y y 8,r M^ o' WFvLI Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/academicarithmetOOwellrich * WELLS' MATHEMATICAL SERIES. Academic Arithmetic. Academic Algebra. Higher Algebra. University Algebra. College Algebra. Plane Geometry. Solid Geometry. Plane and Solid Geometry. Plane and Solid Geometry. Revised. Plane and Spherical Trigonometry. Plane Trigonometry. Essentials of Trigonometry. Logarithms (flexible covers). Elementary Treatise on Logarithms. Special Catalogue and Terms on application. *- AN Academic Arithmetic ACADEMIES, HIGH AND COMMERCIAL SCHOOLS. BY WEBSTER WELLS, S.B., PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY. LEACH, SHEWELL, & SANBOEK BOSTON. NEW YORK. CHICAGO. COPYBIGHT, 1893. By WEBSTER WELLS. NarbJootJ 3Pre88 : J, S. Gushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. PREFACE rr^HE present work is intended to furnish, a thorough. -*- course in all those portions of Arithmetic which are required for admission to any college or scientific school. The pupil is assumed to have already studied the more elementary parts of the subject in a text-book of lower grade ; and only enough examples are given in the earlier chapters to afford material for a review. Great pains have been taken, in the selection of exam- ples and problems, to illustrate every important arithmet- ical process; and in Chapter XXV. there will be found a set of miscellaneous problems of somewhat greater diffi- culty than those in the preceding chapters, furnishiug a complete review of the entire subject. The chapters on the Metric System have been arranged in such a way that they may be taken after the other portions of the work have been studied, or omitted alto- gether, at the option of the teacher. No examples involv- ing a knowledge of the Metric Systepi are given, except in Chapter XIII., and Arts. 256, 257, and 378. The Appendix contains topics of minor importance to the majority of pupils, but still liable to be called for in college entrance examinations. WEBSTER WELLS. Mass. Institute of Technology, 1893. iii ' O O IG CONTENTS. PAGB I. Notation and Numeration 1 The Arabic System of Notation 1 The Koman System of Notation 5 II. Addition 6 III. Subtraction 9 Parentheses 11 IV. Multiplication 12 V. Division 18 VI. Factoring 24 Casting out Nines 29 Casting out Elevens ■ 32 VII. Greatest Common Divisor 33 VIII. Least Common Multiple 38 IX. Fractions 43 Reduction of Fractions 44 Addition of Fractions 51 Subtraction of Fractions 53 Multiplication of Fractions 55 Division of Fractions 58 Complex Fractions 61 Greatest Common Divisor of Fractions .... 65 Least Common Multiple of Fractions .... 66 Miscellaneous Examples 66 Problems 68 V vi CONTENTS. PAGE X. Decimals 74 To reduce a Decimal to a Common Fraction . . 76 Addition of Decimals 76 Subtraction of Decimals 77 Multiplication of Decimals 78 Division of Decimals 81 To reduce a Common Fraction to a Decimal . . 84 Circulating Decimals 86 To multiply or divide a Number by an Aliquot Part of 10, 100, 1000, etc 88 Miscellaneous Examples . . • 89 United States Money 90 Problems 93 XI. Measures . 98 XII. Denominate Numbers . 103 Reduction of Denominate Numbers 103 Addition of Denominate Numbers , 107 Subtraction of Denominate Numbers 109 To find the Difference in Time between Two Dates 110 Multiplication of Denominate Numbers . . . . Ill Division of Denominate Numbers 112 To express a Fraction or Decimal of a Simple Num- ber in Lower Denominations 115 To express a Denominate Number as a Fraction or Decimal of a Single Denomination 116 To express One Denominate Number as a Fraction or Decimal of Another 117 Longitude and Time 118 Problems 120 XIII. The Metric System 125 Metric Numbers . • 130 Miscellaneous Problems" 135 XIV. Involution and Evolution 138 Square Root 140 Cube Root 144 CONTENTS. Vll PAGE XV. Mensuration 151 Plane Figures 151 Solids 160 Application of Mensuration 171 Capacity of Bins, Tanks, and Cisterns .... 171 Carpeting Rooms 172 Plastering and Papering 174 Board Measure 176 Measurement of Round Timber 177 Specific Gravity 178 Geometrical Explanation of Square and Cube Root 179 Problems in Mensuration Involving the Metric Sys- tem 181 XVI. Ratio and Pkoportion 191 Proportion 192 Properties of Proportions 193 Problems 194 Compound Proportion 197 Partitive Proportion 201 Similar Surfaces and Solids 203 XVII. Partnership 206 Simple Partnership 206 Compound Partnership 207 XVIII. Percentage 210 Applications of Percentage 221 Trade Discount 221 Commission and Brokerage 224 Insurance 227 Taxes 229 Duties 231 XIX. Interest 234 Simple Interest 234 The Six Per Cent Method 236 Exact Interest 245 Promissory Notes 246 Partial Payments 248 Compound Interest 252 Annual Interest 258 viii CONTENTS. PAQB XX. Discount 259 True Discount 259 Bank Discount 260 XXI. Exchange 265 Domestic Exchange 267 Foreign Exchange 270 XXII. Equation of Payments 274 Average of Accounts 276 The Interest Method 282 XXIII. Stocks and Bonds 284 XXIV. Progressions 292 Arithmetical Progression 292 Geometrical Progression 295 Compound Interest 298 Annuities 299 XXV. Miscellaneous Examples 303 Miscellaneous Examples Involving the Metric Sys- tem 317 Appendix 321 Measures 321 Difference in Time between Two Dates .... 323 Comparison of Thermometers 323 Money and Coins 325 Legal Rates of Interest 328 Special State Kules for Partial Payments . . . 328 The Connecticut Rule 328 The New Hampshire Rule for Partial Payments on a Note or Other Obligation, Drawing Annual Interest 330 The Vermont Rule 331 To compute Interest on English Money .... 332 Business Forms 333 Savings Bank Accounts 334 Scales of Notation 336 ARITHMETIC. I. NOTATION AND NUMERATION. 1. Let us consider a collection of things of the same kind ; for example, a collection of books. In order to find out how many books there are in the collection, we proceed to count them, as follows : We take any book, and call it one; we then take another and call it two; the next we call three; the next /o?^r; then Jive, six, seven, eight, nine, ten, eleven, twelve, and so on until all have been taken. The expressions one, two, three, etc., used in the above process are called Whole Numbers, or Integers. 2. Notation signifies the representation of numbers by means of symbols. Numeration signifies the reading of numbers when ex- pressed in symbols. THE ARABIC SYSTEM OF NOTATION. 3. In the Arabic System, the numbers one, two, three, four. Jive, six, seven, eight, and nine, are represented by the symbols 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. The symbol 0, read zero, cipher, or naught, when standing by itself, signifies nothing. The above symbols are called Figures. Zero, and the numbers one, two, three, four, five, six, seven, eight, and nine, are called Digits. 1 2 ARITHMETIC. 4. Numbers greater than nine are represented by writing side by side two or more of the above figures. The last figure at the right is said to be in the Jirst place, and denotes ones or units. The figure just before the last is said to be in the second place, and denotes tens. Thus, 10 signifies one ten and no ones; that is, ten. 11 signifies one ten and orie 07ie; that is, eleven. 12 signifies one ten and tivo ones; that is, twelve. In like manner, 13, 14, 15, 16, 17, 18, and 19 represent the next seven numbers in order ; that is, thirteen, fourteeyi^ fif- teen, sixteen, seventeen, eighteen, and nineteen. Two tens are called twenty, and represented by 20. The next nine numbers in order are represented by 21, 22, and so on to 29 ; and read twenty-one, twenty-two, and so on to twenty-nine, respectively. Three tens are called thirty; four tens, forty; five tens, fifty; si:s. tens, sixty ; seven tens, seventy ; eight tens, eighty ; nine tens, ninety; and rejjresented by 30, 40, 50, 60, 70, 80, and 90, respectively ; and in each case the next nine num- bers in order are named and represented in a manner similar to that employed for numbers between 20 and 30. 5. Ten tens are called one hundred. A figure in the third place denotes hundreds. Thus, 100 signifies one hundred, no tens, and no ones; that is, one hundred. 101 signifies one hundred, no tens, and one one; and is read one hundi^ed and one. 783 signifies seven hundreds, eight tens, and three ones; and is read seven hundred and eighty-three. In like manner, any number from one hundred to nine hundred and. ninety-nine, may be represented by three figures. 6. The following table gives the signification of each of the first seven places : NOTATION AND NUMERATION. 8 1st; ones. 2d; tens. 3d; hundreds. 4tli ; tens of hundreds, or thousands. 5th ; tens of thousands. 6th ; tens of tens of thousands, or hundreds of thousands. 7th ; tens of hundreds of thousands, or millions. Thus, 7306592 signifies 7 millions, 3 hundreds of thou- sands, no tens of thousands, 6 thousands, 5 hundreds, 9 tens, and 2 ones. Note. It will be understood hereafter that, when the digits of a number are spoken of, we mean the numbers represented by its figures, without regard to the places which they occupy. Thus, the digits of 352 are 3, 5, and 2, and not 300, 50, and 2. 7. The general law exemplified in Art. 6 may be stated as follows : Any place signifies tens of the numbers signified by the next place to the right. 8. For convenience of reading, places are divided into periods of three places each. The first, second, and third places form the first or units^ period ; the fourth, fifth, and sixth form the second or thou- sands^ period; the third three, the millions' period; the fourth three, the billions^ period. The table gives the designation of each of the first four- teen periods : Period. Dbsigkation. Period. Designation. First. Units. Eighth. Sextillions. Second. Thousands. Ninth. Septillions. Third. Millions. Tenth. Octillions. Fourth. Billions. Eleventh. Nonillions. Fifth. Trillions. Twelfth. Decillions. Sixth. Quadrillions. Thirteenth. Undecillions. Seventh. Quintillions. Fourteenth. Duodecillions. 4 ARITHMETIC. Thus, the number 23,016,797,681 is read twenty-three billion, sixteen million, seven hundred and ninety-seven thousand, six hundred and eighty-one. Note. The above is the usual system of numeration. In the Eng- lish system, the second six places form the millions' period, the third six the billions' period, etc. Thus, according to the English system, the number 57,608,351,000,000 would be read fifty-seven billion, six hundred and eight thousand three hundred and fifty-one million. EXAMPLES. 9. Read the following numbers : 1. 2705618. 6. 144710325046728. 2. 6520741869. 7. 9080600713256. 3. 101294705. 8. 280115769001342. 4. 78220615437. 9. 294007386045. 6. 35400986. 10. 48520010964700. Write the following numbers in figures : 11. One million, three hundred and twenty-five thousand, seven hundred and twenty-six. 12. Five billion, seven hundred and eighty thousand, two hundred and five. 13. Seventy-nine million, one hundred and sixty thou- sand, and four. 14. Sixty-five billion, eight hundred and three million, one hundred and eighty-nine thousand, four hundred and fifty. 15. Three hundred and fifty -six million, eighty-one thou- sand, six hundred and twelve. 16. Two hundred and thirty-five billion, nine hundred and twenty-seven. 17. Eighty-five billion, two hundred and seventy-seven million, six thousand, one hundred. 18. Four trillion, one hundred and sixty billion, twenty- seven million, one hundred and sixteen thousand, and eighty- ihree. NOTATION AND NUMERATION. 5 19. Seven quadrillion, eight hundred and twenty-five trillion, four hundred and sixy-three million, four hundred and forty-five. 20. Nine hundred quintillion, five hundred and twenty billion, seventy thousand, three hundred and fourteen. THE ROMAN SYSTEM OF NOTATION. 10. In the Roman System, the numbers one, Jive, ten, fifty, one hundred, five hundred, and one thousand are represented by the letters I, V, X, L, C, D, and M, respectively. Numbers other than the above are represented by writ- ing side by side two or more of the above letters. When thus expressed, if a letter is written after another letter of the same or of greater value, the sum of their values is represented ; if a letter is written before another letter of greater value, the difference of their values is represented. The following table shows the methods usually employed for representing numbers up to five thousand : Roman. Arabic. Roman. Arabic. Roman. Arabic. I 1 XV 15 xc 90 II 2 XVI 16 c 100 III 3 XVII 17 CI 101 IV 4 XVIII 18 cc 200 V 5 XIX 19 ccc 300 VI 6 XX 20 cccc 400 VII 7 XXI 21 D 500 VIII 8 XXII 22 DC 600 IX 9 XXX 30 DCC 700 X 10 XL 40 DCCC 800 XI 11 L 50 DCCCC 900 XII 12 LX 60 M 1000 XIII 13 LXX 70 MD 1500 XIV 14 LXXX 80 MM .2000 Note. The Roman Method is now rarely used except iornuin- hering chapters of books, hours on clock-dials, etc arithmp:tic. II. ADDITION. 11. To Add two whole numbers is to count upwards from either of the numbers as many units as there are in the other. Thus, to add 3 and 5, we count upwards from ^Jive units as follows: 4, 5, 6, 7, 8; the result is 8. In like manner, we may add three or more numbers. Thus, to add 7, 4, and 8, we first count upwards from 7 four units, and then count upwards 8 units from the result. The answer is 19. Note. The order in which the numbers are taken is immaterial. 12. The result of addition is called the Sum. 13. The symbol +, read ''plus'^ or ^' and/' signifies addition. 14. The symbol = is read ^'equals,'' '•'is equal to" or ''are.'' Thus, 3 + 5=8 is read " three and five are eight." 15. 1. Find the sum of 396 and 842. 396 is the same as 3 hundreds, 9 tens, and 6 units, and 842 the same as 8 hundreds, 4 tens, and 2 units. ^'*'^ We write the numbers so that units, tens, and 1238 Ans hundreds shall be in the same vertical columns. The sum of 2 units and 6 units is- 8 units. We then write 8 under the column of units. The sum of 4 tens and 9 tens is 13 tens, or 1 hundred and 3 tens. We then write 3 under the column of tens, and camj the 1 hundred mentally to the column of hundreds. The sum of 1 hundred, 8 hundreds, and 3 hundreds is 12 hundreds, or 1 thousand and 2 hundreds. We then write 2 under the column of hundreds, and 1 in the thou- sands' place of the answer. Then the required result is 1 thousand, 2 hundreds, 3 tens, and 8 units, or 1238. ADDITION. 7 2. Add 546, 97, 384, and 780. Q_ It IS customary in practice to name results only when adding columns ; thus, in Ex. 2, we say " 4, 11, ^84: 17 '' . write the 7, and carry 1. 780 Then, " 1, 9, 17, 26, 30 " ; write the and carry 3. W7,Ans. Then, "3, 10, 13, 18." From the above examples, we derive the following RULE. Write the numbers so that units, tens, hundreds, etc, shall he in the same vertical columns. Add the digits in the units' column. If the residt is less than 10, write it under the column of units; hut if it is just 10, or more than 10, write the units of the sum under the column of units, and carry the tens mentally to the next column to the left. Proceed in a similar manner with each of the remaining columns., and write under the last column its entire sum. Note. The work may be proved by performing the example a second time ; adding the columns from top to bottom, instead of from bottom to top. Another method of proof is to separate the numbers into two parts by a horizontal line. Adding the numbers above the line, then the numbers below, and then these two sums, the result should agree with that previously obtained. 16. In practice, computers frequently add two columns at once ; thus, in the following example, 3527 8448 1759 2872 16606 we should say 72, 131, 179, 206; write 06, and carry 2; then, 2, 30, 47, 131, 166. 8 ARITHMETIC. 17. Only quantities of the same kind can he added. Thus, the sum of 7 hooks and 8 hooks is 15 books; but it is not possible to add 7 hooks and 8 miles. EXAMPLES. 18. Add the following 1. 2. 3. 4. 5. 1789 5403 4529 7854 1827 6543 786 7992 6215 4329 2177 9230 467 9448 25070 915 1157 8920 4007 6118 6783 898 3508 651 2522 325 7526 7. 2463 8. 5869 9. 3909 6. 10. 79856 45340 43765 59864 77167 35117 10087 89140 86723 63489 32949 76322 35174 48213 68791 18817 36450 64385 54876 74153 85622 78809 79160 . 83538 84375 16304 39713 23099 7176^ 97561 III. SUBTRACTION. 19. To Subtract one whole number from another is to count downwards from the second number as many units as there are in the first. Thus, to subtract 5 from 8, we count downwards from 8 Jive units, as follows : 7, 6, 5, 4, 3 ; the result is 3. 20. The number to be subtracted is called the Subtrahend. The number from which the subtrahend is to be sub- tracted is called the Minuend. The result is called the Remainder or Difference. 21. The symbol — , read '^minus'^ or "less," signifies subtraction. Thus, 8 — 5 = 3 is read " eight less five are three." 22. It is evident from Art. 19 that, if we count upwards from the Eemainder as many units as there are in the Sub- trahend, we shall obtain the Minuend. That is, the Minuend is the sum of the Subtrahend and Eemainder. 23. 1. Subtract 483 from 758. 758 is the same as 7 hundreds, 5 tens, and 8 units, and 483 the same as 4 hundreds, 8 tens, and 3 units. 483 ^j^Q write the subtrahend under the minuend so that 275 Ans ^^i^s, tens, and hundreds shall be in the same vertical columns. 3 units from 8 units leave 5 units. We cannot take 8 tens from 5 tens ; but we can take 1 hundred, or 10 tens, from the 7 hundreds of the minuend, leaving 6 hundreds ; and adding the 10 tens to the 5 tens, we have 15 tens ; then 8 tens from 15 tens leave 7 tens. Finally, 4 hundreds from 6 hundreds leave 2 hundreds. Then the required result is 2 hundreds, 7 tens, and 5 units, or 275. 10 ARITHMETIC. Now instead of taking 1 hundred from the 7 hundreds of the minuend, in the above example, we may get the same result as follows : Adding 1 hundred to the 4 hundreds of the subtrahend, we have 5 hundreds. Then, 5 hundreds from 7 hundreds leave 2 hundreds. This second method is far preferable to the first. From the second method of the above example, we derive the following RULE. Write the subtrahend under the minuend, so that units, tens, hundreds, etc., shall he in the same vertical columns. Subtract the right-hand digit of the subtrahend from the digit above it, and write the result under the column of units. If the right-hand digit of the subtrahend is greater than the digit above it, increase the latter by 10 before subtracting; and add 1 mentally to the digit in the tens' place of the sub- trahend. Proceed in a similar manner with each of the remaining digits of the subtrahend in order. Note 1. If the minuend has more places than the subtrahend, we may make the nuniber of places in the latter the same as in the .former by mentally supplying ciphers in the missing places. 2. Subtract 3728 from 571000. 571000 In this case we say, "8 from 10 leaves 2; 3 3728 from 10 leaves 7 ; 8 from 10 leaves 2 ; 4 from 11 567272, Ans. l^^v^s 7 ; 1 from 7 leaves 6." Note 2. Since the minuend is the sum of the subtrahend and remainder (Art. 22), the work may be proved by adding the subtra- hend to the remainder ; the result should equal the minuend. 24. Only quantities of the same kind can be subtracted. Thus, 16 books less 7 books are 9 books; but it is not possible to subtract 7 books from 16 miles. SUBTRACTION. 11 EXAMPLES. 25. Subtract the following : 1. 2. 3. 4. 5. 6372 5034 8000 9037 48609 2177 786 1256 3409 9085 6. 7. 8. 9. 10. 75816 40709 58000 88713 64751 38912 36090 49374 73536 10968 PARENTHESES. 26. A Parenthesis, ( ), signifies that the numbers enclosed by it are to be taken collectively. Thus, 17— (8 + 4) signifies that 8 and 4 are to be added together, and their sum subtracted from 17. The Vinculum, , has the same force as a parenthesis. Thus, 33 — 11 + 19 — 5 signifies that 11 is to be sub- tracted from 33, then 5 from 19, and the second result added to the first. EXAMPLES. Eind the values of the following : 1. 32 -(13 + 7). 4. 122 -(97 -69). 2. 50-17 + 35-16. 5. 171-119 + 137-88.. 3. (45 + 18) -(22 + 9). 6. (926 -265) -(284 + 198). 7. (823 -486) -75^ -515. 8. (132 -74) + 115 + 97 -(183 -66). 9. 1088 - 905 - 323 - 479 - 741 - 262. Note. Brackets, [ ], and Braces, { }, liave the same force as parentheses. 12 ARITHMETIC. IV. MULTIPLICATION. 27. To Multiply one whole number by another is to take the first number as many times as there are units in the second. Thus, to multiply 3 by 5, we take 3 five times, as follows : 3 + 3 + 3 + 3 + 3; the result is 15. 28. The number taken is called the Multiplicand. The number which shows how many times the multipli- cand is taken, is called the Multiplier. The result of multiplication is called the Product. 29. The symbol x, read ''times/' signifies multiplication. Thus, 3 X 5 = 15 is read " three times five are fifteen." 30. To multiply 5 by 3, we take 5 three times, as follows : 5 + 5 + 5 ; the result is 15. That is, 3 X 5 is equal to 5 x 3. It is evident from this that, in finding the product of two numbers, either may be regarded as the multiplicand, and the other as the multiplier. 31. To multiply together three or more numbers, we multiply the first number by the second, the product by the third number, and so on until all have been taken. It is evident, as in Art. 30, that the order in which the numbers are multiplied is immaterial. 32. If any number of tilings of the same kind be multi- plied by a whole number, the product will be things of the same kind as the multiplicand. Thus, 6 times 7 hooks are 42 books.' But it is not possible to take 6 books times 7 books, nor to multiply 6 by 7 books. 33. The products of the numbers from 1 to 12 inclusive, taken two and two, are given in the following table. MULTIPLICATION. 13 MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 60 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 11 22 33 44 55 66 77 88 99 110 12 24 36 48 60 72 84 96 108 120 11 12 22 24 33 36 44 48 55 60 66 72 77 84 88 96 99 108 110 120 121 132 132 144 The arrangement of tlie table is as follows : To find the product of 7 and 9, look in the left-hand ver- tical column for 7 ; then the product required will be found in the corresponding horizontal line in the column headed 9 ; the result found is 63. 34. 1. Multiply 462 by 7. Multiplicand, 462 462 is the same as 4 hundreds, 6 Multiplier, 7 ^ens, and 2 units. T> J i. oooT A We write the multiplier under the Product, 3234, Ans. .^ , _ . ^. .;. ,. , ' ' units' figure of the multiplicand. 7 times 2 units are 14 units, or 1 ten and 4 units. We then write 4 under the' column of units. 7 times 6 tens are 42 tens ; adding to this the 1 ten reserved from the 14 units, we have 43 tens, or 4 hundreds and 3 tens. We then write 3 under the column of tens. 14 ARITHMETIC. 7 times 4 hundreds are 28 hundreds ; adding to thi^ the 4 hundreds reserved from the 43 tens, we have 32 hundreds, or 3 thousands and 2 hundreds. Writing 2 under the column of hundreds, and 3 in the thousands' place of the product, the required result is 3234. It is customary to use the following words only in ex- plaining the above process : 7 times 2 are 14 ; write the 4, and "carry" 1 ; 7 times 6 are 42, and 1 are 43 ; write the 3, and carry 4 ; 7 times 4 are 28, and 4 are 32. 2. Multiply 743 by 685. Multiplicand, 743 We multiply 743 first by. 5 Multiplier 685 units, then by 8 tens, and finally 1st partial' product, 3715 ^^ ^ hundreds, and add the oj i.- 1 3 J ^f^AAf^ partial products. 2d partial product, 59440 743 ^^^^^ ^ ^^^^ ^^^ 37^5 3d partial product, 445800 units. Product, 508955, Ans. 743 times 8 tens are 5944 tens, or 59440. 743 times 6 hundreds are 4458 hundreds, or 445800. Adding 3715, 59440, and 445800, the required result is 508955. It is customary to arrange the written work as follows : 743 685 3715 5944 4458 508955, Ans. From the above example, we derive the following RULE. WHte the multiplier under the multiplicand, so that units, tens, hundreds, etc., shall be in the same vertical columns. Multiply the midtiplicand by the digit in the units^ place of the multiplier, and write the result under the multiplier so that its right-hand Jigure shall be under the units' figure of the multiplier. MULTIPLICATION. 15 Multiply the multiplicand by the digit in the tens' place of the multiplier, and write the result under the first partial product so that its right-hand figure shall be under the tens^ figure of the multiplier. Proceed in a similar manner with each of the remaining digits of the multiplier, and add the partial products. Note. The work may be proved by interchanging the multiplicand and multiplier. If any digit of the multiplier is 0, the corresponding partial product is 0, and is not expressed in the work. 3. Multiply 1725 by 309. 1725 We say in this case : 9 times 1725 are 15525, which 309 we write so that its right-hand figure shall be under 15525 the 9 of the multiplier ; then, 3 times 1725 are 5175, 5175 which we write so that its right-hand figure shall be 533025 A71S ^^^^^ ^^^ ^ o^ the multiplier. 35. To Multiply by 10, 100, 1000, Etc. To multiply a whole number by 10, 100, 1000, etc., we annex to the multiplicand as many ciphers as there are in the multiplier. Example. Multiply 356 by 1000. Annexing three ciphers to the multiplicand, we have 356 X 1000 = 356000, Ans. 36. To Multiply by any Number of Tens, Hundreds, Etc. Any number of ciphers at the right of the multiplier may be omitted during the operation of multiplication, and annexed to the result. 1. Multiply 2734 by 2600. 2734 2600 We say in this case : 6 times 2734 are 16404 ; 2 16404 times 2734 are 5468; adding, and annexing two 5468 ciphers to the result, the product is 7108400. 7108400, Ans. 16 ARITHMETIC. In like manner, ciphers at the right of the multiplicand may be omitted during the operation. 2. Multiply 63000 by 580. 68000 *^^^ We say in this case : 8 times 63 are 504 ; 5 504 times 63 are 315 ; adding, and annexing four 315 ciphers to the result, the product is 36540000. 36540000, Ans. 37. When two numbers are to be multiplied together, circumstances will often determine which to take as the multiplier. In general, the number having the least number of places should be taken as the multiplier; thus, to multiply 852 and 27, we should take 27 as the multiplier. If one of the numbers has two or more digits alike, it may be easier to take it as the multiplier ; thus, to multiply Q&Q and 329, it would be easier to take the former number as the multiplier. Again, if one of the numbers has ciphers or ones for digits, it may be shorter to take it as the multiplier; thus, to multiply 394 and 2001, it would be shorter to take the latter number as the multiplier. 38. Short Methods in Multiplication. To multiply by 99, 999, etc, we may proceed as follows : Example. Multiply 1652 by 999. 1652000 Since 999 is 1000 - 1, we multiply 1652 by 1000, ^bo^ and then by 1, and subtract the second result from 1650348, Ans. the first. In like manner, we may multiply by 98, 97, 998, 997, or by any number a little less than 100, 1000, 10000, etc. The same artifice may be employed when the multiplier is a little less than 200, 300, 2000, 3000, etc. Thus, to multiply 867 by 698, we multiply it by 700, and then by 2, and subtract the second result from the first. MULTIPLICATION. IT* EXAMPLES. 39. Multiply the following: 1. 873 by 956. 8. 15063 by 9874. 2. 2600 by 3950. 9. 54189 by 7998. 3. 487 by 8009. • 10. 7677 by 4912. 4. 4067 by 997. 11. 2946 by 5335. 5. 3476 by 625. 12. 82821 by 7269. 6. 6872 by 599. 13. 93247 by 2461. 7. 60507 by 3784. 14. 35895 by 6927. 15. Find the value of (17 + 15) x (19 - 3). Note. This signifies that 17 and 15 are to be added together, then 3 subtracted from 19, and then the first result multiplied by the second. (Compare Art. 26.) Find the values of the following : 16. (38 - 15) X 26. 19. (92- 36) x (88 -29). 17. (13 + 21) X (32+ 7). 20. 103 x (186- 115 + 137). 18. (77 + 43) X (56 -19). 21. 463 - (35 + 29) x 5 + 148. 22. (391 - 274 - 89) X (96 -37). 23. (127 - 98) X (101 + 66) - (103 - 79) x (S6 - 47) . 24. (856 - 614 - 477) x (982 - 378 -f- 249). 25. (387 - 35 + 123) x (458 - 129-75+48) + (713 x 294). IB ARITHMETIC. V. DIVISION. 40. To Divide one whole humber by another is to find a number which, when multiplied by the second number, will produce the first. Thus, to divide 15 by 5 is to find a number which when multiplied by 5, will produce 15. The number is 3 ; hence, 15 divided by 5 are 3. We also say, " 5 is contained in 15 three times." 41. The number which is divided is called the Dividend. The number by which the dividend is divided is called the Divisor. The result is called the Quotient. 42. It is evident that the Dividend is the product of the Divisor and Quotient. 43. The symbol -f-, read ^^ divided hy,^^ signifies division. Thus, 15 -^ 5 = 3 is read " fifteen divided by five are three." 44. If one number does not exactly contain another, its excess above the next smaller number that does exactly contain the second number is called the Remainder. Thus, 5 is contained in 17 three times, with a remainder 2. 45. 1. Divide 852 by 3. Divisor, 3)852, Dividend. We write the divisor at the left of Quotient, 284, Ans. *^^ dividend, with a ) between them. 3 is contained in 8 hundreds 2 hundreds times, with a remainder of 2 hundreds, or 20 tens. We then write 2 under the hundreds' figure of the dividend. 20 tens and 5 tens are 25 tens ; 3 is contained in 25 tens 8 tens times, with a remainder of 1 ten, or 10 units. We then write 8 under the tens' figure of the dividend. 10 units and 2 units are 12 units ; 3 is contained in 12 units 4 times. We then write 4 under the units' figure of the dividend. The required result is 284. DIVISION. 19 The following words are used in explaining the above process : 3 in 8 twice, with 2 to carry ; 3 in 25 eight times, with 1 to carry ; 3 in 12 four times. 2. Divide 22236 by 68. First Process. 68 is contained in 222 three times, with a remain- 68") 22236 ^^^ ^^ ^^ > ^^ ^^ contained in 183 twice, with a re- ^^ A mainder of 47 ; 68 is contained in 476 seven times. ' * The required result is 327. In order to avoid the labor of calculating the remainders mentally, it is customary to arrange the written work as follows : Second Process. We say, 68 is contained in 222 three times, 68) 22236 (327, Ans. ^^^h a remainder. orv^ Multiplying 68 by 3, the product is 204, which we write under the 222 ; subtracting 183 204 from 222, the remainder is 18 ; annex 135 to this the next dividend figure, 3. 68 is contained in 183 twice, with a re- 4< 6 mainder ; multiplying 68 by 2, the product 476 is 136, which we write under the 183 ; sub- tracting 136 from 183, the remainder is 47 ; annex to this the last dividend figure, 6. 68 is contained in 476 seven times ; multiplying 68 by 7 the product is 476, which we write under the 476 ; subtracting, there is no remainder. Hence, the required result is 327. It will be seen that the second process is essentially the same as the first ; the only difference being that, in the first, certain operations are performed mentally, which are written out in full in the second. Note 1. The operation is called Short Division when the remain- ders and partial products are obtained mentally, as in Ex. 1, and the first process of Ex. 2 ; and Long Division when they are written out in full, as in the second process of Ex. 2, The method of Short Division should always be used when the divisor is 12 or less. 20 ARITHMETIC. From the second process of Ex. 2, we derive the following RULE FOR LONG DIVISION. Write the divisor at the left of the dividend. Take, at the left of the dividend, the smallest number of digits that will form a number equal to or greater than the divisor. Divide this number by the divisor, arid write the quotient as the first digit of the quotient ; subtract from the number the product of the divisor by the first digit of the quotient, and annex to the remainder the next figure of the dividend. Divide this partial dividend by the divisor, and proceed as before; continuing the process until all the figures of the divi- dend have been taken. Note 2. If any partial dividend is less^than the divisor, write for the corresponding digit of the quotient, and annex to the partial dividend the next figure of the dividend. Note 3. If, on making trial of any number as a digit of the quo- tient, its product by the divisor is greater than the preceding partial dividend, the number tried is too great, and one less must be substi- tuted for it. If any remainder is equal to or greater than the divisor, the digit of the quotient last obtained is too small, and one greater must be sub- stituted for it. 3. Divide 80791 by 386. In this case, the smallest number of (209, Quotient. ^^^^^^ ^^ ^he left of the dividend that will 386) 80791 form a number greater than the divisor, 772 is three. 386 is contained in 807 twice, with a ^^^^ remainder ; 2 times 386 is 772, which, sub- 3474 tracted from 807, leaves 35 ; annex to this ~117, Remainder. ^^^ ^^^^ dividend figure, 9. Since 359 is less than the divisor, we write as the second digit of the quotient, and annex to 359 the last dividend figure, 1. 386 is contained in 3591 nine times, with a remainder ; .9 times 386 is 3474, which, subtracted from 3591, leaves 117. Hence, the quotient is 209, and the remainder 117. DIVISION. 21 Note 4. The work may be proved by multiplying the divisor and quotient, and adding the remainder, if any, to the product ; the result should equal the dividend. 46. To Divide by 10, 100, 1000, Etc. To divide a whole number by 10, 100, 1000, etc., we cut off from the right of the dividend as many digits as there are ciphers in the divisor. The result will be the quotient, and the digits cut off will form the remainder. 1. Divide 360000 by 1000. Cutting ofE three digits from the right of the dividend, we have 360000 - 1000 = 360, Ayis. 2. Divide 7298 by 100. Cutting off two digits from the right of the dividend, we have 7298 ^ 100 = 72, with a remainder of 98, Ans. 47. To Divide by Any Number of Tens, Hundreds, Etc. It is evident that, if both dividend and divisor be divided by the same number, the value of the quotient is not changed ; for the new divisor is contained in the new divi- dend just as many times as the old divisor is contained in the old dividend. It follows from the above that we may remove the same number of ciphers from the right of both the dividend and divisor, and find the quotient of the resulting numbers ; for this is the same as dividing both dividend and divisor by the same number. 1. Divide 400400 by 7700. 77)4004(52, Ans. ggg In this case, we remove two ciphers from ^f,. the right of both the dividend and divisor, and Jg| divide 4004 by 77. 22 ARITHMETIC. 2. Divide 3358617 by 65000. First Process. 65000)3358617(51, Quotient. 325000 108617 65000 43617, Remainder. The process may be shortened by omitting the ciphers at the right of the divisor, and the same number of digits at the right of the dividend, finding the quotient of the result- ing numbers, and annexing to the remainder the digits omitted from the right of the dividend. The written work will then stand as follows : Second Process. 65) 3358 I 617 (51, Quotient. 325 108 65 43617, Remainder. From the above example, we derive the following RULE. If the divisor has, and the dividend has not, ciphers at its right, cut off the ciphers from the right of the divisor, and the same number of digits from the right of the dividend. Divide the resulting numbers, and annex to the remainder the digits omitted from the right of the dividend. 48. If the dividend has, and the divisor has not, ciphers at its right, we proceed as follows : Example. Divide 476000 by 56. 56)476000(8500, Ans. in this case, 56 is contained in 4760 85 448 times ; annexing to this the two ciphers 280 remaining at the right of the dividend, the 280 quotient is 8500. DIVISION. 28 49. If any number of things of the same kind be divided by a whole number, the quotient will be things of the same kind as the dividend. Thus, 42 books divided by 7 are 6 books. Again, if any number of things of the same kind be divided by another number of things of the same kind as the dividend, the quotient will be a number. Thus, 42 books divided by 7 books are 6. EXAMPLES. 50. Divide the following : 1. 488304 by 12. 8. 53803998 by 10629. 2. 517668 by 964. 9. 58161020 by 1357. 3. 63102024 by 6008. 10. 41675206 by 492000. 4. 24884574 by 49082. 11. 23510372 by 73700. 6. 68515100 by 69700. 12. 20301129888 by 25376. 6. 3545000 by 587. 13. 499107840 by 53760. 7. 83057629 by 10000. 14. 626104565 by 7247. 15. Find the value of (59 - 11) - (25 - 17). Note. This signifies that 11 is to be taken from 59, then 17 from 25, and the first result divided by the second. (Compare Art. 26.) Find the values of the following : 16. (78-12-11) X (15-7). 20. (322-106) --(48-21). 17. (13 X 5 + 26) -f- 7. 21. (132-6) x (143- 13). 18. 81 - (108 ^ 9) + 43. 22. (297 + 279) -(6x8). 19. (148-15+31) -34. 23. (33x57) -(442-17). 24. (8567 - 12073 - 7988) ^ (8051 ^ 97) , 24 ARITHMETIC. VI. FACTORING. 51. One whole number is said to be divisible by another when the first number can be divided by the second without a remainder. In the above case, the first number is said to be a Multiple of the second, and the second a Factor of the first. Thus, 15 is a multiple of 5, and 5 is a factor of 15. 52. An Even Number is one that is divisible by 2 ; as 2, 4, 6, 8, etc. An Odd Number is one that is not divisible by 2 ; as 1, 3, 5, 7, etc. 53. A Prime Number is one that is divisible only by itself and 1 ; as 1, 2, 3, 5, 7, etc. A Composite Number is one that is divisible by other numbers than itself and 1 ; as 4, 6, 8, 9, 10, etc. 54. The Prime Factors of a number are the prime num- bers which, when multiplied together, will produce the given number. Thus, the prime factors of 12 are 2, 2, and 3. Note. It is usual to exclude 1 in giving the prime factors of a number. To Factor a number is to find its prime factors. 55. If a number be multiplied by itself any number of times, the result is called a power of the first number. An exponent is a number written at the right of, and above another, to indicate what power of the latter number is to be taken ; thus, 2^, read "2 square^^ or "2 to the 2d power,^^ denotes 2x2; 5^, read " 5 cube " or "5 to the M power, ^^ denotes 5x5x5; 7^ read " 7 fourth " or " 7 to the 4th poicer,'^ denotes 7x7x7x7; and so on. FACTORING. 26 56. The following principles will be found of great use in factoring numbers : Any number of tens is divisible by 2 ; hence, I. Any number is divisible by 2 if its last digit is or an . even number. Thus, 738 is divisible by 2, because 8 is an even number. Any number of hundreds is divisible by 4 ; hence, II. Any number is divisible by 4 if the number formed by its last two digits is divisible by 4. Thus, 3568 is divisible by 4, because 68 is divisible by 4. Any number of thousands is divisible by 8 ; hence, III. A7iy number is divisible by 8 if the number formed by its last three digits is divisible by 8. Thus, 47352 is divisible by 8, because 352 is divisible by 8. Any number of tens is divisible by 5 ; hence, ly. Any number is divisible by 5 if its last digit is or 5. Thus, 120 and 8295 are divisible by 5. Any number of tens is divisible by 10 ; hence, V. Any number is divisible by 10 if its last digit is 0. Thus, 3790 is divisible by 10. VI. Any number is divisible by 3 if the sum of its digits is divisible by 3. Thus, 582 is divisible by 3, because the sum of its digits, 15, is divisible by 3. VII. Any number is divisible by 6 if its last digit is or an even number^ and the sum of its digits is divisible by 3. This follows from I and VI. VIII. A7iy number is divisible by 9 if the sum of its digits is divisible by 9. 26 ARITHMETIC. Thus, 864 is divisible by 9, because the sum of its digits, 18, is divisible by 9. IX. Any number is divisible by 11 if the sum of the digits in the odd places is equal to the sum of the digits in the even places, or differs from it by a number divisible by 11. Thus, 4785 is divisible by 11, because the sum of the digits in the first and third places, 12, is equal to the sum of the digits in the second and fourth places. Again, 39182 is divisible by 11, because the sum of the digits in the first, third, and fifth places, 6, differs from the sum of the digits in the second and fourth places, 17, by 11 ; a number divisible by 11. Note. Principles VI, VIII, and IX may be proved as follows : Proof of VI and VIII. Any number of lO's is equal to the same number of 9's, plus the same number of units ; any number of lOO's is equal to the same num- ber of 99's, plus the same number of units ; etc. Thus, 783 is equal to 7 99's plus 7 units, 8 9's plus 8 units, and 3 units. But the sum of 7 99's and 8 9's is divisible by both 3 and 9. Hence, 783 is divisible by 3 or 9 if the sum of 7 units, 8 units, and 3 units is divisible by 3 or 9, respectively ; that is, if the sum of its digits is divisible by 3 or 9, respectively. Similar considerations hold with respect to any number. Proof of IX. Any number of lO's is equal to the same number of ll's, minus the same number of units ; any number of lOO's is equal to the same number of 99's j9Z?/s the same number of units ; any number of lOOO's is equal to* the same number of lOOl.'s, minus the same number of units ; etc. Thus, 4829 is equal to 4 lOOl's minus 4 units, 8 99's plus 8 units, 2 ll's minus 2 units, and 9 units. But the sum of 4 lOOl's, 8 99's, and 2 ll's, is divisible by 11. Hence, 4829 is divisible by 11 if the sum of 8 units and 9 units, minus the sum of 4 units and 2 units, is divisible by 11 ; that is, if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by 11. Similar considerations hold with respect to any number. FACTORING. 27 57. 1. Find the prime factors of 51480. 2^ )51480 51480 is divisible by 8, or 23, because the number 5)6435 fornied by its last three digits is divisible by 8 (Art. 3^)1287 ^^' ^^^^' IvTZq Dividing 51480 by 8, the quotient is 6435. ll)14o g435 |g divisible by 5, because its last digit is 5. 13 Dividing 6435 by 6, the quotient is 1287. 1287 is divisible by 9, or 3^, because the sum of its digits, 18, is divisible by 9 (Art. 56, VIII). Dividing 1287 by 9, the quotient is 143. 143 is divisible by 11, because the sum of the digits in the first and third places is equal to the digit in the second place (Art. 56, IX). Dividing 143 by 11, the quotient is 13, a prime number. Then, 51480 = 2^ x 3"^ x 5 x 11 x 13, Ans. From the above example, we derive the following RULE. Divide the number by any one of its factors; then divide the quotient by any one of its factors; and so on, continuing the process until the quotient is a prime iiumber. The several divisors and the last quotient are the factors required. Note 1. In determining the prime factors of a number, divisors should be tried in the following order : 2, (22, 23); 5 ; 3, (32); 11 ; then, 7, 13, 17, 19, 23, 29, etc. 2. Prove that 373 is a prime number. Since the last digit is 3, neither 2 nor 5 is a factor. Since the sum of the digits is 13, 3 is not a factor. Since the sum of the digits in the first and third places differs from the digit in the second place by 1, 11 is not a factor. Trying in order the prime numbers 7, 13, 17, and 19, we find that neither of them is contained in 373. There is no need of trying 23, nor any greater prime number ; for if one factor of 373 could be 23, the other would have to be 23, or some greater prime number ; for it has already been proved that no prime number less than 23 is a factor. But 23 X 23 = 529, a number greater than 373 ; and hence 23 can- not be a factor. Therefore 373 is a prime number. 28 ARITHMETIC. Note 2. Li trying prime numbers as divisors, the process need not be continued after the square of tlie prime number next greater than the last one tried exceeds tlie given number. Tims, in Ex. 2, no prime number greater than 19 need be tried, because the square of the next greater prime, 23, is greater than 378. Tlie following table of squares of prime numbers will be found of use : No. 8q. No. Sq. No. Sq. No. Sq. No. Sq. 13 169 29 841 43 1849 61 3721 79 6241 17 289 31 961 47 2209 67 4489 83 6889 19 361 37 1369 53 2809 71 5041 89 7921 23 529 41 1681 59 3481 73 5329 97 9409 58. The following table gives, for convenient reference, a list of the prime numbers from 1 to 997, inclusive : 1 41 101 167 239 313 397 467 569 643 733 823 911 2 43 103 173 241 317 401 479 571 647 739 827 919 3 47 107 179 251 331 409 487 577 653 743 829 929 5 53 109 181 257 337 419 491 587 659 751 839 937 7 59 113 191 263 347 421 499 593 661 757 853 941 11 61 127 193 269 349 431 503 599 673 761 857 947 13 67 131 197 271 353 433 509 601 677 769 859 953 17 71 137 199 277 359 439 521 607 683 773 863 967 19 73 139 211 281 367 443 523 613 691 787 877 971 23 79 149 223 283 373 449 541 617 701 797 881 977 29 83 151 227 293 379 457 547 619 709 809 883 983 31 89 157 229 307 383 461 557 631 719 811 887 991 37 97 163 233 311 389 463 563 641 727 821 907 997 EXAMPLES. 59. Find the prime factors of the following : 1. 684. 5. 3003. 9. 1729. 13. 53625. 2. 686. 6. 4459. 10. 8395. 14. 48204. 3. 2520. 7. 1331. 11. 24108. 15. 32292. 4. 4305. 8. 8085. » 12. 19635. 16. 68364. FACTORING. ^§ Prove that each of the following numbers is prime : 17. 1019. 19. 1367. 21. 1787. 23. 2203. 18. 1193. 20. 1531. 22. 2081. 24. 2999. 60. Verification of Addition, Subtraction, Multiplication, and Division by Casting out Nines. The excess of any number above the next less multiple of 9 is called its Excess of Nines. Thus, since 3527 is equal to 391 x 9, plus 8, its excess of nines is 8. It was shown in the proof of YIII (Art. 56, Note) that any number is equal to a multiple of 9, plus the sum of its digits. Thus, 3527 is equal to a multiple of 9, plus 17 ; or since 17 = 9 + 8, it is equal to a multiple of 9, plus 8. It follows from the above that the excess of nines of any number may be found by subtracting from the sum of its digits the next less multiple of 9. Thus, since the sum of the digits of the number 4619 is 20, its excess of nines is 20 — 18, or 2. Addition. 3527... 8 4619... 2 1...8146 10... 1 The excess of nines of 3527 is 8, and of 4619 is 2. Then since 3527 is equal to a multiple of 9, plus 8, and 4619 to a multiple of 9, plus 2, their sum, is equal to a mul- tiple of 9, plus 8, plus 2 ; or since 10 = 9 + 1, it is equal to a multiple of 9, plus 1. But the excess of nines of 8146 is 1; that is, 8146 is equal to a multiple of 9, plus 1. This agrees with the statement made in the preceding paragraph with respect to the sum of 3527 and 4619. 80 ARITHMETIC. Then, to verify the result of addition, we place to the right of each of the numbers to be added its excess of nines. Adding these excesses, we place to the right of their sum its excess of nines. If this equals the excess of nines of the result, the work may be considered to be correct. Subtraction. 6782... 5 2235... 3... 12 2946. ..3 1167 ... 6 2... 3836 2 6... 1068 6 To the right of the minuend and subtrahend we place their excess of nines. We then subtract the excess of the subtrahend from that of the minuend, increasing the latter by 9 if it is less than the excess of the subtrahend. If the remainder equals the excess of nines of the result, the work may be considered to be correct, Multiplication. 498... 3 376... 7 2988 21 ...3 3486 1494 3... 187248 Since 498 is equal to a multiple of 9, plus 3, and 376 to a multiple of 9, plus 7, their product is equal to the result obtained by multiplying a multiple of 9 plus 3, first by a multiple of 9, and afterwards by 7, and adding the results. It is evident from this that the product is equal to a multiple of 9, plus 3 times 7 ; or since 21 = 18 + 3, it is equal to a multiple of 9, plus 3. To the right of each of the numbers to be multiplied we place its excess of nines. FACTORING. 31 Multiplying the excess of the multiplicand by that of the multiplier, we place to the right of the product its excess of nines. If this equals the excess of nines of the result, the work may be considered to be correct. Division. (768.. .3 3 329)252784.. .1 5 2303 15 2248 4 1974 19 2744 2632 112 ...4 Since the dividend is equal to the product of the divisor and quotient, plus the remainder, we proceed as follows : Multiply the excess of nines of the quotient by that of the divisor. Add to the product the excess of nines of the remainder, and place to the right of the sum its excess of nines. If this equals the excess of nines of the dividend, the work may be considered to be correct. Note. The above methods are not always .to be depended upon as tests of the accuracy of operations. Suppose, for instance, that, in the illustrative example under Addi- tion, we had taken the sum of 2 and 1 as 4, and the sum of 5 and 6 as 10: 3527 4619 8056 The excess of nines in the sum will still be 1, and yet the work is not performed correctly. A balance of errors like this is, however, unlikely to occuj. 32 ARITHMETIC. 61. Casting out Elevens. The excess of any number above the next less multiple of 11 is called its Excess of Elevens. Thus, since 3527 is equal to 320 x 11, plus 1, its excess of elevens is 7. It was shown in the proof of IX (Art. 5Q>, Note) that any number is equal to a multiple of 11, plus the sum of the digits in the odd places, minus the sum of the digits in the even places. Thus, 3527 is equal to a multiple of 11, plus 12, minus 5 ; or, to a multiple of 11, plus 7. It follows from the above that the excess of elevens of any number may be found by subtracting the sum of the digits in the even places from the sum of the digits in the odd places^ the latter being increased by 11, or a multiple of 11, if necessary. Thus, for the number 9484, the sum of the digits in the odd places is 8, and the sum of the digits in the even places is 17. Increasing the former by 11, we have 19. Then the excess of elevens is 19 — 17, or 2. The methods for verifying Addition, Subtraction, Multi- plication, and Division by casting out elevens are precisely similar in theory and practice to the methods by casting out nines. GREATEST COMMON DIVISOR. 33 VII. GREATEST COMMON DIVISOR. 62. A Common Divisor, or Common Factor, of two or more whole numbers is a number that will divide each of them without a remainder. Thus, 3 is a common divisor of 18, 24, and 30. 63. The Greatest Common Divisor (G. C. D.) of two or more whole numbers is the greatest number that will divide each of them without a remainder. Thus, 6 is the greatest common divisor of 18, 24, and 30. 64. Two numbers are said to be prime to each other when they have no common divisor except 1. Thus, 8 and 9 are prime to each other. 65. In determining the G. C. D. of numbers, we may dis- tinguish two cases : 66. Case I. When the numbers can he readily factored as in Art- 57. 1. Find the G. C. D. of 144, 264, and 540. 144 = 2^ X 3^ Factoring each number by the method 264 = 2^ X 3 X 11 of Art. 57, it is evident that the greatest 540 = 2^ X 3^ X 5 number that will exactly divide 144, 264, G. C. D. = 2^ X 3 and 540, is 22 x 3. = 12, Ans. Hence, the required G. C. D. is 12. From the above example we derive the following RULE. Factor each of the numbers. Take every prime number which is a common divisor of all the given numbers, the least number of times that it occurs in any one of the numbers. The product of these numbers will be the G. C. D. required. 34 ARITHMETIC. Note. If any prime number is a common divisor of all the given numbers, its exponent in the G. C. D. will be the lowest exponent with which it occurs in any one of the numbers. Thus, m Ex. 1 , we have in the given numbers 2*, 2^, and 2^ respec- tively, and m the G. C. D., 22. If one of the numbers is exactly contained in another, the latter need not be considered in the operation of finding the G. C. D. ; for since every factor of the first number is also a factor of the second, the result is not affected by omitting the second number from the process. Thus, in finding the G. C. D. of 28, 49, 140, and 196, it would be sufficient to find the G. C. D. of 28 and 49. EXAMPLES. Find the G. C. D. of: 2. 165 and 210. 12. 104, 182, and 351. 3. 288 and 648. 13. 180, 264, and 378. 4. 306 and 476. 14. 16, 52, 160, 224, and 260. 5. 36, 144, and 234. 15. 320, 640, and 1008. 6. 128, 192, and 384. 16. 390, 910, and 1365. 7. 675 and 1125. 17. 360, 750, and 2700. 8. 105, 385, and 455. 18. 432, 1944, and 2592. 9. 96, 108, 132, and 156. 19. 525, 3375, and 7425. 10. 240, 336, and 480. 20. 1540, 5005, and 6545. 11. 81, 117, 126, and 135. 21. 6804, 7056, and 8232. 22. A farmer has three pieces of timber whose lengths are 63, 84, and 105 feet, respectively. What is the length of the longest logs, all of the same length, that can be cut from them ? 23. Two schools, containing 480 and 672 pupils, respec- tively, are divided into classes, each containing the same number of pupils. What is the greatest number of pupils that each class can contain, and how many classes of this size are there in each school ? GREATEST COMMON DIVISOR. 35 24. Three rooms are 168, 196, and 224 inches wide, re- spectively. What is the width of the widest carpeting that is contained exactly in each room ? 25. How many quarts are there in the largest receptacle that will exactly measure the contents of three jars, holding 216, 288, and 312 quarts, respectively ? 26. I have three fields containing 392, 504, and 616 square rods, respectively. Find the size of the largest house-lots, all of the same size, into which the fields can be divided. 27. The sides of a field are 110, 154, 198, and 264 feet, respectively. What is the length of the longest fence-rail that is contained exactly in each side ? 67. Case II. Whe^i the numbers cannot be readily fac- tored as in Art. 57. 1. Find the G. C. D. of 221 and 493. Dividing the greater number by the less, we have 221)493(2, Quotient. 442 51, Kemainder. Now whatever factors occur in 221 must also occur in twice 221, or 442. Hence, any factors which are common to 221 and 493 must occur in the result obtained by subtracting 442 from 493 ; that is, they must occur in 51. Again, any factors which are common to 221 and 51 must occur in the result obtained by adding twice 221 to 51 ; that is, they must occur in 493. Then, since every factor common to 221 and 493 occurs in 51, and every factor common to 221 and 51 occurs in 493, it follows that 221, 493, and 51 have the same common factors. 36 ARITHMETIC. Hence, the G. C. D. of 221 and 493 must be the same as the G. C. D. of 221 and 51. That isj the G. C. D. of any tiuo numbers is the same as the O. C. D. of the less number, and the remainder obtained by dividing the greater number by the less. Dividing the divisor, 221, by the remainder, 51, we have 51)221(4 204 17, Eemainder. Then, by the principle just stated, the G. C. D. of 221 and 51 is the same as the G. C. D. of 51 and 17. Dividing the divisor, 51, by the remainder, 17, we have 17)51(3 51 That is, 17 is the G. C. D. of 51 and 17. Then, 17 is also the G. C. D. of 221 and 51, and is conse- quently the G. C. D. of 493 and 221. From the above example, we derive the following RULE. Divide the greater number by the less. If there be a remainder, divide the divisop by it; and con- tinue thus to make the remainder the divisor, and the preceding divisor the dividend, imtil there is no remainder. The last divisor is the O. C. D. required. 2. Find the G. C. D. of 377 and 667. 377)667(1 377 290)377(1 290 87)290(3 261 29)87(3 87 Then, 29 is the G. C. D. required, Ans. GREATEST COMMON DIVISOR. 37 EXAMPLES. Find the G. C. D. of: 3. 559 and 817. 10. 3703 and 6923. 4. 391 and 598. 11. 1591 and 2183. 5. 589 and 899. 12. 5605 and 6785. 6. 703 and 893. 13. 6059 and 7446. 7. 533 and 1271. 14. 5312 and 10043. 8. 731 and 1247. 15. 2291 and 3713. 9. 3658 and 4602. 16. 10057 and 11659. 68. The G. C. D. of three numbers which cannot be readily- factored by the method of Art. 57, may be found as follows : Let A, B, and C represent the numbers. Let G represent the G. C. D. of A and B ; then every common factor of G and C is also a common factor of A, B, and C. But every common factor of A and B exactly divides G. Whence, every common divisor of A, B, and C is also a common divisor of G and C. Therefore, the greatest common divisor of A, B, and G is the same as the greatest common divisor of G and C. Hence, to find the G. C. D. of three numbers, find the G. C. D. oj two of them, and then of this result and the third number. We proceed in a similar manner to find the G. C. D. of four or more numbers. 1. Find the G. C. D. of 741, 1653, and 7163. We first find the G. C. D. of 741 and 1653, which is 57. We then find the G. C. D. of 57 and 7163, which is 19, Ans. EXAMPLES. •- Find the G. C. D. of: 2. 663, 741, and 4199. 4. 969, 1653, and 9367. 3. 442, 782, and 5083. 5. 5083, 5681, and 7429. 88 ARITHMETIC. VIII. LEAST COMMON MULTIPLE. 69. A Cominon Multiple of two or more whole numbers is a member that will exactly contain each of them. Thus, 72 is a common multiple of 6, 9, and 12. 70. The Least Common Multiple (L. C. M.) of two or more whole numbers is the smallest number that will exactly contain each of them. Thus, 36 is the least common multiple of 6, 9, and 12. 71. In determining the L. C. M. of numbers, we may dis- tinguish two cases : 72. Case I. When the numbers can be readily factored. Example. Find the L. C. M. of 40, 84, and 144. 40 _ 93 y 5 Factoring each of the numbers by Q/l _ 92 Q 7 *^^ method of Art. 57, it is evident »4 = J X o X 7 ^-^^^ ^-^^ smallest number that will ^^^ — 2^^X^3^ exactly contain 40, 84, and 144, is L. C. M. = 2^ X 32 X 5 X 7 24 X 32 X 5 X 7. w^.^ . Hence, the required L. C. M. is = 5Q^,Ans. ^^^ From the above example, we derive the following RULE. Factor each of the numbers. Take every prime number, which is a factor of any one of the given numbers., the greatest number of times that it occurs in any one of the numbers. The product of these numbers will be the L. C. M. required. Note. If aj;iy prime number is a factor of any one of the given numbers, its exponent in the L. C. M. will be the greatest exponent with which it occurs in any one of the numbers. Thus, in the above example, we have in the given numbers 2^, 22, and 2*, respectively, and in the L. C. M., 2*. LEAST COMMON MULTIPLE. 39 73. Second Method. The following rule will be found preferable to that of Art. 72 in the solution of examples : Arrange the numbers in a horizontal line. If two or more of the numbers have a common prime factor, divide them by it, and ivrite the quotients, together with the undivided numbers, in the next line. Continue in this way until a line is obtained m which the numbers have no common factor. The prodiLCt of the divisors and the numbers in the last line will be the L. C. M. required. 1. Find the L. C. M. of 24, 60, and 105. 2 )24 60 105 Dividing 24 and 60 by the com- mon prime factor 2, the second line becomes 12, 30, 105. Dividing 12 and 30 by the com- mon prime factor 2, the third line 2 17 becomes 6, 15, 105. L. C. M. = 2x2x3x5x2x7 Dividing 6, 15, and 105 by the = 840 Ans. common prime factor 3, the fourth line becomes 2, 5, 35. Dividing 5 and 35 by the common prime factor 5, the fifth line becomes 2, 1, 7. Since the numbers 2 and 7 have no common prime factor, the required L. C. M. is the product of the divisors, 2, 2, 3, 5, and the numbers in the last line, 2, 7 ; the result is 840. It is evident that, *in the above process, every prime num- ber which is a factor of any one of the given numbers, is taken the greatest number of times that it occurs in any one of the numbers. Hence, the result is the L. C. M. of the given numbers. If one of the given numbers exactly divides another, the former need not be considered in the operation of finding the L. C. M. ; for since every factor of the first number is also a factor of the second, the result is not affected by omitting the first number from the process. 012 30 105 3)6 15 105 5)2 5 35 40 ARITHMETIC. Tims, in finding the L. C. M. of 15, 26, 78, and 90, it would be sufficient to find the L. C. M. of 78 and 90. Note. If two numbers are prime to each other (Art. 64), their product is their L. C. M. EXAMPLES. Find the L. CM. of: 2. 4, 6, 9, and 10. 14. 28, 49, 147, and 196. 3. 28 and 63. 15. 24, 42, 72, 84, and 112. 4. 24, 112, and 160. 16. 15, 35, and 77. 5. 108 and 144. 17. 36, 104, and 351. 6. 110 and 165. 18. 115, 138, 230, and 345. 7. 231 and 770. 19. 33, 44, 55, and 132. 8. 18, 38, 54, and 57. 20. 288, 324, 432, and 648. 9. 20, 75, 180, and 300. 21. 189, 243, and 405. 10. 176 and 264. 22. 98, 126, 140, and 168. 11. 32, 88, and 121. 23. 52, 81, 117, and 120. 12. 87, 116, and 192. 24. 119, 136, 252, and 280. 13. 144, 216, and 324. 25. 315, 1350, and 1500. 26. How many quarts are there in the smallest vessel whose contents can be exactly measured by measures con- taining 10, 15, and 18 quarts, respectively ? 27. Two horse-cars make round trips in 48 and 60 min- utes, respectively. If they set out at the same time, after how many minutes will they meet again at the starting- point ? 28. What is the smallest sum of money with which I can pjirchase cows at $ 45 each, oxen at $ 54 each, or horses at f 72 each ? 29. Three men. A, B, and C, can walk around a race- course in 9, 12, and 14 minutes, respectively. If they all set out at the same time, after how many minutes will they all meet at the starting-point, and how many times will each have been around the course ? LEAST COMMON MULTIPLE. 41 74. Case II. When the numbers cannot be readily factored. 1. Find the L. C. M. of 221 and 247. If we divide 221 by the greatest common divisor of 221 and 247, the quotient will be the product of those factors of 221 which are not found in 247. Then, if we multiply this quotient by 247, the product will be divisible by both 221 and 247 ; and it is evidently the smallest number that is divisible by both of them. That is, the product is the L. C. M. of 221 and 247. 221)247(1 13)221(17 247 221 13_ _17 26)221(8 91 1729 208 91 247^ G. C. D. = 13)26(2 L. C. M.=4199, Ans. 26 • We first find the G. C. D of 221 and 247 by the rule of Art. 67 ; the result is 13. Dividing 221 by 13, the quotient is 17. Multiplying 247 by 17, the product is 4199. Then, the required L. C. M. is 4199. From the above example, we derive the following RULE. Find the O. C. D. of the given numbers. Divide one of the numbers by their G. C. />., and multiply the quotient by the other number. EXAMPLES. Find the L. C. M. of : 2. 289 and 323. 8. 361 and 437. 3. 629 and 703. 9. 391 and 493. 4. 551 and 589. 10. 403 and 961. 5. 667 and 713. 11. 533 and 1189. 6. 841 and 899. 12. 1403 and 1817. 7. 299 and 529. 13. 6649 and 7957. 42 ARITHMETIC. 75. The L. C. M. of three numbers which cannot be readily factored, may be found as follows : Let Ay B, and C represent the numbers. Let M represent the L. C. M. of A and B ; then, every common multiple of M and C is also a common multiple of A, B, and C. But every common multiple of A and B exactly con- tains M. Whence, every common multiple of A, B, and C is also a common multiple of M and C. Therefore, the least common multiple of A, B, and C is the same as the least common multiple of M and C. Hence, to find the L. C. M. of three numbers, fiyid the L. C. M. of two of them, and then of this result and the third number. We proceed in a similar manner to find the L. C. M. of four or more numbers. 1. Find the L. C. M. of 713, 1081, and 1395. We first find the L. C. M. of 713 and 1081, which is 33511. We then find the L. C. M. of 1395 and 33511, which is 1607995, Ans. EXAMPLES. ' Find the L. C. M. of: 2. 1271, 1674, and 1968. 3. 1505, 1591, and 2109. FRACTIONS. 48 IX. FRACTIONS. 76. If unity is divided into 4 equal parts, and 3 parts are taken, the result is expressed by j ; read " three-fourths.^^ If unity is divided into any number of equal parts, and any number of parts are taken, the result is called a Fraction. 77. The Denominator of a fraction is the number which shows into how many equal parts unity is divided, and the Numerator is the number which shows how many parts are taken. Thus, in the fraction f , the denominator is 4, and the numerator is 3. The numerator and denominator are called the Terms of the fraction. 78. A fraction is usually expressed by writing the numer- ator above, and the denominator below, a horizontal line; and when thus expressed, it is called a Common Fraction. 79. A Mixed Number is the sum of a whole number and a fraction. Thus, 5 + 1, or, as it is usually written, 5f , is a mixed number. 80. Let each of the lines AB, BC, and CD, in the follow- ing figure, represent one unit; then AD will represent 3 units. A E F G B G D I I I [ I I I I I 1 I I I Let AB be divided into 4 equal parts ; AE, EF, FG, and OB\ then the fraction f will be represented by AG. Now it is evident that, if AD be divided into 4 equal parts, one of these parts will be AG. 44 ARITHMETIC. Hence, the fraction J represents the result obtained by- dividing 3 units by 4. And in general, any fraction is an expression of division; the numerator answering to the dividend, and the denomi- nator to the divisor. 81. It follows from Art. 80 that an integer may be ex- pressed in a fractional form by writing 1 for a denominator. Thus, 3 is the same as f . 82. A Proper Fraction is one whose numerator is less than its denominator ; as J. An Improper Fraction is one whose numerator is equal to or greater than its denominator ; as |-, or ^. REDUCTION OF FRACTIONS. 83. To Reduce an Improper Fraction to a Whole or Mixed Number. 1. Eeduce ^- to a whole number. Since a fraction is an expression of division (Art. 80), ^ = 54-9 = 6,^718. 2. Reduce ^^ to a mixed number. Since 290 is equal to the sum of 276 and 14, we have W = W + M = 12 + i|,orl2i|, ^ns. It is customary to perform the work as follows : 23)290(12^, Ans. 23 60 46 14, Remainder. From the above examples, we derive the following RULE. Divide the numerator by the denominator. If there is a remainder, write it over the divisor, and add the fraction thus formed to the quotient. FRACTIONS. 45 EXAMPLES. Reduce each of the following to a whole or mixed number : 3. M- 7. ifF- 11. ^iF- 15- ^W^. 4. ^. 8. m^. 12. ^ffi. 16. ifll-^- 6. ^^. 10. ^A. 14. i^||5. 18. 1^403. 84. To Reduce a Whole Number to a Fraction having a given Denominator. 1. Reduce 5 to sevenths. Since 1 is equal to 7 sevenths, 5 is equal to 5 times 7 sevenths, or 35 sevenths ; whence, 5 = Y-, Ans. From the above example, we derive the following RULE. To reduce a whole number to a fraction having a given denominator, multiply the whole number by the denominator, and write the result as the numerator of the required fraction. EXAMPLES. 2. Reduce 6 to 8ths. 6. Reduce 22 to 18ths. 3. Reduce 13 to 6ths. 7. Reduce 19 to 15ths. 4. Reduce 11 to 9ths. 8. Reduce 31 to 24ths. 5. Reduce 16 to 12ths. 9. Reduce 48 to 37ths. 85. To Reduce a Mixed Number to an Improper Fraction. 1. Reduce 9|- to an improper fraction. Since 9 is equal to 72 eighths, 9| is equal to the sum of 72 eighths and 7 eighths, which is 79 eighths ; whence, 9| = ^^-, Ans. From the above example, we derive the following RULE. Multiply the whole number by the denominator of the fraction; add to the product the numerator of the fraction, and write the result over the given denominator. 46 ARITHMETIC. EXAMPLES. Reduce each of the following to an improper fraction 2. 81^. 6. 7|f. 10. 54i|. 14. 74U. 3. lOi 7. 9«. U. 29fi. 15. 96|f. 4. IIJ,. 8. 40tV. 12. 58f|. 16. 127fJ. 5. 13,^. 9. 79ff . 13. 37^. 17. 156J|. 86. To Reduce a Fraction to its Lowest Terms. A fraction is said to be in its Lowest Terms when its numerator and denominator have no common factor. 87. Let the line AO, in the following figure, represent one unit ; and let it be divided into 6 equal parts, AB, BC, CD, DE, EF, and FG. A B C T> :E F Q b— ^— M^M^W^— lid I I tori 1 Then the fraction ^ will be represented by AE. But since the divisions AC, CE, and EG are all equal, the line AE also represents the fraction -|. Hence, the fraction f is equal to |. Now the fraction f may be obtained from ^ by dividing both numerator and denominator by 2. Hence, if both numerator and denominator of ^ be divided by 2, the value of the fraction is not changed. And in general, if both numerator and denominator of any fraction he divided by the same number, the value of the frac- tion is not changed. 88. In reducing fractions to their lowest terms, we may distinguish two cases : 89. Case I. When the numerator and denominator can be readily factored. FRACTIONS. 47 Since both numerator and denominator can be divided by the same number without changing the value of the fraction (Art. 87), we have the following RULE. Divide both numerator and denominator by any common factor. The greater the common divisor used, the more rapid will be the process. 1. Eeduce ^-|f to its lowest terms. i9^_ 9 9 _ 33 _ 11 J^o Dividing both terms of ^|| by 2, 1Z2-T2Q-T2- TT' ^^«- the result is j%\. Dividing both terms of y^g by 3, the result is ||. Dividing both terms of f | by 3, the result is }|. If all the factors of the numerator be removed by division, 1 remains to form a numerator. If all the factors of the denominator be removed, the result is a whole number, this being a case of exact division. EXAMPLES. Reduce each of the following to its lowest terms : 2- AV 6. in. 10. ^^. 14. mi- S 22 7 288 11 6 50 4- W- 8. m- 12- 6- m- 9- W/- 13- 1936 2662- Tl 15. 2700 1125' 16. 2592 5832- 17. 6615 iJ646- 90. Cancellation. Cancellation is the process of dividing both numerator and denominator by striking out their common factors. It is useful in cases when either the numerator or de- nominator is expressed in the form of a product. 48 ARITHMETIC. 1. Reduce ^l^]^^ ^^ to its lowest terms. 20 X 14 X 36 Cancelling 7 from 21 and 14, ^ S we write 3 above 21, and 2 be- nxX^xl^ ^ 3 _3 ^^^ 10W14. ^flXl^X^^ 4x2 8' ' Cancelling 5 from 15 and 20, 4 2^ we write 3 above 15, and 4 below 20. Cancelling 12 from 12 and 36, we write 3 below 36. We then cancel the 3 above 21 with the 3 below 36. 3 3 The result is , or — 4x2 8 EXAMPLES. Reduce each of the following to its lowest terms : » 49 X 88 y 8 X 12 X 14 X 15 55 X 112 46x63 7 X 23 X 30' 8. 10 Xl6 X 18 X 21 34 X38 x39 57 X85 x91 27 X77 Xl05 135 X 165 21 x26 x52 39 x56 Xll7 54 x84 x270 . 20 X 108 g ■ 18x28x36 K 16 X 95 X 96 .Q 114x128 g 21 X 39 X 55 * .. 20 X 26 X 33* * 50 x 162 x 196 91. Case II. When the numerator and denominator can- not be readily factored. Since the G. C. D. of the numerator and denominator is the greatest number that will exactly divide each of them, we have the following RULE. Divide both numerator and denominator by their greatest common divisor. FRACTIONS. 49 1. Reduce ||-| to its lowest terms. 247)323(1 247 76)247(3 228 ^6 fi^d' ^y t^6 rule of Art. 67, that "^rr^^ Dividing 247 by 19, the quotient is — 13. 19)247(13 19)323(17 ^^^^^^^^^ ^23 by 19, the quotient is 19_ 19^ 57 133 57 133 1^, Ans. Then the required result is ^f . EXAMPLES. Eeduce each of the following to its lowest terms : 2- m- 4- m- 6. iflf . 8. Vi^. 10. Mff. 5. Iff 7. Itti- 9- liH- 11- Mfi- 92. To Reduce Fractions to their Least Common Denomi- nator. Fractions are said to have a Common Denominator when they all have the same denominator. To reduce fractions to their Least Common Denominator (L. C. D.) is to express them as equivalent fractions, having for their common denominator the least common multiple of the given denominators. 93. It was shown in Art. 87 that the fraction | is equal tof. But the fraction f may be obtained from -| by multiplying both numerator and denominator by 2. Hence, if both numerator and denominator of any fraction be multiplied by the same number, the value of the fraction is not changed. 50 ARITHMETIC. 94. 1. Keduce f, -^y and {^ to their least cominon de- nominator. By Art. 73, the L. C. M. of 6, 10, and 15 is 30. Now, by Art. 93, both terms of a fraction may be multiplied by the same number without changing the value of the fraction. Multiplying both terms of f by 5, both terms of j'^ by 3, and both terms of {^ by 2, the given fractions become 2 5 21 nrirl 2 2 J,, o It will be observed that the terms of each fraction are multiplied by a number which is obtained by dividing the least common denominator by its own denominator ; hence the following RULE. Mnd the L. C. M. of the given denominators. Divide this by each denominator separately, multiply the respective numerators by the quotients, and write the results over the common denominator. If the given denominators are prime to each other (Art. 64), the least common denominator is the product of all the denominators ; and each numerator is multiplied by all the denominators except its own. 2. Reduce |, J, and -^ to their least common denominator. The L. C. D. is 3 X 4 X 5, or 60. Multiplying each numerator by all the denominators except its own, the fractions become fj, Mj and 48^ Ans. EXAMPLES. Reduce to their least common denominator : 3. i,iandf 7. A, ^ and ff- 4. I,!, and 44. 8. 3^, A, and i|. 6. |,f,and|. 9. H. M. and ff. 6. fi, tt and if. 10. A. ih tt, and «• FRACTIONS. 51 11- A.H.tt.andf^. 13. A. A. If , if . and 23. 12. T^, tt. U^ and f|. 14. fl, 11, il, if, and ^. The relative magnitude of fractions may be determined by reducing them, if necessary, to their least common de- nominator. 15. Which of the fractions, i and f , is the greater ? We have, i = A? and | = 2^. It is evident from this that f is greater than i. Arrange in order of magnitude : 16. ^ and ^. 18. f if, and |f . 17. ^, f, and |. 19. ^, |, and ^. 95. To reduce a fraction to an equivalent fraction having any required denominator, divide the required denominator by the given denominator, and multiply both terms of the given fraction by the result. 1. Reduce ^ to 165ths. Dividing 165 by 15, the quotient is 11. Multiplying both terms of {\ by 11, the result is |fj, Ans. EXAMPLES. 2. Reduce y\ to 78ths. 5. Reduce |i to 375ths. 3. Reduce ^ to 126ths. 6. Reduce ff to 504ths. 4. Reduce |f to 224ths. 7. Reduce |f to 576ths. ADDITION OF FRACTIONS. 96. 1. "Find the sum of |, |, and |. The L. C. M. of 3, 4, and 6 is 12. Keducing eacli fraction to 12ths, we have 52 ARITHMETIC. From the above example, we derive the following RULE. To add two or more fractions, reduce them, if necessary, to their least common denominator. Add the numerators of the resulting fractions, and write the result over the common denominator. The final result should be reduced to its lowest terms. To add two or more mixed numbers, first add the whole numbers, and then the fractions, and then find the sum of these results. 2. Find the sum of 3J, 1^, 5, and 2\\. 3-1-1 + 5+2 = 11. The sum of the whole numbers 1 -|_ _^ _j_ 1 1 = _5 _|_ _9_ _i_ 2_2 3, 1, 5, and 2, is 11. 3g g '-|j The sum of the fractions |^, -^^, 11 4_1 1-121 Aii^ ana T^, IS ^, or i^. -^-^ + -^^ — -^^3-' ^^^' Then the sum of 11 and 1| is 12^. EXAMPLES. Find the values of the following : 3. J + H. 6. 5H + 3«. 9. I + A + A- 4. ^ + ^. 7. i^ + 1^. 10. ^^ + ^ + ^. 5. lH + 2tt. 8. f + l + A- !!■ A + « + M-^ 12. l| + lf + l|. 21. ll^ + 21H + 1444..^7i^ 13. 1^ + 21 + 3,%. 22. 4 + 3| + 2A + l^. /; ^ 14. 7J + 3f + 5f. l'( 23. 2 + 61 + 81 + 4A. >l >^^ 15. f + f + ^ + T^j. a,. 24. ^ + ^ + ^ + l^. ,j,V^ 16. A + A + iV + A- . -.'25. 6i + 5i + 4| + 3A. ? ;J 17. A + T^ + M + «- ^ -» 26. 3i + 9i + 7| + 5A. .,^»^Vt^ 18- i + | + A + /T- 27. 2A + 4A + 6^ + 8||. vc'-jj 19. 6ii + 4||+8|i. 28. J + l + l + l + J. l:^ 20. 4^ + 9H + 13^. 29. A + A + 1^ + A + A- '^l^ ■3 ^l' FRACTIONS. -^ 53 30. 7f4-9/^ + 2A + 12fi + 34|. 31. 13i + 16| + 191 + 22,7^ + 25^3. 32. 34i + 171 + 28^3^ + 40,^ + 52^. ) 7> ^b^ SUBTRACTION OF FRACTIONS. 97. 1. Subtract y\ from f Reducing each fraction to 42(is, we have 5. 9_ — SA 2 7 — 8 4 y(„ o 6 1 4 — 42^ 4 2 — ¥2" — 2"TJ ^***- From the above example, we derive the following RULE. To subtract one fraction from another, reduce them, if neces- sary, to their least common denominator. Subtract the numerator of the subtrahend from that of the minuend, and write the result over the common deyiominator. The j&nal result should be reduced to its lowest terms. To subtract one mixed number from another, first sub- tract the integers, and then the fractions, and then find the sum of these results. 2. Subtract 3|- from 5|. ^ ~ 3 = 2. Subtracting 3 from 5, the result is 2. |- — |- = \^ — T g" = T^' Subtracting | from f , the result is y\. 2 + ^ = 2^, Ans. Then the sum of 2 and ^\ is 2^^- If the fractional part of the subtrahend exceeds the frac- tional part of the minuend, increase the latter by 1, subtract- ing 1 from the integral part of the minuend to compensate. 3. Subtract 3| from 5f . Q _ i ^ Since f is greater than |, we subtract 1 ~ ~ * from 5, leaving 4, and then add f to the \, ■6~¥~ is^TF^ T8- giving I ; thus, 5^ is the same as 4|. IW, Ans. 64 ARITHMETIC. 4. Subtract 2f from 7. 7-2| = 6|-2| = 4|,^n5. EXAMPLES. 5. Lu uiit; values T2~T(J- Ul UJf 14. ! iUiiuwiiig ; 23. 18A-8«- 7 7; 6. i^—h- 15. li-A- 24. 16if-9|i- 7. A-1%- 16. 6T~Tg"* 26. 23if-17H. ^'3^- 8. 2-tV- 17. M-H- 26. 12i^-if 9. 5-3tf. 18. 8i^-2A. 27. 33A-25A. -r- y 10. If-A- 19. 7tt-6,V 28. 27A-19H. 11. 9-H- 20. lA-l*- 29. 17if-4H. 12. 11-7M. 21. 12i*-5tt- 30. 24A-15tt- 13. A-A- 22. UJI-lOi^. 31. 31A-18«- In finding the value of a series of fractions connected by plus and minus signs, it is better to add all those fractions which are preceded by minus signs, and subtract their sum from the sum of the other fractions. 32. Find the value of 4iJ - 3 ^l + 2|| - 1 1|. % + ltt = 4^Wi = 4|| 2f| = 2H, Ans. The sum of the fractions 4|| and 2|| is 6f f . The sum of the fractions 3y\ and \\\ is 4||. Subtracting 4|f from 6f |, the result is 2f f . Find the values of the following : 33. 5J-3i-2J,. 34. 4i-li + 8|-7i. 37. 9ii- 35. 91 + 2^-5,2,- -3f 36. l^-\\-ll. -If 3if + 7f + A-6i. FRACTIONS. 55 38. 253L 4. 20f - 1711 - IS^V - 8^. 39. 52| -15^-31- 2611 + 9f|. 40. 12H-6A-3H + 7ti-H + 8^. 41. S2^ + 17/3 - 18,^ - U^ + 20H - lOif. 42. 2113 - 11 J3 - 5ff + 58|| - 12e + t\\ - 19if MULTIPLICATION OF FRACTIONS. 98. To Multiply a Fraction by a Whole Number. Let the line AD, in the following figure, represent one unit ; and let it be divided into 8 equal parts. f or f 1 Then the fraction |- will be represented by AB, and f by AG. But AC is twice AB ; hence, the fraction | is twice f . Now f may be obtained from |^ by multiplying its numer- ator by 2 ; hence, if the numerator of a fraction be multiplied by any number, the fraction is multiplied by that number. Again, since f is equal to f , the fraction f is twice f. But f may be obtained from | by dividing its denomina- tor by 2; hence, if the denominator of a fraction be divided by any number, the fraction is multijMed by that number. 99. We derive from Art. 98 the following rule for mul- tiplying a fraction by a whole number : If possible, divide the denominator by the whole number; otherwise, multiply the numerator by the whole number. 1. Multiply 3^ by 5. Dividing the denominator by 5, we have -53^X0 = 1, Ans. 56 ARITHMETIC. 2. Multiply f by 4. Multiplying the numerator by 4, we have 1x4 = \% Ans. Common factors in the whole number and the denomina- tor of the fraction should be cancelled (Art. 90) before per- forming the multiplication. 3. Multiply If by 18. 2 1^ V tc_ 1^ V 9 2^ A^o 111 this case, we cancel 9 from 3 Note. To multiply a whole number by a fraction is the same as multiplying the fraction by the whole number. Thus, 5 X f is the same as | x 5. To multiply a mixed number by an integer, multiply the whole number and the fraction separately, and then find the sum of these results. 4. Multiply 3i| by 12. 3 X 12 = 36. 2 Multiplying 3 by 12, the product is 36. li y 791 = — =81 Multiplying \\ by 12, the product is ^^-^ ^"^^^ 4 ^* or 8^. 4 Then the sum of 36 and 8J is 44J. 36-f 8^ = 44^, Ans. EXAMPLES. Find the values of the following : 5. If X 9. 9. 117 X \%. 13. 42=V X 3. 17. 15-\ x 18. 6. -A- X 8. 10. IJ X 80. 14. 82 X 6. 18. 35 x 17-^:. 7. ^ X 42. 11. 66 X \\. 15. 9 x 7J^. 19. 25 x 22^^. 8. 75 X i|. 12. ^ X 72. 16. 14^ x 33. 20. 16| x 64. 21. 80 X 18|^. 22. 30|f x 84. FRACTIONS. 57 100. To Multiply a Fraction by a Fraction. To multiply | by f is to take | 0/ | j that is, we divide f into 5 equal parts, and take 4 of them. Let the line AC, in the following figure, represent one unit ; and let it be divided into 15 equal parts. A B E F G B G I of \f ^ 1 Then AB will represent if, or |. Now since ^5 is divided into 5 equal parts, AD, DE, EF, FG, and GB, AG will represent | of |. But AG also represents -^-^. Hence, f of |, or | x |, is equal to y%. 101. We derive from Art. 100 the following rule for multiplying one fraction by another : Multiply the numerators together for the numerator of the product, and the denominators for its denominator. Common factors in the numerators and denominators should be cancelled before performing the multiplication. 1. Multiply! by ^x^='^,Ans. To multiply any number of fractions, we multiply their numerators together for the numerator of the product, and their denominators for its denominator. 2. Find the value of f x | of \\. 2 0^11_22 ^ In this case, we cancel 3 from 6 and 9, 7 ^ ^ " 63' * and then 5 from 5 and 15. 3 3 Mixed numbers should be reduced to a fractional form (Art. 85) before applying the rule. 58 ARITHMETIC. 3. Find the value of || of 1 J^ x 2^ x 9. If of 1^^ X 2^2^ X 9 By Art. 85, 1^\ is equal to f^, 5 9 2 3 and 2t?5 toff. cm of» QO OT- ^^® ^^^* cancel 16 from 48 and = fE X ^ X^ X ^ = —, Ans. 32 ; then 5 from 25 and 15 ; then ^ ^^ X^ 2 5 from 5 and 20 ; then 3 from 3 ^ ^ ^ and 27 ; then 2 from 2 and 4 ; and 2 finally 3 from 3 and 9. EXAMPLES. Find the values of the following : 4- Ax J. 6. lHx2Jf. 8. ^of-V/. 6. « X 2^. 7. if of 2^. 9- 3tt x IM- 10. if X H X «. _ 16. I X A X if X 28. -> 11. li X H of 5f. ' _ , 17. if of ^\ of 26 X 4^. :- 12. ifxi|x5f. _ 18. ^o^xMxIfof^. Jt; 13. if of 4,V X 2|f . . ■ 19. ii of If of li of T%. 14- T*A of U of ff • 20. t7jV X A x ifi X 35. 3> 15. lA X 2A x 1^. \ 21. Ifi x l|f x 2H X in-^ 22- A x'^ X 20 X 2^ X lOif. ^ 23. il of ^ of 2if X 24 X 3^. j^ •"-^ DIVISION OF FRACTIONS. 102. To Divide a Fraction by a Whole Number. B . c D i or I 1 It is evident from the above figure that tlie fraction |. is equal to the fraction ^ divided by 2. But f may be obtained from | by dividing its numerator by 2. FRACTIONS. 59 Hence, if the numerator of a fraction he divided by any number, the fraction is divided by that number. Again, the fraction f is equal to f divided by 2. But I may be obtained from J by multiplying its denom- inator by 2; hence, if the denominator of a fraction be multi- plied by any number, thefractioyi is divided by that number'. 103. We derive from Art. 102 the following rule for dividing a fraction by a whole number : If possible, divide the numerator by the whole number; otherwise, multiply the denominator by the whole number. 1. Divide f by 3. Dividing the numerator by 3, we have . f-r-3 = f, Ans. 2. Divide | by 5. Multiplying the denominator by 5, we have To divide a mixed number by an integer, the dividend should first be reduced to a fractional form (Art. 85). 3. Divide ^ by 8. By Art. 85, 6| = V- ; and ^- ^ 8 = |, Ans. If the integral part of the mixed number is equal to or greater than the divisor, it is better to proceed as follows : 4. Divide 86f by 12. 19^Sfi3 ^^ ^^ contained in 86f seven times, with a re- ^ J* mainder of 2f , or J/-. 1^, Ans. Dividing -V- by 12, the quotient is ^f EXAMPLES. Find the values of the following : 5. A2^7. 8.^-4-9. 11. 2H-6. 14.401-^11. 6. i|-^5. 9. l2%-^2. 12. 16|-^7. 15. 9H-4. 7. \3^-f-12. 10. 83^-^10. 13. lOi-5-3. 16.29^-^8. 60 ARITHMETIC. 104. To Divide a Whole Number or a Fraction by a Fraction. 1. Divide 3 by f We have, 3 - f = -V" -^ I- But the quotient of 21 sevenths divided by 5 sevenths is the same as the quotient of 21 divided by 5, which is V- Therefore, 3 ^ f = V. -^^i*- We observe, in the above example, that the quotient may be obtained by multiplying 3 by ^, which is the fraction ^ inverted; whence the following RULE. To divide a whole number or a fraction by a fraction, invert the divisor, and proceed as in multiplication. 2. Divide ^ by |. 9 _i_8 — 9 v9 — 81 Art'i If the numerator and denominator of the divisor are ex- actly contained in the numerator and denominator of the dividend, we divide the numerator of the dividend by that of the divisor for the numerator of the quotient, and the denominator of the dividend by that of the divisor for the denominator of the quotient. 3. Divide ff by ^. Since 35 -=- 7 = 5, and 44 h- 11 = 4, we have If the divisor is an integer, it must be written in a frac- tional form (Art. 81) before applying the rule. 4. Divide f| by 63. 4 36_^63__^1 By Art. 81, 63 may be written in the form 25 * 1 25 0^ ^T^- ; which, when inverted, becomes ■^^. 7 We cancel 9 from 36 and 63. FRACTIONS. 61 If either the dividend or divisor is a mixed number, it must be expressed in a fractional form before applying the rule. 5. Divide 21 by 3{\. 21 -^ 3|| = 21 -7- f f ^1 is tije same as ||. 3 ^ ^ ^ ^ We cancel 7 from 21 and 56. 8 EXAMPLES. Find the values of the following : 6- n-^H- 14. V-2f. 22. 18^ + 4«. 7. 27-f-i^. 15. fl^A- 23. fl + if. 8. Hh-100. 16. 28 ^«. 24. ^A»-.^51^. 9. 45H-6J,-. 17. 3H ^ 96. 25. Hf^^. 10. ll^lSj^j. 18. lH^7i| 26. Itt^l^j. U. 3tV-2^. 19. 3»-fi. 27. T%^lif. 12. 2^15.^12. 20. W : 88. 28. W-W- 13. iH-¥- 21. 85^ Iff. 29. 2TS^^2i|. 30. n^(i-i). 35. (|H -|) + (i^2T^). 31. n^axfi). 36. (« 0ffi)x(if+ff). 32. (iofT7j)x(|- Ht). 37. a- -|) + (Y + A)- 33. (A + A)-lf 38. (|ofl}xH)-ll- 34. (| + i)^(f- f). 39. (If ^4J,) + (ii + 2T^) COMPLEX FRACTIONS. 105. A Complex Fraction is one having a fraction in its numerator or denominator, or in both. A fraction both of whose terms are integers is called a Simple Fraction. 62 ARITHMETIC- A complex fraction may be regarded as a case in divi- sion of fractions; and it may be reduced to a simple fraction by the rule of Art. 104. 5 1. Reduce ^ to a simple fraction. -^ is the same as f -^-^ ; inverting the denominator, we have "^5 3 10 = p^^ = 4'^''^- Another method is to multiply both numerator and denomi- nator by the least common multiple of their denominators. Thus Ex. 1 may be solved as follows : The L.C.M. of 6 and 9 is 18 ; then multiplying both numerator and denominator by 18 (Art. 93), we have 5 6 1^ 18 15 3 A 10" 9 i"^ 18 20~ 4'^' 2. Simplify 2^. 42 63 3 • 42 29 ~42' 1 "29 1 2 3 = 58' 3. SimDlifv 3i + H ' •' 5f-3f The L. C. M. of the denominators 3, 8, 6, and 4, is 24. Multiplying each term of the fraction by 24, we have 80 + 45 125 5 . - =-, Ans. 140 - 90 50 2 FRACTIONS. 63 EXAMPLES. Simplify the following : ' S- '■ 56 10. 51 1 1 13. m «• 1 «• H n 11. IH 6A 14. «. «• 1- «• 2^ 12. if. 15. T%. 2fJ ,6. t + ¥. 17. 5*- 4|- -31 18. 2^ + li 8-5| 19. ^°ft. ffof2f 22. 6i.-4i + 2| 7i + 3A-8J 20 lA..^. • 2A • lOi 23. 3H + 2f + 5T^ 10A-lf-3f 21 foff-T\ofJ^ fofA + ttofil 24. TO" ~~ TO + "60" ~ Tl" H + tt-fV-A 25. A' ofW' + l °f2TV-f ofl^ ofyV -• 106. The Reciprocal of a number is 1 divided by t^hat number. Thus, the reciprocal of 5 is ^. The reciprocal of |- is -, or ^. "g" That is, the reciprocal of a fraction is the fraction inverted. 107. To find what Fraction one Number is of another. 1. What fraction of 21 is 14 ? Since 1 is Jy of 21, 14 is 14 times ^\ of 21, or ii of 21. Result, if, or |. From the above example, we derive the following RULE. Make the first number the denominator^ and the second the numerator, of a fraction. 64 ARITHMETIC. EXAMPLES. What fraction of : 2. 36 is 27 ? 7. 6i is 11 j ? 12. 2^ is 2-jV ? 3. 49 is 70 ? 8. 2^ is 24 ? 13. 90 is 4:j\ ? 4. IJ is H? 9- 40 is 5^^? 14. H is IM? 5. 41- is 3f ? 10. ff is II ? 15. 13if is 68 ? 6. H is if? 11. 43V is 7f ? 16. 1^ is Iff? 17. | + |is| + ^? 21. f ofiis|xl|? 18. 9 + 4iisll-3i? 22. I|x2|is fof 31? 19. 5f-2f is4A_|-i|? 23. 6i + 3| is 5^2-21? 20. 6|-2| is 5I-41-? 24. if of 2^ is ^ of 1|? 25. f+A-«is| + f + i? 26. 3|-l| + 5His53^ + 2|-3|f? 108. To find a Number when one of its Fractional Parts is given. 1. 7 is f of what number ? K 7 is I, one-ninth of the required number will be i of 7, or |. Then the required number is 9 times |, or -*'/, Ans. It is evident from the above that the required result may- be obtained by multiplying the first number by the second number inverted. 2. 2iV is f of what number ? 5 3 s EXAMPLES. 3. 8 is If of what number ? 4. 28 is |-J of what number ? FRACTIONS. 65 5. -^ is f§- of what number ? 6. Sj\ is fl of what number ? 7. f^ is y\ of what number ? 8. 1^ is ff of what number ? 9. f|- is II of what number ? 10. I of i| is ^ of what number ? GREATEST COMMON DIVISOR OF FRACTIONS. 109. The Greatest Common Divisor of two or more frac- tions is the greatest fraction that is contained in each of them an integral number of times. In order that one fraction may be contained in another an integral number of times, its numerator must be a divisor of the numerator, and its denominator a multiple of the denominator, of the second fraction. Thus, I is contained an integral number of times in |, since 2 is a divisor of 4, and 9 a multiple of 3. Now, the greater the numerator of a fraction, and the smaller its denominator, the greater is the value of the fraction. Hence, the greatest common divisor of two or more frac- tions is the greatest common divisor of their numerators^ divided by the least common multiple of their denominators. 1. Find the G. C. D. of -^, |f, and ^<>. The G. C. D. of 24, 16, and 40, is 8. The L. C. M. of 5, 15, and 9, is 45. Then, the required G. C. D. is ^r,^ Ans. EXAMPLES. Find the G. C. D. of: 2. A. M. and ^. 6. 33f , 94|, and 37i. 3. 41., 4^^, and 6^. 7. |, if, 4f , and 13f 4. ^, 1^, and f|. 8. 73^, 1248, and 39||. 5. 2^, fi, and 2^. 9. |4, |f , ^, and f|. 66 ARITHMETIC. LEAST COMMON MULTIPLE OF FRACTIONS. 110. The Least Common Multiple of two or more frac- tions is the smallest number that will contain each oi them an integral number of times. In order that one fraction may contain another an inte- gral number of times, its numerator must be a multiple of the numerator, and its denominator a divisor of the denom- inator, of the second fraction. Thus, f contains f an integral number of times, since 4 is a multiple of 2, and 3 a divisor of 9. Now, the smaller the numerator of a fraction, and the greater its denominator, the smaller is the value of the fraction. Hence, the least common multiple of two or more fractions is the least common multiple of their numerators, divided by the greatest common divisor of their denominators, 1. Find the L. C. M. of ^, ^j, and ^. The L. C. M. of 3, 4, and 9, is 36. The G. C. D. of 14, 21, and 35, is 7. Then, the required L. C. M. is -^^, Ans. Find the L. C. M. of : EXAMPLES. \ 2. 1, U, and If 3. 2|, If, and 3f 4. T-\, f , and ,V 5. ^, ii, and 2^. 6. 25.V,A,and^. 7. 2f , 3i|, and 3^. 8- A. A> A> and ^. 9- ^A,T¥r.T¥r,an'i*k- MISCELLANEOUS EXAMPLES. 111. 1. Reduce -^f^ to a mixed number. 2. Reduce 28 to a fraction having 137 for a denominator. 3. Multiply 2^ by 18. 5. Divide 4f by 11. 4. Multiply m by 23. 6. Divide lif by 19. FRACTIONS. 67 7. Reduce 123f | to an improper fraction. 8. Arrange in order of magnitude fj, ||, and |f . 9. Reduce ff to 768tlis. 10. Divide 33^^ by 12. 11. Divide 37^V ^J ^i- 12. J^ of ^ is f I of what number ? 13. Reduce |-fi-ff to its lowest terms. 14. Find the value of 3i X 5^ x 7^ x OJ. 15. Reduce -VAV" *^ ^*^ lowest terms. 16. Add together 2j\, 5\i, 8if , and llif . 17. Find the value of (2 - fi) X (2 - fj). 18. Subtract ^j9^ from Iff . 19. Find the value of 51 -f- (3 — ^) . 20. Find the value off-f + i-f + f • 21. Divide 2j^ by l^^,.. 22. Add together J^, |f , {i, and if^. 23. What fraction of ^ of 2i is 2^^ ? 24. Reduce f |ff to its lowest terms. 25. Subtract 13i| from 23if . 26. Multiply together |f , f §, 2-^, and 2^. 27. Find the value of (3 - ^) -^ (1 - 1%). 28. Add together 5|, 6|, 7f , 8f , and 9f . 29. Arrange in order of magnitude ^, If, ||. 30. Multiply together ^^ 2^% m, and 1^^^. 31. Find the G. C. D. of 2f |, 4^^ 5^, and 9f 32. Find the L. C. M. of ^, H, H, ff , and ff . 33. Simplify ^^^. 34. Find the value of (3^ + 4|) - (2^ + 1 A)- „^ ^. .... 539x637x66 3^- ^^^P^^^^ 34^^^23r^^^* 68 ARITHMETIC. 36. Simplify ^-H + H-H 37. Simplify (| of 2^) + (f of 8^) - (| of 2^) - (f of 1|). 38. Simplify i^ of i? - §^ + ^ of 3f . 39. Simplify -^^^jy-^j^^^Q^. 40. Simplify ||^ + (| of 3f ) - (1/^ ^ H) - fi- 41. Simplify (4AxlH)-(3A^5i) . 42. Simplify (H X W) + (H X _ii) . VT¥ • 6 3/ ^16 • TUf PROBLEMS. 112. 1. If a man can do f of a piece of work in 2^ hours, in how many hours can he do the whole ? If he can do five-ninths of the work in f f hours, he can do one- ninth in one-fifth of f f hours ; and he can do nine-ninths, or the whole, in 9 times i, or f of f f hours. 3 5 ? X ^ = — = 3| hours, A71S. ^ X^ 4 4 2. A tank can be filled by one pipe in 8 minutes, and by another in 12 minutes. How many minutes will it take to fill the tank, if both pipes are opened ? The first pipe in one minute will fill I of the tank, and the second in one minute will fill ^2 of the tank. Then both together will fill i + tV» o^ 2T of the tank in one minute. Then it will take as many minutes for both pipes together to fill the tank as /^ is contained times in f| ; that is ^-^% or 4| minutes, Ans. FRACTIONS. 69 3. If 4| tons of coal is worth $ 311, how much is 7^ tons worth ? If y- tons is worth ^- dollars, one ton is worth as many dollars as -Li is contained times inA^. 9 2 3 2 ;^ 4' 2 Then, if one ton is worth ^^ dollars, -y- tons will be worth y^ times ^ dollars. 16 3 9 i 4. A can do a piece of work in 12 days, B can do the same work in 14 days, and C in 21 days. How many days will it take all of them together to do the work ? 5. A tank can be emptied by one pipe in 9f minutes, and by another in lOf minutes. How many minutes will it take to empty the tank if both pipes are opened ? 6. A man walked 63 miles. He performed the first half of his journey at the rate of 4^ miles an hour, and the last half at the rate of 5^ miles an hour. How many hours did it take him ? 7. If 3^ of a ton of hay is worth $8|-, how much is 10 tons worth ? 8. A man having lost ^ of his money, and then spent y% of the remainder, found that he had $ 112 left. How much had he at first ? 9. How many pecks of apples, at 25J cents a peck, must be given for 12|^ pounds of sugar, at 4f cents a pound ? 10. I sold a house and lot for $ 3125, which was |f of what they cost me. How much did I lose by the opera- tion? 11. The circumference of the hind- wheel of a carriage is 9f feet, and of the fore-wheel 8f feet. How many times does each wheel turn in travelling 5280 feet ? 70 ARITHMETIC. 12. A merchant who owned J of a ship, sold |^ of his share for f 15625. What was the value of the whole ship at the same rate ? 13. A man sold a horse and carriage for $ 624, receiving f as much for the horse as for the carriage. What did he receive for each ? ■ 14. If a horse travels 7|- miles an hour, how long will it take him to travel 20|- miles ? 15. If 3-j^ is |-| of a certain number, what is J|- of the same number ? 16. What number must be multiplied by 3|^, so that the product may be 20|- ? 17. A man spent ^ of his money, and then received $105, when he found that he had f of his original amount. How much had he at first ? 18. My income is $ 8f a week, and my expenses are $ 5^j a week. How many weeks will it take me to save $ lOOi ? 19. A bale of cloth contains 75 pieces, each piece contain- ing 23|- yards. What is the whole worth at $ If a yard ? 20. What number is that if of which exceeds ^^ of it by 111? 21. If j^ of a piece of land is worth $604|, how much is ii of it worth ? 22. A dealer has 58f tons of coal in his yard. On each of six successive days he puts in 9|- tons, and sells on each day 5| tons. How many tons has he in his yard at the end of the sixth day ? 23. If a rod 2 feet long casts a shadow | of a foot long at 12 o'clock, how high is a flag-pole which casts a shadow 35| feet long at the same time ? 24. If 3f pounds of sugar cost 18 cents, how much will 6| pounds cost ? FRACTIONS. 71 25. If a town pays $ 480 for the supi)ort of 14 paupers for 15 weeks, how much should it pay for the support of 25 paupers for 21 weeks ? 26. A, B, and C found a purse containing money. A took ^ of the money ; B then took | of what remained, and C ^ the remainder, which was $ 10|^. How much money did the purse contain ? 27. In a certain school, -^ of the pupils are in the fourth class, ^ in the third class, ^ in the second class, and the remainder, 27, in the first class. How many pupils are there in each class ? 28. I have three fields containing, respectively, 5-| acres, 4^^ acres, and 111- acres. Find the size of the largest house lots, all of the same size, into which the fields can be divided. 29. If 8f tons of coal can be bought for $ 37|, how many tons can be bought for $ 22f ? 30. If a man can walk 26|- miles in 6^ hours, how far can he walk in 8^ hours ? 31. A merchant sold goods for $451, and gained f of what they cost him. How much did he gain by the opera- tion? 32. If 20f acres of land cost $ 8000, how much will 13J acres cost ? 33: If a man can do a piece of work in 7^ days, working 11| hours a day, how many days will it take him working 9^ hours a day ? 34. A tank has two pipes. One fills it at the rate of 13^ gallons an hour, and the other discharges the contents at the rate of 5| gallons an hour. If the tank holds 18|^ gallons, how many hours will it take to fill it ? 35. A can mow a field in 5 days, and A and B together can mow it in 3^ days. How many days will it take B alone to mow the field ? 72 ARITHMETIC. 36. I have $39 in the bank. If my income is $7f a week, and my expenses $ 9^ a week, how many weeks will my fund last me ? 37. If a man can do a piece of work in If^ days, what part of it can he do in ly^^ days ? 38. If the dividend is f of 21|, and the quotient | of 6J, what is the divisor ? 39. If a man can do a piece of work in 5f days, working 8^ hours a day, how long will it take him working 9f hours a day ? 40. A dealer bought a number of bales of silk, each con- taining 135 yards, at $ If a yard, and sold it at $ 2^ a yard, gaining $ 792 by the transaction. How many bales did he buy? 41. Two pendulums beat once in |-| of a second, and once in If of a second, respectively. If at any time the beats occur together, after how many seconds will they again occur together ? 42. The product of three numbers is 1|| ; if two of them are 1^ and 2-^-^, what is the third ? - 43. A can do a piece of work in 15 hours, B in 20 hours, and C in 30 hours. B and C worked alone for 5 hours, when A joined them. How many hours will it take all of them together to hnish the work ? 44. If a man travels 3i miles an hour, and 9^ hours a day, how many days will it take him to travel 906f miles? 45. A body falls 16^2" ^^^^ *^® ^^*^* second, and in each succeeding second 321 feet more than in the next preced- ing. How far does it fall in 5 seconds ? 46. Three men. A, B, and C, can walk around a circular race-course in 8i, 7^, and 6y\ minutes, respectively. If they all set out together, after how many minutes will they all meet at the starting-point, and how many times will each have gone around the course ? FRACTIONS. 73 47. A leaves Boston at a certain time, and travels at the rate of 3^ miles an hour. After he has been gone 2|- hours, B sets out to overtake him, and travels at the rate of 4| miles an hour. How far apart are A and B 5f hours after B sets out? 48. A can reap a field in 9 days, working 8 hours a day ; B can reap the same field in 8 days, working 7^ hours a day. How long will it take both together to reap the field, working 9 hours a day ? - 49. The sides of a field are 23| rods, 60| rods, 23^ rods, and 58|- rods, respectively. What is the length of the long- est pole that will be contained exactly in each side ? ^ 50. Multiply ^ of f of 9i by one-half of itself, and divide the product by -^. 51. A can do a piece of work in 9 hours ; A and B to- gether can do it in 6 hours, and B and C together can do it in 4 hours. How many hours will it take A and C together to do the work? ■-- 52. What number is that i% of |-| of which exceeds f of i^of itby 2|4? -' 53. A pole stands ^ in the mud, -^ in the water, and the remainder, 12J feet, above water. Find the length of the pole. - 54. A sum of money was divided between A, B, C, and D, in such a way that A received -f^, B -f-^, C -^j, and D the remainder, which was f Q5^. What was the sum divided, and how much did each receive ? ""^ 55. If 17 horses consume S\ bushels of oats in 3J days, how many bushels will 12 horses consume in 6f days ? " 56. A can do a piece of work in 12 days, B in 14 days, C in 18 days, and D in 21 days. How long will it take all of them together to do the work, and what part of the work does each perform ? 74 ARITHMETIC. X. t)EOIMALS. 113. A fraction whose denominator is a power of ten is usually expressed by placing a point at the right of the numerator, and then moving it to the left as many places as there are ciphers in the denominator. When thus expressed, the fraction is called a Decimal Fraction, or simply a Decimal. The point is called a Decimal Point. 114. Consider, for example, the fraction f§^|-. In this case there are three ciphers in the denominator. Placing a point at the right of the 2305, and then moving it three places to the left, we have HM = 2.305 If the number of digits in the numerator is less than the number of ciphers in the denominator, ciphers may be written in the places to the left of the first digit of the numerator. Thus, consider the fraction y^VV?r- Placing a point at the right of the 16, moving it four places to the left, and writing two ciphers at the left of the first digit, we have ■n?^ = .0016 115. The figure immediately to the right of the decimal point is said to be in the first decimal place; the next one to the right in the second decimal place ; etc. The following table gives the signification of each of the first six decimal places : 1st ; tenths. 4th ; ten-thousandths. 2d ; hundredths. 5th ; hundred-thousandths. 3d J thousandths. 6th j millionths. DECIMALS. 76 116. To read a decimal, first read the number to the left of the decimal point, if any ; then the number to the right of the point, regarded as an integer, followed by the name of the right-hand decimal place. Thus, 2,305 is read "two, and three hundred and five thousandths." .0016 is read "sixteen ten-thousandths." In order to avoid ambiguity, it is better to make a pause at the decimal point, and another before pronouncing the name of the right-hand decimal place. EXAMPLES. 117. Kead the following : 1. .5. 6. .0039. 11. 257400009. 2. .17. 7. 8.028. 12. .0004859. 3. 15.3. 8. 24.0071. 13. 863.108642. 4. .461„ 9. .84072. 14. 5.9085495. 5. 90.06. 10. .689313. 16. .00003287. Write the following as decimals : 16. Forty-nine hundredths. 17. Fifty-two, and four tenths. 18. One hundred and fifty-eight thousandths. 19. Nine, and thirteen ten-thousandths. 20. Thirty-seven, and two hundred-thousandths. 21. Fifty-nine thousand three hundred and ninety-eight millionths. 22. Eight hundred and thirty-two, and forty thousand one hundred and two hundred-thousandths. 23. Twenty-six, and eight hundred and five thousand three hundred and three millionths. 24. Seven thousand four hundred and twenty-five ten- millionths. 76 ARITHMETIC. TO REDUCE A DECIMAL TO A COMMON FRACTION. 118. A decimal may be expressed in the form of a com- mon fraction by writing the decimal without its decimal point for a numerator, and for a denominator 1, followed by as many ciphers as there are places to the right of the decimal point. Thus, 11.28 = iji^=.^/; •0523 = ^f|fo; etc. EXAMPLES. Express as common fractions in their lowest terms : 1. 2.8. 6. 8.512. 11. .0376. 16. .01375. 2. .005. 7. 30.75. 12. .0096. 17. 4.4375. 3. 75.44. 8. .1975. 13. 3.0875. 18. .15625. 4. .684. 9. 68.461. 14. .08309. 19. .008128. 6. 1.85. 10. .025. 16. .00128. 20. 2.109375. ADDITION OF DECIMALS. 119. 1. Add 7.89, 31.4, and .086. 7.89 We write the numbers so that their decimal points 31.4 shall be in the same vertical column. QgQ The sum of 8 hundredths and 9 hundredths is 17 hundredths, or 1 tenth and 7 hundredths. 39.376, Ans. The sum of 1 tenth, 4 tenths, and 8 tenths is 13 tenths, or 1 unit and 3 tenths. The sum of 1 unit, 1 unit, and 7 units, is 9 units. Then the required result is 3 tens, 9 units, 3 tenths, 7 hundredths, and 6 thousandths, or 39.376. EXAMPLES. Add the following : 2. 25.5, .00076, 1.7862, and .084. 3. 2.601, .9693, 35.08, and .00745. DECIMALS. 77 4. 165, .94468, .0051, and 59.226. 5. .4085, 8.62, .03947, and 2.139. 6. 5.0902, .00007, .637, and .014961. 7. .39665, 9.9, 72.1508, and .004052. 8. .000616, 93.38967, .0562, and 807.74. 9. .06212, 35.49, 87.56, 4920.04, and 297.868. SUBTRACTION OF DECIMALS. 120. 1. Subtract 89.725 from 162.0738. 162.0738 We write the numbers so that their decimal 89 725 points shall be in the same vertical column, ■"ZT^rTTT . 5 thousandths from 13 thousandths leave 8 thou- 72.3488, ^ns. ^^„^^^^, 3 hundredths from 7 hundredths leave 4 hundredths. 7 tenths from 10 tenths leave 3 tenths. 10 units from 12 units leave 2 units. 9 tens from 16 tens leave 7 tens. Then the required result is 7 tens, 2 units, 3 tenths, 4 hundredths, 8 thousandths, and 8 ten-thousandths, or 72.3488. If the subtrahend has more places than the minuend, we may make the number of places in the latter the same as in the former by mentally supplying ciphers in the missing places. 2. Subtract .008504 from .0162. .0162 .008504 .007696, Ans. EXAMPLES. Subtract the following : 3. .4169 from 5.2705. 8. .005341 from .0091291. 4. .0726 from .32933. 9. .0623907 from 10. 5. .318 from 1. 10. .08194812 from 2.52866. 6. .00986 from .0204. 11. 48.6007 from 830.352. 7. 2.08429 from 11.352. 12. .0002584 from .0T683. 78 ARITHMETIC. MULTIPLICATION OF DECIMALS. 121. 1. Multiply 30.84 by 2.516. Writing the decimals as common fractions, we have 30.84 X 2.516 = ^^M X ?515 = 3084 x 2516. 100 1000 100000 3084 2516 18504 3084 15420 6168 7759344 Then, 30.84 x 2.516 = ^^j^^^- = 77.59344, Ans. It is customary to arrange the work as follows : 30.84 2.516 18504 3084 15 420 6168 77.59344 ^ It will be observed that the number of decimal places in the result is the sum of the number of decimal places in the multiplicand and the number of decimal places in the multiplier ; hence the following RULE. Multiply the numbers as if they were integers, and point off as many decimal places in the result as the sum of the num- ber of decimal places in the multiplicand and multiplier. If the number of digits in the product is not sufficient for this purpose, ciphers may be written in the places to the left of its first digit. DECIMALS. 79 2. Multiply .764 by .0108. .764 .0108 In this case, we point off seven decimal places 6112 in the product, writing two ciphers at the left of 764 the first digit. .0082512, Ans. EXAMPLES. Multiply the following : 3. 8.27 by 29.3. 10. .5114 by .4053. 4. .0966 by .561. 11. .068022 by .16. 5. .00708 by .0365. 12. .4486 by 5.83. 6. .6581 by 9.7. 13. 18.052 by .75. 7. .05648 by .082. 14. 21.96 by 4.78. 8. 1.821 by 34.5. 15. .07819 by 63.05. 9. 45.66 by .00207. 16. .009256 by .08219. 122. To Multiply a Decimal by 10, 100, 1000, Etc. To multiply a decimal by 10, 100, etc., we move its deci- mal point one, twd, etc., places to the right. Or in general, to multiply a decimal by 1 followed by any number of ciphers, we move its decimal point to the right as many places as there are ciphers in the multiplier. Example. Multiply 87.35 by 10000. Moving the decimal point /owr places to the right, we have 87.35 X 10000 = 873500, Ans. 123. To Multiply a Decimal by .1, .01, .001, Etc. To multiply a decimal by .1, .01, .001, etc., we move its decimal point one, two, three, etc., places to the leji. Or in general, to multiply a decimal by .1, or by 1 pre- ceded by any number of ciphers and then a decimal point, we move its decimal point as many places to the left as there are places in the multiplier. 80 ARITHMETIC. Example. Multiply 6.294 by .001. Moving the decimal point three places to the left, we have . ^ 6.294 X .001 = .006294, Ans. 124. To Multiply a Decimal by Any Number of Tens, Hundreds, Etc. Any number of ciphers at the right of the multiplier may be omitted, if the decimal point of the multiplicand be moved to the right as many places as there are ciphers omitted. Example. Multiply 32.851 by 5200. 3285.1 52 We move the decimal point of 32.851 two 6570 2 places to the right, and multiply 3285.1 by 52. 164255 The result is 170825.2. 170825.2, Ans. In like manner, ciphers at the right of the multiplicand may be omitted, if the decimal point of the multiplier be moved to the right as many places as .there are ciphers omitted. EXAMPLES. 125. Multiply the following : 1. 85.2 by 10. 9. 5839 by .0001. 2. 377 by .01. 10. 1.417 by 10000. 3. 4.14 by 300. 11. 368.8 by .00001. 4. .00695 by 1000. 12. .04854 by 89000. 5. .000208 by 100. 13. 937620 by .001. 6. .753 by 1620. 14. .060239 by 100000. 7. .1261 by .1. 15. 14537000 by .0985. 8. 87900 by .0743. 16. 27.405 by 5240000. DECIMALS. 81 DIVISION OF DECIMALS. 126. Example. Divide 4742.66 by 754. Writing the decimal as a common fraction, 4742.66 -- 754 = ^^\^-^ -f- 754. 754)474266(629 4524 2186 1508 6786 6786 Then, 4 7^ y^6 6 _^ 754 = fj 9 ^ 5.29, Ans. It is customary to arrange the work as follows : 754)4742.66(6.29 4524 218 6 150 8 6786 67 86 It will be observed that the number of decimal places in the quotient is the same as the number of decimal places in the dividend. 127. If the divisor is not an integer, it may always be made so by moving the decimal points of both dividend' and divisor as many places to the right as there are decimal places in the divisor. 1. Divide .0275918 by .7261. (.038, Ans. 7261)275.918 ^^ *^^^ ^^^^' ^^ move the decimal points 217 83 ^^ ^*-**^ dividend and divisor four places ro ACQ to the right, and point of£ three places in fjQ Qoo the quotient. 82 l^RITHMETIC. t It is convenient, in Long Division of Decimals, to write the quotient above the dividend in such a way that each of its digits shall be directly over the right-hand digit of the corresponding partial product. Thus, in Ex. 1, the digit 3 of the quotient is directly over the right-hand digit of the first partial product, 21783, and the digit 8 is directly over the right-hand digit of the second partial product, 58088. In this case, the decimal point of the quotient will always be directly over the decimal point of the dividend. If the number of decimal places in the dividend is less than the number of decimal places in the divisor, ciphers may be written in the missing places. 2. Divide 318.68 by .257. (1240, Ans. 257)318680 ^^^ In this case, we move the decimal points of 616 both dividend and divisor three places to the 514 right, annexing one cipher to the dividend. 1028 1028 If the dividend is not exactly divisible by the divisor, it may sometimes be made so by annexing ciphers. 3.. Divide 211.347 by 40.84. (5.175, Ans. 4084)21134.700 20420 714 7 AQo * In this case, we annex two ciphers to the o^^ o/^ dividend to make it divisible by the divisor. oOb oU 285 88 20 420 20 420 DECIMALS. 83 EXAMPLES. Divide the following : 4. 4.361 by .7. 14. .0201474 by .054. 5. .11504 by .0004. 15. .00113291 by .193. 6. .005088 by .06. 16. 19.7635 by 8.41. 7. .9588 by 9.4. 17. .31236 by .00685. 8. 284.24 by .038. 18. 46.290881 by .9107. 9. 19.3752 by 20.7. 19. .487578 by .00665. 10. 4.6292 by 2.84. 20. 6.618015 by 7.174. 11. .492453 by .549. 21. 609.4429 by .001243. 12. .313048 by 87.2. 22. 332.45 by .0488. 13. 29379.7 by .47. 23. .35363808 by 89.28. 128. To Divide a Decimal by 10, 100, 1000, Etc. To divide a decimal by 10, 100, etc., we move its decimal point one, two, etc., places to the left. Or in general, to divide a decimal by 1 followed by any number of ciphers, we move its decimal point to the left as many places as there are ciphers in the divisor. Example. Divide 87.35 by 10000. Moving the decimal point four places to the left, we have 87.35 -- 10000 = .008735, Ans. 129. To Divide a Decimal by .1, .01, .001, Etc. To divide a decimal by .1, .01, .001, etc., we move its decimal point one, two, three, etc., places to the right. Or in general, to divide a decimal by .1, or by 1 preceded by any number of ciphers and then a decimal point, we move its decimal point as many places to the right as there are places in the divisor. Example. Divide 6.294 by .0001. Moving the decimal point four places to the right, we have 6.294 -f- .0001 = 62940, Ans. 84 ARITHMETIC. 130. To Divide a Decimal by Any Number of Tens, Hun- dreds, Etc. To divide a decimal by any number of tens, hundreds, etc., we omit the ciphers at the right of the divisor, and move the decimal point of the dividend as many places to the left as there are ciphers omitted. Example. Divide 4716.28 by 62800. (.0751, Ans. 628)47.1628 ^^ ^^ In this case, we move the decimal point 3 202 of 4716.28 two places to the left, and divide 3140 the result by 628. 628 *628 EXAMPLES >• 131. Divide the following : 1. 542 by 100. 8. .463 by .0001. 2. 2.99 by .01. 9. 7.815 by 1000. 3. 630 by 5000. 10. .0008171 by .001. 4. .426 by 10. 11. 855.36 by 6480000. 5. 2697.5 by 8300 12. 5156.8 by 10000. 6. .00724 by .1. 13. .001807 by .00001. 7. 6.8337 by 270. 14. 30.82 by 100000. TO REDUCE A COMMON FRACTION TO A DECIMAL. 132. 1. Reduce -^^ to a decimal. We have -^-^ = 57 -=- 125 ; dividing 57 by 125, we obtain (.456, Ans. 125)57.000 50 7 00 6 25 750 750 DE(tMi4ISl6yERS/TY ) 86 2. Eeduce | to a decimal, (.6666^, Ans. 3)2.0000 18 20 18 20 18 20 18 2 In this case, the division never terminates, no matter how far the operation may be carried. Whenever the process has been carried as far as desired, the remainder may be written over the divisor, and the fraction thus formed added to the quotient. In many numerical computations, only an approximate value of the quotient is required ; in such a case, the frac- tion which expresses the remainder may be omitted provided that, if it is equal to or greater than i, the last digit of the quotient is increased by 1. Thus, .725^ would be taken as .725, approximately ; .62|- as .63 ; and .6666| as .6667. The result .6667 is said to be the approximate value of j to the nearest fourth decimal place, or to the nearest ten-thou- sandth. Note. It may sometimes happen that neither the expression of the remainder, nor the nearest approximate value is necessary ; in such a case, the incompleteness of the quotient is denoted by a + sign. Thus, .6666+ would be written in place of Mm^. EXAMPLES. Reduce the following to decimals : f 6- W- 9-^- 12. tt- A- 7. AW 10. /j. 13. im- H- S-Tittrr- ll-rfir- 14. T3VW 86 ARITHMETIC. Find the approximate value of each, of the following to the nearest fifth decimal place : 15. f 17. ^. 19. ^^. 21. Hf. 16. {i. 18. ||. 20. ^. 22. HJ. CIRCULATING DECIMALS. 133. In expressing a common fraction as a decimal by the method of Art. 132, if the factors of the denominator are not all 2's and 5's, a single figure, or a set of figures, will be found to recur indefinitely in the quotient. Thus, if 17 be divided by 54, the quotient is .3148148+, where the digits 148 recur indefinitely. Such a decimal is called a Circulating Decimal, or simply a Circulate ; and the digit, or set of digits, which is repeated is called the Repetend. 134. A circulate is usually expressed, when a single digit is repeated, by writing a dot over it; and when a set of digits is repeated, by writing dots over the first and last of the set. Thus, .3148148 + is expressed .3148. 135. A Pure Circulate is one which has no digits except the ones which are repeated ; as .38. A Mixed Circulate is one which has one or more digits preceding the ones which are repeated ; as 4.3157. 136. To Express a Common Fraction as a Circulate. 1. Express -^J as a circulate. (.3i48, Ans. ^ao Dividing 17 by 54, the first four digits of — — - the quotient are .3148. Z/\ At this point a remainder, 80, is obtained which is the same as the first remainder. 260 It is evident from this that the digits 148 ^1^ will recur indefinitely in the quotient. 440 Then the required result is .3148. 432 80 DECIMALS. 87 In any case, the division must be carried ont until a remainder is obtained which is the same as the dividend^ or some preceding remainder. EXAMPLES. Express each of the following as a circulate : 2. A- 5. ^. 8. 3-V\V 11. tA^. 3. 3|. 6. m. 9. m. 12. iUi. 4. if 7. Idj^^. 10. «i. 13. 7^\. 137. To Find the Common Fraction which will produce a Given Circulate. 1. What fraction will produce .53 ? Let F represent the fraction. Then, one-hundred times F is equal to 53.53. Therefore, one-hundred times F, minus F, is equal to 53.53, minus .53, or 53. That is, ninety-nine times F is equal to 53. Whence, F is equal to 53 divided by 99, or -Jf , Ans. From the above example, we derive the following RULE. To find the common fraction which will produce a given pure circulate, divide the repetend by a number having for its digits as many nines as there are digits in the repetend. Thus, the fraction which will produce .353 is ||^. 2. What fraction will produce .5185 ? By the above rule, the fraction is •^^-•^^^-10 "270" 27'^ 3. .39. 8. .327. 4. .47. 9. .0945. 5. .8i. 10. 48.573. 6. .407. 11. .eosi. 7. 2.675. 12. .3726. 88 ARITHMETIC. EXAMPLES. rind the common fraction which will produce each of the following : 13. .5243. 18. .27128. 14. .9482. 19. .51378. 15. .2075. 20. .24259. 16. 7.8456. 21. 5.072i6. 17. .60405. 22. .36138. TO MULTIPLY OR DIVIDE A NUMBER BY AN ALIQUOT PART OF 10, 100, 1000, ETC. 138. An Aliquot Part of a number is a number that is exactly contained in it. Thus, S^ is an aliquot part of 10, for it is contained in 10 three times. 139. 1. Multiply 5.736 by 16|. Since 16| is one-sixth of 100, we may multiply 5.736 by 100, and divide the result by 6. Multiplying 5.736 by 100, the product is 573.6 (Art. 122). Dividing 573.6 by 6, the quotient is 95.6, Ans. 2. Divide 1.056 by 125. Since 125 is one-eighth of 1000, we may divide 1.056 by 1000, and multiply the result by 8. Dividing 1.056 by 1000, the quotient is .001056 (Art. 128). Multiplying .001056 by 8, the product is .008448, Ans. EXAMPLES. Multiply the following : 3. 72 by ^. 7. 2.46 by 166|. 4. .84 by 25. 8. .00047 by 125. 6. .0393 by 6}. 9. .976 by 6J. 6. .00525 by 33^. 10. 175.2 by 83^. DECIMALS. 89 Divide the following : 11. 35 by 2f 15. 413 by 250. 12. 10.1 by 50. 16. .98 by 12f 13. .76 by 333i. 17. 6524 by 8^. 14. 257 by 16|. 18. 802.9 by 62f MISCELLANEOUS EXAMPLES. 140. 1. Add .38752, 12.893, .008245, and 5.0169. 2. Subtract .000695367 from .00518224. 3. Express .03616 as a^common fraction, and reduce tbe result to its lowest terms. 4. Multiply 5629070 by .000001. 5. Express i-| and ff as decimals, and find their sum. 6. Divide 1.0642 by 78.25. 7. Express -jy|-| as a circulating decimal. 8. Express .25625 as a common fraction, and reduce the result to its lowest terms. 9. Multiply 84.175 by .0007302.' 10. Divide .015271938 by .4521. 11. Multiply .0297 by 111|. 12. Eeduce yf g to a decimal. 13. Multiply .00935 by 668000. 14. Divide 473.1 by .0166. 15. What common fraction will produce .9702 ? 16. Divide .0603712 by .00000001. 17. Eind the approximate value of ^-^ to the nearest fifth decimal place. 18. Divide 6^6.6 by 7450000. 19. Divide 85.29 by 3333f 20. Express ^ as a circulating decimal. 90 ARITHMETIC. 21. Express ^^^ and -^^-^ as decimals, and subtract the second result from the first. 22. Divide .48868466 by .005407. 23. What common fraction will produce .51296 ? 24. Simplify -QQ^^ + 'Q^i 2f-1.7 25. Simplify -^^-('^Q^ + 'QQ^^Q^^) . •^ .9 -(.12 -.063) 26. Simplify ^ X -^ X :^535. ^ "^ .049 .0045 .96 27. Simplify ^f:5^-i??Y ^ -^ 21.5V. 034 -85; 28. Simplify :^^^^ + :508+.5004. ^ -^ .5 - .025 .03 + .0015 29 ^ir^pi^fy (-38 X .00027)+ (.057 x .0036) ^ *^ 1-.9487 30. Simplify li+iM-iI^^H^. .16 + .8 .71 - .436 UNITED STATES MONEY. 141. Money is that which is used to measure value. 142. A Denomination is a unit of measure ; as for example, a dollar, or a cent. The denominations of United States Money are given in the following TABLE. 10 mills (m.) = 1 cent. (c.) 10 cents = 1 dime, (d.) 10 dimes = 1 dollar. ($) 10 dollars = 1 eagle, (e.) DECIMALS. 91 143. The only denomijiations used in ordinary business transactions are dollars and cents; eagles and dimes are usually expressed in terms of dollars and cents, respectively, and mills as the fraction of a cent. Thus, 5 eagles, 3 dollars, 7 dimes, 9 cents, and 2 mills, is the same as 53 dollars, and 79|- cents. Since a cent is one-hundredth of a dollar, any sum of money expressed in dollars and cents may be expressed as a decimal of a dollar by writing the number of cents in the hundredths' place. Thus, 53 dollars and 79^ cents may be expressed as 53.79^ dollars, or $ 53.79^. 144. A sum of money expressed as a decimal of a dollar may be expressed as a decimal of a dime by multiplying by 10 ; as a decimal of a cent by multiplying by 100 ; and as a decimal of a mill by multiplying by 1000. Hence (Art. 122), a sum of money expressed as a decimal of a dollar may be expressed as a decimal of a dime by mov- ing its decimal point one place to the right ; as a decimal of a cent by moving the decimal point two places to the right ; and as a decimal of a mill by moving the decimal point three places to the right. Thus, f 7.32 = 73.2 dimes = 732 cents = 7320 mills. 145. A sum of money expressed as a decimal of a mill may be expressed as a decimal of a cent by moving its decimal point one place to the left ; as a decimal of a dime by moving the decimal point two places to the left ; and as a decimal of a dollar by moving the decimal point three places to the left. Thus, 7852.1 mills = 785.21 cents = 78.521 dimes = $7.8521. 146. If two sums of money be expressed as decimals of the same denomination, they may be added or subtracted by the methods of Arts. 119 or 120 ; the result being a decimal of the same denomination. 92 ARITHMETIC. They may also be divided by the method of Art. 127. Again, a sum of money expressed as a decimal of any denomination may be multiplied or divided by the methods of Arts. 121 or 127 j the result being a decimal of the same denomination. 1. Add together $43.29, 1106.7 cents, 7480 mills, and 615 dimes. Reducing each sum to the decimal of a dollar, by the rule of Art. 145, we have, $43.29 11.067 7.48 51.5 $113,337, Ans. 2. Divide .0118364 dimes by .932 mills. Reducing the dimes to the decimal of a mill by moving the decimal point two places to the right, we have .932)1.18364(1.27, Ans. 932 2516 1864 6524 6524 EXAMPLES. 3. Express $ 19.29 as a decimal of a mill. 4. Express .000382 eagles as a decimal of a cent. 5. Express 453.7 dimes as a decimal of an eagle. 6. Express 29 mills as a decimal of a dollar. 7. Find the sum of $ 115.28, 6325.2 cents, 8.3 dimes, and 47101 mills. 8. Find the sum of .437 dimes, 1055.4 mills, $7.2195, and 543.96 cents. DECIMALS. 98 9. Subtract 3845 mills from $9.63. 10. Subtract 115.28 cents from 39.07 dimes. 11. Subtract $59,223 from 20091 cents. 12. Subtract $ .092183 from 489.56 mills, and express the result as a decimal of an eagle. 13. Multiply $320.16 by 100. 14. Multiply $ 95.78 by .0001. 15. Divide $187.25 by 10000. 16. Divide $42.56 by .001. 17. Multiply $73.29 by 380. 18. Multiply $53.08 by 72.9. 19. Multiply $ 216.273 by .414. 20. Multiply 302.8 mills by 967, and express the result as a decimal of a dime. 21. Divide $308,238 by 86.1. 22. Divide $210.60 by 5400. 23. Divide $ 669.90 by 2175 cents. 24. Divide .410652 mills by .0561 cents. 25. Divide 15.2513 dimes by $ .349. 26. Divide 23.39888 cents by 34.01 dimes. PROBLEMS. 147. 1. What is the cost of 27 pounds of tea at 43^ cents a pound ? 27 g1 We multiply 27 first by 3, then by 4, and finally -jl Q g by i, and add the results. 9 $11.70, ^ns. 94 ARITHMETIC. 2. How many yards of cloth at 66^ cents a yard can be bought for ^8.00? 800 _ 800 _ ^^^ _3_ $ 8.00 is the same as 800 cents. 66f "" ^^ ~" ^^^ ^00 Dividing 800 cents by 66 1 cents, the = 12, Ans. quotient is 12. 3. What is the cost of 3| tons of coal at $ 5.76 a ton ? 4. What is the cost of 147 yards of cloth at 16f cents a yard? 5. Find the sum of |i of $ 5.67, and f of $ 8.75. 6. How many pounds of coffee, at 41| cents a pound, can be bought for ^121.25? 7. If 35352 yards of silk cost $ 59.78, how much will one yard cost ? 8. What is the cost of 18 tons of coal at $5.83J a ton ? 9. A grocer received on Monday $135.25, on Tuesday $84.40, on Wednesday $106.65, on Thursday $122.70, on Friday $ 93.62, and on Saturday $185.56. What were his total receipts for the week ? 10. What is the cost of a barrel of sugar weighing 276 pounds, at $ .04f a pound ? 11. What is the cost of 17| acres of land at $ 238.45 an acre? 12. If 12| cords of wood cost $116.66, how much will one cord cost ? 13. If bricks be sold at $ 6.75 a thousand, how much will 8960 bricks cost ? 14. How many barrels, each containing 32.75 gallons, can be filled from 1200 gallons of wine, and how many gallons will remain ? 15. How many barrels of flour, at $ 7.35| a barrel, can be bought for $551.87^? 16. If 13f yards of cloth can be bought for $33.20, how much will 15| yards cost ? DECIMALS. 95 17. If 24 pounds of butter be given for 100.8 pounds of sugar, bow many pounds sbould be given for 91 pounds of sugar ? 18. A man having ^100, spent $28.59, then received $15.75, then spent $48.98, and finally received $37.37. How much money then had he ? 19. A farmer sold 15 loads of wheat, each containing 8.75 bushels, for 92 cents a bushel. How much money did he receive ? 20. If 51 tons of coal cost $ 50, how much will fl- of a ton cost ? 21. The product of three numbers is 3.9701556. If two of them are 92.7 and 5.16, what is the third ? 22. If 13| barrels of flour cost $ 89.11, how much will || of a barrel cost ? 23. A, B, and C received $17.40 for a piece of work. If A did -f^ of the work, B -^-^, and C the remainder, how much money should each receive ? 24. A man worked 25f days. He paid out | of his earn- ings for board, and had $ 12.40 left. Find his daily wages. 25. A and B start at the same time, from the same place, and walk in the same direction at the rates of 3.4798 and 4.1263 miles an hour, respectively. How far apart are they at the end of 7.255 hours ? 26. A dealer sold goods for $124.25, and gained | of what they cost him. How much did they cost him ? 27. Which is the greater, -J of $16.12, or | of $17.67, and how much ? 28. Find the cost of 18| feet of steel rod, at $ .35 a foot, and 241 feet at $ .456 a foot. 29. A merchant sold goods for $ 87.95, and lost f of what they cost him. How much did he lose by the operation? 30. If 15f yards of cloth can be bought for $12.73, how many yards can be bought for $ 18.76 ? 96 ARITHMETIC. 31. A man bought 7 car-loads of wheat, each car con- taining 165 bushels, at ^ .84^ a bushel. What was the cost ? 32. If 5f pounds of coffee can be bought for $ 2.36^, how many pounds can be bought for $6.54^? 33. If A can do a piece of work in 2.4 days, and B in 3.2 days, how long will it take both of them together to do the work? 34. Find the cost of 237 bales of silk, each containing 125| yards, at $1.26 a yard. 35. A man spent -f of his money for provisions, ^ of the remainder for clothing, and had f 12.87 left. How much had he at first ? 36. If .488 of a ton of coal be worth $ 3.05, how much will 7.56 tons cost? 37. A man left ^ of his estate to his wife, f of the re- mainder to his son, and the rest to his daughter. The wife received $ 546.75 more than the daughter. What did each receive ? 38. A farmer sold 35 tubs of butter, each weighing 43^ pounds, at 21|- cents a pound. He bought 21 barrels of flour at $6f a barrel, and received the balance in cash. How much money did he receive ? 39. Three men. A, B, and C, can do a piece of work in 18, 24, and 36 hours, respectively. How long will it take all of them together to do the work? If they receive $ 11.25 for the work, how should the money be divided ? 40. A merchant bought goods to the amount of $ 228.60. He kept y^-g- of them for his own use, and sold the remainder for -^ more than they cost him. How much did he gain ? 41. A grocer bought 10 pounds of tea at $ .38^ a pound, 12 pounds at $.41|- a pound, and 15 pounds at $.433 a pound. He sold the whole at $ .44^ a pound. How much did he gain ? DECIMALS. 97 42. A gentleman divided f 22.05 between his two sons in such a way that the younger received f as much as the elder. How much did each receive ? 43. If a horse travels 8.3 hours a day, at the rate of 6.25 miles an hour, how many days will it take him to travel 830 miles ? 44. Find the cost of 12.6 bales of silk, each containing 73.625 yards, at $ 1.20 a yard. 45. A tank has two pipes. One of them can fill it in 8.5 minutes, and the other can empty it in 12.75 minutes. How many minutes will it take to fill the tank, if both pipes are opened together ? 46. A man invests f of his property in real estate, -^ of the remainder in railway shares, and the balance in city bonds. The amount invested in city bonds exceeds by $93.75 the amount invested in real estate. Find the amount of each kind of investment. 47. A gentleman left ^ of his property to his wife, ^ to his elder son, i to his younger son, i to his daughter, and the balance, $369.75, to a charitable institution. How much did each receive ? 48. A, B, and C can do a piece of work in 6, 14, and 21 days, respectively. B and C worked alone for 5 days, when they were joined by A, and the work was completed by all of them together. If they received $ 54 for the work, how should the money be divided ? 49. Three pipes can empty a tank in 2.25, 3.375, and 5.0625 minutes, respectively. How many minutes will it take to empty the tank if all the pipes are opened ? 98 ARITHMETIC. XI. MEASURES. 148. Measures of Length. Measures of Length, or Linear Measures, are those used in measuring lengths or distances. TABLE. 12 inches (in.) =1 foot, (ft.) 3 feet =lyard. (yd.) 5 J yards = 1 rod. (rd.) 320 rods = 1 mile, (mi.) It follows from the above that 1 mile = 1760 yards = 5280 feet. Surveyors use, in the measurement of land, a chain (ch.) whose length is 4 rods, divided into 100 links (li.) of 7.92 inches each. 80 chains are equal to one mile. 149. Measures of Area. Measures of Area, Surface Measures, or Square Measures, are those used in measuring areas. TABLE. 144 square inches (sq. in.) = 1 square foot. (sq. ft.) 9 square feet = 1 square yard. (sq. yd.) 30^ square yards = 1 square rod. (sq. rd.) 160 square rods = 1 acre. (A.) 640 acres = 1 square mile. (sq. mi.) It follows from the above that 1 acre = 43560 square feet. 10 square chains are equal to one acre. MEASURES. 99 150. Measures of Volume. Measures of Volume, or Cubic Measures, are those used in measuring volumes. TABLE. 1728 cubic inches (cu. in.) == 1 cubic fpot. (cu. ft.) 27 cubic feet = 1 cubic yard. (cu. yd.) The following is used in measuring wood : 128 cubic feet = 1 cord, (cd.) 151. Measures of Capacity. Liquid Measures are used in measuring liquids. TABLE. 4 gills (gi.) = l pint. (pt.) 2 pints = 1 quart, (qt.) 4 quarts = 1 gallon, (gal.) The following are less frequently used : 31J gallons = 1 barrel. 63 gallons = 1 hogshead. Dry Measures are used in measuring grain, vegetables, fruit, etc. TABLE. 2 pints (pt.) =1 quart, (qt.) 8 quarts = 1 peck. (pk.) 4 pecks =1 bushel, (bu.) The quart liquid measure contains 57f cubic inches, and the quart dry measure 67-|- cubic inches ; the gallon contains 231 cubic inches, and the bushel 2150.42 cubic inches. 100 ARITHMETIC. Apothecaries' Liquid Measures are used in compounding medicines. TABLE. 60 minims (n\^)=l fluid dram. (f3) 8 fluid drams = 1 fluid ounce, (f 5 ) 16 fluid ounces = 1 pint. (0.) 152. Measures of Weight. Avoirdupois Weight is used in weighing all common articles. TABLE. 16 drams (dr.) = 1 ounce, (oz.) 16 ounces = 1 pound, (lb.) 100 pounds = 1 hundred-weight, (cwt.) 20 hundred-weight = 1 ton. (T.) The following are used at the United States Custom Houses, and in weighing iron and coal at the mines : 112 pounds = 1 long hundred-weight. 2240 pounds = 1 long ton. Troy Weight is used in weighing gold, silver, and jewels. TABLE. 24 grains (gr.) = 1 pennyweight, (pwt.) 20 penny weights = 1 ounce, (oz.) 12 ounces = 1 pound, (lb.) Apothecaries' Weight is used in compounding medicines. TABLE. 20 grains (gr.)=l scruple. (3) 3 scruples = 1 dram. ( 3 ) 8 drams = 1 ounce. ( 5 ) 12 ounces = 1 pound. (tb) MEASURES. 101 A pound avoirdupois contains 7000 grains, and a pound troy 5760 grains; the pound, ounce, and grain, in apothe- caries' weight, have the same weight as in troy weight. 153. Measures of Time TABLE. 60 seconds (sec.) = 1 minute, (min.) 60 minutes = 1 hour. (h.) 24 hours = 1 day. (d.) 7 days = 1 week. (wk.) 365 days = 1 common year, (y.) 366 days = 1 leap year. The Calendar Months are as follows : January, 31 dayj 3. July, 31 iays February, 28 oi 29 " August, 31 March, 31 " September, 30 April, 30 " October, 31 May, 31 " November, 30 June, 30 " December, 31 The Solar Year is 365 days, 5 hours, 48 minutes, and 49.7 seconds, or very nearly 365^ days. If the number of any year is divisible by 4, the month of February has 29 days, and the year is called a leap year; but if the number of the year is divisible by 100, it is not a leap year, unless it is divisible by 400. Thus, 1600 is a leap year, but not 1700. 154. English Money. TABLE. 4 farthings (far.) = 1 penny, (d) 12 pence = 1 shilling, (s.) 20 shillings = 1 pound. (£) 102 ARITHMETIC. The following are also used : 1 florin = 2 shillings. 1 crown = 5 shillings. 1 guinea = 21 shillings. The English coin of the value of 20 shillings is called a sovereign. 155. Angular Measures. Angular Measures are used in measuring angles and arcs of circles. TABLE. 60 seconds (") = 1 minute. (') 60 minutes = 1 degree. (°) The following are also used : A quadrant is an arc of 90°. A circumference is an arc of 360°. A right-angle is an angle of 90°. The length of a degree (-jj^) of the earth's equator is about 69|- miles. 156. Miscellaneous Tables. Numbers. Paper. 12 units = 1 dozen. 24 sheets = 1 quire. 12 dozen = 1 gross. 20 quires = 1 ream. 12 gross = 1 great gross. 2 reams = 1 bundle. 20 units = 1 score. 5 bundles = 1 bale. DENOMINATE NUMBERS. 103 XII. DENOMINATE NUMBERS. 157. A Denomination is a unit of measure; as for ex- ample, a mile, a pound, or a bushel. (Compare Art. 142. ) 158. Two or more denominations are said to be of the same kind when each can be expressed in terms of the others. Thus, a mile and a rod are denominations of the same kind. 159. If two or more denominations are of the same kind, the product of the first by any number, plus the product of the second by any number, and so on, is called a Com- pound Number. For example, 5 miles + 31 rods + 4 yards + 2 feet, or as it is usually expressed, 5 mi. 31 rd. 4 yd. 2 ft., is a com- pound number. Note. The number by which the denomination is multiplied may be an integer, a mixed number, or a fraction. Thus, 3 ft. 5^ in. is a compound number. In contradistinction, the product of a single denomination by any number is called a Simple Number. For example, 5 mi. is a simple number. 160. Simple and Compound Numbers are called Denomi- nate Numbers. REDUCTION OF DENOMINATE NUMBERS. 161. Reduction Descending. Reduction Descending is the process of expressing denomi- nate numbers in terms of lower denominations. 1. Express dB 4 in pence. £4 20 Since £1 = 20s., £4 = 4 x 20s., or 80s. 80s. Since Is. = 12d., 80s. =: 80 x 12d., or 9600. EXAMPLES. Find the values of the following : '3. V121. 5. v'1296. 7. VW- 4. ^125. 6. ^1024. 8. m 9. 225. Areas of Polygons. The proofs of the following principles may be found in any text-book on Geometry : 1. The area of a triangle is equal to one-half the product of its base and altitude. 2. The area of a parallelogram (or rectangle) is equal to the product of its base and altitude. 3. The area of a square is equal to the square of one of its F "^^~" Q E H B -E C \ 1. \ 4. The area of a trapezoid is equal to one-half the sum of its bases, multiplied by its altitude. It is important to observe that, in finding the product of two lines, such as the base and altitude of a triangle, their lengths must be expressed in terms of the same unit, and the area is obtained in terms of the square of this unit. Thus, to multiply 3 feet by 7 inches, we must first express 3 feet in inches. Kow 3 feet = 36 inches ; and multiplying 36 inches by 7 inches, the product is 252 square inches. EXAMPLES. 226. 1. Find the area of a trapezoid whose bases are 43 in. and 35 in., respectively, and altitude 21 in. By Art. 225, 4, the area of the trapezoid is J x (43 + 35) x 21, or 819 sq. in., Ans. 164 ARITHMETIC. 2. Th^ area of a triangle is 1\ sq. yd. ; if its altitude is 40 in., what is its base in feet ? By Art. 225, 1, the base of a triangle is equal to twice its area, divided by its altitude. We have, IJ sq. yd. = -\5 sq. ft., and 40 in. = Y- ft- Now 45^10^|^A = 27^ 2 3 2 ;p 4 ^ 2 Whence, the required base is 6| ft., Ans. 3. What is the side of a square whose area is 7^ sq. rd. ? By Art. 225, 3, the side of a square is equal to the square root of its area. Now V7i=V^=| = 2f. Hence, the required side is 2| rd., Ans. 4. Find the area in square inches of a triangle whose base is 3^ it, and altitude 2^ ft. 6. Find the area in square feet and square inches of a square, each side of which is 3 ft. 10 in. 6. Find the area in square yards of a parallelogram whose base is 7|- ft., and altitude 64 in. 7. Find the base in feet of a triangle whose area is 14|- sq. ft., and altitude 87 in. 8. Find the altitude in yards of a rectangle whose area is 1782 sq. ft., and base 3 rd. 9. The area of a square is 20 sq. rd. 20 sq. yd. ; what is its side in feet ? 10. Find the area in square inches of a trapezoid whose bases are 8| ft. and 5|- ft., respectively, and altitude 3| ft. 11. Find the altitude in yards of a triangle whose area is 3 sq. ft., and base 27 in. 12. Find the base in inches of a rectangle whose area is 3|- sq. yd., and altitude 4|- ft. MENSURATION. 155 13. Find the area in acres of a square field whose side is 396 ft. 14. Find the area of a floor whose length is 15 ft. 4 in., and width 12 ft. 10 in. 15. A triangular house-lot contains 3 acres. If its base is 500 feet, what is its altitude in rods ? 16. Find the width of a rectangular field whose area is 15 acres, and length 75 rods. 17. If the side of a square field is 14 rd. 3 yd., how much is the field worth at $ 242 an acre ? 18. Find the altitude in feet of a trapezoid whose area is 139J sq. ft., and bases 19 ft. and 12 ft., respectively. 19. How many bricks, each 8 inches long and 4^ inches wide, will be required to lay a sidewalk 7^ feet wide and 220 feet long? 20. A rectangular garden, 64 feet long and 35 feet wide, is surrounded by a walk 3 feet wide. How many square feet are there in the walk ? 21. A map is 1-J ft. long, and 1 ft. wide. If the scale of the map is 2^ miles to an inch, how many square miles of country does it represent ? 22. Find the length in yards of a rectangular field whose area is 7 acres, and width 363 feet. 23. Find the lower base in rods of a trapezoid whose area is 1 A. 50 sq. rd., upper base 99 yd., and altitude 165 ft. 24. The dimensions of a rectangular floor are 22 ft. 6 in., and 16 ft. 9 in. What will it cost to cover it with oil-cloth, at $ 1.52 a square yard ? 25. How many paving-stones, each 7 inches long and 4 inches wide, will be required to pave two miles of street, 63 ft. in width ? 26. A field is 10 rd. 5 yd. long, and 13 rd. 3J yd. wide. How much is it worth at $ 605 an acre ? 156 ARITHMETIC. 27. A man sold a rectangular field for 3J cents a square foot, receiving the sum of $ 5288.40 ; if the field was 28 rd. 2 yd. long, what was its width ? 28. If the area of a field is 2 A. 32 sq. rd. 26 sq. yd. 2 sq. ft., and its width 15 rd. 2 yd. 2 ft., what is its length ? 29. A field is 18 rd. 3 yd. 2 ft. wide, and 33 rd. 1 yd. 1^ ft. long. How much is it worth at $ 484 an acre ? 227. It is proved in Geometry that In a right triangle, the square of the hypot- enuse is equal to the sum of the squares of the other two sides. Note. This means that, if the sides are all ex- ^ pressed in terms of the same unit, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It follows from the above that In a right triangle, the square of either side about the right angle is equal to the square of the hypotenuse, minus the square of the other side. EXAMPLES. 228. 1. The sides about the right angle of a right triangle are 5 in. and 1 ft., respectively; find the hypotenuse in inches. We have 1 ft. = 12 in. But VP + 12^ = V25 + 144 = V169 = 13. Whence, the hypotenuse is 13 in., Ans. 2. The hypotenuse of a right triangle is 2f in., and one of the sides about the right angle is 1\ in. Find the other side. We have V(2|)2 - (4)2 = V(-V-)2 - (|)2 — ■\/l3J'. _ J_6 — •v/225 _ 15 — 2 1 Whence, the required side is 2^ in., Ans. MENSURATION". 157 3. The diagonal of a square is 2 feet ; find the approxi- mate value of its side to three places of decimals. By Art. 227, the square of the diagonal of a square is equal to twice the square of its side. Then 4 sq. ft. is equal to twice the square of the side. Hence, tlie side is equal to the square root of 2 sq. ft. The approximate value of V2 to three decimal places is 1.404. Whence, the required side is 1.404+ ft., Ajis. 4. The sides about the right angle of a right triangle are 7 in. and 2 ft., respectively. Find the hypotenuse in inches. 5. The sides about the right angle of a right triangle ar^ 1 rd. 31 yd., and 7 rd. li yd., respectively. Find the hypotenuse. 6. The hypotenuse of a right triangle is 3^ ft., and one of the sides about the right angle is -| yd. Find the other side in inches. 7. The hypotenuse of a right triangle is 1 yd. 2 ft. 1 in., and one of the sides about the right angle is 11 in. Find the other side. 8. The diagonal 'of a square is 15 in. Find the approxi- mate value of its side to four places of decimals. 9. The sides about the right angle of a right triangle are 10 in. and 7 in., respectively. Find the approximate value of the hypotenuse to four places of decimals. 10. What is the length of the longest straight line that can be drawn on a floor whose length is 17 ft. 8 in., and width 13 ft. 3 in. ? 11. How far from a tower 35 ft. high must the foot of a ladder 37 ft. long be placed, so as to exactly reach the top of the tower ? 12. A tree was broken off 12 ft. above the ground, and fell so that its top lay 47 ft. 3 in. from the foot of the tree, the end where it was broken resting on the stump. What was the height of the tree ? 158 ARITHMETIC, 13. A vessel sails due east at tlie rate of 6f miles an hour, and another sails due south at the rate of V2 miles an hour. How far apart are they at the end of 7 h. 45 min. ? 14. A ladder 37^ feet long is placed so that it just reaches a window 22J ft. above the street ; and when turned about its foot, just reaches a window 36 ft. above the street on the other side. Find the width of the street. 15. If the area of a square is 33 sq. ft. 50 sq. in., what is the length of its diagonal ? . 229. A Circle is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within called the centre; as ABD. The bounding curve is called the cir- cumference of the circle. A radius is a straight line drawn from the centre to the circumference, as CD ; a diameter is a straight line drawn through the centre, having its extremities in the circumference ; as AD. 230. Measurement of the Circle. It is proved in Geometry that, approximately, 1. The circumference of a circle is equal to its diameter multiplied by 3.1416. 2. The area of a circle is equal to the square of its radius multiplied by 3.1416. Note. The above rules give the circumference in terms of the unit in which the diameter is expressed, and tlie area in terms of the square of the unit in which tlie radius is expressed. The following rules are also useful ; 3. The circumference of a circle is approximately equal to twice its radius multiplied by 3.1416. 4. The area of a circle is approximately equal to one-fourth the square of its diameter multiplied by 3.1416. MENSURATION. 159 EXAMPLES. 231. 1. What is the circumference of a circle whose diameter is 7 inches ? By Art. 230, 1, the required circumference is 7 X 3.1416 = 21.9912 in., Ans. 2. Eind the diameter of a circle whose area is 35 sq. ft. By Art. 230, 4, the square of the diameter of a circle is equal to four times the area divided by 3.1416. We have -^ = 44.5632+. o.l41o ' The square root of 44. 5632 + is 6.67 + . Whence, the required diameter is 6.67+ ft., Ans. 3. The radius of a circle is 7 inches. Find its circumfer- ence and area. 4. The diameter of a circle is 50 ft. Find its circumfer- ence in yards. 5. The circumference of a circle is 33 rods. Find its radius. 6. Find the diameter in inches of a circle whose area is one square foot. 7. If the diameter of the earth is 7912 miles, what is the distance around it ? 8. A wheel is 2 ft. 3 in. in diameter. How many miles does it travel in revolving 2000 times ? 9. How many acres are there in a circular field whose diameter is 500 feet ? 10. A horse is tied by a rope 31 ft. 6 in. long ; over how many square yards of ground can he graze ? 11. A wheel turns 29 times in travelling 154 yd. 2 ft. Find its diameter in inches. 12. The floor of a room 12 ft. 3 in. long, and 10 ft. 8 in. wide, has two circular openings whose radii are 2 ft. 1 in., and 1 ft. 5 in., respectively. Find the area of floor remaining. 160 ARITHMETIC. 13. A circular pond, 100 ft. in diameter, is surrounded by a walk 4 feet wide. Find the area of the walk. 14. If a wheel is 5 feet in diameter, how many times does it revolve in running 27 miles ? 15. The diameter of a circle is 10 inches. What is the side of an equivalent square (Art. 216) ? 16. The side of a square is 8 feet. What is the circum- ference of an equivalent circle ? 17. Two plots of ground, one a square, the other a circle, contain each 70,686 sq. ft. How much longer is the perim- eter of the square than the circumference of the circle ? SOLIDS. 232. The volume of a solid is the number of times that it contains another solid, adopted as the unit of measure- ment. Thus, the statement that the volume of a solid is 6 cu. ft., means that a cubic foot is contained in the solid 6 times. Two solids are said to be equivalent when they have equal volumes. 233. A Polyedron is a solid bounded by planes. The bounding planes are called the /aces of the polyedron; their intersections are called the edges, and the intersections of the edges are called the vertices. 234. A Prism is a polyedron two of whose faces are equal and parallel, the other faces being parallelograms ; as A-I. The equal and parallel faces, ABCDE and FQHIK, are called the bases of the prism, and the remaining faces the lateral faces. The lateral faces taken together form the lateral surface of the prism ; and their intersections, AF, BG, etc., are called the lateral edges. MENSURATION. 161 /]— / dI — -}C The lateral area is the area of the lateral surface. The altitude is the perpendicular distance LM between the planes of the bases. 235. A Right Prism is one whose lateral edges are perpendicular to its bases; as ABG-DEF. The lateral faces are rectangles, and the lateral edges are equal to the altitude. 236. A Rectangular Parallelepiped is a right prism whose six faces are all rectangles ; as A-G. The dimensions are the three edges which meet at any vertex. A Cube is a rectangular parallelopiped whose six faces are all squares. 237. A Pyramid is a polyedron bounded by a polygon and a series of triangles having a com- mon vertex ; as 0-ABCD. The polygon ABGD is called the base of the pyramid ; and the common vertex of the triangular faces is called the vertex. The triangular faces are called the lateral faces, and taken together form the lateral surface. The intersections OA, OB, etc., of the lateral faces are called the lateral edges; and the area of the lateral surface is called the lateral area. The altitude is the perpendicular OE drawn from the vertex to the base. 238. A Regular Pyramid is one whose base is a regular polygon, and whose vertex lies in the perpendicular erected at the centre of the base j as 0-ABCD, 162 ARITHMETIC. The slant height of a regular pyramid is the altitude of any one of its lateral faces ; that is, it is the straight line drawn from the vertex to the middle point of any side of the base : as OF. 239. A Frustum of a pyramid is that portion of a pyramid included between the base and a plane parallel to the base ; as ABC-DEF. The altitude of the frustum is the per- pendicular distance between the planes of its bases ; as GH. 240. The slant height of a frustum of a regular pyramid is the straight line joining the middle points of the parallel sides of any lateral face ; as LM. ■D'K Z^ / ^ /i V Je^-^^ ^ °I\!A' / 1 M 1 ~U 241. Lateral Polyedrons. Areas and Volumes of The proofs of the following principles may be found in any text-book on Solid Geometry : 1. The lateral area of a right prism (or rectangular paral- lelopiped) is equal to the perimeter of its base multiplied by its altitude. 2. The volume of a prism (or rectangular parallelopiped) is equal to the area of its base multiplied by its altitude. 3. The volume of a cube is equal to the cube of one of its edges. 4. The lateral area of a regular pyramid is equal to the perimeter of its base multiplied by one-half its slant height. 5. The volume of a pyramid is equal to the area of its base midtiplied by one-third its altitude. 6. The lateral area of a frustum of a regular pyramid is equal to one-half the sum of the perimeters of its bases, multi- plied by its slant height. MENSURATION. 163 7. The volume of a frustum of a pyramid is equal to the sum of the areas of its bases, plus the square root of the product of the areas of its bases, multiplied by one-third its altitude. To multiply an area by a length, the area must be expressed in terms of the square of the unit in which the length is expressed, and the product is obtained in terms of the cube of this unit. Thus, to multiply 2 sq. ft. by 10 in., we must first express the 2 sq. ft. in square inches. Now 2 sq. ft. = 288 sq. in. ; and multiplying 288 sq. in. by 10 in., the product is 2880 cu. in. EXAMPLES. 242. 1. Find the lateral area of a regular pyramid, the perimeter of whose base is 17 in., and slant height 8 in. By Art. 241, 4, the required lateral area is J X 17 X 8, or 68 sq. in., Ans. 2. The volume of a rectangular parallelepiped is 693 cu. ft., and the dimensions of its base are 11 ft. and 7 ft. Find its altitude, and the area of its entire surface. The area of the base is 7 x 11, or 77 sq. ft. By Art. 241, 2, the altitude of the parallelepiped is equal to its volume, divided by the area of its base. Hence, the required altitude = -^^^/-, or 9 ft. By Art. 241, 1, the lateral area is equal to 2 X (7 + 11) X 9, or 324 sq. ft. The area of the two bases is 2 x 77, or 154 sq. ft. Hence, the area of the entire surface is 324 sq. ft. + 154 sq. ft., or 478 sq. ft., Ans. 3. Find the volume of a frustum of a pyramid whose lower base is a square 7 in. on a side, upper base 4 in. on a side, and altitude 6 in. The areas of the bases are 49 sq. in. and 16 sq. in., respectively. Now, V49 x 16 = V78i = 28. 164 ARITHxMETIC. Hence, the square root of the product of the areas of the bases is 28 sq. in. Then by Art. 241, 7, the required volume is ^ X (49 + 16 + 28) X 6, or 186 cu. in., Ans. 4. Find the lateral area and volume of a prism whose altitude is 11 in., having for its base a right triangle whose sides are 5 in., 12 in., and 13 in. 5. Find the volume and area of the entire surface of a cube whose edge is 3:^ inches. 6. Find the lateral area of a regular pyramid whose base is a square 6 ft. on a side, and slant height 12 feet. 7. The volume of a prism, whose base is a square, is 637 cu. ft., and its altitude is 13 ft. Find the length of each side of the base, and the lateral area. 8. What is the volume of a pyramid whose altitude is 21 in., having for its base a right triangle whose sides are 8 in., 15 in., &,nd 17 in. ? 9. Find the lateral area of a frustum of a regular pyra- mid whose lower base is a square 9 ft. on a side, upper base 5 ft. on a side, and slant height 14 ft. 10. The volume of a pyramid, whose base is a square, is 847 cu. in., and its altitude is 21 in. Find the length of each side of the base. 11. Find the volume of a frustum of a pyramid whose lower base is a rectangle 15 in. by 6 in., upper base 5 in. by 2 in., and altitude 15 in. 12. The lateral area of a regular pyramid is 1680 sq. in. The base is a triangle whose sides are all equal, and the slant height is 35 in. Find the length of each side of the base. 13. A box is 13 in. long, 12 in. wide, and 7 in. deep. Find its volume, and the area of its entire surface. 14. The volume of a box is 84 cu. ft., and the dimensions of its bottom are 7 ft. and 4 ft. Find its depth, and the area of its entire surface. MENSURATION. 165 15. A wagon 7 ft. long and 4 ft. wide is piled with wood to a depth of 5^ ft. What is the value of the wood at $7.04 a cord? 16. The lateral area of a frustum of a regular pyramid is 936 sq. in. The lower base is a square 18 in. on a side, and the upper base is 6 in. on a side. Find the slant height of the frustum. 17. The volume of a cube is 4^ cu. ft. What is the area of its entire surface in square inches ? 18. What will be the cost of a pile of wood 35|- ft. long, 6i ft. high, and 4 ft. wide, at $ 7.68 a cord ? 19. A monument whose height is 12 ft., is in the form of a pyramid with a square base, 2 ft. lOi in. on a side. Find its weight, at 180 lb. to the cubic foot. 20. How many bricks, each 8 in. long, 2J in. wide, and 2 in. thick, will be required to build a wall 18 ft. long, 3 ft. high, and 11 in. thick ? 21. What must be the length of a pile of wood that is 3 ft. 9 in. wide, and 5 ft. 4 in. high, to contain 5 cords ? 22. A monument is in the form of a frustum of a square pyramid 8 ft. in height, surmounted by a square pyramid 2 ft. in height. If each side of the lower base of the frus- tum is 3 ft., and each side of the upper base 2 ft., find the volume of the monument. 23. The volume of a frustum of a pyramid is 210 cu. in. The lower base is a right triangle whose sides are 6 in., 8 in., and 10 in., and the upper base a right triangle whose sides are 3 in., 4 in., and 5 in. Find the altitude of the frustum. 24. A box made of 2 in. plank, without a cover, measures on the outside 3 ft. 2 in. long, 2 ft. 3 in. wide, and 1 ft. 6 in. deep. How many cubic feet of material were used in its construction ? 25. The base of a square pyramid is 14 in. on a side, and the altitude is 24 in. Find its lateral area and volume. 166 ARITHMETIC. 26. The water in a certain ditch flows at the rate of 2j miles an hour. If the ditch is 3 ft. wide, and 2 ft. deep, how many cubic feet of water pass through it in one day ? 243. A Cylinder is a solid bounded by two parallel circles, and a curved surface all points of which are equally distant from a straight line within called the axis; as A-D. The parallel circles AB and CD are called the bases of the cylinder, and the curved surface is called the lateral surface. The lateral area is the area of the lateral surfacCj and the altitude is the perpendicular distance EF between the bases. 244. A Cone is a solid bounded by a circle, and a curved surface which tapers uniformly to a point called the vertex; as OAB. The circle AB is called the base of the cone, and the curved surface is called the lateral surface. The lateral area is the area of the lateral surface, and the altitude is the perpendicular OC from the vertex to the base. The slant height is the straight line drawn from the vertex to any point in the circumference of the base ; as OD. 245. A Frustum of a cone is that portion of a cone in- cluded between the base and a plane parallel to the base ; as A-D. The altitude of the frustum is the perpendicular distance between the bases ; as EF. The slant height is that portion of the slant height of the cone included between the bases of the frustum ; as GH. MENSURATION. 167 246. A Sphere is a solid bounded by a curved surface, all points of which are equally distant from a point within called the centre ; as ABD. A radius is a straight line drawn from the centre to the surface, as OA ; a diameter is a straight line drawn through the centre, having its extremi- ties in the surface, as AC. 247. Measurement of the Cylinder, Cone, and Sphere. It is proved in Geometry that : 1. The lateral area of a cylinder is equal to the circumfer- ence of its base multiplied by its altitude ; or, approximately, to twice the radius of its base, times its altitude, times 3.1416. 21 The volume of a cylinder is equal to the area of its base multiplied by its altitude; or, approximately, to the square of the radius of its base, times its altitude, times 3.1416. 3. The lateral area of a cone is equal to the circumference of its base multiplied by one-half its slant height; or, approxi- mately, to the radius of its base, times its slant height, times 3.1416. 4. The volume of a cone is equal to the area of its base multiplied by one-third its altitude; or, approximately, to the square of the radius of its base, times one-third its altitude, times 3.1416. 5. TTie lateral area of a frustum of a cone is equal to one- half the sum of the circumferences of its bases, multiplied by its slant height; or, approximately, to the sum of the radii of its bases, times its slant height, times 3.1416. 6. TTie volume of a frustum of a cone is equal to the sum of the areas of its bases, plus the square root of the product of the areas of its bases, multiplied by one-third its altitude; or, approximately, to the sum of the squares of the radii of its bases, plus the product of the radii of its bases, times one- third its altitude, times 3.1416. 168 ARITHMETIC. Also, approximately, 7. The area of a sphere is equal to the square of its di- ameter, or four times the square of its radius, multiplied by 3.1416. 8. The volume of a sphere is equal to one-sixth the cube of its diameter, or four-thirds the cube of its radius, multiplied by 3.1416. Note. The second paragraph on page 163 applies with equal force to the above rules. EXAMPLES. 248. 1. Find the lateral area and volume of a cylinder whose altitude is 9 in., and radius of base 4 in. By Art. 247, 1, the required lateral area is 8 X 9 X 3.1416, or 226.1952 sq. in. By Art. 247, 2, the required volume is 16 X 9 X 3.1416, or 452.3904 cu. in. 2. Find the radius of a sphere whose area is 452.3904 sq. in. By Art. 247, 7, the square of the radius of a sphere is equal to its area divided by 4 times 3.1416, or 12.5664. Hence, the square of the required radius is 452:3904 ^^3, ^^^ 12.5664 The square root of 36 is 6 ; whence the required radius is 6 in., Ans. 3. The volume of a cone is 1005.312 cu. ft., and the radius of its base is 8 ft. Find its altitude and slant height. By Art. 247, 4, the altitude of a cone is equal to three times its volume, divided by 3.1416 times the square of the radius of its base. Hence, the required altitude is , or 15 ft. 3.1416x64 In the right triangle ABD, by Art. 227, the square of AB is equal to the sum of the squares of AD and BD. But 82 + 152 = 64 + 225 = 289, which is the square of 17. Hence, the required slant height is 17 ft. u MENSURATION. . 169 4. What is the volume of a frustum of a cone whose altitude is 12 in., and radii of bases 6 in. and 2 in., respec- tively? By Art. 247, 6, the required volume is (36 + 4+12) X 4 X 3.1416, or 663.4528 cu. in., Ans. 6. Find the lateral area and volume of a cylinder whose altitude is 7 in., and radius of base 3 in. *^ 6. Find the area and volume of a sphere whose radius is 4 in. 7. Find the lateral area of a frustum of a cone whose slant height is 14 ft., and radii of bases 9 ft. and 2 ft., respectively. 8. Find the lateral "area and volume of a cone whose altitude is 12 in., and radius of base 5 in. 9. Find the volume of a frustum of a cone whose alti- tude is 9 ft., and radii of bases 10 ft. and 6 ft., respectively. 10. The lateral area of a cone is 188.496 sq. in., and the radius of its base is 6 in. Find its slant height and altitude. 11. The area of a sphere is 314.16 sq. in. Find its di- ameter and volume. 12. The volume of a cylinder is 2412.7488 cu. in., and its altitude is 12 in. Find the radius of its base. 13. The lateral area of a frustum of a cone is 603.1872 sq. in., and the radii of its bases are 5 in. and 11 in., respec- tively. Find its slant height. 14. Assuming the earth to be a sphere 7900 miles in diameter, find its area and volume. 15. A tent, in the shape of a cone, has a slant height of 16 feet, and a diameter at the base of 24 feet. How many square yards of material were used in its construction ? 16. The volume of a frustum of a cone is 779.1168 cu. ft., and the radii of its bases are 10 ft. and 2 ft., respectively. Find its altitude. 170 • ARITHMETIC. 17. What will it cost to gild a ball 25 inches in diameter, at $ 13.50 a square foot ? 18. The altitude of a cone is 9 in., and the radius of its base is 7 in. Find the altitude of an equivalent cylinder (Art. 232), the diameter of whose base is 10 in. 19. How many cubic feet are there in a log of wood 20 feet long, whose girth is 3 feet ? 20. A basin is in the shape of a hemisphere whose diameter is 2f yards. How many cubic feet of water will it hold? 21. How many cubic feet of metal are there in a hollow iron tube 18 ft. long, whose outer diameter is 7 in., and thickness 1 in. ? 22. Find the radius of a sphere, whose surface is equiv- alent to the lateral surface of a cylinder, whose altitude is 8 ft., and radius of base 4 ft. 23. The volume of a sphere is 7238.2464 cu. in. Find its radius. 24. A cylindrical vessel, 8 in. in diameter, is filled to the brim with water. A ball is immersed in it, displacing water to the depth of 2^ in. Find the diameter of the ball. 25. The outer diameter of a spherical shell is 9 in., and its thickness is 1 in. What is its weight, if a cubic inch of the metal weighs i lb. ? 26. How many cubic feet are there in a column whose length is 22 ft., diameter of larger end 10 in., and diameter of smaller end 7 in. ? 27. The altitude of a frustum of a cone is 6 ft., and the radii of its bases are 3 ft. and 2 ft., respectively. What is the diameter of an equivalent sphere ? 28. If a gallon contains 231 cu. in., what must be the depth of a cylindrical measure, 3 in. in diameter, to hold a quart ? MENSURATION. 171 APPLICATIONS OF MENSURATION. 249. Capacity of Bins, Tanks, and Cisterns. The following equivalents are to be used in the examples of the present article : 1 bushel = 2150.42 cu. in. 1 gallon = 231 cu. in. 1 bushel = 1\ cu. ft. 7| gallons = 1 cu. ft. 1. How many bushels of grain can be put into a bin 5 ft. 3 in. long, 3 ft. 6 in. wide, and 4 ft. 1 in. deep ? We have 5 ft. 3 in. = 63 in., 3 ft. 6 in. = 42 in., and 4 ft. 1 in. = 49 in. Then the volume of the bin = 63 x 42 x 49, or 129654 cu. in. Since one bushel contains 2150.42 cu. in., as many bushels can be put into the bin as 2150.42 is contained times in 129654, which is 60.29+, Ans. EXAMPLES. 2. How many bushels of grain can be put into a bin 7 ft. long, 3 ft. wide, and 4 ft. deep ? 3. If a ton of coal occupies 38 cu. ft., how many tons can be put into a bin 8 ft. long, 5^ ft. wide, and 6^ ft. deep ? 4. How many gallons will a tank hold which is 2 ft. 6 in. long, 2 ft. wide, and 1 ft. 9 in. deep ? 5. How many gallons of water can be put into a cylindrical tank whose diameter is 50 in., and depth 65 in. ? 6. How deep must a bin be that is 6 ft. long and 4 ft. wide, to hold 84 bushels of grain ? 7. What must be the depth of a cubical bin to hold 100 bushels of wheat ? • 8. If a tank is 66 inches wide and 42 inches deep, how long must it be to hold 1000 gallons ? 9. How deep must a cistern be, whose diameter is 60 in., to hold 800 gallons ? 172 ARITHMETIC. 10. How many bushels of oats can be put into a bin 4 ft. 7 in. long, 3 ft. 5 in. wide, and 3 ft. 10 in. deep ? 11. If a tank 8 ft. long, 5 ft. wide, and 3 ft. deep, is tilled with oil, how much is the oil worth at 13 cents a gallon ? 12. What must be the diameter of a cylindrical tank, whose depth is 55 inches, to hold 640 gallons ? 13. How deep must a bin be that is 7 ft. 2 in. long, and 5 ft. 7 in. wide, to hold 150 bushels of rye ? 14. A cubical bin, 5 ft. 3 in. deep, is filled with wheat. What is its value at $0.96 per bushel ? 15. To what depth must a cistern 38 inches in diameter be filled, to hold 304 gallons ? 16. A well is 3 ft. in diameter, and 32 ft. deep. How many barrels of 31|- gallons each will it contain ? 17. How many bushels can be put into a cylindrical re- ceptacle whose diameter is 3 ft. 6 in., and depth 5 ft. 4 in. ? 18. What must be the diameter of a cistern, of depth 4 feet, to hold 400 gallons ? 19. If a tank holds 500 gallons of water, how many bushels of grain can be put into it ? 250. Carpeting Rooms. 1. A floor is 15 ft. 4 in. long, and 11 ft. 9 in. wide. How much will it cost to cover it with carpeting, each length 2 ft. 8 in. wide, at 63 cents a yard, no allowance being made for waste in matching the pattern ? If the strips are laid lengthioise of the room, as many strips will be required as 2 ft. 8 in. is contained times in 11 ft. 9 in. 32 in. is contained in 141 in. 4 times, with a remainder of 13 in. • Then Jive strips will be required. The total length of the five strips is 15 ft. 4 in. x 5, or 76 ft. 8 in. ; that is, 15 1 yd. Then the cost will be 15| x |0.63, or $9.80, Ans. MENSURATION. 173 Note. It will be understood, in the following examples, that the strips are laid lengthwise of the room, unless the contrary is specified. EXAMPLES. 2. How many yards of carpeting |- of a yard wide will be required for a floor 17 ft. long and 14 ft. wide ? 3. How much will it cost to cover a floor 15 ft. long and 11 ft. 3 in. wide with oil-cloth, at 40 cents a square yard ? 4. How many yards of carpeting 2 ft. 7 in. wide will be required for a floor 20 ft. 11 in. long and 16 ft. 4 in. wide, if the strips run lengthwise of the room ? How many if the strips run across the room ? 5. How much will it cost to cover a floor 14 ft. 3 in. square with straw matting, in strips one yard wide, at 44 cents a yard ? 6. A floor is 16 ft. 11 in. long and 12 ft. 1 in. wide. How much will it cost to cover it with carpeting, each length 2 ft. 5 in. wide, at 75 cents a yard ? 7. How many yards of carpeting 2 ft. 10 in. wide, will be required for a floor 16 ft. 9 in. long and 13 ft. 8 in. wide, if there is a waste of 4|- inches in each strip in matching the pattern ? 8. Which way should the strips run to carpet most eco- nomically a floor 18 ft. 3 in. long and 15 ft. 4 in. wide, the strips being 2 ft. 9 in. wide ? 9. How much will it cost to cover a floor 21 ft. 6 in. long and 18 ft. 4 in. wide with carpeting 2 ft. 6 in. wide, at 87 cents a yard, if there is a waste of |- of a yard in each strip in matching the pattern ? 10. Which will be the cheaper, to cover a floor 19 ft. 7 in. long and 14 ft. 9 in. wide with matting, in strips 2 ft. 8 in. wide, laid lengthwise, at 42 cents a yard, or to cover it with oil-cloth at 48 cents a square yard ? 174 ARITHMETIC. 251. Plastering and Papering. 1. How much will it cost to plaster a room 16 ft. long, 13 ft. wide, and 9 ft. high, at 40 cents a square yard, allow- ing 64 sq. ft. for doors and windows ? The area of the four walls is 2 x (16 + 13) x 9, or 522 sq. ft. The area of the ceihng is 16 x 13, or 208 sq. ft. Then the total area to be plastered is 522 + 208 - 64, or 666 sq. ft. ; that is, 74 sq. yd. Hence, at 40 cents a square yard, the total cost will be 74 X $0.40= $29.60, ^ns. 2. How many rolls of paper, IJ ft. wide, 9 yards to a roll, will be required to paper a room 19 ft. long, 14 ft. wide, and 9J ft. high, allowing for two doors, each 3 ft. wide and 7J ft. high, four windows, each 3 ft. wide and 5^ ft. high, and a base-board 9 in. wide ? The area of the four walls is 2 x (19 + 14) x 9^^, or 605 sq. ft. The area of the two doors is 2 x 3 x 7J, or 44 sq. ft. The area of the four windows is 4 x 3 x 5 J, or 66 sq. ft. The length of the base-board is the distance around the room, less the width of the two doors ; that is, 6Q — 6, or 60 ft. Then the area of the base-board is 60 x f , or 45 sq. ft. Thus, the total area to be deducted from the area of the four walls is 44 + 66 + 45, or 155 sq. ft. Hence, the area to be papered is 605 — 155, or 450 sq. ft. The area of each roll is If x 27, or 45 sq. ft. Then, as many rolls will be required as 46 is contained times in 450 ; that is, 10 rolls, Ans. EXAMPLES. 3. How much will it cost to plaster a room 18 ft. long, 15 ft. wide, and 10 ft. high, at 39 cents a square yard, allow- ing 102 sq. ft. for doors and windows ? 4. How much will it cost to plaster a room 20 ft. 8 in. long, 16 ft. 3 in. wide, and 9 ft. 6 in. high, at 48 cents a square yard, allowing 128 sq. ft. 36 sq. in. for doors and windows ? MENSURATION. 175 5. How many rolls of paper, 1| ft. wide, 12 yards to a roll, will be required to paper a room 17 ft. long, 12 ft. wide, and 9 ft. high, no allowance being made for doors or windows ? 6. How much will it cost to paper a room 14 ft. square, and 8|- ft. high, with paper 1 ft. 10 in. wide, 10 yards to a roll, at 75 cents a roll; allowing 80 sq. ft. for doors and windows ? 7. A room 16 ft. long, 12 ft. wide, and 9 ft. high, has three doors, each 3 ft. wide and 7 ft. high, two windows, each 3 ft. wide and 5 ft. 9 in. high, and a base-board 9 inches wide. How much will it cost to plaster it at 44 cents a square yard ? 8. How much will it cost to paper a room 15 ft. long, 11 ft. wide, and 10 ft. high, with paper 1 ft. 11 in. wide, 11 yards to a roll, at $ 1.10 a roll, allowing 175 sq. ft. for doors, windows, and base-board ? 9. What will it cost to plaster a hemispherical dome, whose diameter is 60 ft., at 50 cents a square yard ? 10. How many rolls of paper, 1 ft. 11^ in. wide, 12 yards to a roll, will be required to paper a room 21 ft. long, 15^ ft. wide, and 8f ft. high, allowing for three doors, each 3 ft. wide and 6f ft. high, three windows, each 2f ft. wide and 5i ft. high, and a base-board 1 ft. wide ? 11. A room 23 ft. 6 in. long, 15 ft. 4 in. wide, and 9 ft. 9 in. high, has three doors, each 3 ft. wide and 6 ft. 8 in. high, four windows, each 2 ft. 9 in. wide and 5 ft. 9 in. high, and is surrounded by a base-board 9 in. wide. How much will it cost to plaster it at 36 cents a square yard ? 12. Find the cost of papering a room 18 ft. 4 in. long, 14 ft. 6 in. wide, • and 9 ft. 6 in. high, with paper 1 ft. 9 in. wide, 111 yards to a roll, at $ 1.19 a roll; allowing for two doors, each 3 ft. wide and 7 ft. high, and two windows, each 2 ft. 9 in. wide' and 6 ft. high ? 176 ARITHMETIC. 252. Board Measure. A board one inch or less in thickness is said fco have as many Board Feet as there are square feet in its surface. If it is more than an inch in thickness, the number of board feet is found by multiplying the number of square feet in its surface by the number- of inches in its thickness. In measuring a board that tapers, the width is taken as one-half the sum of the widths of the two ends. Boards are usually sold at a certain price per hundred (C.) or per thousand (M.) board feet. 1. Find the cost of 24 planks, each 22 ft. 8 in. long, 21 in. wide, and 2 J in. thick, at $ 25 per M. 22 ft. 8 in. = %8- ft., 21 in. = | ft., and 2^ in. = f in. Then the total number of board feet is 17 3 6 ^ X ^ X ^ X ?^, or 2142. 3 ^ ^ At $25 per M., the total cost will be 2.142 X $25, or $53.55, Ans. EXAMPLES. 2. Find the number of board feet in a board 16 ft. 6 in. long, 14 in. wide, and 1 in. thick. 3. Find the number of board feet in a board 10 ft. long, 11 in. wide, and f in. thick. 4. Find the number of board feet in a piece of timber 25 ft. 9 in. long, 9 in. wide, and 8 in. thick. 5. Find the number of board feet in a plank 18 ft. 8 in. long, 1 ft. 5 in. wide, and 3^ in. thick. 6. Find the number of board feet in a tapering plank 15 ft. 4 in. long, 2 ft. 3 in. wide at one end, and 1 ft. 11 in. wide at the other, and 3f in. thick. 7. Find the cost of 45 spruce joists, each 14 ft. long, 6 in. wide, and 4 in. thick, at $ 14 per M. MENSURATION. 177 8. Find the cost of 150 boards, each 11 ft. 8 in. long, 5 in. wide, and |- in. thick, at $ 18.30 per M. 9. Find the cost of 30 planks, each 17 ft. 4 in. long, 1 ft. 10 in. wide, and 2f in. thick, at $ 2.55 per C. 10. Find the cost of a plank-walk 75 ft. long, 2 ft. 6 in. wide, and | in. thick, at $ 31.50 per M. 11. Find the cost of 75 planks, each 12 ft. 10 in. long, 1 ft. 7 in. wide at one end, and 1 ft. 1 in. wide at the other, and If in. thick, at $ 15 per M. 253. Measurement of Hound Timber. To find the side of the squared timber that can be sawed from a log, multiply the diameter of the smaller end by .707. To find the number of board feet in the squared timber that can be sawed from a log, multiply together one-half the length in feet, the diameter of the smaller end in feet, arid the diameter of the smaller end in inches. 1. Find the side, and the number of board feet, in the squared timber that can be sawed from a log whose length is 15^ ft., and diameter at the smaller end 18 in. By the first of the above rules, the side is 18 in. X .707, or 12.726 in. By the second rule, the number of board feet is i X 15i X - X 18 = 1 X — X - X X^ =§15 = 204|. 2 '2 2^2 4 ^ EXAMPLES. Find the side, and the number of board feet, in the squared timber that can be sawed from a log whose length is : 2. 18 ft., and diameter 1 ft. 3. 21 ft., and diameter of smaller end 1\ ft. 4. 17^ ft., and diameter 15 in. 5. 15 ft. 9 in., and diameter 1 ft. 2 in. 6. 23 ft. 10 in., and diameter of smaller end 1 ft. 7 in. 178 ARITHMETIC. 254. Specific Gravity. The Specific Gravity of a substance is the number of times that the weight of a certain portion of the substance con- tains the weight of an equal bulk of water. For example, a cubic foot of copper weighs 8.8 times as much as a cubic foot of water ; hence, the specific gravity of copper is 8.8. In the following examples, the weight of a cubic foot of water is taken as 1000 oz., or 62.5 lb. 1. What is the weight of a cubic foot of iron, if its specific gravity is 7.53 ? Since a cubic foot of water weighs 62.5 lb., and iron is 7.53 times as heavy as water, a cubic foot of iron will weigh 62.5 lb. X 7.53, or 470.625 lb., Ans. 2. A mass of granite (specific gravity 2.6) weighs 7800 lb. ; how many cubic feet does it contain ? Since one cubic foot of granite weighs 62.5 lb. x 2.6, or 162.5 lb., to weigh 7800 lb, will take as many cubic feet of granite as 162.5 is contained times is 7800. Dividing 7800 by 162.5, the result is 48 cu. ft., Ans. EXAMPLES. 3. Find the weight in pounds of a cubic foot of copper (specific gravity 8.81) . 4. Find the w^eight in pounds of a cubic yard of brick- work (specific gravity 1.8). 5. Find the weight in pounds of 5 cu. ft. 288 cu. in. of yellow pine (specific gravity .46). 6. Find the weight in ounces of a cubic inch of mer- cury (specific gravity 13.596). 7. If a mass of iron (specific gravity 7.68) "weighs 6 T., how many cubic feet does it contain ? MENSURATION. 179 8. If a mass of tin (specific gravity 7.5) weighs 18750 lb., how many cubic feet does it contain ? 9. If a certain bulk of alcohol (specific gravity .791) weighs 197f oz., how many cubic inches does it contain ? 10. If a piece of gold (specific gravity 19.4) weighs 37 lb. 14 oz. 4 dr., how many cubic inches does it contain ? 11. If a cubic foot of glass weighs 170 lb., find its spe- cific gravity. 12. If a cubic yard of oak weighs 1485 lb., find its specific gravity. 13. If 3 cu. ft. 432 cu. in. of silver weighs 2132 lb. 13 oz. avoirdupois, find its specific gravity. 14. If a cubic inch of brass weighs 77.2864 dr., find its specific gravity. 255. Geometrical Explanation of Square and Cube Root. Square Root. Let it be required to find the square root of 1296. Let A CEG be a square containing 1296 sq. in. To find its side in inches. Since a square whose side is 30 in. contains 900 sq. in., and a square whose side is 40 in. contains 1600 sq. in., the side of the given square must be between 30 and 40 in. Thus the tens' figure of the root is 3. . Fig. 1 D Fig. 2 K P B 30 30 Q H Fig. 3 p Q R 30 30 M N Removing from the given square the square ABKH, whose side is 30 in., there remains an irregular figure, shown in Fig. 2, composed of two rectangles P and Q, and a square i?, whose united area is 1296 — 900, or 396 sq. in. The rectangles and the square may be arranged as shown in Fig. 3, forming a rectangle LM, whose altitude is the units' figure of the root. Now the altitude of a rectangle is equal to its area divided by its base. 180 ARITHMETIC. Since the base of each of the rectangles P and Q is 30 in., the base of the rectangle LM is something more than 60 in. If we divide the area of LM, 396 sq. in., by its approximate base, 60 in., we obtain something more than 6 in. as the approximate alti- tude. If, now, we make trial of 6 in. as the altitude of the rectangle, the base ZriVis 60 in. + 6 in., or 66 in. ; and multiplying this by the alti- tude, 6 in., the result is 396 sq. in. But this is just the area of the irregular figure of Fig. 2. We then conclude that the units' figure of the root is 6 ; whence, the required root is 30 + 6, or 36. The above process is exactly in accordance with the Rule of Art. 200. Cube Eoot. Let it be required to find the cube root of 13824. Fig. i Fig. 2 A / ) — / / / / 7 / 20 / E F / / / «/ «. / Let ABhe s, cube containing 13824 cu. in. To find its edge in inches. Since a cube whose edge is 20 in. contains 8000 cu. in,, and a cube whose edge is 30 in. contains 27000 cu. in., the edge of the given cube must be between 20 and 30 in. Thus the tens' figure of the root is 2. Removing from the given cube a cube whose edge is 20 in., there remains an irregular solid CD, whose volume is 13824 -8000, or 5824 cu. in. Removing from CD the three solids E, F, and 6r, there remains an irregular solid, shown in Fig. 3, composed of three rectangular paral- lelepipeds, H, K, and i, and a cube. The solids E, F, 6r, H, K, and i, and the cube M, may be arranged as shown in Fig. 4, forming an irregular solid iVP, whose altitude is the units' figure of the root. Now the altitude of this solid is equal to its volume divided by the area of its base. MENSURATION. 181 Since the area of the base of each of the solids E, F, and G is 20^, or 400 sq. in., the sum of the areas of their bases is 3 x 400, or 1200 sq. in. Fig. % ./ / / //// E F G H K L M N 210 20 20 4 444 Then the area of the base of the solid NP is something more than 1200 sq. in. If we divide the volume of iVP, 5824 cu. in., by its approximate area of base, 1200 sq. in., we obtain something more than 4 in. as the approximate altitude. If, now, we make trial of 4 in. as the altitude of the solid, the area of the base of each of the solids H, K, and i is 4 x 20, or 80 sq. in., and the sum of the areas of their bases is 3 x 80, or 240 sq. in. Also, the area of the base of the cube M is 4^, or i6 sq. in. Then the area of the base of the solid NP is 1200 + 240 + 16, or 1456 sq. in. ; and multiplying this by the altitude, 4 in., the result is 6824 cu. in. But this is just the volume of the irregular solid of Fig. 2. We then conclude that the units' figure of the root is 4 ; whence, the required root is 20 + 4, or 24. The above process is exactly in accordance with the rule of Art. 208. PROBLEMS IN MENSURATION INVOLVING THE METRIC SYSTEM. Note. The remainder of the present chapter may be omitted by those who have not previously taken the chapter on the Metric System. 256. Mensuration of Plane Figures. 1. Find the area in square decimeters of a rectangle whose base is 8.9™ and altitude 735'^'". 2. Find the altitude in meters of a triangle whose area is 16.8«'J^% and base .096^'". 3. The hypotenuse of a right triangle is 41™, and one of the sides about the right angle is .9^". Find the other side in hektometers. 182 ARITHMETIC. 4. Find the base in dekameters of a parallelogram whose area is 89.1'"»*=™, and altitude SS*"™. 5. Find the circumference in meters of a circle whose radius is 23^'". 6. Find the area in square hektometers of a triangle whose base is 528'*™ and altitude S-SQ"""™. 7. The sides about the right angle of a right triangle are IS*" and SS**™, respectively. Find the hypotenuse in dekameters. 8. The diameter of a circle is IS**". Find its area in square centimeters. 9. Find the side in decimeters of a square whose area is 143641**1™'". 10. Find the area in square decimeters of a trapezoid whose bases are 3.51™ and 4852™™, respectively, and altitude .0295^™. 11. Find the radius in dekameters of a circle whose area is .19635«'i«™. 12. Find the diameter in centimeters of a circle whose circumference is 12™™. 13. Find the lower base in centimeters of a trapezoid whose area is 3.6"^™, upper base .13°™, and altitude 18^™. 14. Find the area in ars of a rectangular field 253.8'*™ long and 13.9™ wide. 15. Find the area in centars of a floor .5498*^™ long and 467'=™ wide. 16. A circular grass-plot, 17™ in diameter, is surrounded by a walk 18'*™ wide. Find the area of the walk in centars. 17. A triangular house-lot contains .241853^*. If its base is 7.34*^™, what is its altitude in meters ? 18. The area of a square field is 77.2641"^ Find its side in dekameters. MENSURATION. 183 19. What is the length of the longest straight line that can be drawn in a rectangular field whose length is 204™ and width 85™? 20. If the diameter of a wheel is 75*^™, how many times will it revolve in travelling 12.9591^'" ? 21. The side of a sq uire field is 87.2™ How much is the field worth at $ 8750 a hektar ? 22. A vessel sails due north at the rate of 9.3^™ an hour, and another sails due west at the rate of 12.4^"' an hour. How far apart are they at the end of 4 h. 12 min. ? 23. Two circles whose radii are 25^™ and 173'" have the samfe centre. How many square hektometers of area are included between their circumferences ? 24. How many acres are there in a rectangular field whose length is 8.9^™, and width .75^" ? 25. A cow, tied by a rope to a stake, can graze over 415.4766^'' ™ of ground. What is the length of the rope ? 26. Find the diagonal in hektometers of a square whose area is .1764'^^". 27. A man sold a rectangular field for 28 cents a centar, receiving the sum of $ 915.04. If the field was .43°™ wide, what was its length in dekameters ? 28. The floor of a room is 6™ long and 43^™ wide, and has a circular opening .17''" in diameter. Find the number of square centimeters in the floor. 29. A rectangular garden is surrounded by a walk 1.2™ wide, containing 248.16^*^™. If the garden is 64'" long, what is its width ? 30. A circular field contains a hektar. What is its di- ameter in dekameters ? 31. The diagonal of a square is 76™. Find the approxi- mate length of its side. 184 ARITHMETIC. Mensuration of Solids. 1. Find the lateral area and volume of a prism whose altitude is 7'^'", having for its base a right triangle whose sides are 3^"", 4^"\ and 5^"". 2. Find the area and volume of a sphere whose radius is 3cm^ 3. Find the lateral area and volume of a cone whose altitude is 24™, and radius of base 7™. 4. The volume of a rectangular parallelopiped is 1920"=" ^"\ and the dimensions of its base are 15*^™ and 8^"\ Find its altitude and the area of its entire surface. 5. The volume of a pyramid is 1320'="^". The base is a right triangle whose sides are 10^"^, 24^™, and 26^™. Find the altitude of the pyramid. 6. Find the lateral area of a frustum of a cone, whose slant height is 8™, radius of lower base 9™, and radius of upper base 3™. 7. Theareaof a sphere is 201.0624^^^™. Find its radius and volume. 8. The base of a regular pyramid is a square whose area is 67.24''^*^'", and its slant height is 579'™. Find its lateral area in square meters. 9. Find the volume of a frustum of a pyramid whose lower base is a rectangle 12^™ by 3°™, upper base 8^™ by 2°'", and altitude 16.8^™. 10. The volume of a frustum of a cone is 1894.3848'="™ and the radii of its bases are 11™ and 5™. Find its altitude. 11. Find the lateral area in square millimeters, and the volume in cubic centimeters, of a cylinder whose altitude is 81™™, and radius of base 9^=™. 12. The lateral area of a cone is 565.488"^ ™, and the radius of its base is 12™. Find its slant height, altitude, and volume, "^ MENSURATION. 185 13. Find the lateral area of a frustum of a regular pyra- mid whose lower base is, a square 15™ on a side, upper base 9™ on a side, and slant height 8.49™. 14. A basin is in the shape of a hemisphere whose diameter is 4.3*^™. How many deciliters of water will it contain ? What is the weight of this water in decigrams ? 15. A wood pile is 2.5'" long, 13*^™ wide, and 125*=™ high. How much is it worth at $ 3.36 a ster ? 16. What will be the cost of gilding a ball 65*^™ in diam- eter, at $1.50 a square decimeter ? 17. A room is 5™ long, 48^^™ wide, and 326^™ high. How many dekaliters of air does it contain ? 18. How much will it cost to dig a ditch 7°™ long, 18*^™ wide, and 12**™ deep, at 75 cents a ster ? 19. A wood pile, 19"^™ long and 142^=™ wide, contains 3.72324^*. Find its height in meters. 20. Find the number of cubic meters in a tapering piece of timber 9™ long, one of which is 5*^™ square, and the other 32*=™ square. 21. A bar of iron is 6.2™ long, 8^™ wide, and 13™™ thick. Find its weight in kilograms, if iron is 7.53 times as heavy as water. 22. A cylindrical boiler is 4™ long, and 13'^°' in diameter. How many liters of water will it contain ? 23. A trench is 42™ long, 8*^™ deep, 2.1™ wide at the top, and 1.7™ wide at the bottom. How many hektoliters of water will it contain ? 24. How many bricks, each 2^™ long, 82™™ wide, and 4.8*=™ thick, will be required to build a wall 37™ long, 41*'™ wide, and 19.2*^™ high ? 25. A spherical shell, 5*=™ thick, has an outside diameter of 4*™. How many cubic centimeters of metal does it con- tain? 186 ARITHMETIC. 26. A brass rod is 3.5™ long and 4^"" in diameter. Find its weight in dekagrams, if brass is 8.3 times as heavy as water. 27. Find the weight in kilograms of the water that can be put into a pail in the shape of a frustum of a cone, whose depth is 24*=™, diameter at the top 32""', and diameter at the bottom 22*=™. 28. A cannon-ball, 23^"" in diameter, is dropped into a cubical box filled with water, whose depth is 23*="". How many centiliters of water will be left in the box ? 29. A hopper, in the form of an inverted frustum of a pyramid, holds 1.364^^^ of grain. It is 1™ square at the top, and 2*^™ square at the bottom. Find its depth in meters. 30. A projectile consists of two hemispheres, connected by a cylinder. If the altitude and diameter of the cylinder are 2*^™ and 13^°', respectively, find the number of cubic decimeters in the projectile. Capacity of Bins, Tanks, and Cisterns, Carpeting, Plas- tering, and Papering. 1. A tank is 3.7"" long, .98^" wide, and 1.6™ deep. How many liters of water will it contain ? 2. How many hektoliters of wheat can be put into a bin 2.3™ long, 12^™ wide, and 18*^™ deep ? 3. How many meters of carpeting 8^™ wide will be re- quired for a floor 6.59™ long and 5™ wide ? 4. How much will it cost to plaster a room 5™ long, 4™ wide, and 2.9™ high, at 45 cents a square meter, allowing j^Q gsqm jpQj. (joors and windows ? 5. What must be the depth in meters of a tank 24^™ long and 78''™ wide, to hold 35.9424™ of water ? 6. What must be the length in decimeters of a bin 1.88" deep and .13^™ wide, to hold 657.436°^ of grain ? MENSURATION. 187 7. A cylindrical cistern whose diameter is 3"" is filled with, water to a depth of 19*^"". How many hektoliters of water does it contain, and what is the weight of the water in hektograms ? 8. How many rolls of paper 7^"^ wide, ll*" to a roll, will be required to paper a room 5.8'° long, 4.7™ wide, and 2.8" high, allowing 8.75'^™ for doors and windows ? 9. A cubical tank holds 491.3^^ of water. What is its depth in meters ? 10. A hemispherical dome is 25^ in diameter. How much will it cost to plaster it at 48 cents a square meter ? 11. What will it cost to cover a floor 7.9™ long and 5.8™ wide with carpeting 78*"" wide, at 95 cents a meter, if there is a waste of 5*^™ in each strip in matching the pattern ? 12. A tank 5.1™ long, 2.3"^ wide, and 1.7™ deep, is filled by a pipe through which pass 289*^^ of water a minute. How long will it take to fill it ? 13. How many rolls of paper 68*=™ wide, 10™ to a roll, will be required to paper a room 7™ long, 5.5™ wide, and 3.2™ high, with two doors, each 88*=™ wide and 22*^™ high, and four windows, each 86*=™ wide and 17.5*^™ high ? 14. A well is 13*^™ in diameter, and 12™ deep. How many kiloliters of water will it hold ? 15. A cylindrical tank, 16*^™ deep, holds 15393.84^2 of water. What is its diameter in meters ? 16. How much will it cost to plaster a room 7.2™ long, 4.9™ wide, and 3™ high, at 42 cents a square meter, allow- ance being made for two doors, each 9^™ wide and 21*^™ high, three windows, each 8*^™ wide and 18^™ high, and a base- board 2*^™ wide ? 17. If a metric ton of coal occupies 1.184*="™, how many metric tons can be put into a bin 37^™ long, 12**™ wide, and 36^™ deep ? 188 ARITHMETIC. 18. Which way should the strips run to carpet most economically a floor 7.8™ long and 6.2™ wide, the strips being 84^^™ wide ? 19. A bin 2.3™ long, 1.1™ wide, and 1.48™ deep is filled with grain. How much is it worth at ^ 2.75 a hektoliter ? If the grain weighs .83 times as much as an equal bulk of water, what is the weight of the contents in kilograms ? 20. A cylindrical cistern, 25'^™ deep, holds 636.174°^ of water. Find the diameter of the cistern in meters. 21. How much will it cost to cover a floor 5.6™ long, and 4.9™ wide, with carpeting 75*^™ wide, at 87 cents a meter, if the strips run lengthwise of the room ? How much if the strips run across the room ? 22. A tank 84*^™ deep, with a square bottom, contains 869.4^^ of sulphuric acid. If the acid is 1.84 times as heavy as water, what is the length of each side of the bottom in centimeters ? 23. To what depth must a cylindrical cistern 113<=™ in diameter be filled, to hold a metric ton of water ? 24. How much will it cost to paper a room 6.4™ long, 5.4™ wide, and 3.1™ high, with paper 58*^™ wide, 10.7™ to a roll, at 84 cents a roll, allowing for three doors, each 85*^™ wide and 2" high, two windows, each 82*^™ wide and 17*^™ high, and a base-board 3'^'" wide ? 25. A cylindrical tank, 18*^ deep, contains 2862.783^« of oil. If the oil is .9 as heavy as water, find the diameter of the tank in meters. 257. Specific Gravity. If the specific gravity of any substance is 8.7, a cubic centimeter of the substance will weigh 8.7 times as much as a cubic centimeter of water ; that is, it will weigh 8.7^. A cubic decimeter (or liter) of the substance will weigh 1000 X 8.7S or 8.7^«. MENSURATION. 189 A cubic meter of the substance will weigh 1000 x 8.7^^, or 8.7^. It follows from the above that the specific gravity of any substance is : 1. The number of grams in the weight of a cubic centimeter of the substance. 2. The number of kilograms in the weight of a cubic deci- meter {or liter) of the substance. 3. The number of metric tons in the weight of a cubic meter of the substance. EXAMPLES. 1. Find the weight in kilograms of a bar of aluminum (specific gravity 2.57) 8^™ long, 2*=™ wide, and 7'"'" thick. The volume of the bar is 8770 1.316809 1.368569 9 1.143390 1.195093 1.248863 1.304773 1.362897 1.423312 10 1.160541 1.218994 1.280085 1.343916 1.410599 1.480244 11 1.177949 1.243374 1.312087 1.384234 1.459970 1.539454 12 1.195618 1.268242 1.344889 1.425761 1.511069 1.601032 13 1.213552 1.293607 1.378511 1.468534 1.563956 1.665074 14 1.231756 1.319479 1.412974 1.512590 1.618695 1.731676 15 1.250232 1.345868 1.448298 1.557967 1.675349 1.800944 16 1.268985 1.372786 1.484606 1.604706 1.733986 1.872981 17 1.288020 1.400241 1.521618 1.652848 1.794676 1.947901 18 1.307341 1.428246 1.559659 1.702433 1.857489 2.025817 19 1.326951 1.456811 1.598650 1.753506 1.922501 2.10(5849 20 1.346855 1.485947 1.638616 1.806111 1.989789 2.191123 Yre. 5 per cent. 6 per cent. 7 per cent. 8 per cent. 9 per cent. 10 per cent. 1 1.050000 1.060000 ♦ 1.070000 1.080000 1.090000 1.100000 2 1.102500 1.123600 1.144900 1.166400 1.188100 1.210000 3 1.157625 1.191016 1.225043 1.25i)712 1.295029 1.331000 4 1.215506 1.262477 1.310796 1.360489 1.411582 1.464100 5 1.276282 1.338226 1.402552 1.469328 1.538624 1.610510 6 1.340096 1.418519 1.500730 1.586874 1.677100 1.771561 7 1.407100 1.503630 1.605781 1.713824 1.828039 1.948717 8 1.477455 1.593848 1.718186 1.850930 1.992563 2.143589 9 1.551328 1.689479 1.838459 1.999005 2.171893 2.357948 10 1.628895 1.790848 1.967151 2.158925 2.367364 2.693742 11 1.710339 1.898299 2.104852 2.331639 2.580426 2.853117 12 1.795856 2.012197 2.252192 2.518170 2.812665 3.138428 13 1.885649 2.132928 2.409845 2.719624 3.065805 3.452271 14 1.979932 2.260fK)4 2.578534 2.937194 3.341727 3.797498 15 .2.078928 2.396558 2.759031 3.172169 3.642482 4.177248 16 2.182875 2.540352 2.952164 3.425943 3.970306 4.594973 17 2.292018 2.692773 3.158815 3.700018 4.327633 5.064470 18 2.406619 2.854339 3.379932 3.996019 4.717120 5.659917 19 2.526950 3.025600 3.616527 4.315701 5.141661 6.116909 20 2.653298 3.207136 3.869684 4.660957 5.604411 6.727600 INTEREST. 257 1. What is the compound interest of $ 400 for 15 y. 6 mo., at 6%, interest being compounded annually ? Amount of $1 for 15 y. at 6%, $2.396558 400 Amount of $ 400 for 15 y., $ 958.6232 Interest of $ 958.6232 for 6 mo., 28.759 Compound Amount, $ 987.38 400.00 Compound Interest, $ 587.38, Ans. Note 1. If the given time extends beyond the limits of the table, find the amount for any convenient length of time, and then use this amount for a new principal. Note 2. If the interest is compounded semi-annually, take one- half the given rate, and ticice the given time. Thus, to find the compound interest of any sum of money for 8 y. at 6 %, interest being compounded semi-annually, we find by the table the compound interest of the sum for 16 y. at 3 %. If the interest is compounded quarterly, take one-fourth the given rate, and four times the given time. EXAMPLES. 2. What is the amount of $95 for 5 y. 10 mo., at 6%, interest being compounded annually ? 3. What is the amount of $1250 for 3 y. 9 mo., at 6%, interest being compounded quarterly ? 4. What is the compound interest of $ 800 for 8 y. 7 mo., at 5%, interest being compounded semi-annually ? 5. What is the compound interest of $ 480 for 11 y. 5 mo. 10 d., at 4%, interest being compounded annually ? 6. What principal, at 5% compound interest, will amount to $ 500 in 13 y., interest being compounded annually ? 7. What sum of money, at 6 % compound interest, will gain $ 230 in 9 y. 8 mo., interest being compounded semi- annually ? 268 ARITHMETIC. ANNUAL INTEREST. 329. If, when interest is payable annually on a note or other obligation, the payments are not made when due, the amount due at the time of settlement may in certain cases be found by reckoning simple interest on the principal, and on each annual interest after it becomes due. This is called Annual Interest. 1. What amount is due at the end of 3 y. 6 mo. 12 d., on a note for $500, with 6% interest payable annually, on which no payments have been made ? Principal, $500.00 Int. of $ 500 f or 3 y . 6 mo. 12 d., at 6 % , 106.00 Int. of $ 30 for 4 y. 7 mo. 6 d., at 6%, 8.28 Amount due, $ 614.28, Ans. The interest of the principal for 3 y. 6 mo. 12 d., is f 106. Each annual interest is $ 30 ; the first draws interest for 2 y. 6 mo. 12 d. ; the second for 1 y. 6 mo. 12 d. ; and the third for 6 mo. 12 d. In all, this is equivalent to $30 drawing interest for 4 y. 7 mo. 6 d., which is $8.28. Then the amount due is $614.28. EXAMPLES. 2. What amount is due at the end of 2 y. 9 mo. 18 d., on a note for $ 900, with 6% interest payable annually, on which no payments have been made ? 3 What amount is due June 27, 1888, on a note for $385, dated Feb. 5, 1885, with 6% interest payable annually, on which no payments have been made ? 4. What is the annual interest of $ 763 for 2 y. 5 mo. 10 d., at 5% ? 5. What interest is due on a note for $6450, with 4% interest payable annually, at the end of 3 y. 8 mo. 24 d. ? 6. What amount is due Nov. 12, 1892, on a note for $ 898, dated Nov. 29, 1887, with 6% interest payable annually, on which no payments have been made ? DISCOUNT. 269 XX. DISCOUNTo TRUE DISCOUNT. 330. Discount is a reduction made from a debt that is paid before it becomes due. The Present Worth of a sum of money due at some future date without interest, is that sum which, if put at interest for the given time, will amount to the given sum. The True Discount is the difference between the given sum and its present worth ; that is, it is the interest of the present worth for the given time. EXAMPLES. 331. 1. What is the present worth and true discount of f 300, due 1 y. 5 mo. hence, at 5% ? f 1.00 This is the same as 05 finding what principal 17 <8insfi ^^^^ amount to $300 $0.05 X i^ = ^^. in 1 y. 5 mo. at b% ^^ ^^ (Art. 312). 0.85 f 12.85 ^^ find the interest "h^2 ^^ To * of $1 for 1 y. 5 mo. at 5% by multiplying $300 .05 of $1 by II; the 12 $300 result is ^<>-8^- 1 + 12.85) $ 3600 ( $ 280.16, present worth. 12 oe^n-A ""^"TrToT 4. J' i. Then the amount of 2570 $ 19.84, true discount. ^^ . ^^ ' $ 1 for the given time • i-???9 and rate is 10280 $0.85 _$ 12.85 $1+-*^^^^^, or 2000 7^"^1^'"'~T^ 1^^^ ' To divide $300 by 7150 $12.85 ,,. , .^ , we multiply it by 12, and divide the product by $ 12.85 ; the result to the nearest cent is $ 280.16, which is the present worth. Subtracting the present worth from the given sum, $ 300, the trii^ discount is $19.84. 260 ARITHMETIC. EXAMPLES. Find the present worth and true discount of : 2. $400 due 2 y. 11 mo. hence at 6%. 3. $890 due 6 mo. hence at 5%. 4. $ 725 due 4 y. 10 mo. hence at 4|^%. 5. $ 1730 due 3 y. 4 mo. hence at 3i%. 6. $682 due 5 mo. 27 d. hence at 6%. 7. $269.20 due 2 mo. 18 d. hence at 3%. 8. $2500 due 3 mo. 6 d. hence at 7%. 9. $950 due 9 mo. 11 d. hence at 6%. 10. $ 135.75 due 1 y. 8 mo. 15 d. hence at 4%. 11. $347.68 due 7 mo. 20 d. hence at 2i%. BANK DISCOUNT. 332. Bank Discount is a sum of money charged .by a bank for the payment of a negotiable note (Art. 319) before it becomes due. It is reckoned as the simple interest of the face of the note from the day of discount to the day of maturity (Art. 320). The time from the day of discount to the day of maturity is called the Term of Discount, and the rate of interest is called the Rate of Discount. The Proceeds or Avails of a discounted note is its face less the bank discount. 333. If a note is discounted on the day of its date, the term of discount is the time specified in the note, plus three days of grace (Art. 320). If a note is due a certain number of months after date, the term of discount is found in months and days (Art. 165) ; if it is due a certain number of days after date, the term of discount is found in exact days. n DISCOUNT. 261 Note. It will be understood, in the examples of the present chap- ter, that the note is discounted on the day of its date if nothing is said to the contrary. EXAMPLES. 334. 1. Find the proceeds of a 4-months' note for $485, discounted on the day of its date at 6%. The term of discount Face of note, $ 485.00 '^l"""' f '■'i^;'- f ^>; ; By Art. 332, the bank Int. 4 mo. 3 d., at 6%, 9.94 discount is the interest Proceeds, $ 475.06, Ans. of $ 485 for 4 mo. 3 d. , at 6%, which is $9.94. Subtracting this from the face of the note, the proceeds is $475.06. If an interest-bearing note is discounted, the discount is reckoned on the amount due at maturity. 2. Find the proceeds of a note for $ 1050, dated May 12, 1892, payable 90 days after date, and bearing interest at 4%, if discounted June 23, 1892, at 5%. Face of note, $1050.00 Interest of $ 1050 for 93 d., at 4%, 10.85 Amount due at maturity, $ 1060.85 Interest of $ 1060.85 for 51 d., at 5%, 7.51 Proceeds, $ 1053.34, Ans. The day of maturity is 93 d. after May 12, 1892, or Aug. 13, 1892. The interest of $ 1050 for 93 d., at 4 %, is $ 10.85 ; whence the amount due at maturity is $ 1060.85. The term of discount is the exact number of days from June 23, 1892, to Aug. 13, 1892, or 51 d. Then the bank discount is the interest of $ 1060.85 for 51 d. at 5 % ; that is, $7.51. Subtracting $ 7.51 from $ 1060.85, the proceeds is $ 1053.34. Note. The term of discount in Ex. 2 may be found without obtaining the day of maturity, by counting the exact number of days from May 12, 1892, to June 23, 1892, and subtracting the result from 93 days. 3. Find the proceeds of a 3-months' note for ^ 500, dis- counted at 65^?. 262 ARITHMETIC. 4. What is the proceeds of a 60-day note for ^950, discounted at 5% ? 5. What is the bank discount on a note for ^2000, due 90 days hence, at 7% ? 6. Find the bank discount on a note for $ 620, payable in 4 months, and discounted 2 mo. 9 d. after date, at 6%. 7. How much money should a bank pay to the holder of a note for $ 1000, due in 30 days, if discounted at 4% ? 8. What is the bank discount on a note for $ 700, dated July 13, 1891, and payable 90 days after date, if discounted Aug. 25, 1891, at 5i%? 9. Find the proceeds of a note for $6000, payable 2 months after date, and bearing interest at 6%, if discounted at 5%. 10. A note for $425.30, dated Oct. 4, 1890, and payable 6 months after date, was discounted Jan. 24, 1891, at 6%. What was the proceeds ? 11. What is the proceeds of a 60-day note for $800, dated March 16, 1892, and bearing interest at 4%, if dis- counted April 22, 1892, at M%? 12. What charge will be made by a bank for discounting, at 4%, a note for $384.50, due 75 days hence? 13. A note for $ 275, payable 5 months after date, was discounted at 5^%. What was the bank discount ? 14. What sum will be realized by discounting a 30-day note for $3000 ; the rate of discount being 7% ? 15. Find the bank discount on a note for $190, dated Aug. 26, 1890, payable 90 days after date, with interest at 41%, and discounted Nov. 3, 1890, at 6%. 16. A 15-day note for $ 503.70 was discounted at a bank at 41% ; how much did the bank receive ? 17. A note for $ 1100, dated Sept. 10, 1892, and payable 6 months after date, with interest at 6%, was discounted Feb. 7, 1893, at 4^%. What was the proceeds ? DISCOUNT. 263 18. Find the bank discount on a note for $ 9000, due in 4 months, the rate of discount being 6^%. 19. A 60-day note for $ 750, dated June 28, 1889, was discounted July 22, 1889, at 5f%. What was the bank discount ? 20. Find the proceeds of a note for § 280, dated May 21, 1888, payable 45 days after date, with interest at 3J%, and discounted June 7, 1888, at 6%. 21. A 3-months' note for $ 400, dated Nov. 19, 1892, was discounted Jan. 3, 1893, at 6^%. What sum was received by the holder ? 22. How much should a bank receive for discounting a note for f 847.25, payable 30 days after date, with interest at 5% ; the rate of discount being 4f % ? 335. To find the Face of a Note to yield a given Proceeds. 1. What must be the face of a note, due 90 days hence, which, when discounted at 4%, will yield $ 500? ^ 0.001 93 iM93^fM31x?=:iM31 = int.of$lfor93d.at4%. 6 2 3 3 ^ ^ ^ _g0p^$3--$0.031^g2|69^ p^^^^^^3 ^f ^ ^1 ^^,3 1500 3 $ 2.969) f 1500.00 ($ 505.22, Ans. 1484 5 15 500 14 845 6550 5938 6120 We first find the interest of $ 1 for 93 d. at 6 % by multiplying .001 of $ 1 by 93, and dividing the product by 6 ; the result is '^ ' ■ 264 ARITHMETIC. Multiplying this by f , the interest of $ 1 for 93 d. at 4% is ^^^^. 3 ^2 969 Subtracting this from $1, the remainder is ^^— ^ — , which is the o proceeds o/ a $ 1 note for 90 days. Then to yield $ 500, the face of the note must be as many dollars as iM§2 is contained times in $ 500. o To divide $ 500 by iM^, we multiply $ 500 by 3, and divide the o product by $2,969; the result to the nearest cent is $505.22, which is the face of the note required. EXAMPLES. 2. What must be the face of a note, due 3 months hence, which, when discounted at 6%, will yield $ 600 ? 3. The proceeds of a 30-day note, discounted at 5%, was $ 340. What was the face of the note ? 4. Wishing to borrow ^225 at a bank, for what sum must my note be drawn at 60 days to obtain that amount, the rate of discount being 6% ? 5. For what amount must a 5-months' note be drawn, so that, when discounted at 7%, the proceeds may be $ 8000? 6. The holder of a 75-day note received $ 550 as pro- ceeds, when the note was discounted at 4%, What was the face of the note ? 7. For what sum must a note, payable in 6 months, be drawn, to yield $ 1500 when discounted at 4^% ? 8. If the rate of discount is 7%, what must be the face of a note, payable 2 months hence, to yield $ 425 when discounted ? 9. For what amount must my note, due in 90 days, be drawn, in order that I may receive $ 908.70 when the note is discounted at 5|-% ? 10. A merchant received a 60-day note in payment for goods sold ; he at once had it discounted at 3|%, and realized ^ 375. For what amount were the goods sold ? EXCHANGE. 265 XXI. EXCHANGE. 336. A Draft is a written order from one person to another, directing him to pay a specified sum of money to a third person. The Drawer of a draft is the person who signs it. The Drawee is the person to whom it is addressed. The Payee is the person to whom it is payable. The Face of a draft is the sum named in it. 337. A Sight Draft is one which is payable on presenta- tion to the drawee. FORM OP A SIGHT DRAFT, f eOOj^. Philadelphia, Feb. 12, 1892. At sight, pay to the order of Henry F. Sears six hundred dollars, value received, and charge the same to the account of E. B. Hart& Co. To Stone & Morison, Cleveland, Ohio. The above draft may be supposed to have been drawn under the following circumstances : A merchant in Philadelphia wishes to pay $ 600 to Henry F. Sears, in Cleveland. He proceeds to a banking-firm, E. B. Hart & Co., who have an account with Stone & Morison, of Cleveland. They sell him a draft for the above amount, which he forwards to Mr. Sears at Cleveland. When Mr. Sears receives it, he carries it to Stone & Morison, who pay it on presentation. 338. A Time Draft is one which is payable at the expi- ration of some specified time after presentation, or after the date of the draft. It is usual to allow three days of grace. 266 ARITHMETIC. FORM OP A TIME DRAFT. f 845y0^. • Chicago, Nov. 23, 1892. Sixty days after date, pay to the order of Charles H. Jack- son eight hundred and forty-Jive dollars, value received, and charge the same to the account of William Rogers. To the Erie National Bank, Buffalo, NY. 339. The Acceptance of a time draft by the drawee is an agreement on his part to pay it. To accept a draft, the drawee writes the word "Accepted" across its face, with the date, and his signature. The draft is then called an Acceptance, and the drawee an Acceptor. An acceptance may be negotiated in the same manner as a promissory note ; if discounted at a bank, the term of dis- count is the time specified in the draft, plus three days of grace (Art. 333). 340. Exchange is the system of making payments by remitting drafts. 341. Exchange is said to be at a certain per cent Premium, or Above Par, when a draft sells for the specified per cent more than its face value. Thus, if exchange is at 1 % premium, a draft for $ 100 will sell for $100, plus 1% of $ 100, or $ 101. Exchange is said to be at a certain per cent Discount, or Below Par, when a draft sells for the specified per cent less than its face value. Thus, if exchange is at 1% discount, a draft for $ 100 will sell for $ 100, less 1% of f 100, or $ 99. The Rate of Exchange is the per cent which a draft costs more or less than its face value. EXCHANGE. 267 DOMESTIC EXCHANGE. 342. Domestic or Inland Exchange is exchange between persons m the same country. EXAMPLES. 343. 1. Find the cost of a sight draft for $ 452, when exchange is at 1^% premium. $452 $452.00 •Qlj 6.78 1^% of $452 is $6.78. 4 52 $458.78, Ans. Then, since the draft sells for $6.78 2 26 more than its face, the cost is $458.78. $6.78 In finding the cost of a time draft, the interest of the face of Me draft for the specified time, plus three days of grace, must be deducted from the cost; for the drawer has the use of the money for that length of time before the drawee pays the draft. 2. What will be the cost of a draft for $ 1000, due 30 days after sight, with interest at 5%, exchange being at f % discount ? «innn V ^ ^30.00 ^-. k^ We find fo/^ of $1000 by * 1^.^^ X - = ^ = * 7.50. jnuitipiying .01 of $ 1000 by $ 1.00 $ 1000.00 ^ ' *^^ "^"''^^ '^ ^ ^•^^•. oo 7 (lO Then, since exchange is at a discount, a siaht draft for 6)$^33^ $ 992.50 ^ ^^^O will cost $ 1000 - $ 7.50, 6)$5^ 4.58 or $992.50. .92 $ 987.92, Ans. The interest of the face of $4.58 the draft for 33 days, at 5%, is$4.-58. Deducting this from the cost of the sight draft, the required cost is $987.92. 3. Find the cost of a sight draft for $500, when ex- change is at 1\% premium. 268 ^ ARITHMETIC. 4. What will be the cost of a sight draft for $ 280, if exchange is at |% discount ? 5. What will be the cost of a draft for $ 8000, due 60 days after sight, with interest at 4%, exchange being at ■|% discount? 6. What must be paid for a draft on New York for $ 472, due 30 days after sight, with interest at 5%, if exchange is at 1 % premium ? 7. I wish to purchase a sight draft on San Francisco for $1965. If exchange is at 1^% discount, how much must I pay for it ? 8. A merchant purchased a draft for $ 700, due 90 days after sight, with interest at 4^%, exchange being at \% premium. What was the cost ? 9. How much must be paid for a draft for $ 344, due 30 days after sight, with interest at 3J%, exchange being at ■|% premium ? 10. What will be the cost of a draft for $ 632, due 60 days after sight, with interest at 3^%, if exchange is at 3f % discount ? 11. How much must be paid for a draft on Cincinnati for $1378, due 3 months after sight, with interest at 41%, if exchange is at a discount of 2i% ? 344. To find the Face of a Draft when the Cost is given. 1. What is the face of a sight draft which can be bought for $ 248.85, when exchange is at 1^% discount ? $1.00 .011 100 25 $1.00 .0125 $ .9875 $.9875)$ 248.85 ($252, Ans. 197 50 51350 49 375 $ .0125 19750 1 9750 EXCHANGE. 269 li % of $ 1 is $ .0125. Subtracting this from $ 1, the cost of a sight draft for $ 1 is $ .9875. -Then if the cost of the given draft is $ 248.85, its face will be as many dollars as $.9875 is contained times in $248.85. The result is $262.00. 2. What will be the face of a draft, due 60 days after sight, with interest at 6%, which can be bought for $ 1000, when exchange is at a premium of f % ? $ 0.01 X ^ = ^^ = $ 0.00625 8 8 -| $.001 $1.00625 63 .0105 6 ) $.063 $ 0.99575)$ 1000.00($ 1004.27, Ans. $ .0105 995 75 4 25000 3 98300 267000 199150 678500 We find 1% of $1 by multiplying .01 of $1 by | ; the result is $.00625. Then, since exchange is at a premium, a sight draft for $1 will cost $ 1 + $ .00625, or $ 1.00625. The interest of $ 1 for 63 days at 6 % is $ .0105. Deducting this from $1.00625, the cost of a draft for $1, due 60 days after sight, with interest at 6%, will be $0.99575. Dividing the given cost by this, the result to the nearest cent is $ 1004.27. 3. What is the face of a sight draft which can be bought for $ 435.60, if exchange is at 1% discount ? 4. A merchant bought a sight draft on Pittsburg for $ 657.72, when exchange was at l\fo premium. What was the face of the draft ? 270 ARITHMETIC. 5. What will be the face of a draft, due 30 days after sight, with interest at 6%, which can be bought for $ 918, when exchange is at a premium of J% ? 6. A sight draft on Philadelphia was bought for $ 240, when exchange was at J% discount. What was the face of the draft ? 7. Find the face of a draft on Detroit, due 60 days after sight, with interest at 4%, which can be bought for $ 760, exchange being at 1J% premium. 8. I purchased a sight draft on St. Louis for $ 189.84, when exchange was at 1^% discount. What was the face of the draft ? 9. How large a draft, due 90 days after sight, with inter- est at 3%, can be bought for $ 575, when exchange is at a discount of J% ? 10. A merchant paid $ 3000 for a 60-day draft, with interest at 5%, exchange being at If % premium. What was the face of the draft ? 11. How large a draft on Baltimore, due one month after sight, with interest at 2^%, can be purchased for f 2380, when exchange is at a premium of 1|% ? FOREIGN EXCHANGE. 345. Foreign Exchange is exchange between persons in different countries. 346. A draft, in foreign exchange, is usually called a Bill of Exchange. A Set of Exchange is a series of three bills, all of the same date and tenor, called the First, Second, and Tliird of exchange, respectively. They are sent by different mails to avoid the delay which might arise from the loss of a single draft. If any one of the three is paid, the others become void. EXCHANGE. 271 FORM OF A FOREIGN BILL OF EXCHANGE. £200. Boston, Dec. 3, 1892. At sight of this First of Exchange, second and third of the same date and tenor unpaid, pay to the order of George Lewis two hundred pounds sterling, value received, and charge the same to the account of Kidder^ Peabody, & Co. To Messrs. Baring Brothers, London, England. 347. Exchange on Great Britain or Ireland is quoted at the value of a pound sterling in United States dollars ; exchange on Erance, at the value of a dollar in francs ; exchange on Germany, at the value of 4 reichsmarks in cents. Thus the statement "Bankers' Sterling, sight, 4.86 J; Commercial bills, 60 days, 4.82i; Francs, sight, 5. 16 J; Reichsmarks, sight, 95|," would be interpreted as follows : Sight drafts on a bank or banker in Great Britain or Ireland, f 4.86J to the pound sterling; drafts drawn on merchants, due 60 days after sight, $4.82^ to the pound sterling ; sight drafts on France, 5.16^ francs to the dollar ; sight drafts on Germany, 95f cents per 4 marks. Sterling Exchange is exchange on Great Britain or Ireland. EXAMPLES. 348. 1. What will be the cost of a bill of exchange on London for £ 160 8s., when exchange is quoted at 4.85 ? £ 160 8s. = £ 160.4 4.85 £ 160 8s. is the same as £ 160.4. 80 20 Since each pound sterling costs 1283 2 $4.85, 160.4 pounds will cost 160.4 6416 times $4.85, or f 777.94. ^777.94., Ang. 272 ARITHMETIC. 2. Find the cost of a bill of exchange on Paris for 632 francs, exchange being quoted at 5.16J. 5.1625) 632. 00 ($122.42, Ans. 516 25 115 750 Since 5.1625 francs can be 103 250 bought for $ 1, to buy 632 francs 12 5000 ^^^^ ^^^^ ^^ many dollars as 5. 1625 10 S250 '^^ contained times in 682. o i7KrtA '^^^ result to the nearest cent 2 06500 • ''''''■''' 110000 3. How large a draft on Berlin can be bought for $ 100, if exchange on Germany is quoted at 95^ ? $100 4 $.955)$ 400.00(418.85 marks, Ans. ^M^ Since 4 marks cost 18 00 $ .955, as many marks can 9 55 be bought for flOO as 8 450 955 is contained times in 7 640 4 times $ 100, or $ 400. 8100 7640 4600 4. A merchant bought a bill of exchange on London for £348, when exchange was quoted at 4. 86 J. What was the cost? 5. Find the cost of a draft on Berlin for 848 marks, exchange on Germany being quoted at 95^. 6. How much must be paid for a bill of exchange on Paris for 3000 francs, exchange at 5.16 ? 7. How large a draft on London can be bought for $190.12, exchange at 4.85 ? 8. How large a draft on Paris can be bought for $ 5786, exchange at 5.17^ ? » EXCHANGE. 273 9. Find the cost of a bill of exchange on Liverpool for £16 5s., exchange at 4.85^. 10. If exchange on Germany is quoted at 94|, how much must be paid for a draft on Bremen for 2175 marks ? 11. If exchange on Paris is quoted at 5.181, how much must be paid for a bill of exchange for 725 francs ? 12. A merchant paid $ 7085.03 for a draft on Berlin, ex- change on Germany at 96|. What was the face of the draft ? 13. How much must be paid for a bill of exchange on Bristol for £523 17s., exchange at 4.88i? 14. What is the face of a bill of exchange on Glasgow costing $ 858, exchange at 4.87|- ? 15. If exchange on France is quoted at 5.19J, how large a draft on Marseilles can be bought for $ 946.50 ? 16. Find the cost of a bill of exchange on Belfast for £ 95 12s. 6d., exchange at 4.87f . 17. How large a draft on London can be bought for $646.29|, exchange at 4.88 ? 18. If exchange on Germany is 95^, what will be the face of a draft on Hamburg costing f 358.37 ? 19. A merchant imported 2650 yards of silk, invoiced at 7.20 francs a yard, and 1200 yards of woollens, invoiced at 6.70 francs a yard. Find the cost of a draft on Paris for the amount of the bill, exchange at 5.16J. 274 ARITHMETIC. XXII. EQUATION OF PAYMENTS. 349. Equation of Payments is the process of finding at what time several payments, due at different times, may all be paid at once, without injustice to either debtor or creditor. The time thus found is called the Equated Time. EXAMPLES. 350. 1. A owes B $150, of which $25 is due in 3 months, $ 50 in 4 months, $ 35 in 5 months, and $ 40 in 7 months. What is the equated time of payment ? 25x3= 75 50x4 = 200 35 X 5 = 175 40 X 7 = 280 150 ) 730(411 mo. = 4 mo. 26 d., Ans. 600 130 A is entitled to 3 months' use of $ 25, or 25 x 3 months' use of $ 1 " " "4 " " " $50, or 50x4 " " " $1 " " " 5 " " " |35, or 35 X 5 " " " $1 u u u7 u u "$40, or 40x7 " " " $1. In all, A is entitled to 730 months' use of $ 1. Then he is entitled to the use of $ 150 for as many months as 150 is contained times in 730, which is 4if months, or 4 mo. 26 d. 2. What is the average time of paying $ 200 due May 4, ^300 due June 12, and $400 due July 24 ? 200 X = We select the earliest date, 300 X 39 = 11700 May 4, as a convenient date 400 X 81 = 32400 from which to reckon times. 900 ) 44100 From May 4 to June 12 is Aq J 39 d. , and from May 4 to July May 4 + 49 d. = June 22, Ans. -p,.^«„^!q:^„ „„ ,-„ -p^ i „,« •^ ' Proceeding as m iiX. 1, we find the equated time to be 49 d. after May 4, or June 22. EQUATION OF PAYMENTS 275 Note 1. The date from which times are reckoned is called the Focal Date. Note 2. If there is a common term of credit, we may find the average time without regard to that term, and add it to the result. Thus, if goods are bought on 60 days' credit as follows : July 5, $ 600 ; Aug. 15, $ 400 ; Sept. 10, $ 500 ; we find the equated time of payment without regard to the term of credit, and add 60 days to the result. Note 3. If, in any result, the fraction of a day is |, or more than ^, it is reckoned as one day ; but if it is less than ^, it is disregarded. 3. A owes $ 250 due in 9 months ; if he pays $ 75 in 4 months, and $55 in 8 months, when should he pay the balance ? 75 >^ 5 _ 375 By paying $ 75 5 months before it is 55 X j^ _. 55 due, and $ 55 1 month before it is due, A loses the use of $ 1 for 75 x 5 + 55 x 1, 120) 430 (3 i^g. mo. or 430 months. 360 Hence, he is entitled to the use of the balance, $ 120, long enough after it be- ' comes due to be equivalent to 430 months' 3 mo. 18 d., Ans. use of $1. Then he is entitled to the use of the balance for as many months after it becomes due as 120 is contained times in 430 ; that is, S/^ months, or 3 mo. 18 d. 4. What is the average time of paying $55 due in 3 months, $ 170 due in 9 months, and $ 135 due in 7 months ? 5. A owes B $300, of which $45 is due in 6 months, $ 85 in 8 months, $ 75 in 9 months, and $ 95 in 11 months. What is the equated time of payment ? 6. Find the equated time of paying $ 420 due in 30 days, $ 720 due in 60 days, $ 120 due in 90 days, and $ 540 due in 120 days. 7. What is the equated time of paying $ 15 due March 15, $ 135 due April 6, and $90 due May 25 ? 8. Find the average time of paying $ 175 due Aug. 29, $340 due Sept. 23, $225 due Oct. 5, and $410 due Nov. 13. 276 ARITHMETIC. 9. On May 23, I bought a piece of land for $ 1300, on four months' credit. If I pay $625 on July 23, when should I pay the balance ? 10. If goods are bought on three months' credit as fol- lows : Sept. 19, 1889, $ 1150 ; Oct. 3, 1889, $ 925 ; Nov. 12, 1889, $ 775 ; what is the equated time of payment ? 11. Four sixty -day notes bear date as follows : Jan. 4, 1891, f 565; Feb. 27, 1891, $350; June 18, 1891, $495; July 30, 1891, $ 210. What is the average date of payment ? Note. A sixty-day note falls due 63 days after its date. 12. On Oct. 21, Henry Williams owed $ 157.25 due in 40 days, $223.75 due in 60 days, and $186 due in 90 days. What is the equated date of payment ? 13. A bill of $1200 is due in 5 months from Jan. 13, 1892. If payments are made as follows : $ 259, March 20, 1892 ; $ 248, May 5, 1892 ; when is the balance due ? 14. A tradesman owes $ 300 due in 5 months, and $ 750 due in 9 months ; if at the end of 5 mo. 20 d. he pays $ 450, wl)en should the balance be paid ? 15. Hooker and Ingalls bought merchandise on 60 days' credit as follows : June 28, $ 72.30 ; Aug. 1, $ 156.75 ; Sept. 4, $ 95.10. What is the average date of payment ? 16. A merchant owes $2350 due in 10 months. If he pays $400 in 2 months, $350 in 4 months, and $525 in 8 months, when should he pay the balance ? 17. Four ninety-day notes bear date as follows : March 9, 1892, $388.05; May 24, 1892, $254.75; Aug. 13, 1892, $525; Oct. 30, 1892, $409.20. What is the average date of payment ? AVERAGE OF ACCOUNTS. 351. Average of Accounts is the process of finding at what time the balance of an account may be paid, without injustice to either debtor or creditor. EQUATION OF PAYMENTS. 277 EXAMPLES. 352. 1. Find the equated time for paying the balance of the following account : Dr. Charles Stuart. Cr. 1892. 1892. May 28 To Mdse. 30 d. $350 June 8 By Draft, 30 d. $275 June 16 (( u 125 July 15 " Cash, 250 July 7 " " 2 mo. 275 Aug. 21 U (( 125 Solution. June 21, 350x11= 3850 July 11, 275 X 25 = 6875 June 16, 125x0 = July 15, 250 X 29 = 7250 Sept. 1, 275 X 83 = 22825 Aug. 21, 125 X 66 = 8250 750 650 26675 22375 650 ^2375 $100bal. 4300 bal. 4300 - 100 = 43 d. June 16 + 43 d. = July 29, Ans. The sum of the items on the debit side of the account is ^ 750, and on the credit side f 650. Subtracting $ 650 from ^ 750, there is a balance of $ 100 on the debit side of the account, showing the amount still due from Charles Stuart. Payment for the merchandise sold May 28 is due 30 days after May 28, or June 27 ; and payment for that sold July 7 is due 2 months after July 7, or Sept. 7. The 30-day draft dated June 8 is due 33 days after June 8, or July 11. Thus, June 16 is the earliest date at which any item becomes due. Using June 16 as the focal date, we find that the sum of the products on the debit side of the account is equivalent to the use of $ 1 for 26675 days. 278 ARITHMETIC. Also, the sum of the products on the credit side is equiva- lent to the use of $ 1 for 22375 days. Subtracting 22375 from 26675, there is a balance on the debit side of the account, equivalent to the use of $ 1 for 4300 days. Now in order to make the sum of the products on the credit side equal to the sum of the products on the debit side, it is evident that the balance of $ 100 must be paid at some date after the focal date. That is, Mr. Stuart can settle the account equitably by paying $ 100 as many days after June 16 as is equivalent to the use of $1 for 4300 days. But the use of $ 1 for 4300 days is equivalent to the use of $100 for as many days as 100 is contained times in 4300; that is, 43 days. Hence, he can settle the account equitably by paying f 100 43 days after June 16 ; that is, on July 29. If, in the above example, the sum of the products on the a-edit side of the account had been greater by 4300 than the sum of the products on the debit side, an earlier focal date could have been found, which would have made the sum of the products on the credit side equal to the sum of the products on the debit side. If, for example, the focal date had been 43 days earlier than June 16, the sum of the products on the debit side would have been greater by 750 x 43, while the sum of the products on the credit side would have been greater by 650 X 43 ; and since 750 x 43 exceeds 650 x 43 by 4300, this would have made the sum of the products on the credit side equal to the sum of the products on the debit side. This earlier focal date would then have been the equated time of payment. It is evident from the above that if the two balances are on the same side of the account, the equated time is after the focal date ; but if they are on opposite sides of the account, the equated time is before the focal date. EQUATION OF PAYMENTS. 279 From the above example we derive the following RULE. Write to the left of each item of the account its date of maturity ; and select as a focal date the earliest date at which any item becomes due. Multiply each item by the number of days betiveen its date of maturity and the focal date, and add the products on each side of the account. Divide the balance of the sums of the products by the balance of the account, giving the number of days between the focal date and the equated time of payment. If the balaiices are on the same side of the account, the equated time is after the focal date; if they are on opposite sides, the equated time is before the focal date. 2. Find the equated time for paying the balance of the following account : Dr. George Adams. Cr. 1891. 1891. Sept. 16 To Mdse. 30 d. $550 Oct. 4 By Cash, $500 Oct. 28 U (( 375 Nov. 9 u u 325 3. Find the equated time for paying the balance of the following account : Dr. Henry Cole. Cr. 1890. 1890. Nov. 12 To Mdse. 30 d. $620 Dec. 24 By Cash, $840 Dec. 8 " • "2 mo. 475 1891. 1891. Feb. 7 " Mdse. 760 Jan. 25 u u 745 280 ARITHMETIC. 4. Find the equated time for paying the balance of the following account : Dr. William Blake. Cr. 1892. 1892. Jan. 18 To Mdse. 1 mo. ^450 Jan. 31 By Draft, 30 d. $300 Feb. 6 » " 60 d. 600 April 4 " Cash, 550 5. At what date should the balance of the following account begin to draw interest ? Dr. Edward Dodge. Cr. 1892. 1892. May 5 To Mdse. 1 mo. fl65 May 18 By Cash, $115 June 20 u u 280 June 29 (( u 150 July 11 " " 60 d. 105 July 10 " Draft, 60 d. 175 6. Find the equated time for paying the balance of the following account : Dr. Richard Hayes. Cr. 1891. 1891. Feb. 23 To Mdse. 30 d. i$275 Mar. 19 By Draft, 3 mo. $200 Mar. 15 U (( 420 April 6 " Canh, 175 May 9 " " 2 mo. 355 May 31 " Draft, 30 d. 450 7. Find the equated time for paying the balance of the following account : Dr, John Evans. Cr. 1892. 1892. Aug. 23 To Mdse. 90 d. $345 Sept. 2 By Draft, 90 d. $275 Sept. 16 " " 2 mo. 775 Nov. 18 " Cash, 450 Nov. 13 " " 3mo. 530 Dec. 7 " Draft, 2 mo. 325 Dec. 28 " Cash, 210 EQUATION OF PAYMENTS. 281 8. When should the balance of the following account begin to draw interest ? Dr. James French. Cr. 1890. 1890. Dec. 21 ToMdse.eOd. $380 Dec. 19 By Draft, 2 mo. $350 1891. 1891. Jan. 15 u u 560 Jan. 3 " Mdse. 1045 Feb. 27 " 2 mo. 420 Mar. 16 " Draft, 60 d. 1000 April 1 " " 30 d. 700 Note 1. The latest date at which any item becomes due may be taken as a focal date ; in such a case, if the balances are on the same side of the account, the equated time is before the focal date ; if they are on opposite sides, the equated time is after the focal date. Note 2. In settling an account in which the sum of the items on the debit side equals the sum of the items on the credit side, it may happen that the sum of the products on one side of the account is greater than the sum of the products on the other side. Thus,' in settling the following account : Daniel Green. Dr. Cr. 1892. 1892. May 5 To Mdse. $130 May 17 By Cash, $150 June 12 (4 U 210 June 25 it ii 190 with May 5 as the focal date, we find : 130 X = • 210 X 38 = 7980 $340 7980 340 Obal. 150 X 12 = 1800 190 X 51 = 9690 $340 11490 7980 3510 bal. It appears from the above that there is still due from Mr. Green the use, ov interest, of $1 for 3510 days ; which, at 6%, is $0.59. If the balance of 3510 had been on the debit side of the account, settlement would be made by paying Mr. Green $ 0.59. 282 ARITHMETIC. THE INTEREST METHOD. 353. Another method of averaging accounts is known as tiie Interest Method. " We will select for illustration the example which is solved by the product method on page 277. We will suppose that the account is settled on Sept. 7, the latest date at which any item becomes due. Solution. Int. of $350 for 72 d. = $4.20 " " 125 '' 83 d. = 1.7292 " " 275 " Od. = $750 $5.9292 650 5.2625 $100bal. $0.6667 ba Int. u 1. of $ 275 for 58 d. " 250 " 54 d. " 125 " 17 d. $660 = $2.6583 = 2.25 = .3542 $ 5.2625 Int. of $ 100 for 1 d. = $.0167. .6667 -^ .0167 = 40 d. Sept. 7 - 40 d. = July 29, Ans. The balance of the account is $ 100 on the debit side. Payment for the merchandise bought May 28 is due June .27 ; if it is not made until Sept. 7, Mr. Stuart should pay interest on the amount for 72 days; which, at 6%, is 14.20. In like manner, he should pay interest on $ 125 for 83 days, which is $ 1.7292. Hence, if the account is settled Sept. 7, he should pay $ 5.9292 interest in addition to the sum of the items on the debit side. Now on June 8 he gave his draft, at 30 days, for $ 275, which became due July 11 ; in paying this sum 58 days before Sept. 7, he is entitled to interest on the amount for 58 days, which is $ 2.6583. In like manner, he is entitled to interest on $ 250 for 54 days, which is $ 2.25, and on $ 125 for 17 days, which is $0.3542. EQUATION OF PAYMENTS. 283 Hence, if the account is settled Sept. 7, he is entitled to interest to the amount of $ 5.2625. This leaves a balance of interest due from him of $ 5.9292 — $ 5.2625, or $ .6667 ; that is, if he settles the account Sept. 7, he must pay $ .6667 in addition to the balance of the account. But by paying the $ 100 a little earlier than Sept. 7, he can offset the interest charge ; and the question now is, how many days will it take $ 100 to gain $ .6667 at 6% ? In one day, $ 100 gains $ .0167. Then, to gain $ .6667 will take as many days as .0167 is contained times in .6667 ; which is 40. Then, by paying the $ 100 40 days before Sept. 7, that is, on July 29, he can settle the account with equity. It is evident that if the balances had been on opposite sides of the account, in the above example, the equated time of payment would have been after Sept. 7. From the above example, we derive the following RULE. Select as a focal date the latest date at ivhich any item becomes due. Compute interest at Q^o on eacJi item for the number of days from the date when it becomes due to the focal date. Divide the balance of interest by the interest of the balance of the account for one day, giving the number of days from the focal date to the equated time of payment. If the balances are on the same side of the account, the equated time is earlier than the focal date; if they are on opposite sides, it is later than the focal date. The teacher may have the examples of Art. 352 performed by the interest method. 284 ARITHMETIC. XXIII. STOCKS AND BONDS. 354. Stock is the capital of a corporation ; it is divided into a certain number of equal parts called Shares. 355. The original value of a share is usually $ 100 ; it will be so considered in the present chapter, unless the con- trary is stated. 356. The original value of a share of stock is called its Par Value, and the price at which it sells is called its Market Value. 357. If a share of stock sells for more than its par value, the stock is said to be above par, or at a premium; if it sells for less than its par value, the stock is said to be below par, or at a discount. 358. The market value of a stock is usually quoted at a certain per cent of the par value. Thus, if a stock is quoted at 105, it is selling for 5% above its par value ; if it is quoted at 95, it is selling for 5% below its par value. 359. A Certificate of Stock is a document issued by a corporation, specifying the number of shares owned by the holder, and the par value of each share. A Dividend is a sum divided among the stockholders from the profits of the business. An Assessment is a sum required of the stockholders to meet the losses or expenses of the business. Dividends and Assessments are generally reckoned at a certain per cent of the par value of the stock. 360. A Bond is the interest-bearing note of a government or corporation. The interest on bonds is usually paid semi-annually. A Coupon is a certificate of interest attached to a bond. STOCKS AND BONDS. 285 361. Bonds are usually named according to their rate of interest and date of maturity. Thus, " U. S. 4i-'s. '91 " signifies Bonds issued by the United States governmentj bearing 4J% interest, the prin- cipal payable in 1891. 362. Brokerage is the commission received by a broker for buying or selling stocks and bonds. It is usually |^% of the par value of the stock or bond. Note. It will be understood, in the following examples, that the brokerage is not included in the quoted price of a stock. Thus, if a man buys stock at 112 J, and the brokerage is ^%, he pays the broker 112| ; if he sells stock at 112^, and the brokerage is J%, he receives only 1 12 1 from the broker. EXAMPLES. 363. 1. Find the cost of 20 shares New York Central stock, at 112J, brokerage ^%. Since the cost of one share, 1121+ |-=112|-. including brokerage, is $112^, $ 112|- X 20 = $ 2257.50, Ans. the cost of 20 shares will be 20 X $112|, or $2257.50. 2. How much will be received from the sale of 16 shares Chicago, Burlington, and Quincy stock at 103|, brokerage Since the price received for one 103| — I" = 103|-. share is $ 103^, the price received $ 103^ X 16 = $ 1656, Ans. for 16 shares will be 16 x $ 103^, or $ 1656, 3. What amount of Mexican Central 4's, at 64i, can be bought for $16062.50, including a brokerage of i% ? 641- -I- 1- = 641 = 64.25. Since the cost of one dollar's 16062.5 -.6425 = 125000, .In.. Zf.ot"'"'^'"^ ^\f""^^^"'' It ' $ 0. 6425, as many dollars' worth can be bought for 1 16062.50 as .6425 is contained times in 16062.5. Dividing 16062.5 by .6425, the quotient is 25000. Hence, $25000 worth of bonds can be bought for $ 16062.50. 286 ARITHMETIC. 4. If 48 shares Union Pacific stock are sold for $ 1992, brokerage J%, what is the quoted price of the stock ? $ 1992 ^ 48 = $ 41.50 = 1 41f Dividing $ 1992 by 48, the 4i I , ^ I -1 Q A price received for one share is Then the quoted price is 41^, plus the brokerage of I, or 41|. 5. Find the cost' of 95 shares Old Colony stock, at 186f, brokerage ^%. 6. Find the cost of 84 shares of telegraph stock at 95J, brokerage J%. 7. How much will be received from the sale of 37 shares of bank stock at 147f, brokerage i-% ? 8. A man sold 5S shares of New York and New England stock at 44|, brokerage i%. How much did he receive ? 9. Find the cost of $ 12000 U. S. 4's, when at a premium of 12|%, brokerage i%. 10. Find the cost of 72 shares of railway stock, when 31|-% below par, brokerage J%. 11. How many shares of Missouri Pacific stock, at 56|-, can be bought for $2604.75, including a brokerage of |^% ? 12. A gentleman sold bonds at 88|, brokerage i%, re- ceiving the sum of $ 6637.50. What amount of bonds did he sell ? 13. I sold mining stock at 126J, brokerage ^%, receiving the sum of $ 5450.25. How many shares did I sell ? 14. How many shares of railway stock, at $151.75 a share, can be bought for $95073.75, including a brokerage of i% ? 15. What amount of Missouri, Kansas, and Texas 5's, at a discount of 51|%, can be bought for $7237.50, including a brokerage of |% ? 16. How many shares of bank stock, at 27|% above par, can be bought for $ 9581.25, including a brokerage of J% ? STOCKS '^^tt BONDS. 287 17. If 12 shares of Elevated Eailway stock are sold for f 1837.50, including a brokerage of i%, what is the quoted price of the stock ? 18. If 45 shares Boston and Maine stock can be bought for $7627.50, including a brokerage of i%, what is the quoted price of the stock ? 19. If $6000 worth of bonds are sold for $5497.50, including a brokerage of ^%, how much below par are the bonds quoted ? 20. If 18 shares of railway stock can be bought for $1460.25, including a brokerage of ^%, what is the quoted price of the stock ? 21. If $7500 Iowa Central bonds can be bought for $6581.25, including a brokerage of i%, at what per cent discount are the bonds quoted ? 22. If 298 shares Delaware and Hudson stock can be bought for $40788.75, including a brokerage of ^%, how much above par is the stock quoted ? 23. If the brokerage, at ^%, for selling stock is $23.25, how many shares were sold ? 24. What annual income is received from mining stocks whose par value is $11300, paying lf% dividends semi- annually ? 25. A corporation declared a dividend of 7J%, paying to the stockholders a total amount of $ 5437.50. What was its capital stock ? 26. A man purchased 358 shares of a certain stock at 3|% below par, and sold it at a premium of 7f % ; if he paid ^% brokerage on each transaction, how much did he gain ? 27. A corporation whose capital is $225000, levies an assessment of $ 4218.75 on its stockholders. What is the rate per cent of the assessment ? 28. What par value of stocks paying 1|% dividends quarterly, will produce an annual income of $ 513.50 ? 288 . ARITHMETIC. 29. If $ 441.75 is lost by buying stocks at 5J% premium, and selling them at 6J^% below par, in each case paying a brokerage of -J-^, how many shares were bought ? 30. A man sold 204 shares of stock at 80, and with the proceeds bought stock at 106|. If the brokerage on each transaction was i%, how many shares did he buy ? 31. A man sold 285 shares of railway stock at 103 J, and invested the proceeds in bank stock at 78J. If the broker- age on each transaction was ^%, how many shares of bank stock did he receive ? 32. What annual income will be realized from investing ^2145 in a 5% stock at 1071, brokerage -|-% ? The cost of the stock, in- iO'^i + i = ^•^'^i = 107.25. eluding brokerage, is 107.25. ' $ 2145 H- 1.0725 = $ 2000. Then stock to the par value $2000 X .05 = $ 100, A71S. of $2145- 1.0725, or $2000, can be bought for $2145. The annual income from $2000 at 5% is $2000 x .05, or $ 100. 33. What amount must be invested in a 5i% stock, at 93f, no allowance being made for brokerage, to realize an annual income of $ 374 ? ^ 374 -^ .055 = $ 6800. K the annual income is to be <1^_±_29) qj, hq ^^g 2 Observing that 7 is the number of terms, 5 the first term, and 29 the last term, we have the following RULE. To find the sum of the terms of an arithmetical progression, multiply the sum of the first and last terms by the number of terms, and divide the result by 2. EXAMPLES. 367. 1. Find the last term and the sum of the terms of the arithmetical progression 19, IS^, 17|, etc., to 24 terms. The common difference is 19 — 18}, or |, Since the number of terfhs is 24, the last term is the 24th term. By Art. 365, the 24th term is 19 - (23 x f), or 3|. By Art. 366, the sum of the terms is 24 x (19 + 3|)^ ^^ 272. 2. Find the 11th term of the progression 2, 9, 16, etc. 3. Find the 38th term of the progression 4|, h\, 6^, etc. 4. Find the 23d term of the progression 327, 316, 3*05, etc. 294 ARITHMETIC. 5. Find the 20th term of the progression 120, 116.4, 112.8, etc. 6. Find the 47th term of the progression -^q, j\, -^-^, etc. Find the last term and the sum of the terms of : 7. 4, 10, 16, etc., to 12 terms. 8. 9, 23, 37, etc., to 21 terms. 9. 293, 285, 277, etc., to 29 terms. 10. 486, 473, 460, etc., to 36 terms. 11. 3J, 5, 6J, etc., to 48 terms. 12. 97, 90.3, 83.6, etc., to 14 terms. 13. 2^ 3^ 5^ etc., to 59 terms. 14. I, I, If, etc., to 23 terms. 15. Find the sum of the integers beginning with 1, and ending with 99. 16. Find the sum of the even integers beginning with 2, and ending with 100. 17. Continue the progression y^g, ■^, -J, etc., to four more terms. 18. Find the sum of the first 18 integers which are multiples of 7. 19. Find the sum of all the multiples of 11, from 110 to 990, inclusive. 20. A body falls 16^2 ^^^^ *^® ^^^ second, and in each succeeding second 32i- feet more than in the next preceding one. How far will it fall in the 16th second ? How far will it fall in 16 seconds ? 21. A man travelled 43^ miles the first day, and on each succeeding day 2| miles less than on the next preceding. How £ar did he travel on the 11th day ? How far did he travel in 11 days ? PROGRESSIONS. 295 22. If a person saves ^ 100 a year, and puts this sum at simple interest at 4|^% at the end of each year, to how much will his property amount at the end of 25 years ? GEOMETRICAL PROGRESSION. 368. A Geometrical Progression is a series of numbers which increase or decrease by a constant multiplier, called the Ratio. Thus, 1, 3, 9, 27, 81 is an increasing geometrical progres- sion, in which the ratio is 3. Again, 64, 32, 16, 8, 4, is a decreasing geometrical pro- gression, in which the ratio is ^. The numbers which compose the progression are called its Terms. 369. To find any Term of a Geometrical Progression. Example. Find the 6th term of the geometrical progres- sion 2, 6, 18, etc. Dividing the second term, 6, by the first Ratio = 1 = 3. term, 2, the ratio is 3. 6th term = 2x3^ Now the second term is equal to the first = 2 X 243 term times the first power of the ratio ; the = 486 Ans ^^*'^^ term is equal to the first term times the second power of the ratio ; etc. Hence, the sixth term will be equal to the first term times the fifth power of the ratio ; that is, 2 x 3^, or 486. From the above example, we derive the following RULE. To find any term of a geometrical progression, multiply the first term by that power of the ratio whose exponent is less by 1 than the number of the required term. 370. To find the Sum of the Terms of a Geometrical Pro- gression. 1. Find the sum of the terms of the geometrical pro- gression 2, 6, 18, 54, 162. 296 ARITHMETIC. The ratio is 3. The sum of the terms =2 + 6+18 + 54 + 162. (I) Multiplying each term of the result by the ratio, 3, we have 3 X the sum of the terms = 6 + 18 + 54 + 162 + (162 x 3). (2) Subtracting (1) from (2), we have 2 X the sum of the terms = (162 x 3) - 2. rr^u .V. *.!, . (162x3)- 2 (162x3)-2 Then, the sum of the terms = ^ ^ = ^ — ^ — ^ Observing that 162 is the last term, 3 the ratio, and 2 the first term, we have the following rule : The sum of the terms of an increashig geometrical progres- sion is equal to the product of the last term by the ratio, minus the first term, divided by the ratio minus 1. 2. Find the sum of the terms of the geometrical progres- sion 768, 192, 48, 12, 3. The sum of the terms = 768 + 192 + 48 + 12 + 3. (1) Multiplying each term of the result by the ratio, ^, we have ^ X the sum of the terms = 192 + 48 + 12 + 3 + (3 x \). (2) Subtracting (2) from (1), we have I X the sum of the terms = 768 - (3 x I). 768 - (3 X i) 768 - (3 x i) Then, the sum of the terms = f 1- Observing that 768 is the first term,' 3 the last term, and i the ratio, we have the following rule : The sum of the terms of a decreasing geometrical progres- sion is equal to the first term, minus the product of the last term by the ratio, divided by 1 minus the ratio. 371. In Ex. 1, Art. 370, we have 162 X 3 = (2 X 30 X 3 = 2 X 3«. That is, the product of the last term of a geometrical progression by the ratio is equal to the first term, multi- plied by that power of the ratio whose exponent is equal to the number of terms. PKOGRESSIONS. 297 Then the rules of Art. 370 may be stated as follows : Tlie sum of the terms of an increasing geometrical progres- sion is equal to the first term, multiplied by that poiver of the ratio whose exponent is equal to the number of terms, minus the first term, divided by the ratio minus 1. The sum of the terms of a decreasing geometi^ical progres- sion is equal to the first term, minus the first term multiplied by that power of the ratio whose exponent is equal to the num- ber of terms, divided by 1 minus the ratio. EXAMPLES. 372. 1. rind the last term and the sum of the terms of the geometrical progression 3, 12, 48, etc., to -6 terms. The ratio is -L2, or 4. By Art. 369, the last or 6th term is 3x45, or 3072. By the first rule of Art. 370, the sum of the terms is 4 X 3072 - 3 ^^ .^Q. , or 4095. 4-1 ' 2. Find the sum of the terms of the geometrical pro- gression 2, i, f , etc., to 7 terms. The ratio is | ^ 2, or i. By the second rule of Art. 371, the sum of the terms is 2 - 2 X ay _ 2- ^tW _ im_2186 ^^^ 1 - i ~ I "" I - 729 ' 3. Find the 5th term of the progression 4, 20, 100, etc. 4. Find the 6th term of the progression f , i, |, etc. 5. Find the 9th term of the progression 6|, 41 3, etc. 6. Find the 7th term of the progression 600, 150, 37|, etc. Find the last term and the sum of the terms of : 7. 1, 2, 4, etc., to 11 terms. 8. 6, 18, 54, etc., to 8 terms. . 9. 20, 10, 5, etc., to 10 terms. 298 ARITHMETIC. 10- i A» H? etc., to 6 terms. 11. -J-, |-, |, etc., to 7 terms. 12. 12|, 5, 2, etc., to 5 terms. 13. Find the sum of the terms of the progression 12|, 16, 20, etc., to -5 terms. 14. Find the sum of the terms of the progression 15, 10, 6|, etc., to 8 terms. 15. A man agreed to work for 14 days on condition that he should receive 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on. How much did he receive in all ? 16. A man travelled 384 miles the first day, and on each succeeding day one-half as many miles as on the next pre- ceding. How far did he travel on the 10th day ? How far did he travel in 10 days ? 17. Continue the progression 4|f, 3|, 2|, etc., to three more terms. 18. The population of a certain city at the end of each year is 1.04 times as great as at the beginning of the year. If the population on Jan. 1, 1890, was 15625, what will it be on Jan. 1, 1893 ? 19. If the first term is $ 100, the ratio 1.05, and the number of terms 5, what is the last term ? COMPOUND INTEREST. 373. Problems in compound interest may be solved by aid of the principles of geometrical progression. * Thus, let $ 100 be put at compound interest at 6%. The amount at the end of one year is $ 100 x 1.06. The amount at the end of two years is $ 100 x 1.06 x 1.06, or $100 X (1.06)2; and so on. PROGRESSIONS. 299 Hence, the amount at the end of any number of years is equal to the principal, multiplied by that power of 1 plus the rate, whose expo7ient is equal to the number of years, EXAMPLES. 1. Find the amount of $1600 for 4 years, at 5% com- pound interest. By the above rule, the required amount is $1600 X (1.05)*, or $1944.81, Ans. 2. What principal will gain $15,608 in 3 years at 4% compound interest ? The amount of $1 for 3 years at 4% compound interest is (1.04)* dollars, or $1.124864. Then, $ 1 will gain $ .124864 in 3 years at 4% compound interest. Then, to gain $15,608 will take as many dollars as .124864 is con- tained times in 15.608, which is $ 125, Ans. 3.. Find tha^ amount of $6400 for 4 years, at 3% com- pound interest. 4. Find the amount of $8000 for 5 years, at 4% com- pound interest. 5. Find the compound interest of $300 for 4 years, at^%. 6. Find the compound interest of $760 for 3 years, at 7. What principal will amount to $ 25585.35 in 2 years, at 3J% compound interest ? 8. What principal will amount to $5477.5974»in 3 years, at 41% compound interest ? 9. What principal will gain $274.07131 in 5 years, at 5% ^compound interest? ANNUITIES. 374. An Annuity is a specified sum of money payable at equal intervals of time. 300 ARITHMETIC. Note. We shall consider in tlie present chapter those cases only in which the payments are annual. The Amount or Final Value of an annuity is the sum of all the payments, together with the interest on each pay- ment from the time it becomes due until the annuity ceases. The Present Worth of an annuity is that sum of money , which, at the specified rate of interest, will amount to the final value. 375. Annuities at Simple Interest. Problems in annuities at simple interest may be solved by aid of the principles of arithmetical progression. 1. Find the amount of an annuity of $400 for 6 years, at 5% simple interest. The first payment draws interest for five years ; the second pay- ment for four years ; etc. Now the amount of the first payment at the en^ of five ye-ars is $400 X 1.25, or 1 500. The amount of the second payment at the end of four years is $400 X 1.20, or $480; etc. We then have an arithmetical progression, whose first term is $ 500, last term $ 400, and number of terms 6. Therefore, by Art. 366, the sum of the terms is . f X ($500 + $400), or $2700, Ans. From the above example, we derive the following RULE. Find the amount of the annual payment for a number of years less by 1 than the given time. Add this amount to the annual payment, and multiply the result by one-half the given number of years. 2. Find the present worth of the annuity of Ex. 1. By Art. 331, the present worth of $2700 due 6 years hence, at 5 %, is |2700^ or $2076.92 + , Ans. PROGRESSIONS. 301 3. What annuity to continue for 4 years, at 6% simple interest, can be purchased for f 1090 ? An annuity of $ 1 to continue for 4 years, at 6 % simple interest, will amount to | x ($ 1.18 + $ 1), or $4.36. The amount of $ 1090 for 4 years, at 6 %, is ^ 1351.60. Then, an annuity of as many dollars can be purchased for $ 1090 as 4.36 is contained times in 1351.60 ; which is $310, Ans. EXAMPLES. 4. Find the amount and present worth of an annuity of $ 300 for 5 years, at 4% simple interest. 5. Find the amount and present worth of an annuity of $ 250 for 3 years, at 4^% simple interest. 6. Find the amount and present worth of an annuity of $800 for 8 years, at 5% simple interest. 7. Find the amount and present worth of an annuity of $720 for 7 years, at 3f%. simple interest. 8. What annuity to continue for 6 years, at 6% simple interest, can be purchased for $2070? 9. What annuity to continue for 8 years, at 3% simple interest, can be purchased for $ 4420 ? 10. What annuity to continue for 9 years, at 4% simple interest, can be purchased for $ 2088 ? 11. What annuity to continue for 11 years, at 5^% sim- ple interest, can be purchased for $ 5610 ? 376. Annuities at Compound Interest. ^ Problems in annuities at compound interest may be solved by aid of the principles of geometrical pt-ogression. 1. Find the amount of an annuity of $500 for 4 years, at 3% compound interest. The fourth or last payment draws no Interest. The amount of the third payment at the end of one year, at 3 %, is $ 500 X 1.03. 302 ARITHMETIC. The amount of the second payment at the end of 2 years at 3 % compound interest is $500 x (1.03)^ (Art. 373) ; etc. We then have an increasing geometrical progression, whose first term is $ 500, ratio 1.03, and number of terms 4. Therefore, by Art. 371, the sum of the terms is $500x[(1.03)^-11 ^ or $2091.81 + , Ans. .03 2. Find the present worth, of th^ annuity of Ex. 1. By Art. 373, the amount of $ 1 for 4 years, at 3 % compound inter- est, is (1.03)4 dollars, or $1.12550881. Then to amount to $2091.81+ will take as many dollars as 1.12550881 is contained times in 2091.81 + , which is $1858.54 + , Ans. 3. What annuity to continue for 3 years, at 5% compound interest, can be purchased for ^ 2522 ? An annuity of $ 1 to continue for 3 years, at 5 % compound interest, will amount to C^-^^^- ^ dollars, or $3.1525. .05 ' ^ The amount of $2522 for 3 years, at 5% compound interest, is $2522 X (1.05)3, or $2919.53025. Then an annuity of as many dollars can be purchased for $ 2522, as 3.1625 is contained times in 2919.53025 ; which is $926.10, Ans. EXAMPLES. 4. Find the amount and present worth of an annuity of $200 for 3 years, at 5% compound interest. 5. Find the amount and present worth of an annuity of f 300 for 4 years, at 6% compound interest. 6. Find the amount and present worth of an annuity of $400 for 5 ye^s, at 4% compound interest. 7. What annuity, to continue for 2 years, at 6% compound interest, can be purchased for $ 515 ? 8. What annuity to continue for 3 years, at 4% compound interest, can be purchased for $ 975.50 ? 9. What annuity to continue for 4 years, at 5 % compound interest, can be purchased for $ 600 ? MISCELLANEOUS EXAMPLES. 303 XXV. MISCELLANEOUS EXAMPLES. 377. 1. Find the value of (7263 - 34242 - 439) x (61143 ^ 837 - 748). 2. Divide 1661^ by 17, and reduce the result to a mixed number. 3. Eeduce f If to 91767ths. 4. A man spent ^ of his money for provisions, -f of the remainder for clothing, -^-^ of the remainder for charity, and had f 9.10 left. How much had he at first ? 5. Find the interest of $3528.75 from Nov. 25, 1887, to Sept. 11, 1891, at 4i%. 6. Find the proceeds of a 3-months' note for $ 576, dis- counted on the day of its date at 3f %. 7. The area of a square field is 2 A. 77 sq. rd. 17 sq. yd. 4| sq. ft. ; find its side in rods, yards, and feet. ^ ^. ,., 396 X 425 X 1274 »• ^^^P^-^^y 4896x325x1078; 9. The circumference of the hind- wheel of a carriage is 9 ft. 2 in., and of the fore-wheel 7 ft. 9 in. How many times does each wheel turn in travelling 9 mi. 220 rd. ? 10. Extract the square root of .729275008576. 11. A bin 8 ft. 3 in. long, 5 ft. 8 in. wide, and 4 ft. 2 in. deep is filled with wheat. If a bushel is equal to 1^ cu. ft., how much are the contents worth, at 96 cents a bushel ? 12. Multiply 7 mi. 113 rd. 4 yd. 2 ft. 11 in. by 27. 13. When it is 3.08 p.m. at St. Petersburg, Ion. 30° 19' 48" E., it is 4 hr. 6& min. 51| sec. a.m. at San Francisco. What is the longitude of San* Francisco ? 14. Simplify 64-5i + 4t-3i 304 ARITHMETIC. 15. A can do a piece of work in 8i hr. ; A and B together can do it in 4^8_ ^j.^ . ^nd A and C together can do it in 4 hr. How many hours will it take B and C together to do the work ? 16. Subtract y\% from i|f , and reduce the result to its lowest terms. 17. rind the exact number of days from May 15, 1873, to March 12, 1892. 18. A gentleman bequeathed 37|-% of his property of 1 15120 to his wife, 44|% of what remained to his son, 71f % of the balance to his daughter, and the remainder to charity. How much did he leave to charity ? 19. How long must $487 be on interest at 3^% to gain $99.43? 20. Express iff as a circulating decimal. 21. Find the side, and the number of board feet, in the squared timber that can be sawed from a log whose length is 19 ft. 7 in., and diameter at the smaller end 16 in. 22. Multiply 83fff by 59. 23. A wheel revolves 5000 times in travelling 8 mi. 296 rd. What is its radius in inches ? 24. Express .008171875 as a common fraction in its lowest terms. 25. If it costs $ 98.55 to plaster a hemispherical dome whose diameter is 34 ft. 9 in., how much will it cost to plaster a hemispherical dome whose diameter is 57 ft. 11 in. ? 26. What per cent above cost must a merchant mark an article, in order to be able to sell it at a discount of 16% from the list price, and still make a profit of 11% ? 27. Find the L. C. M. of 1656, 3087, and 8316. . 28. Divide 31 sq. mi. 114 A. 132 sq. rd. 21 sq. yd. 3 sq. ft. 90sq. in. by 18. MISCELLANEOUS EXAMPLES. 305 29. A tradesman sold merchandise for $ 1101.75, and gained |-f- of what it cost him. How much did he gain by the operation ? 30. When it is 7.36 a.m. at Washington, Ion. 77° 3' 37" W., what time is it at Calcutta, Ion. 88° 19' 2" E. ? 31. Divide 298i| by 23. 32. A train of 54 cars is loaded with coal, each car con- taining 4 long tons, 18 long hundred-weight. What is the value of the coal, at f 4.95 a short ton ? 33. Find the amount of $ 2893.40 from March 27, 1885, to March 18, 1889, at 2i%. 34. I paid $ 65, including $ 1.75 for the policy, for insur- ing $ 4600 on a house. What was the rate of insurance ? 35. Express 8 mi. 289 rd. 3 yd. 2 ft. 11 in. in inches. 36. If a train performs a certain journey in 5 h. 36 min., travelling at the rate of 56 feet a second, how long will it take it, travelling at the rate of 1085 yards a minute ? 37. Find the present worth and true discount of $478.95 due 1 yr. 9 mo. 10 d. hence, at 4%. 38. How large a draft on Bremen can be purchased for $ 500, if exchange on Germany is quoted at 95} ? 39. Divide 2J^ by 1|^, and reduce the result to a mixed number. 40. Find the lateral area and volume of a pyramid with a square base, each side of whose base is 26 in., and whose altitude is 84 in. 41. A merchant sold goods for $ 17212.50, and lost if of what they cost him. How much did the goods cost him ? 42. A rectangular field is 15 rd. 4 yd. 1^ ft. long, and 12 rd. 3 yd. 1 ft. wide. How much is it worth at $592.90 an acre ? 306 AKITHMETIC. . 43. Find the G. C. D. of 54432, 63504, and 98784. 44. Express £ 84 9s. 2d. 1 far. as a fraction of £ 123 Ss. 9d. 3 far. 45. The population of a town- decreased 6^% from 1870 to 1880, and increased 13f % from 1880 to 1890. If the population in 1890 was 4305, what was the population in 1870? 46. A merchant imported 875 sq. yd. of rugs, invoiced at 139.2 francs a square yard. What was the duty, at 55 cents a square yard, and 45% ad valorem? 47. Prove that 4057 is a prime number. 48. Express 2-Hto" ^^^ tA" ^^ decimals, and divide the first result by the second. 49. How much will it cost to cover a floor 20 ft. 3 in. long and 15 ft. 7 in. wide, with carpeting 28 in. wide, at $1.16 a yard, if the strips run lengthwise of the floor? How much if the strips run across the room ? 50. What number is that ^ of |^ of which exceeds ||- of If of it by 1^9^? 51. A broker receives $ 1000 to invest, after deducting a brokerage of 1J%. What sum can he invest, and what is the amount of his commission ? 52. Divide 1081 lb. 1 oz. 13 pwt. 9 gr. by 2 lb. oz. 16 pwt. 3 gr. 53. At what rate per cent will $ 540 amount to $ 732.39 in 6 y. 5 mo. 22 d. ? 54. Eeduce ^^Vtt *^ ^^^ lowest terms. 55. Eind the cost of a pile of wood 29 ft. 7 in. long, 5 ft. 3 in. high, and 4 ft. wide, at f 7.16|- a cord. 56. Simplify ( 1.37 5 + IJ x -^^ + -^V ^M^ ^^^ express \ .028 J the result as a decimal. MISCELLANEOUS EXAMPLES. 307 57. If a bushel of wheat weighs 59 lb., wh^^t is the value of nine carloads of wheat, each weighing 8 T. 17 cwt., at $1.03| a bushel? 58. Simplify (8^ - 4f3 ) _ (76 _ 5f ) . 59. Extract the cube root of 456.266246971625. 60. A bankrupt owes to A, $398.75; to B, f 508.75; to C, $316.25; and to D, $563.75. If his resources are $ 1115.40, what is each creditor's share ? 61. Find the G. C. D. of 7429, 11339, and 12673. 62. The hypotenuse of a right triangle is 28J ft., and one of the sides about the right angle is IJ yd. ; find the other side in inches. 63. If a tank whose length is 6 ft. 5 in. contains 578f| gallons of water, what is the length of a similar tank which contains 2246^^- gallons ? 64. Express 158324 in. in terms of higher denomi- nations. 65. An agent sells 415 yards of woollens, at $ 1.52 a yard, charging 2i% commission. He invests the net proceeds in silks at $1.95 a yard, charging 3f% commission. How many yards can he buy ? 66. A can do a piece of work in 15 days, B in 18 days, C in 21 days, and D in 24 days. How many days will it take all of them together to do the work ? If they receive the sum of $79.95 for the work, how should the money be divided ? 67. Find the cost of a draft on Chicago for $1500; due 60 days after sight, with interest at 3|-%, if exchange is at 1J% discount. 68. If five men can do a piece of work in 4 d. 5 h. 21 min. 45 sec, how long will it take nine men to do the work ? 69. Find the exact interest of $769.50 from June 11, 1891, to March 23, 1892, at ?>\%. 808 ARITHMETIC. 70. Express 0.79839 lb. troy in lower denominations. 71. If $14513.75 is realized from the sale of $17000 worth of bonds, including a brokerage of i%, what is the quoted price of the bonds ? 72. Find the last term and the sum of the terms of the arithmetical progression ^, -^, if, etc., to 63 terms. 73. Keduce -^-f^j to its lowest terms. 74. Find the last term and the sum of the terms of the geometrical progression 1-J, 2|-, 4, etc., to 10 terms. 75. Add 6 sq. mi. 313 A. 152 sq. rd. 21 sq. yd. 8 sq. ft. 137 sq. in., 13 sq. mi. 602 A. 67 sq. rd. 14 sq. yd. 3 sq. ft. 122 sq. in., 34 sq. mi. 447 A. 112 sq. rd. 9 sq. yd. 5 sq. ft. 64 sq. in., and 19 sq. mi. 296 A. 89 sq. rd. 28 sq. yd. 7 sq. ft. 98 sq. in. f no) What sum must be invested in a 6^% stock at 117, brokerage i%, to yield an annual income of $487.50 ? 77. Find the price in pounds, shillings, pence, and far- things, of an article worth $21.84, if the sovereign be worth $4,871 , 78. A sum of money was divided between A, B, C, and D, in such a way that A received -^,^-^,C ^^, and D the remainder, which was $ 35.55. What was the sum divided, and how much did each receive ? 79. If a bushel = 1 J cu. ft., what must be the depth of a bin 5 ft. 4 in. long, and 4 ft. 9 in. wide, to hold 98 bushels of grain? ^ 80. Express f ^ and JJ)_9^ as decimals, and find the prod- uct of the results. 81. Find the proceeds of a note for S 875, dated Nov. 25, 1892, payable 60 days after date, and bearing interest at 4%, if discounted Dec. 11, 1892, at 5i%. MISCELLANEOUS EXAMPLES. 309 82. Find the cost of a draft on Boston for $ 2875, due 90 days after sight, with, interest at 5%, if exchange is at 2|% premium. 83. If a man can do a piece of work in 8|- days, working 10 h. 55 min. a day, how many days will it take him, work- ing 8 h. 36 min. a day ? / 84. At what rate per cent will ^1125 gain $198.75 from Sept. 20, 1885, to June 5, 1890 ? ' 85. A merchant sold goods for $658.35, losing. 21f% of what they cost him. At what price should the- goods have been sold so as to gain 13^% ? 86. What is the duty, at 40% ad valorem, on an importa- tion of crockery, invoiced at £ 896 5s. 6d., if the pound ster- ling be valued at $ 4.8665 ? 87. Express 225 rd. 4 yd. 2 ft. 6f| in. as a fraction of a mile. 88. A gentleman left .3 of his property to his wife, .4 of the remainder to his son, .65 of the remainder to his daughter, and the balance, $ 1543.50, to charitable institu- tions. How much did each receive ? 89. How many shares of stock, at a premium of 13|^%, can be bought for $93854.25, including a brokerage of i% ? 90. What is the equated time of paying $ 519 due in 30 days, $ 348 due in 60 days, $ 497 due in 90 days, and $286 due in 120 days ? 91. Arrange in order of magnitude |-|, -f-f, and i|-|-. 92. How long must $ 926.50 be on interest at 21% to amount to $1200? 93. Divide 35 d. 17 h. 41 min. 59 sec. by 2.84. 94. How many cubic feet are there in a tapering column 17 ft. in height, whose lower base is a square 16 in. on a side, and upper base 14 in. on a side ? 310 ARITHMETIC. 95. Subtract 39if f from 57^^^^, and reduce the result to its lowest terms. 96. Find the cost of 384 boards, each 12 ft. 10 in. long, 7 in. wide, and | in. thick, at 1 18.75 per M. 97. The local time- at two places on the equator differs by 9 h. 37 min. 33 sec. What is the distance between them in miles, if a degree of the equator be taken as 69^ miles ? 98. Find the L. C. M. of 4199, 7429, and 12673. 99. If a cubic foot of water weighs 1000 oz., and a cubic inch of gold weighs ij lb., what is the specific gravity of gold ? 100. Find the cost of 938 shares of railway stock at 125f, brokerage i%. 101. A tank contains 8 cu. yd. 21 cu. ft. 1048 cu. in. of water. If it can be emptied by a pipe in 1 h. 24| min., how many cubic inches pass through the pipe in one second Z 102. Divide 87843 into parts proportional to li, 2i 3J, ^, and 5i. 103. Find the amount and present worth of an annuity of f 540 for 8 years, at 4J% simple interest. 104. Extract the square root of -f^ to six places of deci- mals. ^105. How much above cost must a tradesman ' mark an article costing ^ 24.64, so as to be able to sell it at a dis- count of 8|^% from the list price, and still make a profit of 15f % ? 106. 1^ of 3iff is 1^ times what number ? 107. What is the length of the longest straight line that can be drawn on a square floor whose area is 237 sq. ft 97 sq. in.? MISCELLANEOUS EXAMPLES 311 108. Express 83|f|- lb. troy in grains. 109. What will be the face of a draft, due 30 days after sight, with interest at 41%, which can be bought for $1000, when exchange is at a discount of 3^% ? iin a- ^^f 5.3-2.45 + 7.83 ^^ 110. Simplify 1.613 -- 6.05 ^ 111. Subtract 7 times £ 136 19s. lOd 1 far. from 12 times £ 87 6s. 3c?. 21 far. 112. What is the average time of paying $ 186 due March 12, $ 155.25 due April 7, $ 414 due May 29, and $ 258.75 due July 20 ? •f(il^ What principal at 5% interest will amount to $ 600 iiriry. 11 mo. 5 d.? 114. If 18|% is gained by selling goods for $126.54, what per cent would be lost by selling them for $ 76.96 ? 115. What must be the face of a note due 90 days hence, which, when discounted at 51%, will yield $850 ? 116. Express 3 lb. 9 oz. 17 pwt. 14 gr. in avoirdupois weight. ^ 117. Simplify 1?-— . 9+ — ^13 118. What is the compound interest of $1260 for 9 mo. at 7%, interest being compounded quarterly ? 119. What is the face of a sight draft which can be bought for $869.35, when exchange is at a premium of |%? 120. If a tank 6 ft. 9 in. long, 3 ft. 4 in. wide, and 2 ft. 2 in. deep, holds 365f gallons of water, how deep must a tank be that is 5 ft. 10 in. long, and 3 ft. 9 in. wide, to hold 437^- gallons of water ? 312 AHITlIiMETlC. .095 -.0005 3.42 -.006 121. Simplify .0087 - .3 .000812 - .04 122. I bought 5 A. 136 sq. r.d. of land at the rate of $ 1200 an acre, and sold it at a profit of 21%. At what price per square foot did I sell it ? 123. Find the G. C. D. of 62^2., 55ff, 47|-ff, and 633%. 124. A town whose taxable property is valued at $ 975400, wishes to raise f 14130.25. There are 479 polls, each paying $1.50. What is the rate of taxation? What tax will be paid by an individual who i)ays for 3 polls, and has taxable property to the amount of $ 8600 ? 125. If a bushel contains 2150.42 cu. in., what must be the depth of a cylindrical measure 12 in. in diameter, to hold a peck ? 126. What principal, at 6% compound interest, will gain $325 in 2 y. 2 mo., interest, being compounded semi- annually ? 127. Multiply together 2^«, 2^\, 2J«|, and 2J||. 128. Express -^-^ sq. mi. in lower denominations. 129. Find the cost of a draft on Paris for 2016.80 francs, exchange on France being quoted at 5.17^^. 130. If the weight of a cubic foot of water is 62i lb., how many cubic inches of mercury (specific gravity 13.596) does it take to weigh 103 lb.? 131. Express .07593 cd. in cubic inches. 132. Extract the cube root of j^ to four places of deci- mals. 133. A triangular plot of ground contains 2.875 A. If its altitude is 25^ rods, what is its base in feet ? 134. Express 11 oz. 13 dr. as a decimal of a pound troy. 135. Express 5 lb. 13 oz. 7 dr. in troy weight. MISCELLANEOUS EXAMPLES. 313 136. What principal at 1\% interest will gain $52.75 from Feb. 8, 1888, to Jan. 4, 1892 ? y 137. What common fraction will produce .38076923 ? 138. Express 9 cu. ft. 771.3792 cu. in. as a decimal of a cord. 139. How many spherical bullets, each \ in. in diameter, can be formed from five pieces of lead, each in the form of a cone, whose altitude is 21 in., and radius of base 4 in. ? 140. Find the L. C. M. of ^-^^Q^, J?//^, -fi^^^ and \\^. 141. A provision dealer uses a false measure of 3 pk. 7 qt. instead of a bushel. What per cent does he gain by his dishonesty ? What per cent do his customers lose ? 142. How much will be realized from the sale of 354 shares of mining stock at a discount of 27-|-%, brokerage 143. A man travelled 69f miles the first day, and on each succeeding day three-fifths as many miles as on the next preceding. How far did he travel on the 8th day ? How far did he travel in 8 days ? 144. If 69.84 pounds of cofPee can be bought for % 30.55^, how many pounds can be bought for % 76.89|- ? 145. A merchant owes $2143.50 due in 10 months. If he pays $425 in 3 months, $580 in 5 months, and $278.50 in 8 months, when should he pay the balance ? 146. If ^ of the price received for an article is gain, what is the gain per cent ? 147. Multiply 11 T. 17 cwt. 81 lb. 9 oz. 13 dr. by ||. 148. If a gallon contains 231 cu. in., what must be the diameter of a cistern whose depth is 9 ft., to hold 1080 gallons ? 149. Find the cost of a bill of exchange on Liverpool for £ 224 16s., when exchange on England is quoted at 4.86}. 314 ARITHMETIC. 150. If the pint liquid measure contains 28.875 cu. in., how many cubic feet are there in a barrel of 31|^ gallons ? - 151. Extract the sixth root of 177210755.074809. 152. A circular garden, 85 ft. in diameter, is surrounded by a walk 6 ft. wide. How many square yards are there in the walk ? 153. If 30 men can dig a trench 108 feet long, 8| feet wide, and 9 feet deep, in 10|^ days of 6| hours each, how many days of 8 hours each will it take 24 men to dig a trench 96 feet long, 12|- feet wide, and 12 feet deep ? 154. For what amount must a vessel worth $ 12325, and her cargo worth $ 8709.36, be insured at 4J%, in order that, in case of loss, the owner may recover the value of the vessel and cargo, and the premium ? 155. A, B, and C formed a partnership for one year. A put in $-925, and at the end of 5 months added $ 250 ; B put in $ 1075, and at the end of 10 months withdrew $ 325 ; G put in ^ 1250, and at the end of 8 months withdrew $ 475. They gained $ 1337. What was each partner's share of the gain ? 156. What annuity to continue for 13 years, at 3J% simple interest, can be purchased for f 6292 ? 157. What amount is due April 2, 1891, on a note for $458, dated April 24, 1886, with 4^% interest payable annually, on which no payments have been made ? 158. Find the lower base in rods of a trapezoid whose area is 66885 sq. in., altitude 16^^ ft., and upper base 7| yd. 159. Express 2 sq mi. in square inches. ' 160. Simplify 7^y;^_ . X Jf^ - H- (^1 g- — ^5 ; 3- oy g — y g- 161. Add together iff, ifi, m, and ^Vo^^ and reduce the result to its lowest terms. MISCELLANEOUS EXAMPLES. 315 162. At what per cent below par must a 4J% stock be quoted, to yield the same per cent on the investment as a 5|% stock at a premium of 23|^%, brokerage in each case |-% ? 163. Find the amount and present worth of an annuity of $500 for 4 years, at 5% compound interest. / 164. Simplify '-r-. — '-^ + '- '—t-., and reduce the result .69 + .2 .5 -.246 to a mixed number. 165. A grocer bought 36 bu. pk. 3 qt. of nuts, at $ 3.20 a bushel, and sold them at 12 cents a liquid quart. If a quart liquid measure contains 57f cu. in., and a quart dry measure 67^ cu. in., did he gain or lose, and how much ? 166. Express 6 sq. rd. 19 sq. yd. 3 sq. ft. 57 sq. in. as a decimal of 9 sq. rd. 7 sq yd. 2 sq. ft. 12 sq. in. 167. Simplify (if of 18f ) + 8f - 4^ ^ ^ 4i-(344^14A)+U 168. If the specific gravity of lead is 11.4, and a cubic foot of water weighs 1000 oz., find the weight in pounds of a sphere of lead 11 in. in diameter. 169. What amount is due Oct. 7, 1892, on a note for $2725, dated Nov. 18, 1891, and bearing interest at 4%, on which the following payments have been made: Jan. 11, 1892, $520; April 17, 1892, $790; June 25, 1892, $655; and Aug. 12, 1892, $480? 170. A man sells 4i% stock to the par value of $65400 at 92 J, and invests tlie proceeds in 3J% stock at 81|, brok- erage i% on each transaction. Is his income increased or diminished, and how much ? -- 171. A tapering hollow iron column, one inch thick, is 24 ft. long, 10. in. in outside diameter at the larger end, and 8 in. in diameter at the smaller. Find its weight, if a cubic inch of the metal weighs .27 lb. 316 ARITHMETIC. . 172. Simplify |^ + ^-li,-(2.,.2H). 173. A cistern oa-n be filled by three pipes in 9 h. 20 mi n., 8 h. 45 min., and 12 h. 36 min., respectively. How many hours and minutes will it take to fill the cistern if all the pipes are opened together ? 174. A railway embankment, 226 rods in length, is 8 ft. 6 in. wide at the top, 21 ft. 6 in. wide at the bottom, and 6 ft. 4 in. high. How many cubic yards of earthwork does it contain ? 175. On a note for $ 1000, dated Feb. 24, 1887, and bear- ing interest at 6%, the following indorsements were made: Jan. 5, 1888, $ 175 ; May 16, 1889, $ 30 ; Dec. 8, 1889, $ 250; Aug. 28, 1890, $ 200. How much was due March 19, 1891 ? 176. If a rail weighs 75 pounds to the yard, how many tons of rails will be required to lay a piece of double track railway 11 mi. 272 rd. in length ? 177. A train leaves A for B, 44 miles distant, travelling at the rate of a chain in 1-J- seconds. Eighteen minutes later a train leaves B for A, travelling at the rate of | of a chain a second. How many miles from B will they meet ? ^178. Simplify (M ^ /A) + (5|f-f- l^V). (6Hx3A)-(4Hx 2f) 179. A room 22 ft. 4 in. long, 15 ft. 9 in. wide, and 9 ft. 6 in. high, has three doors, each 3 ft. wide and 6 ft. 9 in. high, four windows, each 3 ft. wide and 5 ft. 5^ in. high, and is surrounded by a base-board 9 in. wide. How much will it cost to plaster it, at 42 cents a square yard ? How much will it cost to paper it with paper 21 in. wide, 10 yards to a roll, at $1.12 a roll? 180. What annuity to continue for 5 years, at 3% com- pound interest, can be purchased for $ 1000 ? MISCELLANEOUS EXAMPLES. 317 181. Find the equated time for. paying the balance of the following account : Dr. William Lewis. • Cr. 1889 1889 Oct. 29 To Mdse. 30 d. $500 Dec. 20 By Cash. $750 Nov. 24 1.1, It 250 1890 1890 Jan. 19 " Draft, 90 d. 600 Jan. 6 " 2 mo. 600 Feb. 9 " " 1 mo. 450 Feb. 27 " " GO d. 1000 Mar. 28 " Cash. 200 MISCELLANEOUS EXAMPLES INVOLVING THE METRIC SYSTEM. 378. 1. Express .00527*^^ of water in cubic centimeters, and find its weight in dekagrams. 2. Divide .0072321273'="'^'" by 7.489, and express the re- sult as a decimal of a cubic millimeter. 3. Find the altitude in dekameters of a trapezoid whose area is 149878^'^'='", and bases .05787""' and 29.65*^"', re- spectively. 4. A wood-pile is 32 ft. long, 4 ft. wide, and 5 ft. 6 in. high. How much is it worth, at $ 2.50 a ster ? 5. If a train travels at the rate of 36 miles an hour, what is its rate in hektometers a minute ? 6. Find the weight in dekagrams of a ream of paper, each sheet 24*='" long and 15'='" wide, if a sheet of the same thickness 125""™ long and 8"" wide weighs 8.2*^«. 7. Express 2 mi. 223 rd. 4 yd. 1 ft. 9 in. in kilometers. 8. How much do I lose by buying 5 A. 135 sq. rd. of land at $ 300 an acre, and selling it at $ 675 a hektar ? ^ 9. Express 2 gal. 3 qt. 1 pt. 3 gi. in deciliters. 10. How many chains are there in a hektometer ? 818 ARITHMETIC. 11. How mauy decimeters are there in a link ? 12. A rectangular garden, 35.4'" long and .289"'" wide, is surrounded by a wQ,lk 19.6*^'" in width. Find the area of the walk in ars. 13. Express .874^^ in liquid measure, and also in dry measure. 14. If it costs $ 9.54 to trc.vel 397^ miles by rail, what is the rate of fare in cents per kiiometer ? 15. , How many hektoliters of grain can be put into a receptacle 4.38'" long, 27.9^"^ wide, and 185"" deep ? How many myriagrams of water can be put into it ? 16. How many bricks, each 2.3*^"' long, 86'"™ wide, and 5.04*="" thick, will be required to build a wall .645"™ long, .46™ wide, and 1.89^™ high ? 17. Find the value, at 12 cents a square meter, of a circu- lar piece of land, whose circumference is one hektometer. 18. How many silver half-dollars can be coined from ten bars of silver, each 55*='" long, 36'"™ wide, and 25""" thick, if each coin weighs 12.5^, and the specific gravity of silver is 10.5 ? 19. A merchant imports 3500™ of silk, invoiced at 6.5 francs a meter, and sells it at $ 1.55 a yard. If the franc is valued at 19.3 cents, how much does he gain ? 20. If a cubic foot of granite weighs 167 lb., what is the weight of a cubic meter in metric tons ? 21. How much will it cost to cover a floor 6.68™ long and 5.84™ wide, with carpeting 78*=™ wide, at -75 cents a meter, if the strips run lengthwise of the room ? How much if the strips run across the room ? 22. Find the length in hektometers of the longest straight line that can be drawn in a rectangular field, whose area is 14.44908^'^^™, and width .02776^'". MISCELLANEOUS EXAMPLES. 319 » 23. Find tlie volume in cubic decimeters of a frustum of a square pyramid, whose altitude is 825""", lower base .28™ on a side, and upper base 22. 6*"™ on a side. 24. How many yards of fence will be required to enclose a circular grass-plot whose area is 6.157336* ? 25. A wood-pile is 8.2™ long, 14*^ wide, and 165'"" high, rind its value at f 6.70 a cord. ' 26. A cubical tank holds 563975.2°^ of water ; what is its depth in dekameters ? 27. How many dekaliters of petroleum (specific gravity .8778) does it take to weigh 84663.81"^ ? 28. What is the depth in decimeters of a cylindrical tank, 83.5'^'" m diameter, which holds 9856809.27*^- of water ? 29. Express 9.38^*^ in ordinary square measure. 30. A ditch is 142*^'" long, 18.3'^"^ wide, and 876'"'" deep. How many metric tons of water will it contain ? 31. A rod of steel (specific gravity 7.8) is 5.5'" long, and 18™"* in diameter. Find its weight in hektograms. 32. Express 15 lb. 9 oz. 17 pwt. 21 gr. in dekagrams. 33. A tank is 2.28™ long, 1.75'" wide, and 1.625™ deep. How many kilograms of oil (specific gravity .898) will it contain ? 34. What is the surface in square decimeters of a sphere whose volume is 2572446.8^"™™ ? 35. A cubical piece of lead, whose entire surface is ;[3 5sqdm^ is melted, and formed into a cone the radius of whose base is 15*=™. Find the altitude of the cone in meters. 36. What is the diameter in dekameters of a cylindrical cistern 32^™ deep, which holds 769.692^' of water? 37. The volume of a cone is 33250.6944^"''™, and its alti- tude is 7.2*^™. Find the radius of its base in meters, and its lateral area in square dekameters. 320 ARITHMETIC. 38. What is the depth in millimeters of a tank that is 188'='" long and 12.5'^'" wide, and holds 2.5949546'^ of alcohol (specific gravity .791)? 39. How many cubic centimeters of metal are there in a hollow iron tube of uniform diameter, whose length is 3.2™, thickness 2"'°, and outside diameter 1.2*=™ ? 40. A bar of aluminum, '9.2*^'" long and 2.5'"^ in diameter, weighs 11.6062485°^. What is its specific gravity ? 41. Express .378^^ in avoirdupois, and in troy weight. 42. If a cubic decimeter of silver weighs 104.93°^, what is the weight of a cubic foot in pounds avoirdupois ? 43. A river, whose current flows at the rate of 6.4^™ an hour, is .16°™ deep and 84.6°™ wide. How many metric tons of water pass a given point in 43 min. 20 sec. ? 44. If a rail weighs 36^^ to the meter, how many pounds does it weigh to the yard ? 45. A tank containing 20457.2°^ of water has two taps. One fills it at the rate of 342'="'=™ a second, and the other empties the contents at the rate of a dekaliter in .9765625 minutes. How many hours and minutes will it take to fill the tank if both taps are opened ? 46. The cross-section of a tunnel 2.5^™ in length, is in the form of a rectangle 4.2™ wide and 3.8™ high, surmounted by a semicircle, whose diameter is equal to the width of the rectangle. How much did it cost to excavate it, at $ 6 a cubic meter ? 47. A cannon is in the form of a frustum of a cone, joined to a hemisphere whose diameter is equal to that of the larger end of the cannon. The diameter of the larger end of the cannon is 4.2^™, and of the smaller end 32*'™ ; and its entire length is 2.1™. The interior bore of the cannon is 24em ^j^ diameter, and 2™ deep. How many cubic decimeters of metal were used in its construction ? APPENDIX. 321 APPENDIX. Measures of Length. A furloDg A league A fathom MEASURES. = 40 rods. = 3 miles. = 6 feet. A cable-length = 120 fathoms. A 7iautical mile, geographical mile, or k7iot, is equal to 1.15 miles. A marine league is equal to 3 nautical miles. A line = yL i^c^- A hand = 4 inches. A palm = 3 inches. A span = 9 inches. A cubit = 18 inches. Cloth Measure. 2\ inches = 1 nail. 4 nails = 1 quarter of a yard. 4 quarters = 1 yard. 3 quarters = 1 Ell Flemish. 5 quai'ters = 1 Ell English. Measures of Area. A rood = 40 square rods. A square = 100 square feet. Measures of Volume. A cord foot = 16 cubic feet. A perch = 24f cubic feet. 322 ARITHMETIC.' A perch of masonry is 16^ ft., or 1 rod long, IJ feet wide, and 1 foot high. Measures of Weight. A quarter = 25 pounds av„ A stone = 14 pounds av. A pig = 21-1- stone. A fother = 8 pigs. Diamond Weight. 4 quarters = 1 grain. 4 grains = 1 carat. A grain, diamond weight, is equal to four-lifths of a grain troy, and a carat is equal to 3|- grains troy. Measures of Time. A solar year is the time required for the sun, after leav- ing either equinox, to return to it again ; and is 365 days, 5 hours, 48 minutes, and 49.7 seconds. A sidereal year is the time required for the earth to make a complete revolution about the sun; and is 365 days, 6 hours, 9 minutes, and 9.6 seconds. A common, civil, or legal year is one of 365 days ; and a leap, or bissextile year is one of 366 days. By the Julian Calendar, instituted by Julius Caesar, every fourth year is made a leap year ; which produces an error, as compared with the solar year, of 44 minutes and 41.2 seconds in four years, or about one day in 129 years. By the Gregorian Calendar, instituted by Pope Gregory XIII., in 1582, those years only are leap years whose num- bers are divisible by 4, and not by 100, unless they are also divisible by 400. The error of the Gregorian Calendar amounts to about one day in 3866 years. APPENDIX. 323 The reckoning of time by the Julian Calendar is termed Old Style, and by the Gregorian Calendar New Style. The latter is used by most civilized nations, and was adopted by England in 1752 ; in that year, by Act of Par- liament, the day following Sept. 2d was counted as Sept. 14. The present difference between the two styles is 12 days. To find the Difference in Time between Two Dates. The following method is largely used by business men : The year is regarded as consisting of 12 months of 30 days each. The months are represented by their numbers; thus, Jan- uary is the first month, February the second, etc. The difference in time is then found as in subtraction of denominate numbers. Example. Find the difference in time between Oct. 15, 1883, and June 7, 1892. 1892 6 7 June is the sixth month of the 188S 10 15 year, and October the tenth month. — 15 d. from 37 d. leaves 22 d. 18 y. 7 mo. 22 d., Ans. 10 mo. from 17 mo. leaves 7 mo. 1883 from 1891 leaves 8 y. Then the required result is 8 y, 7 mo. 22 d. In some States, the actual " umber of days in the preceding month must be used, when the number of days in the sub- trahend time is greater than the number in the minuend time. Thus, in the above example, the month preceding June has 31 days. AVe should then say, 15 d. from 38 d. leaves 23 d. ; and the time would be 8 y. 7 mo. 23 d. Comparison of Thermometers. In the Fahrenheit Scale (F.), the temperature of the freez- ing point of water is marked 32°, and the temperature of the boiling point 212° ; the difference being 180°. 324 ARITHMETIC. In the Centigrade Scale (C), the freezing point is marked 0°, and the boiling x)oiiit 100°. In the Reaumur Scale (R.), the freezing point is marked 0°, and the boiling point 80°. Thus, 1° F. = ; 0, or 1° C, and j\%, or 4° R. r C. = -;-|0-, or f F, and 3% or f R- r R. = \%o_, or 1° F., and i^%o, or |° C. 1. Express 59° F. in the Centigrade scale, and in the Reaumur scale. 59° F. is 27° above the freezing point. But 27° F. is equal to 27 x |, or 15° C, or to 27 x ^, or 12° R. Hence, 59° F. is equivalent to 15° C, or to 12° R., Aiis. 2. Express 35° C. in the Fahrenheit scale, and in the Reaumur scale. 35° C. is 35° above the freezing point. But 35° C. is equal to 35 x |, or 63° F., or to 35 x |, or 28° R. Hence, 35° C. is equivalent to 95° F., or to 28° R., A71S. 3. Express — 18° R. in the Fahrenheit scale, and in the Centigrade scale. — 18° R. is 18° below the freezing point. But 18° R. is equal to 18 x f, or 40p F., or to 18 x f , or 22^° C. Hence, - 18° R. is equivalent to - 8|° F., or to - 22^° C, A71S. EXAMPLES. Express each of the following in the Centigrade, and in the Reaumur scale : 4. 77° F. 5. 140° F. 6. 8° F. 7. -34°F. Express each of the following in the Fahrenheit, and in the Reaumur scale : 8. 55° C. 9. 70° C. 10. -12°. 11. -25°C. Express each of the following in the Fahrenheit, and in the Centigrade scale : 12. 52° R. 13. 25° R. 14. - 10° R. 15. - 22° R. APPENDIX. 325 Miscellaneous Terms. A sheet of paper folded into : 2 leaves, forms d^ folio ; 4 leaves, forms a quarto, or 4to ; 8 leaves, forms an octavo, or 8vo ; 12 leaves, forms a duodecimo, or 12mo; 18 leaves, forms an 18mo j 24 leaves, forms a 24mo. MONEY AND COINS. United States Money. For the denominations of United States money, see Art. 142. The coins of the United States are as follows : Gold; the quarter-eagle, half-eagle, eagle, and double-eagle. Silver; the dime, quarter-dollar, half-dollar, and dollar. Nickel ; the five-cent piece. Bronze ; the ceyit. Note. The coinage of gold dollars, gold three-dollar pieces, and nickel three-cent pieces, was suspended by act of Congress, approved Sept. 26, 1890. The monetary system of Canada is the same as that of the United* States. English Money. For the denominations of English money, see Art. 154. The coins of Great Britain are : Gold ; the half-sovereign and sovereign. Silver; the three-penny piece, six-pence, shilling, florin, half-crown, and crown. Copper ; the halfpenny and penny. Note. The silver four-penny piece is no longer coined. 326 ARITHMETIC. The value of the pound sterling in United States money is ^4.8665. French Money. 100 centimes (c.)= 1 franc. (/) The coins of France are : Gold; the Jive-franc, ten-franc, twenty-franc, forty-franc, and hundred-franc pieces. Silver; th^ franc, two-franc, and Jive-franc pieces, and the tiuenty-Jive centime awd fifty-centime pieces. Bronze ; the one-centime, two-centime, five-centime, and ten- centime pieces. Note. The gold hundred-franc piece is called a Napoleon. The monetary systems of Belgium and Switzerland are the same as that of France. The value of the franc in United States money is 19.3 cents. German Money. 100 pfennigs (p/.) = lmark. (m.) The coins of Germany are : Gold ; the five-mark, ten-mark, and twenty-mark pieces. Silver ; the one-mark, two-mark, and three-mark pieces ; and the twenty-pfennig Sindfifty-2)fennig pieces. Nickel; the five-pfennig and ten-pfennig pieces. Bronze ; the one-pfennig and tico-pfe7inig pieces. Note. The silver three-mark piece is called a Thaler. The value of the mark in United States money is 23.8 cents. The following table gives the values, in United States money, of Foreign Coins, as proclaimed by the Secretary of the Treasury, Oct. 1, 1891 : APPENDIX. 827 Country. Monetary Unit. Value in U. S. Money. Argentine Republic. Peso. ^0.965. Austria- Hungary. Florin. .357. Belgium. Franc. .193. Bolivia. Boliviano. .723. Brazil. Milreis. .546. British Possessions, N. A., except Newfoundland. Dollar. 1.00. Central American States ; Costa Rica, Guatemala, Honduras, Nicaragua, Salvador. Peso. .723. Chili. Peso. .912. China. f Tael, Shanghai. \ " Haikwan (customs). 1.068. 1.189. Colombia. Peso. .723. Cuba. Peso. .926. Denmark. Crown. .268. Ecuador. Sucre. .723. Egypt. Pound (100 piasters). 4.943. Finland. Mark. .193. France. Franc. .193. German Empire. Mark. .238. Great Britain. Pound sterling. 4.8665. Greece. Drachma. .193. Hayti. Gourde. .965. India. Rupee. .343. Italy. Lira. .193. Japan. r Yen (gold). 1 " (silver). .997. .779. Liberia. Dollar. 1.00. Mexico. Dollar. .785. Netherlands. Florin. .402. Newfoundland. Dollar. 1.014. Norway. Crown. .268. Peru. Sol. .723. Portugal. Milreis. 1.08. Russia. Rouble. .578. Spain. Peseta. .193. Sweden. Crown. .268. Switzerland. Franc. .193. Tripoli. Mahbub of 20 piasters. .652. Turkey. Piaster. .044. Venezuela. Bolivar. .145. Note. The francs of Belgium, France, and Switzerland, the mark of Finland, the drachma of Greece, the lira of Italy, and the peseta of Spain, have all the same value. 328 ARITHMETIC. The crowns of Denmark, Norway, and Sweden have all the same value. The boliviano of Bolivia, the sucre of Ecuador, the sol of Peru, and thejpesos of Costa Rica, Guatemala, Honduras, Nicaragua, Salvador, and Colombia, have all the same value. LEGAL RATES OF INTEREST. The Legal Rate of interest is the rate which is established by law. The following table gives the legal rate of interest in each state and territory of the Union. When no rate is mentioned, the legal rate is that given in the left-hand column; if specified in writing, any rate not exceeding that in the right-hand column is legal. State. Rate. State. Rate. State. Rate, Alabama. 8 8 Kentucky. 6 6 Nevada. 7 Any. Arkansas. 6 10 Louisiana. 5 8 Ohio. 6 8 Arizona. 10 Any. Maine. 6 Any. Oregon. 10 12 California. 10 Any. Maryland. 6 6 Pennsylvania. 6 6 Connecticut. 6 6 Massachusetts. 6 Any. Rhode Island. 6 Any. Colorado. 8 Any. Michigan. 7 10 South Carolina. 7 8 Dakota. 7 12 Minnesota. 7 10 Tennessee. 6 6 Delaware. 6 6 Mississippi. 6 10 Texas. 6 10 Florida. 8 10 Missouri. 6 8 Utah. 10 Any. Georgia. 7 8 Montana. 7 Any. Vermont. 6 6 Idaho. 10 18 N. Hampshire. 6 6 Virginia. 6 6 Illinois. 7 7 New Jersey. 6 6 West Virginia. 6 8 Indian Ter. 6 Any. New Mexico. 6 Any. Washington. 8 Any. Indiana. 6 8 New York. 6 6 Wisconsin. 6 10 Iowa. 6 8 North Carolina. 6 8 Wyoming. 12 Any. Kansas. 6 8 Nebraska. 7 10 Dist. ofCol. 6 10 SPECIAL STATE RULES FOR PARTLAL PAYMENTS. The Connecticut Rule. When at least one yearns interest has accrued at the time of a payment, and in the case of the last payment, follow the United States Rule (Art. 324). APPENDIX. 329 When less than a year's interest has accrued at the time of a payment, except the last, find the amount of the principal for an entire year, and the amount of the payment for the re- mainder of the year after it is made, and subtract the amount of the payment from the amount of the principal for a new principal; but if the payment is less thaii the interest which is due at the time that it is made, no interest is allowed on the payment. Example. A note for ^ 2000, dated Oct. 8, 1888, and bearing interest at 6%, had the following indorsements : March 2, 1889, $500; Aug. 18, 1890, $20. What was due Dec. 23, 1890 ? Solution. Principal, $2000.00 Int. for 1 yr.. 120.00 Amount, Oct. 8, 1889, $2120.00 Amount of 1st payment to Oct. 8, 1889, 7 mo. 6 d., 518.00 New Principal, Oct. 8, 1889, $1602.00 Int. for 1 yr.. 96.12 Amount, Oct. 8, 1890, $1698.12 2d payment. 20.00 New principal, Oct. 8, 1890, $1678.12 Int. to Dec. 23, 1890, 2 mo. 15 d.. 20.98 Amount due, Dec. 23, 1890, $1699.10 In the above example, the first payment is made less than a year after the date of the note. We find the amount of the principal for an entire year to be $ 2120. The remainder of the year after the first payment is made is 7 mo. 6 d. ; and the amount of the first payment for this time is $ 518. Subtracting $ 518 from $ 2120, the new principal is $ 1602. The amount of this principal for one year is $ 1698.12. The second payment, being less than the interest which is due at the time that it is made, draws no interest ; then subtracting $ 20 from $ 1698.12, the new principal is $ 1678.12. The amount of this principal, to Dec. 23, 1890, is $ 1699.10. 330 ARITHMETIC. The New Hampshire Rule for Partial Payments on a Note, or other Obligation, drawing Annual Interest. If in any year, reckoning from the time when the annual interest began to accrue^ payments have been made, compute interest on them to the end of the year. Find also the accrued annual interest on the principal, and any simple interest which may be due upon . unpaid annual interest^ at the end of the year. Then the amount of the payment, or payments, is subtracted from the amount due on the note at the end of the year. But if the payment, or payments, are less than the sum of the simple and accrued annual interests due at the end of the year, no interest is allowed on the payments. In such a case, simple interest is computed on the balance of interest due, unless the payment, or payments, are less thayi the simple interest due on unpaid annual interest, in which case the balance of simple interest draws no interest. Example. A note for $ 2500, dated July 10, 1887, and bearing interest at 6%, had the following indorsements: April 4, 1889, $ 600 ; May 26, 1890, $ 100. What was due Nov. 25, 1891 ? Solution. Principal, f 2500.00 1st Ann. Int., to July 10, 1888, 150.00 2d Ann. Int., to July 10, 1889, 150.00 Int. on 1st Ann. Int. to July 10, 1889, 9.00 Amount due, July 10, 1889, 1st payment, April 4, 1889, Int. on 1st payment to July 10, 1889, New principal, July 10, 1889, 3d Ann. Int., to July 10, 1890, 2d payment, Bal. of Int. due July 10, 1890, ^31.96 $ 600.00 9.60 $2809.00 609.60 $ 2199.40 131.96 100.00 APPENDIX. • 331 Bal. of Int. due July 10, 1890, $ 31.96 Int. to Nov. 25, 1891, 2.64 New principal, July 10, 1890, 2199.40 4th Ann. Int., to July 10, 1891, 131.96 Int. on principal from July 10, 1891, to Nov. 25, 1891, 49.49 Int. on 4th Ann. Int., to Nov. 25, 1891, 2.97 Amount due, Nov. 25, 1891, f 2418.42 In the above example, the first payment is made during the second year after the date of the note. The accrued annual interest on the principal, at the end of the second year, is $ 300 ; and the simple interest due upon the unpaid annual interest of the first year is $ 9. Hence, the amount due on the note, at the end of the second year, is $2809. The first payment is made 3 mo. 6 d. before the end of the second year. The amount of $600, for 3 mo. 6 d., is $609.60. Subtracting this from $ 2809, the new principal at the end of the second year is $ 2199.40. The second payment is made during the third year after the date of the note. The annual interest due on the principal at the end of the third year is $131.96. f The second payment being less than this, draws no interest ; then subtracting .$100 from $ 131.96, the balance of interest due is $31.96. Simple interest is then reckoned on this balance to Nov. 25, 1891, amounting to $2.64. The annual interest due on the principal at the end of the fourth year is $131.96; and the simple interest due on this, Nov. 25, 1891, is $2.97. The interest due on the principal from July 10, 1891, to Nov. 25, 1891, is $49.49. Adding the last five sums to the new principal, July 10, 1890, the amount due Nov. 25, 1891, is $2418.42. The Vermont Rule. If in any year, reckoning from the time when the annual interest began to accrue, payments have been made, compute interest on them to the end of the year. 332 • arithmj:tic. The amount of the payments is then applied : Firstf to cancel any simple interest which may he due upon unpaid annual interest. Second, to cancel the accrued annual interest. Third, to reduce the priricipal. The Vermont Rule is the same as the first three parar graphs of the New Hampshire Eule. Note. At the option of the teacher, the examples given under the United States Rule (Art. 324) may be performed by the Connecticut Rule, the New Hampshire Rule, or the Vermont Rule. TO COMPUTE INTEREST ON ENGLISH MONEY. To compute interest on English money, reduce the shil- lings, pence, and farthings, if any, to the decimal of a pound, and then proceed as in United States money. The decimal of a. pound in the result should be reduced to shillings, pence, and farthings. 1. Find the interest of £ 83 13s. 9d. for 3 y. 6 mo., at 5%. We have, £83 13s. 9d. = £83.6875. The interest of £83.6875 for 3 y. 6 mo. at 5% is £ 14.6453125. Reducing £.6453125 to lower denominations, the result is 12s. lOd. 3.5 far. Hence, the required interest is £ 14 12s. lOd. 3.5 far. , Ans. EXAMPLES. Find the interest and amount : 2. Of £56 5s. for 3 y. 11 mo., at 3J%. 3. Of £ 31 14s. 6d for 8 mo. 18 d., at 4%. 4. Of £ 27 8s. 3d for 5 mo. 25 d., at 6%. 5. Of £40 19s. 2d. 1 far. for 1 y. 1 mo. 10 d., at 41%. 6. At what rate per cent per annum will £ 36 17s. 4d gain £ 3 9s. Id. 2 far. in 1 y. 6 mo. ? 7. In what time will £ 190 gain £ 12 2s. Sd., at 3% ? 8. What principal will gain 17s. 7d. 2 far. in 1 y. 2 mo. APPENDIX. 333 BUSINESS FORMS. Receipt in Full. ^ 271y%%. Cincinnati, July 3, 1891. Received from Henry Clark two hundred and seventy-one T%r dollars, in full of all demands to date. Edward H. Perry. Receipt on Account. $ 100. Buffalo, Nov. 12, 1890. Received from James E. Hoyt one hundred dollars on account. William G. Faxon. Bank Checks. A Check is a written order addressed to a bank by a per- son having money on deposit, requesting the payment, on presentation, of a certain sum to the person named therein, or his order. Form of a Bank Check. $ 73^. iVew? York, Feb. 21, 1892. The National Park Bank. Pay to J. H. Crocker, or order, seventy-three ^^ dollars. No. 815. W. E. Martin & Co. A Certified Check is one on the face of which the Cashier or Paying Teller of the bank has written the word " Certi- fied," and under it his signature ; the bank in this way guarantees the payment of the check. Form of a CeHiJied Check. j^^. . ,^ Boston, Sept. \7,IS91. The iflfg^iKet J^^onal nJ^ank. Pay to George f^^^i^es^^ ordmfone hundred and fifty three y% dollar^, ft/^ ^>^ \ a /> No. 349. P 1 • yV/ William Breck 334 ARITHMETIC. Certificates of Deposit. A Certificate of Deposit is a statement made by a bank, certifying that the person named therein has deposited in the bank a specified sum of money. It is often used in place of a draft in making a remit- tance. Form of a Certificate of Deposit. f 250i%. Philadelphia, May 4, 1892. The National Bank of Commerce. David A. King has deposited in this hank two hundred and fifty -f^Q dollars, to the credit of himself payable on the return of this certificate, properly indorsed. No. 1047. F. F. Hill, Cashier. SAVINGS BANK ACCOUNTS. A Savings Bank receives small sums of money on deposit, paying interest therefor. Money depositedjon or before certain specified dates draws interest from those dates. Interest is computed either monthly, quarterly, or semi- annually, on the smallest balance that has been on deposit dur- ing the entire term of interest; but no interest is allowed on any sum which is withdrawn, for the time between the date of its withdrawal and the date of the last dividend^ nor is interest allowed on fractional parts of a dollar. If interest is not drawn when due, it is added to the prin- cipal, and draws interest as a new deposit; thus, savings banks pay compound interest. A depositor in a savings bank receives a bank-book, in which all deposits and amounts withdrawn are entered. He may usually withdraw his entire deposit, or any por- tion of it, at any time when he sees fit ; but some banks require a week's notice before paying money to a depositor. APPENDIX. 335 Example. In a certain savings bank, money deposited on or before the first days of January, April, July, and Octo- ber, draws interest from those dates at 4% ; the interest being payable on the above dates. A depositor, whose balance Jan. 1, 1891, was $152.43, deposited on Feb. 3, 1891, $75, on May 20, 1891, $30, and on Aug. 12, 1891, $ 100. He drew on April 27, 1891, $ 46, and on Oct. 7, 1891, $ 151. What was his balance on Jan. 1, 1892 ? Solution. Dates. Deposits. Drafts. Interest. Balances. 1891 Jan. 1 $ 152.43 Feb. 3 $ 75.00 227.43 April 1 $1.52 228.95 " 27 $46.00 182.95 May 20 30.00 212.95 July 1 1.82 214.77 Aug. 12 100.00 314.77 Oct. 1 2.14 316.91 " 7 151.00 165.91 1892 Jan. 1. 1.65 167.56 On making the deposit Feb. 3, the balance becomes $ 227.43. On April 1, interest is paid on the smallest balance that has been on deposit since Jan. 1. No interest being paid on fractional parts of a dollar, the interest due April 1 is 1 % of $ 152, or $ 1.52 ; and the balance becomes $ 228.95. On April 27, .$46 is withdrawn, and on May 20, $30 is deposited, making the balance $212.95. On July 1, interest is paid on the smallest balance that has been on deposit since April 1, which is $ 182.95. 1 % of $ 182 is $ 1.82 ; which makes the balance July 1 $214.77. 336 ARITHMETIC. Aug. 12, $ 100 is deposited, and the balance jDecomes $314.77. On Oct. 1, interest is paid on a balance of $ 214, amounting to $ 2. 14 ; which makes the balance Oct. 1 $316.91. Oct. 7, $151 is withdrawn, leaving $165.91. On Jan. 1, 1892, interest is paid on $165, amoimtingto $1.65 ; and the balance due on that date is $167.56, Ans. SCALES OF NOTATION. In the ordinary method of expressing whole numbers, a figure in any place represents a number ten times as great as if it stood in the next place to the right. This method of representing numbers is called the Com- mon, or Decimal Scale of Notation; and the multiplier 10 is called the Radix. It is possible, however, to represent numbers by taking as a radix any whole number except 1. The following table gives the name and radix of each of the first eleven scales : SCALB. Radix. SCALB. Radix. Scale. Radix. 10 11 12 Binary Ternary Quaternary Quinary 2 3 4 5 Senary Septenary Octary Nonary 6 7 8 9 Decimal Undenary Duodecimal To express a number in any uniform scale, as many dis- tinct symbols are required as there are units in the radix of the given scale. Thus, in the decimal scale, 10 symbols are required; in the binary scale, 2 symbols, and 1 ; in the ternary scale, 3 symbols, 0, 1, and 2 ; the cipher being a symbol in every scale. In the duodecimal scale, 12 symbols are required, and the numbers 10 and 11 are represented by the symbols t and e, respectively. APPENDIX. 837 In the decimal scale, a digit in the second place represents tens ; in the third place, squares of tens ; in the fourth place, cubes of tens ; and so on. Thus, 3548 represents 3x10^ + 5x102 + 4x10 + 8. In like manner, in any scale, a digit in the second place represents so many times the radix ; in the third place, so many times the square of the radix ; and so on. Thus, in the nonary scale, 7524 represents 7x9^ + 5x92 + 2x9 + 4 Example. Write in the senary scale the numbers from 1 to 19 inclusive in the common scale. The symbols used in the senary scale are 0, 1, 2, 3, 4, 5. The numbers from 1 to 5 inclusive are expressed in the same way in each scale. The number 6, being 1 six and no owes, is expressed 10. The number 7, being 1 six and 1 one, is expressed 11 ; etc. Kesult : 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 26, 30, 31. To change from the Decimal to any other Scale. Example. Change 77609 to the duodecimal scale. 12) 77609 The radix of the duodecimal scale is 12. 12)6467 5 Dividing 77609 by 12, the quotient is 6467, 1 o\KQQ 1 i ^^ ^ and the remainder 5. S In That is, 77609 = 6467 x 12 + 5. 1^)44 10 or t Dividing 6467 by 12, the quotient is 538, and ^ ^ the remainder 11. 38^e5 AyiS. That is, 77609 = (538 x 12 + 11) x 12 + 5 ' * = 538 X 122 + 11 X 12 + 5. Dividing 538 by 12, the quotient is 44, and the remainder 10. That is, 77609 = (44 x 12 + 10) x 122 + 11 x 12 + 5 = 44 X 123 + 10 X 122 + 11 X 12 + 5. Dividing 44 by 12, the quotient is 3, and the remainder 8. That is, 77609 = (3 x 12 + 8) X 123 + 10 x 122 + 11 x 12 + 5 = 3 X 12* + 8 X 128 + 10 X 122 + 11 X 12 + 5. Thus, 77609 is expressed by 38«e5 in the duodecimal scale. ANSWERS. Note. In the following collection of answers, all those are omitted which, if given, would destroy the utility of the example. Art. 26. Page 11. 18. 4440. 22. 12. 1. 12. 19. 3304. 23. 1855. 2. 52. 20. 21424. 24. 54. 3. 32. 4. 94. 5. 101. 6. 179. 7. 100. 8. 153. 9. 506. 21. 291. 22. 12154. 23. 3907. 24. 255245. 25. 291146. Art. 50. Page 23. 2. 537. 2. 3. 4. 5. 6. Art. 66. Pages 34, 35. 15. 72. 34. 18. 64. . Art. 39. Page 17. 3. 10503. 7. 225. 1. 834588. 4. 507. 8. 35. 2. 10270000. 5. 983. 9. 12. 3. 3900383. 6. 6039; 107, Rem. 10. 48. 4. 4054799. 8. 5062. 11. 9. 5. 2172500. 9. 42860. 12. 13. 6. 4116328. 10. 84 ; 347206 , Rem. 13. 6. 7. 228958488. 11. 319; 72, Rem. 14. 4. 8. 148732062. 12. 800013. 15. 16. 9. 433403622. 13. t)284. 16. 65. 10. 37709424. 14. 86395. 17. 30. 11. 15716910. 15. 6. 18. 216. 12. 602025849. 16. 48. 19. 75. 13. 229480867. 17. 13. 20. 385. 14. 248644665. 18. 26. 21. 84. 15. 512. 19. 3. 22. 21 feet. 16. 598. 20. 8. 23. 96 ; in first, 5 ; in 17. 1326. 21. 242. second, 7. ACADEMIC ARITHMETIC. 24. 28 inches. 16. 1155. 10. 98||. 25. 24. 17. 2808. 11. 69H. 26. 56 square rods. 18. 690. 12. 65||. 27. 22 feet . 19. 660. 13. 91f|. 20. 2592. 14. 85. Art. 67. Page 37. 21. 8505. 15. 145tf. 3. 43. 22. 17640. 16. 82/^. 4. 23. 23. 42120. 17. mu- 5. 31. 24. 42840. 18. 53ie. 6. 19. 25. 94500. 7. 41. 26. 90. Art. 84, Page 45 8. 43. 27. 240 niinutes. 6. ¥/. 9. 118. 28. $1080. 7. w- 10. 161. 29. 252; A , 28 times ; 8. -w. . . . 11. 37. B, 21 times ; C, 9. 'II-- " ' 12. 295. 18 times. 13. 73. Art. 85. Page 46 14. 83. Art. 74. Page 41. 6. -¥/• 15. 79. 2. 5491. 7. W- 16. 89. 3. 11951. 8. -W- 4. 17081. 9. '-IF- Art. 68. Page 37. 5. 20677. 10. ^iP- 2. 13. 6. 26071. 11. -11^- 3. 17. 7. 6877. 12. "M-- 4. 19. 8. 8303. 13. ^H-- 5. 23. 9. 11.339. 14. 10. 12493. 15. Art. 73. Page 40. 11. 15457. 16. ^W^- 2. 180. 12. 1 10837. 17. 82 • 3. 252. 13. 485377. 9 3 4. 3300. Art. 75. Page 42. Art. 89. ■ Pago 47 5. 432. 2. 549072. 7. 1 6. 330. 3. 3174045. 8. i- 7. 2310. 9. u- 8. 1026. Art. 83. Page 45. 10. tV- 9. 900. 3 3/3- 11. If. 10. 528. 4. 9. 12. t\. 11. 3872. 5. 14f^ 13. ff 12. 5568. 6. 4711. 14. t\- 13. 1296. 7. 35i|. 15. V-. 14. 688. 8. mh 16. f. 15. 1008. 9. 46. 17. i- ANSWERS. Art. 90. Page 48. 17. f , f , ^V- 31- 97|. 2. tV. 18- M, If, f 32. 172/,. 5. 10. Art. 97. Art. 95. Page 51. Pages 54, 55. 6. V. 2. 4f. 5. tV 7. 1. 3. ^3^^. 6. H- 8. 3^^. 4. Mf. 7. f|. 9. V-. ^- tM- 10. if. 10. ^. 6. f§f. 13. If. 11. H- 7- m- 14. M- 15. if. Art. 91. Page 49. Art. 96. 16. ^V 2. / ^. Pages 52, 53. 17. ^5^. 3. If. 3. jif. 18. 6i|. ^^f- 4. if. ■ 4. if) 5. f |. 5. 3f. 20. l^V 6. If. 6. m 21. 7if 7. AV 7. IHH. 22. 4tVV 8. fi. 8. |. 23. 9fi. 9. f|. 9. f|. 24. 6fi. 10. -If. 10. ^%. 25. 5if. 11. t\V 11. Iff. 26. llff. 12. 5/^. 27. 8ff. Art. 94. 13. 7|. 28. 7ff. Pages 50, 51. 14, ^jj^, 29 13_2^ 15. y-. 30. 9/^. 16. If. 31. 12|f. 5. lit, iff, HI . 17. W- 33. i. 6. -W, If, M- 18. If. 34. 4^. 7. fi, H, M- 19. 20^^. 35. 2|f. 8- M» H, If- 20. 27i. 36. 3f. 9- H. If. M- 21. 471. 37. 93V 10. t¥o. t¥o.t¥o't¥o. 22. lW:f. 38. 6jV 11- fi, W, M, U' 23. 21ff. 39. I7/3V 12. tV'o.tVo.tVo.t^^o. 24. 10,V - 40.. IS^Viy. 13. fit, Hh IM, Iff. 25. 1911. 41. 26ff. Iff. , 26. 24ff. 42. 32^\. 14. Vo\S M, in. 27. 20^%%. Uh ^%%- 28. f If . Art. 99. Page 56. 16. y\ is greater than 29. f |f . 7. V-. A. X-Q ACADEMIC ARITHMETIC. 10. V- 16. 2tV 7. Y. 11. V- 16. 3f 8. if. 12. V. 9. if 13. 12|. Art. 104. Page 01. ^^' fl' 14. 49|. 15. 64J,. 6. If. 8. /.. 9. 3^3. 10. ,^7- 11. ¥• 12. H- 11. ^^. 12. If. 16. 480. 13. H- 17. 275|. 14. H. 18. 617i 15. 2%- 19. 561f. 20. 1080. 16. i^. 17. ¥-• 21- 1^1^^ u. %. 22.-2593^ ^g l^,^ Art. 101. Page 58. ^'^' tI(J- 18. |. 19. }t. 20. If. 21. if. 22. f. 6. f. 18. V-. 19. W- 20. /,. 21. if^. 22. ^, 6. V. 23. V/. 7. |. 8. if- 9. M. 24. ^. 25. W. 10 5 23. e%. Art. 107. Page 64. XV. g. 11. V. 12. 1. 24. t¥^. 25. If. 4. if 6. f. 13. \^ 14. H- 15. f. 16. M. 17. %«-. 26. if. 27. If. 28. ff. 29. ff. 30. 2^*. 31. H. 32. ^\. 33. T^^. 34. V-. 36. A. 6. ^^. 7. ff. • 8. ^K 9. /^. 10. f. 18. ^h- 11. -V. 19. A. 20. W- 12. If. 13. ,%. 21. V-- 14. !f. 22. 18. 15. V. 23. *^. 36. H. 37. xV 16. If 17. f?. Art. 103. Page 59. ^®- at- • 11 18. j%. 19. y. 9. 1|. 39. If. 10. A. 20. |. 11. f|. Art. 105. Page 63. 21. f^. 12. 2f. 4. f. 22. If. 13. 3f. 5. If. 23. A. 14. 3H. 6. /t- 24. If ANSWERS. 5 26. J^. 8. If, ih e. 8. $234. 26. J. 9. Uh 9. 2|. 10. 2\l 10. $750. Art. 108. 11. f. 11. Hind-wheel, 560 ; Pagres 64, 65. 12. U. fore-wheel, 600. 3. V-. 13. f. 12. $25000. 4. -%*. 14. 1144. 13. Carriage, $.390 ; 5- If. 16. ft. horse, $234. 6. -V-. 16. 26^^. 14. 2^^^ hours. 7. |. 17. If 15. 3j. 8- tV 18. xVrV 16. 5i|. 9. f. 19. V. 17. $450. 10. U- 20. ff^. 18. 36. 21. Y-. 19. $2535. Art. 109. Page 65. 22. W. 20. 216. 2- tV 23. M. 21. $594. 3. ii. 24. Hf 22. 83/j. 4. M- 25. 9H. 23. 106f feet. 5. A- 26. U. 24. 32 cents. 6. -V- 27. ff. 25. $1200. 7. ^f^. 28. S9,%. 26. $87. 8. AV 29. n, If, If. 27. Second class, 42 ; 9- ^ih- 30. If third, 48 ; fourth, 31. ^\\- 63. Art. 110. Page 66. 32. V^. 28. If acre. 2. -3^-. 33. fi. 29. 5J. 3. 48. 34. 3^3^. 30. 37^ miles. 4. -V- 35. If. 31. $123. 5. -y/-. 36. U. 32. $5400. 6. -W. 37. -Vt^-. 33. 9|. 7. ^fi 38. H. 34. 2f. 8. V/-. 39. -3//. 35. 10. 9. W. 40. ^%. 36. 26. 41. Iff. 37. U- Art. 111. 42. If. 38. 3|. Pages 66-68. 39. 44 days. 1. leofff Art. 112. 40. 11. 2. wf . Pages 69-73. 41. 37f. 4. 104|. 4. 4if. 42. If 5. ^. 5. 5jV 43. 3f. 6. /r. 6. 13^. 44. 32. 7. ^^i^. 7. .$204. 45. 402rV feet. ACADEMIC ARITHMETIC. 46. 163^ minutes; 1 A Art. 119. Art. 127. Page 83. will have gone Pages 76, 77. 7. .102. around 20 times, 2. 27.3709G. 8. 7480. B 21 times"; i and 3. 38.65775. 9. .936. C 25 times. 4. 225.17578. 10. 1.63. 47. 82^^ miles. 5. 11.20697. 11. .897. 48. 3rV days. 6. 5.742231. 12. .00359. 49. -H rod. 7. 82.451502. 13. 62510. 50. M- 8. 901.186486. 14. .3731. 51. 3A. 9. 5341.02012. 15. .00587. 52. 4|. 16. 2.35. 53. 44f feet. Art. 120. Page 77. 17. 45.6. 54. The sum divided 3. 4.8536. 18. 50.83. was 1252 ; A re- 4. .25673. 19. 73.32. ceived .$105, B 6. .01054. 20. .9225. $33|, and C$48. 7. 9.26771. 21. 490300. 55. 10|. 8. .0037881. 22. 6812.5. 56. 3f 1 days ; A per- 10. 2.44671188. 23. .003961. forms f^, B Mr 11. 781.7513. C i|, and D bo' 12. .0765716. Art. 131. Page 84. Art. 121. Page 79. 5. .325. 3. 242.311. 7. .02531. Art. 118. Page 76. 4. .0541926. 11. .000132. 5. .00025842. 1. V- 6. 6.38357. 2. jhs- 7. .00463136. Art. 132. 3. iff^. 8. 62.8245. Pages 85, 86. 4. hU- 9. .0945162. 4. .08. 5. u- 10. .20727042. 5. .6875. 6. ~w/-. 11. .01088352. 8. .01176. 7. 123. 12. 2.615338. 9. .1424. 8. tVV. 13. 13.539. 10. .078125. 10. iV 14. 104.9688. 11. .00672. 11. rfk. 15. 4.9298795. 12. .96875. 12. ^h- 16. .00076075064. 13. .3.3828125. 13. -w- 14. .004672. 15. JTli'S' Art. 125. Page 80. 17. .38462. 16. j\'u' 6. 1219.86. 18. .85185. 17. n- 8. 6530.97. 19. .03123. 18. ■h- 12. 4320.06. 20. .07209. 19. lUh- 15. 1431894.5. 21. 1.58416. 20. w- 16. 143602200. 22. .55008. answp:rs. 7 Art. 136. Page 87. 9. 6.1. Art. 146. 4. .513. 10. 14600. Pages 92, 93. 5. .124. 11. 14. 7. .$226,463. 6. .684. 12. .202. 8. $13.7582. 7. 19.7227. 13. .00228. 9. ^5.785, ' 8. .32074. 14. 15.42. 10. 275.42 c. 9. .46125. 15. 1.652. 11. $141,687. 10. .84653. 16. .0784. 12. .0397377 e. 11. .00692307. 17. 782.88. 17. .127850.2. 12. .6421296. 18. 12.8464. 18. $3869.532. 13. 7.1984126. 19. ■'$89..537022. 20. 2928.076 d. Art. 140. 21. •'5*3.58. Art. 137. Page 88. Pages 89, 90. 22. $0.39. 3. M. 1. 18.305665. 23. 30.8. 4- U- 2. .004486873. 24. .732. 5. tV 6. if 3. .v^. 25. 4.37. 5. .615625. 26. .0688. 7. 2|f. 6. .0136. 8- i|. 7. .28621. Art. 147. 9. tWt- 8. tVo- Pages 94-97. 10. 48ff. 9. .061464585. 3. $21.60. 11. n. 10. .03378. 4. $24.50. 12. t¥(jV 11. 3.3. 5. $11.76. 13. tV^. 12. .0390625. 6. 291. u. laf. 13. 6245.8. t $1.68. 15. m- 14. 28500. 8. $105. 16. 7|tf|. 15. ifT- 9. $728.18. 17. fH. 17. .14087. 10. $12.07^. 18. Ul 18. .000868. 11. $4149.03. 19. tffi 19. .025587. 12. $9.21. 20. Mi 20. .123456790. 13. .$60.48. 21. ^dlh- 21. .09671875. 14. 36, and 21 gallons 22. ^Vj. 22. 90.38. remaining. 23. m- 15. 75. Art. 139. 24. ^'oV 16. $37.50. Pages 88, 89. 25. .0459. 17. 21 f. 4. 21. 26. -i. 18. .$75.55. 6. .262. 27. If. 19. $120.75. 6. .175. 28. H- 20. $6.66|. 7. 410. 29. .006. 21. .0083. 8. .05875. 30. tVV. 22. .$5.25. ACADEMIC ARITHMETIC. 23. A, $7.25 ; 48. A, $12.75; B, $4.64 ; B, $24.75 ; C, $5.51. C, $16.50. 24. $1.44. 49. lA- 25. 4.6903575 miles. 26. $88.75. Art. 161. 27. 1 of $17.67 is the Pages 104, 105. greater by $.62. 5. 6720 pt. 28. $17.46. 6. 6480 pwt. 29. $35.18. 7. 6930 in. 30. 231. 8. 1105920 cu. in. 31. $974.05. 9. 4088 gi. 32. 141. 10. 7476 far. 33. mdays. 11. 916960 dr. 34. $37493.40. 12. 453912 gr. 35. $171.60. 13. 67073 gr. 36. $47. 2r.. 14. 146832^ sq. ft. 37. Wife, $874.80; 15. 31556929.7 sec. son, $546.75 ; 16. 176957 in. daughter, $328.05. 17. 438 gi. 81920 dr. 38. $180.81. 18. 39. 8 hours ; A, $5.00 ; 19. 1518|". B, $3.75 ; 20. 88704 in. C, $2.50. 21. 274909^1 sec. 40. $52.07. 22. 627264 sq. in. 41. $.97. a 23. 787^ far. 42. Elder, $12.60 ; 24. 4672 nv. younger, $9.45. 25. 59405 f gr. 43. 16. 26. 609.92 pt. 44. $1113.21. 27. 338688 sec. 45. 25i. 28. 127.7952 dr. 46. Real estate, 29. 7.09038 in. $562.50 ; 30. 2342.439 sq. ft. railway shares, $281.25; city bonds, Art. 162. Page 106. $656.25. 3. 15 bu. pk. 7 qt. 47. Wife, $2465 ; 1 pt. eld. son, $1848.75 ; 4. lcwt.47 1b. 15oz. younger son. 5. £6 16s. 10d.3far. $1479; 6. 1 111 4 5 5 3 2 3 dau., $1232.50. 7gr. 7. 4 0. 10 f 5 4 f 3 38 rn^. 8. 92 rd. 1 yd. 2 ft. 9. 81 gal. 2 qt. pt. Igi. 10. £31. 11. 3 d. 17 h. 51 min. 55 sec. 12. 34° 19' 9". 13. 8 lb. 6 oz. 15 pwt. 6gr. 14. 228 rd. 5 yd. 1 ft. 6 in. 15. 2 A. 96 sq. rd. 4 sq. yd. 16. 1 wk. 2 d. 6 h. 13 min. 20 sec. 17. 2 cd. 47 cu. ft. 1502 cu. in. 18. 1 T. 17 cwt. 14 lb. 13 oz. 8 dr. 19. 5 mi. 89 rd. 1 yd. 1 ft. 6 in. 20. 1 sq. rd. 1 sq. yd. 5 sq. ft. 33 sq. in. 21. 1 mi. 33 rd. 2 yd. 2 ft. 10 in. 22. 2 A. 9 sq. rd. 1 sq. yd. 23 sq. ft. Art. 163. Page 108. 3. 44 gal. 3 qt. 1 pt. Igi. 4. 19 cwt. 1 lb. 5 oz. 4 dr. 5. 164 bu. 3 pk. 7 qt. 1 pt. 6. 275 d. 6 h. min. 50 sec. 7. 313° 27' 26". 8. 80 It) 5 3 2 3 10 gr. ANSWERS. 9 9. 26mi. 119rd.0yd. 15. 19A.0sq.rd. 7sq. Art ;. 167. Page 113. 1 ft. 6 in. yd. 3 sq. ft. 9 sq. 2. 11 bu. 3 pk. 7 qt. 10. £451 Os. 6d. 1 far. in. 1 pt. 11. 69 T. 13 cwt. 67 3. 5 gal. 1 qt. 1 pt. lb. 10 oz. 5 dr. Art. 165. Page 111. 3gi. 12. 146 cd. 30 cu. ft. 4. 258. 4. 2° 48' 45". 1142 cu. in. 5. 295. 5. 23 T. 9 cwt. 78 lb. 13. 68 lb. 9 oz. pwt. 6. 265. 15 oz. 10 gr. 7. 695. 6. 25 cu. yd. 16 cu. 14. 14 sq. ml. ij6 A. 8. 627. ft. 843 cu. in. 5 sq. rd. 12 sq. 9. 3513. 7. 5 lb. 9 oz. 15 pwt. yd. sq. ft. 108 10. 7 y. 10 mo. 16 d. 12] gr. sq. in. 11. 5 y. 3 mo. 20 d. 8. 136 rd. 4 yd. 2 ft. 15. 129 mi. 223 rd. 3 12. 1 y. 10 mo. 22 d. 8 in. yd. ft. 2 in. 13. 5 y. 8 mo. 5 d. 9. £9 16s. 7d. 3i far. 14. 8 y. 11 mo. 15 d. 10. 34 gal. 2 qt. 1 pt. Art. 164. 15. 1 y. 4 mo. 19 d. 3gi. Pages 109, 110. 11. 6 d. 13 h. 41 min. 3. 10 bu. 2 pk. 6 qt. Art. 166. Page 112. 33.2 sec. Ipt. 2. 41 bu. 3 pk. qt. 12. 4tb9§6323 4. £17 19s. Ud. 1 pt. 12 gr. 2 far. 3. 213° 10' 50.50". 13. 2 mi. 45 rd. 3 yd. 5. 37° 46' 41". 4. 154 gal. 1 qt. 1 pt. 2 ft. 10 in. 6. 9 gal. 3 qt. 1 pt. 2gi. 14. 3 A. 132 sq. rd. 3gi. 5. 610. 15 f 5 6 f 3 10 sq. yd. 8 sq. 7. 9 lb. 11 oz. 1 pwt. 37 m.. ft. 31 sq. in. 20 gr. 6. 75 mi. 144 rd. yd. 8. 7 cu. yd. 9 cu. ft. 2 ft. 976 cu. in. 7. 87 d. 6 h. 25 min. Art. 168. Page 114. 9. 9 0. 7f 5 4 f 3 30 sec. 2. £18 2s. M. 1 far. 35 n^. 8. £1563 2s. Od. 3. 46 lb. 3 oz. 17 pwt. 10. 7 mi. 203 rd. 4 yd. 3 far. 22 gr. ft. 6 in. 9. 82 T. 9 cwt. 13 lb. 4. 5 'J\ 1 cwt. 42 lb. 11. 14 cwt. 18 lb. 2 oz. 6 oz. 8.5 dr. 9 oz. 14 dr. Idr. 10. 40 cd. 64 cu. ft. 5. 5 bu. 2 pk. 6 qt. 12. 5 A. 12 sq. rd. 10 987 cu. in. 1 pt. sq. yd. 6 sq. ft. 11. 253 lb. 2 oz. 10 6. 1 d. 23 h. 37 min. 36 sq. in. pwt. 20 gr. 15 sec. 13. 4 d. 19 h. 16 min. 12. 6 mi. 7. 3 mi. 20 rd. 1 yd. 4 sec. 13. 56 A. 17 sq. rd. 13 2 ft. 10.8 in. 14. 16 mi. 114 rd. 5 sq. yd. 2 sq. ft. 8. 7 gal. 2 qt. 1 pt. yd. 1 ft. 66 sq. in. 3gi. 10 ACADEMIC ARITHMETIC. 9. 6 cd. 124 cii. ft. 8. ;; gal. 6. 9h. 15min. 40| 760 cu. in. 9. >|cd. sec. 10. 22° 55' 27 f.". 10. 4rd. 7. 51i. 33min.28sec. 11. 10 lb 1 5 6 3 13 11. f cwt. A.M. 5gr. 12. .^0. 8. 91i. 5 min. 29 sec. Art. 169. Page 115. 13. A A. A.M. 2. 25. 3. 31. 4. 8. 5. 12. 6. 13. 14. 15. .8bu. .4275". 9. 9h. 59 min. 3^ sec. P.M. 16. .96875 gal. 10. 10 h. 53 min. 57f 17. .69 wk. sec. P.M. 18. £.148. 11. 89° 15'. 19. .625 cu.yd. 12. 38° 40' 30". Art. 170. Page 115. 20. .8551b. 13. 168° 16' 15". 3. 6 d. 3 h. 21. .7532 T. 14. 131° 24' 30". 4. 3 oz. 6 pwt. 16 gr. 22. .3 sq. rd. 15. 74° 1' W. 6. 2 qt. 1 pt. 3j5jgi. 23. .81 mi. 16. 88° 12' W. 6. 1 pk. 1 qt. 1.2 pt. Art. 172. 17. 18° 30' 6" E. 7. 52' 15.6". Pages 117, 118. 18. 73° 25' 57" W. 8. 12 cwt. 10 lb. 3. 1 19. 72° 53' 10" E. 9. 6f5 5f3 20in^. 4. A. 10. 4 yd. 2 ft. 8 in. 5. 6. 1 6' 7 Art. 174. 11. 21 sq.yd. 1 sq. ft. Pages 120-124. 82.8 sq. in. 7. 1. 17. 12. 3 5 4 ?i 3 8. i. 2. $97.44. 7.68 gr. 9. 10. 5 3. A train that runs 13. 9s. 2d. 3 Jy far. 9- _8_ a mile in 85 sec. 14. 58 cu. ft. ZU^\ 11. 1 1* 4. Ih. 30 min. 50 sec. cu. in. 12. .52. .7. 5. 1 y. 8 mo. 18 d. 15. 163 rd. 1 yd. ft. 13. 6. 5 A. 133sq.rd. 10 3.6 in. 14. .9616. sq. yd. sq. ft. 16. Is. 10c?. 3.68 far. 15. 16. .75. 108sq. in. 17. 91 sq. rd. 12 sq. .875. 7. 40 rd. 1yd. Oft. 9 yd. 8 sq.ft. 97f 17. .625. in. sq. in. 18. .27. 8. 5. 19. .4. 9. $3470.625. Art. 171. 10. 15. Pages 116, 117. Art. 173. 11. $160. 3. |lb. Pages 119, 120. 12. 16tV. 4. £/,. 3. 3h. 13min. 48 sec. 13. 7. 5. f|°. 4. 6h. 48min.4sec. 14. 18H. 6. t|bu. 5. llh. 54min. 241 15. £4 9s. lO^d. 7. Hd. sec. 16. 2023. ANSWERS. 17. Urd. 3 yd. 1ft. Art. 185. Page 133. 25. .842265^1. 18. $13590.72. 3. 361.24091. 26. 2.309136 sq. rd. 19. 21b 115 73 03 4. 63.43126?. 27. 37.5441D1. 2gr. 5. 980.203'". 28. .51562i>ni. 20. 17tVV 6. 314.41948^^ "'. 29. 7.22624«'J m. 21. $315.15jV 7. 191. 94D?. 30. .589932«t. 22. $31.25. 8. .499198'". 31. 128283.75CK. 23. 2.6. 9. 934.37c" '^'". r^R. 34.681 5<=" dm. 24. 89f cents. 10. 4246.567'!'. 33. 163.23675'ii. 25. 52f ft. per sec. 11. 1.3581Hn,. 34. .046206*=" Dm. 26. 41t) 85 03 13 12. .2543772sq'". 35. 3.09479825Kg. 6Jgr. 13. 2.26803«^?. 36. 2.1964347"^ 27. Hh 14. .2086118^'. 37. 2523.1344'". 28. 95^^jf cents. 15. .037. 29. $39.69. 16. .1821i>g. Art. 187. 30. 155. 17. .568. Pages 135-137. 31. 8 oz. 1511 dr. 18. 93,6cumm. 1. 2.0116. 32. 3* 2. 395.4. 33. 8||. Art. 186. 3. .164"^"!. 34. 86631 miles. Pages 134, 135. 4. $1092. 35. 87° 48' 45" W. 3. 9.144g. 11 12 ACADEMIC ARrjHMP:TIC. 27. IGO. 0208 rods a 6. .3162 + . 8. .854+. minute. 7. 1.4443 + . 9. 1.077 + . 28. 102.30912. 8. 19.3864 + . 10. .637 + . 29. 2.09+ cents per 9. 26.1653 + . 11. .873 + . mile. 11. 1.6583 + . 12. .912 + . 30. 10.35 + . 12. .4330 + . 13. 1.1304 + . 31. 6.103. 13. 1.2018+ . 14. .7862 + . 32. 264.17. 14. .4472 + . 33. 32.362512. 15. .6454+. Art. 214. Page 150. 34. $35.07. 16. .9045+. 1. 33. 35. 2.076 + Kg. 17. .6236 + . 2. 76. 36. .0839+. 18. .4249 + . 3. 88. 37. 13.63 + Kg. 19. 1.1319+. 4. 514. 38. 1.27 + . 20. .8552 + . 5. 49. 39. 34.22 + . 6. 65. 40. 13490. 7619Kni. Art. 212. Page 149. 1. 31. Art. 226. Art. 203. Page 143. 2. 4.6. Pages 154-156. 1. 78. 3. .88. 4. 588sq.in. 2. .97. 4. 123. 5. 14 sq.ft. lOOsq.in. 3. 21.4. 5. 1.14. 6. 4isq.yd. 4. 523. 6. .098. 7. 4 ft. 5. .286. 7. 2.02. 8. 12 yd. 6. 80.9. 8. 372. 9. 75 ft. 7. .497. 9. 21.6. 10. 3780 sq. in. 8. .0722. 10. .803. 11. §yd. , 9. 5.76. 11. 4.89. 12. 75 in. 10. .1082. 12. .317. 13. 3|A. 11. 21.12. 13. .898. 14. 196 sq.ft. 112 sq. 12. .8253. 14. 101.3. in. 13. 900.8. 15. .0534. 15. 3U|rd. 14. 5783. 16. 73.4. 16. 32 'rd. 15. 7.641. 17. 5.815. 17. 1320. 16. .04738. 18. .6523. 18. Oft. 17. 859.35. 19. 6600. 18. .98657. Art. 213. 20. 630. Pages 149, 150. 21. 1200. Art. 204. Page 144. 2. 1.259 + . 22. 280 yd. 2. 2.6457 + . 3. 1.817 + . 23. 24 rd. 3. 5.5677 + . 4. 1.930+ . 24. 163.65. 4. 4. 1593+ . 5. 3.448+. 25. 3421440. 5. .2828+. ■ 6. 5.528 + . 26. $562.50. ANSWERS. 13 27. 20 rd. 3 yd. 28. 22 rd. 4 yd. 1 ft. 29. .$1878.80. Art. 228. Pages 157, 158. 4. 25 in. 5. 7rd. 2.iyd. 6. 32 in. 7. 1yd. 2 ft. 8. 10.6066+ in. 9. 12.2065+ in. 10. 22 ft. 1 in. 11. 12 ft. 12. 60 ft. 9 in. 13. 105.4 mi. 14. 40.5 ft. 15. 8 ft. 2 in. Art. 231. Pages 159, 160. 3. 43.9824 in.; 153.9384 sq. in. 4. 52.36 yd. 5. 5.25+ rd. 6. 13.54+ in. 7. 24856.3392 mi. 8. 2.67 + . 9. 4.507 + . 10. 346.3614. 11. 61.11+ in. 12. 110 sq.ft. 104.5776 sq. in. 13. 1306.9056 sq.ft. 14. 9075 + . 15. 8.862+ in. 16. 28.3372.32 ft. 17. 120.96 ft. Art. 242. Pages 164:-166. 4. 330sq. in.; 330 ca in. 5. 34|| cu. in. ; 63| sq. in. 6. 144 sq.ft. 7. 7 ft.; 364 sq.ft. 8. 420 cu. in. 9. 392 sq.ft. 10. 11 in. 11. 050cu.in. 12. 32 in. 13. 1092 cu. in. ; 662 sq. in. 14. 3ft.; 122 sq.ft. 15. ^8.47. 16. 19Hn. 17. 2400 sq. in. 18. -$55.38. 19. 5951^ lb. 20. 1944. 21. 32 ft. 22. 53|cu. ft. 23. 15 in. 24. 31-lf. [cu.in. 25. 700sq.in.; 1568 26. 2090880. Art. 248. Pages 169, 170. 5. 131.9472 sq. in.; 197.9208 cu. in. 6. 201.0624 sq. in.; 268.0832 cu. in. 7. 483.8064 sq. ft. 8. 204.204 sq. in. ; 314.16 cu.in. 9. 1847.2608 cu. ft. 10. 10 in.; 8 in. 11. 10 in.; 523.6 cu.in. 12. 8 in. 13. 12 in. 14. 196067256 sq. mi.; 258155220400 cu. 16. 67.0208. 16. 6 ft. 17. $184.07 + . 18. 5^1 in. 19. 14.32 + . 20. 134.0416. 21. 2.3562. 22. 4 ft. 23. 12 in. 24. 6 in. 25. 67.3698+ lb. 26. 8.7593+. 27. 6.109 ft. 28. 8.169+ in. Art. 249. Pages 171, 172. 2. 67i. 3. 7. 4. 65|. 5. 552.5. 6. 4i ft. 7. 5 ft. 8. 833^ in. 9. 65.35+ in. 10. 48.23+. 11. $117. 12. 58.5+ in. 13. 4 ft. 7.98+ in. 14. $111.13 + . 15. 61.91+ in. 16. 53.856+. 17. 41.0502 + . 18. 4.12+ ft. 19. 53.71 + . Art. 250. Page 173. 2. 34. 3. $7.50. 4. 48|f ; 49. 5. $10.45. 6. $21.14j:V 14 ACADEMIC ARITIIIMETIC. 7. 28H. 5. 148 'lib. 28. 235301.94. 8. Across the room. 6. 7[||oz. 29. 37m. 9. 651.04. 7. 25. 30. 11.2 + Din. 10. With oil-cloth. 8. 40. 31. 53.74 +m. 9. 432. Art. 251. 10. 54. MENSURATION OF Pages 174, 175. 11. 2.72. SOLIDS. 3. $35.88. 12 .88. 1. g^sq dm • 42*^" ^'^, 4. $48.48. 13. 10.5. 2. 113.097689 cm. 6. 8rV 14. 8.3469312. 113.0976<="cm. 6. $5.40. 3. 549.78 sqm. 7. $27.53§. Art. 256. 1231.5072cum. 8. $6. Pages 181-188. 4. 16cni; 976^1 cm. 9. $314.16. MENSURATION 5. 33dm. 10. 6r. OI ^ PLANE FIGURES. 6. 301 .5936*1 m. 11. $37.71^. 1. 6541.5«qdm. 7. 4«»; 268.0832cucm. 12. $10.97§. 2. 35™. 8. 9.4956«qm. 3. .4Hm, 9. 425.6c^i Dm^ Art. 252. 4. .0162Dm. 10. 9m. Pages 176, 177. 5. 144513.6ra. 11. 45804.528^1 mm. 2. lOJ. 6. .094776«q Hm. 2061.20376cucm. 3. 9^. 7. 1.83Dm. 12. 15m; 9m; 4. 154^. 8. 13273.26^1 cm. 1357.1712<-"m. 5. 92|. 9. 3.79dm. 13. 407.52«im. 6. 107if. 10. 1233.395«qdm. 14. 208.149326 ; 7. $17.64. 11. 2.5D'". 208149.326dg. 8. $13,343. 12. .3819+ c'". 15. $13.65. 9. $66.85^. 13. 270<=m 16. $199.0989. 10. $5,901. 14. 3.52782^ 17. 7824. 11. $33.68|. 15. 25.67566ca. 18. $113.40. 16. 106.311741ca. 19. 1.38m. Art. 253. Page 177. 17. 65.9m. 20. 1.5372. 2. 8.484 in.; 108. 18. 87.9Dm. 21. 48.55344KK. 3. 11.312 in.; 224. 19. 221m. 22. 5309.304. 4. 10.605 in.; 162 ^ 20. 5500. 23. 638.4. 5. 9.898 in.; 128 -. 21. $6653.36. 24. 37000. . 6. 13.433 in.; 358/,V 22. 65.1Kni. 25. 19373.2. 23. 10.23250536. 26. 3650.5392Dg. Art. 254. 24. 1.6493925. 27. 13.8984384Kg. Pages 178, ,179. 25. 11.5m. 28. 579.63588. 3. 550.6251b 26. 5.939 + Hm. 29. .33m. 4. 3037.51b. 27. 7.6Dni. 30. 3.8050012. ANSWERS. CAPACITY OF BINS, 11. 11568.72d8. 10 . 49^. TANKS, AND CIS- 12. 65cni. 11 . $331.25. TERNS, CARPETING, 13. 7.1. 12 . $1125. PLASTERING, AND 14. 2.08. 13 , 280. PAPERING. 15. 9dm. 14. , $20.40. 1. 5801.6. 16. 20.5633428Kg. 15. 159 ft. 41 in. 2. 49.68. 3. 46.13. 4. $27.63. Art. 263. Pages 191, 192. 1. 2:5. 2. 8 : 15. 16. 17. 18. 8|. 42 mi. an hour. $29.80. 5. 1.92"'. 19. 8^- 6. 26.9dn\ 3. 23:37. 4. 9:10. 5. 9:13. 6. 22:15. 7. 11:14. 8. 5:3. 9. 7:9. 10. 25 : 27. 11. 8:5. 12. 2 : 3. 13. 16:25. 14. 64:45. 20. 10 min. 30 sec. 7. 134.3034; 21. 2Jh. 134303.4Hg. 22. 51. 8. 6.5. 9. 1.7'". 23. 24. 276 lb. 32 oz. 105 d. 10. $471.24. 25. $151t. 11. $60.42. 12. 11 h. 30 mm. Art. 273. 13. 10.31. Pages 198-201. 14. 15.927912. 15. 3.5'». 16. $40,026. 17. 13.5. 2. 3. 4. 5. 135. 8. 20. 18. Across the room. Art. 270. Page 194. 6. 46i. 19. $102,971; 3. 112. 7. 5. 3l07.852Kg. 4. 36. 8. 12. 20. 1.8'». 5. 24. 9. 2f.^ 21. $34,104 ; $34,104. 6. If. 10. 9. 22. 75cn'. 7. i-|. 11. 216 mi. 23. .997+"'. 2 5 8. -V-. 12. 42i. 24. $7.98. 9. 1§. 13. ■Sj% oz. 25. 1.5"'. 10. V. 14. ^■ 15. 72. Art. 257. Art. 271. 16. 660 lb. Pages 189, 190. Pages 196, 197. 17. 10. 4. 12008. 3. $4.51. 18. m ft. 5. 1317Hg. 4. $15.96. 19. 7^. 6. 18.7. 6. 511. 20. 78| lb. 7. 12.9. 6. 65^. 21. 13|. 8. .92. 7. 648f. 22. Oft. 9. 13.596. 8. 78f 23. 4. 10. 2.30679. 9. $4.60. 24. 4^ ft. 15 16 ACADEMIC ARITHMETIC. 25. 20}. 9. 4 ft. () in. 4. A, $26.40; 26. 1150 1b. 10. 9 in. B, $13.20; 27. 28. 11. $37.80. C, $33 ; D, $44. 28. 81. 12. 218 lb. 12 oz. 5. Adams, $1200; 29. 8. 13. 7.93 -f in. Burke, $1120. 30. 5,^ 14. 2 h. 6 min. 6. A, $1275; 15. 1 ft. 4 in. B, $1020; Art. 275. C, $1360. Pages 202, 203. Art. 280. 7. Hand, $704 ; 3. 14, 35, 56. Pages 206, 207. Sears, $720 ; 4. 140, 184. 2. A, $600; B, $860. Thomas, $768. 5. 39,05,91,117,143. 3. Allen, $1250 ; 8. A, $532 ; 6. 672, 630, 588, 576. Brown, $1500 ; B, $148. 7. 50|, 355}. Cole, $800. 9. A, $48.75; 8. 510 lb. copper. 4. A, $192.75; B, $39; C, $22.75. 306 lb. zinc. B, $64.25. 10. P^uller, $200 ; 119 lb. tin. 5. A, $49.24 ; Gray, $250. 9. A, .$375 ; B, $480 ; B, $61.55; 11. A, $145; C, $612. C, $73.86. B, $185 ; 10. 182f parts salt- 6. A, $17120; C, $200. petre, B, $8560 ; 12. A, $1500; 34} parts charcoal, C, $1712. B, $1800 ; 57Y\r parts sul- 7. Hale, $1725 ; C, $3375. phur. Hunt, $2070. 13. Lowe, $2035 ; 11. 42, 84, 252. 1008. 8. A, $1077.36 ; Martin, $2255; 12. $23.36, $35.04, B, $1258.02 ; Neal, $776. $43.80, $01.32. C, $3145.05. 13. 26 II cu. ft. oxy- 9. A, $5.25 ; Art. 287. Page 210. gen, 99|^ cu. ft. B, $3.75 ; 8. .056. nitrogen. C, $6.00 ; 9. .008. 14. 30, 20, 16. D, $4.50. 10. .001875. 15. $115. 10. A, $867.84; 11. .0041. 16. 15, 30, 55, 90. B, .$578.56; C, $1084.80. Art. 288. Page 211. Art. 276. 1. if- Pages 204, 205. Art. 282. 2. I' 3. 540 sq. in. Pages 208, 209. 3. iV- 4. 8 in. 2. A, $162 ; 4. tV 5. 708J cu. in. B, $194.40. 5. hh 6. 6 ft. 3. A, $82.50 ; 6. If 7. 1 ft. 4| in. B, $247.50 ; 7. 8 8. 7 in. C, $137.50. 8. jh' ANSWERS. 17 9. tU- 13. tVo- 24. $ 10.38.54 j-V 10. tI^- 14. 3 ft. 6^ in. 26. f- 11- ?VV 16. $413. 26. .$45.75. 12. 3^. 16. $415. 27. 146ft. Sin. 13. ^',. 17. $953.04. 28. 7608. 14. ^h- 18. .$498.27. 29. $1286.40. 15- Th- 19. $1132.25. 30. $107.80. 20. $378.42. 31. 2928. Art. 289. Page 211. 21. $2257.57. 32. 135 girls, 162 boys. 2. 57^%. 22. 14093. 33. $17500. 3. 1H%. 23. $891. 34. 1241, 816. 4. n2i%. 24. .$9.45. 35. $2220.35. 5. 58J%. 25. $.03. 36. $5688. 6. 85%. 26. 10401b. lead; 37. $1000.50. 7. 16%. 16 lb. silver. 38. $1015.30. 8. 7^/0. 27. $1815. 39. 57615. 9. 131io/„. 28. $344.75. 40. $1250. 10. 73.^0/^. 29. 703. 11. 171^0/,. 12. 54p/o. 30. lin. Art. 292. 31. $.46^0. Pages 219-221. 13. fi%. 32. $1573. 4. 6,1%. 14. 1%. 5. 40%. 15. if %. 16. f%. Art. 291. 6. 54^/0. Pages 215-218. 7. 74%. 17. f%. 3. .32.4. 8. 9ir/o. 18. 123i\%. 4. $18.50. 9. 75%. 19. l3\%. 6. ¥• 10. 108% 20. 2l5V%. 6. 17° 45'. 11. 63«%. 21. 49||o/o. 7. 2^,. 12. 31i%. 8. 47 ft. 8 in. 13. 1%. Art. 290. 9. W- 14. 42f%. Pages 212-214. 10. 3. 16. i%. 3. 4.291. 11. ¥• 16. 4ir/o. 4. .17.77. 12. 187 lb. 4 oz. 17. 12]%. 5. 62 bu. 13. 2^f. 18. 76% silver; 24% 6. ^,3.. 14. $193.75. copper; 31^%. 7. f. 18. $83.50. 19. 19|%- 8. 1. 19. $1328. 20. 26tr/o. 9. .$2.27. 20. 384. 21. 221%. 10. tI^. 21. 656.25. 22. 181%. 11. £100 2^, 22. $425. 23. 87^%. X2. f-, 23. $62.50. 24. 20%. 18 ACADEMIC ARITHMETIC. 25. 1%. 7. $520; $31.20. 3. , $17.50 on $1000. 26. 2.i%. 8. $990,099+ ; 4. $54.28. 27. m%- $9,901. 5. $4200. 28. 28%. 9. 3^ %. 6. $7000. 29. 170/0. 10. 400. 7. $215.79. 30. l()f%. 11. $2496. 8. $12.75 on $1000; 31. 19%. 12. $7840. $107.10. 32. 8.9%. 13. $2364.18. 9. $35700. 33. 6|%; 61%. 14. $438 18. 10. $332. 34. 15^%. 15. $2444.98+ ; 11. $11.30 on $1000. 35. 2.^0/^. $55.02. 12. $16 on $1000. 36. 19i%. 16. $8400. 13. $22401. 37. 37^ %. 17. .$2157.40. 18. 336. Art. 303. Art. 294. 19. $632. Pages 232, 233. Pages 222-224. 20. $6390. 3. $7.98. 3. $45.75. 21. 3.50. 4. $100.36. 4. $203.28. 22. 13 ; $8.84. 5. $157.6746. 5. .|14.70. 23. 96. 6. $151.20. 6. $im. 7. $82.40. 7. 18.00. Art. 298. 8. $61.11. 8. .S4..35. Pages 228, 229. 9. $1614.60. 9. {$14.28. 3. $47.97. 10. $690. 10. Loses 3| %. 4. $415.25. 11. 55%. 11. $1.26. 5. $1860. 12. $77.50494. 12. $2653.56. 6. h%- 13. $293.443375. 13. .$2.10. 7. $1720. 14. $904.8553125. 14. 9fo/„. 8. $3475.50. 15. $24.75. 9. $2009.60. Art. 307. 16. $13.32. 10. -1%. Pages 235, 236. 18. 33|o/,. 11. $5760. 4. $217.50. 19. 50%. 12. $9600. 5. $33.78. 20. 27|o/o. 13. $2625. 6. $16.96. 21. 36i%. 14. $2653.75. 7. $4.74. 22.37^0/0. 15. $108.50. 8. $108.35. 16. $163.50. 9. $2.67. Art. 296. 17. $7980. 10. $21.28. Pages 225-227. 18. 3|%. 11. $14.11. 3. $18.72. 12. $50.36. 4. $4964.82. Art. 301. 13. $18.08. 5. 50/0. Pages 230, 231. 14. $2.61. 6. $21.25. 2. $13,50 ou $1000, 15. $14.44. ANSWERS. 19 16. $197.95. 10. $197.45. Art. 312. Page 244^ 17. $76.39. 11. $845.54. 3. $310. 18. $276.37. 4. $107.75. 19. $2214.36. Art. 310. 6. $196. 20. $467.99. Pages 240, 241. 6. $297.20. 21. $3794.82. 2. 21%. 7. $178.80. 22. $1102.58. 3. 31%. 8. $657.16. 23. $636.46. 4. 6%. 9. $86.34. 5. 2%. 10. $947.93. Art. 308. Pages 238, 239. 6. H%- 11. $74.16. 4. $161.35. 6. $10.33. 6. $6.42. 7. $1.05. 8. $8.12. 9. $7.08. 10. $86.68. 11. $20.17. 7. 8. 9. 2r/o. 7%. 1|%- 12. $305.48. 13. $.325.85. 14- $250.91. 10. 11. 12. 2r/o. 4%. 3%. 16. $342.86. 16. $40.74. 17. $573.25. 13. 14. 4J%. 18. $4191.04. 19. $589.66. 12. $36.80. 15. 8%- 20. $1.36.08. 13. $333.60. 16. 17. n%. 3%. Art. 314. Page 246. 14. $5183.46. 2. $188.24. 15. $1047.72. Art .-^11 3. $27.21. 17. $162.17. Pages 242, 243. 4. $31.06. 18. $13.65. 19. $1.24. 20. $24.25. 21. $38.23. 22. $1.73. 23. $142.06. 3. • 4. 6. 6. 7. 4 y. 3 mo. 2 y. 10 mo. 6 mo. 3 y. 1 mo. 5 mo. 24 d. 5. $103.02. 6. $52.88. 7. -$24.38. 8. $37.95. 9. $38.43. 24. $157.11. 8. 11 mo. 12 d. Art. 323. 25. $9179.74. 9. 1 y. 8 mo. 12 d. Pages 249, 250. 26. $328.47. 10. 6 mo. 3 d. 2. $318.35. 11. 9 mo. 6 d. 3. $244.24. Art. 309. Page 240. 12. 4 mo. 7 d. 4. $606.91. 2. $921.18. 13. 2 mo. 11 d. 5. $90.66. 3. $1.26. 14. 8 mo. 6. $371.65. 4. $19.38. 15. 21 d. 7. $183.05. 6. $62.59. 16. 9 mo. 29 d. 8. $521.25. 6. $10.53. 17. 5 y. 5 mo. d. 7. $1.92. 18. 7 mo. 12 d. Art. 324. 8. $6689.71. 19. 16 y. 8 mo. Pages 251, 252. 9. $250.61. 20. 22 y. 2 mo. 20 d. 2. $309.85. 20 ACADEMIC ARITHMETIC. 3. $251.39. 4. !$754.70. 6. $766.71. 6. $244.06. 7. $436.79. 8. $327.70. 9. $345.68. Art. 326. Pages 253, 254. 2. $156.63. 3. $258.81. 4. $59.85. 6. $167.24. 6. $464.16. 7. $180.89. 8. $801.87. 9. $624.48. 10. $143.42. 11. $1790.82. Art. 327. Page 255. 2. $566.34. 3. $1696.18. 4. $429.25. 6. $639.00. 6. $788.13. 7. $696.71. 8. $771.80. Art. 328. Page 257. 2. $133.49. 3. $1562.79. 4. $422.37. 5. $272.07. 6. $265.16. 7. $298.30. Art. 329. Page 258. 2. $1059.62. 3. $469.21. 4. $96 86. 5. $1016.86. 6. $1196.76. Art. 331. Page 260. 2. $340.43 ; $59.57. 3. $868.29; $21.71. 4. $595.47 ; $129.53. 5. $1549.25; $180.75. 6. $662.46; $19.54. 7. $267.46; $1.74. 8. $2454.19; $45.81. 9. $907.50; $42.50. 10. $127.07 ; $8.68. 11. $342.21 ; $5.47. Art. 334. Pages 261-263. 3. $492.25. 4. $941.69. 5. $36.17. 6. $5.58. 7. $996.33. 8. $5.35. 9. $6009.95. 10. $420.05. 11. $802.98. 12. $3.33. 13. $6.43. 14. $2980.75. 15. $191.44. 16. $1.13. 17. $1128.73. 18. $199.88. 19. $4.67. 20. $279.76. 21. $396.59. 22. $3.71. Art. 335. Page 264. 2. $609.45. 3. $341.57. 4. $227.39. 6. $8245.30. 6. $554.81. 7. $1535.12. 8. $430.27. 9. $921.80. 10. $377.48. Art. 343. Pages 267, 268. 3. $506.25. 4. $278.60. 5. $7874. 6. $474.56. 7. $1928.16. 8. $693.61. 9. $344.97. 10. $606.80. 11. $1333.59. Art. 344. Pages 260, 270. 3. $440. 4. $649.60. 5. $918.46. 6. $241.81. 7. $752.10. 8. $192. 9. $584.65. 10. $2985.07. 11. $2346.71. Art. 348. Pages 272, 273. 4. $1693.89. 5. $201.93. 6. $581.40. 7. £39 4s. 8. 29942.55 fcs. 9. $78.89375. 10. $515.20. 11. $139.83. 12. 29368 mks. 13. $2557.697625. 14. £176. ANSWERS. 21 15. 4914.70125 fcs. 16. $466.4109375. 17. £132 8s. 9d. 18. 1501.03 mks. 19. .$5248.19. Art. 350. Pages 275, 276. 4. 7 mo. 10 d. 5. 8 mo. 27 d. 6. 73 d. 7. April 23. 8. Oct. 10. 9. Nov. 18. 10. Jan. 8, 1890. 11. June 5, 1891. 12. Dec. 24. 13. July 29, 1892. 14. After 9 mo. 15 d. 15. Oct. 2. 16. 5 mo. 27 d. after it becomes due. 17. Oct. 14, 1892. Art. 352. Pages 279-281. 2. Nov. 12. 3. Jan. 11, 1891. 4. Feb. 13. 5. April 26. 6. Nov. 5, 1890. 7. Dec. 2. 8. Dec. 27, 1890. Art. 363. Pages 286-291. 5. $17741.25. 6. $8011.50. 7. $5466.75. 8. $2588.25. 9. $1354.50. 10. $4977. 11. 46. 12. $7500. 13. 43. 14. 626. 15. $15000. 16. 75. 17. 153. 18. 169|. 19. 8^o/o. 20. 80|. 21. 12|%. 22. 36|%. 23. 186. 24. $395.50, 25. $75000. 26. $3982.75. 27. 1|%. 28. $7900. 29. 38. 30. 153. 31. 375. 36. $160. 37. $150. 38. $271.25. 39. $484.50. 40. $4431. 41. $5797.50. 42. $12060, 43. $7449.75. 44. 5%. 45. 4^o/o. 46. 6.4%. 47. Sh%- 48. 224. 49. 74. 50. 146|. 51. 67|. 52. 150i. 53. A 4|% stock at 90. 54. Increased $11. 65. 61^. 66. The investments are equally good. 67. Diminished $23.50. 58. $6757.50 invested in 6% bonds at 112f. 59. 4io/„. 60. A 3 1 % stock at a disc't of 28 1 %. 61. 129. 62. Increased $36.75. Art. 367. Pages 293-295. 2. 72. 3. 32^. 4. 85. 5. 51.6. 6. V-- 7. 70, 444. 8. 289, 3129. 9. 69, 5249. 10. 31, 9306. 11. 81|, 2040. 12. 9.9, 748.3. 13. V-, -Hl^- 14. h W- 15. 4950. 16. 2550. 17. The last term is 18. 1197. 19. 44550, 20. 498^2 ft., 4117^ ft. 21. 15f mi., 325r\mi. 22. $3850. Art. 372. Pages 297, 298. 3. 2500. 22 ACADEMIC ARITHMETIC. 4. W- $1780.73. 31. 123-1. 6. ^%- 7. $280.90. 32. $1466.9424. 6. z\'-2' 8. $351.52. 33. $3151.82. 7. 1024,2047. 9. $169.21. 34. 11%. 8. 13122, 19680. 35. 564245 in. 9- jh> -\W- Art. 377. 36. 5 h. 46ff min. 10. ,V¥^, WiV- Pages 303-317. 37. .$447.15; $31.80. 11. -W-. ¥#-. 1. 4936095. 38. 2088.77 mks. 12- /3» HF. 2. 9Mf. 39. 1,VV 13. H-gJ-. 3. VtVW-. . 40. 4420 sq. in. ; 14. HIF- 4. $39.20. 18928 cu. in. 15. $163.83. 5. $602.98. 41. $21356.25. 16. 1 mi., 767} mi. 6. $570.42. 42. $738.92. 17. The last term is f . 7. 19 rd. 5 yd. 1 ft. 43. 1008. 18. 17576. 8. i. 44. if. 19. $121.550625. 9. Hind-wheel, 5580; 45. 4032. fore-wheel, 6600. 46. $11059.58. Art. 373. Page 299. 10. .853976. 48. .0875. 3. $7203.26. 11. $149.60. 49. .$54.81 ; $54.23. 4. $9733.22. 12. 198 mi. 195 rd. 50. 3|i. 5. $78.74. 2 yd. Oft. 9 in. 51. $987.65; $12.35. 6. $132.42. 13. 122" 27' 17" W. 52. 523. 7. $24000. 14. m- 63. 5^%. 8. $4800. 15. 3i-f 54. fli. 9. $992. 16. U- 55. .$34.79. 17. 6876. 56. 2.375. Art. 375. Page 301. 18. $1500. 57. $2801.25. 4. $1620 ; $1350. 19. 6 y. 3 mo. 12 d. 20. .73142857. 58. l|f-f. 59. 7.6985. 5. $783.75 ; $690.53. 6. $7520; $5371.43. 7. $5607; $4441.19. 8. $408. 9. $620. 10. $272. 21. 11.312 in. ; 208|. 22. 49.34if. 23. 18 in. 60. A, $248.82 ; B, $317.46 ; C, $197.34; 24. ^nh- D, $351.78. 25. .$273.75. 61. 23. 26. 32f%. 27. 18744264. 62. 3.36 in. 11. $642. 63. 10 ft. 1 in. 28. 1 sq. mi. 468 A. 96 64. 2 mi. 159 rd. 3 yd Art. 376. Page 302. sq. rd. 7 sq. yd. Hft. 4. $630.50 ; $544.65. 8 sq. ft. 29 sq. in. 65. 304. 5. $1312.38; 29. .$253.50. 66. 4ff|; A, .$25.20 $1039.53. 30. 6 h. 37 min. 30| B, $21 ; C, $18 6. $2166.53; sec. P.M. D, $15.75. ANSWERS. 23 67. $1472.06, 68. 2 d. 8 h. 18 min. 45 sec. 69. $19.60. 70. 9 oz. 11 pwt. 14.7264 gr. 71. 85J. 72. 9J3 ; 2991^. 73. tVo- 74. 68ii; 201 Hi. 75. 74 sq. mi. :]80 A. 102 sq. rd. 14 sq. yd. 3 sq. ft. 61 sq. in. 76. $9185.75. 77. £4 9s. Id. I far. 78. A, $39.50; B, $38.71; C, $52.14. 79. 4 ft. fOgV in. 80. .06444625. 81. $874.80. 82. $2906.14. 83. lOii. 84. 3|. 85. $949.62. 86. $1744.688915. 87. if mi. 88. Wife, $3150; son, $2940; dauglit'r,$2866.50. 89. 826. 90. 70 d. 91. ft, Vih f^. 92. 11 y. 9 mo. 21 d. 93. 12 d. 14 h. min. 25 sec. 94. 26r«o\- 95. 174|. 96. $53.90. 97. 9986J|mi. 98. 2800733. 99. 19.584. 100. $118070.75. 101. 81^Y3- 102. 8010, 12460, 17355, 22428, 27590. 103. $5038.20; $3650.87. 104. .860855. 105. $6.44. 106. IxV/i- 107. 21ft. 9.629+ in. 108. 479332^ gr. 109. $1040.72. 110. 40,^^. 111. £88 16s. 7c?. 3 far. 112. May 20. 113. $547.18. 114. 271%. 115. $861.69. 116. 31b.2oz.5.376dr. 117. 2_43^2^ 118. $67.31. 119. $863.95. 120. 2 ft. 8 in. 121. /o- 122. $.03J. 123. ^\\h- 124. $13.75 on $1000; $122.75. 125. 4.753+ in. 126. $2376.36. 127. 37J3. 128. 393 A. 135sq.rd. 11 sq. yd. 5 sq.ft. 102^*^3 sq. in. 129. $389.91. 130. 209 rV 131. 16794.50112 cu. in. 132. .8796 + . 133. 605 ft. 134. .897216796875 1b. 135. 7 lb. 1 oz. 3 pwt. 6||gr. 136. $179.96. 137. ,\V 138. .0738 cd. 139. 26880. 140. ^H!^. 141. 3,Vo/^; Zn. 142. $25709.25. 143. IHfini-; 144. 175.76. 145. 7 mo. 14 d. after it becomes due. 146. 29^0/^. 147. 19 T. 12 cwt. 91 lb.5oz. 12ifdr. 148. 4 ft. 6.2+ in. 149. $1093.09. 150. 4^V 151. 23.7. 152. 190.5904. 153. 19|. 154. $21968. 155. A, $449.75; B, $428.75; C, $458.50. 156. $582. 157. $668.91. 158. 2~jV rd. 159. 8028979200 sq.in. 160. iV,. 161. \\K 162. 8i%. 163. $2155.06; $1772.98. 164. 2/^. 165. Gained $15.78, 166. .71875. 167. 1|. 168. 287.354- lb. 24 ACADEMIC ARITHMETIC. 169. .1335.95. 170. Diminished $179.25. 171. 1954.3265281b. 172. IJ. 173. 3 h. 19141 min. 174. 13120|. 175. .1546.09. 176. 3128.4. 177. 14|. 178. 1,1^. 179. $41.94|; $11.67^. 180. $218.35. 181. March 7, 1890. Art. 378. Pages 317-320. 1. .527^" cm; .0527^8. 2. 965.7cumm. 3. .34250™. 4. $49.83. 6. 9.6558Hm a min. 6. 141.696Dg. 7. 4..3443906Km. 8. $156.77 + . 9. 112.373125^1. 10. 4.97096. 11. 2.01168. 12. 2.674224a. 13. 230 gal. 3 qt. 1 pt. .30656 gi. ; 24 bu. 3 pk. 1 qt. .90432 pt. 14. 1.491 + c. per kilo- meter. 15. 226.0737 ; 2260.737. 16. 562500. 17. $95,496. 18. 4158. 19. $1542.03. 20. 2.6753T. 21. .$40.08; $39.42. 22. .5899H™. 23. 53.0079c" d™. 24. 96.19830528. 25. $35.01485526. 26. .178Dtn. 27. 964.5. 28. 18dm. 29. 23 A. 28 sq. rd. 14 sq. yd. 3 sq. ft. 116.4672 sq. in. 30. 2.2763736. 31. 109.1674584Hg. 32. 590.64552Dg. 33. 5822.4075. 34. 9.0792248qdm, 35. .1432 -}-m. 36. .175D«i. 37. .21m; .00494802'"! Dm. 38. 1396mm. 39. 201.0624. 40. 2.57. 41. 8 lb. 5 oz. 5.347328 dr. ; 10 lb. 1 oz. 10 pwt. 12.96 gr. 42. 655.02+ lb. 43. 62566400. 44. 72.572 + . 45. 3 h. 19 min. 46. $343308.42. 47. 156.0872544. Appendix. Page 324. 4. 25° C. ; 20° R. 5. 60° C. ; 48° R. 6. -13\°C.; -105° R. 7. -36S°C. ; -29^° R. 8. 131° F. ; 44° R. 9. 158° F. ; 56° R. 10. 10|°F. ; -9f°R. 11. -13° F. ; -20° R. 12. 149° F. ; 65° C. 13. 88J°F.; 31^° C. 14. 9^°F.; -12^° C. 15. -17^° F.; -27^° C. Page 332. 2. £7 14s. m. 2.5 far. ; £63 19s. 2c?. 2.5 far. 3. 18s. 2d. 1.072 far.; £32 12s. 8d 1.072 far. 4. 15s. IM. 3.65 far.; £28 4s. M. 3.55 far. 5. £2 Os. lid. 2.05 far. ; £43 Os. Id. 3.05 far. 6. 6io/„. 7. 2 y. 1 mo. 15 d. 8. £14 1.3s. 9d. Page 339. 7. 10010111. 8. 114144. 9. 1576^2. 10. 100120100112; 557663; 90e?e. 11. 3230. 12. 1826. 13. 16046. 14. 51692. 15. 100000000000. 16. 547771. 17. 1433423. 18. 1032e. 19. 13122^5. 20. 8536. THIS BOOK IS DUE ON THE T.AST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. Stl- 2- ^^^ i)'"" UL LD 21-100m-8,'34 ' O I /H#C> V