKS AND PAPER. 
 
 NAMES AND SIZES OF PAPER MADE BY MACHINERY. 
 
 Double Imperial 32 by 44 inches. 
 
 Double Superroyal 27 by 42 " 
 
 Double Medium 23 by 26 " 
 
 " " 24 by 37i « 
 
 " " 25 by 38 " 
 
 Royal and Half 25 by 29 " 
 
 Imperial and Half 26 by 32 « 
 
 Imperial 22 by 32 « 
 
 Superroyal 21 by 27 « 
 
 Royal 19 by 24 « 
 
 Medium ^. ^ 181 by 23| « 
 
 Demy ." 17 by 22 " 
 
 Folio Post ,. ♦ 16 by 21 " 
 
 Fo?4^5ap.....^...':..:?^^ ,| 14 by 17 « 
 
 fo-wn 
 
 Cro-wl....c:r:../ ^!V.... 15 by 20 " 
 
 ^ sheet folded in 2 leaves is called -dT folio. 
 
 A sheet '' 4 ' "" " ^^ quarto or 4to, 
 
 A sheet " 8 ''■" "an octavo or 8vo. 
 
 ' A'jshee'f , " 12 " "^ " a 12rao. 
 
 A sheet " 18 " " an 18mo. 
 
 A sheet " 24 " ** a 24mo. 
 
 A sheet « 32 » " a 32mo. 
 
 THE END. 
 
s 
 
 m 17439 
 
 M305987 
 
 T 5'9 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
 m ■ 
 
 liliiiililiill' 
 
■n 
 
 -3^ 
 
 
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