FROM-THE-UBRARY- OF WILLIAM A HILLEBRAND Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/arithmeticforpreOOhobbrich AN ARITHMETIC PREPARATORY SCHOOLS, HIGH SCHOOLS, AND ACADEMIES. BY CHARLES A. HOBBS, A.M., Mastbk of Mathematics in the Belmont School, Belmont, Mass. NEW YORK: A. LOVELL & CO. 1896. hsf COPTBIGHT, By Chablbs A. Hobbs, W'nW^No ikl^'^-'^ p^^ PREFACE. This book is designed particularly for pupils in prepara* tory schools, and it is likewise adapted for the use of all pupils who desire a thorough knowledge of Arithmetic. The four fundamental operations of Arithmetic should be learned by a thorough drill in the elementary schools. This cannot be too strongly insisted upon, as accuracy and readiness of work in all parts of Arithmetic are depend- ent on these fundamental operations. After such a drill the pupil is ready for rapid advancement. Since this fundamental work belongs to the elementary schools, it has been omitted in this book, and the space has been given to examples for more advanced pupils. Special attention has been paid to the selection of exam- ples, over a thousand of which have been taken from entrance papers given at various universities and colleges. The 275 miscellaneous examples at the end of the book are all taken from such entrance papers and from entrance papers given at the United States Military and Naval Academies at West Point and Annapolis. An abundance and variety of examples, sufficient to render the pupil master of the subject, will be found in all parts of the book. In the selection of illustrative examples, great care has been taken to present those which will make clear to the pupil all the difficulties he is liable tg meet. The solu- 995888 IV PREFACE. tions are given in full in order that the principles involved may be clearly understood with but little aid from the teacher. The adaptability of a text-book to school purposes can be determined only by actual use in the school-room. This treatise has already stood this test, since nearly every part of it has been used by the author in classes, whose mem- bers have without exception passed successfully their col- lege entrance examinations in Arithmetic. JSTo attempt has been made to introduce novel methods, but in all cases methods are given which experience has shown to be well adapted to the needs of the pupil. In the arrangement of subjects, no important departure has been made. The Metric System, a thorough knowledge of which is required by all first-class universities and colleges, is given directly after Compound Numbers, and thereafter the two systems are used side by side, thus giving the pupil a thoroughly practical as well as theoretical knowledge of the system. It is expected that the teacher will use his judgment with regard to omissions. In the endeavor to make the book complete, certain subjects have been included which are not necessary to a good knowledge of Arithmetic ; a careful study of college entrance papers shows that these subjects are insisted on by some colleges. The author desires to express his thanks to the many persons who have aided him by valuable suggestions, and also to the many institutions which have responded so promptly and often to requests for entrance papers. CHARLES A. HOBBS= Belmont, July Ist, 1889. CONTENTS Chapteb Paob I. Introduction 1 Casting out the Nines 1 Casting out the Elevens 3 Principles of Multiplication and Division 4 Separation of Terms 4 II Decimal Fractions 7 Exercises in Numeration 8 Exercises in Notation 8 Addition of Decimals 9 Subtraction of Decimals 10 Multiplication and Division by 10, 100, 1000, etc 11 Multiplication of Decimals 12 Contracted Multiplication 14 Division of Decimals 15 Contracted Division 19 United States Money 23 III. Factoks 26 Factoring 28 Greatest Common Divisor 29 Least Common Multiple 32 Cancellation 36 IV. Common Fractions 38 Reduction of Fractions to Lowest Terms 39 Reduction of Improper Fractions to Whole or Mixed Numbers 40 Reduction of Whole or Mixed Numbers to Improper Fractions 41 Least Common Denominator 4? VI CONTENTS. Aud'ition of Fractions 43 Subtraction of Fractions 44 Addition and Subtraction of Fractions Combined 45 Multiplication of Fractions 46 Division of Fractions 49 Short Methods of Multiplication and Division 50 Complex Fractions 52 To Find a Number when a Fractional Part of it is Known 55 To Find what Fractional Part one Number is of Another 57 Reduction of Common Fractions to Decimal Fractions. . 58 Reduction of Decimal Fractions to Common Fractions. . 59 Reduction of Common Fractions to Circulating Decimals 59 Reduction of Circulating Decimals to Common Fractions 61 ■ Greatest Common Divisor of Fractions 62 Least Common Multiple of Fractions 63 V. Compound Numbers 75 Long or Linear Measure 75 Surveyors' Measure 75 Square Measure 75 Cubic Measure 76 Wood Measure 76 Liquid Measure 76 Apotliecjiries' Fluid Measure 77 Dry Measure 77 Troy Weight 77 Apothecaries' Weight 77 Avoirdupois Weight 78 Circular or Angular Measure 78 Measures of Time 79 English or Sterling Money 81 Miscellaneous Tables ^ 81 Reduction Descending 81 Reduction Ascending 83 Addition of Compound Numbers 86 Subtraction of Compound Numbers . , 87 Difference between Dates 89 Multiplication of Compound Numbers 90 Division of Compound Numbers .... 91 CONTENTS. Vll To Multiply or Divide a Compound Number by a Fraction 93 To Reduce a Fraction of one Denomination to Lower Denominations 96 To Reduce Lower Denominations to a Fraction of a Higher Denomination 96 To Find what Fractional Part one Compound Number is of Another 98 To Reduce a Decimal of one Denomination to Lower Denominations 99 To Reduce Lower Denominations to a Decimal of a Higher Denomination 100 To Find what Decimal one Compound Number is of Another 102 Comparison of Weights 102 Comparison of Money 103 Rectangular Surfaces 105 Rectangular Volumes 109 VI. The Metric System 116 Measures of Length 116 Measures of Surface 116 Measures of Volume 110 Measures of Capacity 117 Weight ■. 117 Reduction of Metric Numbers 118 Rectangular Surfaces and Volumes 121 The Metric System Compared with the Common System 124 VII. Special Problems 132 Carpeting Rooms 132 Plastering Rooms 135 Papering Rooms 137 Board Measure , 139 Work Problems 141 Clock Problems 144 Comparison of Thermometers 146 Specific Gravity 148 Longitude and Time 151 VIII. Ratio and Proportion 157 Ratio , 157 VUl CONTENTS. Simple Proportion U>9 Rule of Three 161 Compound Proportion 164 Cause and Effect 167 Partitive Proportion 171 Simple Partnership 173 Compound Partnership 174 Averages or Alligation 177 IX. Percentage 185 To Express a Rate Per Cent as a Common Fraction .... 185 To Express a Common Fraction as a Rate Per Cent .... 186 To Find any Per Cent of a Number 187 To Find the Base when any Per Cent of it is Known . . . 189 To Find what Per Cent one Number is of Another 192 Profit and Loss 194 Commercial Discount 198 Commission 201 Insurance 205 Taxes 208 Duties 211 X. Interest and Discount 216 Simple Interest 216 Exact Interest 221 To Find the Rate Per Cent 222 To Find the Time 224 To Find the Principal ,. 226 Promissory Notes , 228 Partial Payments „ 231 Compound Interest 236 Annual Interest 239 True Discount 240 Bank Discount 242 To Find the Face of a Note to Yield a given Proceeds. . 245 Exchange 247 Domestic or Inland Exchange 249 Foreign Exchange 252 Equation of Payments , 254 Average of Accounts 258 XI. Stocks .,....,...,..,..,,,.,.,,..,.,,.,.,.,,.. 26^ . CONTENTS. IX XII. Involution and Evolution 271 Involution 271 Evolution 272 Square Root 273 Cube Root 278 Higher Roots 283 XIII. Series 284 Arithmetical Progression 284 Geometrical Progression 288 Compound Interest 292 Annuities 294 Annuities at Simple Interest 295 Annuities at Compound Interest 296 XIV. Mensuration 299 Definitions • 299 Triangles 301 Right Triangles 303 Quadrilateraii, 306 Circles 307 Prisms and Cylinders 310 Pyramids and Cones 312 Spheres 316 Similar Surfaces and Solids 316 Miscellaneous Examples 319 ARITHMETIC. CHAPTER I. INTRODUCTION. ' 1. Before studying this book the student should be perfectly familiar with the four fundamental operations of Arithmetic — Addition, Subtraction, Multiplication, and Division. In order to ensure accuracy it is always advisa- ble to test each step of work. In Addition add the column downwards and then up- wards, and if the results are alike, they may be considered correct. In Subtraction the best test is to add the subtrahend and the remainder, and if the work is correct, the sum is the same as the minuend. In Multiplication and Division the easiest way to test the work is simply to repeat each step ; however, if further tests are desired, the following can be used: in Multipli- cation divide the product by the multiplier, and if the work is correct, the result is the same as the multiplicand ; in Division multiply the quotient by the divisor and add the remainder, if any, and if the work is correct, the result is the same as the dividend. 2. The fundamental operations can also be tested by the method known as casting out the nines. The excess of 2 ARITHMETIC. nines is a term used to denote the remainder arising from dividing a number by 9. 10 = 9 +1, 1000 = 9 X 111 +1, 100 = 9 X 11 + 1, 10000 = 9 X 1111 + 1, etc. We thus see that a unit of any order equals one more than 9 multiplied by some number. From this it follows that any number of units of any order equals that number of units aiddedtjo .'9 multiplied by some number. For example, .50 c=.9.x. 5 + 5, 600 = 9x66 + 6, 7000=9x777 + 7, etc. "'i'hemfore ei'ery ijUmber consists of a certain number of nines increased by the sum of its digits. For exa;mple, 7654 equals a number of nines increased by 7 + 6 + 5+4; this excess equals 9x2 + 4, so that the excess of nines is 4. This process can be applied to any number ; hence the excess of nines in any number equals the excess of nines in the sum of its digits. Addition. 82765 ... 1 The excess of nines in the first number 4912 ... 7 is 1, in the second number 7, in the third 25754 ... 5 number 6, and in the fourth number 0. 6732 ... The sum is 13, and the excess of nines is 4 . . . 120163 13 ... 4 4. The sum of the numbers is 120163, in which the excess of nines is 4. The ex- cess is the same in each case ; therefore the work may be considered correct. Subtraction. 89643 3 The minuend equals the sum of the 3598 ... 7 subtrahend and remainder. The excess 86045 ... 5 o^ nines in the minuend is 3, The excess To Q of nines in the subtrahend is 7, and in the remainder 5. The excess in the sum of these two excesses is 3, the same as in the minuend ; therefore the work may be considered correct. INTRODUCTION. Multiplication. 857 ... 2 The excess of nines in the multiplicand is 62 ... 8 2; in the multiplier 8. The product is 16, and 1714 16 ... 7 *^® excess of nines is 7. The product of the ^14.2 numbers is 63134, in which the excess of nines y is 7. The excess is the same in each case; •therefore the work may be considered correct. 53134 Division. 563)87614(155 87614 ..8 563 563... 5 3131 155 ... 2 2815 349 ... 7 3164 5x :2 + 7 = 17.. ..8 2815 349 The dividend equals the product of the divisor and quotient plus the remainder. The excess of nines in the dividend is 8. The excess of nines in the divisor is 6, in the quotient 2, and in the remainder 7. The product of 5 and 2 plus 7 equals 17, in which the excess is 8, the same as in the dividend ; therefore the work may be considered correct. 3. The excess of elevens is a term used to denote the remainder arising from dividing a number by 11. 10 = 11-1, 1000 = 11 X 91 -1, 100 = 11 X 9 + 1, 10000 = 11 X 909 + 1, etc. We thus see that a unit in an odd place equals one more than 11 multiplied by some number, and a unit in an even place equals one less than 11 multiplied by some number. From this it follows that any number of units in an odd place equals that number of units added to 11 multiplied by some number, and any number of units in an even place equals that number of units subtracted from 11 multiplied by some number. Therefore every number consists of a 4 ARITHMETIC. certain number of elevens increased by the sum of the digits in the odd places, decreased by the sum of the digits in the even places. For example, 45316 equals a number of elevens increased by 4 -f- 3 + 6, decreased by 5 + 1, and the excess of elevens is 13 — 6, or 7. 18394 equals a num- ber of elevens increased by 1 + 3-1-4, decreased by 8 -f- 9 ; 17 cannot be subtracted from 8, so one of the elevens is added to 8, and the excess of elevens is 11 -f- 8 — 17, or 2. This process can be applied to any number ; hence the excess of elevens in any number can be found by subtracting the sum of the digits in the even places from the sum of the digits in the odd places (increased, if necessary, by 11 or some number of IV s). The processes of testing the fundamental operations by casting out the elevens are similar to those by casting out the nines. 4. There are a few principles of Multiplication and Divis- ion which should be borne in mind. (1) Multiplying either multiplicand or multiplier by a number multiplies the product by the same number. (2) Dividing either multiplicand or multiplier by a num- ber divides the product by the same number. (3) Multiplying the dividend or dividing the divisor by a number multiplies the quotient by the same number. (4) Dividing the dividend or multiplying the divisor by a number divides the quotient by the same number. (5) Multiplying or dividing both dividend and divisor by the same number does not change the quotient. 5. The student should always take great care in his use of signs. The signs + and — always denote the points of separation in a problem, and the parts between these signs INTRODUCTION. 5 are called terms. Each term should be simplified by itself before the operations of addition and subtraction are per- formed. In the expression 15 + 12^3—5x2 there are three terms, 15, 12 -^ 3, and 5x2; hence the value is 15_|.4-10 = 9. The parenthesis, (), is used to denote that the expression contained therein should be used as a whole. The expres- sion should be simplified at first, and its value substituted. For example, (18+7-5) -i-(10-6) + 3x8 = 20-!-4 + 3xB = 5 + 24=29. Note. Brackets, [], braces, {}, and the vinculum, , may b« used with the same meaning as a parenthesis. EXAMPLES. Find the value of 1. 28x4 + 32-i-8-16. 2. 10 + 6x3-24-*-6 + 12. 3. 56-J-7 + 12X 3-52-2. 4. 22-6x3 + 2x5 + 50-h25. 5. 28x6^14 + 9x8-12 + 42-^-7x3. 6. 99x8 + 51x10-7x104 + 26. 7. 99x(8 + 51)xl0-(7xl04 + 26). 8. (99 X 8) + (51 X 10 - 7) X 104 + 26. 9. (99x8 + 51)xl0-7x(104 + 26). 10. (99x8) + (51x10) -(7x104) + 26. 11. 99 X 8 + 51 X (10 -7) X 104 + 26. 12. 99 X (8 + 51) X (10 - 7) X (104 + 26). 13. (105 -i- 21 + 80 -5- 5) X (81 + 36 -9). 14. (3146 + 279 - 2141) - (370 - 263) + 91 x 3. 6 ARITHMETIC. 15. (2142 - 1729) X (3666 -2514) --(354 -5- 6). 16. 327 X 6 -- 109 + 52 X 5 - (42 + 8 X 4). 17. (46-8)xll+17xl5-f(83x4-327)xl0-39xl4. 18. 864 -J- 12 - 124 -- (775 --- 25) + 54 -f- (61 - 34). 19. 949 - 13 - (119 -f- 7 + 1176 -i- 21) + 3648 ^ 32 - (306 H51-5 + 672--6). 20. From 126+ (16+4) X 2 take(48-5-2) +34 x 6--(17-5). DECIMAL FRACTIONS. CHAPTER II. DECIMAL FRACTIONS. 6. When a number is nsed to express whole units, it is called an integer, or integral number. As in the decimal system of notation a unit of any order has one tenth the value of a un^t of the next order to the left, we can continue our notation toward the right beyond units' place. To do this, we place a decimal point (.) imme- diately after the number in units' place, and the next place to the right is called tenths; a digit in this place has one tenth the value of the same digit in units' place. For example, .3 is read three tenths. The places are continued to the right indefinitely, and the % i names are given in the table an- I I ^ § nexed. The places at the right of the decimal point are called decimal 5^ S-.D- 2'<» 9 :5 a 3 ja 0)3"- § 3 « places, and the numbers thus writ- n n n n n ^ ten are called decimal fractions, or decimals. They are read like whole numbers, adding the name of the right-hand place. For example, .67 is read sixty-seven hundredths. It is to be noticed that the number consists of six tenths and seven hundredths, but as six tenths is the same as sixty hun- dredths, the entire decimal is sixty-seven hundredths. .5432 is readj^-ye thousand four hundred thirty-two ten-thousandths. When whole numbers and decimal fractions are written together, the word "and" should be used to connect the 8 ABITHMETIC. two parts. For example, 362.459 is read three hundred sixty- two and four hundred Jifty -nine thousandths. Note. Zeros may be annexed or omitted at the right of a decimal fraction without altering the value, for either with or without th«se zeros the significant digits are in the same decimal places. When a decimal is written without an integer, a zero may be pu* m units' place, or the place may be left vacant. EXERCISES IN NUMERATION. Read the following numbers : 1. .6. 2. .572. 3. .05072. 4. .090067. 5. .31402. 6. .00000041. 7. 27.8. 8. 2.78. 9. .6008. 10. 6000.0008. 11. 41.002. 12. 750.0081. 13. 75000.81. 14. 3.14159. 15. 28.0097. EXERCISES IN NOTATION. Write the following in figures : 1. Nine tenths. 2. Eighty-two hundredths. 3. Eighty-two ten-thousandths. 4. Three hundred sixty-one thousandths. 5. Three hundred sixty-one millionths. 6. Nine thousand two hundred nine hundred-thousandths. 7. Thirty thousand six hundred-thousandths. 8. Thirty thousand and six hundred-thousandths. 9. Sixteen and three hundred forty-one thousandths. 10. One hundred fifty-five millionths. DECIMAL FRACTIONS. \f 11. One hundred and fifty-five millionths. 12. Six and five ten-millionths. 13. Sixty-five ten-millionths. 14. Forty-five and fifty-eight hundredths. 15*. Three hundred twent}^-nine thousandths. 16. Three hundred and twenty-nine thousandths. Addition of Decimals. 7. Units of the same order should be written in the same column. This can easily be accomplished by making the decimal points fall under each other. The process of adding is precisely the same as in whole numbers. I. Find the sum of 18.47, 159.363, 70.00451, and 0.926. 18.47 159.0DO ^j^Q begin at the right and add as in whole numbers. 7U.UU4ol The decimal point comes under the decimal points of ^•^^^ the problem. 248.76351 EXAMPLES. 1. Add together 12.613, 0.00175, 257.8425, ar-d 0.001345. 2. Add together 17.429, 0.0173, 1156.8, and 0.0001723. 3. Add together 0.20765, 0.00631, 6758.13247, and 5.973. 4. Add together 3107.8192, 0.0624, 0.00414, and 47.2875. 5. Add together 371.87007, 0.00731, 5768.45321, and 0.0093. 6. Add together 11.431, 0.00101, 243.342, 400, and 1.3734. 7. Add together 16.41215, 9.736, 0.00304, 188.24, and 29.03069. 10 ARITHMETIC. 8. Add together 0.61692, 243.734, 901, 68.45213, and 8.386. 9. Add ten thousand and one millionth, four hundred- thousandths, ninety-six hundredths, and forty-seven million sixty thousand and eight billionths. 10. Find the sum of the following numbers : fifty-seven and three thousandths, three hundred and sixty-four hun- dred-thousandths, forty-seven thousand and eight thousand seven hundred-thousandths, eighty-seven millionths, and four hundred and twenty-seven ten-thousandths. Subtraction of Decimals. 8. The process is the same as in whole numbers, taking care to have the decimal points under each other. I. From 934.2963 subtract 47.794. 934.2963 in the subtrahend there is no digit in the place of ten- 47.794 thousandths, so we annex a zero mentally, and this zero 886 5023 subtracted from the 3 of the minuend leaves 3. 11. From 35.2 subtract 24.543. 35.2 In the minuend there are no digits in the places of 24.543 hundredths and thousandths, so we annex two -zeros men- 10 657 tally, and then subtract as in whole numbers. EXAMPLES. 1. Subtract 284.7654 from 321.07659. 2. Subtract 17.2398 from 27.06. 3. Subtract 29.9189 from 240.775. 4. Subtract 84.736568 from 100.3064231. 5. Subtract 49.934 from 500.39. 6. Subtract 70.2574 from 365.71. DECIMAL FRACTIONS. 11 7. Sul)tract 876.351 from 1000.01. 8. Subtract 185.939131 from 186.847. 9. Find the difference between 0.0000005 and 0.00005. 10. From ten take six millionths. 11. From two hundred and six thousandths take two hundred six thousandths. 12. What is the value of thirty-six million minus thirty- six millionths ? Multiplication and Division by 10, 100, 1000, etc. 9. Since a unit of any order is ten times as large as a unit of the next order to the right, when we move a digit one place to the left, we multiply it by 10. This is the same as moving the decimal point one place to the right. For example, 70 X 10 = 700 ; 85.43 x 10 = 854.3. If the dec- imal point be moved two places to the right, the number is multiplied by 100; to multiply by any number of lO's, the decimal point is moved as many places to the right as there are zeros in the multiplier. For example, 0.95 x 100 = 95 ; 2.5 x 1000 = 2500 ; 0.00043 x 10000 = 4.3. For the same reason, when we move a digit one place to the right, we divide it by 10. This is the same as moving the decimal point one place to the left. For example, 26 H- 10 = 2.6 ; 0.49 ^ 10 = 0.049. To divide by any num- ber of lO's, the decimal point is moved as many places to the left as there are zeros in the divisor. For example, 195-5-100 = 1.95; 4.53 -- 10000 = 0.000453. In like manner, to multiply by 0.1, 0.01, 0.001, etc., move the decimal point as many places to the left as there are decimal places in the multiplier. To divide by 0.1, 0.01, 0.001, etc., move the decimal point as many places to the right as there are decimal places in the divisor. 12 ARITHMETIC. Find the value of 1. 8.7x10. 2. 0.0069x10. 3. 95.6x100. 4. 0.0453 X 100. . 5. 4.069 X 1000. 6. 0.00094x10000. 7. 9.2-10. 8. 7.49-^100. 9. 0.036-100. 10. 854.3 -r- 1000. 11. 1.00182 -J- 1000. 12. 76.541-10000. EXAMPLES. 13. 4.7x0.1. 14. 8.76x0.01. 15. 0.0469x0.01. 16. 0.037x0.001. 17. 4.62x0.001. 18. 573.7x0.00001. 19. 10 - 0.1. 20. 53.4 - 0.01. 21. 97.42-0.001. 22. 0.48-0.001. 23. 0.1-0.0001. 24. 7.32-0.00001. Multiplication of Decimals. 10. I. Multiply 4.92 by 0.3. 4.92 If units of any order be multiplied by an integer, the 0.3 product consists of units of the same order. If 4.92 be multiplied by 3, the product is 14.76. But the multiplier is 0.3, a number one tenth as large ; hence the product is one tenth as large, and the decimal point must be moved one place to the left, giving 1.476 for the answer. 1.476 II. Multiply 0.718 by 0.028. 0.718 0.028 5744 1436 0.020104 0.718 multiplied by 28 would give 20.104. But the multiplier is only one thousandth of 28 ; hence the deci- mal point must be moved three places to the left, giving 0.020104 for the answer. DECIMAL FKACTIONS. 13 Fvom tuese two examples we see that we multiply as in whole numbers, pointing off as many decimal places in the product as there are in both midtiplicand and multiplier. EXAMPLES. 1. Multiply 6.4 by 1.5. 2. Multiply 0.64 by 0.15. 3. Multiply 0.09 by 0.0016. 4. Multiply 0.427 by 345. 5. Multiply 76000 by 1.05. 6. Multiply 0.076 by 0.0105. 7. Multiply 37900000 by 2.005. 8. Multiply 0.0379 by 0.2005. 9. Multiply 34.27 by 60000. 10. Multiply 200.043 by 2.021. 11. Multiply 0.785 by 0.0191. 12. Multiply 2.708 by 0.007005. 13. Multiply 947.36 by 0.00423. 14. Multiply 2.708 by 70050000. 15. Multiply 8.764 by 40.015. 16. Multiply 25.3784 by 12.567. 17. Multiply 0.0400268 by 0.260075. 18. Multiply 3 hundredths by 300 thousandths. 19. Multiply five thousand and three ten-thousandths by five thousand three ten-thousandths. 20. Multiply twelve thousand five hundred and six hun- dred seventy-five millionths by four thousand sixteen ten- thousandths, 14 ARITHMETIC. 21. Multiply six hundred twenty-five ten-millionths by three hundred and eight thousandths. Contracted Multiplication of Decimals. 11. In many examples in multiplication of decimals, only a certain number of accurate decimal places are required in the product. All extra work involving figures beyond the required degree of accuracy can be avoided by the use of the following method : Invert the order of the figures of the multiplier, and place them so that the tenths' figure may be under that order of decimals to which it is proposed to limit the prod- uct. Multiply the multiplicand by each figure of the mul- tiplier, beginning at the figure immediately above it, and taking in the number carried from the right hand. Place the first figure of each partial product in the same column, and add the partial products, rejecting the sum of the right- hand column, after carrying the nearest ten. I. Multiply 29.637842 by 85.916, the result to be correct to three decimal places. 29.637842 In order to ensure accuracy to tliree decimal places, 61958 the partial products should be accurate to four decimal 23710274 places. Hence four decimal places must be multiplied 1481892 ^y *^® units' figure, and 5 is placed under the figure 266740 in the fourth decimal place ; this brings the figure 2964 in tenths' place under the order of decimals to which 1778 the product is to be limited. As the multiplier is now 2546.365 arranged, each partial product obtained by beginning to multiply at the figure directly above contains four decimal places ; hence the first figures of the partial products are to be placed in the same column. Begin at the right to multiply. 8 X 4 = 32 ; however, 2 must be carried from the right, for 8x2 = 16, which is nearer 20 than 10; hence we set down 4 and carry 3 to the next place. The process is t^hen continued as in ordinary multiplication^ DECIMAL FKACTIONS. 16 EXAMPLES. 1. Multiply 3.7185625 by 2.2134125, the result to be correct to two decimal places. 2. Multiply 8.170663 by 1461.203, the result to be cor- rect to three decimal places. 3. Multiply 78.5126 by 37.8759, the result to be correct to four decimal places. 4. Multiply 375.76843 by 3.14159, the result to be cor- rect to four decimal places. 5. Multiply 13.50629 by 0.36472, the result to be cor- rect to five decimal places. 6. Multiply 5.7203716 by 2.71728, the result to be cor- rect to five decimal places. 7. Multiply 87.896397 by 3.5298875, the result to be correct to five decimal places. 8. Multiply 0.86858896 by 1.0986123, the result to be correct to six decimal places. 9. Multiply 0.69314718 by 0.43429448, the result to be correct to seven decimal places. 10. Multiply 3.1415926 by itself, the result to be correct to seven decimal places. Division of Decimals. 12. Since units of any order multiplied by a whole num- ber yield units of the same order in the product, units of any order divided by units of the same order yield a whole number as the quotient ; in other words, with equal decimal plac^$ ill dividerid and divisor the quotient is (i whole number. 16 ARITHMETIC. I. Divide 38.4 by 6. Since the divisor is a whole number, the quotient 6 )oo.4 jg j^ y^]iQ\Q number as far as the dividend is a whole 6.4 number. Hence the decimal point in the quotient is directly below the decimal point of the dividend. II. Divide 0.58961 by 0.07. Mark off by a star as many decimal places in 0.07 )0.58^961 the dividend as there are in the divisor. Then the 8.423 quotient is a whole number as far as this star, and the decimal point is directly below the star. III. Divide 3.12 by 8000. 8000 ) ^003.12 If the decimal point be moved three places to 0.00039 *^^® ^^^^ ^^ ^°*^^ dividend and divisor, the quotient is not changed, and the problem becomes .00312 divided by 8, which equals .00039. If, however, we cross off the zeros in the divisor, and place a star three places to the left of the decimal point in the dividend, the effect is the same. In Long Division the quotient can be written above the divi- dend, and then the decimal point is directly above the star. IV. Divide 82.32 by 2.1. 39.2 2.1)82.3^2 63 Since there is one decimal place in the divisor, 193 the star comes after the 3, and in the quotient the 189 decimal point is directly above the star. "~42 42 V. Divide 0.03969 by 4900. 0.0000081 Since there are two zeros at the end of the di- 4900)^00.03969 visor, the star should be placed two places to the 392 left of the decimal point. Then divide by 49, and ' 49 the decimal point in the quotient is directly above 49 the star. DECIMAL FRACTIONS. 17 VI. Divide 403920 by 0.00108. 374000000. 0.00108)403920.00000^ In order to make decimal places even, 324 zeros must be annexed to the dividend. 799 Then the division is as before, and the 756 decimal point in the quotient is directly 432 above the star. 432 VII. Divide 0.05 by 4.3 to five decimal places. 0.01162 -}- In this example the divisor is not con- 4 3'iO 0^50000 tained an exact number of times in tlie 43 dividend. In all such cases it is cus- tomary to carry the division to a certain *|^ number of decimal places, and then stop. ^^ The answer can be written 0.01 162 +, the 270 + sign denoting that there is a remainder. 258 If, however, the + sign is omitted, the answer should be written 0.011 Go, which would be nearer correct than 0.01162, because the next figure would be greater 34 than 5. From these examples we see that the method for Division of Decimals is as follows : If necessary^ annex zeros to the dividend in order to make the decimal places equal in dividend and divisor. Then mark off by a star as many decimal places in the dividerid as there are in the divisor. Divide as in whole numbers, and in the quotient place the decimal point directly belotv or above the star. When the divisor is a whole number ending in zeros, cross off the zeros, and place the star in the dividend as many places to the left of the decimal point as there are zeros in the divisor. Note. In problems where the divisor is not contained an exact number of times in the dividend, five decimal places in the quotient is ordinarily far enough to carry the division. If the next digit is to be less than 5, keep the last digit as it comes in the division ; if the next diffit is to he 5, or more, increase the last digit hy one. 120 86 18 ARITHMETIC. EXAMPLES. 1. Divide 769.428 by 200. 2. Divide 76.9428 by 0.0002. 3. Divide 0.000064 by 0.008. 4. Divide 9.00081 by 900. 5. Divide 0.000144 by 120000. 6. Divide 0.01625 by 0.000025. 7. Divide 0.000744 by 0.62. 8. Divide 67.56785 by 0.035. 9. Divide 0.09 by 0.0016. 10. Divide 287.1 by 3300. 11. Divide 0.0002548 by 0.0364. 12. Divide 0.000647808 by 6.72. • 13. Divide 0.00309824 by 0.376. 14. Divide 2926.5 by 0.3902. 15. Divide 29.265 by 390.2. 16. Divide 1.096641 by 1521. 17. Divide 0.0018891 by 3.75. 18. Divide 190.914 by 270800. 19. Divide 10.85 by 0.0775. 20. Divide 3336.894963 by 72530-. 21. Divide 0.00091471 by 9.43. 22. Divide 189695.4 by 2.708. 23. Divide 76.125 by 463000. 24. Divide 8.21 by 0.41. 25. Divide 0.821 by 410. DECIMAL FRACTIONS. 19 26. Divide 0.314 by 1.785. 27. Divide 0.10724 by 0.003125. 28. Divide 2.838913 by 708.4. 29. Divide 0.011825369 by 5.884. 30. Divide 695.57270875 by 52.35775. 31. Divide four millionths by four million. 32. Divide 300 thousandths by 3 hundred-thousandths. 33. Divide sixteen thousandths by forty-five hundred. 34. Divide eighty-four and eighty-four hundredths by forty-eight thousandths. 35. Divide fifty millionths by six hundred twenty-five ten-thousandths. 36. Divide two thousand five hundred one and four tenths by four thousand one hundred twenty-five ten-mil- lionths. 37. Divide 59285 ten-millionths by 835 hundred-thou- sandths. 38. Divide four thousand three hundred twenty-two and four thousand five hundred seventy-three ten-thousandths by eight thousand and nine thousandths. Contracted Division of Decimals. 13. When examples in division of decimals are required to be accurate only to a certain number of decimal places, the work can be shortened by the use of the following method : Determine by inspection the position of the decimal point in the quotient, and the number of significant figures in the quotient can at once be determined. Write the divisor so 20 ARITHMETIC. as to contain two more figures than the quotient. Cut off the right-hand figure of the divisor, and then divide; in multiplying the divisor by the figure of the quotient, the product must be increased by the number carried from the right hand. Instead of bringing down each time a figure at the right of the remainder, cut off the right-hand figure of the divisor, and proceed as before. I. Divide 7.97647964 by 3.7876476, the result to be cor- rect to four decimal places. 2.1059 3 is contained in seven twice ; hence 3 787^8^7 976479^64 *^^ quotient contains one integral place, 757530 ^"^ *^^ entire number of figures in the ^rv-j-jo quotient is five. Write the first seven S7X76 figures of the divisor, changing 7 to 8 on account of the 6 dropped. Before . gQ . dividing cut off the 8 in the divisor. Then in multiplying by 2, 2 must be ^'*^ carried from the right hand, for 2 X 8 §^ ^ 16, which is nearer 20 than 10. The first remainder is increased by 1 on account of the 9 in the dividend. Then cut off the 4 in the divisor aM proceed as before. Note. If necessary, zeros can be annexed to the divisor to make tho required number of figures ; zeros occurring before the first significant figure are not to be counted. EXAMPLES. 1. Divide 3698.779375 by 375.625, the result to be cor. rect to three decimal places. 2. Divide 0.046 by 0.00762089, the result to be correct to four decimal places. 3. Divide 0.32165 by 0.003516, the result to be correct to four decimal places. 4. Divide 0.765439 by 359.21, the result to be correct to five decimal places. DECIMAL FRACTIONS. 21 5. Divide 0.22165 by 0.0035216, the result to be correct to five decimal places. 6. Divide 6.38572164 by 0.0752681, the result to be cor- rect to five decimal places. 7. Divide 29.48495554 by 378.6725, the result to be cor- rect to six decimal places. 8. Divide 100.016 by 3.056, the result to be correct to six decimal places. 9. Divide 0.765439 by 359.21, the result to be correct to seven decimal places. 10. Divide 2.71828128 by 3.1415926, the result to be cor- rect to seven decimal places. MISCELLANEOUS EXAMPLES. 1. Simplify 8.763 - 4.12 + 78.326 -t- 1.1126 - 68.0816. 2. Simplify 198.63 + 21.3711 - 100.416 - 45.79 -f 8.3. 3. Simplify 8.72 X 5.4 + 196 x 0.004 - 6.25 x 4.8 - 0.06 x 21.7. 4. Simplify 3.71 X 8 + 2.64 -- 160 -f 7.55 x 0.07 + 0.071 x 25. 5. Simplify 84 X 1.13- (66-1.2 X 2.4) -f-100 x (4 x 0.018+0.189). 6. Simplify 94.5 - 250 + 16^-. (4.5 --0.225) +87.25^ (1.6- 0.35). 7. Simplify (15 - 10 x 0.3) x 6.192 - (7 x 5.4-35.048). 8. The difference between two numbers is 94.32^ and the smaller is 147.631 ; find the larger. 9. Divide the sum of four thousandths and four mil- lionths by their difference. 22 ARITHMETIC. to. Divide 876.196 by 2.12. If the decimal point were moved in the dividend two places to the left, and in the divisor one place to the right, how many times greater or less would the quotient be ? 11. Miiltiply forty-eight ten-thousandths by two and one thousandth, and divide the result by one million. 12. Divide 375 by 0.75 and 0.75 by 375, and find the sum and difference of the quotients. 13. The product of three numbers is 5.76 ; one of them is 0.024, and another is 0.06 ; find the third. 14. The product of three numbers equals 70.04597 ; two fi them equal 3.91 and 3.0005 respectively ; find the third. 15. What number divided by 28.15 will give 1.216 as the quotient and 1.5195 as the remainder ? 16. The dividend is 7423.973, the quotient is 12.13, and the remainder is 0.413 ; what is the divisor ? 17. Find the number of rods of fence necessary to enclose a field, the sides of which are respectively 42.78 rods, 51.3 rods, 27 rods, and 37.22 rods. 18. Two men walk respectively 26.7 miles and 22.94 miles per day ; how much further does the first walk than the second ? 19. If the year is considered as 365.25 days instead of 365.242264 days, how great will be the error in 1880 years ? 20. From a tank containing 1200 gallons, 22.75 barrels of 31.5 gallons each were pumped out; how many gallons remained ? 21. How many barrels, each containing 44.5 gallons, can be filled from 16554 gallons of oil ? 22. 4 cords of wood are worth as much as 13.4 bushels of rye ; how much rye can be obtained for 15 cords of wood ? DECIMAL FRACTIONS. 23 23. A merchant bought 972 bushels of wheat ; how many bins, each containing 16.25 bushels, will be filled, and how much remains ? 24. Two wheels of a carriage are respectively 13.5 feet and 11.75 feet in circumference ; how much oftener does one turn than the other in going 4000 feet ? United States Money. 14. The money in use in the United States is expressed in a decimal system, of which the unit is a dollar. The symbol for dollars ($) is placed before the number used to represent them. One dollar equals 100 cents (cts,), and cents are written as a decimal fraction of a dollar. For example, $6.43 means six dollars and forty -three cents. There are three other denominations sometimes mentioned, which are not in common use, — ten dollars equal one eagle, ten cents equal one dime, and one tenth of a cent is a mill. All operations in United States Money (sometimes called Federal Money) are performed as in decimal fractions. Note 1. In general when the final result in a problem contains mills, if less than 5 they are rejected, if 5 or more they are called another cent. If, however, the final result is the value of one article where a number are considered, any fraction of a cent should be retained. Note 2. In business transactions C is often used for hundred, and M for thousand, when the price is by the hundred or by the thousand. EXAMPLES. 1. A farmer sold 24 cows for $32.25 apiece ; how much did he receive ? 2. A drover sold 42 hogs for $246.75 ; how much apiece did he receive ? 24 ARITHMETIC. 3. A merchant paid $52 for a lot of cloth at 8 cts. a yard ; how many yards did he buy ? 4. A merchant's receipts for a week were as follows : Monday, $102.79; Tuesday, f72.73; Wednesday, $150.65; Thursday, $127.70 ; Friday, $205 ; Saturday, $278.92. Find the amount of his receipts for the entire week. 5. A clerk has a yearly salary of $1000 ; he pays $312 for board, $157.50 for clothing, and $372.25 for all other expenses. How much does he save in a year ? 6. Bought three loads of wood, containing respectively 2.15 cords, 1.98 cords, and 1.625 cords ; find the cost at $3.15 a cord. 7. Find a man's daily wages when he was paid $29.70 for 22 days' work. 8. At $12,375 a ton, how many tons of hay can be bought for $2326.50 ? 9. Bought 3 pounds of tea at 72 cts. a pound, 8 pounds of coffee at 28 cts. a pound, and 15 pounds of rice at 6 cts. a pound ; find the amount of the bill. 10. If $31.75 be paid for 5 barrels of flour, what would 28 barrels cost at the same rate ? 11. If 0.62 of a ton of hay be worth $11.47, find the value of 8.75 tons. 12. A farmer sold in one month 62 pounds of. butter at 28 cts. a pound, 45 dozen eggs at 18 cts. a dozen, and 27 chickens at 55 cts. apiece ; find the amount of his receipts. 13. A lady bought 12 yards of crash at 14 cts. a yard, and 8 yards of cotton cloth at 18 cts. a yard, and gave a $5 bill in payment ; how much change should she receive ? 14. If the price of gas be $1.75 per M, find the amount of a man's bill when 12240 cubic feet have been consumed. DECIMAL FKACTIONS. 25 15. Sold 7250 cigars at $4.20 per C; find the amount received. 16. Paid $10.44 for 1440 bricks j what was the price per M ? 17. A pedler sells beets, six in a bunch, at 10 cts. a bunch, and gains 1 ct. on each bunch ; find the cost per C. 18. How many tons of coal at $4.75 a ton must be given in exchange for 19 barrels of flour at $6.25 a barrel ? 19. How many dozen eggs at 18 cts. a dozen must be given in exchange for 28 pounds of sugar at 11 cts. a pound and 8 pounds of coffee at 29 cts. a pound ? 20. A merchant bought a load of grain for $50, and by retailing it at $1.20 a bushel, he gained $22; how many bushels were there in the load ? 21. To send a telegram from New York to Boston costs 25 cts. for 10 words and 2 cts. for each additional word ; find the cost of a telegram containing 28 words. 22. A grocer bought 16 barrels of sugar, each containing 232 pounds, for $335, and sold it at 10 cts. a pound ; how much was his gain ? 23. Bought a roll of carpet, containing 82 yards, for $45, and sold it for 75 cts. a yard ; find the amount of profit. 24. Bought a horse for $125, a carriage for $140, and a harness for $18 ; kept them a month at an expense of $17.25, and then sold the team for $300. Did I gain or lose, and how much ? 25. A merchant bought 150 barrels of apples for $300 ; he sold seven tenths of them at $2.25 a barrel, and the re- mainder at $1,875 a barrel. Did he gain or lose, and how much? 26 ARITHMETIC, CHAPTER III. FACTORS. 15. A number which can be contained in another without a remainder is called a divisor or factor of that number. When a number has no factor except itself and one, it is called a prime number ; when it has other factors besides itself and one, it is called a composite number. When two numbers have no common factor except one, they are said to be prime to each other. Numbers of which 2 is a factor are called even numbers ; all others are odd numbers. When a number is applied to some particular object or objects, it is called a concrete number; when not applied to any object, it is called an abstract number. For example, 4 and 7 are abstract numbers, but 4 boys and 7 books are concrete numbers. An exponent, or index, is a small figure placed at the upper right-hand corner of a number to show how many times it is used as a factor. For example, 5* =5x5x5x5. 16. For determining at sight whether certain numbers are contained in a given number, the following tests can be used: (1) A number is divisible by 2 if its right-hand figure is zero or an even digit. (2) A number is divisible by 3 if the sum of its digits is divisible by 3. For example, in 741, 7 -f 4 -f- 1 = 12, which is divisible by 3 j hence 741 is divisible by 3. FACTORS. 27 (3) A number is divisible by 4 if the two right-hand figures are zeros, or if the number expressed by them is divisible by 4. (4) A number is divisible by 5 if its right-hand figure is or 5. (6) A number is divisible by 6 if it is an even number, and at the same time is divisible by 3. (6) A number is divisible by 8 if its three right-hand figures are zeros, or if the number expressed by them is divisible by 8. (7) A number is divisible by 9 if the sum of its digits is divisible by 9. (8) A number is divisible by 10 if its right-hand figure is 0. (9) A number is divisible by 11 if the sums of the alter- nate digits are the same, or if the difference between these sums can be divided by 11 . For example, in 7458, 7 + 5 = 4 + 8 ; hence 7458 is divisible by 11. In 19382, l-f-3-f-2 = 6, 9 4- 8 = 17, and the difference between 6 and 17 is 11 ; hence 19382 is divisible by 11. 17. To find whether a number is prime or composite, divide by the prime nup\bers in succession until one of them is contained in the number, or else the quotient is less than the divisor. In the former case the number is compos- ite ; in the latter case t>>iB number is prime. For example, take 491. Dividing in succession by 2, 3, 5, 7, 11, 13, 17, and 19, none of them ar? contained in 491, and in every case the quotient is greater than the divisor. Divide by 23, and the quotient is 21 with a remainder ; hence 491 is a prime number. If the number were composite, both divisor and quotient would be factors. We have seen that 491 has no factor less than 23, and as the divisors grow greater, the quotio^:>ts grow less and will be less than 23. Then since 28 ARITHMETIC. there is no factor less than 23, in no case will the quotient be a whole number, and there are no prime factors for the number. Factoring. 18. The prime numbers which multiplied together pro- duce a given composite number are called the prime factors of that number. I. Find the prime factors of 182. 2^ 182 Divide by 2, the least number that is a ijTKT factor of 182. Then divide the quotient — r^ by 7, and we find that the three prime factors of 182 are 2, 7, and 13. For 182 = 2 X 7 X 13. convenience of work it is best always to divide by the least possible factor. II. Find the prime factors of 3465. 3 )3465 3 )1155 5)385 When the same factor occurs more jyfj than once, it is best to write that factor ~7T with an exponent. 3465 = 32 X 5 X 7 X 11. EXAMPLES. Find the prime factors of the following numbers : 1. 176. 7. 792. 13. 4800. 2. 210. 8. 1221. 14. 6902. 3. 360. 9. 1836. 15. 8364. 4. 384. 10. 1872. 16. 10917. 5. 432. 11. 2310. 17. 37125. 6. 48C 12. 2346. 18. 179487. FACTORS. 29 19. Which of the numbers 5, 9, 13, 18, 21, and 25 are prime numbers ? Which of them are prime to the num- ber 10? 20. Select the prime numbers between 50 and 100. 21. Make a list of all the prime numbers below 40, and use it to prove that 541 is prime. 22. Which of the numbers 293, 371, 385, 440, 524, 617, and 713 are prime ? 23. Of what number are 2, 3, 5, 7, 11, and 13 the prime factors ? 24. How many of the different divisors of 150 are prime, and how many are composite ? 25. Find all the prime factors common to 1001 and 616. Greatest Common Divisor. 19. A common divisor of two or more numbers is a num- ber that will be contained exactly in each of them. The greatest common divisor of two or more numbers is the greatest number that will be contained exactly in each of them. For example, 3 and 4 are common divisors of 24 and 36, but 12 is the greatest common divisor. For convenience G.C.D. is used to represent the greatest common divisor. Greatest common measure (G.C.M.) and highest common factor (H.C.F.) are expressions which have the same mean- ing as greatest common divisor. I. Find the G.C.D. of 56, 84, and 140. The G.C.D. is the product of all the 56 = 2 X 7. common prime factors. 2 and 7 are the 84 = 2 X o X 7. common factors, but 2 occurs twice in each 140 = 2 X O X 7. number, and hence will occur twice in the CCD :=2*X7 = 28 G.C.D. The G.C.D. is thus seen to bp ' ^2^7, which equals 2§. 2)56 84 140 2)28 42 70 7)14 21 35 30 ARITHMETIC. To find the G.C.D. of two or more numbers, resolve th^ numbers into their prime factors, and find the product of the common factors, taking each factor the least number of times it occurs in any number. Th.Q following arrangement of work may be used : Arrange the numbers in a line, and divide by all the prime numbers that will be contained in all the numbers. The "^ o K divisors are the common prime factors, and the product of the divisors is the G.C.D. = 22 X 7 = 28. G.C.D. EXAMPLES. Find the G.C.D. of 1. 48 and 120. 12. 105, 231, and 1001. 2. 84, 126, and 140. 13. 156, 234, and 260. 3. 48, 26, 72, and 24. 14. 189, 243, and 297. 4. 6, 8, 20, 36, and 48. 15. 240, 560, and 616. 5. 45, 75, 90, 135, 150, and 180. 16. 252, 315, 420, and 504. 6. 66, 78, 102, and 114. 17. 256, 480, and 1296. 7. 66, 308, and 506. 18. 432 and 1872. 8. 119 and 231. 19. 936 and 2925. 9. 168, 192, and 216. 20. 720, 336, and 1736. 10. 120, 228, and 720. 21. 927, 342, and 861. 11. 144 and 780. 22. 252, 588, 924, and 1092. 23. 4815, 4905, and 5085. 24. 1209, 1885, 2457, 2691, and 2717. 25. Find all the common divisors of 225, 2025, and 8100. 26. What is the length of the longest boards that will exactly fit three floors 42; 63; and 105 feet long respectively ? FACTORS. 31 27. A Mian has three farms of 56, 72, and 88 acres respec- tively, and wishes to fence them into the largest possible fields, having each the same number of acres. How many acres could he put in each ? 28. How many gallons are there in the largest vessel that will exactly measure the contents of three hogsheads, con- taining respectively 143, 104, and 156 gallons ? 29. rind the length of the longest pole that will exactly measure the sides of a field, which are respectively 72, 126, 162, and 90 feet. 30. Three military companies consisting respectively of 36, 42, and 54 men are divided into squads, each containing the same number; find the largest number of men each squad may contain, and the number of squads in each company. 20. When the numbers cannot be factored easily, a differ- ent method is employed for finding the G.C.D. I. Find the G.C.D. of 161 and 368. 161)368(2 Divide 368 by 161. If 161 were contained 322 exactly, it would be the G.C.D. Howerer, 46'\16ir3 there is a remainder 46. Divide 161 by 46, •j^gg and there is a remainder 23. Divide 46 by 23, '~oQ\Aa/o *"^ there is no remainder. 23, the last divisor, ^^^f^^ is the G.C.D. — Since 23 is a divisor of 46, it is a divisor of 138, which equals 46x3, and is then a divisor of 161, which equals 138 + 23. Since 23 is a divisor of 161, it is a divisor of 322, which equals 161 X 2, and is then a divisor of -368, which equals 322 + 23x2. Hence 23 is a common divisor of 161 and 368. Furthermore, 161 and 368 are each a certain number of times the G.C.D. 161 X 2 = 322, which is a certain number of times the G.C.D. 368 — 322 =: 46, which must be a certain number of times the G.C.D., because if a number is a divisor of two other numbers, it is a divisor of their difference. 46x3=138, which js Vk certaip nwmber pf tiwes tb« 32 ARITHMETIC. G.C.D. Then 161 — 138, which equals 23, is a certain number of times the G.C.D., and the G.C.D. cannot be greater than 23. But it has been proved that 23 is a common divisor ; hence it is the G.C.D, The method may be stated as follows : Divide the greater 7iumher by the less, and the divisor by the remainder, and so on till there is no remainder, each time dividing the last divisor by the last remainder. The last divisor is the G.C.D. When there are more than two numbers, find the G.C.D. of any two of them, then of that divisor and a third 7iumber, and so on till all the numbers have been used. The last G.C.D. is the one required. EXAMPLES. Find the G.C.D. of 1. 187 and 153. 9. 3432 and 4760. 2. 323 and 374. 10. 4939 and 3143. 3. 434, 539, and 616. 11. 3696 and 1440. 4. 1235 and 1495. 12. 9249 and 10920. 5. 1181 and 2741. 13. 2618, 39039, and 1771. 6. 1417 and 1469. 14. 43700 and 9430. 7. 630, 840, and 2772. 15. 13860 and 38500. 8. 13212 and 1841. 16. 17640 and 18375. 17. 4994, 12485, and 16117. 18. 36864, and 20736. 19. 156, 585, 442, and 1287. 20. 1274, 2002, 2366, 7007, and 13013. Least Commo^st Multiple. 21. A multiple of a number is any number in which it is contained exactly. A common multiple of two or more numbers is a number that will exactly contain eacjj of thejft. FACTORS. 33 The least common multiple of two or more numbers is the least number that will exactly contain each of them. For example, 48 and 72 are common multiples of 6 and 8, but 24 is the least common multiple. " For convenience L.G.M. is used to represent the least common multiple. I. Find the L.G.M. of 36, 42, and 88. 36 = 22 X 32. The L.C.M. must consist 42 = 2 X 3 X 7. of all the different factors 88 = 2^ X 11. that are in the numbers, L.C.M. = 2« X 32 X 7 X 11 = 5544. ^"'^ '*'^' ^'"''"'' ™"«* ^' present as many times as it is in any one number. The L.C.M. is thus seen to be 2'*x3'^x7 xH, which equals 6644. To find the L.C.M. of two or more numbers, resolve the numbers into their prims factors, and find the product of all the different factors, taking each factor the greatest number of times it occurs in any number. If the numbers are prime to each other, their product is their L.C.M. The following arrangement of work may be used : 2') 36 42 Rf^ Arrange the numbers in a 9vr« — 91 — JT ^^^^ ^^^ divide by all the ^' — prime numbers that will be o )J 21 22 contained in any two of the 3 7 22 numbers, bringing down in L.C.M. = 22 X 32 X 7 X 22 = 5544. ^^"^ ^"'^ ^^^"'"^ *'^^ quotients and the numbers that can- not be divided, and so on until the numbers in the line are prime to each other. The product of the divisors and the numbers in the last line is the L.C.M. EXAMPLES. Find the L.C.M. of 1. 15, 18, and 35. 3. 36, 48, and 72. 2. 20, 24, and 36. 4. 10, 14, 15, 21, 3(3, and 42. 34 ARITHMETIC. 6. 48, 98, 21, and 27. 14. 84, 126, and 140. 6. 18, 32, 48, and 52. 15. 156, 234, and 260. 7. 48, 26, 72, and 24. 16. 105, 476, and 306. 8. 21, 36, 50, and 64. 17. 144 and 780. 9. 14, 36, 108, and 144. 18. 240, 560, and 616. 10. 72, 80, 84, and 96. 19. 740, 333, and 296. 11. 91, 52, 39, 28, and 21. 20. 945 and 1485. 12. 3, 91, 78, 182, and 231. 21. 936 and 2925. 13. 108, 217, 54, and 31. 22. 504, 924, and 2184. 23. 1209, 1885, 2457, 2691, and 2717. 24. Find the L.C.M. of the nine digits. 25. Find the L.C.M. of the even numbers from 10 to 20 inclusive. 26. What is the shortest length that can be measured by either of three measures, which are respectively 9, 15, and 24 inches long ? 27. Find the contents of the smallest cistern that can be exactly measured by either one of three casks containing respectively 18, 25, and 30 gallons. 28. What is the width of the narrowest walk that can be paved with blocks each 12 inches long and 15 inches wide, allowing the blocks to run either lengthwise or across the walk? 29. What is the smallest sum of money that can be made up either of 2-cent, of 3-cent, of 5-cent, of 10-cent, or of 25- cent pieces ? 30. Four boys start together to run around a square ; the first can run around in 12 minutes, the second in 15 minutes, the third in 16 nrinutes, and the fourth in 18 minutes ; how long will it be before they all meet at the starting-point ? FACTORS.. 35 22. When the numbers cannot be factored easily, find the G.C.D. of two or more of the numbers, and use it as a divisor in the first line, and then proceed as before. I. Find the L.C.M. of 368, 483, and 532. 368)483(1 23)368(16 23)483(21 23)532(23 368 23_ 46_ 46_ 115)368(3 138 23 72 345 138 23 69 23)115(5 3 115 23)^68^8^432 J^^^^a" .e .f Vh" divide all the numbers by 23. It is contained in the first two, but not in tlie third. 4 3 19 However, the numbers are L.C.M. = 23 X 22 X 7 X 4 X 3 X 19 """"^ '"* '"'''^^ simplified that __ 145832 *he method as previously given can be applied. When there are but two numbers, the L.C.M. can be found by dividing one number by the G.C.D. and multiplying the quotient by the other number. 2) 16 21 532 2) 8 21 266 7) 4 21 133 Find the L.C.M. of EXAMPLES. 1. 187 and 153. 9. 1217, 1422, and 1611. 2. 391 and 493. 10. 3150 and 2310. 3. 209, 247, and 253. 11. 9249 and 10920. 4. 187, 539, and 847. 12. 4939 and 3143. 5. 630, 840, and 2772. 13. 2618, 39039, and 1771. 6. 1417 and 1469. 14. 43700 and 9430. 7. iOll, 1685, and 2359. 15. 13860 and 38500. 8. 1517 and 1763. 16. 2520, 2772, and 30888. 17. 17640 and 18375. 18. 340200, 583200, and 2268000. 36 ARITHMETIC. . Cancellation. 23. Division may be indicated by writing the dividend above a line and the divisor below the same line. For ex- ample, \2_ means 42 divided by 6. This method of indicat- ing division is commonly used when there are several factors in either dividend or divisor. Such an example in division can be simplified by striking out, or cancelling, like factors in dividend and divisor. This does not affect the result, be- cause when both dividend and divisor are divided by the same number the quotient remains the same. I. Find the value of ^ X 5 x 30 x 12 20 X 7 X 6 5 3 Cancel 7 and 7. Then cancel 5 in the J X p X TJf!) X 7^ __ -j^K dividend and 20 in the divisor, writing 4 . ^0 X y X ^ ' below the 20, as 20 divided by 5 equals 4. ^ Then cancel 4 and 12, writing 3 above the 12 ; cancel 6 and 30, writing 5 above the 30. The result is 5 X 3, which equals 15. II. How many pounds of sugar worth 9 cents a pound must be given in exchange for 18 dozen of eggs worth 1^ cents a dozen ? fy The value of the eggs is 18 X 16 cents, ^g, -tn and as many pounds of sugar can be ob- ^ = 32 pounds. tained as 9 is contained times in 18 X 16. r Simplify by cancellation. EXAMPLES. Find the value of 1 5 X 8 X 3 X 16 3 9 X 25 X 64 * 8x15x4 ' * 5x18x32 2 11x33x21x13 ^ 20 X 56 X 12 * 13x7x11x3" '21x10x8' FACTORS. 37 792 rj 54x84x99 11x4x9 9x22x63 g 108 X 132 g 57 X 119 X 16 * 99x144* '17x12x19* 9. How many yards of cloth worth 22 cents a yard must be given in exchange for 11 bushels of potatoes worth 60 cents a bushel ? 10. If 50 oranges cost 75 cents, find the cost of 30 oranges. 11. How many barrels of flour can be bought for $247, when 8 barrels cost $52 ? 12. If 16 men can dig a ditch in 21 days, how long will it take 24 men ? 13. If the work of 11 men equals the work of 17 boys, how many men's work will equal the work of 68 boys ? 14. Allowing 17 bushels of wheat to make 4 barrels of flour, how many bushels will be necessary to make 68 barrels ? 15. When 15 barrels of pork, each containing 200 pounds, are worth f 250, find the value of 60 pounds. 16. How many dresses, each containing 16 yards, can be made from 20 pieces of cloth, 52 yards in each piece ? 17. If 54 men can build a wall in 35 days, working 10 hours a day, how many men will be necessary to build it in 15 days, working 9 hours a day ? 18. A gardener sells 75 crates of berries, 24 boxes in a crate, at 8 cents a box, and receives in return 12 rolls of matting, 40 yards in a roll ; find the price of the matting a yard. 38 ABITHMETIC. CHAPTER IV. COMMON FRACTIONS. 24. A unit may be divided into any number of equal parts, and any number of these parts may be taken to- gether. For example, -f- means that the unit is divided into seven equal parts, of which five are taken; it is read jive sevenths. As has been stated in § 23, ^ also means five divided by seven, but these two meanings are really the same, because the quotient arising from dividing five by seven is five sevenths. An expression used to denote one or more of the equal parts of a unit is called a fraction. When it is represented by two numbers, one written above the other with a divid- ing line between, it is called a common fraction, or vulgar fraction. When it is represented by figures at the right of the decimal point, as shown in Chapter II., it is called a decimal fraction. In common fractions the number below the line, which shows into how many equal parts the unit is divided, is called the denominator. Tho number above the line, which shows how many of the equal parts are taken, is called the numerator. The two numbers are called the terms of the fraction. A proper fraction is one whose numerator is less than its denominator ; as f . An improper fraction is one whose numerator is equal to or greater than its denominator; as f, ^. When the numerator is greater than the denominator, more than one unit must be divided into equal parts ; for example, ^ means COMMON FK ACTIONS. 39 that three or more units have each been divided into eight equal parts, and nineteen of these parts are taken. A mixed number is an integer and a fraction expressed together ; as 6^-^, which is read six and seven fifteenths. A compound fraction is a fraction of a fraction ; as J of -^. A complex fraction is one which has a fraction in one or both of its terms; as -, -^ . The reciprocal of a number is the quotient arising from dividing 1 l)y that number. For example, the reciprocal of 7 is f 25. Since a fraction is an expression of division, tlu; last three principles of § 4 may be restated for fractions as follows : (1) Multiplying the numerator or dividing the denomi- nator by a number multiplies the fraction by the same number. {2) Dividing the numerator or multiplying the denomi- nator by a number divides the fraction by the same number. (3) Multiplying or dividing both numerator and denom- inator by the same number does not change the, value of the fraction. Eeduction of Fractions to Lowest Terms. 26. A fraction is in its lowest terms when the numerator and denominator are prime to each other. 1. Reduce ^^^ to its lowest terms. 2^5 21 3 Since dividing both numerator and denominator by ^iT 6 3 y ^\^Q same number does not change the value of the fraction, we can divide both terms by 5 and thus obtain || for a value of the fraction in lower terms. Then divide both terms by 9, and we obtain f . Since 3 and 7 are prime to each other, the fraction is in its lowest terms. 40 ARITHMETIC. To reduce a fraction to its lowest terms, divide numerator and denominator' successively hy their common factors. Since the product of all the common factors is the G.C.D., in ail cases where the common factors are not easily seen, divide numerator and denominator hy their G.G.D. EXAMPLES. Beduce the following fractions to their lowest terms . 14 2 » 625 IK 4 3 4 3 TW 392 2. ^. 9. 3. ff. 10.. m 4. i^. 11. Ill 5. ^. 12. m 6. iff. 13- t\%V 20. 7. |M. 14. -MA. • 21. "JT5T- 16- im- 17- im- 18- Mm 1^- Hf'ST- 23820 1 84800 22. Ascertain whether the fraction -exWr ^^ "^ ^*^ lowest terms or not, and explain the process you employ. Reduction of Improper Fractions to AViiole or Mixed Numbers. 27. I. Reduce -L3^ to a whole number. 13^ = 11. W- is the same as 132 h- 12, which equals 11. II. Reduce i^- to a mixed number. J.J ^^ g ^^3 -lyY" is the same as 141 -f- 12, which equals lly'^ T^ 12 ^' and this reduces to 11|. EXAMPLES. Reduce to whole or mixed numbers 1. ¥• 3. -VV-. 5. -w. 2. ||. 4. y/. 6. ^. COMMON FRACTIONS. 41 8. ^^. 11. le^. 14. i^. 9. V/. 12. i-fp. 15. -V//- Reduction of Whole or Mixed Numbers to Improper Fractions. 28. I. Reduce 8 to sevenths. 8 = Af-. Since there are 7 sevenths in 1, in 8 there are 8 times 7 sevenths, whicli equals ^y". A whole number may be written as a fraction with 1 for the denominator. For example, 13 = X^. II. Reduce 12|^ to an improper fraction. 12 J = 108 + 7 = i^. 12 = 101 ; adding | to this, the result is i^; EXAMPLES. Reduce to improper fractions 4. 12^. 7. 9|i. 5. 16,^. 8. 12tt. 6. 21,^. 9. 15^. 10. Reduce 9 to eighths. 11. Reduce 12 to elevenths. 12. Reduce 19 to thirteenths. 13. Reduce 25 to fifteenths. 14. Reduce 42 to twenty-fourths. Least Common Denominator. 2^. When several fractions have the same denominator. they are said to have a common denominator. It is always possible to reduce two or more fractions to equivalent frac- 1. 3|. 2. 7*. 3. 10i|. 42 ARITHMETIC. tions having a common denominator ; but the most useful common denominator is the least common multiple of the denominators, which is known as the least common denomi- nator. For convenience L.C.D. is used to represent the least common denominator. I. Eeduce |, J, and \^ to equivalent fractions having the L.C.D. 5 _ _5^ _ 30, We find the L.C.M. of 6, 9, and 12 to be 36. 6 7 7x4 28 must be multiplied by 6 to obtain 36; hence 5 must be multiplied by 6 in order to keep the frac- 11 --11 X3.— 33 I' J i- 12 3 6 3 6" tion of the same value, because multiplying both numerator and denominator by the same number does not change the value of the fraction. A similar process is applied to the other two fractions. To reduce fractions to equivalent fractions having the L.C.D., in each fraction multiply both terms by the quotient arising from dividing the L.C.D. by the denominator. Note. Fractions should always be in their lowest terms before find- ing the L.C.D. EXAMPLES. Reduce to equivalent fractions having the L.C.D. 1. 1 f, and f . 8. 1 \, \, I, and \. 2. I, f,andTV 9. i, |, ^V. and |i. 3. if, ^2. and If. 10. f , A, 2\. and ^3^. 4. f,T\,andif. 11. A. A. i*. and if . 5. I, I, and ^. 12. ^-^^ ||, _2_ and ^. 6. A. \i. and ^. 13. A, \^, ^, ^, and ^. 7. |,A,and3-V 14. A, H. H. If . and ||. 15. t\, II, ^3_ _9_9_ 100^ and ^\. 16. ih U^ n, T^A. T^B^. and AV COMMON FRACTIONS. 48 Addition of Fractions. 30. I. Find the sum of 3^, 3^, and ^. 2 j^ 5 , g _ 2-^5+8 — 15 Quantities to be added together tV-f-TV + lT— "tY""— T7- jnust ^g ^f the same kind. Since the denominators are alike, we have merely to add the numerators. II. Find the sum of f, f, and ^^. When the denominators are unlike, the fractions must first be reduced to equivalent fractions having the L.C.D., and then added. III. Find the sum of 2\, 3|, 6|^, and ^. 2i + 3| + 6ii + A = lU^±&|F±i^ = lift = 12H = 12i. The sum of the whole numbers is 11, and the sum of the fractions is II; the two sums taken together equal 12§J, which reduced to its lowest terras becomes 12|. To add fractions, reduce the fractions to equivalent fractions having the L.C.D., and ivrite the sum of the numerators over the L.C.D. When there are mixed numbers, add the whole numbers and fractions separately, and combine the results. Improper fractions should be reduced to mixed numbers be- fore adding. EXAMPLES. Find the sum of 1. 4,f,and-^. 7. f,|,H.and4f. 2. t, I, and f 8. |, A, {i, and Jf. 3. i,^,2indj\. 9. 1%, H. A. A. and ^. 4. A s% A and H- 10. i, ^, ^, and ^. 5. f,|,f,and|i 11. I, I, 2^3^, and S^V 6. I, I, J, and ^^. 12. 5f , 3 A, 2^, H> and 21 44 AHITHMETIC. " 13. H. 4i, \% and 1,%. 19. 5^, 2f , ^, and ^f . . 14. 5f , Jf , I, and if . 20. 5|i, f, 3^, and 1^^. 15. 2f , H, 3i, and f 21. «, ^V, and 1|S|. 16. i, A. tV. and 4J. ■ 22. 4^, 7^, \\, |f, and Iff. 17- 5f If, il, and 3^. 23. |, A, Iff, and ff . 18. I, il, 11 A. and iJ. 24. if, ^, and 2^^. 25. 12^, 13^^, 17|, and J^^^. 26. 18 A, 12i|, 104,1^, and 29^. Subtraction of Fractions. 31. I. Subtract I from jL. The fractions must first be reduced to ^ — 1^ = ^ ^^^ ^ = g^g-. equivalent fractions having the L.C.D., and then subtracted. II. Subtract 42^ from 7yV The difference between the whole num- 7 9 A 5 Q 27-10 Q17 ^^^^ ^^ ^> ^"^ ^^^^ difference between the ^^ 2^ ~T^ ¥Y' fractions is ^|; these two differences taken together equal Z\\. III. Subtract 5f from llf . 112-5| = 68^ = 5^^#i = 5i4. '^^ ''^''"''* subtract fi ' * ^ ** 2 ^ 2 ® from /g, so we take 1 from 0, which makes f f to be added to aV J the example thus becomes S^-Vg— , which equals h\\. To subtract fractions, reduce the fractions to equivalent fractions having the L.G.D., and ivrite the difference between the numerators over the L.C.D. When there are mixed num- bers, subtract the whole numbers and fractions separately, and combine the results. Improper fractions should be reduced to mixed numbers before subtracting. COMMON FRACTIONS. 45 EXAMPLES. Find the difference between 1. I and f . 12. 3j\ and 2^. 2. ^andf 13. 9yV and 8f 3. |andf 14. 4f and 3f 4. -y-and^?^. 15. 15^2^ and 12^. 5. T^and^^. 16. 36y9^ and f 6. 3^and^. 17. 7^ and 6^. 7. ^and/^. 18. 20^ and 19^^. 8. i^and^. 19. 4/^ and 2|^. 9. T^^andii^. 20. 19^ and 9^. 10. TfJ^and^Hr. 21. 20^ and 15^- 11. 18f andl5i. 22. lO^i^ and 9^- Addition and Subtraction of Fractions Combined, 32. I. Simplify 8/^ -2J-3J + 635^ -5{. 8A + 6A = 14it^ = 14ff. 2i + 3i 4- 5| = 10^i:^+-i^ = 10|i = llif . 14i| - llif = 31^:^ = 211^ = 2}f = 2H. The three terms preceded by the minus sign are to be subtracted from the sum of the remaining terms. Subtracting the sum of these terms gives the same result as if they were subtracted separately. EXAMPLES. Simplify 1. 9j_6A + 2|-H-4A. 2. 7i-3j%-TV-i + H- a 28^^-7^ + 16^ + 41-14^. 46 ARITHMETIC. 5. 15-2H-4A-6f-A. 6. 7|--6f + 5f-4f+3|-2| + lf 7. 40ff- 3^^-5^-14^- 8 + 12101 -16^. 8. 13|-2,^-6A + 3-l,% + 8f-ff-10Jf. 9. Add together f, ^, and y\, and from their sum sub- tract ^. 10. -^^ of a pole is in the mud, -^ of it is in the water, and the rest of it is in the air ; what part of it is in the air ? 11. In a school J of the scholars are Germans, | Irish, ^ English, Jg- Swedes, and the remainder Americans; what part of the scholars are Americans ? 12. A man performs a journey of 84 J miles in five days. He travels 12| miles the first day, 18J miles the second day, 16f miles the third day, and 20|- miles the fourth day ; how far does he travel the fifth day ? 13. A merchant bought two pieces of cloth containing 46|- yards and 53|- yards respectively. He sold 12| yards, 15| yards, 18|- yards, and 14y^2 yards; how many yards were there remaining ? 14. A painter receives $15 for painting a room. He ex- pends $6\ for labor, $4y^^ for paint, and $2-^-^ for varnish ; find the amount he gained. Multiplication of Fractions. 33. I. Multiply I- by 3. Since multiplying the numerator by a number multi- ^ X 3 = y. plies the fraction by the same number, we multiply the numerator by 3, and obtain f as the result. •COMMON FRACTIONS. 47 II. Multiply fxf If we multiply the numerators together, we obtain 4 X # = f §• 5X4. Since dividing either multiplicand or multiplier by a number divides the product by the same number, if we divide one by 7 and the other by 9, we divide the product by 7 X 9. 5 divided by 7 equals f , and 4 divided by 9 equals f ; hence the product of f and I is 5x4 divided by 7 XO, which may be written ^^, and is equal to f §. III. Multiply I by H- g *2 4 f XT5 = fx'3 5- '^^^ principle of cancellation can ^ X — = — • then be applied, but it gives the same result to apply 3 7 the cancellation at once to the example. IV. Multiply together f , ^, and 5\. ft M il ^'i Reduce 5} to an improper ^ X ii X 6^ = ^ X J^ X ^ = Jl = 2 r^j. fraction, and proceed as in the 9 do 9 ^p 4 15 J, , 3 5 precedmg example. To find the product of a whole number and a fraction, ivriie the product of the whole number and the numerator over the denominator. • To find the product of several fractions, write the product of the numerators over the product of the denominators, first can- celling the factors common to a numerator and denominator. Mixed numbers should be reduced to improper fractions before multiplying, and a whole number should be treated as a numerator. Compound fractions are simplified by multiplying to- gether the simple fractions. For example, | of |- = f x f . ^. , ., , . EXAMPLES. Find the product of 1. j\Xi. 4. iof^. 7. H X H- 2. 48xA- 5. ix3i. 8. Hx6|. 3. ^off 6. e^xff. 9. ^x^. 48 ARITHMETIC. 10. 8^x22V . 18. 2ixl^xl^xA- 11. I off off 19. ^ot3^xlj\Xj\. 12. AxAxil. 20. 5,VXT%of||oflJ,. 13. l^XfiofTf 21. i|-x2|XTteXlA- 14. J X f X If X If 22. 3f X iJ X 2ii X If of 14. 15. 101x21x^x3^. 23. lAx2ifx2i|xlxVxfi. 16. Hx3fx3|Xii' '24. lAxl7Jxlifxl9Vx2A. 17. 2ix4fx5fxf 25. |x||xAx7ixl6eV 34. In finding the product of a mixed number and a whole number, or of two mixed numbers, we can use another method, which is particularly useful when the integral parts of the mixed numbers are large numbers. I. Multiply 23 by 6^. 23 Q 7 Multiply 23 by 7 and divide the result by 11, which is il'ilfil *'^^ same as multiplying by y\ ; this gives 14 j7p Then multiply 23 by 6, and add 138 thus obtained to 14^7^. J4yy fjjjg gjj^jj.^ product is 152^. , loo 152^ II. Multiply 18| by8i. 18| 8i .tV |xi= = tV; 18Xi = 4i; 1X8 = = 3^; 18X8 = = 144. T ^^ sum of these partial products is 151|. H 144 151f EXAMPLES. Mnd the product of 1. 41x3f 3. 25x6f 5. 29 X 21tV 2. 23x6J. 4. 32x10jV 6. 18 X 11^. * COMMON FRACTIONS. 49 7. 8ix6J. 11. 14|x8|. 15. 31ifxl4|. 8. 8fx5|. 12. 12tVx43^. 16. G6| x 37J. 9. 22ix8f. 13. 18|xl2f. 17. 112^ x 31J. 10. lly^xSf 14. 25^x16^. 18. 168ix83i. Division of Er actions. 35. I. Divide I by f 5.6 — 5 1— M. ^ divided by 1 equals f . If we diyide the ^ ' T'~"^ b -~ Ti' divisor by 7, we multiply the quotient by 7 ; hence | divided by } equals ^§^. If we now multiply the divisor by (), we divide the quotient by 6 ; hence ^ divided by f equals f^. This is the same as | X |, which equals ||. To divide a fraction by a fraction, inveii, the divisor and proceed as in multiplication. II. Divide A'of 2^. by -^ of 5f ;2 2 ? _ 3 "14* When the divisor contains more than o factor, each factor should be inverted. EXAMPLES. Find the quotient of 1- l^f. 9. H^H- 17. 10^^13. 2. 18-1- 10. 4|^6|. 18. 3H-^iff. 3. 3-^^15. 11. 100 -4f. 19. 3A-2«. 4. H-^ff. 12. 4fJ^10^. 20. m-^m- 5. 5^^ If. 13. tVo^A- 21. 23-T • 241- 6. If ^21 14. m^m- 22. lOM^irfr- 7. 5^-^21. 15. u^n- 23. 5f-^^- 8. 161 --42. 16. lOi^iH- 24. 7*^31^. 50 ARITHMETIC. 25. foflH-fof^. 29. 3^of2i-JVof 5i|. 26. J of l| - A of If. 30. i of f of 2i - f^ of If 27. I of 91 -- 1^ of 637. 31. f of ^\ -- 3% of 3^ of 4^. 28. I of 7i-| of 11^. 32. I of I of A^f ^^ i of I- 36. When the divisor is either a whole or a mixed num- ber, a different method may often be used to advantage. I. Divide 29f by 4. ^ /-^"t 29 divided by 4 equals 7 with a remainder of 1. If 7^ divided by 4 equals V X5, which equals |f. II. Divide 52| by 3|. "^^z A-"^ Since multiplying both dividend and divisor by the ^ ^ same number does not affect the quotient, we can multi- 11 )157^ ply both dividend and divisor by 3, the denominator of 14|4. the divisor, and then proceed as in the preceding example, EXAMPLES. Find the quotient of 1. 22i-r-3. 7. 156J-25. 13. 481-51 2. 19| -- 5. 8. 128^ --30. 14. 561-^81 3. 283-9^ -7. 9. 22 -6|. 15. 104yi--^8f 4. 33^-12. 10. 21 - 5|. 16. 906-^111 5. 44J : 15. 11. 64-9^. 17. 115|-21f 6. 87-5- -J- 21. 12. 13t^2i. 18. 402^-30^. Short Methods of Multiplication and Division. 37. Any exact fractional part of a number is called an aliquot part of that number. For example, 2, 2^, 3^, and 5 are aliquot parts of 10. COMMON FRACTIONS. 51 To multiply by aliquot parts of 10, 100, 1000, etc., multi- ply hy 10, 100, 1000, etc., as the case may j-equire, and then find the required part. For example, since 16J = ^ of 100, 24 X 16| = I- of 2400 = 400. To divide by aliquot parts of 10, 100, 1000, etc., divide by 10, 100, 1000, etc., as the case may require, and then multiply hy the denominator of the fraction expressing the aliquot part. For example, since 12^ = i of 100, 225 ^ 12^ = 2.25x8 = 18. To multiply by a number a little less than 10, 100, 1000, etc., multiply by 10, 100, 1000, etc., and from the product subtract the product of the multiplicand by the difference be- tween the midtiplier and 10, 100, 1000, etc., as the case may require. For example, 184 x 99 = 18400 - 184 = 18216 ; 184 X 98 = 18400 - 368 = 18032. EXAMPLES. 1. Multiply 423 by 5. 14. Divide 11150 by 25. 2. Multiply 2918 by 2f 15. Divide 2700 by 16f . 3. Multiply 57162 by 3^. 16. Divide 42125 by 12f 4. Multiply 3143 by 25. 17. Divide 1172 by 331- 5. Multiply 4890 by 16J. 18. Divide 87320 by 250. 6. Multiply 12792 by 33J. 19. Divide 1183^ by 166J. 7. Multiply 804320 by 12^. 20. Divide 33625 by 125. 8. Multiply 84322 by 250. 21. Multiply 64 by 9. 9. Multiply 7614 by 125. 22. Multiply 82 by 99. 10. Multiply 5436 by 166|. 23. Multiply 127 by 999. 11. Divide 7165 by 5. 24. Multiply 7342 by 9999. 12. Divide 8775 by 2^. 25. Multiply 138 by 98. 13. Divide 876| by 3J. 26. Multiply 72 by 997. 52 ARITHMETIC. Complex Fractions. 5 ^ 38. I. Reduce -^ to a simple fraction. 5 A complex fraction indicates that the nu- 5/^ _ ^ 2 _ 5 merator is to be divided by the denominator. 6^ X^ X^ 6 Hence we perform the division by inverting the ^ divisor and proceeding as in multiplication. 3 II. Reduce 5 to a simple fraction. 5 a o III many cases the simplest solution is to ^ = — • multiply both numerator and denominator by their least common denominator. In this ex- ample we multiply both terms by 4. III. Reduce -5 — ^ to a simple fraction. 6+-t 3i-2J = ll^=9_2 = 7 The numerator and denomi- 2f + If = 3-^ = 3f = 4| = 4|-. nator must each be simplified, J_ _ 7 w 2 _ _Z.. ^^^^ ^^^" ^^® ^'^'^ proceed as in 4| ^ 9 36 the preceding examples. 4 EXAMPLES. Reduce the following complex fractions to simple frac- tions : 2. 4. ^ 6i 3f 5fi-5i 10 6. 18^ 10. 4]t-2i 16| 1 of 1 of 1 6i-2| 9f 2* 7. i of f of 7f 19A 11. 2i + 2f 4f-3f 6i 33i" 8. 2i + 5i i 12. 3i-2i. COMMON f'l^ACTIO^fs. 53 13. 3|-2i H + 3^ 14. 3i + ii-| 6*-fxi 15 2|^4x2 .-|.5 16 4* + 24-^| 6i-l|x| 17 lT^_9} + 4^ |x9i^ 19. 4* '^^n 20 ItziM. 45 149 U 20 21 i + f + i-A 3t-2i 22. 23. (3i-2i)^l| l| + 2i f off ^f of A 7i-Hx5f • 18 (4i + 7i)^3i 24 ^ + i + i ' " "" ^- ' i + i + - 2i^3i^4i |x2^x5i • }_^1__^X_ 39. When in a series of fractions we have only the signs of multiplication and division, one operation is sufficient to obtain the result. . I. Simplify tQ^TT jj. 5f-2ii 12f 5* - 2H = 3^^^ = 3A = 3f • We must first perform the f of |-^ 1^ f of ^ 12f subtraction in the denominator 3i ~ 12^ ~~ 3I II of the first fraction. We can 3 then invert the second fraction — §X — X— X — X — = -• *"^ obtain the result by one ^ m %'S 4 ^^4 process of cancellation. 64 ARITHMETIC. EXAMPLES. 112 Q 1. Find in its simplest form the value of — ^ -e — ^ 12f 9 2. Multiply || by ^. 3. Multiply || by 3-«j of 2i. 4. Multiply f of p by I of i. 5. Multiply I of y_ of 4i by :y^%^' 6. What is the product of f of ^^ of 15 and |f of llf ? 7. Reduce to its simplest form A of \8- of 31 ^ — ?ii— , 8. Divide I by il. 6 "^F 9. Divide |i x 72| by | of f of 9|. 10. Divide iof 12|- by X of 8f. TT '^^ 11. Divide A of ^ of 8i by t^f*. 12. Divide -^^ of ^ of 7^ by — iM — 13. Divide 10 times f- of -il- of 9A^ by -t. V9 12^ ^y ^ 1\ 14. Reduce -| x -| h- A to its simplest form. ^^4 '*t^ ^6 15. Reduce ^ ^ ^, -=- , \ to its simplest form. COMMON FKACTIONS. 55 17. Reduce to its simplest form - of ^- of -^ -*- 1^ 18. Divide ^ of ^ of 13f by ^ . 19. Divide ?i by i±i. 21.Divide|xA,yg^X^^^. 22. Simplify (i + ^)x^^?^. 23. Simplify 2} x^^:^^j^ 9* 24. Simplify 54 of r-^— - -^ ^2L±_^. ^ -y ^ 4 + 2^ 4i+3f To Find a Number when a Fractional Part of it IS Known. 40. I. 5f is J of what number ? 5 3 I of some number means the same as 1 f-5_^7_?^ £_16_«i times some number. If 5| is | times a cer- '9^72 ^ tain number, the whole number is as much * as I is contained in 5|, which equals 7^. II. A boy after spending | of his money has $7^ left ; how much had he at first ? 5 If he spent |, he had | remaining. tri^^ — lEx- = — = 12}f ^^"^^ t^^ tot^l sum is as much as f is "62^2 "' contained in $7 J, which is $12 J. Ans. $12J. 5ti ARITHMETIC. EXAMPLES. 1. 19^ is ■§■ of what number ? 2. 32| is 3§j of what number ? 3. 1^ of II is ^^g. of what number ? 4. Y^:f of ^ of ly\ is f of what number ? 5. ^ of 5y\- is 6J times what number ? 6. Of what number is f the -J part ? 7. From Boston to Worcester is 44 miles, which is f of the distance from Boston to Springfield ; find the distance from Boston to Springfield. 8. Find the cost of a barrel of flour when Jg- of a barrel costs $3.50. 9. A man can dig i|- of a ditch in 2| days ; how long will it take him to dig the whole ditch ? 10. A man after selling ^ of his farm has 28| acres remaining ; how many acres were there in the entire farm ? 11. ^ of a basket of eggs were broken, and there were 66 left ; find the original number in the basket. 12. A grocer sold f of a barrel of sugar to one customer and ^ of it to another customer, and had 45 pounds left ; how many pounds were there in the barrel when full ? 13. In an orchard J of the trees bear apples, J bear pears, ^ bear peaches, and the remainder, 39 in number, bear plums ; find the number of trees in the orchard. 14; A boy after losing \ of his money has 10 cents given him, and then finds that he has f of the original amount ; what was the original amount. ? COMMON FRACTIONS. 57 To Find what Fractional Part one Number is of Another. 41. I. 8 is what part of 13 ? r^ 1 is y^j of 13, and 8 is 8 times j^^ of 13, which is y\. The — • number of which a part is taken is the denominator, and the part taken is the numerator. II. If is what part of 3| ? ^ 5 ^ o o q^ We first form a complex fraction exprcss- ^ = — X ~ = ^' ing the fractional part, and then reduce this ^ complex fraction to a simple one. EXAMPLES. What part of 1. 12 is 7? 5. 15| is i? 9. 7Jis3i? 2. lOi is 3 ? 6. ttisi? 10. 25Jis2|? 3. 17is4i? 7. *ist? 11. 4 is 1 of 6? 4. 12isf? 8. 2|isli? 12. H-l' 13. A of ^ of 12 is T^ of 2^? 14. l + A + iis^-F 15. 12-7f is li + 2f ? 16. 37f is 2ix3f + iof 5i? 17- i + i + i + iisi-i + i-i? 18- (J-i)x(4-3f) is (2+1)^(3 + 1)? 19. If a tank can be filled by a pipe in 11 hours, what part can be filled in 3^ hours ? 20. If a man can build a wall in 3J days, what part can he build in 2^ days ? 58 ARITHMETIC. 21. A man owning f of a ship's cargo sells f of the cargo ; what part of his share does he sell ? 22. A boy had $15^ given him, and he spent $8 ; what part of the money did he spend ? Reduction of Common Fractions to Decimal Fractions. 42. I. Reduce J to a decimal fraction. 8)7.000 Since | equals 7-^-8, we can perform the division 0.875 decimally and obtain a decimal fraction for the value. If, when the fraction is in its lowest terms, the denomi- nator contains any factor besides 2 and 5, the quotient cannot be obtained exactly. In such cases, as in division of decimals, five decimal places are ordinarily enough for the answer. EXAMPLES. Reduce the following common fractions to decimal frac- tions : 1. f. 4. ^. 7. A. 10. m- 2. i. 5. ^. a U. 11. 15^. 3. A. 6. 3U. ' 9. 10,%. 12. 62iKt- 13. Reduce -^ to a decimal fraction. T 14. Express ^ decimally to three places. 15. Reduce to decimals and add |, |-J, and 9|-J. 16. Write 1-^ and 2^ in decimal form. Qive the divis- ion in decimals of the first by the second. COMMON FRACTIONS. 59 Reduction of Decimal Fractions to Common Fractions. 43. I. Reduce 0.0375 to a common fraction. n nw^ — _a7 s _ a 0.0375 can be expressed as a common — Toxriyu ~ 'SU- fraction in the form x^J^^y. This common fraction reduced to its lowest terms equals /j. The denominator of the common fraction is always 1 with as many zeros annexed as there are decimal places in the decimal fraction. EXAMPLES. Reduce the following decimal fractions to common frac- tions : 1. 0.7. 7. 0.0625. 13. 0.00096. 2. 0.24. 8. 0.0806. 14. 21.1875. 3. 0.625. 9. 0.98. 15. 0.05128. 4. 0.440. 10. 12.043. 16. 14.06225. 5. 0.0016. 11. 0.03125. 17. 42.030125. 6. 5.082. 12. 8.65. 18. 0.0007648267. Reduction of Common Fractions to Circulating Decimals. 44. When the result cannot be obtained exactly in re- ducing a common fraction to a decimal, if the division be carried far enough, the quotient will be found to contain the repetition of a figure or series of figures. For example, 2 = 0.6666 + ; 2V = 0.3181818 -f . Such decimals are known 60 ARITHMETIC. as circulating decimals, repeating decimals, or infinite deci- mals. The figure or series of figures which is repeated is called the repetend. In the case of a single figure the repe- tend is denoted by a dot over the figure, and in the case of a series of figures by dots over the first and last figures. For example, | = 0.6 ; ^^ = O.SiS. I. Reduce ^ to a circulating decimal. 0.3428571 ,35)12.0 105 150 140 100 70 300 280 We must continue the division until the re- mainder is the same as some preceding remainder ; from this point the figures will continue in series as before. In the present example 15 is the same as the second remainder. The repetend begins with 4 and ends with 1 ; hence dots are placed 200 -^'^ over these figures. 250 245 50 35 15 EXAMPLES. Reduce the following common fractions to circulating decimals : 1. i. 5. f 9. SA- 13. 2^. 2. A- 6. Jj. 10. ,1^. 14. ^. 3. f 7. S^. 11. ^j. 15. 12^. 4. 5Jj. 8. n. 12. 18^. 16. 5^1^. 17. What circulating decimal is equivalent to the sum of 1 1 oTirl 1 9 3, y, ana yy .'■ COMMON FRACTIONS. 61 18. Find the sum of 6J, 1^, and 8^%, and express the result as a circulating decimal. 19. Reduce ^ ^TF + it) to a repeating decimal. Reduction of Circulating Decimals to Common Fractions. 45. When the repetend comprises all the decimal places, a circulating decimal is equal to a common fraction which has the repetend for the numerator and as many nines as there are decimal places for the denominator. Take 0.324 as an example to show this. 1000 times 0.324 = 324.324324+. Once 0.324 = 0.324324 +. By subtraction we obtain 999 times 0.324 = 324. Hence 0.324 = ff|. I. Reduce 0.72 to a common fraction. n 79 _ 72 _ 8 '^^^ denominator is 99. Then ^f must be ~"9T ~~ TT- reduced to its lowest terms, which is ^. II. Reduce 0.4772 to a common fraction. 0.4772 = 0.47|| = 0.47A = ^ = ^ = |. The repetend reduces to the common fraction j*y. Then the decimal 0.47 y\ can be expressed as a complex fraction, which reduces to \\- When circulating decimals are to be added, subtracted, multiplied, or divided, they should first be reduced to com- mon fractions ; then perform the operations indicated, and reduce the resulting fraction to a decimaL 62 ARITHMETIC. EXAMPLES. Keduce the following circulating decimals to common fractions : 1. 0.3. 2. 0.27. 3. 0.0027. 4. 0.0127. 5. 0.216. 6. 7.0136. 7. 0.20054. 8. 2.00054. 9. 4.608i. 10. 0.4081. 11. 15.i08. 12. 0.225. 13. 0.00225. 14. 0.857142. 15. 3.2343. 16. 0.002343. 17. 0.012343. 18. 10.002343. 19. 12.03405. 20. 0.81247. 21. 1.15479li. 22. What common fraction equals the sum of 0.18 and 0.307692 ? 23. Add 0.03 to 0.462, expressing the result as a circu- lating decimal. 24. Add 0.07 to 0.382, expressing the result as a circu- lating decimal. 25. Multiply 0.145 by 0.297, and give the answer as a circulating decimal. 26. Multiply 0.3461538 by 0.285714, and express the result as a circulating decimal. 27. Multiply 2.604 by 1.234, and divide the result by 0.004. Greatest Common Divisor oe Fractions. 46. I. Find the G.C.D. -of ^, ^, and ||. G.C.D. of 3, 9, and 12 = 3. L.C.M. of 20, 10, and 25 = 100. G.C.D. 3 The G.C.D. of the numera- tors is 3. To be a divisor of 7^^ this number must be divided bj 20; to be a divisor of /^ it COMMON FRACTIONS. 63 must be divided by 10 ; and to be a divisor of ^f it must be divided by 25, If, however, 3 be divided by the L.C.M. of the denominators, it will be divided by the least number containing air the factors of the denominators ; hence this result is the G.C.D. of the fractions. To find the G.C.D. of several fractions, write the G.C.D. of the numerators over the L. CM. of the denominators. The fractions should be in their lowest terms, and mixed num- bers should first be reduced to improper fractions. EXAMPLES. Find the G.C.D. of 1. t,i,andf 5. 3i,2i,andf 2. f,i|,andf^. 6. 6|, IGf, and 6f 3. ^,}i,and|^. 7. GJ, 84, and 12J. 4. 1 \, i, and 4. 8. 1 A, h\. and 1^.- 9. Find the width of the widest stone that can be used in laying three walks which are respectively 3} feet, 3^ feet, and 54 feet wide. 10. What is the largest measure that can be used in measuring the contents of four bins which contain respec- tively 9, 134, 10|, and lOJ- bushels ? Least Common Multiple of Fractions. 47. I. Find the L.C.M. of /^, 3%, and 4|. L.C.M. of 3, 9, and 12 = 36. The L.C.M. of the numera- G.C.D. of 20, 10, and 25 = 5. tors is 36, which is also a mul- J p-vT 36—71 *^P^® °^ *^^ fractions. If this ^ ^' number be divided by 20 or any factor of 20, it is still a multiple of 2% ; if divided by 10 or any factor of 10, it is still a multiple of ^-^ ; if divided by 25 or any factor of 25, it is still a multiple of if. If, then, 36 be divided by the G.C.D. of 20, 64 ARITHMETIC. 10, and 25, it will be divided by the greatest common factor of the denominators ; hence this result is the L.C.M. of the fractions. To find the L.C.M. of several fractions, write the L.C.M. of the numerators over the G.C.D. of the denominators. EXAMPLES. Find the L.C.M. of 1. f,T\, and^. 5. 2f, 31, and 43^. 2. -I, i, and -|. 6. Ill 142, and 33^. 3. I, i, I, and -I-. 7. 4^, 5^^, and 43^- 4. ^, ^V. and A- 8. ^\, ^, 2|, 5, and 6^. 9. Find the capacity of the smallest tank whose contents can be exactly measured by either of three measures which contain respectively 1^, If, and 2^ quarts. 10. A can travel around a certain island in 2^ days, B in 3^ days, and C in 3J days. If they set out at the same time from the same point, and travel in the same direction, in how many days will they all come together at the start- ing point, and how many times will each man have gone around the island ? 11. The pendulum of one clock makes 24 beats in 26 sec- onds ; that of another, 36 beats in 40 seconds. If they start at the same time, when first will the beats occur together ? MISCELLANEOUS EXAMPLES. 1. Add |i, I, and A of f . 3. Add A to ?i^. 2. Add i, I of I, and If- 4. Add ^f of -i" *« '^' 5. Add f of 3i to 4 of i?iof -A. loj COMMON FRACTIONS. 65 G. Add f of 18 j3_ and |i of | of G^. 7. Add i of f of 28|| to 8^. 8. Add |i, I, and I of A. 9. Add^itofof ifof |of (J-.+). 6J 10. Add J to o ^*T)' '5i 11. Add ? of '^ to I of (4i - 2J). "8" .o All ^-^007 . „ ''• ^^'^' ao3^ ""^' *• 13. Add l-LM and ?i + ?i. i + 4i 7i-4| 14. What is the sum of t^ and i^LA ? i J of 2i . P, . . 1 1 of 2i . 0.06 + 0.3^ 16. Subtract J of ^-^ from ^^j^- 17. From } of f take ^ of |. 4i 18. From f of ^ take ^ of 1|. 19. From ^T^ ^ ^^ subtract ^. 5i-4J 2i 4. Ql ,02 20. Find the difference between -3- and -^ 2. If 7 ^±1, 21. Find the difference between 3J x 6|| and ^ 66 ARITHMETIC. 22. Subtract 7-i- -f A of tt ^^^^^ 15^ + -|- + OM. 23. From 34 subtract (— of ii of l*^ h- t^- 24. From 54 subtract ?i ^ /^A of ^ of 4iY ^ 3i VlO 2| V 25. Find the sum and product of -J, ^, and f . 26. Divide | of 7| by f of 12U. 3 4* 27. Divide (±-1) by 28. Divide A _ 1 by A 6^ 7 -^ 11 29. Divide it by — of (^-l\ 30. Diviflp 14 of 9 nf 12 hv "^ TOt,eOtlTby^_^^^^ 31. Divide.^|-l|by|-of(^^ + |). 32. Divide 0.75 by ?i X 0.081. lo 33. Reduce to a common denominator and add f X | X |, A. i, and j%. 34. Find the simplest expression for 1: + _j: _ JI_. ^ ^ 3^ 9 2 44. 35. Add ^, y\, -^, and ^, and reduce the sum to a deci- mal fraction carried to three decimal places. 36. Add f , 2|-, f, and \^, and divide the sum by fifty-six thousandths. 37. From ^\ of 1| subtract ^ of ^, and reduce the answer to a decimal. COMMON FRACTIONS. 67 38. From } of |-| subtract ^ of 2J, and reduce the answer to a decimal. 39. Divide (2| x j\) by (2J - 1^), and reduce the result to a decimal. 40. Divide J of ^ of f by j^, and add the quotient to -f. 41. Divide f of ^^ of |- by ~^, and add the quotient to 5A' f-A- 42. Divide (— — --\-~\ by f , and reduce the resnlt to an equivalent decimal fniction. 43. From \ of 1^ take ^, add to the remainder |, and divide the result by 6f . 2" on 44. From the sum of --^ and — subtract 44, and divide 13J 1 ^' the result by the product of 3J and 2^. 45. To f of f add y\- -^ ^, multiply the sum by , and 2. of 6 ^^ divide the product by ^— — ^• 46. Add -^ and ^; divide the result by 7^, and change the quotient to a decimal. 47. From ^ of 2f subtract the product of 0.075 and 1^ and divide the remainder by 12; reduce the result to a decimal form. 31. 48. From f of f subtract -| of —2-, add to the remainder ^, divide the result by 6J, and change the quotient to a decimal. 68 ARITHMETIC. 49. Eeduce ^^ of 4^ of f to a simple fraction. 5 + t 50. Reduce ^ ^ "^ to a decimal fraction. 31 + 1.125 2 ^ — ^ X — - 51. Express as a decimal ^ x ^ '^ ^ • 52. What decimal is equivalent to | of ^ X 0.021 ? 53. Simplify (l + i±i) ^ (l + ^). 54. M4.5:5?_Zi=what? 3-V^ 33 27 55. (1 J + H - ^-^24) - (151 - 1.209) = what ? 02 29 12 56. Reduce ; to its simplest decimal form. 1300 41.64 57. Snnphfy3j^X^-+3|j. 58. Reduce ^^ ^ 1^^ X ^ X ^ to a decimal. V41- 2y 5 2 59. Simplify L^ 2 + - 4H «+i 60. What is the exact value of ('2|+'f of ^ + i") .^ i^-. T? ~ ~ ~ i nf i "■" 61. Simplify ^^ZJE of ^~^ of 1—1 of 585. COMMON FRACTIONS. 69 62. The sum of | and ^^ is diminished by yi^. How many times does the difference contain ^ of the sum of J, ^, and jV '^ 63. The sum of two numbers equals 3^, and one of them is the difference between — '^ and — ^ ; what is the other 11 9 number ? 64. -J- of a number exceeds y^j^ of it by 15 ; what is the number ? G5. What part of ^ is i^? ^ i m. What part of 24 is ^ x ^^~^^ ? ^ ^ 31| "^ ^ X 3f 67. Simplify ^j - 0.042 - 2.4 + Tj 16 J^ _j- 60^ 68. Simplify (3.71-1.9Q8)x7.03, 2.8 of 2.27 69. Simplify 1.136 70. Simplify ^±^^ + _ii_ « A ^ 1. 13^ Q, 1 12 17 71. Simplify (2^of 3yV)+t-(liof l,^)-(l|of 4^of ,^> 72. Find the G.C.D. and the L.C.M. of J, 1|^, and 3.60. 73. The sum of ^ ^'^•^H and ?i^ is how many times 0.5 31 their difference ? 74. The sum of M-^^ and i^^i is how many times .^ . -..^ « 0.5 I X 2.25 -^ their difference ? 70 ARITHMETIC. 75. What is the value of (^-^^ + i^J^ - ^ -7^^ ? rra o- r* 54-!-f ^ 2 . U of 41 1,2 76. Simpnfy 2^ — ^ — X - of —^ ^ 77. Simplify ^,^^^+ffi~_ff° and 3 X (3| x 5f ) x 17f , and find their sum. 78. Find the vahie of (^4| -- ^i±i^ x 0.3G x 0.236. 79. Simplify 0.6 of 3.3 -f ^— of 17 + 0.4 of 5.75 - ^•^^^^^^ . ^.625 2.095238 80. By what must ^ be multiplied to give the product 1? 81. What number is that, -f of which exceeds \ of it by Hi? 82. Find the cost of 81| acres of land at f 28|- per acre. 83. Find the weight of 8^ reams of paper at 14-f^ pounds per ream. 84. Find the cost of |^ of a ton of coal if 3 tons cost $20. 85. If a man saw 3 J cords of wood in one day, how much will he saw in | of a day ? 86. Find the price of flour per barrel when 9f barrelf cost $65f . 87. At %2\ per barrel, how many barrels of apples can be bought for f 55 ? 88. If a man travel 28-f^ miles in one day, how many days will it take him to travel 177| miles ? COMMON FRACTIONS. 71 89. If a man walk 3J miles in | of an hour, at what rate does he walk per hour ? 90. What number divided by ^ equals 6^ ? 91. Find the cost of 8 rolls of carpet, 42\ yards in a roll, at 91J cents a yard. 92. If f of a yard of cloth cost f 3^, what is the cost of 4| yards ? 93. A farmer sold 4|^ tons of hay at the rate of 2| tons for f 44 ; what did he receive for it ? 94. rind the number of square yards in the surface of three floors measuring respectively 16J, 21-}-|, and 28^ square yards. 95. A farm is divided into four fields which contain re- spectively 18 1, 22|^, 19i^j and 29^^ acres; find the number of acres in the entire farm. 96. What number is that, to which if you add J of 19|, the sum will be 150 ? 97. A man bought 95 bushels of corn at 33J cents a bushel and sold it at 37^ cents a bushel ; find the amount gained. 98. What number is that, ^ of which exceeds 2\ by 13| ? 99. A merchant sold 38 yards of cloth at the rate of 2^ yards for $3 ; what did he receive for it ? 100. What is the price of land per acre when ^ of an acre costs $44.25 ? 101. The product of three numbers is 453 J; two of them are 5f and 11 J ; find the third number. 102. If J of a ton of hay will pay for 8 barrels of apples worth $2 J per barrel, what is the value of the hay per ton ? 72 ARITHMETIC. 103. If j^ of a yard of velvet cost f T-f, how many yards can be bought for $68if ? 104. A clerk spends $425 a year for board, which is ^J of his salary ; what is his salary ? 105. If 5^ tubs of butter cost $103|, how many bar- rels of flour worth f 8J per barrel will pay for one tub of butter? 106. If I of I of a ship cost $70000, what is -^ of it worth ? 107. How many pieces of cloth, each containing 2-^ yards, can be cut from a piece 50^ yards in length ? 108. rind the cost of 8J tons of hay when 2\ tons cost $31J. 109. A farmer exchanged 10 pounds of butter worth 31J cents a pound for sugar worth 7|- cents a pound ; how much sugar did he obtain ? 110. If a certain number is increased by |- of i|- of itself, the result is 246 ; find the number. 111. A boy spent -^ of his money one day, and f of it the next day, and then had 65 cents left j how much had he at first ? 112. A, owning -J of a farm, sold J of his share to B, and -J- of what he then owned to C for $420 ; what was the value of the entire farm at the same rate ? 113. A tailor has 97^ yards of cloth, from which he wishes to cut an equal number of coats and vests; how many of each can he cut if they contain 4J and 1|- yards respectively ? 114. If I of a ton of coal cost $6}, what will -f^ of a ton cost? COMMON FRACTIONS. 73 115. A horse and cow were bought for ^180, and* the cow cost -J as much as the horse ; find the price of each. 116. If I of a bushel of corn be worth ^ of a bushel of wheat, and wheat be worth $1.40 a bushel, how many bushels of corn can be bought for $27 ? 117. ^ of ^ of 28 times what number equals 50|? 118. A man, owning |- of a mill, sold -^ of his share for $2750 ; find the value of the whole mill at the same rate. 119. A man has ^ of his property invested in real estate, J in state bonds, ^ in bank stock, and the remainder, $5500, in business ; find the value of his entire property. 120. A owns ^ of a mill, and B the remainder; | of the difference between their shares is $10500; find the value of the whole mill. 121. A farmer sold 21^ dozen eggs at 18| cents a dozen, and bought 14J yards of cloth at 12^ cents a yard; how much money did he have left ? 122. If 19 pounds of butter cost $6.33^, what part of a pound can be bought for 25 cents ? 123. What is the smallest sum of money that can.be exactly paid either in pieces of money worth 6^ cents or in pieces worth 8J cents ? 124. Find the width of the widest blocks that will exactly fit either of three walks which are respectively 6J, 7^, and 10 feet wide. 125. If 6 be added to both terms of the fraction ^, is the value of the fraction increased or diminished, and how much ? 126. If 6 be subtracted from both terms of the fraction ■^, is the value of the fraction increased or diminished, and how much ? 74 ARITHMETIC. 127. If 6 men can do a piece of work in | of f of ^ of 6J days, how many men could do it in one day ? 128. $48| are to be divided among 5 men and 3 boys so that each boy will have half as much as a man ; how much will each have ? 129. A grocer sold f of a barrel of flour to one customer, |- of the remainder to another customer, and had 24-|- pounds left; how many pounds were there in the barrel when full? 130. A merchant owned ^ of a stock of goods ; -f- of the whole stock was destroyed by fire, and -^ of the remainder damaged by water. How much did the merchant lose, pro- vided the uninjured goods were sold at cost for $4200, and the damaged at half the cost ? COMPOUND NUMBERS. 76 CHAPTER V. COMPOUND NUMBERS. 48. When the value of anything is expressed in different units of the same nature, it is called a compound number ; as 3 bushels 2 pecks 5 quarts. 49. Long or Linear Measure is used in measuring lengths and distances. TABLE. 12 inches (in.) = 1 foot (ft.). 3 feet =lyard (yd.). 5 J yards or 16|- feet = 1 rod (rd.). 320 rods or 5280 feet = 1 mile (mi.). Note. A line = ^^ in. ; a furlong = 40 rd. ; a fathom = 6 ft. 50. Surveyors' Measure is used in measuring dimensions of land. TABLE. 7.92 inches = 1 link (li.). 100 links = 1 chain (ch.). 80 chains = 1 mile (mi.). Note. A surveyors' chain is 4 rods long and contains 100 links. Engineers use a chain, or measuring tape, 100 feet long. 51. Square Measure is used in measuring the area of surfaces. 76 ARITHMETIC. TABLE. 144 square inches (sq. in.) = 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 301 square yards or 7 ^ j / j \ 0T01 -p 4. C = ^ square rod (sq. rd.). 272J square feet ) 160 square rods = 1 acre (A.). 640 acres =1 square mile (sq. mi.). Note. A perch (P.) is a square rod, and a rood (R.) = 40 sq. rd. 10 square chains = 1 acre. A section of land is a square mile ; 36 sections = 1 township. 52. Cubic Measure is used in measuring things which haVe length, breadth, and thickness. TABLE. 1728 cubic inches (cu. in.) =1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.). 53. "Wood Measure is used in measuring wood and other merchandise. TABLE. 16 cubic feet = 1 cord foot (cd. ft.). 8 cord feet or 128 cubic feet = 1 cord (cd.). Note. A cord of wood, as generally piled, is 8 ft. long, 4 ft. wide, and 4 ft. high. 54. Liquid Measure is used in measuring liquids. TABLE. 4 gills (gi.) =lpint (pt.). 2 pints =1 quart (qt.). 4 quarts =1 gallon (gal.). Note. A gallon contains 231 cu. in. 31 1 gallons are considered a barrel (bbl.), and 63 gallons a hogshead (hhd.) ; but barrels and hogs- heads are made of various sizes. COMPOUND NUMBERS. T7 55. Apt/cliecaries' Fluid Measure is used in compounding medicines. TABLE. 60 minims {%) =1 fluid dram (f 3). 8 fluid drams = 1 fluid ounce (f S ). 16 fluid ounces = 1 pint (0.). 56. Dry Measure is used in measuring dry articles. TABLE. ? pints (pt.) =1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). Note, a oushel contains 2160.42 cu. in. 57. Troy Weight is used in weighing gold, silver, and precious stones. TABLE. 24 grains (gr.) =1 pennyweight (pwt.). 20 pennyweights = 1 ounce (1 oz.). 12 ounces = 1 pound (lb.). Note. 1 lb. Troy = 5760 grains. In weighing diamonds 1 carat = 3^ Troy grains, and is divided into quarters, which are called carat grains. The word carat applied to gold indicates the number of parts in 24 that are pure gold. For example, 18 carats fine means that || is pure gold, while the rest is alloy. 58. Apothecaries' Weight is used in compounding medi- cines TABLE. 20 grains (gr.) = 1 scruple (3). 3 scruples = 1 dram (3). 8 drams = 1 ounce ( 5 ) • 12 ounces =1 pound (lb.). Note. The pound, ounce, and grain have the same weight as those of Troy Weight. 78 ARITHMETIC. 59. Avoirdupois Weight is used in weighing all articles except gold, silver, and precious stones. TABLE. 16 drams (dr.) =1 ounce (oz.). 16 ounces = 1 pound (lb.). 100 pounds =1 hundred-weight (cwt.). 20 hundred-weight or ) _ .. , .rp ^ 2000 pounds ) on (^ .;. Note. 1 lb. Avoirdupois — 7000 gr. The long ton of 2240 lb., and the long hundred-weight of 112 lb., are used at United States Custom Houses and in wholesale transactions in coal and iron. The ton of 2000 lb. is often called the short ton. 1 quarter (qr.) = 25 lb. ; when the long ton is the standard, 1 qr. = 28 lb. 60. Circular or Angular Measure. A circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within called the centre. The bounding line is called the circumference, and any part of the circumference is called an arc. A straight line pass- ing through the centre and having its extremities in the circumference is called a diameter; a straight line drawn from the centre to the circumference is called a radius, and it is equal to one half a diameter. The circumference is divided into 360 equal parts, called degrees, each degree into 60 minutes, and each minute into 60 seconds. The opening between two straight lines which meet at a point is called an angle, and the point where the lines meet is called the vertex of the angle. An angle with its vertex at the centre of a circle is measured by the arc included between its sides. The length of an arc of one degree varies with the size of the circle, but an angle of one degree always has the same size opening between the two lines. COMPOUND NUMBERS. 79 The annexed diagram represents a circle ; G is the centre, DB an arc, AB a diameter, and CD a radius. The angle DCB contains the same number ^ of degrees as the arc DB. TABLE. 60 seconds (")= 1 minute ('). 60 minutes = 1 degree (°). 360 degrees = 1 circumference. Note. An arc of 90° is called a quadrant, and an angle of 90° is called a right angle. A degree of longitude at the equator, or a degree of latitude, equals 69.16 miles. 61. The measures of time are determined by the revolu- tion of the earth on its axis and around the sun. TABLE. 60 seconds (sec.) = 1 minute (min.). 60 minutes =lhour(hr,). 24 hours = 1 day (da.). 7 days = 1 week (wk.). 365 days =lyear(yr.). 366 days = 1 leap year. 100 years = 1 century. The length of a solar day is the interval between two successive transits of the sun over the same meridian. The length of a civil day is the interval between two successive midnights, and is the average length of all the solar days in the year. The exact time in which the earth revolves about the sun is 365 da. 5 hr. 48 min. 49.7 sec. For convenience in reck- oning it is necessary to have an integral number of days in 80 ARITHMETIC. a year, so it lias been arranged to let the common year con- sist of 365 days, while certain years, called leap years, consist of 366 days. When the number denoting the year is divisible by 4 and not by 100, or is divisible by 400, the year is a leap year. For example, 1884 and 2000 are leap years, but 1885 and 1900 are common years. By this method of reckoning the error is less than 1 day in 3600 years. The year is divided into 12 months (mo.). Their names and the number of days in each are given in the following table : January (Jan.) 31. February (Feb.) . . 28; in leap year 29. March (Mar.) 31. April (Apr.) 30. May . 31. June 30. July 31. August (Aug.) . 31. September (Sep. or Sept.) . . . .30. October (Oct.) 31. November (Nov.) 30. December (Dec.) 31. In business it is customary to reckon 30 days to a month, which makes an error of 5 days a year. Note. The number of days in each month can easily be remembered by the following stanza : Thirty days hath September, April, June, and November ; All the rest have thirty-one, Except February alone, To which we twenty-eight assign, Till leap year gives it twenty-nine. COMPOUND NUMBERS. *81 62. English or Sterling Money is the currency of Great Britain and many of its colonies. TABLE. 4 farthings (far.) = 1 penny (d.). 12 pence = 1 shilling (s.). 20 shillings = 1 pound (£). Note. A florin = 28.; a crown = 58.; a sovereign = 20 s. ; a guinea = 21 8. 63. Miscellaneous Tables. NUMBERS. PAPER. 12 units = 1 dozen (doz.). 24 sheets = 1 quire. 12 dozen = 1 gross (gro.). 20 quires = 1 ream. 12 gross = 1 great gross. 2 reams = 1 bundle. 20 units = 1 score. 5 bundles,= 1 bale. BOOKS. in 2 leaves is a folio, in 4 leaves is a quarto or 4to. in 8 leaves is an octavo or 8vo. in 12 leaves is a 12mo. in 16 leaves is a 16mo. in 18 leaves is an 18mo. . in 24 leaves is a 24mo. Note. These names are based on sheets measuring about 18 in. X 24 in. A book formed of sheets folded Reduction Descending. 64. The process of changing a compound number from one denomination to another without altering its value is called reduction. When the reduction is from a higher to a lower denomination, it is called reduction descending. 82 ARITHMETIC. I. Reduce 8 lb. 6 oz. 8 pwt. 21 gr. to grains. 8 lb. 6 oz. 8 pwt. 21 gr. 12 96 6 Since there are 12 oz. in 1 lb., in 302 OZ. ^ lb- there are 8 times 12 oz., which 20 equals 96 oz. ; 6 oz. added to this oqTq gives 102 oz. In 1 oz. there are 20 o pwt. ; in 102 oz. there are 102 times 20 pwt., or 2040 pwt.; 8 pwt. added gives 2048 pwt. In 1 pwt. there are 24 gr. ; in 2048 pwt. there are 2048 times 24 gr., or 49162 gr.; 21 gr. added gives 49173 gr. 2048 pwt. 24 8192 4096 49152 21 49173 gr. EXAMPLES. 1. Eeduce 5 yd. 2 ft. 7 in. to inches. 2. Reduce 27 gal. 2 qt. 1 pt. 3 gi. to gills. 3. Reduce 8 bu. 3 pk. 4 qt. 1 pt. to pints. 4. Reduce 29 cu. yd. 8 cu. ft. 999 cu. in. to cubic inches. 5. Reduce 145° 6' 33" to seconds. 6. Reduce £24 18 s. 9 d. 2 far. to farthings. 7. Reduce 19 lb. 6 oz. 3 pwt. 20 gr. to grains. 8. Reduce 11 lb. 4 5 4 3 1 3 15 gr. to grains. 9. Reduce 4 T. 2 cwt. 1 qr. 11 lb. to ounces. 10. Reduce 3 0. 7 f 5 4 f 3 40 rn^ to minims. 11. Reduce 8 cd. 2 cd. ft. 13 cu. ft. to cubic feet. 12. Reduce 2 mi. 51 rd. 4 yd. 2 ft. 7 in. to inches. 13. Reduce 8 mi. 3 fur. 15 rd. 1 ft. 9 in. to inches. COMPOUND NUMBERS. 83 14. Keduce 5 A. 101 sq. rd. 25 sq. yd. 112 sq. in. to square inches. 15. Reduce 8 A. 2 R. 21 P. 17 sq. yd. 6 sq. ft. 89 sq. in. to square inches. 16. Reduce 4 mi. 65 ch. 72 li. 5 in. to inches. . 17. Reduce 3 yr. 7 wk. 6 da. 21 hr. to seconds. 18. Reduce 11 yr. 3 wk. 4 da. 18 hr. to minutes, allowing for three leap years. 19. How many units are there in 8 gro. 8 doz. ? 20. How many sheets are there in 2 bundles 1 ream 15 quires 10 sheets ? 21. Find the number of ounces in a long ton. 22. Find the number of gills of molasses in a barrel which contains 86 gal. 23. What is the value of a silver cup weighing 10 oz. 16 pwt. at 12^ cents a pennyweight ? 24. What is the value of 50 lb. 8 oz. of gold at $20.59J per ounce ? Reduction Ascending. 65. When a compound number is reduced from a lower to a higher denomination, the process is called reduction ascending. I. Reduce 766 gi. to higher denominations. 4)766 gi. Since there are 4 gi. in 1 pt., in 2)191 pt. 2 si. "^^^ S^- t^^re are as many pints as A\qn nt 1 -nt ^ ^'^ contained times in 766, which —^ , o 1. equals 191 pt and 2 gi. remaining. Z6 gal. 6 qt. rj,^^^^ ^^g 2 pt. in 1 qt. ; in 191 pt. Ans. 23 gal. 3 qt. 1 pt. 2 gi. there are as many quarts as 2 is contained times in 191, which equals 96 qt. and 1 pt. remaining. There are 4 qt. in 1 gal. ; in 96 qt. 84 ARITHMETIC. there are as many gallons as 4 is contained times in 95, which equals 23 gal. and 3 qt. remaining. The entire result is 23 gal. 3 qt. 1 pt. 2 gi. II. Eeduce 104037 in. to higher denominations. 12) 104037 in. The method is the same as that 3)8669 ft. 9 in. ^^®^ ^^ *^® preceding example. KixoSftQ vd 2 ft When the divisor is 5|, both divi- o 2 dend and divisor are multiplied by 2 to avoid fractions. The divi- ^ dend and divisor thus obtained are 320)525rd.f yd. = 1J yd. half-yards; hence the remainder 1 mi. 205 rd. is 3 half -yards, which equals 1^ 1 mi. 205 rd. 1 yd. 2 ft. 9 in. ^^ 1 ft. 6 in. 1 mi. 205 rd. 2 yd. 1 ft. 3 in. added to the rest of the answer. The sum of 9 in. and 6 in. is 15 in., which equals 1 ft. 3 in. Write the 3 in., and carry 1 ft. to the column of feet. The sum of 2 ft., 1 ft., and 1 ft. is 4 ft., which equals 1 yd. 1 ft. Write the 1 ft. and carry 1 yd. to the column of yards. The sum of 1 yd. and 1 yd. is 2 yd. Note, In square measure, when the divisor is 30|, multiply both dividend and divisor by 4. III. Reduce 4840371 min. to higher denominations. 60)4840371 min. When the subject of time is 24)80672 hr. 51 min. being considered, proper allowance oaK^ZooF^ -, o 1, must be made for leap j^ears. 365)3361 da. 8 hr. ^. . ^, K \ — Smce every fourth year is a leap o year, in 9 years there are at least — 2 leap years, and hence 2 da. must • ^ d^' be taken from the 76 da. remain- Ans. 9 yr. 74 da. 8 hr. 51 min. ^»g' ^^^^h leaves 74 da. In order to ensure absolute ac- curacy with regard to the number of days, the exact number of leap years in the given time must be known. Note. When the divisor is a large number, it is more convenient to perform the long division at one side of the work and then tabulate the results as if it had all been done by short division. COMPOUND NUMBERS. 85 EXAMPLES. 1. Eeduce 34718 far. to higher denominations. 2. Eeduce 2763 gi. to higher denominations. 3. Eeduce 935923 cu. in. to higher denominations. 4. Eeduce 67421" to higher denominations. 5. Eeduce 49328 rri to higher denominations. 6. Eeduce 677653 in. to higher denominations. 7. Eeduce 10075 li. to higher denominations. 8. Eeduce 147655 sq. yd. to higher denominations. 9. Eeduce 1286 pt. to bu., pk., etc. 10. Eeduce 54321 gr. to lb., oz., etc. (Troy Weight). 11. Eeduce 87634 gr. to lb., S, etc. 12. How many cords and cord feet are there in 2224 cubic feet ? 13. How many bales, bundles, etc., are there in 10379 sheets ? 14. Eeduce 8256120 sec. to higher denominations. 15. In 372483 oz. how many T., cwt., qr., etc. ? 16. Change 106760 ft. to mi., rd., etc. 17. Eeduce 8868097 sq. ft. to A., E., P., etc. 18. In 8476321 in. how many mi., fur., rd., etc. ? 19. In 1320765 sq. in. how many sq. rd., sq. yd., etc. ? 20. In 80937864 sq. in. how many A., sq. rd., etc.? 21. Eeduce 7963721 min. to yr., da., etc. 22. A box contains 12579 buttons ; find the number esti- mated in great gross, gross, etc. 4rd. ^yd. 1ft. 10 in. 6 2 7 14 2 1 9 21 3 2 6 46 3i 2 8 1 6 86 ARITHMETIC. Addition of Compound Numbers. 66. I. Find the sum of 4 rd. 2 yd. 1 ft. 10 in., 6 rd. 2 ft. 7 in., 14 rd. 2 yd. 1 ft. 9 in., and 21 rd. 3 yd. 2 ft. 6 in. Since only units of the same kind can be added, write the numbers so that units of the same kind shall be in the same column. Then begin at the riglit to add. The sum of the inches is 32 in., which equals 2 ft. 8 in. Write the 8 in., and carry 2 ft. 46 rd. 4 yd. 1 ft. 2 in. to the column of feet. The sum of the feet, including 2 ft. previously obtained, is 8 ft., which equals 2 yd. 2 ft. Write the 2 ft., and carry 2 yd. to the column of yards. The sum of the yards, including 2 yd. previ- ously obtained, is 9 yd., which equals 1 rd. 3| yd. Write the 3^ yd., and carry 1 rd. to the column of rods. The sum of the rods, including 1 rd. previously obtained, is 46 rd. Cross out the | yd., and write its equivalent 1 ft. 6 in. Then add again, and the entire result is 46 rd. 4 yd. 1 ft. 2 in. EXAMPLES. 1. Find the sum of 18 gal. 3 qt. 1 pt. 1 gi., 5 gal. 1 pt. 3 gi., 16 gal. 2 qt. 2 gi., and 4 gal. 1 qt. 1 pt. 2. Find the sum of 101 bu. 3 pk. 5 qt., 27 bu. 2 pk. 6 qt. 1 pt., 14 bu. 1 qt. 1 pt., and 33 bu. 3 pk. 7 qt. 3. Find the sum of 27° 30' 54", 32° 24' 58", 62° 47' 25", and 75° 29' 47". 4. Find the sum of 16 cd. 6 cd. ft. 14 cu. ft., 22 cd. 2 cd. ft. 4 cu. ft., and 19 cd. 1 cd. ft. 2 cu. ft. 5. Find the sum of 74 cu. yd. 20 cu. ft. 918 cu. in., 29 cu. yd. 15 cu. ft. 1000 cu. in., and 14 cu. yd. 2 cu. ft. 323 cu. in. 6. Find the sum of 4 wk. 2 da. 17 hr. 48 min. 37 sec, 6 da. 2 hr. 29 min. 13 sec, and 12 wk. 1 da. 11 hr. 16 min. 4 sec COMPOUND NUMBERS. 87 7. Find the sum of £5 15 s. 6 d. 2 far., £7 8 s. 1 far., 19 s. 7d. 3 far., and £12 10 s. 8. Find the sum of 3 lb. 9 oz. 15 pwt. 12 gr., 6 lb. 16 pwt. 8 gr., and 3 lb. 11 oz. 7 pwt. 4 gr. 9. Eind the sum of 5 lb. 9 5 6 3 2 3 15 gr., 7 lb. 11 5 73238 gr., and 11 lb. 7 5 G 3 1 3 3 gr. 10. Find the sum of 18 T. 7 cwt. 1 qr.. 18 lb. 12 oz., 18 cwt. 3 qr. 21 lb. 6 oz., and 9 T. 14 cwt. 15 lb. 15 oz. 11. Find the sum of 5 T. 18 cwt. 52 lb. 8 oz. 6 dr., 15 T. 7 cwt. 44 lb. 10 oz. 12 dr., and 15 cwt. 78 lb. 12 oz. 14 dr. 12. Find the sum of 3 mi. 7 fur. 19 rd. 4 yd, 1 ft., 7 mi. 1 fur. 32 rd. 1 yd. 2 ft., and 9 mi. 6 fur. 25 rd. 3 yd. 13. Find the sum of 15 mi. 110 rd. 4 yd. 1 ft. 6 in., 22 mi. 15 rd. 5 yd. 2 ft. 10 in., 17 mi. 214 rd. 2 yd. 7 in., and 63 rd. I yd. 1 ft. 11 in. 14. Find the sum of 28 A. 120 sq. rd. 15 sq. yd. 7 sq. ft. 120 sq. in., 6 A. 91 sq. rd. 21 sq. yd. 4 sq. ft. 32 sq. in., and 65 A. 11 sq. rd. 12 sq. yd. 6 sq. ft. 14 sq. in. 15. Find the sum of 3 A. 2 R. 17 P. 22 sq. yd. 8 sq. ft. 28 sq. in., 4 A. 1 R. 35 P. 17 sq. yd. 4 sq. ft. 92 sq. in., and II A. 26 P. 26 sq. yd. 7 sq, ft. 116 sq. in. Subtraction of Compound Numbers. 67. I. Subtract 7 T. 13 cwt. 78 lb. 5 oz. from 18 T. 6 cwt. 31 lb. 12 oz. 18 T. 6 cwt. 31 lb. 12 oz. Write the subtrahend under the 7 13 78 5 minuend so that units of the same TTZZ TZ ] ZZT-, Z kind shall be in the same column. 10 T. 12 cwt. 53 lb. 7 oz. rp, , . ^ ^, . w ^ , ^ ^ Then begm at the right to subtract. Subtracting 5 oz. from 12 oz., we have 7 oz. remaining. We cannot subtract 78 lb. from 31 lb., so we take 1 cwt. from 6 cwt. and add it. 88 AEITHMEO'lC. reduced to pounds, to 31 lb., making 131 lb. ; subtracting 78 lb. from 131 lb., we have 53 lb. We cannot subtract 13 cwt. from 5 cwt. ; so we take 1 T. from 18 T. and add it, reduced to hundred-weight, to 5 cwt., making 25 cwt. ; subtracting 13 cwt. from 25 cwt., we have 12 cwt. Subtracting 7 T. from 17 T., we have 10 T. The entire result is 10 T. 12 cwt. 53 lb. 7 oz. EXAMPLES. 1. Subtract 8 cwt. 72 lb. 9 oz. from 13 cwt. 90 lb. 14 oz. 2. Subtract 5 cd. 6 cd. ft. 8 cu. ft. from 76 cd. 3 cd. ft. 12 cu. ft. 3. Subtract 35 gal. 3 qt. 2 gi. from 57 gal. 2 qt. 1 pt. 1 gi. 4. Subtract 2 bu. 3 pk. 2 qt. 1 pt. from 7 bu. 3 pk. 5. Subtract £13 12s. 5d. 3far. from £21 9s. 7d. 2far. 6. Subtract 54° 42' 30" from 90°. 7. Subtract 12 lb. 5 oz. 16 pwt. 15 gr. from 18 lb. 6 oz. 14 pwt. 3 gr. 8. Subtract 9 lb. 10 5 5 3 1 3 16 gr. from 12 tb. 8 5 4 3 2 3 7 gr. 9. Subtract 10 lb. 8 oz. from 5 T. 10. Subtract 2 yr. 202 da. 21 hr. 54 min. 25 sec. from 5 yr. 71 da. 15 hr. 45 min. 45 sec. 11. Subtract 3 da. 16 hr. 18 min. 45 sec. from 1 wk. 12. Subtract 4 mi. 220 rd. 4 yd. 2 ft. 7 in. from 5 mi. 115 rd. 2 yd. 13. Subtract 6 fur. 32 rd. 5 yd. 1 ft. 6 in. from 3 mi. 4 fur. 32 rd. 3 yd. 10 in. 14. Subtract 7 sq. yd. 139 sq. in. from 1 sq. rd. 5 sq. ft. 16. Subtract 3 A. 3 E. 12 P. 18 sq. yd. 8 sq. ft. 87 sq. iiL from 4 A. 2 E. 16 P. 11 sq. yd. 7 sq. ft. 54 sq. in. 16. What is the difference between 2S mi. and 27 mi. 7 fur. 39 rd. 5 ft. 11.9 in.? COMPOUND NUMBERS. 89 Difference between Dates. 68. I. Find the difference of time between Aug. 25th, 1872 and Mar. 12th, 1887. 1887 — 3 — 12 In subtracting dates it is customary to 1872 — 8 — 25 reckon 30 da. to a month. Since March — ■ — and August are respectively the third and 14 yr. 6 mo. 17 da. ^j^j^^,^ months, we write 3 and 8 instead of the names of the months. The subtraction is then performed like that of ordinary compound numbers. II. Find the exact number of days from Dec. 23rd, 1887 to Apr. 13th, 1888. When the exact number of days is wanted, we must take the actual number of days in eacli month. We must not count in both dates; it is customary to omit the former and count the latter. From Doc. 23rd to the end of the month there are 8 da. January has 31 da. Since 1888 is a leap year, February has 29 da. March has 31 da. There are 13 112 da. da. to be counted in April. By addition we find the result to be 112 da. 8 31 29 31 13 EXAMPLES. 1. Find the difference of time laetween Feb. 18th, 1856 and Oct. 25th, 1874. 2. Find the difference of time between Nov. 19th, 1874 and Apr. Srd, 1888. 3. Find the difference of time between Aug. 31st, 1876 and May 10th, 1886. 4. Washington was born Feb. 22nd, 1732, and died Dec. 14th, 1799 ; what was his age ? 5. A ship started on a whaling voyage Apr. 24th, 1875, and returned Sept. 8th, 1880 ; how long was it gone ? 90 ARITHMETIC. 6. A mortgage was dated Oct. 21st, 1879, and was paid July 1st, 1887 ; how lon^ did it run ? 7. On the 1st of January, 1888, how much time had passed since the Declaration of Independence, July 4th, 1776? 8. Find the exact number of days from 'Apr. 5th, 1883 to Dec. 9th, 1883. 9. Find the exact number of days from Aug. 19th, 1883 to June 25th, 1884. 10. Find the exact number of days from May 14th, 1884 to Nov. 5th, 1884. 11. Find the exact number of days from Dec. 27th, 1884 to July 5th, 1885. 12. A note was dated July 30th, 1886, and was paid May 9th, 1887 ; how many days did it run ? 13. A man started on a business trip Kov. 14th, 1887, and returned Mar. 9th, 1888 ; how many days was he gone ? 14. If the spring term of a school ends June 22nd, and the fall term begins Sept. 19th, how many days are there in the summer vacation ? Multiplication of Compound Numbers. 69. I. Multiply £4 13 s. 10 d. 3 far. by 7. £4 13 s. 10 d. 3 far. Write the multiplier under the lowest 7 denomination of the multiplicand, and be- £32 17s. 3d. Ifar. S^^ ** *^^^ right to multiply. 7 times 3 far. are 21 far., which equals 5 d. 1 far. Write the 1 far., and reserve 5 d, to be added to the product of the pence. 7 times 10 d. are 70 d. ; the sum of 70 d. and 5 d. is 75 d., which equals 6 s. 3 d. Write the 3 d., and reserve 6 s. to be added to the product of the shillings. 7 times 13 s. are 91 s. ; the sum of 91 s. and 6 s. is 97 s.. COMPOUND NUMBERS. 91 which equals £4 17 s. Write the 17 s., and reserve £4 to be added to the product of the pounds. 7 times £4 are £28 ; the sum of £28 and £4 is £32. The entire result is £32 17 s. 3 d. 1 far. EXAMPLES. 1. Multiply 12 T. 14cwt. Iqr. 181b. 10 oz. by 4. 2. Multiply 4 yr. 78 da. 18 hr. 15 min. 30 sec. by 6. 3. Multiply 8 A. 2 R. 22 P. 6 sq. yd. 5 sq. ft. 42 sq. in. by 8. 4. Multiply 18 bu. 3pk. 5qt. 1 pt. by 9. 5. Multiply 3 A. 104 sq. rd. 25 sq. yd. 8 sq. ft. by 12. 6. Multiply 37 gal. 3 qt. 1 pt. 2 gi. by 14. - 7. Multiply 24° 36' 50" by 15. 8. Multiply 15 cu. yd. 12 cu. ft. 227 cu. in by 18. 9. Multiply 21 tb. 9 5 2 3 1 3 16 gr. by 20. 10. Multiply 15 mi. 128 rd. 1 ft. by 23. 11. Multiply £9 17 s. 6d. 1 far. by 28. 12. Multiply 5 T. 8 cwt. 64 lb. 8 oz. 6 dr. by 37. 13. Multiply 8 mi. 5 fur. 16 rd. 3 yd. 2 ft. 8 in. by 48. 14. Multiply 4 lb. 8 oz. 16 pwt. 20 gr. by 72. Division of Compound Numbers. 70. I. Divide 41 bu. 1 pk. 7 qt. 1 pt. by 9. 9 )41 bu. 1 pk. 7 qt. 1 pt. Write the divisor at the left of the 4 bu. 2 pk. 3 qt. 1 pt. dividend, and begin at the left to divide. 41 bu. divided by 9 equals 4 bu., with a remainder of ^ bu. Write the 4 bu., and reduce 5 bu. to pecks; the re- sult, after adding 1 pk., is 21 pk. 21 pk. divided by 9 equals 2 pk., with a remainder of 3 pk. Write the 2 pk., and reduce 3 pk. to quarts; the result, after adding 7 qt., is 31 qt. 31 qt. divided by 9 equals 3 qt., with a remainder of 4 qt. Write the 3 qt., and reduce 4 qt. to pints; the result, after adding 1 pt., is 9 pt. 9 pt. divided by 9 equals 1 pt The entire result is 4 bu. 2 pk. 3 qt. 1 pt. 92 ARITHMETIC. II. Divide 161 cd. 4 cd. ft. 13 cu. ft. by 21. 21)161 cd, 4 cd. ft. 13 cu. ft. (7 cd. 147 14 _8 112 4 ?l)116cd.ft. (5cd.ft. The arrangement of work _ '^ here given can be used when 11 the divisor is a large number. J6 176 J3 . . 21)189(9 cu. ft. 189 Ans. 7 cd. 5 cd. ft. 9 cu. ft. III. Divide 245° 34' 12" by 9° 26' 42". 9° 26' 42" 245° 34' 12" 60 60 540 14700 34002)884052(26 26 34 68004 566' 14734' 204012 60 60 204012 33960 884040 42 12 34002" 884052" When both dividend and divisor are compound numbers, reduce both to the lowest denomination mentioned in either, and divide as in simple numbers. Notice tliat the answer is 26, not 26". EXAMPLES. 1. Divide 3 wk. 6 da. 14 hr. 17 min. 57 sec. by 3. 2. Divide 38 A. 114 sq. rd. 11 sq. yd. 4 sq. ft. 72 sq. in. by 5. COMPOU]ST> NUMBERS. 93 3. Divide £32 16 s. 3 d. by 7. 4. Divide 5 cwt. 12 lb. 4 oz. by 7. 5. Divide 85 gal. 2 qt. 1 pt. 3 gi. by 11. 6. Divide 32 lb. 10 oz. 7 pwt. 18 gr. by 13. 7. Divide 4° 3' 17".06 by 15. 8. Divide 347 bu. 1 pk. 7 qt. 1 pt. by 19. 9. Divide 6 mi. 7 fur. 30 rd. 2 ft. by 48. 10. Divide 152 lb. 5 5 1 3 2 3 14 gr. by 61. 11. Divide 3 rd. 3 yd. 2 ft. 10 in. by 2 ft. 8 in. 12. Divide 35 gal. 3 qt. 1 pt. 1 gi. by 4 qt. 1 pt. 3 gi. 13. Divide 15 cwt. 2 qr. 19 lb. 12 oz.. by 1 qr. 12 lb. 6 oz. 14. Divide 92 cd. 1 cd. ft. by 5 cd. 5 cd. ft. 6 cu. ft. 15. Divide £131 Is. lOd. 2 far. by £8 14s. 9d. 2 far. 16. How many house lots, each containing 1 A. 89 sq. rd., are there in a piece of land containing 56 A. 4 sq. rd. ? 17. How many coins, each weighing 15 pwt. 18 gr., can be made out of 16 lb. 10 oz. 7 pwt. 18 gr. of metal ? 18. How many bags, each containing 2 bu. 1 pk. 3 qt., will be required to hold 111 bu. 2 pk. 4 qt. of grain ? To Multiply or Divide a Compound Number by a Fraction. 71. I. Multiply 4 gal. 1 pt. 3 gi. by f . 4 gal. qt. 1 pt. 3 gi. 3 Multiplying by | is the same as 5 )12 gal. 2qt. 1 pt. 1 gi. multiplying by 3 and dividing the 2gal. 2qt. Igi. result by 6. 94 ARITHMETIC. To multiply a compound number by a fraction, multiply by the numerator and divide by the denominator. If the multiplier is a mixed number, multiply by the integral and fractional parts separately, and add the results. II. Divide 2 lb. 8 oz. 2 pwt. 6 gr. by ^. 2 lb. 8 oz. 2 pwt. 6 gr. 9 Dividing by | is the same as mul- 7 )24 lb. 1 oz. pwt. 6 gr. tiplying by f . 3 lb. 5 oz. 5 pwt. 18 gr. To divide a compound number by a fraction, midtiply by the denominator and divide by the numerator. If the divisor is a mixed number, reduce the mixed num- ber to an improper fraction, and proceed as before. EXAMPLES. 1. Multiply 5 T. 6 cwt. 48 lb. 5 oz. by f . 2. Multiply 13 cd. 2 cd. ft. 9 cu. ft. by ^. 3. Multiply 7 bu. 2 pk. 3 qt. 1 pt. by j\. 4. Multiply 14 lb. 8 5 5 3 1 3 12 gr. by ff . 5. Multiply 4 mi. 112 rd. 1 yd. 1 ft. 8 in. by 5|. 6. Multiply 23° 6' 41" by 13^. 7. Multiply 2 wk. 4 da. 19 hr. 7 min. 24 sec. by 8JJ. 8. Divide £4 14s. 3d. 3 far. by j%. 9. Divide 7 sq. rd. 4 sq. yd. 6 sq. ft. 41 sq. in. by ^. 10. Divide 23 gal. 3 qt. 1 pt. 3 gi. by if. 11. Divide 2 mi. 3 fur. 16 rd. 2 yd. 2 ft. 6 in. by 3f 12. Divide 13 T. 13 cwt. 3 qr. 9 lb. 6 oz. by 8|. 13. Divide 14° 58' 8" by ll^^. COMPOUND NUMBEh-S. 95 To KEDX;t;i!,' A FRACTION OF ONE DENOMINATION TO LoWER Denominations. 72. I. Keduce -^ of a yard to the fraction of an inch. - « 3 times the number of yards equals the num- — X3xX?=Ti>i' ber of feet, and 12 times the number of feet 4 equals the number of inches. Hence :^ yd. equals jV X 3 X 12 in., which equals f in. II. Kednce ^ cwt. to lower denominations. 7 25 ,-. ±XX^0 = ^ = 29^ lb. Since there are 100 lb. in 1 cwt, in ^^ ^:i cwt. there are ^^^ of 100 lb., which equals 29^ lb. In 1 lb. there are 16 oz. ; i X 10 = - = 22 oz ^" s ^^* *^'®'*^ ^^ i ®^ 1^ ®2'' which equals ^ 3 2§ oz. In 1 oz. there are 16 dr. ; in jj oz. ^ there are i? of 16 dr., which equals lOJ ?Xl6 = — = 10f dr. dr. The entire result is 29 lb. 2 oz. 3 3 lOfdr. Ans. 291b.2oz.l0|dr. When fractions of different denominations are to be added or subtracted, reduce the fractions to lower denominations, and then perform the operations indicated. EXAMPLES. 1. Eeduce ^ro- ^^ ^ cord to the fraction of a cubic foot. 2. Eeduce -g^^ of a shilling to the fraction of a farthing. 3. Keduce -^ of a gallon to the fraction of a gill. 4. Eeduce ^ of a bushel to the fraction of a pint 5. Reduce -ff of a week to minutes. 6. Reduce ^^ of a furlong to inches. 7. Reduce -^^ of a pound Troy to grains. 8. Reduce y|^ of a rood to square feet. 96 AEITHMETIC. 9. Eeduce ^ bu. to lower denominations. 10. Keduce f tb. to lower denominations. 11. Reduce f rd. to lower denominations. 12. Reduce -^ mi. to lower denominations. 13. Reduce i yr. to lower denominations. 14. Reduce ^^ bu. to lower denominations. 15. Reduce -f^ A. to lower denominations. 16. Reduce ^ A. to lower denominations. 17. Reduce ff gal. to lower denominations. 34 18. Find the value of -^ of J of an acre at $1.36 per square foot. ^'^ 19. Add ^ of a furlong, ^ of a rod, and ^ of a yard. 20. From f of a gallon subtract 1-| of a pint. 21. Add |- of a pound, | of a shilling, and -f- of a penny. 22. Add -^ lb., m oz., and 9|- pwt. 23. Express in rods, yards, etc., -f-^ mi. +| of 40 rd. + |yd. 24. From | of 4| bu: subtract f of 2^ pk. 25. Add together f of £13, | of i- of f of £2 12 s., and fof9d. ^^ To Reduce Lower Denominations to a Fraction OF A Higher Denomination. 73. I. Reduce -| of a grain to the fraction of an ounce Troy. ^1 y J_ _ _1_ ^V ^f *^® number of grains equals the 3 24 ^0 720 ' number of pennyweights, and ^V of the 1^ number of pennyweights equals the num- ber of ounces. Hence f gr. equals f X ^V ^ ^V o*=> which equals j^-^ oz. 20 _ = lhr. 60" 3 2 M- = ^X 1 _2 da. 24 3 3 "" 9 2| = = '^x 1_ 20 wk. COMPOUND KUMBEKS. 97 IL KeduL'e ij? da. 5 hr, 20 min. to the fraction of a week. Since there are 60 min. in 1 iir., 20 min. equal |g of an hour, whioii equals \ hr. In 1 da. there are 24 hr. ; 5^ hr. equals — of a day, which equals f da. In 1 wk. there are 7 da.; 2f da. e^ual -2 of a week 9" "^ 7 ~ 63 ""^ which equals l^ wk. EXAMPLES. 1. Keduce ^ of a gill to the fraction of a gallon. 2. Keduce ^ of an inch to the fraction of a rod. 3. Reduce ^ of a pound to the fraction of a ton. 4. Reduce 2| qt. to the fraction of a bushel. 5. Reduce 9^ sq. ft. to the fraction of a rood. 6. What part of a mile is one inch ? 7. Reduce 12 s. 6d. to the fraction of a pound. 8. Reduce 9 hr. 20 min. to the fraction of a week. 9. Reduce 7 qt. 1 J pt. to the fraction of a bushel. 10. Reduce 9 rd. 1 ft. 6 in. to the fraction of a furlo/ig. 11. Reduce 6 cwt. 2 qr. 24 lb. to the fraction of a ton. 12. Reduce 15 5 3 1 3 16 gr. to the fraction of a pound. 13. Reduce 6 rd. 5 ft. 9 in. to the fraction of a mile. 14. Reduce 40 sq. rd. 27 sq. yd. 4 sq. ft. 72 sq. in. to the fraction of an acre. 15. Reduce 3 bundles 9 quires 4 sheets to the fraction of a bale. 16. What part of a hogshead is 3 gal. 1 qt. 2^ gi. ? 98 ARITHMETIC. To Find what Fractional Part one Compound Number is of Another. 74. I. What part of 32 bu. 2 pk. 4 qt. is 8 bu. 3 pk. 2 qt. ? 8 2^ 8 4^ 21 5 31 13 Reduce both compound numbers to the 4 ~ § ■ 4 ~ 16 same denomination, and then by the method .« shown in § 41, find what fractional part one 814 X^X ^ 47 number is of the other. 2 87 EXAMPLES. 1. What part of 6 A. 1 E. is 3 E. 5 P. ? 2. What part of 3 bu. 2 pk. is 5 pk. 6 qt. ? 3. What part of 51° 25' 20" is 3° 51' 24" ? 4. What part of 37 sq. yd. 2 sq. ft. 116 sq. in. is 10 sq. yd. 5 sq. ft. 136 sq. in. ? 5. What part of 2 A. 71 sq. rd. is 1 A. 7 sq. ch. ? 6. What part of 4 da. is 1 da. 9 hr. 13 min. 50f| sec. ? 7. What part of 11 mi. 156 rd. 5 yd. is 1 mi. 69 rd. 1 ft. 6 in.? 8. What part of 22 gal. 3 qt. 1 gi. is 4 gal. 2 qt. 1 pt. 2 gi.? 9. What part of 4 lb. 9 5 1 3 12 gr. is 10 5 636 gr. ? 10. What part of £18 15s. 4d. 2far. is £4 3s. lOd. 2far. ? 11. What part of 5 fur. 3 rd. 3 yd. 1 ft. 6 in. is 4 fur. 17 rd. 4 yd. 10 in. ? 12. What part of 8 long tons is | of 720 lb. ? /-» 2 13. What part of -^ yards is |- of an inch ? TT 14. What part 12 yd. 1 ft. 6 in. is -^^ of a mile ? COMPOUND NUMBERS. 99 To Eeduce a Decimal of one Denomination to Lower Denominations. 75. I. Reduce 0.4375 gal. to lower denominations. 0.4375 gal. 4 Since there are 4 qt. in 1 gal., in 0.4.375 gal. 1.750P qt. there are 0.4375 of 4 qt., which equals 1.76 qt. ^ In 1 qt. there are 2 pt. ; in 0.75 qt. there are 0.75 ■'^•^^P*- of 2pt., which equals 1.5 pt. In 1 pt. there are — — - , 4 gi. ; in 0.5 pt. there are 0.5 of 4 gi., which ■'^ ^^' equals 2 gi. The entire result is 1 qt. 1 pt. 2 gi. Ans. 1 qt. 1 pt. 2 gi. When decimals of different denominations are to be added or subtracted, reduce them to the same denomination, and then perform the operations indicated. EXAMPLES. 1. Eeduce 0.7375 lb. Troy to lower denominations. 2. Eeduce 0.5625 da. to lower denominations. 3. Eeduce 0.3125 bu. to lower denominations. 4. Eeduce 0.4125 lb. to lower denominations. 5. Eeduce 0.0625 A. to lower denominations. 6. Eeduce 0.795 wk. to lower denominations. 7. Eeduce 0.445 A. to lower. denominations. 8. Eeduce 0.845 mi. to lower denominations. 9. Eeduce 0.428 T. to lower denominations. 10. Eeduce 0.333 A. to lower denominations. 11. Eeduce 0.984375 bu. to lower denominations. 12. Eeduce 0.758762 A. to lower denominations. 13. Subtract 10.869 oz. from 1.203 lb. Troy. 100 ARITHMETIC. 14. Add together 1.001 cwt. and 0.039 qr., and gire fh^ answer in ounces and the decimal of an ounce. 15. Subtract 0.335 gal. from 12.51 qt., and give the answer in pints and the decimal of a pint. 16. Subtract 7.3125 fur. from 1.03125 mi., and give the answer in yards and the decimal of a yard. ' 17. What is the cost of 0.33 bbl. of wine at ^1.15 per pint ? 18. A man bought a piece of ground containing 0.316 A. at 53 cents a square foot ; what did he pay for the piece ? To Eeduce Lower Denominations to a Decimal of a Higher Denomination. 76. I. Reduce 7 5 7 3 1 3 16 gr. to the decimal of a pound. Since there are 20 gr. in 1 3, the number of scruples equals 2V of t^e number of grains ; ^^ of 16 is 0.8,^ which, added to 1 3, equals 1.8 3- Since there are 3 Q in 1 3 , the number of drams equals \ of the number of scruples ; I of 1.8 is 0.6, which, CC2^ ft added to 7 3 > equals 7.6 3 • Since there are 8 3 in 1 5 J the number of ounces equals I of the number of drams; 1 of 7.6 is 0.95, which, added to 7 5, equals 7.95 §. Since there are 12 ^ in 1 lb., the number of pounds equals ^^ of the number of ounces ; j\ of 7.95 is 0.6625. EXAMPLES. 1. Reduce 4 s. 9 d. to the decimal of a pound. 2. Reduce 5 yd. 2 ft. 6 in. to the decimal of a rod. 3. Reduce 3 R. 13 P. 8 sq. ft. to the decimal of an acre. 4. Reduce 12 s. 9 d. 2 far. to the decimal of a pound. 5. Reduce 15 lb. 5 oz. 4 dr. to the decimal of a ton. 20 16.0 gr. 3 IS 3 8 7.60 3 12 7.9500 5 COMPOUND NUMBERS. 101 6. Eeduce 5 fur. 33 rd. 9 ft. 10.8 in. to the decimal of a mile. ', ' ;- : : ,: 7. Keduce 2 cwt. 3 qr. 3 lb. 8 oz. to tl^e decimal of i^.tjcn 8. Reduce 1 hr. 25 min. 30 sec. to the decimal' of a day. 9. Reduce 1 oz. 8 pwt. 19.2 gr. to the decimal of a pound. 10. Reduce 6 fur. 30 rd. 6 ft. 7.2 in. to the decimal of a mile. 11. Reduce 38 sq. rd. 21 sq. yd. 5 sq. ft. 108 sq. in. to the decimal of an acre. 12. Reduce 30 rd. 4 yd. 2 ft. 10 in. to the decimal of a mile. 13. Reduce 71 sq. rd. 54 sq. ft. 64.8 sq. in. to the decimal of an acre. 14. What decimal part of a degree is 52' 43".5 ? 15. Reduce 1 fur. 25 rd. 12 ft. 11 in. to the decimal of a mile. 16. Express to the nearest millionth 34 A. 7 sq. rd. 3 sq. yd. 7 sq. ft. 104 sq. in. as a decimal of a square mile. 17. Reduce 2 yr. 5 mo. 12 da. to years and the decimal of a year. 18. Reduce 18216 ft. to miles and the decimal of a mile. 19. Reduce ^ of a farthing to the decimal of a pound. 20. At 6 cents a pound, what decimal part of a ton of nails can be bought for f 4.20 ? 21. Find the value of 21 A. 3 R. 12 P. of land at $45 per acre. 22. Find the value of 7 A. 35 sq. rd, 127 sq. ft. of land at f 108.15 per acre. 23. What is the value, at $4500 an acre, of a piece oi land containing 30 sq. rd. 19 sq. ft. 89 sq. in. ? 102 AKITHMETIO. To FiND^ WHAT Decimal one Compound Number is op :« ; ./ Another. .:77:." £ What, decimal of 8 bu. 3 pk. is 2 bu. 1 pk. 5 qt. ? 8 bu. 3 pk. 2 bu. 1 pk. 5 qt. _4 4 280)77.000(0.275 32 8 5^ _3 1 2100 35 pk. 9pk.- I960 8 _8 • 1400 280 qt. 72 1400 _5 77 qt. 8 bu. 3 pk. = 280 qt. 2 bu. 1 pk. 5 qt. = 77 qt. 77 qt. is ^^^ of 280 qt., and this can be expressed as a decimal by dividing 77 by 280. Note. It is merely necessary to reduce both compound numbers to the same denomination, and then divide. Choose the denomination in which the two numbers can be most simply expressed. EXAMPLES. 1. What decimal of 132 bu. is 8 bu. 1 pk. ? 2. What decimal of 2 gal. is 1 qt: 1 pt. 2 gi. ? 3. What decimal of 19 s. 6 d. is 13 s. 41 d. ? 4. What decimal of 12° 51' 20" is 3° 51' 24"? 5. What decimal of 2 lb. 1 oz. 6 pwt. is 1 lb. 16 pwt. ? 6. What decimal of 4 da. 14 hr. 36 min. is 9 hr. 5 min. ? 7. What decimal of 4 T. 14 cwt. 56 lb. is 11 cwt. 82 lb. ? 8. What decimal of 300 yd. is 1 fur. 2 rd. 6 yd. 2 ft. ? Comparison of Weights. 78. The Troy pound and the Apothecaries' pound are equal in weight, and each contains 5760 grains ; the Avoir- dupois pound contains 7000 grains. The grain is the only COMPOUND NUMBERS. 103 denomination that is the same in all three weights ; hence, when Avoirdupois Weight is to be compared with either Troy Weight or Apothecaries' Weight, the comparison can be made by first reducing the given denominations to grains. EXAMPLES. 1. What part of a pound Avoirdupois is a pound Troy ? 2. What part of an ounce Avoirdupois is an ounce Troy ? 3. What part of 1 3 is 1 dr. ? 4. Express 4 lb. Avoirdupois as the decimal of 8 lb. Troy. 5. Find the number of pennyweights in a pound Avoir- dupois. 6. Reduce 5 lb. 6 oz. Troy to Apothecaries' Weight. 7. Reduce 8 lb. Avoirdupois to Apothecaries' Weight. 8. Reduce 12 lb. Troy to Avoirdupois Weight. 9. Reduce 18 lb. 4 oz. Avoirdupois to Troy Weight. 10. A miner obtained $9600 worth of gold. At $16 an ounce Troy, what was the weight of the metal in Avoirdu- pois Weight ? 11. Find the value of a silver cup, weighing 1 lb. 11 oz. Avoirdupois, at $1.95 per ounce Troy. 12. Find the amount gained by a druggist, who buys 16 lb. Avoirdupois of drugs at $2.75 per pound, and sells the same at 20 cts. per dram. Apothecaries' Weight. Comparison of Money. 79. The relations between United States Money and the moneys of other countries vary from time to time. It is customary for the Secretary of the Treasury to publish annually a table giving the values of the standard coins in United States Money. The values given Jan- 1st, 1889 are as follows : 104 ARITHMETIC. Value in COUNTRY. Monetary Unit. U.S. Money. Divisions of Units. Argentine Rep. Peso . . . . •10.965 100 centavos = 1 peso. Austria . . . Florin . . . .336 100 kreuzers =^1 florin. Belgium . . . Franc .193 100 centimes = 1 franc. Bolivia . . . Boliviano . . .68 100 centavos = 1 boliviano Brazil .... Milreis . . . .546 1000 reis = 1 milreis. British Posses- sions NA. . . Dollar . . . LOO 100 cents = 1 dollar. Chili .... Peso . . . . .912 100 centavos = 1 peso. Cuba .... Peseta . . . .926 100 centimos = 1 peseta. Denmark . . . Crown . . . .268 100 ore = 1 crown. Ecuador . . . Sucre . . . . .68 100 centavos = 1 sucre. Egypt .... Pound . . . 4.943 100 piastres —1 pound. France , . . Franc . . . .193 100 centimes = 1 franc. German Empire Mark . . . . .238 100 pfennig = 1 mark. Great Britain . Pound Sterling 4.8665 20 shillings = 1 pound. Greece . . . Drachma . . .193 100 lepta = 1 drachma. Guatemala . . Peso . . . . .68 100 centavos = 1 peso. Hayti .... Gourde . , .965 100 centavos = 1 gourde. Honduras . . Peso . , .68 100 centavos = 1 peso. India .... Rupee .323 16 annas = 1 rupee. Italy .... Lira . . . .193 100centesimit=l lira. Japan .... Y^n Silver . .997) .734/ 100 sens = 1 yen. Liberia . . . Dollar . . . 1.00 100 cents = 1 dollar. Mexico . . . Dollar .739 100 centavos = 1 dollar. Netherlands . . Florin .402 100 cents = 1 florin. Nicaragua . , Peso . .68 100 centavos = 1 peso. Norway . . . Crown .268 100 ore := 1 crown. Peru .... Sol . .68 100 centavos = 1 sol. Portugal . . . Milreis LOS 1000 reis = 1 milreis. Russia .... Rouble .544 100 copecks = 1 rouble. Spain .... Peseta .193 100 centimos = 1 peseta. Sweden . . . Crown .268 100 ore = 1 crown. Switzerland . . Franc . .193 100 centimes = 1 franc. Tripoli . . . Mahbub .614 20 piastres = 1 mahbub. Turkey . . . Piastre .044 40 paras = 1 piastre. U.S.of Columbia Peso . .68 100 centavos = 1 peso. Venezuela . . Bolivar • • .136 100 centimos = 1 bolivar. Note. In all answers in English Money reject fractions of a farthing less than one half, and when the fraction of a farthing equals or ex- ceeds one half, reckon it as another farthing. In the moneys of all countries having a decimal system, treat like United States Money, retaining only two decimal places in the answer (for Brazil and Portu- gal, three decimal places). COMPOUND NUMBERS. 105 EXAMPLES. 1. Eeduce £S 12 s. 6 d. .to United States money. 2. Reduce 16 guineas to United States money. 3. Eeduce $128.42 to English money. 4. Eeduce 172.46 francs to United States money. 5. Eeduce $45.36 to French money. 6. Eeduce 64.35 marks to United States money. 7. Eeduce $75.50 to German money. 8. Eeduce 32 roubles to United States money. 9. Eeduce 75 crowns to United States money, 10. Eeduce $114.25 to Austrian money. 11. Eeduce £16 15 s. to French money. 12. Eeduce 22.25 marks to English money. 13. Eeduce £17 9s. 3d. to Federal money, taking 4s. 6d. = $1.00. 14. If the value of a pound sterling is $4.85, and of a franc is 19^ cts., what is the equivalent in francs of 2 s. 4 d.? Eectangular Surfaces. 80. Any part of a flat surface taken by itself is called a plane figure. The extent of surface of a plane figure is called the area, and the distance around it is called the perimeter. A rectangle is a plane figure having four straight sides and four right angles. When the four sides of a rectangle are all equal, it is called a square. A rectangle has two dimensions — length and breadth. 106 ARITHMETIC. I i I I The unit of surface is a square, each side of which is a unit of length. For example, a square foot is a square 1 ft, long and 1 ft. wide. Suppose we have a rectangle' 4 in. long and 3 in. wide. Divide the length into four equal parts, and the width into three equal parts, and draw lines through the points of division as represented in the figure. The rectangle is thus divided into square inches. Upon each inch of length there is constructed a square inch, mak- ing a row of 4 sq. in. ; since the rectangle is 3 in. wide, there are 3 rows, each containing 4 sq. in., making 3 times 4 sq. in., which equals 12 sq. in. Hence, to find the area of a rectangle, multiply together the length and breadth expressed in the same linear units, and the result is the area exp)ressed in square units of the same name. The quotient arising from dividing the product of two factors by one of the factors is the other factor ; hence, if the area of a rectangle he divided by one dimension, the quo- tient is the other dimension. I. Find the area of a rectangular table whose length is 6 ft. 4 in. and width 4 ft. 9 in. 6 ft. 4 in. 12 72 _4 76 in. 4 ft. 9 in. 12 48 9 §7 in. 76 _57 532 380 144) 4332 (30 sq.ft. 432 12 sq. in. A71S. 30 sq. ft. 12 sq. in. The two dimensions must be reduced to inches, and their product denotes the number of square inches. Since the dimensions are given in feet and inches, the area should be expressed in square feet and square inches. COMPOUND NUMBERS. 107 II. The area of a rectangular floor is 224 sq. ft., and its length is 16 ft. ; find its width. 1fi^224. "^^^ number of square feet in the area divided by .. „ the number of feet in length equals the number of ^^ "• feet in breadth. EXAMPLES. 1. How many square yards are there in a floor 24 ft. long and 14 ft. wide ? 2. Find the cost of oil-cloth to cover a floor 15 ft. long and 10^ ft. wide at 45 cts. per square yard. 3. If a floor contains 35 sq. yd., and is 21 ft. long, what is its width ? 4. How many blocks 1 ft. square will it take to pave an alley 54 rd. long and 8 ft. wide ? 5. How many acres are there in a field 72 rd. long and 60 rd. wide ? 6. Find the area of a square field each of whose dimen- sions is 65 rd. 7. Find the value of a field 180 rd. long and 94-J- rd. wide at $18 an acre. 8. The area of a field is 10 A., and its width is 20 rd. ; what is its length ? 9. Find the cost of paving a street 1028 ft. long and 63-J- ft. wide at $3.25 a square yard. 10. What length of road 38^ ft. wide will contain 3 A. ? 11. A field is 38-^ rd. long and 37i rd. wide ; find its area in acres and square rods. 12. A path is 26 ft. 8 in. long and 5 ft. 3 in. wide j find its area in square feet. 108 ARITHMETIC. 13. A garden, containing | of an acre, measures 198 ft. on one sid5 ; find the length of the other side. 14. How many acres are there in a field 15.72 ch. long and 8.95 ch. wide ? 15. A building lot, containing |- of an acre, has a frontage of 90 ft. ; how far back does it extend ? 16. Find the difference in area between two lots of land, one of which is 30 rd. square, and the other contains 30 sq. rd. 17. A field, containing 14 A., is 56 rd. long ; what is its width ? Find the cost of building a fence around it at 45 cts. a rod. 18. Find the cost of slating a roof 40 ft. long and each of the two sides 20 ft. wide, at ^10 per square of 100 sq. ft. 19. At $37.50 per acre, find the cost of a field 55.33 ch long and 148 rd. 3 yd. 1 ft. 6 in. wide. 20. How many bricks 7^ in. long and 3^ in. wide will it take to lay a walk 462 ft. long and 6^ ft. wide ? 21. How many tiles 9 in. square will it take to pave a court 114 ft. long and 48 ft. wide ? 22. How many boards, each 14 ft. long and 7^ in. wide, will it take to build a platform 42 yd. long and 30 yd. wide ? 23. The area of a field is 49 sq. rd. 22 sq. yd. 6 sq. ft. 108 sq. in., and the length is 7 rd. 4 yd. 1 ft. 6 in. ; find the width. 24. Find the cost, at 60 cents a square yard, of making a gravel path 5 ft. wide around a garden 78 ft. long and 42 ft. wide : (i) when the path is outside the garden, (ii) when the given dimensions include both garden and path. COMPOUND NUMBERS. 109 JIectangular Volumes. 81. A rectangular parallelepiped is a volume bounded by six rectangular surfaces. The bounding surfaces are called faces, and the bounding lines are called edges. The faxies taken together constitute the surface, and the lower face is called the base. When the faces are six equal squares, the volume is called a cube. A rectangular parallelopiped has three dimensions — length, breadth, and thickness. The unit of volume is a cube, each dimension of which is a unit of length. For example, a cubic foot is a cube 1 ft. long, 1 ft. wide, and 1 ft. thick. Suppose we have a rectangular parallelopiped 4 in. long, 3 in. wide, and 2 in. thick. The upper face contains 3 times i sq. in., or 12 sq. in. ; and if fche parall-elopiped were 1 in. thick, there would be as many cubic inches as there are square inches on the upper face. But the paral- lelopiped is 2 in. thick, and must, therefore, contain twice as many cubic inches as it would if it were only 1 in. thick, or 2 times 3 times 4 cu. in., which equals 24 cu. in. Hence, to find the cubic contents of a rectangular parallelopiped, multiply together the three dimen- sions expressed in the same linear units, and the result is the cubic contends expressed in cubic units of the same name. The quotient arising from dividing the product of three factors by the product of two of the factors is the third factor ; hence, if the cubic contents of a rectarigular parallelo- piped be divided by the product of two dimensions, the quotient is the third dimension. llrik \ ^ \ ^ \ Ik \ \ ^-^- Bi ^ ^" ^ "^ ■ ■':: iilltll IIMItilil ill no ARITHMETIC. I. Find the number of gallons in a cistern 7 ft. long, 6 ft. wide, and 5 ft. deep. » 7 X 6 X 5 = 210 cu. ft. 1728 210 17280 3456 231)362880 cu. in.(1570if gal. 231 1318 1155 1638 1617 The product of the three dimensions gives 210cu. ft. as the cubic contents, and this equals 362880 cu. in. Since there are 2.31 cu. in. in 1 gal., there are as many gallons in 362880 cu. in. as 231 is con- tained times in 362880, which equals 1570}^ gal. 210 231 10 11 II. A rectangular solid, whose cubic contents are 924 cu. ft., is 22 ft. long and 7 ft. wide ; what is its thickness ? 22 X 7 = 154 The product of the two known dimensions is 154)924(6 ft. 154, and dividing 924, the cubic contents, by 154 924 gives 6, the number of feet in thickness. EXAMPLES. 1. Find the number of cubic feet of air in a room 24 ft. long, 18 ft. wide, and 10^ ft. high. 2. Find the cost, at 30 cts. a cubic yard, of digging a cellar 56 ft. long, 28 ft. wide, and 9 ft. deep. 3. A reservoir 30 ft. wide and 12 ft. deep contains 960 cu. yd. ; what is its length ? 4. A vat 12 ft. square contains 1368 cu. ft. ; find its depth. 5. Find the volume of a cube whose edge measures 2 ft. 9 in. COMPOUND NUMBERS. Ill 6. How many cubic feet are there in a stick (n timber 17 ft. long, 15 in. wide, and 8 in. thick ? 7. What must be the length of a stick of timber, 1^ ft. square at the end, to contain 100 cu. ft. ? 8. If a box 5 ft. 4 in. high contains 36 cu. ft., what is the area of the base ? 9. How many cords of wood are there in a pile 36 ft. long, 6 ft. high, and 4 ft. wide ? 10. Find the value of a pile of wood 28 ft. ?.ong, 5^ ft. high, and 4 ft. wide, at $3.25 a cord. 11. What must be the length of a pile of wood, 4Jft. high and 3^ ft. wide, to contain 2 cords ? 12. If a cubic foot of ice weighs 58.1 lb., how many tons will be contained in an ice-house 45 ft. long, 32 ft. wide, and 20 ft. high ? 13. Find the number of gallons in a tank 3 ft. 6 in. long, 2 ft. 4 in. wide, and 1 ft. 10 in. deep. 14. Find the number of gallons in a cistern 5^ ft. square and 7 ft. deep. ' 15. Find the value, at 90 cts. a bushel, of the grain that will be contained in a bin 14 ft. long, 10 ft. wide, and 5 ft. deep. 16. Find the depth of a bin necessary to hold 160 bu., if its length is 9 ft. and its width 5 ft. 17. How many bricks will it take to build a wall 56 ft. long, 9 ft. high, and 4 ft. thick, each brick being 8 in. long, 4|- in. wide, and 2\ in. thick ? 18. How many stones, 10 in. long, 9 in. broad, and 4 in. thick, would it require to build a wall 80 ft. long, 20 ft. high, and 2^ ft. thick ? 112 ARITHMETIC. 19. How many square feet are there on the surface of a box 2^ ft. long, 2 ft. wide, and 3 ft. deep ? 20. How many square feet are there on the surface of a cubical box, each of whose dimensions is 2| ft. ? 21. A river, 30 ft. deep and 20 yd. wide, flows 4 mi. an hour. Find the number of cubic feet of water which pass a given point in a minute. 22. How many cords of stone will it take to build a wall, 2 ft. thick and 6 ft. high, about a cellar whose interior di- mensions, when the wall is completed, shall be 20 ft. long and 16 ft. wide ? MISCELLANEOUS EXAMPLES. 1. What is the cost of 5 T. 7 cwt. 24 lb. of hay at $16 per ton ? 2. Find the cost of 1 qt. of olive oil when 1 doz. pints cost $3.45. 3. What is the value of 1 doz. silver spoons, each weigh- ing 2 oz. 16 pwt. 16 gr., at $1.15 per ounce ? 4. How many times will a wheel, 9 ft. 4 in. in circum- ference, turn in crossing a bridge, the length of which is 54 rd. 4 yd. 2 ft. 4in. ? 5. Reduce 44920.9025 hr. to years (of 365 days), days, hours, minutes, and seconds. 6. When coal is worth $6.25 a long ton, what is the expense of a coal fire for the month of January, allowing 351b. a day? 7. If one man performs a piece of labor in 2 da. 13 hr. 41 min., how long would it take 10 men to perform the same work ? COMPOUND NUMBERS. 113 8. A rectangular field measures 30 rd. 6 ft. by 21 rd. 11 ft. ; find the area in acres, square rods, and square feet. 9. What is the value of a piece of ground, 16^ rd. long and 27^ yd. wide, at 1 s. 4 d. per square foot ? 10. If 12^ yd. of silk that is f yd. wide will make a dress, how many yards of muslin that is If yd. wide will be re- quired to line it ? 11. A cellar is to be dug 30 ft. long and 20 ft. wide ; at what depth will 50 cu. yd. of .earth have been removed ? 12. It takes 8 hr. 40 min. to fiU a certain cistern ; what part of it has been filled after water has been running in 2 hr. 45 min. 45 sec. ? 13. How long will it take a man to walk 48 mi. 210 rd. 12 ft., if he walks 15 mi. in 4 hr. 15 min. ? 14. How many silver dollars, each weighing 412^ gr., can be coined from a bar of silver weighing 8|- lb. Avoirdupois ? 15. A man earns f 325 in 2^ months, and spends in 6 months what he earns in 4^ months ; what does he save in a year ? 16. A regiment of troops enlisted for 9 months and was discharged May 25th, 1863, which was 1 mo. 12 da. after the term of service had expired. Find the- date when they enlisted. 17. A cable that weighs one ton per mile weighs how much per foot ? 18. The velocity of a body is 40 mi. per hour; what is it expressed in feet per second ? 19. If a train travels 40 ft. in a second, how far will it travel in 1 hr. 31 min. 18 sec. ? 20. Divide 2 gal. 1 qt. 1.02 pt. by 17. Express the result in pints ; also in the decimal of a barrel. 114 AKITHMETIC. 21. Find ^ of 3 mi. 2 fur. 25 rd. 3 yd. 2 ft. 6 in. as a com- pound number; reduce it to chains and the decimal of a chain. 22. Express as a fraction of an acre the ground taken up by a path 3 ft. broad round a house, the front of which is 57 ft., and side is 37 ft. 23. If 2 cu. in. of iron weigh as much as 15 ri\x. in. of water, and 1 cu. ft. of water weighs 1000 oz., find the weight in tons of 1 cu. yd. of iron. 24. If a grocer's scales give only 15 oz. 4 dr. for a pound, out of how much money is a customer cheated who buys sugar to the amount of $55.04 ? 25. If 2 A. 3E. 4 P. be multiplied by 2f, what part is the product of 15 A. 1 R. 2 P. ? 26. A owns -j^ of a field, and B owns the remainder; | of the difference between their shares is 5 A. 3R. 16^ ¥. What is B's share in acres ? 27. Thirty-six persons buy 2766 A. 3 R. 12 P. of land on equal shares. What does one man receive, who sells f of his share at Is. 9 d. 2 far. per square rod? [Give the answer in pounds and the decimal of a pound.] 28. Pind the weight of 500000 bricks at 4 lb. 2 oz. each, and the cost in dollars and cents, at 27 s. 6 d. a thousand, allowing 4 s. 2 d. to make a dollar. 29. Reduce 12 T. 8 cwt. 551b. 3 oz. 3|dr. Avoirdupois Weight to pounds and the decimal of a pound ; then reduce to Troy Weight. 30. If a body revolves uniformly in the circumference of a circle at the rate of 12° 15' 25" per minute, how long is it in performing a complete revolution ? THE METRIC SYSTEM. 115 CHAPTER VI. THE METRIC SYSTEM. 82. The metric system is a system of weights and meas- ures based upon the decimal system of notation. It has been adopted by nearly all civilized nations, and its use has been legalized in the United States and Great Britain. The unit of length is the meter, and from it are derived the units of surface, volume, capacity, and weight. The length of the meter is defined by a bar kept at Paris. This length was adopted in 1799, and is one ten-millionth of the distance from the equator to either pole, as calculated at that time. However, later calculations have proved that the meter is a very small fraction shorter than one ten- millionth of this distance on the earth's surface. From the different units are derived other denominations by adding prefixes. The prefixes for the fractional parts of the unit are derived from Latin numerals, and those for the multiples of the unit are derived from Greek numerals. Deci means tenth. Centi " hundredth. Milli " thousandth. Deka means ten. Hgkto " hundred. Kilo " thousand. Myria " ten thousand. In the following tables the denominations in common use are printed in full-faced type. Abbreviations beginning with a small letter denote a fractional part of the principal unit ; abbreviations beginning with a capital letter denote a multiple of the unit. 116 ARITHMETIC. Measures of Length. 83. TABLE. 10 millimeters C""") = 1 centimeter (•""). 10 centimeters = 1 decimeter (^'"). 10 decimeters = 1 meter ('"). 10 meters = 1 dekameter (^"'). 10 dekameters = 1 hektometer (^'"). 10 hektometers = 1 kilometer (^'"). 10 kilometers = 1 myriameter {^^), Measures of Surface. 84. The units of surface are squares whose dimensions are the corresponding linear units ; hence it takes 10 times 10, or 100, of one denomination to make one of the next higher. For measuring small surfaces the principal unit is the square meter. TABLE. 100 square millimeters (««"»'") = 1 square centimeter (*'i*™). 100 square centimeters = 1 square decimeter (sq^"'). 100 square decimeters = 1 square meter ("'i'"). 100 square meters = 1 square dekameter (^i^m), 100 square dekameters = 1 square hektometer (^^Hm^. 100 square hektometers = 1 square kilometer (^q ^m^ In measuring land the square meter is called a centar (^), the square dekameter is called an ar (*), and the square hektometer is called a hektar (°*). Measures of Volume. 85. The units of volume are cubes whose dimensions are the corresponding linear units ; hence it takes 10 times 10 THE METRIC SYSTEM. 1x7 times 10, or 1000, of one denomination to make one of the next higher. The principal unit is the cubic meter. TABLE. 1000 cubic millimeters (<="'"'") = 1 cubic centimeter («="«^«). 1000 cubic centimeters = 1 cubic decimeter (•=«). 1000 cubic decimeters = 1 cubic meter C^™). Ifi measuring wood the cubic meter is called a ster (**) ; one tenth of a cubic meter is a decister (*^), and ten cubic meters are a dekaster (^^). Measures of Capacity. 86. The unit of capacity is a liter, which equals a cubic decimeter. TABLE. 10 milliliters C"^) = 1 centiliter (<=0. 10 centiliters = 1 deciliter (<"). 10 deciliters = 1 liter ('). 10 liters = 1 dekaliter (»'). 10 dekaliters = 1 hektoliter (™). 10 hektoliters = 1 kiloliter (k>). Weight. 87. The unit of weight is a gram, which equals the weight of a cubic centimeter of water at its greatest density. TABLE. 10 milligrams (™s) = 1 centigram (•«). 10 centigrams = 1 decigram {^^). 10 decigrams = 1 gram (»). 118 ARITHMETIC. 10 grams = 1 dekagram (^«). 10 dekagrams = 1 hektogram (°8). 10 hektograms = 1 kilogram (^«) or kilo (^). 10 kilograms = 1 myriagram (^^). 10 myriagrams = 1 quintal {^). 10 quintals = 1 tonneau or ton (^). A cubic centimeter or milliliter of water weighs a gram. A cubic decimeter or liter of water weighs a kilogram. A cubic meter or kiloliter of water weighs a ton. 88. A metric number can be reduced to another denomi- nation by simply moving the decimal point. Por example, 1945.2^= 1.9452^*^, because dividing by 10 three times is the same as moving the decimal point three places to the left ; 3.726""= 372.6*=™, because multiplying by 10 twice is the same as moving the decimal point two places to the right. In ]:educing from a lower to a higher denomination, move the decimal point to the left as many places as there are in- tervals in the table between the given denomination and the required denomination. In reducing from a higher to a lower denomination, move the decimal point to the right as many places as there are in- tervals in the table between the given denomination and the required denomination. In measures of surface it takes 100 of one denomination to make one of the next higher ; hence the decimal point must be moved two places for every interval. In measures of volume it takes 1000 of one denomination to make one of the next higher ; hence the decimal point must be moved three places for every interval. EXAMPLES. 1. Reduce 6453™ to kilometers. 2. Beduce 4.15™ to centimeters. THE METRIC SYSTEM. 119 3. Reduce 6.45^ to milliliters. 4. In 9780*" how many kilometers ? 5. Write 4^"^, 5"% 2"", and S""" as meters. 6. Write 84°^, 8^, and 92*=^ as liters. 7. Write 1872.6'" as kilometers ; as centimeters ; as mil- limeters. 8. Write 67.43^*^ as tons ; as grams ; as milligrams. 9. Write 7529*=^ as liters ; as hektoliters. 10. Write 96547™^ as grams ; as kilograms. 11. W^ite 7.653^^ as liters ; as centiliters. 12. 43720""" equals how many meters ? how many centi- neters ? what fraction of a kilometer ? 13. Write S^% 6% and 72*=» as hektars. 14. Write 968.32"*! m ^g ^^^.g . g^g square centimeters. 15. Write 546.31^ as square kilometers ; as centars. 16. Write 8915200''^*=™ as square meters; as ars; as hektars. 17. How many cubic millimeters are there in a cubic dekameter ? 18. In 2.15*^""' how many cubic millimeters ? 19. Express 2328000*="*^™ as sters; as dekasters; as deci- sters. 20. What is the value in cubic centimeters of 297^ ? in cubic meters ? in cubic kilometers ? 21. Write 0.853*^"" as hektoliters; as liters; as centi- liters ; as cubic centimeters. 22. Write 81470*^"*=™ as liters; as hektoliters; as cubic meters. 120 ARITHMETIC. 23. Express 29.73^^ as liters ; as centiliters ; as cubic meters ; as cubic centimeters. 24. What is the weight of 276.5'=" «=™ of water? 25. Find the weight in kilograms of 0.0316'="'" of water. 26. How many decigrams does a dekaliter of watei weigh ? 27. What is the weight in kilograms of 12"' of water ? 28. How much will a cubic hektometer of water weigh in kilograms ? Express the same quantity of water in liters. 29. What is the amount of 34789.56^ of water in cubic centimeters ? in cubic meters ? in cubic kilometers ? its weight in grams ? in kilograms ? 30. What is the amount of 294.7361™ of water in cubic meters ? in liters ? in cubic millimeters ? its weight in tons ? in grams ? in milligrams ? 89. All operations in the metric system are performed as in decimal fractions. If metric numbers are expressed in different denominations, they must be reduced to the same denomination before they can be added or subtracted. EXAMPLES. 1. How many grams are there in 23.45^*^ and 15.8°^ ? 2. Add- together 1.23^, 306.7""", 0.5219^™, and 2.91", and express the sum in centimeters. 3. Express the sum of 305"^^, 218°^, and 7"^ in kilograms. 4. Express in square meters l^-^^ 250*- ISO"**- 1500'"i ''™. 5. Eind the sum of 1871'="'=™, 541', and 4.51™, and give the answer in liters. THE METRIC SYSTEM. 121 6. Express in cubic meters 7^"^"+ 54^'+ .03°'+ 5400'="'*'". 7. Multiply 17.28« by 312500, and give the product in kilos. 8. Multiply the sum of 7^, 823", and 125"^ by 5.12. 9. Divide 3035.25*"" by 0.0375. 10. Divide 2700^1 by 90«=l 11. Find the value in cubic decimeters of \^ of 87*^"* gOcudm QQQcucm 12. What will 100' of mercury weigli, mercury being 13.5 times as heavy as water ? 13. What weight of mercury will a vessel contain whose capacity is 20*="<='"? 14. What is the weight of water in a tank if it would take 98 minutes to empty it at the rate of 8.7' a minute ? If it were filled with oil at f 18.75 a hektoliter, what would the contents be worth ? Rectangular Surfaces and Volumes. 90. For principles see sections 80 and 81. EXAMPLES. 1. How many square decimeters are there in a board 4*" long and 0.4"* wide ? 2. How many hektars are there in a strip of land 62*'"' broa^ and 1.7^" long ? 3. How many centars are there in a sidewalk 0.42^™ long and 2.8" wide ? 4. How many ars are there in a field 54" long and 28.4™ wide? / : 122 ARITHMETIC. 5. Two rectangular fields are G^'" long and IQ^*" wide and 7^™ long and IS^"" wide respectively ; how many more hektars are there in the second than in the first ? 6. How many bricks, each 20°™ long and lO^"" wide, will it take to pave a sidewalk 3.3™ wide and 1.7^'™ long ? 7. If a person steps 0.8™ at each step, how many steps will he take in walking around a rectangular field, which contains lOSO^"^, and whose breadth is 1800™ ? 8. A rectangular piece of ground is 32™ 7*^™ long and 19™ S*'" broad. Find the cost of enclosing it with a path 1™ 5*^™ broad at 3 francs 5 centimes a square meter : (i) when the path is outside the ground, (ii) when the path is part of the ground. 9. How many liters are there in a vat 2.8™ long, 2™ wide, and 5*^™ deep ? 10. How many liters are there in a box 1.2™ long, 8°™ wide, and 50™™ deep ? 11. A cistern is 4™ long, 24^™ wide, and 80*=™ deep. How much water will it hold in cubic meters ? in liters ? ' 12. How many cubic meters of air will a room contain whose length is 5™ 2^™, whose breadth is 4™, and whose height is 35*^™ ? What is the amount in liters ? 13. There is a bin 7.6™ long, 4.3™ wide, and 3.86™ deep. How many hektoliters of wheat will it contain ? 14. How many hektoliters of oats can be put into a bin that is 2™ long, 1.3™ wide, and 1.5™ deep ? 15. What is the cost of a pile of wood whose dimensions are 2™, 1.9™, and 42.5™, at f 2 per ster ? 16. What is the cost of digging a cellar 3°'" wide, 5°™ 4™ long, and 2™ 6*^ deep, at the rate of 50 cts. a ster ? THE METRIC SYSTEM. 123 17. How many sters are there in a wall 24"* long, 8"* 5**" high, and 52*='" thick ? What would be the cost of building it at $4.25 a cubic meter ? 18. What weight of water (in kilograms) may be con- tained in a cistern 1.75"' long, l.S™ broad, and 0.8'" deep ? 19. How many liters of water can be contained in a cis- tern 5"™ long, 3"" wide, and 2*" deep ? What would be the weight of the water in kilograms ? 20. A bin is 3.4™ long, 1.36'" wide, and 0.84" deep. How many kilograms of water will it hold ? How many hekto- 1 iters of wheat will it contain ? 21. How many hektoliters will a bin hold that is 3™ long, 22*'" wide, and 0.015"'" deep? How many kilograms of water will it hold ? 22. Required the weight in centigrams of the water in a vessel 1'" 2*='" long, 6*^"* broad, and 5*" 1'""* deep. 23. A cistern is 5"" long, 36'*'" wide, and OO*^'" deep. How much water will it hold in cubic meters ? in liters ? in cubic centimeters ? in grams ? in kilograms ? 24. A box 2.3™ long, 196.7'='" broad, and 901.9*"™ deep con- tains how many liters ? If filled with water, how many grams would the water weigh ? 25. A vat is 6.3™ long, 3™ wide, and 4.2™ deep. How long will it take a water-pipe to fill the vat, if the current flows at the rate of 3.6^^ a minute ? 26. A cubical cistern is 6"** in each dimension. If 1.725^ of water can flow out per minute, how much must flow in per minute to fill it in an hour ? 27. How many grams of a liquid li times as heavy as water will fill a cube whose edge is 20*=™? How many liters ? 124 ARITHMETIC. 28. Find the weight in grams of a bar of gold l*^*" long, 2-i.'^"» wide, and 2'^'" thick, assuming the bar to be 19 times as heavy as its own volume of water. 29. What must be the length of a pile of wood, 2'" high and 1.25'" wide, to contain 12** ? 30. What must be the length of a box, 1™ wide and 1™ deep, to contain 4500^ ? 31. A cistern holding 10i-'=""' is 25*^™ wide and S'^long; find its depth in centimeters. 32. A roof 10.5™ long by 5.4™ wide drains into a tank 1" deep with a base 1.25'" by 2.5"'. What dej)th of water must fall on the roof to fill the tank ? 33. How many hektars of land can be flooded to the depth of 5"'"' from a tank holding 1000 '^ of water? The Metric System Compared with the Common System. 91. The following table of equivalents should be com- mitted to memory, since by its use weights and measures of one system can readily be converted into weights and meas- ures of the other. 1 meter = 39.37 inches. 1 kilometer = 0.62138 mile. 1 square meter = 1550 square inches. 1 hektar = 2.471 acres. 1 cubic meter = 1.308 cubic yards. 1 ster = 0.2759 cord. 1 liter = 1.0567 liquid quarts, = 0.908 dry quart. 1 gram = 15.432 grains. 1 kilogram = 2.2046 pounds Avoirdupois. THE METRIC SYSTEM. 126 I. Reduce 34* to square rods. 2.471 A. 0.34 34a must first be ted need to hektars, the 9884 denomination given in the equivalent. Since 7413 !"«= 2.471 A., 0.34"a equals 0.34 of 2.471 A., 84014 A which equals 0.84014 A. Then reducing to ■1 gA square rods, we have 134.4224 sq. rd. 134.42240 sq. rd. II. Reduce 5 mi. 3 fur. 10 rd. to kilometers. 40 )10.00 rd. 0.62138)5.40625(8.7004'^ 8 )3:25 fur . 497104 5.40625 mi. 435210 434966 244000 The compound number must first be reduced to miles, the denomi- nation given in the equivalent. Since lKm = 0.62138 mi., in 5.40625 mi. there are as many kilometers as 0.62138 is contained times in 6.40625, which equals 8.7004Km. In reducing from metric to common weights and measures, multiply the jiumher, expressed in the denomination of the equiv- alent, by the equivalent. In reducing from common to metric weights and measures, divide the number^ expressed in the denomination of the equiv- alent^ by the equivalent. EXAMPLES. 1. Reduce 600^'" to miles. 2. Reduce 10°^ to feet. 3. Reduce 42.5^ to gallons. 4. Reduce 16. 75H» to bushels. 5. Reduce 18'' to cords. 6. Reduce 50^ to grains. 126 ARITHMETIC. 7. Eeduce 126* to ounces Avoirdupois. 8. Eeduce 20^*^ to pounds Avoirdupois. 9. Find the length of a centimeter in inches. 10. Pind the number of pints in a dekaliter. 11. Eeduce 36^^ to bushels, pecks, quarts, and pint& 12. Eeduce 40.0973^™ to miles, rods, feet, and incheii 13. Eeduce 12^^ to Troy Weight. 14. Eeduce 250^^ to Avoirdupois Weight. 15. Eeduce to pounds Avoirdupois and the decimal of a pound T" 4^« 1658. 16. How many kilometers make a mile ? 17. How many hektars make a square mile ? 18. Eeduce 80 A. 40 sq. rd. to hektars. 19. Eeduce 87 bu. 3 pk. 4 qt. to hektoliters. 20. Eeduce 68| yd. to meters. 21. How many liters are there in 6 gal. of water ? 22. How many meters are there in 25 ft. ? 23. 2 gal. 3 qt. \\ pt. equal how many liters ? 24. 2 mi. 17 ft. 16.2 in. equal how many kilometers ? 25. 3 mi. 2 rd. 10 ft. equal how many meters ? 26. How many kilometers are there in 2 mi. 6 fur. 39 rd. 5 yd.? 27. How many hektoliters are there in 57 gal. 3^ pt. ? 28. If the distance between two places is 1^ miles, what is the number of kilometers ? of centimeters ? 29. Find the equivalent of the rod in the hektometers. THE METRIC SYSTEM. 127 30. A lead pencil is If ^"^ long ; 32186 of them arranged in a line would extend how many miles ? 31. Find the weight in grams of a quart of water. 32. Find the weight in grams of a cubic yard of water. 33. Find the number of kilograms in a cubic foot of water. 34. Find the weight in kilos of 15 gal. of water. 35. A cubic foot is what part of a cubic meter ? 36. Find the weight in kilograms of a block containing 12516 cu. in., assuming the weight of a cubic inch of the material to be 2 oz. 37. A train is 27 min. in passing through a tunnel, the length of which is 11220"' ; find the speed of the train in miles per hour. 38. A platform bears a weight of 100 lb. per square foot ; what is the weight in kilograms per square meter ? 39. Find in acres the area of a plot 300" long and 2°" wide. 40. How many square rods are there in a field 300" long and yig- of a kilometer wide ? 41. How many cubic yards are there in a cistern, the dimensions of which are 64'^'", 225'"", and 3.75" ? 42. How many hektoliters of grain will a bin hold whose interior length, width, and depth are each 6 ft. 6 in. ? 43. Find the value of a pile of wood 4.5" long, 1.4" wide, and 1.8" high, at $3.75 a cord. 44. Find the value of a pile of wood 15 ft. long, 4 ft. wide, and 4^ ft. high, at f 1.10 a ster. 45. Albany, IT. Y., is in 42° 39' 50", and Montreal in 45° 31' north latitude; find the distance between them in kilometers. 128 ARITHMETIC. 92. When an equivalent is given in a problem, that equivalent, and no other, should be used in the solution. In some cases the equivalent is one not given in the pre- ceding section. Other problems are given to show that it is possible to make all such reductions by using merely the two equivalents, 1™= 39.37 in. and 1^= 15.432 gr. EXAMPLES. 1. The distance from Boston to Albany is 320^" ; find the distance in miles, assuming the meter to equal 3-^ ft. 2. If one kilometer equals five eighths of a mile, how many turns will a wheel make in 20 mi., the circumference of the wheel being 4" 5'°'^ ? 3. The deciliter is 0.026 of a gallon ; wha,t will be the weight in grams of a pint of water ? 4. One dekagram is 0.3527 oz. Avoirdupois ; how many pounds Avoirdupois are there in a quintal ? 5. The kilogram equals 2 lb. 8 oz. 3 pwt. ; how many centigrams equal one grain ? 6. What fractional part of -^ of the Avoirdupois ton is 12^K, the ounce being equal to 28.35^ ? 7. A tunnel is 2 mi. 21 ch. 13.2 yd. long. Find its length in meters (1 mi. =1.61^™). 8. If a pound Avoirdupois equals 0.4536^^, how many grains are there in 3f^ ? 9. A dekaliter is 2.6 gal. What will be the weight in grams of a quart of distilled water at its greatest density ? 10. If a quart is ||- of a liter, how many quarts are there in a box 2" long, 17'*" wide, and 80*^™ deep ? 11. Given that a meter equals 3.2809 ft. ; find how many square meters there are in 1000 sq. yd. THE METRIC SYSTEM. 129 12. Find the number of cubic inches (to the nearest *nith) in the British imperial gallon, which contains 10 lb. A water. (Given l^^-^zzrSS.S cu. ft. and 1*^k=2.2 lb.) 13. The gram contains 15.432 gr. How many pounds Avoirdupois make a myriagram ? 14. The meter equals 39.37 in. ; compare the kilometer with the mile. 15. The meter equals 39.37 in. ; compute from this datum the value of 4 mi. in kilometers. 16. The meter equals 39.37 in. ; express in the metric system 1 ft. and 1 sq. ft. 17. The meter equals 39.37 in. ; find how many hektars make an acre. 18. One centimeter equals 0.3937 in. ; find how many cubic meters there are in a cord of wood. 19. Find the weight in grams of a cubic yard of water (1™=39.37 in.). 20. How many liters are there in 10 gal. 3 qt. 1 pt. 3 gi., the gallon being 231 cu. in., and the meter 39.37 in. ? 21. A cubical vat measures 9 ft. in each direction ; what is its capacity in liters ? (Given 1™=39.37 in.) 22. How many liters are contained in a cubical box 13 in. long, 13 in. wide, and 13 in. deep on the inside ? (Given l'»= 39.37 in.) How many grams of water will such a box hold? MISCELLANEOUS EXAMPLES. 1. A tank can be emptied in 86 min. by a pipe flowing at the rate of 10.8^ a minute ; find the value of the contents of the tank when filled with oil worth $15.25 a hektoliter. 130 ' ARITHMETIC. 2. If a ream of paper is 11.76°'" thick, find the thickness in millimeters of a single sheet. 3. A speculator bought 18.54* of land for $2500, and sold it for $4.50 a square meter ; find the amount he gained. 4. The distance between two places measured on a map is 156'"'". What is the distance in kilometers if the scale of the map is 1 to 80000 ? 5. The scale of a map is 4*^"" to a hektometer. The dis- tance between two points, measured on the map, is 354.2'"'" ; what is the actual distance between the points in kilometers ? 6. The area of a court is 50.7**1 "'. How many square slabs of marble, each 150"^ *'"' on the surface, will pave it ? 7. A man bought a piece of land 52.5"* square for $120 y at what price per ar must it be sold to gain $56.40 ? 8. A man bought 30" of cloth at $2.50 per meter; at what price per yard must it be sold to gain $25 ? 9. A man buys 454 bu. of wheat for $3 a bushel, and sells the wheat at $8.75 a hektoliter; how much does he gain? 10. A merchant buys 2|Hm of silk for $480, and sells the silk at $1.95 a yardj does he gain or lose, and how much ? 11. If a man pays 600 francs for 75^ of wine, what is the price per gallon in United States money ? 12. A field is 5^^™ long and 300"° wide ; find its area in hektars and in acres. 13. Find how many cubic centimeters there are in a deka- liter of water. Find also how many pounds Avoirdupoi'i there are in the same water. 14. The dimensions of a box are 3.1'", 1.5™, and 0.6™; what is the contents in cubic yards ? also in dekasters ? THE METRIC SYSTEM. 181 15. A rectangular vessel is 7™ long, 4.5'*"' wide, and 20'^'" deep; find its capacity in cubic meters, hektoliters, and gallons. 16. A vessel is 3*™ long, 20^™ wide, and 100""° deep ; how many liters of water will it contain ? How many grams ? How many cubic inches ? How many pounds ? 17. The water contained in a vessel 2*" long, 30°° wide, and 300™"" deep, would weigh how many kilograms ? would measure how many cubic inches ? how many gallons ? 18. A cistern is 4™ long, 24*" wide, and 80"" deep. (i) How much water will it hold in cubic meters ? (ii) How much in liters 1 (iii) Find approximately the amount in gallons, (iv) What would the contents weigh in kilograms » 132 ARITHMETIC. CHAPTER VII. SPECIAL PROBLEMS. Carpeting Rooms. 93. Carpeting comes in rolls, and is sold by the yard or meter. In estimating the amount of carpeting necessary for a floor, we must ascertain the number of strips, and then multiply the length of a strip by the number of strips. In making a carpet, the strips may run either lengthwise or across the room; the former way is the more usual, and should be used in solving problems, except when stated to the contrary. Oil-cloth and some other materials for floors are sold by the square yard or square meter. I. How many yards of carpet |- yd. wide does it require to cover a floor 22 ft. long and 16 ft. wide ? How much will be turned under ? 5 y(j _ ;[7 f^^ It takes as many ^rf4. ^if^ 1^., 8 198 o« 4.' strips as 1| ft. is con- 16ft.-ll.ft. = 16XT«^=-W-=8,8^strips. ,^.^^^ tj„^^^ i„ 16 22 ft. X 9 = 198 ft. = 66 yd. of carpet. ft., which equals S^V g g_8 __ _7 strips. But a strip is never split ; hence 15°*^^^ii^8^8^*''^ ^^i ^^' ^^^^^^^ under, i* is necessary to buy ^^ 9 strips, and then turn under (or cut off) the excess. Since the length of the room is 22 ft., each strip is 22 ft. long ; for 9 strips it takes 9 times 22 ft., which equals 198 ft., or 66 yd. Since 9 strips are bought and only 8^j strips are necessary to cover the floor, the remainder, f^ of a strip, is turned under (or cut off). SPECIAL PROBLEMS. 133 Since the carpet is 1| ft. wide, /j of a strip is ^5 of 1| ft., which equals I ft., or 10^ in. II. How many meters of carpet 80"*" wide will it require to cover a floor 5.48™ long and 4.6™ wide, if the strips are to run across the room ? .8" ') 5.48"* Since the strips run across the room, the width 6.85 strips. ^^ *^*® carpet runs in the same direction as the length of the room, and we must divide the 4.6"* length of the room by the width of the carpet to * - find the number of strips. Otherwise the work is 32.2™ the same as in the preceding example. Note. In examples dealing with the common system of weights and measures, common fractions are used. In examples dealing with the metric system, use decimal fractions. EXAMPLES. 1. How many yards of carpet f of a yard wide does it require to cover a floor 17 ft. long and 16 ft. 6 in. wide ? 2. How many yards of carpet 22 in. wide will it take to cover a floor 16 ft. by 17^ ft. ? 3. A person, purchasing a carpet for a room 21 ft. long and 15 ft. 9 in. wide, chooses a material which is | of a yard wide, and the pattern of which is complete in each yard of length. How much carpet must he buy in order that the pattern may be unbroken ? 4. How many yards Of carpet f of a yard wide will it require to cover a floor 18^ ft. long and' 14 ft. wide, if the strips are to run across the room ? 5. A room is 24 ft. 3 in. long and 16 ft. 4 in. wide. Find the cost of covering it with a carpet f of a yard wide at $1.25 per yard. 134 ARITHMETIC. 6. How many yards of carpet J of a yard wide will it take to cover a floor 28 ft. long and 17 ft. 9 in. wide, if there is a waste of 4 in. in each strip in matching patterns ? How much will be turned under ? 7. How many yards of carpet 1^ yd. wide does it require to cover a floor 22 ft. 8 in. long and 13 ft. 6 in. wide, if the strips run across the room ? How much will be turned under ? Find the cost of the carpet at 69 cts. a yard. 8. How many meters of carpet 90'^'" wide does it require for a floor 7™ long and 5.4™ wide ? 9. How many meters of carpet 70*='" wide will it take to cover a floor 5.56™ long and 4.7™ wide, if the strips are to run across the room ? 10. How many meters of carpet 75*^" wide will it require to cover a floor 6™ long and 4.8™ wide ? How much will be turned under ? 11. How many meters of carpeting 75*=™ wide will be needed for a room 5|™ square ? How wide a strip must be turned under ? 12. A floor 10™ by 6.5™ is to be covered with a carpet 90*=™ wide ; find the cost at |1.25 per meter. 13. The floor of a room is 5.25™ by 4.75™ ; the carpet is 75cm ■y^ri(Je and is f 4.25 a meter. Find the cost of the carpet if 3™ is wasted in matching the pattern. 14. How many meters of carpet 9*^™ wide will cover a floor 6™ long and 5™ 4*^™ wide ? What would be the cost of the carpet at $2.50 a centar ? 15. It takes 54 yd. 2\ ft. of carpet to cover a floor 23|- ft. long and 15f ft. wide ; find the width of the carpet. 16. It takes 37.5™ of carpet to cover a floor 6.25™ by 5.1".; find the width of the carpet. SPECIAL PROBLEMS. 135 17. What will it cost to floor a room 17-J- ft. long and 16 ft. wide at J^l.lO per square yard ? 18. Find the cost of covering with oil-cloth a floor 7.56" long and 5.5™ wide at 62^ cts. a square meter. 19. Eind the cost of covering with linoleum a floor 19^ tt. square at 75 cts. a square yard. Plastering Booms. 94. In estimating the amount of plastering for a room, we must take the area of the ceiling and the four walls, and from it subtract the area of the doors, windows, etc. The area of the four walls is the same as that of a rectangle whose dimensions are the perimeter and height of the room. Note. The length of a base-board equals the perimeter of the room minus the width of the doors. I. Mnd the cost of plastering a room 32 ft. long, 21 ft. wide, and 9^ ft. high, if 21 sq. yd. be allowed for doors and windows, at 33 cts. per square yard. Twice the length added to twice the width equals ^'•^^ the perimeter, and this multiplied by the height equals the number of square feet in the four walls. The area of the ceiling equals the prod- uct of the length and breadth. Adding these tdK'i r. J two results we have 1679 1651 sq. yd. sq. ft., or 186Mq. yd. Subtracting from this 21 sq. yd., the allowance for doors and windows, we have 165| sq. yd. If one square yard costs 33 cts., 165| sq. yd. cost 165| times 33 cts., which equals $54.63. 64 32 $.33 42 21 165f 106 32 9)165 ^i 64 18i 53 672 sq. ft. 165 954 1007 198 1007 sq. ft. 9)1679 sq.ft. 33 1864 sq. yd. $54.63 21 136 ARITHMETIC. EXAMPLES. 1. How many square yards of plastering are there in a ^oom 18 ft. 8 in. long, 14 ft. 6 in. wide, and 9 ft. high, mak- ing no allowance for doors and windows ? 2. How many square meters of plastering are there in a room 6.4™ long, 4.8™ wide, and 3™ high, making no allowance for doors and windows ? 3. Find the cost of plastering a room 28 ft. 8 in. long, 18 ft. wide, and 10 ft. high, if 19 sq. yd. be allowed for doors and windows, at 30 cts. a square yard. 4. Find the cost of plastering a room 8.4™ long, 5.2™ wide, and 3.5™ high, if IJ^^ be allowed for doors and windows, at 38 cts. a square meter. 5. Find the cost of plastering the walls of a room 12 ft. 11 in. square and 9 ft. 3 in. high, allowing for two windows and one door each 6 ft. 2 in. by 2 ft. 4 in., at 28 cts. a square yard. 6. Find the cost of plastering a room 6.4™ square and 3.8™ high at 42 cts. a square meter. There is a base-board 30*^™ high, and an allowance of 13.6"^™ is made for doors and windows. 7. Find the cost of plastering a room 17 ft. 4 in. long, 15 ft. 4 in. wide, and 10 ft. 6 in. high, at 35 cts. a square yard. Make allowance for a door 8 ft. by 3 ft. 6 in., three windows each 5 ft. 6 in. by 3 ft., and a base-board 1 ft. 4 in. liigh. 8. Find the cost of plastering the walls of a room 10.5™ long, 8.4™ wide, and 4.2™ high, at 45 cts. a square meter. Make allowance for a door 2.9™ by 1.2™, two windows each 2.4™ by 1™, and a base-board 32''™ high. SPECIAL PROBLEMS. 137 Papering Rooms. 95. Paper is generally sold by the roll. To find the number of rolls necessary for a room, we must divide the area of the walls by the area of one roll. When the ceiling is to be papered, its area should be added to the area of the walls. I. How many rolls of paper, 7.6™ long and 50*"" wide, will be required for a room 6.4*" long, 5.2™ wide, and 3.5" high, deducting 14*1™ for doors and windows ? Find the cost of papering the room at 90 cts. a roll and of putting on a border at 12 cts. a meter. $.90 f .12 The area of the 18 23.2 walls is found to be 12.8 7.5 10.4 .50 23.2 3.750 3.5 1160 696 3.75)67.20(17 + 375 81.20 14. 2970 2625 67.2'«i™ ^45 18 rolls. 16.20 f 2.784 67.2«i'n,and the area 2.78 of one roll of paper $18 98 ^^ 3.75*«™. It will require as many rolls as 3.75*i ™ is con- tained times in 67.2"^™, whicli equals 17 and a fraction. But a fraction of a roll is never sold ; hence it is necessary to buy 18 rolls. If 1 roll costs 90 cts., 18 rolls cost 18 times 90 cts., which equals $16.20. The border runs around the room, and its length equals the perimeter of tlie room. If 1 meter costs 12 cts., 23.2"^ cost 23.2 times 12 cts., which equals $2.78. The entire cost is the sum of 116.20 and $2.78, which equals $18.98. EXAMPLES. 1. How many rolls of paper, 7^ yd. long and 18 in. wide, will be required for a room 26 ft. long, 20 ft. 6 in. wide, and 9 ft. 4 in. high, deducting 17 sq. yd. for doors and windows ? 138 ARITHMETIC. 2. How many rolls of paper, 8*" long and 45^"" wide, Avill be required for a room T.S"" long, 5.4'" wide, and 3™ high, deducting 21"^ ™ for doors and windows ? 3. Find the cost of papering a room 23 ft. 8 in. long, 20 ft. 6 in. wide, and 10 ft. high, with paper, each roll of which is 8 yd. long and 18 in. wide, at 62-^ cts. a roll, allow- ing for a door 7 ft. 6 in. by 3 ft. 3 in., and two windows each 5 ft. 8 in. by 3 ft. 3 in. 4. Find the cost of papering a room 7.5™ square and 3.5™ high, with paper, each roll of which is 8"' long and 50*=" wide, at $1.10 a roll, and of putting on a border at 15 cts. a meter. Make allowance for two doors each 2.8™ by 1.2™, two windows each 2.2™ by 1™, and a base-board 30''™ high. 5. Find the cost of papering a room 19|- ft. long, 16J ft. wide, and 10|- ft. high, with paper, each roll of which is 7^ yd. long and 20 in. wide, at 80 cts. a roll, and of putting on a border at 12|- cts. a yard. Make allowance for a door 8 ft. by S^ ft., three windows each 5|- ft. by 3 ft., and a base- board 1^ ft. high. 6. Find the cost of papering the walls and ceiling of a room 11.2™ long, 9.6™ wide, and 5™ high, with paper, each roll of which is 7.5™ long and 50*=™ wide, at $1.25 a roll. Make allowance for two doors each 3.5™ by 1.5™, three win- dows each 2.5™ by 1.4™, and a base-board 40"''" high. 7. How many yards of paper 1^ yd. wide will be needed to paper the walls of a room 10 ft. high, 18 ft. long, and 12 ft. wide ? 8. Find the cost of papering a room 11 yd. 2 ft. 4 in. long, 6 yd. 2 ft. wide, and 5 yd. 2 ft. 6 in. high, with paper 1 yd. 4 in. wide, at 6 cts. a yard. 9. Find the cost of papering a room 6.78™ long, 4.1™ vade, and 3.64™ high, with paper 96*=™ wide, at 8 cts. a meter. SPECIAL PROBLEMS. 139 Board Measure. 96. The standard thickness for boards is one inch. A board one inch or less in thickness contains as many board feet as there are square feet in its surface. For boards more than one inch thick, and for other kinds of lumber, we must multiply the number of square feet in the surface by the number of inches in thickness in order to find the number of board feet. For example, a board 8 ft. long, 1 ft. wide, and 1 in. or less in thickness, contains 8 board feet ; a plank 8 ft. long, 1 ft. wide, and 3 in. thick, contains 24 board feet. When the metric system is used, the standard thickness is 25""". A board 25™'" or less in thickness contains as many board meters as there are square meters in its surface. For boards more than 25""" thick, and for other kinds of lumber, we must multiply the number of square meters in the sur- face by the number of times 25""° is contained in the thick- ness. For example, a board 4™ long, 50*=™ wide, and 25"*™ or less in thickness, contains 2 board meters j a plank 4"" long, 50*=™ wide, and 10*=" thick, contains 8 board meters. If a board is tapering, we must take the average width, which is one half the sum of its two end widths. In buying and selling boards, it is customary to quote them by the hundred or thousand, meaning a hundred or thousand board feet, or a hundred or thousand board meters. EXAMPLES. 1. How many board feet are there in a board 16 ft. long, 9 in. wide, and -J in. thick ? 2. How many board feet are there in a board 17 ft. 6 in. long, 1 ft. 3 in. wide, and 1 in. thick ? 3. HoAv many feet, board measure, are there in a plank 12 ft. 4 in. long. 2 ft. 3 in. wide, and 4 in. thick ? 140 ARITHMETIC. 4. How many feet, board measure, are there in a plank 16 ft. 4 in. long, 1 ft. 7 in. wide, and 4|- in. thick ? 5. How many feet, board, measure, are there in a plank 12 ft. 4 in. long, 2 ft. 5 in. wide at one end, 2 ft. 1 in. wide at the other, and 4 in. thick ? 6. How many feet of board are there in a plank 17 ft. long, 22 in. wide at one end, 13 in. wide at the other, and 3 in. thick ? 7. Find the number of board feet in a stick of timber 18 ft. long and 8 in. square. 8. Find the cost of 72 boards, each 11 ft. long, 16 in. wide, and f in." thick, at $16.50 per M. 9. Find the cost of 14 joists, each 4 in. by 3 in., and 10 ft. long, at f 13.75 per M. 10. Find the cost of the flooring for two rooms, each 24 ft. by 201 ft., with boards IJ in. thick, at f 27 per M. 11. How many board meters are there in a board 7" long, 18"» wide, and 20°^™ thick ? 12. How many board meters are there in a board 7.5"* long, 20*"" wide, and 30°*™ thick ? 13. How many meters, board measure, are there in a plank 5.8"* long, 30^™ wide, and 75"*'° thick ? 14. How many meters, board measure, are there in a plank 6™ long, 36*^"* wide at one end, 30*=™ wide at the other, and 11*=™ thick ? 15. Find the number of board meters in a stick of timber 9™ long and 30*=™ square. 16. Find the cost of 50 boards, each 4™ long. 36"° wide, and 25™"* thickf at $18 per C. SPECIAL PROBLEMS. 141 17. Find the cost of 24 planks, each 4.5" long, 42^ wide, and 8«'» thick, at 122.50 jjer C. 18. Find the cost of a stick of timber 10.6" long, 30'"' wide at one end, 25*=" wide at the other, and 20*^" thick, at ^18.50 per C. Work Problems. 97. Problems concerning work should be solved by con- sidering the fractional part of the work that can be done in a unit of time. For example, if a man can do a piece of work in 8 days, he can do -J of the work in one day ; if a cistern can be filled by a pipe in 2^ hours, — , or -j^, can be filled in one hour. * I. A can do a piece of work in 8 days, working 10 hours a day, and B can do it in 6 days, working 12 hours a day ; in how many days of 9 hours each can they together do it ? j^ J _ g^j Q _ ^Q Since A can do the work TU~^^ — TTG' — rfU- in 80 hr., he can do ^^ of it 1 "^ T A^ = 1 X ^^ = 37fJ hr. in 1 hr. ; since B can do it in 37i^-F-9 = ^H-9 = ff = 4^da. 72 hr., he can do ^V of it in 1 hr. ; hence both together can do jV+ tV* ^^ jts* ^^ 1 ^^- Since they can do J^j^ in 1 hr., it will take as many hours to do the whole as j^^^ is contained times in 1, which equals 37|| hr. Since they work 9 hr. a day, it will take as many days as 9 is coiltained times in 37}|, which equals 4j\ da. II. A cistern can be filled by a pipe in 4^ hours, and can be emptied by another pipe in 6|- hours ; if both pipes be opened, in what time will the cistern be filled ? By the first pipe /^ of the cistern will be filled in 1100111"^ ^ ^^'' ^"^ ^^ *^^ second J- -^ Tlnr = 1 X V- = 1% ^i"- pipe -^ will be emptied in the ^ajne time; hence, when both pipes are open- /y — 2%» ^^ jwu* ^1 116 3 24-15 9 ^■ 6| 25 " 20 100 100 142 ARITHMETIC. be filled in 1 hr. It will take as many hours to fill the cistern as j^ ig contained times in 1, which equals 11^ hr. EXAMPLES. 1. A can mow a field in 4|- days, and B can mow it in 6 days ; how long will it take them both to mow it ? 2. A can build a wall in 18f days, and B in 31:|^ days ; how long will it take them both together to build it ? 3. A cistern can be filled by a pipe in 3^ hours, and by another in 2^ hours ; how long will it take both together to fill it ? 4. A can do a certain piece of work in 6 days, B in 8 days, and C in 9 days. How long will it take them to do it together ? 5. A can dig a ditch in 5 days, B in 7 days, and C in 9 days ; how long will it take them all together to dig it ? 6. A can do a piece of work in 12 days, B in 15 days, and C in 20 days ; what fractional part of the work can they together do in 3 days ? 7. A cistern has two pipes, one of which can fill it in 3 hours, and the other in 4 hours ; a third pipe can empty it in 2 hours. If all three are opened when the cistern is empty, in what time will it be filled ? 8. A cistern can be filled by a pipe in 75 min., and can be emptied by another pipe in 30 min. ; if the cistern is full, and both pipes are open, in what time will the cistern be emptied ? 9. Pipes A and B can fill a cistern in 3 min. and 5 min. respectively, and C can empty it in 7^ min. In what timt will the cistern be filled when A^ B, and C are all open ? SPECIAL PROBLEMS. 143 10. A and B can do a piece of work in 12 days. A, work- ing alone, can do the same work in 20 days. How long would it take B to do it ? 11. A can do a piece of work in 10 days ; A and B can do the same work together in 7 days ; in how many days can B, working alone, do the work ? 12. A can do a piece of work in 10 days, A and C can do it in 7 days, and A and B can do it in 6 days ; in how many days can B and C together do it ? 13. A can do ^ of a piece of work in 4 days, B ^ in 5 days, C ^ in 3 days, and D ^ in 1^ days ; how long will it take them all to do it ? 14. Three men can do a piece of work in 12 hours ; A and B can do it in 16 hours, and A and C in 18 hours. What part can B and C do in 9^ hours ? 15. A can do a certain piece of work in 10 days, working 8 hours a day. B can do the same work in 9 days, working 12 hours a day. They decide to work together, and to finish the work in 6 days. How many hours a day must they work? 16. A does ^ of a piece of work in 6 days, when B comes along and helps him, and they finish it in 5 days ; how long would it take B alone to do the work ? 17. A can do as much work in 4 hours as B in 6, and B in 3^ as C in 5. A does half a certain piece of work in 12 hours ; in what time can it be finished by B and C, working separately equal times ? 18. A and B can do a piece of work in 6 days, A and C in 7 days, and B and C in 8 days; in what time can all three do it, working together^ and in what time can each one do it alone ? 144 ARITHMETIC. Clock Problems. 98. In twelve hours the minute hand of a clock passes over the face twelve times, and if the hour hand were sta- tionary, the two hands would be together twelve times. But in this interval of twelve hours, the hour hand, instead of remaining stationary, passes over the face once ; hence the two hands are together once less than twelve times, or eleven times. Since the two hands progress with a regular movement, there always is the same interval between two successive times when the hands are together, and this in- terval is ^ of 12 hours, which equals 1 hr. 5^ min. It is also true of any other position of the hands, that there is an interval of 1 hr. 5^ min. between two successive times when the hands have the same relative position. I. At what time betw-een 7 and 8 o'clock are the hands of a clock together ? 1 hr. 5^ min. The hands are together at 12 o'clock, and there 7 are 7 intervals from 12 o'clock to the required time. 7 hr 38 2 min Since each interval is 1 hr. 5y\ min., 7 intervals equal 7 times 1 hr. 5y\ min., which equals 7 hr. Ans. 38 j^ min. 33^2_ min. Hence the required time is 38i\ min. past 7 o'clock, past 7 o'clock. II. At what time between 5 and 6 o'clock are the hands of a clock at right angles ? Ihr. 5,5. min. 1 hr. S^-^min. The hands are at right 2 Q angles at 3 o'clock and at 9 o'clock. There are 2 in- o ' ^ -^ ' Q ' ^ ^ tervals from 3 o'clock to the i:r-, TTTTT r— tt; tttz : — required time; hence we 5 hr. lOi^ mm. 5 hr. 43,^, mm. ^^^ ^ j^^ ^^,, ^.^ ^^ 3 Ans. lOfJ min. past 5 o'clock, o'clock to obtain one an- or 43iV min. past 5 o'clock. ^^^^- ^^''^ ^'^ ^ i"*^^^^^« from 9 o'clock to the re- quired time ; hence we add 8 hr. 43j'^y min. to 9 o'clock to obtain the other, answer, writing: 5 hr. instead of V hr.. because 17 ig j5 pjore than 12- SPECIAL PROBLEMS. 145 III. Find when first after 1 o'clock the hands of a clock make an angle of 60° with each other. ^ g . The hands of a clock make 1 hr. Oyy mm. ^^ ^^^^^ ^^ g^o ^i^^ each other at 2 o'clock and at 10 o'clock. 3 hr. 16y\ min. When first after 1 o'clock they 10 make an angle of 60° with each Z 7^~4 '- other, they have the same rela- 1 hr. 16yy mm. ^j^^ position as at 10 o'clock. A71S. 16y\ min. past 1 o'clock. There are 3 intervals from 10 o'clock to the required time; hence we add .3 hr. 16^^ min. to 10 o'clock to obtain the answer. EXAMPLES. 1. At what time between 4 and 5 o'clock are the hands of a clock together ? 2. At what time between 6 and 7 o'clock are the hands of a clock together ? 3. At what time between 8 and 9 o'clock are the hands of a clock together ? 4. At what time between 12 and 1 o'clock are the hands of a clock opposite each other ? 5. At what time between 3 and 4 o'clock are the hands of a clock opposite each other ? 6. At what time between 9 and 10 o'clock are the hands of a clock opposite each other ? 7. At what time between 2 and 3 o'clock are the hands of a clock at right angles ? 8. At what time between 4 and 5 o'clock are the hands of a clock at right angles ? ^ 9. At what time between 9 and 10 o'clock are the hands of a clock at risrht an^le^ : 146 ARITHMETIC. 10. At what time between 11 and 12 o'clock are the hands of a clock at right angles ? 11. Find when first after 11 o'clock the hands of a clock make an angle of 30° with each other. 12. Find when first after 2 o'clock the hands of a clock make an angle of 60° with each other. 13. Find when first after 6 o'clock the hands of a clock make an angle of 120° with each other. 14. Find when first after 4 o'clock the hands of a clock make an angle of 150° with each other. Comparison of Thermometers. 99. There are two important points to be determined in the graduation of a thermometer, — the freezing point and the boiling point of water. In the Fahrenheit scale the freezing point is marked 32°, and the boiling point 212° ; the intervening space is divided into 180 equal parts called degrees. In the Centigrade scale the freezing point is marked 0°, and the boiling point 100° ; the intervening space is divided into 100 degrees. In the E^aumur scale the freezing point is marked 0°, and the boiling point 80° ; the intervening space is divided into 80 degrees. In expressing temperatures it is customary to indicate the scale referred to by the initial letters F., C, and R. Temperatures below 0° are indicated by the minus sign. For example, -15° C. indicates 15° below 0° ; -10° F. indi- cates 10° below 0° or 42° below the freezing point. Since 180 Fahrenheit degrees = 100 Centigrade degrees = 80 Reaumur degrees, 9 Fahrenheit degrees = 5 Centi- grade degrees = 4 Reaumur degreee SPECIAL PROBLEMS. 147 I. Express 95° T. in Centigrade scale. 95 7 Subtracting 32 from 95, we find that 95° 32 2 of ^^ _ 350 Q p jg 53 degrees above the freezing point. 63 '^ Since 9 Fahrenheit degrees = 5 Centigrade degrees, there are | as many Centigrade degrees as Fahrenheit degrees; hence 95° F. is 35° above the freezing point in the Centigrade scale, or 36° C. II. Express 45° C. in the Fahrenheit scale. Since 9 Fahrenheit degrees = 5 Centigrade 9 -J«_Q-i degrees, there are | as many Fahrenlieit de- ^ * grees as Centigrade degrees ; hence 45° C. is 81 -4- 32 = 113° F ^^° above the freezing point in the Fahrenheit * scale, or 113° F. EXAMPLES. 1. Express 86° F. in the Centigrade scale. 2. Express 68° F. in the Centigrade scale. 3. Express 23° F. in the Centigrade scale. 4. Express —4° F. in the Centigrade scale. 5. Express 60° C. in the Fahrenheit scale. 6. Express 15° C. in the Fahrenheit scale. 7. Express —10° C. in the Fahrenheit scale. 8. Express —30° C. in the Fahrenheit scale. 9. Express 50° F. in the Reaumur scale. 10. Express 14° F. in the E^amnur scale. 11. Express 24° R. in the Fahrenheit scale. 12. Express —16° R. in the Fahrenheit scale. 13. Express 60° C. in the Reaumur scale. 14. Express —25° C. in the Reaumur scale. 15. Express 36° R. in the Centigrade scale, 16. Express — 16" R. in the Centigrade scale. 148 ARITHMETIC. Specific Gravity. 100. The specific gravity (sp. gr.) of any substance is its weight compared with the weight of an equal bulk of water. Since water is the standard, its specific gravity is 1. The specific gravity of any other substance denotes the number of times it is heavier than water. For example, if a bar of silver has a specific gravity of 10.5, it is 10.5 times as heavy as an equal bulk of water. In the metric system of weights and measures, the weight of any bulk of water can readily be found by remembering that 1 cubic meter of water weighs 1 metric ton, 1 liter weighs 1 kilogram, and 1 cubic centimeter weighs 1 gram. In the common system of weights and measures, 1 cubic foot of water weighs 1000 ounces Avoirdupois. When the bulk and specific gravity of a substance are known, the weight of the substance can be found by multi- plying the weight of an equal bulk of water by the specific gravity. When the bulk and weight of a substance are known, the specific gravity of the substance can be found by dividing the weight of the substance by the weight of an equal bulk of water. When the weight and specific gravity of a substance are known, the weight divided by the specific gravity equals the weight of an equal bulk of water, and from this the bulk of the substance can readily be found. All bodies weigh less in water than in air. It can be proved by experiment that the difference between the weight of a body in air and its weight in water equals the weight of the water displaced. Hence, if the weight of a body in air be divided by the difference between its weight in air and its weight in water, the Yesult is the specific gravity. SPECIAL PROBLEMS. 149 J. Find the weight of a bar of copper (sp. gr. 8.79) 2 ft. long and 3 in. square. ?xlxi = |cu.ft. 2 , 125 i of xm= 125 oz. 1125 875 1000 The dimensions expressed in feet are 2 ft., \ ft., and \ ft., and the product of these di- mensions gives ^ cu. ft. as the cubic contents The weight of an equal bulk of water is | of 1000 oz,, or 125 oz. Multiplying this result by 8.79, we find the weight of the copper to be 1098.75 oz., which equals 68.672 lb. 16 )1098.75 o z. 68.672 lb. II. If 650'="*='" of ether weigh 468«, what is its specific gravity ? 650)468.00(0.72 4550 1300 1300 650cucm of water weigh 650 k; hence the spe- cific gravity of ether is as much as 650 is con- tained in 468, which equals 0.72. III. Find the bulk of a piece of coal (sp. gr. 1.8) which weighs 56.88 H 1.8)56.88(31.6 54 28 18 108 108 Since the specific gravity of coal is 1.8, the weight of an equal bulk of water is as much as 1.8 is contained in 56.88^8, which equals 31.6 Kg, or 31600 K. The bulk of 31600 k of water is 31600 c« cm, which is also the bulk of the coal. ^TlS. 31600 «="«=". EXAMPLES. 1. Find the weight of a cubic foot of ice (sp. gr. 0.92). 2. Find the weight of a gallon of milk (sp. gr. 1.03). 3. Find the weight in grains of a cubic inch of iron (sp. gr. 7.21). 150 ARITHMETIC. 4. Find the weight of a bar of platinum 10 in. long, 4 in. wide, and 1^ in. thick, if its specific gravity is 22.07. 5. A tank is 6 ft. long, 4 ft. wide, and 3 ft. deep. How many pounds of sulphuric acid (sp. gr. 1.84) will it contain? 6. Find the specific gravity of a stone, a cubic foot of which weighs 185 lb. 7. Find the specific gravity of a liquid weighing 10 lb. per gallon. 8. A bar of gold 3 in. long, 1^ in. wide, and i in. thick weighs 25 oz. Avoirdupois ; find its specific gravity. 9. A piece of iron weighs 12 lb. in air and 10^ lb. in water ; find its specific gravity. 10. Find the number of cubic inches in a pound of alu- minium (sp. gr. 2.64). 11. Find the number of bushels in a ton of salt (sp. gr. 2.15). 12. A piece of glass weighs 4320 gr. in air and 3195 gr. in water ; what is its specific gravity ? its volume ? 13. Find the weight of 58^ of sand (sp. gr. 1.65). 14. Find the weight of 6.32 ^^ of olive oil (sp. gr. 0.915). 15. A plank is 5°» long, 3^ wide, and 3*=°^ thick; find its weight in grams, if the specific gravity of the wood is 0.8. 16. A tank is 1.5™ wide, 3.2™ long, and 80'"" deep. How many kilograms of alcohol (sp. gr. 0.8) will be required to fill it one third full ? 17. What is the weight in metric tons of a block of stone (sp. gr. 2.5) measuring 12.37" by 7.14™ by 83*=™? 18. Find the specific gravity of an acid weighing 1.58^* per liter. SPECIAL PROBLEMS. 151 19. If 2V of alcohol weigh 22.14^, what is its specific gravity ? 20. A brick 20"=™ long, 11 "^^ wide, and 5.5 <'°' thick weighs 2.904 ^« ; find its specific gravity. 21. A plate of iron 137 «=" long, 643"" wide, and 43"" thick weighs 277.54^^. What is its specific gravity? 22. A stone weighs 8.42 ^^^ in air and 6.32 ^^ in water ; find its specific gravity. 23. A body weighs 460 « in air and 401.16 «^ in water; what is its specific gravity ? 24. Find the number of cubic centimeters in a piece of brass (sp. gr. 8.38) weighing 86.7338. 25. If alcohol (sp. gr. 0.81) costs $1.45 a kilogram, what is the price of a liter ? 26. If salt (sp. gr. 2.15) costs $7.50 a metric ton, what is the price of a hektoliter ? 27. If cork (sp. gr. 0.24) is worth 2\ cts. a cubic deci- meter, find the value of 10^«. 28. If marble (sp. gr. 2.83) is worth $28.50 a cubic meter, find the value of a block weighing 764 ^«. Longitude and Time. 101. The longitude of a place is the arc or portion of the equator between a standard meridian and the meridian of the given place. A place is in east or west longitude, ac- v3ording as it is east or west of the standard meridian, and the longitude is reckoned in degrees, minutes, and seconds up to 180°, or half way round the earth. For example, long. 32° 25' W. indicates a place situated on the meridian which is 32° 25' west of the standard meridian. The meridian of 152 ARITHMETIC. Greenwich, England, is usually taken as the standard by- English-speaking people. When two places are on the same side of the standard meridian, the difference of longitude is found by subtract- ing their longitudes ; when two places are on opposite sides of the standard meridian, the difference of longitude is found by adding their longitudes. If, however, in the lat- ter case, the sum exceeds 180°, it must be subtracted from 360° to obtain the correct difference of longitude. The earth revolves on its axis once in 24 hours, thus making 360° of longitude pass under the sun in that time. In 1 hr. J^ of 360°, or 15°, pass under the sun ; in 1 min. Jg- of 15°, or 15'; in 1 sec. -^ of 15', or 15". Hence a dif- ference of 15° of longitude causes a difference of 1 hr. of time; a difference of 15' of longitude causes a difference of 1 min. of time ; a difference of 16" of longitude causes a difference of 1 sec. of time. I. The difference of time between two places is 2 hr. 15 min. 27 sec. ; what is the difference of longitude ? 2 hr. 15 min. 27 sec. Since 1 hr. of time corresponds to 15° 15 of longitude, 1 min. of time to 15' of 330 gY' 45'' longitude, and 1 sec. of time to 15" of longitude, 15 times the number of hours, minutes, and seconds equals the number of degrees, minutes, and seconds. II. The difference of longitude between two places is 48° 24' 36" ; what is the difference of time ? 15)48° 24' 36" Since 15° of longitude corresponds 3hr.l3min.38|sec. *« ^ ^^- «^ ^^^^' ^^' «^ longitude to 1 min. of time, and 15" ot xongitude to 1 sec. of time, -^-g of the number of degrees, minutes, and seconds equals the number of hours, minutes, and seconds. SPECIAL PROBLEMS. 168 EXAMPLES. 1. The difference of time between two places is 6 hr. 42 min. 22 sec. ; what is the difference of longitude ? 2. The difference of time between St. Petersburg and St. Paul is 8 hr. 13 min. 36 sec. ; what is the difference of longitude ? 3. The difference of time between Boston and St. Louis is 1 hr. 16 min. 47 sec. ; what is the difference of longitude ? 4. The time in Montreal is 4 hr. 53 min. h^\ sec. earlier than in London ; what is the difference of longitude between the two places ? 5. The time in Berlin is 44 min. 14^ sec. later than in Paris ; what is the difference of longitude between the two places ? 6. The difference of longitude between two places is 71° 4' ; what is the difference of time ? 7. Find the difference of time between New York (long. 74° 0' 3" W.) and San Francisco (long. 122° 25' 40" W.). 8. Find the difference of time between Ottawa (long. 75° 42' 4" W.) and Washington (long. 77° 2' 48" W.). 9. Find the difference of time between Bombay (long. 72° 54' E.) and Cape of Good Hope (long. 18° 29' E.). 10. Find the difference of time between Constantinople (long. 28° 59' 14" E.) and Quebec (long. 71° 13' 45" W.). 11. Find the difference of time between Canton (long. 113° 14' E.) and Chicago (long. 87° 37' 30" W.). 12. Find the difference of time between Boston (long. 71° 3' 30" W.) and St. Paul (long. 93° 5' W.). 13. Find the difference of time between Pekin (long. 116° 27' E.) and New York (long. 74° 0' 3" W.) 154 ARITHMETIC. 14. Find the difference of time between Rome (long. 12° 28' 40" E.) and Paris (long. 2° 20' 14" E.). 102. The earth revolves on its axis from west to east, and the sun seems to move from east to west. Of any two places, the sun rises earlier at the place farther east, and since the sun rises earlier, the clock-time is later. Hence, to find the clock-time of a given place when the clock-time of another place and their difference of time are known, add the difference of time to the given time, when the place whose time is to be found is farther east; subtract the differ- ence of time from the given time, when the place whose time is to be found is farther west. I. When it is 12 min. 30 sec. past 2 p.m. at Berlin (long. 13° 23' 45" E.), what is the time at New York (long. 74° y) o W . ) . "Phg difference of time is found 13° 23' 45" to be 5 hr. 49 min. 35i sec. Since 74° 0' 3" New York is west of Berlin, the 15)87° 23' 48" ti™^ is earlier; hence we sub- 5 hr. 49 min. 354- sec. *^^^* ^ ^^- ^^ ™^"' ^^^ '^^- ^^°"^ ^ 2 hr. 12 min. 30 sec. p.m. 2 hr. 2 hr. 12 min. 30 sec. p.m. ^^^^"^ "°°" '^ ^^'^ ^^"^^ as 14 hr. 5 49 35^ after midnight, and as we can- o 1, oo • KAA not subtract 5 hr. from 2 hr., we 8 hr. 22 mm. 54i sec. a.m. , . ^ , , , ' . ^ subtract it from 14 hr., and write Ans. 22 min. 54|- sec. past 8 a.m. a.m. instead of p.m. II. What is the longitude of a place whose time is 48 min. past 8 p.m., when it is half past 6 p.m. at Eome (long. 12° 28' 40" E.) ? 8 hr. 48 min. The difference of longitude is found to be 34° 30^ Since the time is later, the place is east of Rome ; hence we add 34° 30' to 12° 28' 40". 6 30 2hr, . 18 min. 15 34° 12° 30' 28' 40" 46° 58'40"E. SPECIAL PROBLEMS. 165 EXAMPLES. 1. Bangor is 15° 39' east of Cincinnati ; what time is it at Bangor when it is 5 o'clock p.m. at Cincinnati ? 2. The longitude of Berlin is 13° 23' 45" E. ; what time is it at Greenwich when it is midnight at Berlin ? 3. What is the time at Canton (long. 113° 14' E.) when it is noon at Greenwich ? 4. The longitude of Boston is 71° 3' 30" W., and of Paris 2° 20' 14" E. ; when it is 10 o'clock a.m. at Boston, what time is it at Paris ? 5. The longitude of St. Petersburg is 30° 19' E., and of New York 74° 0' 3" W. ; when it is 1 p'elock p.m. at St. Petersburg, what time is it at New York ? 6. When it is 10 o'clock at Boston, what time is it at Amherst, the longitude of Boston being 71° 3' 30" W., and that of Amherst being 72° 31' 50" W. ? 7. The longitude of Boston is 71° 3' 30" W., and that of San Erancisco is 122° 25' 40" W. When it is noon at Bos- ton, what is the time at San Francisco ? 8. Eind what time it is at Cape of Good Hope (long. 18° 29' E.) when it is noon at St. Paul (long. 93° 5' W.). 9. When it is 6 min. 15 sec. past 4 a.m. at Pekin (long. 116° 27' E.), what is the time at London (long. 5' 48" W.). 10. What is the longitude of a place whose time is 42 min. 42 sec. past 8 p.m. when it is midnight at Greenwich ? 11. What is the longitude of a place who§e time is 6 o'clock A.M. when it is quarter past 4 a.m. at Washington (long. 77° 2' 48" W.) ? 12. What is the longitude of a place whose time is 35 min. past 10 a.m. when it is. 5 o'clock p.m. at Paris (long. 156 ARITHMETIC. 13. When it is noon at St. Paul (long. 93° 5' W.), it is 37 min. 12 sec. past 1 p.m. at Bangor ; what is the longitude of Bangor ? 14. When it is 9 o'clock p.m. at Calcutta (long. 88° 20' E.), it is 27 min. 19|- sec. past 5 p.m. at Jerusalem ; what is the longitude of Jerusalem ? Note. The time considered in the preceding problems is the actual local time, but nearly all railroads, cities, and towns of the United States now use standard time, which is the time of some particular meridian. The meridians selected are those which are respectively 75°, 90°, 105°, and 120° west of Greenwich. The time of the meridian 75° W. is known as Eastern standard time ; that of 90° W. is Central standard time; that of 105° W. is Mountain standard time; and that of 120^ W. is Pacific standard time. By this method, when there is any difference of time between two places, the difference is one, two, or three hours, and all confusion arising from different local times is thereby avoided. RATIO AND PROPORTION. 157 CHAPTER VIII. RATIO AND PROPORTION. Ratio. 103. The relation between two numbers is called their ratio, and it is determined by dividing the first by the second. The sign of ratio is the colon (:), which is the sign of division with the line omitted. For example, 6 : 4 is read the ratio of 6 to .4, or 6 is to 4, and its value is 6 -h 4, or f. The two numbers whose values are compared are called the terms of the ratio, and together they form a couplet. The first term is called the antecedent, and the second term is called the consequent. A ratio can exist between two concrete numbers only when they are expressed in terms of the same unit, and the ratio is equal to the ratio of the corresponding abstract numbers. For example, 8 pt. : 15 pt. equals 8 : 15. When each term of a ratio is a single number, it is called a simple ratio. The product of two or more simple ratios is called a compound ratio. A simple ratio having a fraction in either term is also called a complex ratio. For example, 8 : 13 and 2^ : f are simple ratios, the latter of which is complex ; q ! i o r is a compound ratio. Since antecedent and consequent bear the same relation to each other as dividend and divisor, both terms of a ratio may be multiplied or divided by the same number without affecting the value of the ratio. A complex ratio can be 158 ARITHMETIC. simplified by multiplying both terms by their least common denominator. A compound ratio can be simplified by taking the product of the antecedents for a new antecedent, and the product of the consequents for a new consequent. When the antecedent and consequent of a ratio are inter- changed, the resulting ratio is called the inverse of the given ratio. I. Eeduce 2J : 5J to a simple ratio. ^i * ^¥- Multiplying both terms by 12, we obtain 28 : 63 ; and 28 : 63. then dividing both terms by 7, we obtain 4 : 9 as the 4 • 9 simplest value. II. Which is the greater ratio, 7:8 or 8:9? Expressing the ratios in a fractional form, ^^' we have | and *, which, after reducing to their 9 = 1^ = YY' least common denominator, equal f f and f f . 8 = l = ff- Q _ 8 — 64 9 is the larger. than 7 EXAMPLES. 1. Eeduce 3f : 5^ to a simple ratio. 12 2. Eeduce 74 : — to a simple ratio. 3 • 7 ) 3. Eeduce ^^ ' q >■ to a simple ratio. 4. Eeduce of'.al [- to a simple ratio. 5. Find the ratio of 2 pk. to 3 qt. 6. rind the ratio of 4 gal. to 1 cu. ft. 7. Find the ratio of a field 15 rd. long and 11 rd. wide to a field 14 rd. long and 12 rd. wide. 8. Which is the greater ratio, 7 : 11 or 8 : 12 ? RATIO AND PROPORTION. 159 9. "WnicK is the greater ratio, ^ : f or 4| : 4|- ? 10. Which is the greater ratio, $2.50 : |3.75 or 8 ft. : 12 ft. ? Simple Proportion. 104. All expression of equality between the two ratios is called a proportion, and the four terms are called propor- tionals. When a proportion consists of two simple ratios, it is called a simple proportion. A proportion is indicated by putting a double colon (: :) or a sign of equality ( = ) between the two ratios. For example, 4 : 6 : : 10 : 15 is read 4 is to 6 as 10 is to 15; 4 : 6 == 10 : 15 is read the ratio of 4 to 6 equals the ratio of 10 to 15. The first and fourth terms of a proportion are called the extremes, and the second and third terms are called the means. Three numbers are said to be in proportion when the ratio of the first to the second equals the ratio of the second to the third. The second number is called a mean proportional between the other two. For example, in the proportion 2 : 6 : : 6 : 18, 6 is a mean proportional between 2 and 18. The solution of problems in proportion depends on the following principle : — In any projyortion the product of the extremes equals the product of the means. This can be proved in any proportion by expressing the ratios in fractional form, and then multiplying both of them by the product of the denominators. As an illustration, take the proportion 2 : 3 : : 4 : 6. 2 : 3 : : 4 : 6. 2 _ 4_ The proportion written in fractional form becomes | = |. Multiplying both fractions 2x3x6 ^ 4x3x^ , by 3x6, the results are still equal. Hence ^ ^ 2X6 = 4x3, 2X6 = 4X3. 160 . ARITHMETIC. From the above principle it follows that either extreme equals the product of the means divided by the other extreme i and either mean equals the product of the extremes divided by the other mean. I. Find a fourth proportional to 8, 10, and 12. 8 : 10 : : 12 : ic. IJet the required term be represented by X. Then 8 : 10 : : 12 : a;. Since either extreme equals the product of the means divided by 10 X 12 the other extreme, x— — — — = 15. 8 II. Find the second term of a proportion of which the first, third, and fourth terms are respectively 2|, 3f , and 7^. 2f : aj : : 3f : 71. Let the required term be represented X — ^jXi^-^^J' hy X. Then 2| : x : : 3f : 71. Since 3 either mean equals the product of the _ X^ y^lE y^ J_ — ^ — ^1 extremes divided by the other mean, ^ 2 ^1 4 :r = 2fx7i-^3f=5i. EXAMPLES. 1. Find a fourth proportional to 9, 51, and 75. 2. Find the third term of a proportion of which the first, second, and fourth terms are respectively 18, 15, and 100. 3. Find the number which has to 6f the same ratio which llf has to 3f 4. Find the number to which 8^ has the same ratio which 25 has to 37^. ..'5. Find the third term of a proportion of which the first, second, and fourth terms are respectively -^, J, and ^ 6, Find a fourth proportional to 3.75, 0.23, and 0,16. BATTO AND PROPORTION. 161 7. Find the number which has to 0.649 the same ratio which 58 has to 634. 8. Find the fourth term of a proportion of which the first, second, and third terms are respectively 3.81, 0.056, and 1.67. 105. The method of finding either term of a proportion when the other three are known is often called the rule of three. It is customary to represent the required term by x, and then arrange the terms so that x will be the fourth term. The number in the problem which corresponds to the answer must be the third term. I. If 15 yards of silk cost |36, what will 25 yards cost? Since the answer is to be dollars, make S3G the 15 : 25 : : 36 : cc third terra. 25 yards will cost more than 16 6 12 23 x M yards, and the fourth term will be greater than ^ ~- — Y^ = ^^^- the third ; lience the second term must be greater ^ than the first, and the first couplet is 16 : 25. The answer is then found as in the preceding section. II. If 8 men can do a piece of work in 5 days, how long will it take 10 men to do the same work ? 10 : 8 : : 5 : a; Since the answer is to be days, make 6 days the 4 third terra. 10 men can do the work in less time a;z=?il^=4(lays. *^^" ^ "^^"' ^"^ *'^^ fourth term will be smaller ^^ than the third ; hence the second term must be ^ smaller than the first, and the first couplet is 10 : 8. The answer is then found as in the preceding section. In the solution of problems in simple proportion, make that mimber the third term ivhich is of the same kind as the required answer. If from the nature of the question the answer is to be greater than the third term, make the greater of the other two numbers the second term, and the smaller the first; if the answer is to be smaller than the third term, make 162 ARTTHMEtlC. the second term smaller than the first. Divide the product of the means by the first term, and the quotient is the fourth term, or answer. EXAMPLES. 1. If 18 barrels of flour last a garrison 8 weeks, how long will 63 barrels last ? 2. If the rent of 36"* of land is $48, how many hektars* can be rented for $84 ? 3. If a stock of provisions will supply a garrison of 240 men 96 days, how long will the same stock supply 384 men ? 4. If 36 men can do a piece of work in 22 days, how many men can do the same work in 8 days ? 5. If a clock ticks 120 times in a minute, how many times will it tick in 2^ hours ? 6. If 3 lb. 7 oz. of butter cost $1.10, what will 14f lb. cost? 7. If a train runs 160^"' in 3 hours, how long will it take it to run 70«^'" ? 8. If a man earns $16 in 5 days, how much will he earn in 14 days ? 9. If 22 yd. of silk 18 in. wide are required for a dress, how many yards of cloth 30 in. wide would be required for a similar dress ? 10. A field can be mowed in 4 days of 11 hours each ; how many days of 9 hours each will it take ? 11. If 12 men can build a wall 19 rods long in a day, how long a wall will 32 men build in the same time ? 12. If 14 yd. of cloth 32 in. wide will make a dress, how many yards of cambric 24 in. wide will be required to line it? RATIO AND PROPORTION. 163 13. If 10.5"' of wood cost ^12.25, how many sters can be bought for f 50 ? 14. If f of a warehouse is worth $7000, what is ^ of it worth ? 15. If a post 5 ft. 4 in. high casts a shadow 6 ft. 4 in. long, how long a shadow wiH be cast by a steeple 176 ft. high? 16. At the time when a man 5 ft. 9 in. in height casts a shadow 4 ft. 6 in. long, what is the height of a tree that casts a shadow 52 ft. 6 in. long ? 17. If a cistern can be filled in 2 hr. 27 min. by 3 pipes, in what time can it be filled by 7 pipes of the same size ? 18. If 9J yards of cloth cost $23|, how many yards can be bought for |38-/^ ? 19. What is the cost of 60.5 tons of coal when 0.9 of a ton costs ^6.66 ? 20. A merchant failed and paid 60 cents on a dollar; how much would a creditor receive whose bill Avas $1426 ? 21. If 108^1 of oats last 100 horses 9 days, how long will 192H' last them ? 22. If a man travels 64 rods in 0.05 of an hour, how many minutes will it take him to go a mile ? 23. If 18 men can perform a piece of work in 42 days, in how many days can they perform the same work with the assistance of 9 more men ? 24. A piece of work can be done in 50 days by 35 men. After 12 days 16 men strike. In how many days will the rest finish the work ? 25. If 6iT. of coal cost £6 15 s. 5d., what will be the price of 5 T. 3 cwt. ? 164 ARITHMETIC. 26. By a pipe of a certain capacity a cistern can be emptied in 3^^ hours ; in what time can it be emptied by a pipe, the capacity of which is | greater ? Compound Proportion. 106. An expression of equality between a compound ratio and a simple ratio, or between two compound ratios, is called a compound proportion. I. If 4 men dig a trench 84 feet long and 5 feet wide in 3 days of 8 hours each, how many men can dig a trench 420 feet long and 3 feet wide in 4 days of 9 hours each ? 84 : 420 5:3 4:3 9:8 Since the answer is to be men, make 4 men the third term. The : : 4 : 07. number of men required depends upon four conditions — the length of the trench, the width of the ^^ n o Q * trench, the number of days, and the ^^ ^^^X^X>^X8X^ ^g men. number of hours per day; all of '^ ^ these must be considered m statmg the problem. A trench 420 ft. long will require more men than a trench 84 ft. long, and the first ratio is 84 : 420 ; a trench 3 ft. wide will require less men than a trench 5 ft, wide, and the second ratio is 5:3; to complete the work in 4 da. will require less men than to complete it in 3 da., and the third ratio is 4:3; days of 9 hr. each will require less men than days of 8 hr. each, and the fourth ratio is 9:8. Dividing the product of the means by the product of the given extremes, we have 8 men as the answer. In the solution of problems in compound proportion, make that number the third term which is of the same kind as the required answer. Take the other terms in pairs of the same kind, and form a ratio of each pair as in simple propor- tion. Divide the product of the means by the product of the given extremes, arid the quotient is the fourth term, or answer. RATIO AUD PROPORTION. 165 EXAMPLES. 1. If 6 men in 15 days earn $135, how much will 9 men earn in 18 days ? 2. If 6 men can dig 6 rods of a ditch in 6 hours, how; many rods will 12 men dig in 12 hours ? 3. If 16 men build 18 rods of wall in 12 days, how many men will be needed to build 72 rods in 8 days ? 4. If the wages of 12 men for 8 days of 8 hours each are f 135, what will be the wages of 25 men for 12 days of 10 hours each ? 5. If a man travels 117 miles in 15 days, travelling 9 hours a day, how far would he go in 20 days, travelling 12 hours a day ? 6. If 5 men can do a piece of work in 7 days of 10 hours each, in how many days can 12 men do the same, working 8 hours per day ? 7. If 2|- acres of pasturage can support 5 oxen for 3^ days, how many acres would be required to support 26 oxen for 17-^ days ? 8. If 14 horses eat 70 bushels of grain in 20 days, how many bushels will suffice 30 horses 50 days ? 9. If 8 horses consume 3| tons of hay in 30 days, how long will 4^ tons last 10 horses ? 10. If 9 men build 247y23 rods of wall in 28 days, in how many days will 8 men build 51 rods ? 11. If 2 men, working 8 hours, can carry 12000 bricks to the height of 50 feet, how many bricks can 1 man, working 10 hours, carry to the height of 30 feet ? 166 ARITHMETIC. 12. If a six cent loaf weighs 8 ounces when wheat is $1.25 per bushel, how much bread may be bought for 50 cents when wheat is f 1.00 per bushel ? 13. If 49 men can empty a reservoir in 65 days, pumping- 8 hours a day, how many hours a day must 196 men pump to empty it in 26 days ? 14. If 5 horses will consume 8 bu. 1 pk. 6 qt. of oats in 6 days, what quantity of oats will 7 horses consume in 11 days ? 15. If it take 35^^ of wool to make a piece of cloth 25™ long and f "" wide, how long a piece of cloth, i™ wide, can be made from 112^« ? 16. If a family of 9 persons spends $305 in 4 months, how many dollars will maintain it 8 months, if 5 persons were added to the family ? 17. If a man travels 1440 miles in 36 days, travelling 10 hours a day at the rate of 4 miles an hour, in what time will he travel 576 miles, going 8 hours a day at the rate of 3 miles per hour ? 18. If 3 men can build a wall 60 feet long, 8 feet high, and 3 feet thick, in 64 days of 9 hours, how many days of 8 hours will 20 men require' to build a wall 400 feet long, 9 feet high, and 5 feet thick ? 19. If a slab of marble, 8 feet long, 3 feet wide, and 3 inches thick, weighs 1050 pounds, how much will another slab of the same marble weigh which is 6 feet long, 2 feet wide, and 2 inches thick ? 20. If 6 iron bars, 4 feet long, 3 inches broad, and 2 inches thick, weigh 288 pounds, find the weight of 15 bars, each 6^ feet long, 2^ inches broad, and 1^ inches thick. KATIO AND PROPORTION. 167 21. If 25 men, working 8 hours a day, do f of a piece of work in 24 days, in liow many days of 10 hours each will 30 men finish the piece of work ? 22. If 12 pipes, each delivering 12 gallons a minute, fill a cistern in 3 hr. 24 niin., how many pipes, each delivering 16 gallons a minute, will fill a cistern 6 times as large in 6 hr. 48 min. ? 23. A man has a bin 7 ft. long, 2^ ft. wide, and 2 ft. deep, which contains 28 bushels of corn ; how deep must he build another, which is to be 18 ft. long, 1 ft. 10^ in. wide, in order to contain 120 bushels ? 24. If 496 men, in 5 days of 12 hr. 6 min. each, dig a trench of 9 degrees of hardness, 465 ft. long, 3J ft. wide, and 4| ft. deep, how many men will be required to dig a trench of 2 degrees of hardness, 168| ft. long, 7J ft. wide, and 21 ft. deep, in 22 days of 9 hr. each ? Cause and Effect. 107. Problems in proportion can also be solved by the application of the following principle : Like causes produce like effects, and the ratio hetioeen any two causes equals the ratio between the effects produced. Note. As examples of causes may be mentioned men at work, time, and goods bought or sold ; as examples of effects, work done, wages, and cost of goods. I. If 15 yards of silk cost f 36, what will 25 jards cost? 15 : 25 : : 36 : iC Let the required number of dollars be rep- 6 12 resented by x. The first and se lond causes /gsa^iilif =$60. ^re respectively 15 yd. and 25 yd The first '^ and second effects are respectively $36 and x dollars. Hence the proportion is 15: 26:: 36: x 168 ARITHMETIC. II. If 8 men can do a piece of work in 5 days, how long will it take 10 men to do the same work ? 8 : 10 ) ^ ^ Let the required number of days be repre- 5 : ic j ■ * ■ sented by x. The first causes are 8 men and 4 5 days, and the second causes are 10 men and g._. ^X^Xl _.^ days. ^ ^^ys- The effects are the same, and can -^^Xl each be represented by 1. Hence the i)ro- portion is r\ V : : 1 : 1. Since x is a mean, its value is found by dividing the product of the extremes by the product of the given means. III. If 4 men dig a trench 84 feet long and 5 feet wide in 3 days of 8 hours each, how many men can dig a trench 420 feet long and 3 feet wide in 4 days of 9 hours each ? 4:ic) (qj ir)A "^^^ *''^ required number of men 3 : 4 >- : : -< k o ^^ represented by x. The first causes 8:9) I ^ • '^ are 4 men, 3 da., and 8 hr., and the ^^ second causes are x men, 4 da., and ^^ ^X3x8x^^Px3 _g ^^^ 9hr. The first effect is a trench ^XgX^^X^ '84 ft. long and 5 ft. wide, and the second effect is a trench 420 ft, long and 3 ft. wide. Hence the proportion is 3:41::-^ ^io • Note. The illustrative problems are the same three that were ex- plained by the rule of three in the two preceding sections. All prob- lems given under either head can be solved by either method. EXAMPLES. 1. How many hektars of land can be bought for $84, when 3^* can be bought for $26.25? 2. If a horse-car goes 4 miles in 35 minutes, how far will it go in 3 hours ? 3. If a tree 24 feet high casts a shadow 30 feet long, what must be the height of a building to cast a shadow 55 feet long ? BATIO AND PROPORTION. 169 4. If 16™ of silk cost 120 francs, what will 25" cost ? 5. If 14 men can build a wall in 10 days, how many men will it take to build the same wall in 7 days ? 6. If 67.5" of carpeting 80*="* wide will cover a floor, how many meters 90*"" wide will it take to cover it ? 7. If a pasture of 18 acres will feed 8 cows 5 months, how many months will a pasture of 27 acres feed 12 cows ? 8. A man receives f 18 for 6 days' work of 8 hours each ; what should he receive for 5 days' work of 9 hours each? 9. If 8 men spend ^32 in 13 weeks, what will 24 men spend in 52 weeks ? 10. If the wages of 72 men for 5 days is f 450, how many men may be hired for 12 days for $540 ? 11. A man, travelling 9 hours a day, goes 234 miles in 15 days ; how far can he go in 30 days, travelling 8 hours a day? 12. If a man travelling uniformly, 7 hours per day, goes 455 miles in 26 days, how far can he go in 20 days, travel- ling 9 hours per day at the same rate per hour as before ? 13. If 6 men can build 20 feet of a stone wall in 10 days, how many men can build 360 feet of the same wall in 90 days ? 14. If 17 men can reap a field in 9 days, how long would it take to reap half of it when 5 men refuse to work ? 15. If 3 men can reap 8 acres in 5 days, working 8 hours a day, in how many days can 8 men, working 12 hours a day, reap 192 acres ? 16. If it costs f 7.20 to transport 18^ cwt. 5-^ miles, what will it cost to transport 112f tons 62^ miles ? 170 ARITHMETIC. 17. If 27 men, working 10 hours a day, do a piece of work in 14 days, how many hours a day must 12 men work to do the same amount of work in 45 days ? 18. If 24 men can saw 90 cords of wood in 6 days, when the days are 9 hours long, how many cords can 8 men saw in 36 days, when they are 12 hours long ? 19. If a block of granite 8 ft. long, 2 ft. wide, and 1 ft., 6 in. thick, weighs 920 lb., how much will a block of the same kind of granite weigh which is 12 ft. long, 3 ft. wide, and 2 ft. thick ? 20. If 6 men do a certain piece of work in 17 days of 9 hours each, how many days of 8|- hours each will 24 men, working at the same rate, require to do 20 such pieces ? 21. A wall which was to be 36 ft. high was raised 9 ft. in 16 days by 16 men ; how many men will be needed to finish the work in 4 days ? 22. If 8 ounces of bread can be bought for 10 cents when wheat is $1.00 per bushel, what weight of it may be bought for 18 cents when the price of wheat is $1.12 per bushel ? 23. If 30 lb. of cotton will make 3 pieces of muslin 42 yd. long and f yd. wide, how many pounds will it take to make 50 pieces, each containing 35 yd., 1-J yd. wide ? 24. If 6 men can build a wall 80 ft. long, 10 ft. high, and 9 ft. thick in 100 days of 9 hours, how many days of 10 hours will be required by 15 men to build a wall 200 ft. long, 9 ft. high, and 5 ft. thick ? 25. If 5 compositoi-s in 16 days, 11 hours long, can com- pose 25 sheets of 24 pages in each sheet, 44 lines in a page, and 40 letters in a line, in how many days, 10 hours long, can 9 compositors compose a volume (to be printed in the same kind of type), consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters to a line ? RATIO AND PROPORTION. 171 Partitive Proportion. 108. The process of dividing a number into parts which are proportional to given numbers is called partitive pro- portion, and the parts are called proportional parts. I. Divide 168 into four parts which shall be to each other as 3, 5, 7, and 9. 3 + 5 + 7 + 9 = 24 — of ^^^ = 21; The number 168 may be conceived as ^^ divided into a number of equal parts, 3 of r 7 wliicli make up the first part, 5 the second, — of X^^ = 35 ; 7 the third, and 9 the fourth ; thus the num- ber of equal parts is 24. Hence the first — of 108 — 49' P^^* equals -^^ of 168, or 21; the second ^^ ' part equals ,\ of 168, or 35 ; the third part 7 equals ^j of 168, or 49; and the fourth ^ of X^^ = 63. part equals ^% of 168, or 63. ^ns. 21, 35, 49, and 63. II. Divide $580 into three parts which shall be to each other as ^, f, and 1\. 1=3-6^ 6+8 + 15 = 29 2Q Reducing the fractions to their l=T% A of ^^p = 120 ; L.C.D., we have j%, j%, and |f. ^ J J 5 r9 Since the fractions now liave the *~12 20 same denominator, the parts which — of ,*^P=1dO; are proportional to the fractions are proportional to the numerators. Hence — of S80=r3OO ^^ divide $580 into three parts pro. ^9 ' ' portional to 6, 8, and 15. Ans. $120, $160, and $300. 172 ARITHMETIC. EXAMPLES. 1. Divide 324 into two parts which shall be to each other as 19 to 8. 2. Divide 90 into five parts which shall be to each other as 1, 2, S, 4, and 5. 3. Divide 968 into three parts which shall be to each other as 2|-, 3J, and 4J. 4. Divide 420 into three parts, such that they shall be proportional to ^, |, and J. 5. Divide the reciprocal of 8 into two parts which shall be to each other as the reciprocals of 4 and 2|. 6. Coffee is mixed in the ratio of 2 lb. of Java to 1 lb. of Mocha; how much of each kind is there in a mixture weighing 75 lb. ? 7. The cost of a horse and harness was $384, and the horse cost seven times as much as the harness ; find the cost of each. 8. An alloy contains 325 parts of copper to 175 parts of zinc ; how much of each metal is contained in 43^^ 850^ of this alloy? 9. If bell metal is made of 25 parts of copper to 11 parts of tin, find the weight of each metal in a bell weigh- ing 1044 pounds. 10. A father divided f 1550 among three sons in parts proportional to their ages, which were respectively 17, 20, and 25 years ; how much did each receive ? 11. A man said, " I will spend half my income, save a third of it, and devote a fourth to business." His income was $780. Point out his blunder, and divide his income rightly in the proportion intended by him. RATIO AND PROPORTION. 178 12. Gunpowder is composed of nitre, charcoal, and sul- phur, in the proportion of 15, 3, and 2. A certain quantity of gunpowder is known to contain 20 cwt. of charcoal ; find its weight, and also the weight of nitre and sidphur it contains. Simple Partnership. 109. An association of two or more persons for the transaction of business is called a partnership. Such a partnership association is called a firm, company, or house, and the persons associated together are calletl partners. The money and property invested in the business is called capital or stock. The resources or assets of a firm are its property of all kinds together with the amounts due it ; the liabilities of a firm are its debts. When the capital of several partners is invested for the same time, the partnership is called simple partnership. The profits and losses are shared in proportion to the amount of capital each partner has invested in the business, except when some other special agreement has been made. I. A, B, and C formed a partnership ; A put in ^700, B ^800, and C f 1000 ; what was each partner's share of a profit f^ounting to |950 ? ^-^of m = ^QQ; 700 %m The method of partitive propor- 800 ^^^.im=^^^'. tion is used, dividing the profit into 1000 parts proportional to 700, 800, and 2500 1000. ^ of ^^0^380. %m A, %2m', B, $304; C, $380. 174 ARITHMETIC. EXAMPLES. 1. A and B form a partnership, A putting in $3000 and B $2500 ; what is each partner's share of a profit amounting to $2200? 2. A, B, and C invested in trade as follows : A $800, B $600, and C $900. What was each partner's share of a profit amounting to $1350? 3. A and B formed a partnership, and A's capital was equal to J of B's ; what was each partner's share of a profit amounting to $3600 ? 4. A, B, C, and D traded in company. A put in $7500, B $7000, C $9500, and D $8000; what was each partner's share of a profit amounting to $9280 ? 5. A, B, and C formed a partnership, A putting in $1500, B $1800, and C $1400. On closing business they found they had lost $800. What was the loss of each? 6. A, B, and C hired a pasture for $100; A put in 12 cows, B 8 cows, and C 5 cows ; how much should each pay ? 7. A bankrupt owed $550 to A, $675 to B, and $875 to C. His entire property was sold for $1043.28 ; what was each creditor's share? 8. A, B, and C engaged in trade with a joint capital of $9000. At the end of a year A's gain was $1250, B's $1000^ and C's $1500. What was each partner's share of the capital ? Compound Partnership. 110. When the capital of the several partners is invested for unequal times, the partnership is called compound part- nership. The division of the profits a»d losses depends both RATIO AND PROPORTION. 175 on the amount of each partner^s capital and the time for which it is invested. I. A, B, and C invested in trade as follows : A $300 for 10 months, B $400 for 8 months, and C $600 for 6 months. What was each partner's share of a profit amounting to 11960? 20 The use of §300 for OAA . iA OAAA ^^ of ^^^P = 600 ; lOiTio. is equivalent to 300x10 = 3000 Qmo ^^ ,, ^ ,^,. ^^f,,, 400 X 8 = 3200 Zll 20 *^^7,l'"^^^\""''^T' 600 X 6 = 3600 ^ ^^ ^^^^ = ^^^ ' """ ^^ ' ""' '"''• ' ^^P0 use of .$400 for 8 mo. is 9800 3600 . j^„„rt _ Y20 equivalent to the use of 8 times $400, or $3200, A, f 600 ; B, $640 ; C, f 720. ^'„„\:V„:"; Z^. alent to the use of 6 times $600, or $3600, for 1 mo. The amounts invested are thus reduced to the same standard, and the profit is divided into parts proportional to 3000, 3200, and 3600. EXAMPLES. 1. A and B enter into partnership. A contributes $1200 for 13 months, and B $1600 for 10 months. What is the share of each in a gain of $1300? 2. Three partners. A, B, and C, furnish capital as fol- lows : A $500 for 2 months, B $400 for 3 months, and C $200 for 4 months. They gain $600 ; what is each part- ner's share ? 3. Two men hire a pasture for $50; one puts in 20 horses for 12 weeks, and the other 25 horses for 10 weeks. How much should each pay ? 4. A, B, and C hire a pasture for $92. A pastures 6 horses for 8 weeks, B 12 oxen for 10 weeks, and C 50 cows for 12 weeks. If 5 cows are reckoned as 3 oxen, and 3 oxen as 2 horseS; how much shall each man pay ? 176 ARITHMETIC. 5. Three men harvested and thrashed a field of grain on shares, A furnishing 4 hands 5 days, B 6 hands 4 days, and C 5 hands 8 days. The whole crop was 630 bushels, of which they had one fifth ; how much did each receive ? 6. Three men contract to do a piece of work for $8775. The first man employs 20 men, 24 days, 10 hours a day ; the second 25 men, 20 days, 12 hours a day ; the third 30 men, 25 days, 9 hours a day. How much should each of the contractors receive ? 7. A, B, and C contract to build a piece of railroad for $7500. A employs 30 men 50 days ; B employs 50 men 36 days ; and C employs 48 men and 10 horses 45 days (each horse to be reckoned equal to 1 man), and is to have $115.50 for overseeing the work. How much is each man to receive ? 8. A and B rent a pasture for $690 per annum. A putL in 200 sheep, and B 160 ; at the end of 6 months they dis- pose of half their stock and allow C to put in 120 ; what should A, B, and C pay severally towards the rent at the year's end ? 9. A and B entered into partnership for one year. A had $800 in the business during the first 4 months, and $400 more during the remainder of the year ; B had $500 during the first 7 months, and $1300 during the last 5 months. At the end of the year they found they had lost $3800 ; what was each partner's loss ? 10. A, B, and C formed a partnership and cleared $1200. A put in $8000 for 4 months, and then added $2000 for 6 months ; B put in $16000 for 3 months, and then withdraw- ing half his capital, continued the remainder 5 months longer; C put in $13500 for 7 months. How should the profit be divided ? EATIO AND PROPORTION. 177 11. A and B entered into partnership for 3 years, A put- ting in 15000, and B $6000. At the end of a year A put in $3000, and B put in flOOO. At the end of the second year A took out $4000. At the end of the third year they divided a profit of $8140. What was each partner's share ? 12. A and B engaged in trade for 1 year. Jan. 1st A advanced $2400 and B $3600 ; May 1st C was admitted to the firm with $4000; July 1st B withdrew $1000; and Oct. 1st C withdrew $150l). Their profits for the year were $6800 ; what was each partner's share ? 13. A and B began business Jan. 1st, each with a capital of $2500. Apr. 1st A added $500, and Aug. 1st he added $800 more. June 1st B added $1000. What was the share of each, at the year's end, of a profit of $5425 ? 14. A's gain is $840, B's gain is $1125, and C's gain is $1820. A's capital was in trade 7 months, B's 9 months, and C's 14 months. How much of the capital $13875 did each own ? Averages or Alligation. 111. The process of finding the average or mean value of several quantities of different values is called alligation medial. I. A grocer mixed 12 lb. of tea worth 40 cents a pound, 10 lb. worth 65 cents a pound, and 8 lb. worth 75 cents a pound ; what was the mixture worth a pound ? 40 X 12 = 480 12 lb. at 40 cts. a pound are worth 480 cts. ; 65 X 10 = 650 10 lb. at 65 cts. a pound are worth 650 cts. ; 8 lb. 75 X 8 = 600 at 75 cts. a pound are worth 600 cts. Adding, 30 )1730 we find the value of 30 lb. to be 1730 cts. ; hence 67|cts. 1 lb. is worth 3L of 1730 cts., or 57f cts. 178 ARITHMETIC. EXAMPLES. 1. Four children weigh respectively 62 lb., 77 lb., 89 lb., and 102 lb. ; find their average weight. 2. In a certain school there are 15 pupils 10 years old, 6 pupils 9 years old, 10 pupils 8 years old, 8 "pupils 7 years old, and 3 pupils 6 years old ; find their average age. 3. Find the average daily expenses of a travelling sales- man whose expenses for the week were as follows : Monday f 10.50, Tuesday $3.84, Wednesday $5.25, Thursday $4.33, Friday $6.78, and Saturday $9.44. 4. A grocer mixed 16 lb. of coffee worth 25 cents a pound, 24 lb. worth 30 cents a pound, and 10 lb. worth 33 cents a pound ; what was the mixture worth a pound ? 5. Teas are mixed as follows : 40 lb. worth 70 cents a pound, 60 lb. worth 60 cents a pound, 100 lb. worth 50 cents a pound, and 80 lb. worth 40 cents a pound ; for what should the mixture be sold a pound ? 6. Find the value per gallon of the following mixture : 6 gal. of wine worth $1.10 per gallon, 14 gal. of wine worth $1.35 per gallon, 7 gal. of wine worth $1.50 per gallon, and 5 gal. of water. 7. A merchant sold 75 bbl. of flour at $5.60 per barrel, 45 bbl. at $5.95 per barrel, 30 bbl. at $6.10 per barrel, and 25 bbl. at $6.50 per barrel ; what was the average price per barrel ? 8. A goldsmith combined 7 oz. of gold 22 carats fine, 12 oz. 20 carats fine, 10 oz. 15 carats fine, and 5 oz. of alloy ; how many carats fine was the composition ? 9. A miller mixes 18 bu. of wheat at $1.44 with 6 bu. at $1.32, 6 bu. at $1.08, and 12 bu. at $0.84. What will be his gain per bushel if he sells the mixture at $1.50 ? RATIO AND PROPORTION. 179 10. Some sugar is adulterated as follows : y^^ is worth 8 cents per pound, | is worth 10 cents per pound, ^^ is worth 12 cents per pound, and the remainder, 33 pounds, is sand. What is the mixture worth per pound ? , 112. The process of finding the proportion of several quantities that may be used to form a mixture of given average value is called alligation alternate. I. rind the proportion in which teas worth respectively 65 and 80 cents a pound must be taken to form a mixture worth 70 cents a pound. 65 + 5 . . 10 . . 2 Write the given prices in a column 80 — 10 . . 5 . . 1 with the price of the mixture at the left. If tea worth 65 cts. be sold for 70 cts., there is a gain of 5 cts., which is indi- 70 Ans. 2 lb. at ^^ cts and 1 lb. at 80 cts. ^^^^^ \iy\^ annexed to 65 ; if tea worth 80 cts. be sold at 70 cts., there is a loss of 10 cts., which is indicated by — 10 annexed to 80. A gain of 5 cts. a pound on 10 lb. will exactly bal- ance a loss of 10 cts. a pound on 5 lb., and we write 10 opposite 65 + 5, and 5 opposite 80 — 10. This means that the two kinds must be mixed in the ratio of 10 to 5; dividing both numbers by 5, we obtain 2 lb. of the first and 1 lb. of the second. We can take any number of pounds that are in the ratio of 2 to 1. Notice that the number of pounds taken at first of either kind is the same as the number of cents gained or lost on the other kind. II. Find the proportion in which two kinds of vinegar worth respectively 12 and 15 cents a quart must be taken to form a mixture worth 13 1 cents a quart. Using the same process as in the pre- 12 + 1^ • • 1|^ • • 5 ceding problem, we find that the two -IK -12. 11 4 kinds are to be mixed in the ratio of 1| to U. This ratio can be simplified by Ans. 5 qt. at 12 cts. multiplying both numbers by 3, thus and 4 qt. at 15 cts. obtaining 5 qt. of the first kind, and 4 qt. of the second kind. l^ 180 ARITHMETIC. 32 22+10 30+2 35-3 40- 8 J Ans. 4 lb. at 22 cts., 3 lb. at 30 cts., 2 1b. at 35 cts., 5 lb. at 40 cts. 32 22 + 10-1 30+ 2— 35 40 3J 3 8 10 2 III. A grocer wishes to mix coffees worth respectively 22, 30, 35, and 40 cents a pound so as to make a mixture worth 32 cents a pound ; how many pounds of each kind shall he take ? We begin as before, and link them together two by two, always linking one on which there is a gain with one on which there is a loss. Comparing the first and fourth, we find that they must be taken in the ratio of 4 to 5 ; compar- ing the second and third, we find that they must be taken in the ratio of 3 to 2. We can take any quantities provided that the first and fourth are in the ratio of 4 to 5, and the second and third in the ratio of 3 to 2. By linking the first with the third, and the second with the fourth, we ob- tain an entirely different answer. When more than two kinds are to be mixed, they can be taken in an infinite number of ways, for it is only necessary to combine them two by two, so as to make mixtures of the required value, and tliese mixtures may be com- bined in any proportions whatever. In solving problems it is generally best to take the combinations that involve the smallest numbers. IV. How much sugar worth respectively 6 and 10 cents a pound must be mixed with 20 lb. worth 9 cents a pound in order that the mixture may be worth 8 cents a pound ? 1 . . 10, 2 . . 1 ; 10 + 1 = 11. Link the first with 2 . . 20 the second and the 2 . . 1 first with the third. Ans. 11 lb. at 6 cts. and 1 lb. at 10 cts. Comparing the first and second, we find that they must be taken in the ratio of 1 to 2 ; hence 10 lb. of the first must be taken with 20 lb. of the second. Comparing the first and third, we find that they must be taken in the ratio of 1 to 1. Thus we must take 10 lb. of the first, and in addition equal quantities of the first and third. Ans. 3 lb. at 22 cts., 4 lb. at 30 cts., 10 lb. at 35 cts., 1 lb. at 40 cts. 6 + 2^ 9-1 10-2 RATIO AND PROPORTION^ 181 V. A trader mixed oats worth respectively 32, 35, 40, xnd 42 cents a bushel in order to make a mixture of 45 bu. worth 36 cents a bushel ; how many bushels of each kind did he take ? We find that the dif- 32 + 4-6.. 3 3 + 4+1 + 2=10; f^^^„, kinda may be 36 35 + 1 40-4 42 -6J ^ 3 ^ 27 mixed in quantities pro- ] n J^ ^^ ^^ " 2" " ^^^ ' portional to 3, 4, 1, and ^ • • ^ 2 2. Dividing 45 into 2 parts proportional to JL f 4S— IR" these numbers, we ob- ;p ° ^^~ ' tain 13.> bu. of the first ?• kind, 18 bu. of the sec- 9 J_ f/er7_^_Ai. ond, 4 J bu. of the third, X0^ 2~ ^' and 9 bu. of the fourth. 2 If integral answers j_ f 43 — Q *^® desired, the ratios X0 * must be so taken that f the sum of the numbers Ans. 13|bu.at32cts.,18bu.at35cts., representing them is a 4|bu.at40cts.,and9bu.at42cts. ^^"^'°'' ^^ '^'' "°*^" quantity. EXAMPLES. 1. How shall corn at 45 cts. a bushel be mixed with oats at 36 cts. a bushel that the mixture may be worth 40 cts. a bushel ? 2. Find the proportion in which oils worth respectively $1.30 and 85 cts. a gallon must be taken to form a mixture worth $1.10 a gallon. 3. In what proportion shall sugars worth respectively 7 and 12 cents a pound be taken to form a mixture worth 9^ cents a pound ? 4. Find the proportion in which sugars worth respec- tively 5 and 8 cents a pound must be taken to form a mixture worth 6f cents a pound. 182 ARITHMETIC. 5. In what proportion must alcohol (sp. gr. 0.82) be mixed with water to make a mixture having a specific gravity of 0.9 ? 6. A grocer wishes to mix teas worth respectively 40, 55, and 65 cents a pound so as to make a mixture worth 50 cents a pound; how many pounds of each kind shall he take ? 7. Find the proportion in which three kinds of rice worth respectively S^, 11, and 12 1 cents a pound must be taken to form a mixture worth 10|^ cents a pound. 8. A merchant has coats worth $12 each, vests worth $6 each, and hats worth $4^- each ; how many of each must he sell in order that the average price may be ^7^? 9. How much water must be mixed with wine worth 90 cts. per gallon to make a mixture worth 60 cts. per gallon ? 10. A grocer wishes to mix syrups worth respectively 42, 56, 64, and 75 cents a gallon so as to make a mixture worth 60 cents a gallon ; how many gallons of each kind shall he take ? 11. In what proportion shall sugars worth respectively 6, 7^, 8 1", and 9 cents a pound be taken to form a mixture worth 7 cents a pound ? 12. A grocer has wines worth respectively f 1.10, $1.30, $1.35, and $1.50 per gallon, which he wishes to mix with water so as to form a mixture worth $1.25 per gallon ; how many gallons of each shall he take ? 13. Teas at 3 s. 6 d., 4 s., and 6 s. a pound are mixed to produce a tea worth 5 s. a pound ; what is the least integral number of pounds that the mixture can contain ? 14. How ma' y acres of land worth $70 an acre must be added to a fai q of 75 acres worth $100 an acre in order that the whole may average $80 per acre ? RATIO AND PROPORTION. 183 15. How many bushels of corn at 50 cents a bushel must be mixed with 100 bu. of oats at 80 cents a bushel that the mixture may be worth 75 cents a bushel ? 16. How many pounds of chicory at 6 cts. a pound and coffee at 28 cts. a pound must be mixed with 30 lb. of coffee worth 35 cts. a pound in order that the mixture may be worth 20 cts. a pound ? 17. A goldsmith has 4 oz. of gold 20 carats fine and 6 oz. 22 carats fine ; how many ounces of alloy must be combined with it to make a mixture 16 carats fine ? 18. A farmer wishes to mix 20 bu. of oats worth 33 cents a bushel with oats worth respectively 35, 38, and 40 cents a bushel, making a mixture worth 36 cents a bushel; how many bushels of each kind must he take ? 19. A grocer wishes to mix 15 lb. of coffee at 40 cents a pound and 25 lb. at 35 cents a pound with two kinds worth respectively 25 and 28 cents a pound so that the mixture may be worth 30 cents a pound ; how much of the latter kinds must he take ? • 20. A wholesale dealer has an order for 1000 bu. of wheat at 75 cents a bushel ; how shall he mix three kinds of wheat, valued respectively at 72, 76, and 80 cents a bushel, to fill the order ? 21. How many pounds of tea worth respectively 50, 60, and 75 cents a pound must be taken to make a mixture of 70 lb. worth 65 cents a pound ? 22. A man has 100 three-cent pieces, which he wishes to exchange for dimes, half-dimes, two-cent pieces, and cents, and still have the same number of coins ; how many of each kind will he receive ? 184 • ARITHMETIC. 23. A lady bought 100 yd. of cloth for $10, some at 3 cents a yard, some at 8 cents, some at 12 cents, and some at 15 cents ; how many yards of each kind did she buy ? 24. A dealer paid f 182 for 20 barrels of flour, giving $10 for first quality and $7 for second ; how many barrels were there of each ? 25. An alloy, formed from two metals whose specific gravities are 8.29 and 10.35, has a specific gravity of 9.87 ; how many grams of each metal are there in a kilogram of the alloy ? 26. Find how much gold 15, 17, and 22 carats fine must be mixed with 5 oz. 18 carats fine so as to make 12 oz. 20 carats fine. 27. A man paid $70 to three men for 35 days' labor ; to the first he paid $5 per day, to the second $1 per day, and to the third $0.50 per day. How many days did each work ? PERCENTAGE. 186 CHAPTER IX. PERCENTAGE. 113. The process of computing by hundredths is called percentage. Per cent, a contraction of the Latin per centum^ means hy the hundred. For example, 4 per cent of 25 means 4 hundredths of 25. The sign % is used in place of the words per cent. For example, 6% means 6 per cent. 6% has the same value as 0.06 or ^. The number on which the percentage is reckoned is called the base, the number of hundredths taken is called the rate, the fraction denoting the number of hundredths is called the rate per cent, and the part of the base corresponding to the rate per cent is called the per cent or percentage. For example, 5% of 200 = 0.05 of 200 = 10 ; 200 is the base, 5 is the rate, 5% or 0.05 is the rate per cent, and 10 is th9- per cent or percentage. To Express a Rate Per Cent as a Common Fraction. 114. I. Express 31:^% as a common fraction. 5 Since any per cent equals the 3U%=^=3^X — = — • same number of hundredths, 31i% ^'° 100 4 XPP 16 311 4 equals — i, which equals ^^. EXAMPLES. Express as common fractions the following rates per cent : 1. 6i%. 3. ^%. 5. 1H%. 7. 14f%. 2. 6|%. 4. 10%. 6. 12^%. 8. 16|%. 186 ARITHMETIC. 19. 80%. 20. 831%. 21. 87i%. 22. 6%. 23. 71%. 24. 28%. 25. 35%. 26. 68f%. 27. 125%. 28. 1331%. Note. The fractional equivalents of the first twenty-one examples given above should be committed to memory by the student, as they are very often made use of. To Express a Common Fraction as a Rate Per Cent. 115. I. Express f as a rate per cent. 2 _ 28* — 28 4 <^ Since we are to obtain the rate per cent, or 7 / • number of hundredths, to which f is equal, we divide the numerator by the denominator, carrying the division to two decimal places and retaining the remainder as a fraction. EXAMPLES. Express as rates per cent the following common fractions : 9. 20%. 10. 25%. 11. 331%. 12. 371%. 13. 40%. 14. 50%. 15. 60%. 16. 621%. 17. 66io/„ 18. 75%. 29. 140%. 30. 162^%. 31. 266|%. 32. 340%. 33. Wc- 34. -h% 35. %% 36. *%. 37. t\%. 38. \%% 1. \. 6. A- 11. i^. 16. ^f^. 2. J. 7. |. 12. W 17. ^. 3. \. 8. A- 13. A- 18. f. 4. f. 9- «• 14. W 19. |. 6. f 10. tV- 15. T*Tr- 20. ->^. PERCENTAGE. 187 To Find any Per Cent of a Numbeb. U6. I. What is 62-1% of 72 ? 9 62i% = |;|ofr? = 45. ^ „„,_ „„^, ,, ,5. Since 62|% equals f , 62^% of 72 equals II. What is 12% of f210? 210 0.12 Since 12% equals 0.12, 12% of $210 equnl.< 0.12 of $210, which is $25.20. ^25.20 Note 1. When a problem can be done mentally, use the method of example I. ; otherwise use the method of example II. * Note 2. When a number is said to be a certain per cent more or less than another number, the second number should be taken as the base; never use the sum of two numbers thus spoken of as the base. 1. What 2. What 8. What 4. What 5. What EXAMPLES. s8% of 1200? s 76% of 126.25 ? s7i% of $2450? slf% of 1264? s45% off? s75% of 12i? s 33^% of 16 gal. 2 qt. ? 0. What 7. What 8. What 9. What 10. What 11. What 12. What 13.' How many pounds are there in f % of 6 cwt. 1 qr. ? U What is 62^% of 175% of 20% of $576? s 371% of 6 ft. 8 in. ? s 100% of 1760 bu. ? s 120% of 250 mi. ? s 11% of 1280 men ? sf% of 140 books? 1^8 ARITHMETIC. 15. A man bought a house for $6750, paying 25% cash; how much remained due ? 16. A farmer had 320 sheep, and lost 6^% of them; how many had he left ? 17. A man has a yearly income of $1400, and pays 17-f % of it for house rent ; what rent does he pay ? 18. If an ore yields 62% of pure iron, how many pounds of iron can be obtained from a ton of ore ? 19. A miller charges 6% for grinding; how many quarts will he take when he grinds 25 bu. ? 20. A merchant failed and paid 60% of his debts ; how much was received by a creditor to whom he owed $2180 ? 21. If a yard of cloth shrinks 41% in length in spong- ing, what fraction of a yard will it measure after sponging ? 22. The amount of sunshine recorded in a certain city in the month of April was 33% of the possible amount, and the average length of the nights in that month is 10 hr. 30 min. ; find the number of hours of sunshine in the month. 23. A man has a capital of $12500 ; he puts 15% of it in stocks, 33|^% in land, and 25 ^ 230 ARITHMETIC. NOTE PAYABLE AT A BANK. Irf^c^^i- Pittsburg, Pa.,^.^^ y^:^d-ut. jQQ Dollars, at the %\\t)^\xtxi\y^ Rational gawfe. Value received, without defalcation. Note. The law of Pennsylvania requires the words " without defalcation." NON-NEGOTIABLE NOTE. Qle^ Q/<^ld, Ocl y^// '^n^ t^e'T^'C^'^t^^ Cy fiyio^'^'T^'C-d^^ ^ C^^^^ J2^ Qfo^v^i^ C/^/^ /^. ^e^M-e ^^eoei^-ue^/ , ^iij^^ri i-^^Cel-e^d^c tz^ iU^^ /i.-ed' 'C^'^^t /h€4^ t^^n^'Ti.'U^r^. ^o'^ ^. c9 c^TSn" vSf-e^^lt/y (_>ri^^j^^^i^^^^ INTEREST AND DISCOUNT. 231 Partial Payments. 133. When partial payments are made on notes or other obligations bearing interest, allowance is made for these partial payments in computing the amount due at the time of settlement. The amounts of the payments and their dates are written on the back of the obligation ; such pay- ments are called indorsements. The Supreme Court of the United States has adopted a rule for partial payments, which is known as The United States Rule. Find the amount of the principal to the time when the pay- ment, or the sum of the payments, equals or exceeds the interest due. From the amount subtract the payment^ or the sum of the payments^ and, with the remainder as a new principal, proceed as before to the time of settlement. This rule is based on the following principles : 1st, payments must be applied first to the discharge of interest due, and the balance, if any, toward the discharge of the principal. 2nd, unpaid interest must not be added to the principal to draw interest. 3rd, only unpaid principal can draw interest. I. A note for $750 is dated July 11th, 1884 and bears the following indorsements : Feb. 17th, 1885, $225 ; Dec. 24th, 1885, $25; May 19th, 1886, $375. What balance is due Dec. 1st, 1886, reckoning interest at 6% ? 282 ARITHMETIC. 1885— 2 — 17 750. 1884— 7 — 11 .036 7— 6 .036 4500 2250 27.000 750. 1885 — 12 — 24 1885— 2 — 17 777. 225. 552. .0514 The amount of $750 from July 11th, 1884 to Feb. 17th. 1886 is $777, and subtracting the 1st payment from this . ^ amount, we obtain $552 for 1^ — '^ 92 the 2nd principal. The in- '^^H 552 terest of $552 from Feb. 2760 17th, 1885 to Dec. 24th, 1885 28.244 is $28.24, which is more than 4 CO/. K Hq ;-;ro the 2nd payment ; hence we ■tQQK 2 17 OT'^i P^^^ °^ *® ^^^ next date. ^ The amount of $552 from Feb. 17th, 1885 to May 19th, 1886 is $593.58, and subtract- ing the sum of the 2nd - 2 184 .075J 2760 3864 41.584 and 3rd payments from this ^^^- amount, we obtain $193.58 593.58 for the 3rd principal. The 400. amount of $193.58 from May 1886 — 12 — 1 193.58 19th, 1886 to Dec. 1st, 1886 1886— 5 — 19 .032 is $199.77. 6 — 12 38716 .032 58074 6.19456 193.58 $199.77 Note 1. When the rate is any other than 6%, each separate interest must be found at the given rate. Note 2. When the interest is required on a principal on which par- tial payments have been made, find the amount due, and from that amount subtract the difEerence between the principal and the sum of the payments. INTEEEST AND DISCOUNT. 233 EXAMPLES. 1. On a note for f 1400, given Apr. 12th, 1882, two pay- ments were made : Aug. 30th, 1884, ^400 ; Aug. 30th, 1886, $600. At 6% interest, what was due Dec. 30th, 1887 ? 2. A note for $1000, dated Jan. 1st, 1883, and bearing interest at 6%, is indorsed with three payments of $80 each, made on Jan. 1st of 1884, 1885, and 1886. What was due on the note at settlement Oct. 1st, 1886 ? 3. Find what is due Oct. 1st, 1888 on a note for $1750 at 6% interest, dated Dec. 13th, 1884, with three payments indorsed, viz.: June 10th, 1886, $360 j Jan. 1st, 1887, $40; Aug. 10th, 1887, $500. 4. On a note for $2000, dated July 15th, 1885, and bear, ing interest at 7%, there was paid $100 May 5th, 1886, and $200 Jan. 1st, 1887. Find what remained due Aug. 1st, "1887. 5. A note of $600, dated Aug. 10th, 1885, had indorse- ments as follows: Feb. 4th, 1886, $50; July 27th, 1886, $10 ; Oct. 9th, 1886, $75. How much was due Dec. 15th, 1886 at 5% interest? 6. A note for $1580, dated Oct. 19th, 1882, is indorsed Sept. 6th, 1883, with $640; Jan. 30th, 1884, with $20; Oct. 9th, 1884, with $380. What balance is due Feb. 3rd, 1885, interest at 4^%? 7. A note for $300, dated May 15th, 1878, and bearing interest at 5%, is indorsed as follows: Feb. 25th, 1881, $40; Sept. 15th, 1882, $25 ; May 4th, 1883, $150. What balance is due Jan. 1st, 1884 ? 8. On a note for $1000, dated Jan. 1st, 1878, due in one year, and bearing interest at the rate of 6% from the date of maturity, the following payments were made : Aug. 16th, 234 ARITHMETIC. 1879, ^300; Feb. 12th, 1880, $200; Oct. 3rd, 1881, $50; Jan. 17th, 1882, $19; May 31st, 1883, $22. What was due Jan. 1st, 1884 ? 9. What is the interest at 4^% of $360.45 from July 5th, 1883 to Nov. 4th, 1885, allowing a credit of $75 paid Oct. 6th, 1884 ? 134. When the whole period of time is not longer than one year, business men commonly employ The Merchants' Rule. Find the amount of the principal for the whole time the note is on interest; find also the amount of each payment from the time it was made until settlement; from the amount of the principal subtract the amounts of the payments. I. Find the balance due Mar. 1st, 1888 on a note for $875, given Apr. 18th, 1887, on which the following pay- ments had been made: July 7th, 1887, $360; Oct. 28th, 1887, $250. 1888- 3- 1 1887-^ 4-18 1888-3- 1887-7- • 1 ■ 7 1888- 3- 1887-10- 1 28 374.04 255.13 10-13 .052^- 7-24 .039 4- 3 .020^ 629.17 876. .052^ 360. .039 250. .0201 920.65 629.17 145f 1750 3240 1080 125 5000 $291.48 4375 45.645f 14.040 360. 5.125 250. 875. 374.04 255.13 920.65 The amount of $875 from Apr. 18th, 1887 to Mar. 1st, 1888 is $920.65. The amount, of $360 from July 7th, 1887 to Mar. 1st, 1888 is $374.04, INTEREST AND DISCOUNT. 235 and the amount of $260 from Oct. 28th, 1887 to Mar. Ist, 1888 is .$256.13. The sum of the amounts of the payments is $029.17, and subtracting this from $920.66, we find $291.48 to be the balance due. EXAMPLES. 1. On a note for $1500, dated Jan. 1st, 1886, and bear- ing interest at 4^%, there was paid $550 Apr. 1st, 188G, and $725 Oct. 1st, 1886. Find what remained due Jan. 1st, 1887. 2. A lends B $1000 Feb. 12th, 1885; B pays $200 Mar. 27th, 1885, and $50 Dec. 12th, 1885. Find what is due Jan. 18th, 1886 at 6% interest. 3. Find what is due Oct. 1st, 1888 on a note for $2500 at 9% interest, dated Jan. 1st, 1888, on which payments of $600 each have been made : Mar. 1st, May 1st, and July 1st, 1888. 4. Find the balance due Sept. 1st, 1888 on a note for $600, given Sept. 1st, 1887, on which the following pay- ments had been made: Feb. 15th, 1888, $120; May 24th, 1888, $350 ; July 20th, 1888, $100. 5. A note for $1372.50, dated Nov. 10th, 1887, and bear- ing interest at 7%, is indorsed as follows: Jan. 20th, 1888, $321; Mar. 29th, 1888, $490; June 14th, 1888, $275. What balance is due Sept. 10th, 1888 ? 6. Payments were made on a debt of $2470, due May 7th, 1886, as follows : June 24th, 1886, $420 ; Aug. 3rd, 1886, $345; Oct. 20th, 1886, $500; Nov. 29th, 1886, $790. What was due Jan. 1st, 1887 at 5% interest ? 7. What is the interest at 5% of $722.85 from Oct. 19th, 1886 to May 3rd, 3887, allowing a credit of $500 paid Jan. 4th, 1887 ? 236 ARITHMETIC. Compound Interest. 135. Compound interest is interest reckoned on both the principal and the unpaid interest added at regular intervals. The interest may be compounded, or added to the principal, annually, semi-annually, or for any other period of time according to agreement. I. Find the compound interest of $800 for 2 yr. 8 mo. i2 da. at 7%. 800 1.07 5600 800 856. 1.07 5992 856 915.92 .042 183184 366368 5)38.46864 6.41144 44.88008 915.92 960.80 800. The amount of $1 for 1 yr. at 7% is $1.07, and the amount of $800 is 800 times $1.07, which equals $856. Taking this as a new principal, we find the amount at compound interest for 2 yr. to be $915.92, which we take as the principal for the remaining 8 mo. 12 da. Thus the amount of $800 for 2 yr. 8 mo. 12 da. at compound interest is $960.80. The original principal subtracted from this amount gives $160.80 as the com- pound interest. $160.80 Note. When the interest is compounded semi-annually, the inter- est must be found for each half-year at one half the yearly rate, and similarly for any other period of time. Interest is compounded annu- ally if nothing is stated to the contrary. INTEREST AND DISCOUNT. 237 EXAMPLES. 1. What is the compound interest on $1000 for 3 yr. at 7%? 2. Find the amount of $100 at the end of 3 yr. at 4^% compound interest. 3. To how much will $1000 amount in 4 yr. at 20% compound interest ? 4. How much would $350 amount to in 7 yr. at 6% compound interest ? 5. What is the amount, at compound interest, of $500 for2yr. 6 mo. at 7%? 6. What is the compound interest of $25 for 3 yr. 5 mo. at 6%? 7. Find the amount of $1000 for 2 yr. 2 mo. 12 da. at 6% compound interest. 8. Find the compound interest of $200 for 2 yr. 6 mo. 18 da. at 4%. 9. What is the amount of $5216.75 from Jan. 21st, 1885 to July 3rd, 1888 at 8% compound interest ? 10. What is the compound interest on £47 13 s. 6d. for 3 yr. 4 mo. 15 da. at 3i%? 11. Find the compound interest of $720 for 2 yr. at 7%, interest being payable semi-annually. 12. What will be the amount of $103 for 2 yr. 6 mo. at 5%, the interest being compounded semi-annually ? 13. What is the amount of $340 at 8% for 1 yr. 3 mo., the interest being compounded semi-annually ? 14. What is the amount of $450 for 1 yr. 2 mo. 18 da. at 6%, interest compounding quarterly ? 238 ARITHMETIC. • 15. Find the compound interest of $122.50 from Sept. 1st, 1884 to Nov. 25th, 1885 at 4%, interest being payable quarterly. To Find the Principal when the Compound Interest (or Amount), Eate Per Cent, and Time are Given. 136. I. Find the principal on which the compound inter* est for 2 yr. 6 mo. at 6% is f 108.75. 1.06 1.06 .157308)108.750000(1691.32 943848 636 106 1436520 1415772 1.1236 1.03 207480 157308 33708 11236 501720 471924 ' 1.157308 297960 The compound interest of $1 for 2 yr. 6 mo. at 6% is $0.157308. To produce an interest of $108.75 will require as many dollars as $0.157308 is contained times in $108.75, which equals $691.32. Note. When the amount is given in place of the interest, the divi- sor should be the amount of $1 for the given time at the given rate. EXAMPLES. 1. What principal will in 2 yr. at 5% produce a com- pound interest of $350 1 2. Find the principal on which the compound interest for 3 yr. at 4% is $468.24. 3. What sum of money at 6% compound interest will amount to $2703 in 1 yr. 4 mo.? 4. At 4:% compound interest what sum of money will amount in 2 yr. to $594.88? INTEREST AND DISCOUNT. 239 5. What sum of money, at 10% compound interest, will amount to 18651.50 in 3 yr. ? 6. Find what principal will amount to f 1000 in 3 yr. 6 mo. at 3^% compound interest. 7. What principal will produce $250 interest in 1 yr. 8 mo. 24 da. at 6%, the interest being compounded semi annually ? 8. What principal will amount to $2000 in 1 yr. 4 mo. 15 da. at 4%, the interest being compounded quarterly ? Annual Interest. 137. Annual interest is simple interest reckoned on the principal and also on each year's interest after it is due. I. Find the annual interest of $1710 for 3 yr. 4 mo. 12 da. at 5%. The simple interest of $1710 for 3 yr. 4 mo. 12 da. is 1287.85. The interest due at the end of the first year draws interest for 2 yr. 4 mo. 12 da. ; the interest due at the end of the second year draws interest for 1 yr. 4 mo. 12 da.; the interest due at 3420 35 50 the end of the third year draws fiVS45420 -246 interest for 4 mo. 12 da. ; the — K7- rij ' K-j oAQ sum of the interests of the yearly '' — S4200 unpaid interests is equivalent to Zoi.oo 17100 *^® interest of one year's inter- «\oi HQQnn ®^* ^^^ *^^ ^^™ ®^ these periods. oinrr The interest of $1710 for 1 yr. ^-^^^^ is $85.50, and the interest of 17.53 $85.50 for 4 yr. 1 mo. 6 da. is 287.85 $17.5.3; adding $287.85 to this $305.3? amount, we find S305.38 to Ut the entire annual interewki- 3 yr 18 2—4—12 4 mo 02 1_4— 12 12 da. . . .002 4—12 .202 4—1— 6 .246 1710 .202 1710. 3420 .05 240 AEITHMETIC. EXAMPLES. 1. What is the interest for 3 yr. on a debt of $1800 at 6% annual interest? 2. How much interest is due on a debt of $1500, at 6% annual interest, at the end of 3 yr. 6 mo. ? 3. Find the annual interest of $1200 for 4 yr. 3 mo. 10 da. at 5%. 4. A note for $500, with annual interest at 6%, is due 4 yr. 6 mo. after date ; if no interest has been paid, what will be due at maturity ? 5. A note for $2250, with interest payable annually at 8%, was paid 3 yr. 3 mo. 18 da. after date, and no interest had been previously paid ; what was the amount due ? 6. A note was given May 8th, 1883 for $625, interest payable annually at 5%; if no payment is made, what will be due Mar. 8th, 1886? 7. Find the interest due Dec. 20th, 1888 on a note for $725, dated June 11th, 1884, with interest payable annually at 7%, when no interest has been paid. 8. Find the amount due Apr. 4th, 1888 on a note for $1150, dated Nov. 22nd, 1883, with interest payable annually at 4J%, when no interest has been paid. True Discount. 138. Discount is a deduction made for the payment ol a debt before it is due. The present worth of any sum of money due at a future time without interest, is that sum which put at interest for the given time^ will amount to the given sum. The differ- INTEREST AND DISCOUNT. 241 ence between the given sum and its present worth is called the true discount. I. Find the present worth of $800, due in 1 yr. 7 mo. 24 da., at 5%. 1 yr 06 1.0825) 800.0000 ($739.03 7mo....035 75775 24 da. . . .004 42250 6 ). 099 32475 .0165 97750 .0825 97425 32500 The amount of $1 for 1 yr. 7 mo. 24 da. is $1.0825; hence the pres- ent worth of .$1.0825 is $1. The present worth of $800 is as many times $1 as $1.0825 is contained times in $800, which equals $739.03. The process is the same as finding the principal, when the amount, rate per cent, and time are given, as shown in § 131. Note. When the money is at compound interest, the amount of $1 should be found at compound interest. EXAMPLES. 1. Find the present worth and discount of $3230, due in 4 yr. 10 mo. 12 da., at 6%. 2. Find the present worth and discount of $2000, due in 1 yr. 8 mo., at 4|^%. 3. Find the present worth and discount of $2500, due in 3 mo., at 8%. 4. Find the present worth and discount of $1926.94, due in 8 mo. 3 da., at 7%. 5. Find the present worth and discount of $1025, due in 36 da., at 10%. 6. What is the present worth of $3471.50, due 3 mo. 9 da. hence, at 7% ? 7. What is the present worth of $1609.30, due in 10 mo. 24 da., when money is worth 5^ ' 242 ARITHMETIC. 8. Find the present worth of a note for ^313.31, due in 2yr. 2 mo. 2 da., at 3i%. 9. What is the present worth of $10000, due 3 yr. hence, at 5% compound interest ? 10. What is the present worth of $678.75, due 3 yr. 8 mo. hence, at 7% compound interest? Bank Discount. 139. Bank discount is a deduction made by a bank for advancing money on a note before it is due, and it is the interest on the face of the note from the day of discount to the day of maturity ; this period of time is called the term of discount, and the rate of interest is called the rate of discount. The sum of money received for a note when it is dis- counted at a bank is called the proceeds or avails, and it equals the face of the note minus the bank discount. Note. If no day of discount is given, a note is understood to be discounted on the day of its date ; in such a case the term of discount equals the time specified in the note plus three days of grace. When the time a note has to run is designated by months, the term of discount is determined by subtracting dates ; when the time is desig- nated by days, the term of discount is determined by exact days. I. Find the bank discount on a note for f 450, due in 60 days, at 5%. 450. 63 1350 2700 The term of discount is 60 da. + 3 da., which is 63 6000)28350 ^^' "^^^ interest of $450 for 63 da. equals $3.94. 6 )4.725 .7875 $3.94 INTEREST AND DISCOUNT. 243 II. Find the proceeds of a note for $1200, dated Apr. 10th, payable in 90 days, and discounted May 14th at 7%. 90 days after Apr. 10th is July 9th ; hence the note becomes due July ®/i2. The term of discount is from May 14th to July 12th, which equals 59 da. The interest of $1200 for 59 da. at 7% is .^13.77. The proceeds is .$1200 — ^13.77, which equals .$1186.23. Note. As the day of maturity was not needed in this problem, the term of discount could have been determined as follows: from Apr. 10th to May 14th is 20 da. + 14 da., or 34 da., and 93 da. — 34 da. = 59 da. III. A note for $2020, dated Oct. 31st, 1887, and payable in 6 months with interest at 4^%, was discounted Mar. 12th, 1888 at 6% ; find the proceeds. 17 1200 1200. 30 59 13.77 12 10800 $1186.23 59 da. 6000 6000)70800 6)11.80 1.967 13.77 2020 .030^ 1010 60600 4)61.610 15.4025 46.2075 2020. 2066.21 1888- 1888- 3 12 2066.21 When a note " with in- 17.56 terest " is discounted, the $2048.65 discount is computed on the amount due at ma- turity. The amount of $2020 for 6 mo. 3 da. is $2066.21. 6 mo. after Oct. 31st is called Apr. 30th ; since April has no 31st day, the time expires on the last day; hence the note becomes due Apr. 30/|j^ay 3. The term of discount is 1 mo. 21 da., and the interest of $2066.21 for this time is $17.56. The proceeds is $2066.21 -$17.56, which equals $2048.65. 1—21 .008^ 2066.21 .008^ 103310^ 1652968 17.56278^ EXAMPLES. 1. Find the bank discount on a note for $125, due in 3 months, at 5%. 244 ARITHMETIC. 2. What is the discount on a note for $475, due in 75 days, discounted at a bank at 4^%? 3. If you have a note for $1000, payable in 60 days, discounted at a bank at 6%, what sum will you receive ? 4. How much money should be received on a note of $1000, payable in 4 months, discounted at a bank where the rate of discount is 6%? 5. A man buys $800 worth of goods and gives his note for that sum, payable in 90 days. Find the sum realized on the note if it is immediately discounted at a bank at 6%. 6. If the rate of discount is 5%, how much can be ob- tained on a note for $600, payable in 4 months, discounted at a bank ? 7. If the rate of discount is 5^%, how much can be ob- tained on a note for $1000, payable in 60 days, discounted at a bank ? 8. Find the proceeds of a note for $1225, due in 30 days, discounted at 5|-%. 9. Find the proceeds of a note of $620.25, discounted at a bank for 53 days. 10. Find the bank discount and proceeds of a note of $1285^ dated Mar. 28th, 1883, payable Jan. 5th, 1885, and discounted at 4%. 11. Find the proceeds of a four-months' note for $1350, discounted 15 days after date at 7%. 12. Find the proceeds of a note for $25, dated Aug. 17th, payable in 30 days, and discounted Sept. 1st at 5%. 13. Find the proceeds of a note for $250, dated July 31st, payable in 4 months, and discounted Sept. 15th at 4-^%. INTEREST AND DISCOUNT. 245 14. A note for $500, dated Mar. 9tli, at 3 months, is dis- counted Apr. 11th at 8% ; what is received for the note ? 15. Find the bank discount on a note for $400, dated Jan. 12th, 1887, due in 90 days, and discounted Feb. 1st at 6%. 16. Find the proceeds of a note for f 384.22, at 60 days, dated July 17th, and discounted Aug. 2nd. 17. Find the proceeds of a note, dated Oct. 5th, 1886, for $428.50, payable in 6 months, and discounted Jan. 1st, 1887 at 5%. . 18. What is the bank discount on a note for $392, paya- ble in 90 days with interest at 6%, and discounted 15 days after date at 7%? 19. Find the proceeds of a note for $625 at 5% interest, due in 60 days, dated Aug. 1st, and discounted Sept. 21st at 5%. 20. A note for $150, dated June 14th, and payable in 4 months with interest at 5%, was discounted July 20th at 7% ; find the proceeds. 21. A note for $1000, dated Jan. 17th, 1888, and payable in 90 days with interest at 7%, was discounted Mar. 1st at 6% ; find the proceeds. 22. A note for $1875, dated Aug. 30th, 1887, and payable in 6 months with interest at 6%, was discounted Oct. 27th, 1887 at 8% ; find the proceeds. To Find the Face of a Note to Yield a given Proceeds. 140. I. Find the face of a note for 90 days which, when discounted at 4%, will yield $300. 246 ARITHMETIC. 3 ) .0155 .98961)300. The bank discount on $1 0051^ 3 3 for 90 days is the same as 01031 2.969 ) 900.000 (1303.13 the interest for 93 days, ^ g907 which equals $0.01031, and 1 0000 Qono *h^ proceeds of $1 equals 01034 8907 $0.9896|. To produce a !98964 "3930 ^""''^' °^ ^ ,"'" "" oqpq qmre as many dollars as 9610 $0.9896f is contained times in $300, which equals $303.13. EXAMPLES. 1. What must be the face of a note having 4 months to run that it may yield $1959 when discounted ? 2. What must be the face of a note which, when dis- counted at a bank for 60 days at 6%, shall give as its pro- ceeds $500 ? 3. For what sum must a note be drawn at 90 days to net $2050 when discounted at 7%? 4. Find the face of a note at 2 months that would real- ize $4500 when discounted at a bank, interest being 6%. 5. I wish to borrow $560 at a bank ; for what sum must I give my note for 90 days at 8%? 6. What must be the face of a note which, discounted at a bank for 30 days, would realize $200 ? 7. If the rate of discount is 5%, for what amount must a note, payable in 4 months, be given to realize $600 ? 8. Find the face of a note payable in 60 days, so that the proceeds shall be $1200 when discounted at 5%. 9. For what amount must a note, payable in 120 days, be given to. a bank discounting at 6% to obtain $500 ? 10. For what sum must T give my note for 90 days at a bank, in order to receive $1100, money being worth 7%? INTEREST AND DISCOUNT. 247 Exchange. 141. A bill of exchange, or draft, is a written or printed order from one person to another directing the payment of a specified sum of money to a third person. The person signing the draft is called the drawer ; the person to whom the draft is addressed is the drawee ; and the person to whom the money is payable is the payee. A sight draft is a draft which is payable on presentation to the drawee. A time draft is one Avhich is payable at a specified time after presentation, or after date. When the drawee accepts a draft, he writes the word " Accepted " with the date and his signature ; the draft is then called an acceptance, and the drawee, who now be- comes an acceptor, is responsible for its payment. The following are common forms of bills of exchange : SIGHT DRAFT. %^S0~. Buffalo, ^X.,/cm^ y^^ 188/. O^ d-e^oA/ pay to the order of -^-^^^^^^^^^^-^^^(^iz-i^e^ SiXnd^^^ (I^a^€. ^^i^'yit/^e^/ Ztjf/'y, Dol lars , value received, and charge to the account of 248 AKITHMETIC. TIME DRAFT. $J(^^(^Wo' Chicago, III., c^^. Ya^ 188^. d/A^-i^ i^^^uyd ^...v,>^^,^v.^^^^ after date pay tc the order of j^ea^^^ jsC (^d/e^^ c/w.^ ■^--a^ud-tZ'm/-^^^ Dollars, value received, and charge the same to the account of FOREIGN BILL OF EXCHANGE. Exchange \ox£SS. New Yoek, cJu^. JO^A 188^. On demand pay for this Bill of Exchange to the order of C^^^^.^ J't^i^^^ To ^ c;^«^«« <#«^.4 m. c^ J INTEREST AND DISCOUNT. 249 The system of making payments in distant places by transmitting drafts instead of money is called exchange. The business is commonly carried on through bankers, who have credit in distant places, and sell drafts to persons wishing to make payments in those places. When a draft sells for its face value, exchange is said to be at par ; when a draft sells for more than its face value, exchange is above par, or at a premiuin ; when a draft sells for less than its face value, exchange is below par, or at a discount. Domestic or Inland Exchange. 142. Exchange between places in the same country is called domestic or inland exchange. I. Find the cost of a draft on Chicago for $1320 when exchange is l-|-% discount. 1320 1320. Qj^i 14.85 "^^^ discount on .$1 is $0.01|, and on 165^ S1S05 15 $1320 it is $14.86. Subtracting this from -4 QOA ' $1320, we find the cost of the draft to be ^"^"^^ $1305.15. 14.85 II. Find the cost of a draft on Omaha for f 1400, payable in 60 days, when exchange is J% premium, and interest 5%. mrl? ^^^^' The bank discount on $1400 for 60 da. .0105 12.25 ^^ 50/^ jg ^j^g gj^^g j^g jj^g interest for 63 da., which equals ^12.25, and the proceeds is $1387.75; this would be the cost of the draft if bought at par. At ^% the premium on $1400 is $7 ; adding this to $1387.75, we find the cost of the draft to be $1394.75. EXAMPLES. 1. What is the value of a sight draft on Buffalo for f 1800, when exchange is at a premium of 1 Ji^ ? 7000 1400 1387.75 7. 6)14.70 2.45 12.25 $1394.75 250 ARITHMETIC. 2. Find the cost of a sight draft on New York for $1300^ when exchange is 1J% premium. 3. Find the cost of a sight draft on Detroit for $840, when exchange is |-% discount. 4. What will be the cost of a draft for $3000, payable in 30 days after sight, exchange being 1 % discount, and in- terest 6% ? 5. Find the cost of a draft for $700, payable in 60 days, when exchange is at par, and interest 7%\ ' 6. What must be paid for a draft of $925, at 60 days, at 6%, exchange being -|% premium ? 7. What must be paid for a draft of $450 on New Or- leans, at 90 days, exchange being |% discount, and interest 8. What will be the cost of a draft for $750, payable in 60 days after sights exchange being ^% premium, and in- terest 7% ? 9. Find the cost of a draft on Baltimore for $1237.50, payable in 30 days after sight, exchange being -|-% discount, and interest 5%. To^ Find the Face of a Draft when the Cost is Given. 143. I. Find the face of a sight draft bought for $3559.50, when exc>>ange is 1^% discount. 1. .01125 -^^ ^¥^0 discount a sight draft .9887S> ^559.50000 ($3600 296625 • 693250 593250 00 for $1 would cost $0.98875. $3559.50 will buy a draft for as many dollars as .$0.98875 is contained times in $3559.50, which equals INTEREST AND DISCOUNT. 251 II. What is the face of a draft, payable in 90 dayf;', that can be bought for $2000, exchange being 1^% premiun^, and interest 4% ? 3 ). 0155 1.00461) 2000. The bank discount 005^2 3 3 >^ on $1 for 90 days at 'oios! 3.014 )6000.000($1990.71 4%i8.1i0.0103.',and the .yjxyjo^ 3014 proceeds is $0.P896§. 1 29860 ^^ ^^°/" premium the 01034 27126 ^**^* "^ '"^ "^""^^^ ^°^ Vsqfi! "27340 ^ is $1.0046^. 1^2000 :Olf* S will buy a draft for as many dollars as 1.0046i 21400 $1.0046? is contained ^^^^^ times in $2000, which 3020 is $1990.71. EXAMPLES. 1. What is the face of a sight draft that can be purchased for $2351.70, when exchange is ^% premium ? 2. How large a sight draft can be bought for $2500, ex- change being |% discount ? 3. What is the face of a sight draft bought for $1650, when exchange is 3^% discount ? 4. How large a draft, payable in 30 days after sight, can be bought for $4018, exchange being 1% premium, and in- terest 6% ? 5. How large a draft on Cincinnati, at par, at 30 days, can be 'bought for $1989, when money is worth 6% ? 6. What is the face of a draft, payable in 60 days after date, that can be bought for $2386.20, exchange being 1% premium, and interest 9% ? 7. Find the face of a draft on New York, at 90 days sight, bought for $450, exchange at 1|% premium, and interest 5^0 252 ARITHMETIC. 8. What is tlie face of a draft on St. Paul for 60 days which may be bought for f 1000, exchange being J% dis- count, and interest 7% ? 9. Find the face of a draft on Boston, at 90 days sight, bought for $75, exchange at*^% premium, and interest 4%. Foreign Exchange. 144. Exchange between places in different countries is called foreign exchange. Exchange with Europe is carried on principally through large commercial cities, as London, Paris, Antwerp, Geneva, Hamburg, Frankfort, Bremen, and Berlin. Sterling Exchange, as exchange with Great Britain and Ireland is called, is quoted at a certain number of dollars per pound sterling. Exchange with France, Belgium, and Switzerland is quoted at a certain number of francs per dollar. Exchange with Germany is quoted at a certain number of cents per four reichsmarks (marks). I. Find the cost of a bill of exchange on London for £326 16 s., when sterling exchange is quoted at 4.83|-. 326.8 4.83i 1634 ^326 16 s. equals £326.8. Since the value of £1 9804 is $4,831, the value of £326.8 is 826.8 times .$4.83|, 26144 which equals $1580.08. 13072 1580.078 Ans. $1580.08. II. Find the cost of a bill of exchange on Paris for 4730 francs, when Paris exchange is quoted at 5.15^. LNTEREST AND DISCOUNT. 258 5.155)4730.000(1917.56 46395 9050 Since 5.156 francs are worth 5155 |i, 4730 francs are worth as 38950 many dollars as 5.155 francs are 36085 contained times in 4730 francs, 28650 which equals $917.56. 25775 28750 III. Find the cost of a bill of exchange on Berlin for 2760 J narks, when German exchange is quoted at 96^. 2760 .244 o.^ Sjnce .$0.06 J is the Taliie of 4 marks, the value of 11040 ^ "^^^^ ^^ ^ °^ $0,961, or .$0.24J. The value of 2760 5520 marks is 2760 times I0.24J, wliich equals f666.85. $665.85 The face of a bill of exchange can be found when the cost is given by performing the reverse operation to that used in finding the cost when the face is given. EXAMPLES. 1. What must be paid in New York for a bill of ex- change on London for £725 10 s., when sterling exchange is quoted at 4.87:| ? 2. Find the cost of a bill of exchange on Dublin for £296 8 s. 6d., when sterling exchange is quoted at 4.85^. 3. What is the cost of a bill of exchange on Liverpool for £137 15s. 4d., exchange at 4.86 ? 4. Hov/ large a bill of exchange on Edinburgh can be bought for $2500, when sterling exchange is quoted at 4.87 ? 5. When exchange on London is quoted at 4.85, what will be the face of a draft that can be bought for $3889.70 ? 254 ARITHMETIC. 6. What is the face of a bill of exchange on Liverpool for which $4800 was paid, exchange at 4.84| ? 7. What is the cost of a bill of exchange on Paris foi 975 francs, exchange at 5.16 ? 8. Find the cost of a bill of exchange on Geneva for 1822 francs, exchange at 5.17. 9. Find the cost of a bill of exchange on Antwerp for 2025.25 francs, exchange at 5.17:|-. 10. What is the face of a bill of exchange on Paris bought for $2240.25, when Paris exchange is quoted at 5.15 ? 11. How large a bill of exchange on Geneva can be bought for $850, exchange at 5.16^ ? 12. When exchange on Paris is quoted at 5.21:|^, what will be the face of a draft that can be bought for $2046.50 ? 13. How much must be paid in Boston for a bill of ex- change on Hamburg for 2672 marks, exchange at 95 ? 14. Find the cost of a bill of exchange on Bremen for 1685.25 marks, exchange at 94|-. 15. Find the cost of a bill of exchange on Berlin for 1050 marks, exchange at 95|. 16. When exchange on Frankfort is quoted at 95^, what will be the face of a draft that can be bought for $2871.40 ? 17. Find the face of a bill of exchange on Hamburg cost- ing $892.76, when German exchange is quoted at 95^. 18. How large a bill of exchange on Berlin can be bought for $1500, exchange at 96| ? Equation of IPayments. 145. Equation of payments is the process of finding the time when several payments due at different times can all INTEREST AND DISCOUNT. 256 be paid at once without loss to either debtor or creditor. This time is called the equated time. I. A man owes $1200, of which $600 is due in 3 months, $400 in 5 months, and $200 in 6 months ; find the equated time of payment. 600 X 3 = 1800 The use of $600 for 3 mo. is equivalent 400 X 5 = 2000 to the use of $1 for 1800 mo. ; tlie use 200 X 6 = 1200 • of $400 for 5 mo. is equivalent to the use of 1200 )5000 ^^ for 2000 mo.; and the use of $200 for 6 mo. is equivalent to the use of $1 for 1200 mo. This amounts to the use of $1 for 5000 mo., which is equivalent to the use of $1200 for ^^W ^^ ^^^^ ™**'' which equals 4^ mo., or 4 mo. 6 da. 4|- mo. = 4 mo. 5 da. II. A merchant bought the following bills of goods : Jan. 15th, $600 on 2 months' credit ; Feb. 1st, $300 on 3 months' credit ; Mar. 25th, $550 on 30 days' credit ; and Apr. 8th, $400 on 60 days' credit. Find the equated time of payment. Mar. 15. 600 X = Write the dates on which May 1. 300 X 47 = 14100 the several payments are Apr. 24. 550 X 40 = 22000 due with the amounts op- June 7. 400 X 84 = 33600 posite. Take the earliest 1850 )69700(37.7 date. Mar. 15th, as a 6on- 5550 venient date from which to 14200 reckon (sometimes called Ans. Apr. 22nd. 12950 ^'^"^ ^«^^)- The periods ^ornrj of time, reckoning from Mar. 16th, are 0, 47, 40, and 84 days respectively. Computing as in the previous example, we find the equated time to be 38 days after Mar. 15th, which is Apr. 22nd. III. A man owes $2000 due in 8 months ; he pays $500 in 2 months and $800 in 3 months ; when in equity should he pay the balance ? 256 ARITHMETIC. 500 X 6 = 3000 $500 paid in 2 mo. is paid 6 mo, be. 800 X 5 = 4000 fore it is due, and its use is equivalent 2000-1300 = 700)7000 *« ^^^ "se of $1 for 3000 mo. To mo P^^^ ^" ^ ™^' ^^ P^^*^ ^ "^°" ^^f"''^ ^t is due, and its use is equivalent to the use of $1 for 4000 mo. ; tliis amounts to the use of $1 for 7000 mo. To offset these payments made before maturity, the balance of $700 can be retained after maturity for -^ of 7000 mo., which equals 10 mo. EXAMPLES. 1. A man owes $300 due in 4 months, and $600 due in 7 months ; find the equated time of payment. 2. What is the equated time for paying $20 due in 20 days, $60 due in 30 days, $40 due in 50 days, and $80 due in 75 days ? 3. A man buys a house for $2500, and agrees to pay $500 down, and the rest in 4 equal annual instalments; when can he justly pay the whole at once ? 4. A merchant owes $2400, of which $400 is payable in 6 months, $800 in 10 months, and $1200 in 16 months ; what is the equated time of payment ? 5. Find the equated time for the payment of $400 due in 30 days, $250 due in 60 days, and $200 due in 90 days. 6. Of a debt, ^ is to be paid in 2 months, ^ in 6 months, ^ in 10 months, and the balance in a year. Find at what time in equity the whole should be paid if all the pay- ments were converted into one. 7. A debt is to be paid ^ down, :^ in 6 months, |- in 8 months, and the balance in a year ; if the payments are all converted into one, what is the equated time of payment ? 8. Three bills are due as follows : Sept. 5th, $275 ; Oct. 1st, $180 ; and Nov. 20th, $350. Find the equated time of payment. INTEREST AND DISCOUNT. 257 9. What is the equated time for the payment of $170 due Mar. 12th, f 250 due Apr. 12th, 11280 due May 17th, and $325 due June 12th ? 10. I owe three notes bearing interest from date : the first, dated Jane 1st, 1886, is for $450 ; the second, dated Dec. 17th, 1886, is for $750; the third, dated Mar. 15th, 1887, is for $600. I wish to substitute for these a single note for $1800; what should be the date of it ? 11. A merchant bought goods on 6 months' credit as fol- lows: Mar. 20th, $420; May 3rd, $270; and June 12th, $340. When shall a note to settle for the whole be made payable ? 12. Find the equated time of payment for the following bills of merchandise : Oct. 10th, 1887, $625 on 60 days' credit ; Nov. 1st, 1887, $314 on 3 months' credit ; and Jan. 4th, 1888, $266 on 30 days' credit. 13. A merchant bought the following bills of goods : Dec. 23rd, 1887, $428 on 90 days' credit ; Jan. 17th, 1888, $206 on 2 months' credit ; Feb. 3rd, 1888, $90 on 30 days' credit ; and Feb. 8th, 1888, $214 on 60 days' credit. Find the equated time of payment. 14. A man bought a horse and carriage for $500 on 6 months' credit ; if he pays $200 down, when should he pay the balance ? 15. A man owes $1600 due in 9 months ; he pays $300 in. 4 months, $200 in 6 months, and $300 in 8 months; when is the balance due ? 16. A man owes $600 due in 6 months, $900 due in 10 months, and $1200 due in 12 months; at the end of 8 months he pays $1800; when in eo.uity should the re- mainaer be paid? 258 ARITHMETIC. 17. On a debt of $5000 due in 8 months from Jan. 1st, the following payments were made : Apr. 1st, $500 ; June 1st, $600 ; and Aug. 1st, $1000. When is the balance due ? Average of Accounts. 146. Average of accounts is the process of finding the time when the balance of an account can be paid without loss to either debtor or creditor. I. Find the equated time for paying the balance of the following account : Dr. B. R. Harvey. Cr. 1888. 1888. Aug. 2 To Mdse., 30 da. 1400 Aug. 4 By Draft, 60 da. $200 " 29 « (( 350 " 31 " Cash. 300 Sept. 7 « " 2 mo. 250 Sept. 8 (( « 400 Solution. Sept. 1. Aug. 29. Nov. 7. XXX 3 = = 70 = : 1200 = 17500 Oct. 6. 200x38= 7600 Aug. 31. 300 X 2= 600 Sept. 8. 400 X 10 = 4000 1000 900 18700 12200 900 12200 100 )6500 65 da. Ans. Nov. 2nd. Following the method of equation of payments, taking Aug. 29th for the focal date, we find the amount of the debtor side of the account to be $1000, equivalent to the use of $1 for 18700 days. We find the amount of the creditor side to be $900, equivalent to the use of $1 for 12200 days. The balance on the debtor side is $100, equivalent to the use of $1 for 6500 days, and the equated time is 65 days after Aug. 29th, or Not. 2nd. Note. In determining tne maturity of a note or draft, 3 days of grace must be added to the specified time. INTEREST AND DISCOUNT. 259 When the balance of the account and the difference be- tween the sums of the products fall on the same side, the result is of the same nature as a result in equation of pay- ments, and the equated time is later than the focal date. When the balance of the account and the difference between the sums of the products fall on opposite sides, it is readily- seen that an earlier focal date could be taken which would give the same sum of products on each side, and thus this date is the equated time ; hence the equated time is earlier than the focal date regularly taken. The method may be stated as follows : Write each item with its date of maturity on the respective sides of the account, and take as the focal date the earliest date of maturity. Multiply each item by the number of days intei-vening between the focal date and the date of maturity, and find the sums of these products on each side of the account. Divide the differ- ence between the sums of the products by the balance of the account, and the quotient is the number of days between the focal date and the equated time. When the balance of the account and the difference between the sums of the products fall on the same side, count forward from the focal date; ivhen they fall on opposite sides, count backward, EXAMPLES. 1. Find the equated time for paying the balance of the following account : Dr. M. P. Bartlett. Cr. Apr. 20 May 10 To Mdse., 30 da. .$520 135 May 15 By Cash. $600 260 ARITHMETIC. 2. Find the time when a note for the balance of the fol- lowing account should begin to draw interest : Dr. R. J. Miner. Cr. 1888. Mar. 1 June 8 To Cash. " Mdse. 11500 235 1888. May 14 July 10 By Mdse. " Real Estate. 12050 145 3. Find the equated time for the payment of the balance of the following account : Dr. J. H. Adams. Cr. 1888. 1888. May 10 To Mdse., 30 da. $420 May 4 By Draft, 30 da. $750 June 15 « « 380 June 12 " Cash. 400 " 20 « « 450 4. Find the equated time for the settlement of the fol- lowing account ; Dr. B. P. Harper. Cr. 1888. 1888. Jan. 10 To Mdse. $672 Jan. 28 By Cash. $475 Eeb. 7 " 30 da. 428 Apr. 10 " Mdse. 462 " 24 « 2 mo. 550 May 18 " Cash. 250 5. Find the face of a note that will balance the following account, and the date at which it should begin to draw interest : Cr. Dr. A. r. Brackett. 1887. 1887. Sept. 14 To Mdse., 30 da. $1950 Nov. 19 By Cash. $750 Oct. 16 « 3 mo. 532 Dec. 1 1888. " Draft, 60 da. 1000 " 20 2 mo. 1178 Feb. 4 " Cash. 600 STOCKS. 261 CHAPTER XL STOCKS. 147. A corporation is an association of individuals au- thorized by law to transact business as a single person. The capital invested in the business is called stock, and it is divided into equal parts called shares. The owners of the shares are called stockholders, each of whom holds a document called a certificate of stock, which is issued by the corporation and specihes the number of shares owned. The usual value of a share is ^100, although it varies in different corporations. In this book it will be regarded as f 100, unless otherwise stated. The original or face value of a share is called the par value, and the value at which it sells is called the market value. When shares sell for their face value, they are said to be at par ; when they sell for more than their face value, they are above par, or at a premium ; when they sell for less than their face value, they are below par, or at a discount. The market value is quoted at a certain per cent of the par value. For example, when stock is at par, it is quoted at 100 ; when it is 8% above par, it is quoted at 108 ; when it is 15% below par, it is quoted at 85. Stocks are generally bought and sold through the agency of brokers, who receive a commission, called brokerage, reckoned on the par value of the stock. The usual rate of brokerage is -|%, but other rates may be charged. A dividend is a sum paid to stockholders from the profits of the business. An assessment is a sum sometimes required of stockholders to meet losses or pay expenses. Dividends 262 ARITHMETIC. and assessments are generally reckoned at a certain per cent of the par value. Dividends are usually declared an- nually, semi-annually, or quarterly, and the rate per cent is called the rate of dividend. Bonds are interest-bearing notes issued by governments or corporations ; they are bought and sold in the same man- ner as shares of corporations. Bonds are commonly desig- nated according to the rates of interest which they bear. For example, Virginia 6's are bonds issued by the state of Virginia bearing 6% interest. The methods of percentage apply to stocks. The par value of the stock is the base, and the premium, discount, dividend, or assessment is a percentage of the par value. I. Eind the cost of 32 shares of railroad stock at 8^% discount. 32 16 At 8^% discount the cost of one share is $91 J, and 182 the cost of 32 shares is 32 times 91^, or |2928. 273 $2928 II. How much, including brokerage at ^%, must be paid for $15000 U.S.4's at llli? 15000 7500 1875 If the brokerage is |%, 111| -|- 1^, or lllf, represents 15000 the price paid. lllf% of $16000 is $16743.75. 15000 15000 $16743.75 III. How much bank stock at 131|^ must be sold in order to receive $4725, brokerage ^% ? STOCKS. 1,3126) 4725.0000 ($3600 If the brokerage is \%, 131^ - \, or 39375 131 1 , represents the price received. 78750 $4726 is 134% o^ ^^e amount obtained 78750 by dividing $4726 by 1.3126, which is 7^ $3600. IV. Find the quoted price of stock when 15 shares cost $1886.25. 1 p;M SSA 9Pi ^^ ^^ shares cost $1886.26, one share ToK n _ OK8 *^°*** ^'^ ^^ $1886.25, which is $126.76. 125.75 _ 125f . jj^,n^,g t,,g quoted price is 126J. V. A man owns $5000 of the stock of a railroad which declares a dividend of 4%; what is the amount of his dividend ? 5000 .04 4% of $6000 is $200. $200.00 VI. How much 8% stock must be bought to yield an income of $3000 ? .08 )3000.00 ^3QQQ jg go, of $37600. $37500 Note. In this book the brokerage is included in the price of a stock, unless otherwise stated; hence no account should be taken of broker- age when it is not mentioned in the problem. EXAMPLES. 1. How much must be paid for $8500 Iowa 6's at 112|? 2. What is the cost at 63^ of stock having a par value of $2800 ? 3. A man bought 28 shares of railroad stock at 18% discount ; what did they cost him ? 4. Find the cost of 36 shares of bank stock at 121^, brokerage ^%. 264 ARITHKETIC. 5. How much, including brokerage at ^%, must be paid for $3500 Tennessee &s at 88^ ? 6. How much will be received from the sale of $11200 U. S. 3^'s at 107f, brokerage ^% ? 7. A speculator bought 45 shares of stock at 4|^% dis- count, and sold it at 2^% premium ; what was his gain ? 8. A broker bought 84 shares of railroad stock at 19% discount. He sold 35 shares at 27^% discount, and the balance at 8% discount. Did he gain or lose, and how much? 9. How many shares of stock at 78% premium can be bought for $9790 ? 10. How much stock can be bought for $14178, when the quoted price is 208| ? 11. What amount of Union Pacific bonds at 104|- can be bought for $7837.50 ? 12. Find the number of shares of bank stock at 105 that can be bought for $25260, including brokerage at ^%. 13. A broker receives $3762.50 to invest in stocks at $75 per share and cover his brokerage at |-%. How many shares should he purchase ? 14. How much canal stock must be sold at 136f in order to receive $6552, brokerage i% ? 15. A broker received $10.50 for selling stock at 122-^ j how many shares did he sell, brokerage ■^% ? 16. I sent $40100 to a broker for the purchase of bank stock at par. If the brokerage is :^%, what does he pay for the stock, and what is his brokerage ? 17. A man exchanged 72 shares of bank stock at 85 for railroad stock at 136 ; how many shares of railroad stock did he receive ? STOCKS. 265 18. A speculator bought stock at 1^% discount, and gained $495 by selling the same at 6% premium; how many shares did he purchase ? 19. Bought bonds at 115, and sold at 110, losing f 300. How many bonds of f 1000 each did I buy ? 20. Find the quoted price of stock when 35 shares cost $2931.25. 21. What must be the quoted price in order that $6800 stock may be bought for $2941 ? 22. How should U. S. 4^'s be quoted when $10093.75 is paid for bonds having a par value of $8500 ? 23. When the cost of $4500 telegraph stock, including brokerage at ^%, is $7380, what is the quoted price ? 24. Find the quoted price of railroad stock when the cost of 250 shares, including brokerage at -J^, is $30312.50. 25. Find the quoted price of bank stock when $10175 is received for 110 shares, brokerage i%. 26. A man bought stock at 115, and sold the same at 128J ; what per cent of the investment did he gain ? 27. If I buy railroad stack at 20% discount, and sell at 10% premium, what per cent do I gain? 28. A railroad declares a dividend of 3^% ; how much will a man owning 48 shares receive ? 29. The capital of a manufacturing company is $300000, and it declares a semi-annual dividend of 4% ; find the en- tire amount of the dividend. 30. An insurance company calls an assessment of 2f % to meet losses ; how much is the assessment on $7200 stock ? 31. A man owns 150 shares .of mining stock, and the company declares a dividend of 6% payable in stock ; how many shares will he then own ? 266 ARITHMETIC. 32. Find the total income from $4000 9% stocks and 17800 7% stocks. 33. How much 7% stock must a man own in order to receive an income of $4200 ? 34. If a person receives $360 when a 4J% dividend is declared, how many shares, $50 each, does he own ? 35. The net earnings of a corporation are $2625, from which a dividend of 6J% is declared ; find the capital. 36. Mnd the number of shares owned by a person after re- ceiving 12 shares when a stock dividend of 15% is declared. 37. Find the rate of dividend when a man owning 52 shares receives $182. 38. A company with a capital of $325000 calls an assess- ment of $4875 ; what is the rate ? - 39. A company, whose capital is $275000, has $15125 from its earnings to divide. What per cent dividend can it declare ? 40. The capital of a company is $175000 ; the gross re- ceipts are $35930, and the expenses are $19205 ; find the rate of dividend that can be declared after reserving a sur- plus of $2725. 148. In making investments, it is necessary to consider both the market value of a stock and the rate of dividend. I. What income will be realized from investing $10650 in 8% stock at 142? 142)10650.00(7500 7500 J)94_ -08 If .$10650 is invested at 142, 710 $600.00 the par value of the stock is 710 $7500 ; 8% of $7600 is $600. 00 STOCKS. 267 II. What sum must be invested in 6% bonds at 92^ to yield an income of $1500 ? .06 )1500.00 25000 •92^ To yield an income of $1500, the par 12500 value of the stock must be .$25000, and this 50000 amount of stock at 92^ will cost $23126. 225000 $23125.00 III. When 9% stock is quoted at 192, what rate of inter- est does the investment pay ? 64)3.00(.04i^= 4|J% The cost of a share is $192, and the in- 256 come is $9, which is yf j, or ^\^%, of the 44 11 cost. 64""l6 IV. What is the quoted price of a 6% stock which pays 4|% interest on the investment ? .04J)6.00 The income is $6, which is 4|% of the 9 9 market value ; hence the quoted price is as .40)54.00 much as 0.04| is contained times in 6, which 135 is 135. EXAMPLES. ^ 1. What annual income would a man receive from $9810 invested in railroad stock at 109, and paying 5^/o dividend ? 2. What income will $10120 yield if invested in 4% bonds bought at 115 ? 3. What income would a man receive from $9525 in- vested in Mexican Central 4's at 63^ ? 4. How much will be realized yearly from an invest- ment of $7620 in a 5% stock bought at 95, brokerage i% ? 268 ARITHMETIC. 5. A man invests $11459 in telephone stock at 204|-, paying ^% brokerage. What will he receive when a divi- dend of 5% is declared ? 6. How much must be invested in 8% stock at 170f to afford an income of |2000 ? 7. What sum must I invest in 6% bonds, selling at 2|-% premium, to secure an annual income of $840 ? 8. How much mu^t be invested in a stock at 213|-, which pays 5% semi-annual dividends, to realize an annual income of $420 ? 9. What sum must be invested in U. S. 4's at 121^, brokerage ^%, to secure an annual income of $700 ? 10. When Wisconsin Central 5's are selling at 85|-, how much must be invested to produce an income of $750, bro-. kerage i% ? 11. If a 6% stock is at 120, what rate per cent will an investor receive on his money ? 12. Bank stock, which sells at 170, pays an annual divi- dend of 12^% ; what rate of interest does a buyer receive ? 13. Keceived 6% dividend on stock bought at 25% below par ; what rate of interest did the investment pay ? 14. Stock bought at 20% below par paid 7% ; what was the rate on the investment ? 15. What .per cent of income does stock paying 8% divi- dends yield, if bought at 168^ ? 16. Which is the better investment, a 4% stock at 120, or a 5% stock at 166f ? 17. Which is the more profitable stock to invest in, 3% at83i, or3i% at 97 ? STOCKS. 269 18. "Wnich is the more profitable, 88400 invested in 5 per cents at 105, or in 7 per cents at 150 ? 19. Which will yield the better income, a 4% stock at 73, or a 7% stock at 126f, brokerage ^% in each case ? 20. At what price must I purchase 8% stock that the investment shall pay 5% ? 21. What must be paid for 7% bonds that the invest- ment may yield 6% ? 22. A bank declares a semi-annual dividend of 4% ; what could I afford to pay for its shares if I wish to get 6% a year for my money ? 23. At what price must a stock paying 6% dividends be bought to pay the same income as an 8% stock at par ? 24. If money is worth 3%, what is the premium on gov- ernment 3^% bonds ? 25. If I invest $1500 in 3% stock at 75, what is my in- come, and what rate per cent do I get on my investment ? 26. If a man invests $1338 in bank stock at 167^, what is the rate of dividend when he receives $120 ? 27. A 5% stock pays a dividend of $510; if it is sold for $11985, what premium is paid ? 28. What must be the price of a 5% stock in order to yield the same rate of income as a 4% stock at 87 ? 29. When stock is quoted at 120, what rate of dividend must be paid in order to yield the same rate of income as a 6% stock at 144? 30. A man having a certain sum of money to invest has an opportunity of purchasing 7% stock at 95, but delays until it has risen to 110. What per cent is his income less than if he purchased at the first price ? 270 ARITHMETIC, SI. A man seUs $10000 3^% bonds at 109^ and rein- vests the proceeds in 3% bonds at 92; is his income in- creasecf or diminished, and by what amount ? 32. If a man sells $4000 6% bonds at 113f, and invests the proceeds in 4|^% bonds at 91, is his income increased or diminished, and by what amount ? 33. A man sold $6000 of 6% stock at 144|-, and invested the proceeds in 8% stock at 170. How much 8% stock did he buy, and what was the change in his income ? 34. How much 3|-% stock must I sell at 84, to enable me to buy $7700 4% stock, the value of the stock being pro- portional to the dividends they pay ? 35. If I exchange 48 shares of a 9% stock at 176 for U. S. 4's at 116^, how much must I add to my investment to secure the same income ? 36. If I sell $5000 Alabama 6's at 132 and buy sufficient U. S. 4-^'s at 108 to secure an income of $225, how much shall I have left, brokerage ^% for each transaction ? 37. A man has a certain sum of money to invest. He finds that by buying 5% stock at 90 his income will be $10 more than if he bought 8% stock at 150. How much money has he to invest ? 38. A man sold $4500 of 9% stock at 172^, and invested the proceeds in 4% stock, thereby increasing his income by $55. Find the price of the 4% stock. 39. By selling his 6% stock at 147, and investing the proceeds in a 5% stock at 96^, a man increases his income by $54. How much 6% stock did he sell ? 40. A person sells a certain amount of 5% stock for 86, and invests in 6% stock at 103, and by so doing changes his income by $1. Is the change an increase or decrease ? How much stock does he sell ? INVOLUTION AND EVOLUTION. ^71 CHAPTER XII. INVOLUTION AND EVOLUTION. 149. A power of a number is the number itself, or the product obtained by taking the number several times as a factor. A root of a number is one of the equal factors of that number. A power or root receives its name from the number of equal factors. For example, 3^ = 3. 3 is the first power of 3. 3^ = 9. 9 is the second power, or square, of 3 ; and 3 is the second root, or square root, of 9. 3* = 27. 27 is the third power, or cube, of 3 ; and 3 is the third root, or cube root, of 27. 3* = 81. 81 is the fourth power of 3 ; and 3 is the fourth root of 81. The radical sign, V, indicates a root. The name of the root is indicated by a small figure placed in the opening of the sign, called the index of the root. In expressing square root, the radical sign is generally used alone. For example, V49, or V49, denotes the square root of 49 ; V256 denotes the fourth root of 256. Note. A root may also be indicated by a fractional exponent. For example, 25' denotes the square root of 25 ; 64^ denotes the cube root of the square of 64. Involutiojq^. 150. The process of finding a power of a number is called volution. involution, 272 AKITHMETIC. I. Find the fourth power of 6. The fourth power of 6 is the product 6^ =6x6x6x6 = 1296. ^f f^^j. factors, each equal to 6, which equals 1296. EXAMPLES. 1. What is the square of 11? of 0.11? 2. Find the square of 0.9 ; of three millionths. 3. What is the third power of 0.1? of 100? 4. What is the third power of 3 ? of 0.3 ? of 0.03 ? of 30 ? 5. Find the cube of 10.1; of 1.01. 6. What is the cube of |? of 0.006 ? 7. What is the fourth power of 2? of 0.2? of 0.02? 8. Find the fifth power of 5 ; of 50 ; of 0.5. Note. The student will find it advantageous to commit to memory the squares of all the numbers from 1 to 25 inclusive, and the cubes of all the numbers from 1 to 10 inclusive. Evolution. 151. The process of finding a root of a number is called evolution. When the exact root of a number can be found, the number is called a perfect power ; all other numbers are imperfect powers. The roots of perfect powers can readily be found by factoring. I. Find the cube root of 9261. 3 )9261 3 )3087 3)1029 n\oAo The prime factors of 9261 are S^XT^; hence the i^^ cube root of 9261 is 3 X 7, or 21. 7 )49 T u 3 X 7 = 2i, INVOLUTION AND EVOLUTION. 273 EXAMPLES. 1. Find the square root of 3136. 2. Find the square root of 5184. 3. Find the square root of 11025. 4. Find the cube root of 32768. 5. Find the cube root of 91125. 6. Find the cube root of 456533. 7. Find the fourth root of 331776. 8. Find the fourth root of 1185921. 9. Find the fifth root of 1889568. 10. Find the sixth root of 2985984. Square Root. 152. To obtain a general method for finding the square root of numbers, we must investigate the relations between simple numbers and their squares. The first step in extracting the square root of a number is to determine the number of figures in the root. V = 1, 102 ^ 100, 1002 ^ 10000, 10002 ^ 1000000, and so on. Hence the square of any number between 1 and 10 is a number between 1 and 100, the square of any number between 10 and 100 is a number between 100 and 10000, the square of any number between 100 and 1000 is a number between 10000 and 1000000, and so on. Thus we see that the square of a number contains twice as many figures as the number, or twice as many less one. If, therefore, a number be sep- arated into periods of two figures each by placing a dot over every alternate figure, beginning with the units' figure, the number of figures in the root equals the number of periQiis. 2T4: ARITHMETIC. Note 1. The left-hand period has but one figure when the numbei' consists of an odd number of figures. Note 2. The principle applies also to decimals, because the s juare of a decimal contains twice as many decimal places as the decimal itself. 153. The component parts of the square of a number of two figures may be learned from the following multiplica' tion: 47 = 40 + 7 47= 40 + 7 329 = 40 X 7 + V 1880 = 40^+ 40x7 472 = 2209 = 402 + 2 X (4Q x 7) + 7^ In general, the square of any number composed of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. I. Find the square root of 5329. 5329^70 4-3 = 73 Since the number consists 4900 of two periods, the square root 140 -+- S = 143'>429 ^^^^ consist of two figures. 429 ^^^ square of the tens' figure of the root must be the largest ^ 7 . ,T square in 53 hundreds, which Or more oriefly • At^-u a a u *i * ^ ^ IS 49 hundreds ; hence the tens 5329(73 figure of the root is 7. Sub- 49 tracting 4900, the square of 70, 143)429 from 5329, the remainder is , 429 429. Since the square of the. tens has been subtracted, 429 equals twice the product of the tens by the units plus the square of the units ; this is the same as twice the tens plus the units multiplied by the units. Thus the two factors of 429 are twice the tens plus the units and the units. We can find the units' figure by dividing 429 by the other factor; however, this factor is known only in part, so we take twice the tens, tlie part known, as a trial divisor, and we find that 140 is contained in 429 three times. The units* figure, therefore, fs 3, and the complete divisor is 140 + 3, or 143. Multiplying 143 by 3, we obtain INVOLUTION AND EVOLUTION. 275 429, and there is no further remainder. Hence the square root of 5329 is 73. In the shorter arrangement of work, 14 may be considered as the trial divisor, and the units' figure is found by dividing 42 by 14, which gives the same result as dividing 429 by 140. If the number consists of more than two periods, after finding the first two figures of the root, we can consider them as tens in reference to the next figure, and then pro- ceed as before. At each stage of the work the trial divisor is obtained by doubling that part of the root already found. II. Find the square root of 6796449. 6796449(2607 ^ After finding 26 in the root, the trial di- 4.fi^97Q visor is 52; 52 is not contained in 30, and the (yra. next figure of the root is 0. Then the trial K9n7 \ divisor is 520, which can be used at once by bZm) ^^^49 bringing down another period. III. Find the square root of 2.5 to four decimal places. 2.50(1.5811 1 25)150 125 308)2500 2464 3161)3600 3161 A zero must be annexed to 5 to complete the first period after the decimal point. Other periods of two zeros each can be brought down as they are needed. 31621)43900 The square root of a fraction in its lowest terms may be obtained by taking the square root of both terms when they are perfect squares. For example, the square root of ff is f ; the square root of 7^ is the same as the square root of ^-, which equals |, or 2|. When either term is not a per- 276 ARITHMETIC. feet square, reduce the fraction to a decimal and then ex> tract the root. When the denominator of a fraction is the square root of a number, the work may be simplified by first multiplying boHh. terms of the fraction by the denominato-r. For ex- EXAMPLES. Find the square root (to five decimal places when the number is not a perfect square) of 1. 676. 2. 1681. 3. 624100. 4. 46656. 6. 6.7081. 6. 49.2804. 7. 747.4756. 0.005625. 1361610000. 4.190209. 3444736. 12. 0.05331481. 37. Find the value 13. 7.333264. 14. 39.037504.' 15. 0.9. 16. 0.001. 17. 0.196. 18. 530. 19. 3369. 20. 79000. 21. 0.002539. 22. 0.01952. 23. 102.002. 24. 0.001601. 25. AV 27. 4. A- 28. 29. 30. 31. 32. 33. 5|. 34. 24J. 35. 42f 36. 201^. f 30i. 22tV »'aS 23 .625 to four decimal places. 38. Find the value of V(1.06)^ to five decimal places* 39. Find the value of VJ to three decimal places. 40. Find the value of — to three decimal places. V3 41. Calculate the value of \3 + 2 V2 to two decimal places. ■ INVOLUTION AND EVOLUTION. 277 42. Multiply 3.15 by 0.075, and extract the square root of the product to three decimal places. 43. Multiply 903.14 by 0.063, and extract the square root of the product to three decimal places. 44. Divide 3.63 by 2.353, and find the square root of the quotient to three decimal places. 45. Extract the square root of 0.875 -*- 2.63 to three deci- mal places. 46. Multiply V2 by V0.12S, and carry the result to three decimal places. 47. The area of a square is 655.36 sq. ft. ; what is the length of its side ? 48. If a square field contains 10.24^, find the length of its side in meters. 154. Square root can also be explained by the aid of diagrams. The area of a square surface is found by squaring the length of one side; hence the length of one side may be found by extracting the square root of the number denoting the A E B area. Let ABCD represent a square contain- ing 676 square units ; we wish to determine the length of one side. ^ ^ Since the number denoting the area consists of two periods, the square root M N will consist of two figures. The square of \ h \ c \d\ the tens' figure of the root must be the P largest square in 6 hundreds, which is 4 hundreds; hence the tens' figiu-e of the 676(26 root is 2, and the length of a side of 4 the square is 20 plus the units' figure of 46)276 *^^ '°^*- ^^' '^^ ^^ ^^ vimis in length; 276 then a represents a square whose area is 400. Subtracting this from 676, we find the area of the irregular figure EBCDFG to be 276. This figure con- a G b c d 278 ARITHMJJTIC. sists of the two rectangles b and c and the square d; arranging them as in MNOP, we have a rectangle whose width is the units' figure of the root, and whose length is 40 plus the units' figure of the root. When the area and length of a rectangle are known, the width can be found by dividing the area by the length ; in this case there is a small part of the length unknown, so we take 40 as. a trial divisor, and by dividing 276 by 40, we obtain 6 for the units' figure. The entire length, then, is 46, and the breadth is 6. The product of 46 and 6 is 276, and there is no further remainder. Hence 26 is the square root of 676. Cube Koot. 155. A general method for finding the cube root of num- bers can be obtained by pursuing lines of investigation similar to those made use of in determining the method for square root. Since 13 = 1, 10^=1000, 100^=1000000, and so on, the cube of any number between 1 and 10 is a number between 1 and 1000, the cube of any number between 10 and 100 is a number between 1000 and 1000000, and so on. Thus we see that the cube of a number contains three times as many figures as the number, or three times as many less one- or two. If, therefore, a number be separated into periods of three figures each by placing a dot over every third figure, beginning with the units' figure, the number of figures in the root equals the number of periods. Note 1. The left-hand period may contain but one or two figures. Note 2. The principle applies also to decimals, because the cube of a decimal contains three times as many decimal places as the decimal itself. 156. The component parts of the cube of a number of two figures may be learned from the following multiplication : 472= 2209= 402 + 2 X (40 x7) + 72 47= 40 + 7 15463 = - 402 X 7 + 2 X (40 X 72) + 73 8836 = 40" + 2x (40^x7)+ 40 x 7^ 47» = 103823 = 40» + 3 X (40^ x 7) + 3 x (40 x V) + 7» INVOLUTION AND EVOLUTION. 279 In general, the cube of any number composed of tens and units is equal to the cube of the tens, jflus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. I. Find the cube root of 175616. rj(j Since the number consists of two pe- riods, the cube root will consist of two figures. The cube of the tens' figure of the root must be the largest cube in 175 thousands, which is 125 thousands; hence the tens' figure of the root is 5. Subtracting 125000, the cube of 60, from 175616, the re- mainder is 60016. Since the cube of the tens has been subtract- ed, 6061^ equals three times the product of the square of the tens by the units, p'ms three times the product of the tens by the square of the units, plus the cube of the units ; this is the same as three times the square of the tens plus three times the product of the tens by the units plus the square of the units multiplied by the units. Thus the two factors of 50610 are three times the square of the tens plus three times the product of the tens by the units plus the square of the units and the units. We can find the units' figure by dividing 50616 by the other factor ; however, this factor is known only in part, so we take three times the square of the tens, the part known, for the trial divisor, and we find that 7500 is contained in 50616 six times. The units' figure, therefore, is 6, and the complete divisor is 7500 -1- 3 X 60 X 6 -f- 6^, or 8436. Multiplying 8436 by 6, we obtain 50616, and there is no further remainder. Hence the cube root of 175616 is 56. If the number consists of more than two periods, after finding the first two figures of the root, we can consider 175616(50 -f 6 125000 3x502 = 7500 3 X 50 X G = 900 6^= 3G 50616 Or 8436 50616 more briefly 17561( 125 3(56 3 7500 900 36 5061( 8436 5061( 5 280 ARITHMETIC. them as tens in reference to the next figure^ and then pro- ceed as before. II. Find the cube root of 22069.810125. 22069.810125(28.05 After finding 28 in the root, the trial divisor is 3x2802, or 235200. This is not contained in 117810, so the next figure of the root is 0. Then the trial divisor is 23520000, which can be used at once by bringing down another period. The quotient obtained by dividing the dividend by the trial divisor often proves to be too large ; in such a case try a smaller figure in the root. When there is a remainder after using all the periods, further figures in the root can be obtained by continuing the process, bringing down periods of three zeros each. The cube root of a fraction in its lowest terms may be obtained by taking the cube root of both terms when they are perfect cubes. When either term is not a perfect cube, reduce the fraction to a decimal and then extract the root. 1200 14069 480 64 1744 13952 23520000 117810125 42000 25 23562025 117810125 EXAMPLES. Find the cube root (to four decimal places when the num- ber is not a perfect cube) of 1. 4913. 2. 24389. 3. 250047. 4. 636056000. 5. 96702.579. 6. 8365.427. 7. 0.000032768. 8. 0.001295029. 9. 0.000000148877. 10. 225866529. 11. 1027243.729. 12. 12000.8121619. INVOLUTION AND EVOLUTION. 281 25. f 26. tt. 27. 15f. 28. 12f 29. 46f. 30. 81,^. l.t 0.27. 19. 0.61. 14. 10. 20. 51. 15. 1.025. 21. 1729. 16. 3.7. 22. 9358. 17. 0.0093. 23. r%- 18. 18.65. 24. ,7_2_9 1728* 31. Find the Vcilue of \(7.^j-^ to three decimal places. \U.li6U 32. Find the value of yjz^ '^Tr^^ *^^®® decimal places. 33. Find the value of -v^ (1.05 V to three decimal places. 34. Find the value of y5 4-2\/6 to two decimal places. 35. Multiply 2.49 by 22.32, and extract ^he cube root of the product to two decimal places. 36. Divide 3.15 by 0.075, and extract the cube root of the quotient to two decimal places. 37. Divide 6 by 0.89, and extract the cube root of the quotient to two decimal places. 38. Multiply 108 billionths by two thousand, and extract the cube root of the product. 39. Multiply -y/i by ^/0.456, and carry the result to two decimal places. 40. Find the difference between the sum of the cube roots of 32768 and 0.000512 and the cube root of their sum. 41. A cubical vessel contains 1331' ; what is ^he lenjth of its edge in meters ? 42. A cubical block contains 12695.24 ca. in. ; fiiid th^ lenath of one side. 282 ARITHMETIC. /'X4 / / d 9 C / h / Fia. 1. Fig. 2. Fig. 13824(24 8 157. Cube root can also be explained by the aid of dia- grams. The solid contents of a cube are found by cubing the length of an edge; hence the length of an edge may be found by extracting the cube root of the number denoting the solid contents. Let Fig. 1 represent a cube con- taining 13824 cubic units ; we wish to determine the length of an edge. Since the number denoting the cubic contents consists of two pe- //y/i riods, the cube root will consist of y a X h X '^yYcrr ? ^^^ figures. The cube of the tens' ^ figure of the root must be the largest cube in 13 thousands, which is 8 thousands ; hence the tens' figure of the root is 2, and the length of an edge of the cube is 20 plus the units' figure of the root. Let Fig. 2 repre- sent a cube whose edge is 20 units ; then its volume is 8000. Subtracting ^ this from 13824, we have left an irregular solid whose volume is 5824. This ^rfegular solid consists of the three rectangular solids a, h, and c, th*. three smaller rectangular solids d, e, and f, and the cube g. Arranging them as in Fig. 3, we have a series of solids which have for their common thickness the units' figure of the root. The bases of a, 6, and c are each 20^, or 400; the bases of d, e, and/ are each tlie product of 20 and the units' figure of the root ; and the base of g is the square of the units' figure of the root. When the volume and base of a rectangu- lar solid are known, the thickness can be found by dividing the volume by the base ; the common thickness of the solids in Fig. 3 can be found by dividing their entire volume by the sum of their bases, a, b, and c are the only solids whose bases are known, so we take the sum of these bases, 3 X 20^, or 1200, for the trial divisor, and by dividing 5824 by 1200, we have 4 for the units' figure of the root. The sum of the bases, then, is 3 X 202 + 3 X 20 X 4 -f 42, or 1456. The product of 1456 and 4 is 5824, and there is no further remainder. Hence 24 is the cube root of 13824. 1200 240 16 1456 5824 5824 IKVOLUTIOK AND EVOLUTION. 288 Higher Roots. 158. By means of the processes of square and cube root, we can find any root whose index contains only the factors 2 and 3. For example,, the fourth root of a number may be found by taking the square root of the square root; the sixth root of a number may be found by taking the square root of the cube root or the cube root of the square root. EXAMPLES. 1. Find the fourth root of 1874161. 2. Find the fourth root of 8.25 to tliree decimal places. 3. Find the sixth root of 1291467969. 4. Find the sixth root of 0.184 to three decimal places. 5. Find the eighth root of 2562890625. 6. Find the eighth root of 5 to three decimal places. 7. Find the ninth root of 134217728. 8. Find the ninth root of 3^ to two decimal places. 9. Find the twelfth root of 13841287201. 10. Find the twelfth root of 0.75 to two decimal places. 284 ARITHMETIC. CHAPTER XIII. SERIES. 159. A series is a succession of numbers, each of which is derived from one or more of the preceding by a fixed law. The numbers which compose a series are called its terms ; the first and last terms are called the extremes, and the other terms are called the means. An ascending series is one in which the terms increase regularly from the first term ; a descending series is one in which the terms decrease regularly from the first term. Arithmetical Progression. 160. When a series increases or decreases by a common difference, it is called an arithmetical series or arithmetical progression. For example, 2, 5, 8, 11, 14, 17 is an ascend- ing arithmetical progression, in which the common differ- ence is 3. In every arithmetical progression there are five elements to be considered, — the first term, the last term, the common difference, the number of terms, and the sum of the terms. These five elements bear such a relation to each other that when any three are given, the other two can be found. This gives rise to twenty distinct cases, a few of the more important of which will here be illustrated. I. In an ascending arithmetical series the first term is 7, and the common difference is 4 ; find the 10th term. _ „ . „ „ . „ The first term is 7 ; the second term ■" "' ' equals 7 plus the common difference ; the third term equals the second term plus the common difference, or SERIES. 285 7 plus twice the common difference ; the fourth term equals the third term plus the common difference, or 7 plus three times the common difference. In like manner, the tenth term equals 7 plus nine times the common difference. II. Find the first term of an ascending arithmetical series, the last term of which is 47, the common difference 6, and the number of terms 8. ._ _ n _ Arr A(y r '^^^^ ^^^^ \,^xxQ. Hiust bc such a num- 47 — ^ X — 4/ — 4Z _ o. ^^^ ^^^^ .^ ^ ^g ^g ^j^^^ ^^ .^^ ^^^ result will be 47 ; hence, if 7 X 6 be subtracted from 47, the remainder is the first term. III. The extremes of an arithmetical progression are 8 and 63, and the number of terms is 12 ; what is the com- mon difference ? The last term is determined by add- 63 — 8 _ ^ __ 5 ing the common difference to the first 12 — 1 11 * term as many times as there are terms less one; hence 63 — 8, the difference between the extremes, equals the common difference taken 12—1 times, and the common difference equals 63 — 8 divided by 12 — 1. IV. The extremes of an arithmetical progression are 4 and 103, and the common difference is 9 ; what is the num- ber of terms ? ^ ^f^ . The difference between the extremes (- 1 = 11 -j-1 = 12. equals the common difference taken as " many times as there are terms less one ; hence J-^f=^ equals one less than the number of terms, and J-^|^^+ 1 equals the "number of terms. V. The first term of an arithmetical progression is 4, the last term 19, and the number of terms 6 ; find the sum of the terms. A V /'/i _L1 Q\ A v/ OQ Using the method shown in example o X y^-^rd) _ D X ^c» _ go 19_4 9 9 ■ III., the common difference is , which equals V, or 3. The sum of the series can then be writteo 286 ARITHMETIC. 4+ 7+10 + 13 + 16 + 19, or 19 + 16 + 13 + ID + 7 + 4. By addition we find twice the sum to be 23 + 23 + 23 + 23 + 23 + 23, which equals 6 X (4 + 19) ; hence the sum of the terms equals 6x(4+19) 2 Let the first term be represented by a, the last term by ?, the common difference by d, the number of terms by n, and the sum of the terms by s ; then the principles illustrated in the foregoing examples maybe briefly expressed as follows ; 1= ■■a + {n-l)x d. a- : l-{n-l)x d. d = l-a n-1 n = d .= :|x(a + 0. Note. The principles as stated apply to ascending series. They can be stated so as to apply to descending series by interchanging a and /. To solve a problem in arithmetical progression, it is merely necessary to substitute the given values in the proper formula, and then simplify the expression thus ob' tained. EXAMPLES. 1. The first term of an ascending arithmetical series is 6, and the common difference is 5 ; find the 20th term. 2. In an ascending arithmetical series the first term is 8, and the common difference is f ; what is the 30th term ? 3. The first term of a descending arithmetical series is 120, the common difference 6, and the number of terms 15 ; what is the last term ? SBEIES. 287 4. The first term of a descending arithmetical series is 64, and the common difference is 4 ; find the 12th term. 5. The 12th term of an ascending arithmetical series is 60, and the common difference is 2 ; what is the first term ? 6. The last term of a descending arithmetical series is 3, the common difference 4, and the number of terms 11 ; what is the first term ? 7. The extremes of an arithmetical progression are 9 and 49, and the number of terms is 9 ; what is the common difference ? 8. The extremes of an arithmetical progression are l^- and 24, and the common difference is 2^ ; find the number of terms. 9. The extremes of an arithmetical progression are and 150, and the number of terms is 16 ; find the sum of all the terms. 10. Find the sum of the first 12 terms of the series 3, 7, 11, etc. 11. Find the sum of the first 10 terms of the series 24, 22i, 21, etc. 12. Find the sum of the odd numbers from 1 to 49 in- clusive. 13. How many strokes does the hammer of a clock strike in 12 hours ? 14. How far can a man walk in 10 days, going 12 miles the first day, and increasing the rate 3 miles a day ? 15. A man travelled 13 days, travelling each day | of a mile more than the preceding day. If he went 18 miles the last day, how many miles did he travel the first day ? 16. A man going a journey, travelled the first day 5 miles, the last day 32 miles, and each day 3 miles more than the preceding day. How many days did he travel ? 288 ARITHMETIC. 17. A man travels 11 days, travelling 5 miles the first day, and increasing the distance equally each day, so that the last day's journey is 20 miles ; find the daily increase. 18. A man has 8 children, whose several ages differ alike; the youngest is 2 years old, and the oldest 30. What is the common difference of their ages ? 19. A laborer worked for 40 cents the first day, and on each succeeding day his wages were increased 5 cents ; on the last day he received $2.50. How many days did he work? 20. A stone falls 16.08 ft. during the first second, 48.24 ft. during the next second, 80.4 ft. during the third second, and so on ; how far will it fall during the ninth second ? How far will it fall in nine seconds ? 21. Find the sum of an arithmetical series whose first two terms are 6 and 13, and whose last term is 62. 22. Find the amount of $300 for 15 years at 5% simple interest. 23. In how many years will $150 amount to $330 at 6% simple interest ? 24. The amount of $500 for 18 years is $860 ; what is the yearly interest ? Geometrical Progression. 161. When a series increases or decreases by a common ratio, it is called a geometrical series or geometrical pro- gression. For example, 2, 6, 18, 54, 162 is an ascending geometrical progression, in which the ratio is 3. An infinite series is a descending series of an infinite number of terms. For example, 1, J, |, ^, -^L etc. ; the last term is infinitely small and is regarded as zero, SERIES. 289 In every geometrical progression there are five elements to be considered, —^Ae^rs^ term, the last term^ the ratio, the number of terms, and the sum of the terms. As in arithmeti- cal progression, the five elements bear such a relation to each other that when any three are given, the other two can be found. I. The first term of a geometrical series is 2, and the ratio is 3 ; what is the 7th term ? 2 X 3^^ = 2 X 3« = 2 X 729 = 1458. '^^^ ^"* *"'"^ ," f ' *^," second term equals 2 mul- tiplied by the ratio; the third term equals the second term multiplied by the ratio, or 2 multiplied by the square of the ratio; the fourth term equals the third term multiplied by the ratio, or 2 multiplied by the cube of the ratio. In like manner, the seventh term equals 2 mul- tiplied by the sixth power of the ratio. II. The last term of a geometrical series is 640, the ratio 2, and the number of terms 8 ; what is the first term ? 640 640 640 ^^® ^^^^ *^'™ °^"^* ^'^ *"^^ * number 98^ = ~-^ = r^ = ^« that if it be multiplied by 2^, the result will ^ ^"^^ be 640; hence, if 040 be divided by 2^ the quotient is the first term. III. The first term of a geometrical series is 7, the last term 567, and the number of terms 5 ; what is the ratio ? 5-1 1567 S67, the fifth term, equals 7 multiplied by "~= V81 = 3. the 4th power of the ratio ; hence, if 567 be divided by 7, the quotient, 81, is the fourth power of the ratio, and the ratio is the 4th root of 81, or 3. IV. The extremes of a geometrical progression are 8 and 768, and the ratio is 4 ; find the sum of the terms. 768 X 4 - 3 _ 3072-3 _ 3069 ^ ^^q 4-1 - 3^ - ^- -=^^^^- The sum of the series may be written 290 ARITHMETIC. 3 + 12 + 48 + 192 + 768. Four times the sura equals 12 + 48 + 192 + 768 + 3072. Subtracting the upper line from the lower line, we have 3072 — 3, which equals three times the sum of the series ; hence the sum of the series equals or 1023. 3 Note. The last term, 768, equals the first term multiplied by 4^-1 ; hence 768 X 4, or 3072, equals the first term multiplied by 4^, and the value of the sum might be written — — ^^^, which equals 1023. 4 — 1 V. Find the sum of the first 5 terms of the series 768, 192, 48, etc. 768 -3 X i: ^ 768 - | ^ 767^^3069 1 _ 1 3 8 Q 1023. The sum of the series may be written 768 + 192 + 48+12 + 3. One fourth of the sum equals 192 + 48 + 12 + 3 + f . Subtracting the lower line from the upper line, we have 768 — |, which equals three fourths of the sum of the series ; hence the sum of the series equals ^^^^, or 1023. 3 * Note. The value of the sum might be written '^Q — 768x(^) ^^^^i^.]^ equals 1023. ^~? VI. rind the sum of the infinite series 768, 192, 48, etc. 768 _ 768 _ 3072 _ .. ^^^a The last term is regarded as ; 1 _ 1 3 3 hence the expression for the sum as found in the preceding example be- comes 768-0 X^ ^jjj^jjj g ig 768 ^^ JQ24. 1-i f Let the first term be represented by a, the last term by I, the ratio by r, the number of terms by n, and the sum of the terms by s ; then the principles illustrated in the fore- going examples may be briefly expressed as follows : I a — — -• SERIES. 291 T \C f — d Qi ^ 9*" -— CI s = -^ or J for ascending series. r — 1 7' — 1 a — lxr a — axr^'o -, ■,. 8 = or > tor descending series. 1 — r 1—r for infinite series. 1-r To solve a problem in geometrical progression, it is merely necessary to substitute the given values in the proper for- mula, and then simplify the expression thus obtained. EXAMPLES. 1. The first term of a geometrical series is 8, and the ratio is 4 ; find the 8th term. 2. The first term of a geometrical series is 27, and the ratio is ^ ; find the 7th term. 3. The 6th term of a geometrical series is 3888, and the ratio is 6 ; find the first term. 4. The last term of a geometrical series is 60f , the ratio £, and the number of terms 7 ; what is the first term ? 5. The first term of a geometrical series is 10, and the 6th term is 2430 ; what is the ratio ? 6. The first term of a geometrical series is y^, the last term 104J, and the number of terms 7 ; find the ratio. 7. Find the sum of the first 9 terms of the series whose first term is 6 and ratio 4. 8. Find the sum of the first 5 terms of the series whose first term is 100 and ratio I. 9. Find the sum of an infinite series whose first term is 3 and ratio J. 292 ARITHMETIC. 10. Find the sum of the first 8 terms of the series 8, 4, 2, etc. Find the sum of the same series to infinity. 11. The second term of a geometrical progression is 36 ; find the sum of 4 terms when the ratio is 1 J ; also when the ratio is 1|. 12. A merchant doubles his capital every 5 years ; if he begins with $2000, how much has he at the end of 25 years ? 13. If a ball be put in motion by a force which would move it 10 feet the first second. 8 feet the second, 6.4 feet the third, and so on, how far would it move ? 14. What sum of money can be paid by 10 instalments, the first of which is $1, the second $2, the third f 4, and so on in a geometrical progression ? 15. A man worked 8 days on condition that he should receive 1 cent the first day, 5 cents the second day, and so on, the wages of each day being 5 times the wages of the previous day ; how much did he receive ? 16. A man travels 4 miles the first day, 8 miles the sec- ond day, 16 miles the third day, and so on. How far does he travel the 7th day ? How far does he travel in 7 days ? Compound Interest. 162. Problems in compound interest can be solved by the principles of geometrical progression. Let P represent the principal, r the interest of fl for 1 year, n the number of years, and A the amount of the given principal for n years. In computing compound interest / is multiplied by 1+^' as many times as there are years. Thus P is the first term of a geometrical series, of which A is the last term, and l-\-r the ratio ; the number of terms ia" SERIES. 203 one more than the number of years, or n+1. Hence the first three formulae for geometrical progression when applied to compound interest become A = Px(l+r)\ A (l+ry l + r = Sp or r -
  • ' the end of the fifth = 300-H(5-l)xl8 j year is ^00, the = 300 + 72 = 372. = ^ X (300 + 372) payment at the end of the fourth year ^^iin «iAcn amounts to $300 ^ plus the mterest for 1 year, the pay- ment at the end of the third year amounts to $300 plus the interest for 2 years, and so on. These sums form an arithmetical progression, of which $300 is the first term, the interest of $300 for 1 year, or $18, is the common difference, and 5 is the number of terms. By principles roi arithmetical progression, we find the sum of the terms to be $1680. To find the present worth of the annuity of example I., find the present worth of $1680 for 5 years, as shown in § 138. II. Find the annuity whose amount for 6 years at 5% simple interest is $1350. lz=a-{-{n — l)xd = 1 +(6 -l)x. 05=1.25. ^. ,, ., ^, ' \ ^ If the annuity were $1, we Wb-uld s = -(a+l) have a= 1, /= 1 +(6-l)x.06= 1.25, 2 and s = f (1 + 1.26) = 6.75. It takes 6 x^ _, ^ nK\ a TK *^ annuity of as many dollars to "2' ^ ^^ amount to $1350 as $6.75 is con- 6 TS'ilS^O 00<'$200 tained times in $1350, which equals 1350_ 00 $200. 296 ARITHMETIC. EXAMPLES. 1. What is tlie amount of an annuity of $500 for 8 years at 6% simple interest? 2. What is the amount of an annuity of $1000 for 10 years at 5% simple interest ? 3. What is the amount of an annuity of $200 for 5 years at 4|% simple interest? 4. A clerk's salary of $1000 a year, payable quarterly, remained unpaid for three years ; find the amount then due, reckoning interest at 6%. 5. What is the present worth of an annuity of $600 for 6 years at 6% simple interest ? 6. What is the present worth of an annuity of $150 for 12 years at 4% simple interest ? 7. Find the annuity whose amount for 6 years at 6% simple interest is $3450. 8. Pind the annuity whose amount for 10 years at 5% simple interest is $3675. 9. Find the annuity whose amount for 5 years at 4% simple interest is $1500. Annuities at Compound Interest. 165. Problems in annuities at compound interest can be solved by the principles of geometrical progression. I. What is the amount of an annuity of $200 for 4 years at 6% compound interest ? ^ g X r" - g ^ 200 X (1.06)^- 200 ^ r~l 1.06-1 SERIES. 297 The payment at the end of the fourth year is $'200, the payment at the end of the third year amounts to $200 plus the interest for 1 year, the payment at the end of the second year amounts to $200 plus the compound interest for 2 years, and so on. These sums form a geometrical pro- gression, of which $200 is the first term, 1.00 is the ratio, and 4 is the number of terms. By the principles of geometrical progression, we find the sum of the terms to be $874.92. To find the present worth of the annuity of example I., find the sum of money which put on interest for 4 years at 6% compound interest will amount to f 874.92, as shown in § 162. II. Find the annuity whose amount for 3 years at 5^^ compound interest is f 504.40. ^ axr-a ^ l x(1.05)»-l ^ r-1 1.05-1 1.06 1.06 1.26247696 200 636 106 252.49539200 200 1.1236 1.06 67416 11236 .06)52.495392 ^874.92 1.191016 1.06 7146096 1191016 1.26247696 55125 11025 1.157625 L .05 ). 157625 3.1525 1.05 1.05 3.1525)504.4000(1160 31525 525 189150 105 189150 1.1025 1.05 If the annuity were $1, we would have a=l, and ^^1X005)^^3,52, 1.05-1 It takes an annuity of as many dollars to amount to $504.40 as $3.1525 is con- tained times in $504.40, which equals f _60. 298 ARITHMETIC. EXAMPLES. 1. What is the amount of an annuity of $50 for 5 years at 6% compound interest ? 2. What is the amount of an annuity of $200 for 6 years at 7% compound interest? 3. What is the amount of an annuity of $1000" for 8 years at 4% compound interest ? 4. If a man deposits f 100 a year in a savings bank that pays 3% compound interest, how much will he have in the bank at the end of 10 years ? 5. What is the present worth of an annuity of f 60 for 4 years at 6% compound interest ? 6. What is the present worth of an annuity of $200 for 7 years at 5% compound interest ? 7. Find the annuity whose amount for 3 years at 6% compound interest is $95.51. 8. Find the annuity whose amount for 5 years at 6% compound interest is $2818.55. 9. Find the annuity whose amount for 6 years at 5% compound interest is $1000. MENSUKATION. 299 CHAPTER XIV. MENSURATION. 106. Mensuration is the process of finding the lengths of lines, the areas of surfaces, or the volumes of solids. The principles of mensuration that apply to rectangles and rec- tangular solids are given in sections 80 and 81. The present chapter contains such principles as are useful to students of Arithmetic, but the proofs of these principles must be learned from Geometry. Definitions. 167. A point is that which has only position. A line is that which has length without breadth or thick- ness. A straight line is a line which has the same direction throughout its wliole extent. A curved line is a line which changes its direction at every point. Straight Line. Curved Linx. A surface is that which has length and breadth without thickness. A plane surface is a surface such that if any two of its points be joined by a straight line, that line lies wholly in the surface. A curved surface is a surface no portion of which is plane. A solid is that which has length, breadth, and thickness. Parallel lines are lines in the same plane which have the same direction. _"" They are equally distant and can never parallel lineb. meet. 300 ARITHMETIC. Right Aitgle. Obtusb Angle. Acute Angle. An angle is the difference in direc. tion between two lines which meet at a point, called the vertex; the lines are called the sides of the angle. BAC is an angle whose vertex is the point A, and whose sides are AB and AC. A right angle is an angle such that if one of its sides be produced through the vertex, the two angles thus formed are equal. The two sides of a right angle are said to be perpendicular to each other. All angles not right angles are called oblique angles. An angle greater than a right angle is called an obtuse angle, and an- angle less than a right angle is called an acute angle. A plane figure is a portion of a plane surface bounded by straight or curved lines. A polygon is a plane figure bounded by straight lines ; the lines are called the sides of the polygon. The perimeter of a polygon is the sum of its sides. The area of a polygon is the surface included within the perimeter. A diagonal of a polygon is a line joining the vertices of two angles not adjacent. An equilateral polygon has all its sides equal. An equi- angular polygon has all its angles equal. A regular polygon is both equilateral and equiangular. Polygons are named according to the number of sides. A polygon of three sides is a triangle, four sides a quadri- lateral, five sides a pentagon, six sides a hexagon, seven sides a heptagon, eight sides an octagon, nine sides a nona^ gon, ten sides a decagon, and so on. MENSURATION. 301 Triangles. 168. A polygon of three sides is called a triangle. The base of a triangle is the side on which it is supposed to stand. The vertical angle is the angle opposite the base, and its vertex is called the vertex of the triangle. The altitude is the perpendicular distance from the vertex to the base. The base and altitude are called the dimen- sions of the triangle. In the triangle ABC, BG is the base, the angle BAO is the vertical angle, and AD is the altitude. An equilateral triangle has all its sides equal. (Fig. 1.) An isosceles triangle has two of its sides equal. (Fig. 2.) A scalene triangle has no two sides equal. (Fig. 3.) Pio. 1. Fig. 2. Fig. 3. Fig. 4. A right triangle is a triangle having one right angle. The side opposite the right angle is called the hypotenuse or hypothenuse, and the side perpendicular to the base is called the perpendicular. Fig. 4 is a right triangle, in which AB is the hypotenuse, CB the base, and AC the perpendicular. An obtuse triangle is a triangle having one obtuse angle. (Fig. 3.) An acute triangle is a triangle having three acute angles. (Fig. 2.) An ec^niangalar triangle has all its angles eijual. (Fig. 1.) 302 ARITHMETIC. To find the area of a triangle wlien the base and altitude are given, take one half the product of the base by the altitude. To find the area of a triangle when three sides are given, subtract each side separately from half the sum of the three sides; then multiply the continued product of these three re- mainders by half the sum of the sides, and extract the square root of the product. To find one dimension of a triangle when tfte area and the other dimension are given, divide twice the area by the given dimension. Let A represent the area of a triangle, h the altitude, b the base, a and c the other two sides, and s half the sum of the sides ; then A^^Xbxh. A = Vs X {s — a) X {s — b) X {s — c). 2xA b = h ^ = 2x^. EXAMPLES. 1. A triangle has a base of 40 ft., and an altitude of 15 ft. ; how many square feet does it contain ? 2. Find the area of a triangle, the length of whose base is 25 ft., and the height 12 ft. 4 in. 3. Find the number of acres in a trianguJar field whose base is 20.28 ch. and altitude 14.5 ch. 4. How many acres are there in a triangular lot whose base is 432 ft. and altitude 320 ft. ? 5. Find the number of hektars in a triangular field whose base is 196.8"' and altitude 85'". MENSURATION. 303 6. Find the area of a triangle whose sides are respec- tively 4 ft., 6 ft., and 8 ft. 7. Find the number of hektars in a triangular field whose sides are respectively 62.4", 84.2% and 106.8™. 8. At 60 cents a square yard, find the cost of paving a triangular court whose sieves are respectively 80 ft., 75 ft., and 60 ft. 9. Find the altitude of a triangle whose area is 137^ sq. ft. and base 20 ft. 10. Find the altitude of a triangle whose area is 3.25"" and base 502™. 11. Find the base of a triangle whose area is 20 A. and altitude 80 rd. 12. Find the base of a triangle whosp area is 12.6* and altitude 30". 13. Find the area and altitude of an equilateral triangle whose sides are each 12 ft. long. 14. Find the perpendicular distances from the vertices to the opposite sides of a triangle, when the sides are re- spectively 12^™ 15*=™, and 20*=". Eight Triangles. 169. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. This principle is illustrated in the annexed dia- gram. To find the hypotenuse of a right triangle when the other two sides are given, extract the square root of the sum of the squares of the other two sides. I J ■ 304 ARITHMETIC. To find one side of a right triangle when the hypotenuse and other side are given, extract the square root of the differ- ence between the squares of the hypotenuse and the other side. Note. The length and breadth of a rectangle form two sides of a right triangle of which the diagonal of the rectangle is the hypotenuse. To find the diagonal of a rectangular parallelopiped, ex- tract the square root of the sum of the squares of the three dimensions. EXAMPLES. 1. Find the hypotenuse of a right triangle whose base is 30 ft. and perpendicular 16 ft. 2. The hypotenuse of a right triangle is 16:|^ ft., and the base is 15 ft. ; what is the perpendicular ? 3. The hypotenuse of a right triangle is 3.25", and the perpendicular is 3" ; find the base. 4. A flag-pole 140 ft. high casts a shadow 105 ft. in length ; what is the distance from the top of the pole to the end of the shadow ? 5. What is the length of a ladder that will just reach to the top of a house 12" high, when its foot is placed 8.4™ from the house ? 6. Find the height of the eaves of a house that can be reached by a ladder 40 ft. long, when the foot of the ladder stands 24 ft. from the house. 7. A pole was broken 26 ft. from the bottom, and fell so that the end struck 19 ft. 6 in. from the foot ; find the length of the pole. 6. Find the width of a street, from a point in which a ladder 36 ft. long will reach a window 28 ft. high on one side^ and one 25|- ft. high on the other, MENSURATIOK. 80d 9. A steamer goes due north at the rate of 15 miles an hour, and another due west 18 miles an hour. How far apart will they be in 6 hours ? 10. A rectangular field is 96 rd. long and 72 rd. wide ; find the length of the diagonal. 11. Find the longest straight line that can be drawn on a floor 4.5™ long and 3.2™ wide. 12. The side of a square field is 40 rd. ; find the distance between two diagonally opposite corners. 13. The diagonal of a square equals 16 ft. ; find the length of a side. 14. What is the length of the diagonal of a room 20 ft. long, 16 ft. wide, and 12 ft. high ? 15. Find the length of the diagonal of a box 4 ft. 8 in. long, 2 ft. 4 in. wide, and 7 in. deep. 16. Find the diagonal of a cubical block whose edge is 3^ inches. 17. The diagonal of a cube equals 10"=™ ; find the length of an edge. Quadrilaterals. 170. A polygon of four sides is called a qnadrilateral. A parallelogram is a quadrilateral whose opposite sides are parallel. A rectangle is a parallelogram whose angles are right angles ; a square is a rectangle whose sides are all equal. A rhomboid is a parallelogram whose angles are oblique angles ; a rhombus is a rhomboid whose sides are all equal. A trapezoid is a quadrilateral which has two sides par- allel. A trapezium is a quadrilateral which has no two sides parallel. 306 AKITHMETIC. Rkctanglk. Square. Rhomboid. Rhombus. TUAPKZDII). TRAI'EZIUM. The side upon which a parallelogram is supposed to stand and the opposite are called the lower and upper bases. The parallel sides of a trapezoid are called the bases. The per- pendicular distance between the bases of a parallelogram or trapezoid is the altitude. To find the area of any parallelogram, multiply the base by the altitude. To find the area of a trapezoid, multiply half the sum of the parallel sides by the altitude. To find the area of a trapezium, multiply the diagonal by half the sum of the perpendiculars to it from the vertices of opposite angles. The area of any polygon may be found by dividing it into triangles and obtaining the sum of their areas. EXAMPLES. 1. Find the number of square yards in a parallelogram whose base is 25 ft. and altitude 22|- ft. 2. Find the number of hektars in a parallelogram whose base is 640^" and altitude 180°™. 3. Find the area of a rhomboid whose altitude is 1,32'" and base 154.4™. MENSURATION. 307 4. Find the area of a rhombus, of which one of the equal sides is 358 ft., and the perpendicular distance be- tween it and the opposite side is 194 ft. 5. The parallel sides of a trapezoid are 30" and 25.2"*, and the altitude is 18.2™ ; find the area. 6. Two sides of a held, which are parallel, are respec- tively 262 yd. and 486 yd., and the perpendicular distance between them is 440 yd. How many acres does it contain ? 7. Find the area of a trapezium whose diagonal is 21 ft. and the perpendiculars to this diagonal are 9 ft. and 8 ft. 8. What is the area of a trapezium, the length of a diagonal of which is 25 ft., and of the perpendiculars from the opposite vertices to the diagonal 5 ft. and 17^- ft. ? Circles. 171. For definitions, see § 60. The ratio of the circumference to the diameter is the same for all circles, and it is customary to represent this ratio by the Greek letter tt (pi). The numerical value of tt cannot be obtained exactly, but the value tt = 3.1416 is correct to four decimal places. Note. ir=3f is suflSciently accurate for many purposes, and it is used to a great extent. In this book tt — 3.1416 is the value used. To find the circumference of a circle when the diameter is given, multiply the diameter by 3.1416. To find the diameter of a circle when the circumference is given, divide the circumference by 3.1416. To find the area of a circle when the radius is given, mul- tiply the square of the radius by 3.1416. To find the radius of a circle when the area is given, divide the area by 3.1416, and extract the square root of the quotient. 308 ARITHMETIC. Let A represent the area of a circle, C the circumference, D the diameter, and B the radius ; then c= ttXD or 2X7rXR. /)= TV B = G 2X7r A = ■■itXR' or ix 7rXi>^, Ii = ■■^. Note. When a circle is circumscribed about a square, the diagonal of the square is the diameter of the circle. When a circle is inscribed in a square, the diameter of the circle is equal to a side of the square. EXAMPLES. 1. Find the circumference of a circle whose diameter is 22 ft. 2. Find the circumference of a circle whose diameter is SO'". 3. Find the diameter of a circle whose circumference is 284*'™. 4. Find the radius of a circle whose circumference is 82 ft. 4 in. 5. Find the diameter of a tree whose circumference is 25 ft. 10 in. 6. Find the number of acres in a circular field whose radius is 32 rd. 7. Find the number of hektars in a circular field whose radius is 325™. 8. Find the area of a circle whose diameter is 2 ft. 7 in. MENSURATION. 309 9. Find the radius of a circle whose area is 163^ sq. rd. 10. Find the radius of a circle whose area is ISGO*** "°. 11. Find the diameter of a circle whose area is 38 sq. ft. 12. Find the circumference oi^ a circle whose area is 25'. 13. Find the number of acres in a circular park whose circumference is 3^ miles. 14. Find the diameter of a wheel which turns 23 times in going 103.5™. 15. How many times will a wheel whose radius is 0.762" revolve in running 1.6043^" ? 16. How many turns per minute does a pulley 1.3" in diameter make when the belt travels 50^'" per hour ? 17. A horse, tied to a stake, can graze to the distance of 35 ft. from the stake ; find the number of square yards of surface on which he can graze. 18. In a board 6 ft. long and 16 in. wide are two round holes, one of which is 10 in. across, and the other 12 in. across. Find the area remaining. 19. Find the width of the ring between two concentric circles whose circumferences are respectively 225 ft. and 300 ft. 20. What is the area of a circular ring formed by two concentric circles whose diameters are respectively 6 ft. 4 in. and 4 ft. 6 in. ? 21. Find the area of a circle inscribed in a square con- taining 225 sq. ft. 22. Find the area of a circle circumscribed about a square containing 144*'^". 310 ARITHMETIC. 23. Find the side of the largest square that can be laid out in a circular enclosure whose diameter is 10 rd. 24. Find the side of a square inscribed in a circle whose areais78.54«'i™ Pentagonal Prism. Prisms and Cylinders. 172. A prism is a solid whose ends, or bases, are equal and parallel polygons, and whose sides are parallelograms. A prism is triangular, quadrangular, pentagonal, etc., according as its ends are triangles, quadrilaterals, pentagons, etc. A right prism is a prism whose sides are perpendicular to the bases. A cylinder is a solid whose ends, or bases, are circles, and whose lateral sur- face is a uniformly curved surface. The axis of a cylinder is a straight line join- ing the centres of the two bases. A right cylinder is a cylinder whose axis is perpen- dicular to the bases. The perpendicular distance between the bases of a prism or cylinder is called the altitude. To find the lateral surface of a right prism or right cylinder, multiply the perim- eter of a base by the altitude. To find the volume of a prism or cylin- der, multiply the area of a base by the altitude. Let S represent the lateral surface of a right prism or right cylinder, V the volume of any prism or cylinder, B the area of a base, P the perimeter of a base, and H the altitude: then Ctlindkb. MENSURATION. 311 S=FxH. V=BxH. Let a represent the radius of a base of a cylinder, and D the diameter; then the following formulae are true for cylinders : ^ = 2X7rX RXH OT ttXDxH. V=7r X M' X H 01 \ X TT X D' X H, EXAMPLES. 1. How many square feet are there in the lateral sur- face of a right prism whose altitude is 3 ft., and whose base is a regular hexagon, each side of which is 6 in. long ? 2. The radius of the base of a cylinder is 8 in., and the altitude is 2^ ft. ; how many square feet are there in the lateral surface ? in the whole surface ? 3. The sides of the base of a triangular prism are re- spectively 12, 15, and 24 feet, and the altitude is 20 ft. ; find the cubic contents. 4. Find the volume of a prism whose base contains 7^ sq. ft., and the square of whose height equals five times the number of square feet in the base. 5. Find the capacity in gallons of a cylindrical cistern, measuring 16 ft. across and 15 ft. deep. 6. Find the number of liters contained in a cup, measur- ing 20"^"^ across and 31.831*'" deep. 7. Find the number of cubic feet in a log 28^ ft. long and 6 ft. 2 in. round. 8. How many kiloliters must be drawn from a cylindri- cal tank, the diameter of the base being 10™, in order to lower the surface 7*^ ? B12 ARITHMETIC. 9. A cylindrical vessel 1"' high is made of sheet iron 2"" thick, and holds 100\ What is its outer diameter ? 10. The diameter of a cylindrical vessel filled with water is 6 in. An immersed stone displaces 1^ in. in depth of the water. How many cubic inches are there in the stone ? Pyramids and Cones. 173. A pyramid is a solid whose base is a polygon, and whose sides are triangles meeting in a common point, called the vertex. A pyramid is triangular, quad- rangular, pentagonal, etc., according as its base is a triangle, quadrilateral, pentagon, etc. A right pyramid is a pyramid whose base is a regular polygon, and in which the perpendicular from the vertex passes through the centre of the base. A cone is a solid whose base is a circle, and whose lateral surface tapers uniformly to a point, called the vertex. The axis of a cone is a straight line drawn from the vertex to* the centre of the base. A right A cone is a cone whose axis is perpendicular to the base. The altitude of a pyramid or cone is the perpendicular distance from the vertex to the base. The slant height of a right pyramid or right cone is the shortest distance from the vertex to the perimeter of the base. Cone. A frustum of a pyramid or cone is the part of the pyramid or cone that remains after cutting off the upper part by a plane parallel to the base. The altitude of a frustum is the perpendicular distance Quadrangular Pyramid. MENSURATION. 313 between the two bases, and the slant height is the shortest distance between the perimeters of the bases. To find the lateral surface of a right pyramid or right cone, multiply the perimeter of the base by one half of the slant height. To find the lateral surface of the frustum of a right pyra- mid or right cone, multiply one half the sum of the perimeters of the bases by the slant height. To find the volume of a pyramid or cone, multiply the area of the base by one third of the altitude. To find the volume of a frustum of a pyramid or cone, multiply the sum of the areas of the two bases and the square root of their jn'oduct by one third of the altitude. Let S represent the lateral surface of a right pyramid or right cone, V the volume of any pyramid or cone, B the area of the base, P the perimeter of the base, U the alti- tude, and L the slant height ; then S = \xPy.L. V=^ixBxH. Let E represent the radius of the base of a cone, and JJ the diameter; then the following formulae are true for cones : S = Tr X JiX L or ^XttX D X L. V=iXTrxR'xH or ^\X7rxD'xH. The formulae for a frustum, representing the areas of the bases by B and B', and the perimeters of the bases by P and P', are as follows : S = ix{P+P>)xL. V=ix{B-\-B'+ VBxB') X H. Eepresenting the radii of the bases of a frustum of a cone by R and R\ and the^ diameters by D and D\ the formulae foi a frustum of a cone are 314 ARITHMETIC. S = 7r X {R -\- R') X L OT ^ X TT X (D -\- D') X L. V= i X TT X {B' -h M" -\- E X R') X H 0T^X7rX{D'-}-D"4-DxD')xH. EXAMPLES. 1. How many square feet are there in the lateral sur- face of a right pyramid whose slant height is 6 ft., and whose base is a regular octagon, each side of which is 4 ft. long? 2. The radius of the base of a right cone is 16 in., and the slant height is 4 ft. ; how many square feet are there in the lateral surface ? in the whole surface ? 3. The slant height of a frustum of a right pyramid is 5™, and the perimeters of the two bases are 12™ and 8™ re- spectively ; find the lateral area of the frustum. 4. The slant height of a frustum of a right cone is 8 ft., and the radii of the bases are 8 ft. and 5 ft. respectively ; how many square feet are there in the lateral surface ? in the whole surface ? 5. The altitude of a pyramid is 8"", and its base is a rec- tangle 3" by 2" ; find the volume. 6. The altitude of a cone is 18 ft., and the radius of its base is 6 ft. ; find the volume. 7. The base of a right triangular pyramid is an equilat- eral triangle, each side of which is 6 ft., and the altitude is 9 ft. ; find the cubic contents. 8. Find the volume of a pyramid whose base is 5™ square, and whose height equals the diagonal of the base. 9. Find the capacity in liters of a pail 25'^'" deep, meas- uring 28°° across the top and 18*=" across the bottom. MENSURATION. 815 Spheres. 174. A sphere or globe is a solid bounded by a cxirved surface, every point of which is equally distant from a point within called the centre. A straiglit line passing through the centre and having its extremities in the surface is called a diameter ; a straight line drawn from the centre to the surface is a radius. A section of a sphere made by a plane passing through the centre is called a great circle, and the Sphere. circumference of a sphere is the same as a circumference of a great circle. To find the surface of a sphere, multiply the circumference by the diameter. To find the volume of a sphere, multiply the surface by one third of the radius. Let S represent the surface of a sphere, Fthe volume, R the radius, and D the diameter ; then >S' = 4X7rXi?^ or ^XT)*. F= f X TT X 72" or ^ X TT X D*. EXAMPLES. 1. Find the number of square feet in the surface of a sphere whose radius is 8 ft. 2. Find the number of cubic feet in the volume of a sphere whose radius is 6 ft. 3. How many cubic centimeters are there in a cannon ball whose diameter is IS'^" ? 31^ ARITHMETIC. 4. How many square inches of leather will cover a ball 8 in. in circumference ? 5. A ball contains 2144.6656 cu. in. ; what is the diam- eter? 6. The earth's surface contains about 509294630'"^'"; find the radius. 7. A hemispherical vessel measures 2^ ft. across the top ; how many gallons does it hold ? 8. A spherical shell of copper has an outer radius of 2™ and is 5^™ thick. What is the weight of this shell in kilo- grams when it is filled with mercury, the specific gravity of the copper being 8.8 and mercury 13.6 ? Similar Surfaces and Solids. 175. Surfaces or solids which have the same form are said to be similar. Like dimensions of similar surfaces or similar solids are jwoportional. The areas of similar surfaces are to each other as the squares of their corresponding dimensions. The volumes of similar solids are to each other as the cubes of their corresponding dimensions. I. A triangle whose base is 12 ft. has an area of 54 sq. ft. ; find the base of a similar triangle whose area is 96 sq. ft. 54 : 96 : : 12^ .- a^. Let x represent the base of the second \Q A ^ triangle. Since the areas are to each other S^XZ^XX^ OKA ^^ ^'"^ squares of their corresponding dinun- ~ "m ~ sions, 54 : 96 : : 12^ : x^. From this propor- ? tion we find that a-2 = 256. Hence the base ^ of tlie triangle is the square root of 256, or Ans. 16 ft. 16 ft. MENSURATION. 317 II. A cylinder which is 9 ft. high contains 504 cu. ft. ; find the volume of a similar cylinder 6 ft. high. Let X represent the volume of the second cylinder. Since the volumes are to each other as the cubes of their corre- 504 : « : : 9^ : 6^ 66 T6{8 2 2 2 ^ K2if-448_i4Qip„ ft spo^'ling 'Ji'nensions, 2 2 Q 1* y ^ From this proportion we find that a:=149|. EXAMPLES. 1. The bases of two similar triangles are 6 ft. and 8 ft. respectively, and the altitude of the former is 9 ft. ; find the altitude of the latter. 2. The hypotenuse of a right triangle is 26"; find the hypotenuse of a similar triangle which contains twice the area. 3. The area of a trapezoid is 108 sq. ft., and its altitude is 6 ft. ; find the altitude of a similar trapezoid whose area is 192 sq. ft. 4. How many circles, each 4 in. in diameter, will equal in area a circle whose diameter is 2 ft. ? 5. Two farms of exactly similar form contain respec- tively 16 and 25 acres. One side of the former is 60 rd. in length ; find the corresponding side of the latter. 6. If a cistern can be filled in 30 min. by a pipe 1 in. in diameter, in what time can it be filled by a pipe 3 in. in diameter ? 7. If a pyramid 6 ft. high contains 45 cu. ft., what is the height of a similar pyramid that contains 100 cu. ft. ? 8. How many spheres, each 6 in. in diameter, will equal in volume a sphere whose diameter is 2 ft. ? 318 ARITHMETIC. 9. If a man digs a small square cellar, measuring 6 ft, each way, in one day, how long would it take him to dig a similar one measuring 10 ft. each way ? 10. If a stack of hay 5 ft. high weighs 100 lb., find the weight of a similar stack 24 ft. high. 11. If a rope 1 in in diameter weighs 2^ lb., what is the diameter of a rope of the same length which weighs 50 lb. ? 12. How far from the base must a cone whose altitude is 8 ft. be cut off so that the ;Prv*-;t'\m .shaP bp equivalent to one half of the cone ? MISCELLANEOUS EXAMPLES. 319 MISCELLANEOUS EXAMPLES. 1. Divide 3380321 by MDCCXCIX, and express the quotient by the lioman system of notation. 2. Find, by casting out the nines, whether the following is correct : 349761 x 28637 = 10015819397. 3. Multiply 4.32 by 0.00012. 4. Divide 0.002268 by 10.8. 5. Divide the product of 12, 20, and 30 by the product of 15, 24, and 18, by cancellation. 6. Find the factors and the greatest common divisor of 1498, 1582, and 2331. 7. Arrange in order of magnitude J^, |^, and -J-J. 8. Eeduce ^^, -^j and \^ to their least common denominator. 9. Divide | of 47 by ^ of 51. 10. Find the value ofJ-J + 4f+29+3^; reduce the result to its lowest terms, and also to a decimal form. 11. At f 1.75 a rod, what will it cost to fence a piece of ground 63.5 rd. long and 27.75 rd. wide ? 12. From a piece of cloth containing 84f yd. there were sold 4f yd., 26 J yd., and \ of 7J yd. ; how much remains ? 13. Name all the prime numbers in the series of numbers between 1 and 30 inclusive ; resolve all the composite num- bers into their prime factors ; and name all the perfect S(juares, cubes^ and other powers in the same series. 320 ARITHMETIC. 14. How mucli will be paid for 3760 lb. of coal at f 15 a ton ? 15. Pind the product of 157.757 and 15.3254 to two places of decimals. 16. Divide 1728 by 0.00144, and multiply the result by 0.000012. 17. Divide $125 among 4 boys and 3 girls, and give each boy "I as much as each girl. 18. Bought 360 gallons of wine at $2.60 a gallon ; paid for carriage |17.20, and for duties $86.50. If ^ of it be lost by leakage, at what price must the remainder be sold to gain $50 on the whole transaction ? 19. Find the product of three, three hundredths, thirty- three thousandths, three thousand millionths, and two twenty-fifths. 20. Divide ten thousand six hundred twenty-five bill- ionths by seventeen thousandths, and extract the square root of the quotient. 21. Find the value of the following fraction to three 1.0045 X 0.0875 decimal places 0.0016 22. Find the sum of five, five tenths, thirty-seven thou- sandths, one thousand millionths, XIX, MDCCCLXXXI, and O.iS. 1.28 23. Reduce ^1 j- 3 to its simplest decimal form. 24. Reduce 3.36 inches to a decimal fraction of a rod. 25. Reduce a pressure of 22.5 lb. Avoirdupois per square foot to ounces per square inch. MISCELLANEOUS EXAMPLES. 321 26. If either 5 oxen or 7 horses will eat up the grass of a field in 87 days, in what time will 2 oxen and 3 horses eat up the same *? 27. How many liters of water may be contained in a reservoir 10*" long, 6™ wide, and 4™ high ? What will be the weight in kilograms ? 28. A bin is 2^"" high, and contains IG^^'. The base of the bin is a square ; how many centimeters are there in one of its sides ? 29. If 4 men can mow 15 acres in 5 days of 14 hours, in how many days of 13 hours can 7 men mow 19J acres ? 30. I buy 300 bu. of grain consisting of wheat, rye, and oats, in the proportion of 3, 4, and 5. How many bushels of each do I buy ? 31. The sum of two numbers is 100, and i of one of them is f of the other ; find the numbers. 32. How much water must be mixed with 31 gal. of another liquid which cost $45.25, that the mixture may be sold at $1.25 per gallon, and 25^0 be gained? 33. What is the interest on $647.65 for 2 yr. 5 mo. 10 da. at 5% ? 34. What sum will produce $12.50 interest in 20 days at 4% ? 35. Find the selling price of goods by which there is a loss of 2% and an actual loss of $54.50. 36. If 12 barrels of corn will pay for 10 cords of wood, and 48 cords of wood will pay for 8 tons of hay, and 5 tons of hay will pay for 16 kegs of nails, how many barrels of corn will pay for 12 kegs of nails ? 37. Find the reciprocal of 155 carried to five decimal places. 322 ARITHMETIC. 38. Convert into a decimal i-±^ x 0.00025. ■ 0.075 39. What are the prime factors of 1716? How many integral divisors has this number, and what are they ? What is the smallest integer by which this number can be multiplied, so that the product shall be a square ? 40. Reduce to its simplest form the expression - of — ^ 41. Reduce 5f and 10^ to the decimal form, and divide the first by the second. 42. Find the greatest common divisor of 26^, 28|^, and 291 43. Find the least common multiple for the numbers J, 2.1, 5.25, and f. 44. Change 0.013 to an equivalent fraction whose denom- inator is 135. 45. -^ of f of f ft. equals what decimal of a rod ? 46. A block of stone (sp. gr. 2.5) is 1™ long, S**'" wide, and 45'^™ thick. How many kilograms does it weigh ? 47. rind the number of liters in a vat 2™ by 75'='" by 50'=". Also find the weight in kilograms of the sulphuric acid (sp. gr. 1.84) required to fill it. 48. If a meter is 39.37 inches, how many feet are there in a dekameter ? How many square centimeters in a square kilometer ? 49. If 4 masons build 27 yd. of wall in 5 days working 9 hr. a day, in how many days will 32 masons build 81 yd. of a similar wall if they work 10 hr. a day ? MISCELLANEOUS EXAMPLES. 323 50. Separate 772| into three numbers, which shall be in the same proportion as 2|, -^f and y^^. 51. Compute the square of the sum of the cubes of the first twelve prime numbers, and check all the work by cast- ing out the nines. 52. Multiply 34.056 by 0.065043, obtaining the product to four decimal places. 53. Divide 0.0144 by 4800; multiply the quotient by 6.004, and extract the square root of the product. 54. Simplify ^ of -^i^ , and divide the result by 0.0018. li X 3^ 55. Find the least common multiple of 76, 105, 150, and 175. 56. Reduce to a common denominator and add f X J X |, A, f , and -^. 57. If f of a cord of wood cost $3.33J, what would } of a cord cost ? 58. 75 miles equals how many kilometers ? 75 pounds Avoirdupois equals how many kilograms ? 75 quarts equals how many liters ? 59. A man's height is 174'=™. What is his height in feet and inches ? 60. Find the value of 17' of sulphuric acid (sp. gr. 1.84) at 5 cents a kilogram. 61. By selling a horse for $64.75, I lost 7^%; what per cent would I have gained by selling him for $73.50 ? 62. Sold steel at $25.44 a ton with a profit of 6% and a total profit of $103.32. What quantity was sold ? 63. If I buy stocks, par value $187.50, at 15% below par and sell them at 19^% above par, what is the gain per cent on my investment ? 324 ARITHMETIC. 64. If I buy coal at $4.12 per ton on six months' credit, for what must I sell it immediately to gain 10% ? 65. Find the amount of f 342.42 from Feb. 5th, 1879 to Mar. 15th, 1881, with interest at 7%, and reduce it to pounds sterling. 66. Find the interest, discount, and bank discount on $25 for 60 days at 7%. 67. Find the interest, discount, and bank discount on $17.50, due in 30 days, at 4|-%. 68. A merchant bought flour for $1000 cash and sold the same immediately for $1200 on 6 mo. credit, for which he received a note. If he should get the note discounted at a bank at 5%, what would be the gain on the flour ? 69. A and B can do a piece of work in 4 hours, A and C in 3f hours, and B and C in 5| hours. In what time can A do it alone ? 70. Divide 52 into such parts that \ of one part shall equal | of the other. 71. A cubical cistern holds 1331^^ of water ; what is the length of an inner edge ? 72. Arrange in order of magnitude |-|, 4|, and 0.89. 73. Show that the square root of 0.3 lies between |^ and -fj. 74. I have a rectangular lot of land, 64 rd. long and 36 rd. wide, and a square lot of the same area ; how many more feet of fencing will be needed for the former lot than for the latter ? 75. If 144 pounds Avoirdupois be equivalent to 175 pounds Troy, what is the ratio of the pennyweight Troy to the dram Avoirdupois ? MISCELLANEOtTS EXAMPLES. 325 76. Reduce to equivalent fractions having a common denominator | of |, 2f, 5|, and | of ^ of 3J. 77. The number 209.069673692836 is composed of three factors, of which two are 20083.6 and 0.260075 j find the third factor. 78. Simplify H >< 3| of 2^ , l + H-iJ 79. Simplify (3.2 + 0.004 - t.lll)x 0.25, ^ -^ (4 -^ 0.2) -17.907 80. Simplify 1__. 2+ ^^ 4 + ? 6 81. Write in Arabic numerals the value of the expression [MDCCCLXXXIII ^ 16.6] x [(2.5 - 1.25) -- 0.03]. 82. Reduce |^ to its lowest terms ; reduce the result to a decimal, and extract the square root to three figures. 83. What is the length in meters and decimeters of a side of a square which contains 0.1335*? 84. Find the length in dekameters of the side of a square, the area of which equals the area of a rectangle which is ]^Km gin loj^g and 4i|Hm ^^^Jg^ 85. A square field contains 1016064 sq. ft. What is the length of a side expressed in meters ? 86. Find the side to millimeters of a cubical box that contains 1^™. 87. The volume of a sphere is 0.056 cu. yd. What is the length in inches of the side of a cube containing the same volume ? 88. Find the edge of a cubical can which will hold 27.57^» of sulphuric acid, whose specific gravity is 1.8. 826 ARITHMETIC. 89. Find a fourth proportional to 0.37, 8.9, and 4.3, and extract the cube root of it to two decimal places. 90. Find the fourth term in V4.913 : 0.0016 : : 48000 : . 91. If 60 cannon firing 5 rounds in 8 min. kill 350 men in 75 min., how many cannon firing 7 rounds in 9 min. will kill 980 men in 25 min. ? 92. A man travelled 2 days at the rate of 15 miles per day, 4 days at the rate of 20 miles per day, and 5 days at the rate of 30 miles per day ; what was his average rate of travel per day ? 93. A farmer divides among his 3 sons 246 A. 1 E. 32 P., sharing it among them as the numbers 3, 4, and 5 ; what were the shares ? 94. How many kilograms are there in a cubic foot of water ? 95. A rectangular box is 4™ long, 30^"" wide, and 20*" deep. (i) Find its capacity in liters, (ii) What weight of water will it contain "? (iii) What weight of mercury (sp. gr. 13.6) will it contain ? 96. An empty bottle weighs 380^; when filled with water it weighs 0.985^^. How many liters does the bottle hold ? 97. Find the present worth of a note for $1320, due in Syr. 4 mo., money being worth 6%. Find also what could be obtained for the same note at bank discount. 98. Find the interest, discount, and bank discount on 165.33 for 90 days at 41%. 99. What is the difference between the true and bank discount of $250, due 10 mo. hence, at 7% ? 100. How long must a note of $243 at 3|% run that its interest may equal the interest on a note of $125 for 7 mo. MISCELLANEOUS EXAMPLES. 327 101. How many times does the least common multiple of 6, 25, 40, and 75 contain the square of their greatest common divisor ? 102. Reduce -^ and ^^^ to their least common denom- inator ; add the results, and express the sum decimally to four places. 103. Divide (^-jIt) ^^7 (A of 0.00616), carrying the quotient to five places of decimals. 104. Simplify ^+^^ + (|^' . is) ~1 105. 4ill^+(3-2J)-(4-3i) equals what? Ex- ? "^ T tract the square root of the result to two decimal places. 106. Find the value of (^^ of —^ divided by ^, and extract the square root of the quotient to two decimal places. 107. Simplify the expression J^^^x^- 108. How many rods of fence will it take to enclose a 20 acre lot in the form of a square ? 109. If a man can walk 16 rods in | of a minute, in what time can he walk 0.00164 of a mile ? 110. The length of a rectangular field containing 30 acres is 3 times its width. Find the length of the field in feet. 111. A cubic inch of gold is hammered out until it covers 6 acres ; how many leaves of gold of this thickness would it take to make one inch ? 112. A cube contains 79507 cu. in. How many square inches does its surface contain? 113. How many rods of fence will be required to enclose 640 acres of land in a square form ? 828 ARITHMETIC. 114. A man lost ^, ^, and f of his money, and then had ^2600 left ; what sum had he originally, and how much per cent had he lost ? 115. Eeduce J of 6% of 1.05 -j-^\ of -f^ to the simplest form. 116. I earned $10 by collecting bills on which a discount of 10% was allowed for cash. My commission was 5% ; how much did I collect ? 117. If 4:^% Government bonds sell at 116, what sum of money invested in them will yield an interest of f 1 per day? 118. Bought a bill of goods on 6 months' credit for $500. What would be the gain per cent on my bargain if I sold the same at once for $525 cash, interest being reckoned at 6% per annum ? 119. Find the difference between the true and bank dis- counts on a note for $1000, due 3 mo. hence, money being worth 6%. 120. Find the interest, discount, and bank discount on $327.19 for 90 days at 71%. 121. Bought $1500 worth of goods, half on 6 months' and half on 9 months' credit. What sum at 7% interest, paid down, would discharge the bill ? 122. Find the principal that will amount to $962 in 4 yr. 6 mo. at 41%. 123. Find the compound interest on $300 for 2 yr. at 4%, interest being compounded semi-annually. 124. Find the annual interest of $200 for 3 yr. 1 mo. at 6%. 125. Find the greatest common divisor of 113.355 and 3.141592. MISCELLANEOUS EXAMPLES. 329 126. Divide 2| + | by |1. 127. Simplify 5:52^ + ^^, expressing the result in decimal form. 128. [5| X tt X f +1.0176] -5- [3f + 300.003] equals what ? 129. Keduce to a decimal form ^ , and from it subtract 0.01 of |. ^'^^^ 130. Eeduce to a vulgar fraction the decimal 0.0001234. Test your answer by reversing the process. 131. Find the square root, to three places of decimals, of 15.75 10h-{- 1.5 4 of 2i 132. Find the sum of 3^, 6f 8^2^, and 65|, reduce the fractional part to a decimal, and extract the cube root of the result. 133. A and B, 44 miles apart, travel towards each other. A travels -^j of the whole distance, while B travels ^ of the remainder. How far are they then apart ? 134. Two engines, 40 miles apart, approach each other at the rate of 25 and 35 miles an hour. Find the time and place of their meeting. 135. A river 10™ deep and |^™ wide flows 2^*" an hour ; find the number of kiloliters of water that falls into the sea in a minute ; also its weight in kilograms. 136. If a sheet of paper weighs 8^^ per square meter, find the weight in grams of a piece IJ"* long and 25'^^ wide. 137. What is the difference in volume between two blocks of granite, one 1™ long, 6*^" wide, and 0.5" thick, the other 300°" long, 40«° wide, and 0.2" thick ? 330 ARITHMETIC. 138. Find tlie weight in kilograms, and in pounds, of a rectangular block of marble (sp. gr. 2.83) 3.7"' long, 7*^"' wide, and 30'"" thick. 139. If I buy macaroni at 30 cents a kilo, pay f 12 a metric ton for transportation, and sell at 14 cents a pound, what per cent do I gain or lose ? 140. A square field contains 0.08346 of an acre. Find the length of one side of the field in meters, the hektar being equal to 2.4711 acres. 141. The stere contains 1.308 cu. yd. How many meters are there in the side of a cube containing 0.056 cu. yd. ? 142. If 35 men can build a wall 50 ft. long, 2 ft. thick, and 10 ft. high in 8 days, how long will it take 50 men to build a wall 250 ft. long, 3 ft. thick, and 7 ft. high ? If the first wall costs $910, what will the second one cost ? 143. How many men would be required to cultivate a field of 2|- acres in 5^ days of 10 hours each, if each man completed 77 square yards in 9 hours ? 144. An estate is divided among three persons. A, B, and C, so that A has f of the whole, and B has twice as much as C. It is found that B has 27 acres more than C. How large is the estate ? 145. Copper weighs 550 pounds, and tin 462 pounds to the cubic foot. What will be the weight of a cubic foot of a mixture 6 parts copper to 5 parts tin ? 146. A man bought 16 horses and 19 cows for f 1865. He paid upon the average yV as much for a cow as he did for a horse. What was the average price per head he paid for the horses ? 147. By selling a lot of land for $783 Host 13%. What would it have brought if I had sold it at a loss of 8^% ? MISCELLANEOUS EXAMPLES. 331 148. At what rate per cent is the deduction made when 19 s. 10^ d. is taken from an account of £39 15 s. in con- sideration of immediate payment ? 149. At what per cent premium must a 4% perpetual bond be bought in order that it may pay only 3^% on the investment ? 150. Find the bank discount of a note for $25000 for 2yr. 6 mo. at 3^%. 151. Find the principal that will amount to $724.92 in 2yr. 3 mo. at 3J%. 152. Find' the simple interest, the annual interest, and the compound interest of $1200 for 2 yr. 6 mo. 18 da. at 4%. 153. What is the interest, discount, and bank discount on $127.42 for 65 days at 5% ? 154. Find the difference between the amount of $1000 for 3yr. at 6% compounded yearly, and at 3% compounded half yearly. 155. Find the square root of five million five thousand and five tenths to two decimal places. 156. Compute the value of 3 + V3 + ■v^29 to three places of decimals. 157. Multiply the square root of 0.173056 by the cube root of iff If. 158. What is the difference between the square root and the cube root of 1771561 ? 159. On a map whose scale is -^ of an inch to a mile, what would be the area covered by a tract of land contain- ing 720 square miles ? 160. A rectangular tank, with a square base, 3 ft. deep, contains 675 cu. ft. Find the length of a side of the base. 332 AETTHMETIC. 161. A man paints two sides of a wall 7 ft. high in 31 hr. 6 min. 40 sec. If he can paint 4 sq. yd. in an hour, how long is the wall ? 162. A certain square field contains 38.75 acres. Com- pute the length of one side of the field in meters. (Given 1*1'" = 1550 sq. in.) 163. The specific gravity of iron is 7.2; find the volume in cubic decimeters, and the weight in kilograms, of a block of iron whose dimensions are 5, 8, and 11 inches. 164. A railroad train makes a mile in 57 seconds. What is its rate per hour, and what per cent of the hour is occu- pied in its making a single mile ? 165. If 5 horses eat as much as 6 oxen, and 12 oxen eat 12 tons of hay in 40 days, how much hay will 7 horses and 15 oxen eat in 65 days ? 2 \2^ 7-1- 6/ 166. Express the value of -^-^ — :— i exactly as a decimal. 167. Which is the larger, |^ or VJ ? 5j + |i- 0.725 168. Eeduce . , ^ .^ to an equivalent decimal. 4 + 3.45 ^ 169. Simplify 14 of 24 + 6J -5- 2} + (^54 + 0-24 + 0.53 ^ V 2.2-0.64; 170. From the sum of 3| and 4f subtract 6f-, multiply the difference by | of fj of 88, and find what fraction the product is of 999. 171. A cistern 6.84"' long and 2.36™ wide contains 34^^ of wine. What is the depth of the liquid ? 172. How many cubic meters are there in a cord ? MISCELLANEOUS EXAMPLES. 333 173. ixovv many liters of water are there in a full rectan- gular tank 12 ft. long, 6 ft. wide, and 4 ft. deep ? 174. A tank which holds 100 gal. can be filled by one pipe in 25 min., and emptied by another pipe in 45 min. ; if both are opened together, how long will it take to fill, and how much water will have been lost ? 175. If 8 men can build a brick wall 125 ft. long, 2 ft. wide, and 4 ft. high, in 4 days, working 10 hr. each day, how many days will it take 12 men to build a wall 465 ft. long, 3 ft. wide, and 6 ft. high, working 8 hr. each day ? 176. Two men undertake to do a piece of work for $6. One could do it alone in 5 days, and the other in 8 days. With the assistance of a boy, they finish it in 2J days. How should the money be divided ? 177. A gallon contains 231 cu. in., and a bushel 2150.4 cu. in. ; how will a liquid quart compare with a dry quart ? 178. What is the present worth of $1000, due 6 mo. hence, money being worth 6% ? 179. Find the interest, discount, and bank discount on $416.03 for 60 days at 7^%. 180. What is the difference between the true discount and that taken by banks on $1500, due one year hence with- out grace ? The rate of discount in both cases is 5%. 181. What is the difference between the simple and com- pound interest on $700 for 2 yr. 6 mo. at 7%, interest com- pounded annually ? 182. A note for $500 at 60 days without interest is bought for $450. What is the profit if money is worth 1% a month ? 183. How many rods of fence will it take to enclose a square field containing exactly one acre ? 334 AEITHMETIC. 184. The area of a circle is 5 sq. rd. What is the length in inches of one side of a square which contains the same area? 185. Find the depth in meters of a cubical cistern which has a capacity of 300001 Give the result to three decimal places. 186. A is 156 miles ahead of B. A travels 30 and B 42 miles a day. In how many days will B overtake A ? 187. If 4 men or 6 boys can do a piece of work in 27^ days, in how many days will 5 men and 9 boys do it ? 188. If 8 horses consume 3|- tons of hay in 30 days, how long will 4^ tons last 10 horses and 15 cows, each cow con- suming I as much as a horse ? 189. A carriage, at the rate of 8^ miles an hour, coni pletes "I of a certain distance in 3^ days ; in how many days will it complete ^ of the same distance, going at the rate of 10 miles an hour ? 190. There are two casks, one containing 15 gal. of water, and the other 35 gal. of spirits ; how many gallons must be transferred from each to the other in order that the mix- tures in each may be of the same strength ? 191. Find the value to three decimal places of V(0.146)2+ (0.063)2. 3 192. Find the value of — ==: , correct to four places of decimals V19 — 4 .go a» „| -... (2.01 +2.25 X 0.004W(1.0337- 31.09 x 0.03) ^ ^ 4.5-900 2fof lH + 4t 194. Simplify^," MISCELLANEOUS EXAMPLES. 335 195. Find the sum of - — ^ — --—, and ^ ^ ,J^ ^ ,,^}^ * f X|x(ir 1.6 + 0.625 196. Simplify the following expressions : V2J, -v/JU, ^mi, and {2iy. 197. A and B together have f 136, and -| of A's money is equal to f of B's. How much has each ? 198. A certain piece of work can be done by 8 men or 16 boys in 10 days. In how many days can the work be done by 8 men and 16 boys ? 199. If 8 horses in 30 days eat 3^ tons of hay, how long will 4y9^ tons last 10 horses, 15 cows, and 10 sheep, each cow eating | a& much as a horse, and each sheep J as much as a cow ? 200. A pail will hold 5^. The area of its base is 330'^"". Kequired its height in inches. 201. One meter equals 39.4 inches. How many cubic inches are there in one liter ? 202. Leap year is omitted once in every century except in those centuries whose number is divisible by four. What is the average length of a year ? 203. What is the value in pounds sterling of half an acre of land at 9^ pence per square foot ? 204. Separate 280 into two such numbers that -f of one is equal to the other. 205. A milkman bought 40 gal. of new milk at 16 cents a gallon and 60 gal. of skimmed milk at 8 cents a gallon, which he mixed with 12 gal. of water, and sold the whole at 24 cents a gallon. What was his profit ? 206. What sum placed at simple interest for 3 yr. 10 mo. at 7% will amount to the same as $1500 placed at compound interest for the same time at 7|^^ ? 336 ARITHMETIC. 207. I buy goods to the amount of $4978.70, payable in 4 mo. with interest at 5%, and give my note without in- terest. What must be the face of the note ? 208. Compute the value of VS — 1 + V6 to four decimal places. 209. Extract the square root of 2.26 to three places of decimals. Show how you can derive from the square root of this number that of 0.0226. 210. A rectangle is 1.25^"* long and 3.5^™ wide ; find the side of the equivalent square in dekameters. 211. What is the length of a cubical bin which will con- tain 4500 cu. ft. ? 212. Find as circulating decimals the square of 0.4 and the square root of 0.694. 213. A rectangular block of stone, square at the base and 8 ft. high, contains 162 cu. ft. What is the length of one side of the base ? 214. The weight of a cubical block of stone, 2 ft. on each edge, is 1352 lb. What is the weight of a cubical block whose edge is 4 ft. ? 215. A cubical vessel contains 150 lb. of pure water. Find the length of an inner edge of the vessel in decimeters. 216. How many books, each 10|- in. long, 4^ in. wide, and If in. thick, can be packed in a box 5 ft. 3 in. long, 3 ft. wide, and 2 ft. 9 in. thick V 217. Supposing that the driving wheels of a locomotive are 16 ft. in circumference, what number of revolutions must they make per minute so that the locomotive may attain a speed of 60 mi. per hour ? 218. rind wliat decimal part the square root of -ffj is of the square root of 5J. MISCELLANEOUS EXAMPLES. 337 219. A sum of £250 17 s. Gd. is transmitted through Paris to New York ; find the value of the sum in United States money (£1=24.79 francs ; 9.2 francs = f 1.75). 220. What sum of money is the same part of £14 7 s. 9f d. that 4 oz. 7 pwt. 5 gr. is of 8 oz. 10 pwt. 15 gr. ? 221. The wages of A and B together for 22 days amount to the same sum as the wages of A alone for 38^ days ; for how many days will this sum pay the wages of B alone ? 222. If Greenwich time be 5 hi*. 8 min. 12 sec. later than Washington, what is the difference in time between Wash- ington and a point 87° 35' west of Greenwich ? 223. How much carpeting f yd. wide will cover the top and sides of a box 3 ft. 6 in. long, 2 ft. 3 in. wide, and 9 in. high ? 224. What would it cost to paper the walls of a room 30 ft. 8 in. long, 20 ft. 4 in. wide, and 11 ft. high, the paper being 2 ft. 3 in. wide, and costing 63 ots. per roll of 12 yd. ? 225. Two bells commence tolling together, one at the rate of 5 times in 24 sec, the other at the rate of 4 times in 23 sec. ; in what time will they again toll together ? 226. A man bought 200™ of cloth in France at 16^ francs per meter ; he paid 12^ cents a yard for duty and freight, and sold it in Boston at $4.62^ a yard. What was the gain ? 227. A block in the form of a perfect cube contains 12516 cu. in. How many square yards of paper are required to cover it ? 22S. Sold a hundred bushels of wheat, which cost $150, at 50 cents a peck, taking in payment a 6 months' note which was discounted immediately at the bank at 6%. What was the profit ? 338 ARITHMETIC. 229. A note for $1000, with interest at 7% payable an- nually, has run 3 years, but no interest has been paid. What is now the amount of the note at simple interest ? at annual interest ? at compound interest ? 230. What is the compound interest of $1 for 143 yr., allowing it to double once in 11 yr. 11 mo. ? 231. A grocer makes a mixture of which 21.5 lb. contain ^ lb. of rye, 12 lb. of wheat, 5 lb. of oats, and 4 lb. of barley. How much of each ingredient will be contained in 100 lb. of the mixture ? 232. If the pay of a man, a woman, and a boy be in the ratio 3, 2, 1 ; and 24 men, 20 women, and 16 boys receive £20 8 s. a week, what will 27 men, 40 women, and 15 boys receive in 365 days ? 233. Find how many yards of carpet a yard wide must be bought to cover a floor 23 ft. 6 in. by 17 ft. 5 in., suppos- ing that the strips run lengthwise, and that the figure of the carpet is 8 ft. long, and is laid to match. Find also how much of the carpet must be turned under or cut off at the ends and sides. 234. In the Centigrade and Fahrenheit thermometers the freezing points are 0° and 32° respectively, and the boiling points 100° and 212° respectively. When the Centigrade stands at 37°, what will the Fahrenheit read ? 235. A train travels 82 mi. 7 fur. 26 rd. 4 yd. in 3 hr. 48 min. 51|- sec. ; what is the rate per hour ? 236. A person by selling an article, which cost him $60 per 100 pounds, at 67|- cents per pound, makes 5% more than he would by selling the whole for f267.67|; how many pounds were there ? . 237. The amount of a certain principal at a certain rate of interest for 6 mo. is $949.76, and for 1 yr. at the same rate is $1003.52. Eequired the rate per cent and principal MISCELLANEOUS EXAMPLES. 339 238. Lead is 11.4 times, and zinc 7.2 times, as heavy as water. If 3 lb. of lead and 2 lb. of zinc be melted together, compare the weight of the alloy with that of water. 239. There is a rectangular lot of ground 64.8 rd. long and 36.05 rd. widep and a square lot of the same area ; which will require the more feet of fencing, and how much ? 240. A cubic foot of iron weighs 450 lb. ; what will be the weight of a rectangular closed box made of iron ^ of an inch thick, the extreme dimensions of the box being 7 ft. 5 in., 8 ft. 3 in., and 4 ft. 3 in. ? 241. In 100.93*^ of chemically pure saltpetre there are 39.04« of potassium, 14.01« of nitrogen, and 47.88« of oxy- gen ; determine the per cent of each of these elements in the compound, and how many grams of each there are in a kilogram of the latter. 242. A commission merchant sells 28000 lb. of cotton at 12^ cents per pound ; after deducting $35.36 for freight and cartage, $10.50 for storage, and his commission, he remits $3252.89 as net proceeds of the sale. At what rate did he charge commission ? 243. I have bought a farm for $6500 ; $2000 of this is to be paid down, $500 in one year, and the remainder in two years. If a note for the whole amount were preferred, when would it become due ? 244. On the first of January, 1884, A, B, and C enter into a partnership. A and B each furnish $4000, and C $8000. At the end of a year B withdraws $1500, while 6 months later C adds $2000. At the end of 2 years they find their profits are $1580. How shall the profits be di- vided between them ? What per cent do they realize on their capital ? 340 ARITHMETIC. 245. If the diameter of the earth is 7926 mi., what height in inches on a globe 2 ft. in diameter will represent a moun- tain 15000 ft. in height ? 246. The least common multiple of four numbers is 283500. Prove that if three of the iftimbers are 140, 42, and 60, the fourth must contain 225 as a factor. 247. A house costs $5000, and rents for $25 a month, with $25 to pay annually for repairs and f 50 for taxes ; what is the difference in the income from this and from the same money invested in 6% stock at 96 ? 248. A cistern contains 23104^^ of water. What is its volume in cubic meters ? If it has a square base, and its depth is 25™, what is the length of an edge of its base ? 249. How many five-cent coins may be made from a bar of silver O.S'" long, 0.6'^*" wide, and 5*=™ thick, if each coin weighs 5^, and silver is 10.5 times as heavy as water ? 250. Eesolve 21600 into its prime factors ; and use them to find the greatest square number, and also the greatest cube, that will divide 21600 without remainder. 6 29 nf a/Q4 8 251. Divide "^""215 ^y >/67419143. 252. rind the value to three decimal places of the expres- sion 3/ 3^ X 1| + 4yL — 3_^ \5i-7|-28^-Fi 253. The length of a rectangular field is | of the breadth, and the area is 9 acres. Find the diagonal in rods, feet, and inches. 254. If A can row at the rate of 12|^ miles per hour, and B at the rate of llf miles per hour, what start should A give B in a race of 500 yards in order to beat him by one yard ? MISCELLAKEOtJS EXAMPLES. S41 255. A clock gains 3^ min. in 23 hr. 59 min. 45 sec. ; at noon it is 2 min. slow ; when will it indicate correct time ? 256. 5 cu. ft. of gold weigh 98.2 times as much as a cubic foot of water, and 2 cu. ft. of copper weigh 18 times as much as a cubic foot of water ; how many cubic inches of copper will weigh as much as J of a cubic inch of gold ? 257. A wins 9 games out of 15 when playing against B, and IG out of 25 when playing against C. How many games out of 118 should C win when playing against B ? 258. A cube is formed of a certain number of pounds Avoirdupois of a substance, and the same number of pounds Troy of the same substance. What rjftio will a side of the (jube bear to a side of a cube formed of the same number of pounds as before, but all Avoirdupois? (1751b. Troy = 144 lb. Avoirdupois). 259. A Frenchman sells a draft on Pai-is for 10000 francs in New York at 5.15 francs for f 1, and witli the proceeds buys a bill of exchange on London at 8^% premium; what is the amount of the bill in English currency ? 260. A man paid | of his money for stock, J of what re- mained for goods, and ^ of what then remained for tools ; he then found that $26 was ^ of one half of what was left. Find what part of the whole was left, and how much money he had at first. 261. Find in acres the area of a rectangular field of which the longer side is to the shorter as 15 : 8, and which a person walking at the rate of 3^^ miles per hour takes 5 min. 45 sec. to walk around. 262. A rectangular piece of ground is 13 ch. 44 li. by 8 ch. 40 li. How many square feet would be occupied on paper by a plan of the land drawn upon a scale of IJ inches to a chain ? 342 ARITHMETIC. 263. A and B run a race, their rates of running being as 17 to 18. A runs 2 J mi. in 16 min. 48 sec., and B runs the entire distance in 34 min. What was the entire distance ? 264. Mr. A. buys a house for $10000 and rents it for a month, paying $150 per annum for taxes and repairs. He also buys 181 shares of railroad stock (par $50) for $55 each, and receives a dividend of 7%. What is his income and rate of interest from each investment ? What is his total investment, income, and rate of interest ? 265. Find to two decimal places the sixth root of the least common multiple of 899 and 1073. 266. A sum of money was placed at interest at 6% per annum ; at the end of the first year the interest was added to the principal, and at the end of the second year the amount was $842.70 ; what was the original sum ? 267. Six men, working 9 hours a day, can do a piece of work in 15 days. In how many days will a party of men, working 10 hours a day, do the work, the number of men being equal to the number of days ? 268. A square of 25™ on a side is inscribed in a circular walk 5™ wide. This walk is covered with asphalt 10*"" thick. What is the weight of the asphalt in metric tons if its spe- cific gravity is 10 ? 269. A owes B $1080 due July 5th, 1885, $250 due Sept. 15th, 1885, $700 due Dec. 10th, 1885, and $300 due Mar. 20th, 1886; B owes A $500 due Aug. 25th, 1885, $400 due Jan. 1st, 1886, and $350 due June 20th, 1886. Eind the time when the balance due B may be paid without loss to either party. Find also the equitable value of that balance if payment were made Sept. 15th, 1886, the rate of interest being 6%. MISCELLANEOUS EXAMPLES. 343 270. A loaded wagon weighs 2 T. 3 cwt. 48 lb. ; the wagon itself weighs 18 cwt. 75 lb. The load consists of 215 pack- ages, each of the same weight. Find the weight of each, and reduce it to kilograms. 271. A box with- a lid measures externally 16 in. each way, and the wood of which it is made is 1 in. thick ; what would be the weight of the box when filled with paper, a cubic foot of paper weighing 792 oz. and a cubic foot of wood 840 oz. ? 272. A box in the form of a cube is partially filled with water. Half a dozen balls, each 4 in. in diameter, are thrown in, and the water rises i in. in consequence. Find the length of an edge of the box. 273. English shillings are coined from a metal which con- tains 37 parts of silver to 3 parts of alloy ; one pound of this metal is coined into 66 shillings. The United States dollar weighs 412.5 grains, and consists of 9 parts silver to 1 of alloy. What fraction of the United States dollar will con- tain the same amount of silver as one English shilling ? 274. Two bodies let fall at different instants from the same point are found, VW^^^ seconds after the latter of them started, to have fallen, the one 25™, the other 100™. These distances being to one another as the squares of the times during which the bodies have been falling, how many seconds must the one body have started before the other ? 275. Using the prefix "mega-^' for a million times, and "micro-^^ for a millionth part of. show how many megame- ters (roughly) make up the earth's circumference, and how many cubic micrometers of water weigh a microgram. YB I 1^^^ 995888 ^^"^ THE UNIVERSITY OF CALIFORNIA UBRARY