THE SENIOR ARITHMETIC FOlt GRAMMAR SCHOOLS BY CHARLES E. WHITE PRINCIPAL FKANKLIX SCHOOL, SYRA( i ,SK. BOSTON, U.S.A. IX C. HEATH & CO., PUBLISHERS 1897 COPYRIGHT, 1897, BY D. C. HEATH & CO. TYPOGRAPHY UY C. J. PETERS & SON, BOSTON. PBLSSWOKK BY ROCKWELL & CHUKCHI LL. PREFACE. THE SEXIOK ARITHMETIC is intended for use in the higher grammar grades, beginning with the sixth year. The first eight pages are devoted to definitions in review, covering fundamental subjects which should be mastered before the book is taken up. The senior arithmetic proper begins with decimals. Though largely review matter, denominate numbers are treated with considerable fulness. Teachers are expected, however, to use their discretion as to omissions in this subject. It has been the author's aim to employ such definitions, solutions, explanations, and rules as can be readily com- prehended and applied by the pupils, unaided by the teacher. While care has been exercised in selecting a great variety of practical business problems and in arran- ging them progressively, the development of mental power has been kept constantly in view. The arrangement of this book is topical, but subjects previously studied are kept fresh in the minds of the pupils by frequent carefully prepared reviews. The practice of referring percentage problems back to the original questions of relation has proven highly suc- cessful in the author's experience. 85984 IV VKKFAOK. Thanks are due to the various superintendents of city schools who kindly furnished copies of recent examination questions, which largely constitute the test problems of this book. The author has also received invaluable aid from many leading educators in the State of New York, all of whom he desires to thank most cordially. C. E. W. SYRACUSE, N.Y., Oct. 18, 18%. CONTENTS. DEFINITIONS, RULES, AND PRINCIPLES IN REVIEW PAGE Notation, Numeration 1 Addition, Subtraction, Multiplication 2 Division, Factors, Multiples .* 3 Least Common Multiple 4 Greatest Common Divisor 4 Cancellation 5 Common Fractions 5 Addition arid Subtraction of Fractions 7 Multiplication and Division of Fractions 8 DECIMAL FRACTIONS To read a decimal 9 To write a decimal 10 To reduce two or more decimals to a common denominator . . 11 To reduce a common fraction to a decimal 12 Addition of decimals 13 Subtraction of decimals 13 Multiplication of decimals 14 To multiply by 10, 100, 1000, etc 15 Division of decimals 16 To divide by 10, 100, 1000, etc 1.7 Parts of 100 and 1000 18 To multiply by 25 18 Aliquot Parts of a dollar 19 Review of decimals 21 Accounts and Bills 25 Indicated Operations "27 MISCELLANEOUS REVIEW Factors, Multiples, Divisors, and Cancellation 30 Common Fractions 32 Review questions 39 COMPOUND NUMBERS Linear measure 43 Surveyors' measure 44 v VI CONTENTS. COMPOUND NUMBERS (Continued) PAGE Square measure 44 Cubic measure 44 Liquid measure 45 Apothecaries' fluid measure 45 Dry measure 45 Avoirdupois weight 45 Troy weight 46 Apothecaries' weight 46 Measure of time 46 Circular measure 47 Federal money 48 English or Sterling money 48 Counting table 49 Paper table 49 Reduction descending 49 Reduction ascending 51 Review problems 53 To reduce denominate fractions to integers of lower denomi- nations 55 To reduce denominate numbers to fractions of higher denomi- nations 56 To find what part one denominate number is of another . . . ."> Addition of compound numbers 58 Subtraction of compound numbers 61 Difference between dates 63 Multiplication of compound numbers 64 Division of compound numbers 65 Miscellaneous problems 69 MEASUREMENTS, Surfaces 72 Carpeting rooms 77 Plastering and painting 79 Papering walls 81 Board measure 82 Miscellaneous problems 84 MEASUREMENTS, Solids 86 Wood measure 89 Capacity of bins 90 LONGITUDE AND TIME 91 Standard time 92 Review questions 94 THE METRIC SYSTEM Linear measure 96 Surface measure 100 CONTENTS. Vll THE METRIC SYSTEM (Continued) PAGE Volume measure 102 Capacity measure 104 Measure of weight . . . ' 105 Review questions 10j5 General review 107 PERCENTAGE 114 Profit and loss 124 Commission 12(J Insurance 130 Trade discount 132 Taxes 134 Duties 13(5 Review questions 137 Miscellaneous review of percentage 137 SIMPLE INTEREST 143 Exact interest 149 To find the rate, when principal, interest, and time are given . 150 To find time, when principal, interest, and rate are given . . 151 To find principal, when interest or amount, rate and time are given 152 Promissory notes 153 Partial payments, U. S. rule 156 Merchants' rule 159 Compound interest 160 Review of interest 161 TRUE DISCOUNT 165 BANK DISCOUNT 167 To find the face of a note when the proceeds, time, and rate are known 173 Review of discount 174 STOCKS AND BONDS 176 Bonds 179 Miscellaneous .....' 182 AVERAGE OF PAYMENTS 183 Review questions 188 RATIO AND PROPORTION Ratio 189 Simple proportion 190 Compound proportion 195 Partnership 198 Review questions 202 Vlll CONTKNTS. INVOLUTION AND EVOLUTION PACK Involution 203 Evolution 204 Square root 205 . Right-angled triangles 211 Similar surfaces 213 Cube root 214 Similar solids 220 Questions 221 General review 222 TEST QUESTIONS 236 Decimals 244 Denominate numbers 248 Percentage 255 Interest and discount 263 Proportion and partnership 271 Involution and evolution 274 Miscellaneous 276 MENSURATION Surfaces 285 Solids 287 Review of mensuration . . 290 SENIOR ARITHMETIC. DEFINITIONS. 1. A Unit is one, or one thing. 2. A Number is that which tells how many. 3. The Unit of a Number is one of its units. 4. Numbers having the same unit are Like Numbers. 5. A number not applied to any particular object is an Abstract Number ; as 6, 11, 15. 6. A number that is applied to a particular object is a Concrete Number; as 6 men, 11 IV).. 1T> days. 7. An Integer is a whole number. 8. Expressing numbers by figures or letters is called Notation. 9. Arabic Notation is expressing numbers by figures. 10. Roman Notation is expressing numbers by letters. 11. Naming the places of figures and reading numbers is Numeration. 12. A figure standing alone expresses units. 13. When figures stand side by side, the right-hand figure expresses units, the next tens, the next hundreds, etc. 1 2 SENIOR A KITH MET 1C. 14. The value of a figure, without regard to its place, is its Simple Value. The value of a figure with reference to its place in a number is its Local Value. NOTE. In the number 5555, the simple value of each figure is 5. The local value of the right-hand figure is 5. Of the second, 50. Of the third, 500. Of the fourth, 5000. 15. Each group of three figures, beginning with units and counting to the left, is a Period. TO READ NUMBERS. Rule. Begin at tJie right, and separate the numbers into groups of three figures each, using the comma. Begin at the left, and read the number in each group, giv- ing to it the name of that group, Xo name is given to the number in the last f/ro///>. 16. Addition is the process of uniting two or more like numbers into one sum. 17. The result of addition is called the Sum or Amount. 18. Subtraction is the process of finding the difference between two like numbers. 19. The number from which we subtract is called the Minuend, and the number subtracted, the Subtrahend. The result in subtraction is the Difference or Remainder. 20. Multiplication is the process of finding a number that is a given number of times another number. 21. The Multiplicand is the number multiplied. 22. The Multiplier is the number multiplied by. 23. The result of multiplication is called the Product. PRINCIPLES. The multiplier must be an abstract number. The multiplicand and product are like numbers. The prod- uct is the same in whatever order the numbers are taken. DEFINITIONS. 8 24. Division is the process of finding how many times one number is contained in another. 25. The number divided is the Dividend. 26. The number by which the dividend is divided is the Divisor. 27. The result of division is the Quotient. 28. When the divisor is not exactly contained in the dividend, the part of the dividend that is left is the Remainder. 29. PRINCIPLES. The remainder and dividend are like numbers. When the divisor is abstract, the dividend and quotient are like numbers. When the dividend and divisor are concrete, the quotient is abstract. 30. The Sign of Division is -r- , and when placed between two numbers signifies that the first is to be divided by the second. FACTORS AND MULTIPLES. 31. A Factor of a number is any integer that will exactly divide it. 32. A number that has no factors except itself and 1 is a Prime Number. 33. A number that has other factors besides itself and 1 is a Composite Number. 34. A prime number used as a factor is a Prime Factor. 35. What are the prime factors of 1155 ? Rule. Divide the number by any prime fac- tor that will exactly divide it. Divide the quotient in the same manner. Continue the division until a quotient is found that is a prime 5, 3, 7, and 11 number. The divisors and the last quotient are the primp factors. 4 SENIOR ARITHMETIC. 36. Numbers that have no common factor or divisor are Prime to Each Other. 37. A Multiple of a number is a number that exactly contains that number. 15 is a multiple of 5. 38. A number that is a multiple of two or more num- bers is a Common Multiple of them. 24 is a common multi- ple of 4 and 3. 39. The least multiple of two or more numbers is their Least Common Multiple. 12 is the least common multiple of 3 and 4. 40. What is the least common multiple of 18, 27, and 30? Rule. Divide by any prime number that 2/18, 27, 30 is exactly contained in two or more of 3 /9, 27, 15 the numbers, and bring doivn the quo- 3/3, 9, 5 tient and undi- 1, 3, 5 vided numbers. 2x3 X 3 x 3 X 5 = 270. Ans. Divide again as before, continuing th<> dh'ision until the quotients and undivided numbers are prime to each otluT. The product of the divisors, quotients, and undivided numbers is tJie least common multiple. 41. A number that is a factor of two or more numbers is a Common Divisor of them. 5 is a common divisor of 30 and 40. 42. The greatest factor of two or more numbers is the Greatest Common Divisor of them. 10 is the greatest com- mon divisor of 30 and 40. 43. 'What is the greatest common divisor of 324 and 372? Rule. Divide the greater 324/372(JL_ number by the less, then the divisor by the remain- 48 / 324 (_6_ tier, until there is no re- tl_Ji mainder. The last divisor ' ^JR -is the greatest common divisor. When there are more than two numbers, first find the greatest common divisor of two of them, then of this divisor, and a third number, until all the numbers are used. CANCELLATION. 44. Cancellation is a process of shortening indicated di- vision by rejecting the same factors from both dividend and divisor. 45. PRINCIPLES. Rejecting the same factor from divi- dend and divisor divides both by that factor. Dividing both dividend and divisor by the same number does not affect the quotient. COMMON FRACTIONS. 46. A Fraction is one or more of the equal parts of a unit. The unit of which the fraction is a part is called the Unit of the Fraction, and one of the equal parts is called the Fractional Unit. Two or more fractions hav- ing the same fractional unit are Like Fractions. 47. A fraction is written with two numbers, one above the other, with a line between ; as. i. 48. The number below the line in a fraction is the De- nominator, and shows into how many equal parts the unit is divided. 49. The number above the line in a fraction is the Nu- merator, and shows how many of the parts are taken. b SENIOR ARITHMETIC. 50. The numerator and denominator are the Terms of a Fraction. 51. A Proper Fraction is a fraction whose value is less than 1. Its numerator is less than its denominator; as, i, I- 52. An Improper Fraction is a fraction whose value is 1 or more than 1. Its numerator is equal to, or greater than, its denominator ; as, , |, ^f . 53. An integer may be written in fractional form by giving it 1 for a denominator ; as, 5 = f . 54. A Mixed Number is a number composed of an integer and a fraction ; as, Sj, 5|. 55. Reduction of Fractions is the process of changing their forms without changing their values. 56. PRINCIPLE. Multiplying or dividing both terms of a fraction by the same number does not change the value of the fraction. 57. To reduce a fraction to Higher Terms. Rule. Multiply both terms IHJ the same number. 58. | equals how many 60ths ? NOTE. Since the new denominator must be 60, or twelve times the given denominator, the new numerator must be twelve times the given numerator, therefore I -jj-$. 59. To reduce a fraction to its Lowest Terms. Rule. Divide both terms by any common factor ; divide the result in the same way until the terms are prime to each other. If the terms are large, divide by their greatest common divisor. 60. To reduce a Mixed Number to an Improper Fraction. Rule. Multiply the integer by the denominator of the frac- tion, add the numerator to the prvduM, and write the result over the denominator. DEFINITIONS. 7 61. To reduce an Improper Fraction to an Integer or Mixed Number. Rule. Divide the numerator by the denominator. 62. A number that is the denominator of two or more fractions is the Common Denominator of those fractions. 63. XOTE 1. The common denominator of two or more frac- tions is a common multiple of their denominators. . XOTE 2. The least common denominator of two or more frac- tions is the least common multiple of their denominators. 64. To reduce fractions to a Common Denominator. Rule. Multiply the denominators together for tin' common denominator, divide it by the denominator of each frac- tion, and multiply both terms by the quotient. NOTE. To find the least common denominator, find the least common multiple of the denominators, and proceed as above. ADDITION OF FRACTIONS. 65. Rule. If the fractions are not like fractions, reduce them to a common denominator, add their numerators, and place the sum over the common denominator. Re- duce the result to lowest terms. If the result is an improper fraction, reduce it to an integer or mixed number. To add mixed numbers, add the integers and fractions separately, and unite the results. SUBTRACTION OF FRACTIONS. 66. Rule. If the fractions are not like fractions, reduce them to a common denominator, and write the difference of their numerators over the common denominator. To subtract mixed numbers, subtract integers and fractions separately. 8 SENIOR AK1THMKTIC. MULTIPLICATION OF FRACTIONS. 67. Rule. Reduce integers and ////,/w/ mi miters to imj>r<>in't< fractions, ami multiply tin- HI' nu>>'r> ; T fo = .07. Thousandths are written like mills, with three decimal figures ; thus, jfifr = .125 ; T ifo = .016 ; TTy % = .004. Ten-thousandths require four decimal figures ; hundred- thousandths, five ; millionths, six, etc. 77. Xame the denominators in the following : .36 ; .08 ; .294 ; .1406 ; .0001 ; .263402. Change to decimals: -^ ; T Vo 5 o 5 T oon of tlif right-hand figure. Read the following numbers : 1. .7 6. .16984 11. .500 2. .07 7. .10016 12. 4.98625 3. .007 8. .0000054 13. 38694.06 4. .700 9. : W. 18006 14. 9.98463004 5. .03065 10. .0005 15. 235.850062 10 SENIOR ARITHMETIC. 16. 100.000104 18. 3543.4536982 20. 303.303303 17. 9.1632002 19. 30.3303303 21. 9.999999 80. To write a decimal. Rule. Write the numerator, prefixing ciphers when necessary to express the denominator, and place the point at the left. NOTE. There must be as many decimal places in the decimal as there are ciphers in the denominator. Express decimally : 22. Four tenths. Seventeen hundredths. Five hun- dredths. Three hundred twenty-five thousandths.. Five thousandths. Fifteen thousandths. Nineteen, and seven hundred twenty-four thousandths. 23. Seven thousand five hundred four ten-thousandths. Sixteen, and 125 ten-thousandths. Six ten-thousandths. Five thousand ten-thousandths. 24. Seventeen thousand two hundred eleven hundred- thousandths. Four hundred-thousandths. Fifteen hun- dred-thousandths. Eighteen, and two hundred sixteen hundred-thousandths. One hundred twelve hundred-thou- sandths. 25. Twenty-nine hundredths. Twenty -nine thousandths. Twenty-nine ten-thousandths. Twenty-nine hundred-thou- sandths. One and one tenth. One and one hundredth. One and one thousandth. One and one ten-thousandth. One and one hundred-thousandth. 26. 324 and one hundred twenty-six millionth^. 4582 and 36242 hundred-thousandths. Seventeen mil Months. Five hundred-thousandths. Twenty-four, and three thou- sand four hundred six ten-millionths. 27. 10 millionths. 824 ten-thousandths. 31 hundredths. DECIMAL FRACTIONS. 11 216 hundred-thousandths. 7846 hundred-millionths. Four and 15 hundred-thousandths. 28. A 32 ' ToWoU 36. T 3 40 ' 29. T W 33. T $JHi*iF 37. T i n 41. 30- T Vo% 34. T^Wooo 38. 500 t V 42. 31- iVoVo 35. 15 T o%o 39. TT $fo 43. REDUCTION OF DECIMATE. 81. PRINCIPLES. Ciphers annexed to decimals do not change their- value. For each cipher prefixed to a decimal, the value is dimin- ished ten-fold. The denominator of a decimal when expressed is always 1 with as many ciphers as there are decimal places in the decimal. 82. To reduce two or more decimals* to a Common Denomi- nator. Rule. Annex ciphers so that each decimal will have the same number of decimal figures. 83. Reduce to a common denominator : 44. .5, .017, .1256, .000155, 29.803. 45. .80062, 305.24, 70.5, 3.85263. 46. .1. .0001. 1000.001, 1 .0100385. 47. .26, .13682, 9.4, 25., 8.63521. 84. Reduce .375 to a common fraction. .375 as a common fraction is fVo 5 o- This i n lowest terms = 1- Rule. Write the numerator, omitting the point. Supply f//f denominator, lowest terms, 12 SENIOR ARITHMETIC. Reduce to common fractions : 48. 1.24 53. .325 58. 16.144 49. .16 54. .113 59. 28.3695 50. .325 55. .7282 60. 34.000010 51. .098 56. 2.25 61. 25.0000100 52. .875 57. .2425 62. 1084.0025 85. 63. Reduce .371 to a common fraction. a-i-fl.. An*. 64. .12J 65. .061 66. .621 67. .16 68. .33^ 69. .831 70. .87 71. .66f 72. .36| 86. To reduce a common fraction to a Decimal. Reduce f to a decimal. | = 3 times J. 3 = (3.0), 30 tenths. J of 3.0 = (.7), 7 tenths, and 2 tenths remainder. 2 tenths = 20 hun- dredths. | of .20 = .05. Hence = .7 + .05 = .75. Rule. Annex decimal ciphers to the numerator, and divide by the denominator. Point off from the r'ujht of the quotient as many places as there are ciphers annexed. NOTES. A decimal cipher is a cipher at the right of the decimal point. If there are not enough figures in the quotient, prefix ciphers. The division will not always be exact. In such cases write the remainder over the divisor as a common fraction, or place the sign + after the decimal to show that the result is incomplete. Thus, \-= .1425 or .142+. 87. Reduce to decimals : 73. $ 77. A 81. I 85 - iV 89 - 66 it 74. | 78. | 82. o\ 86 . ^5 90 . 25.121 75. | 79. | 83. | 87. 12^ 91. 161 76. 80. J 84. T \ 88. 33^ 92. 16.25^ DECIMAL FRACTIONS. 13 ADDITION. 88. Add .35, 4.375, 28.3&6S. Rule. Write the numbers so that decimal points .35 stand in a column. Add as in integers, and 4.375 ^ place the point in the sum directly under the z^r * points above. Find the sum : 93. 24.36 94. 38.28006 1.358 1.005 .004 2.16 1632.1 1873.148| 96. .175 + 1.75 + 17.5 + 175. + 1750. 97. 145. + 14.5 + 1.45 + .145 + .0145. 98. 32.58 + 28963.1 + 287.531 + 76398.9341. 99. 1. + .1 + .01 + .001 + 100 + 10. + 10.1 + 100.001. 100. 1.923 + .008 + 251.47 -f 1.961 + 0.0543 + .006 + 18.7. 101. Add 750.3521, 698.42001, .005321, 3.5, 749.006984, 36950.06, 875.942, 286.753. 102. Add 5 tenths ; 8063 millionths ; 25 hundred-thou- sandths ; 48 thousandths ; 17 millionths ; 95 ten-mil- lionths ; 5, and 5 hundred-thousandths ; 17 ten-thousandths. 103. Add 24J, 17^, .0058, 7i, 9 T V- SUBTRACTION. 89. Rule. Write the numbers so that the decimal point of the subtrahend stands directly under the decimal point in the minuend. Subtract as in integers, and place the point directly under the points above. NOTE. It is sometimes convenient to give the decimals the same denominator by annexing ciphers. 14 SENIOR ARITHMETIC. 104. From 6.008 105. 38. 106. 26.34 107. 16.2600 Take 3.154 .356 - 1.28983 1.0001 108. 32.90596 - 75 114. .00011 - .000011 109. 9.5 - 3.35006 115. 10 - .1 + .0001 110. 856.2 - 8.562 lie. 8.75 + .95 -f .125 111. .1 .00001 117. 16 .00001 -f 27.69852 112. 1000 - .001 118. 2.5 .09 + 1.85 - 1.283 113. 20 - .00205 119. 83.1 - 8.31 + .831 120. From one thousand take five thousandths. 121. Take 17 hundred-thousandths from 1.2. 122. From 8.5 take eighty-four hundredth s. 123. Find the sum of 500 thousandths and 5 hundred- thousandths and from it subtract y\. 124. From 17.37^ take 14.161. 125. Find the difference between T 3 ^% and T fo. 126. From 10 take T V ; T ^ ; 4.98 ; 1.05. 127. From one million and one millionth take one tenth. 128. From 1 tenth take 1 millionth. 129. Which is the greater and how much, one tenth or 100 thousandths ? 130. Prove tha.t ^ and .500 are equal. MULTIPLICATION . 90. Every decimal equals a corresponding common frac- tion, and for each cipher in its denominator there is a deci- mal figure in the decimal fraction. T$O X fV = Tinb- (Three ciphers in the denominator.) .05 x .3 = .015. (Three decimal places in the decimal.) Rule. Multiply as in integers, and give to the product as many decimal figures as there are in both multiplier and multiplicand. DECIMAL FRACTIONS. 15 ! NOTE. If there are not figures enough, prefix ciphers. Ciphers at the right of a decimal have no value, and may be omitted. Find the products : 1. .38 x 1.6. 11. 1.04 x 6. 2. .015 X .05. 12. 327f X 4f . 3. 7i X 3.4. 13. 58.42 x 20.06. 4. 50 X .304. 14. .0001 X 1000. 5. 2.65 x .104. 15. .325 x 12i. 6. 257 X .354. 16. .333 x .333. 7. .296 X 124. 17. .001542 X .0052. 8. 1.001x1.01. 18. 26x36.82. 9. 13.33 X 1.3. 19. 2.84. x 3J. 10. 25.863 X 4i. 20. 11.11 X 100. 91. To multiply by 10, 100, 1000, etc. 21. Multiply 1.265 by 100. B-emove the point one place to the right for 1.265 each cipher in the multiplier. 100 Do not write the multiplier. 126.500 Oral. 22. 3689.25 X 10. 27. .5 X 100. 23. 38.6422 x 100. 28. .5 X 1000. 24. 269.8342 X 1000. 29. 384.2 x 10. 25. 100 X 23.85. 30. .3659 X 100. 26. 1000 X 1.52. 31. .1000 X .01. 92. To multiply by 200, remove the point to the right and multiply by 2. Oral. 32. 86.44 X 200. 35. 750.5 X 5000. 33. 3.894x3000. 36. 1.892x2000. 34. 88.42 X 20. 37. 156.2 X 200. 16 SUNIOU A1MTHMKT10. 93. Written. 38. Find the product of 1 thousand by one thousandth. 1 million by one millionth. 39. Multiply 700 thousandths by 7 hundred-thousandths. 40. Multiply the sum of 2 millionths and 10 thousandths by their difference. 41. Multiply together .35, 18.5, 28.004. DIVISION. 94. Since in multiplication there are as many decimal places in the product as in both multiplier and multipli- cand, in division the quotient must have as many places as the number of places in the dividend exceeds those in the divisor. 1. Divide 12.685 by .5. SOLUTION. Since there are three decimal places .5/12.685 in the dividend and one in the divisor, there must be 25 37 two in the quotient. Rule I. In all cases divide fs in integers, then place the decimal point. 2. Divide 399.552 by 192. Rule II. When the divisor is an integer, 2.081 place the point in the quotient directly ooli over the point in the dividend in long ~T^^ division (directly under in short divis- 1536 ion). Prove by multiplying divisor by Jcj7> quotient. 192 PRINCIPLE. Multiplying both dividend and divisor by the same number does not change the quotient. 3. Divide 28.78884 bv 1.25. DECIMAL FRACTIONS. 17 Rule III. When the divisor contains 23.031 + (IfcItHftl figures, move the point in 1-25 /28.78'884 bnth divisor and dividend as many orro to the right as there are deci- ^ mal places in the divisor (this, in Ex. 3, in a It i plies both by 100), then place the point in the quotient as if ~134 the divisor were an integer. ^25 Q NOTE 1. The new points may be placed on a line with the tops of the figures, and the original point may stand to preserve the reading of the decimals. NOTE 2. If the quotient does not have a sufficient number of figures, prefix ciphers. NOTE 3. Before commencing to divide, see that there are at least as many decimal places in the dividend as in the divisor. ^ NOTE 4. If there is a remainder after all the figures of the divi- dend are used, annex decimal ciphers and continue the division. NOTE 5. It is not usually necessary to have more than four decimal figures in the quotient. Find the quotients : 1. .288 4- .64. 11. 315.432 -4- .132. 2. .36 -4- 600. 12. 1.5906 -=- 241. 3. 144 -4- .12. 13. 36.25 -j- 1.25. 4. .25 -4- .2500. 14. 75 -4- .0125. 5. .12 -4- 30. 15. 125 -4- .12J. 6. .96 -4- .08. 16. 25 -4- .25. 7. 384.526 -*- 1.16. 17. .25 -5- 25. 8. 1440 -^ .0018. 18. 1000 -j- .001. 9. 1.225 -5- 4.9. 19. .001 -*- 1000. 10. 9.156 -4- 12. 20. 18.65 H- 100. 95. To divide by 10, 100, 1000, etc., remove the point one place to the left for each cipher in the divisor. 18 SKNIOll ARITHMETIC. Oral. 21. 38.64 -h 10. 25. 3.91 -h 1000. 22. .5 -5- 10. 26. 1.155 -f- 100. 23. 558 -f- 100. 27. 398.42 -4- 1000. 24. 1684.32 -f 1000. 28. 2.46 -*- 200. NOTE. To divide by 200, remove the point to the left, and divide by 2. 29. 386.54 -*- 2000. 31. 865.45 -5- 5000. 30. 38.28 H- 400. 32. 2.5 -=- 500. PARTS OF 100 OR 1OOO. 96. 1. What part of 100 is 12$ ? 25 ? 33J ? 2. What part of 1000 is 125 ? 250 ? 333$ ? 3. How much is 1 of 100 ? Of 1000 ? 4. How much is | of 100 ? Of 1000 ? 5. Find i of 100. Of 1000. 6. How much is 25 times 24 ? SOLUTION. 100 times 24 = 2400. 25 times 24 = i as much as 100 times 24, = 600. 97. To multiply by 25, annex two ciphers, and take 1 of the result. 7. Tell how to multiply by 33^ ; by 12^ ; by 250 ; by 125 ; by 333. Oral. 8. 36 X 25. 11. 444 x 25. 14. 333J X 30. 9. 48 X 12i. 12. 320 X 33i. 15. 168 X 250. 10. 24 X 33^. 13. 125 X 80. 16. 12$ X 48. 17. What cost 650 oysters at 50 cents a hundred ? SOLUTION. 650 -f 100 = 6.50 hundred. $.50 x 6.50 = ? DECIMAL FRACTIONS. 19 18. What will be the cost of 3850 laths at 40 cents a hundred ? 19. What is the freight on 685 pounds of baggage at $1.10 per 100 Ib. NOTK. C. means 100 ; M., 1000. 20. What is the cost of 4862 ft. of pine lumber at $30 per M. ? 21. Find the cost of 38,586 bricks at $8.25 a thousand. 22. What will 583 heads of cabbage cost at $3.50 a hun- dred ? 23. At $3.50 a thousand, what will be the cost of 7800 shingles ? 24. At $8.25 per C., what will be the cost of 2864 Ib. of dried fish ? 25. At $50 per M., what will be the cost of 3865 feet of cherry lumber ? 26. What is the cost of laying 5890 bricks at $9.00 a thousand ? To find the cost of merchandise sold by the ton, divide the price by 2 and proceed as above. 27. Three loads of hay weigh 7894 -lb. What will the hay bring at $12 a ton ? NOTE. 1000 lb. will cost i of $12 = $6. $6 x 7.984 = ? 28. What cost 48986 lb. of railroad iron at $35 a ton ? 29. Four loads of coal weigh respectively 3896 lb., 3524 lb., 4106 lb., and 3123 lb. What is the cost of the coal at $4.82 a ton. ALIQUOT PARTS OF $1.OO. 98. The Aliquot Parts of a number are the numbers which are exactly contained in it. The aliquot parts of 100 are 5, 20, 12, 16 j-;, 33, etc. 20 SENIOR ARITHMETIC. 99. The aliquot parts of $ 1, commonly used, are as follows : 6 cents = $ T V 25 cents = #J. 8 cents = $ T V,. 33 J cents = $J. 12 cents = $. 50 cents = $. 16| cents == $. 1. What is the cost of 69 books at 16f / each ? SOLUTION. 60 books will cost 69 times 16f<', or 69 x $& = $-\ 9 $11.50. An*. 100. Oral, Multiply : 2. 33 cents by 36. 5. 25 cents by 40. 3. 12 cents by 24. 6. .75 cents by 4. 4. 61 cents by 32. 7. What is the cost of : 48 Ib. of bacon at 12 / a pound ? 80 hand balls at 50/ each ? 36 yd. of ribbon at 33/ a yard ? 80 Ib. of candy at 25/ a pound ? 101. Written. 8. Find the cost of the following : 66 Ib. of pork at 12 J/. 148 Ib. of veal at 16|^. 48 boxes of strawberries at 25 /'. 48 Ib. of honey at 25/. 64 bars of soap at 6i/. 60 doz. of eggs at 16f/. Find the cost of : 9. 1580 Ib. of sugar at 6^/ a pound. 10. 500 books at 25/ each. 11. 16 yd. of dress-goods at 33^ a yard. DECIMAL FRACTIONS. 21 12. At 25/ a pound, how many pounds of butter can be bought for $8.00 ? SOLUTION. As many pounds as 25^ or $| is contained times in $8 -i- $i = 8 x f = 32 Ibs. Am. 102. Oral. Divide : 13. $5 by 33$?. 16. $3 by 8^/. 14. $6 by 6J/. 17. $4 by 25/. 15. $9 by 12i/. 18. $4 by 66X- 19. At 25^ each, how many hats can be bought for $6 ? 20. At $1 a pound, how many pounds of cheese can be bought for $6 ? 21. At 33$ f a yard, how many yards of linen can be bought for $ 10 ? 103. Written. 22. At 75/ a bushel, how many bushels of barley can be bought for $125 ? 23. When butter is 25 8,j,, 4 T v, 708H4- Change to common fractions, then to simple decimals : 15. .11, .07|, .18$, .107|, .121, .08J, .22|, .045&, .37|, .38f, .54$, .000051, .78|, .38J. Reduce to a common denominator and add : 16. 50.06, 367.41, 200.200, .12304, 40.0056, 7.5620, .096071. 17. 1301.6, 904.02, .547, .0009, .00001, 218.94, 203.410, 1000, .01. 18. 100.101. 82.4, 401.009, .00038, 60702, 10.10, 574.68139. 19. 5.628, 850.002, 9.00256, 37.0005, 724.6811, 3759, 7000.0036, 2.25. 20. $11.78, $347, $5.06, $218, $20.07, $42.0244, $7.104, 837.625. 21. 4.76, .390,.. 0915, .00207, 841, 63.2, .00234, 1.43, .00536. 22. .00908, .0371, 24.5, 7.03, .0127, 354, .000781, .0436, 20.7354. Subtraction. 23. 5.74 3.23 = ? 26. 367. 1.52 = ? 24. .876 .343 = ? 27. 200 .02 = ? 25. 67.5 41.5 = ? 28. Which is greater, f or 4 tenths ? 29. How much more is $20 than $17.84 ? 30. From two million take two millionths. 31. T bought 4 farms ; one contained 19.368 acres ; one, 27.96 acres; one, 473.0008 .acres ; and the last one, 73.7561 acres. I sold 300.25 acres ; how much land had I left ? 32. From 1 inch take one ten-thousandth of an inch. 24 SENIOR ARITHMETIC. Multiply : 33. 7.945 by .3. 37. 7.853 by 23.16. 34. 350 by .42. 38. 1.36 X 20.04 = ? 35. One tenth by one hundredth. 39. 27.27 X 4.0004 = ? 36. 25 units by 25 tenths. 40. If wheat is worth $.38 a bushel, what will 117.75 bushels cost ? 41. Apples sell for 1.28 a bushel ; how much money will 24 barrels bring, each containing 2^ bu. ? 42. Find the cost of 3.325 Ib. of butter at 18.75 cents a pound. 43. What will 6| yd. of broadcloth cost at $1.375 a yd.? 44. A boy paid $.125 a dozen for 1.75 dozen eggs ; what did they cost him ? 45. 3.64 x .0002 x 1.756 x 4.004 = ? Divide : 46. 1738.89 by .00417. 52. 42.475681 by .29. 47. 1237.6 by 26. 53. 40.20 by .000012. 48. 36.11 by .021. 54. $302.03 by 200. 49. 2.38 by .17. 55. 64.64006 by .002. 50. 36.82 by .0003. 56. 12.9643 by 18.4. 51. 437.96 by 2.8. 57. 759.806 by 90.3. 58. 16 + 3.06 - | + .002 - 2.1 + .03 - j + .00 = ? 59. i + | - -65 + .5 + I - J + 3.14 = ? 60. Find the product of .003 multiplied by .06, and divide it by 3. 61. A certain decimal divided by 1000 is 35.002. What is one fifteenth of the decimal ? ACCOUNTS AND BILLS. 25 62. The sum of two numbers is 306.52 ; one of them is 100. What is the other ? 63. A man spent $450, which was .125 of his money. How much money had he ? 64. Mr. A. bought a cow for $45, which was .375 of what he paid for a horse. How much did he pay for the horse ? 65. John spent .75 of his money for a book and had $50 left. How much had he at first ? ACCOUNTS AND BILLS. 105. An Account is a record of indebtedness for articles bought or sold, cash paid or received, or services rendered. 106. A Debtor is a person who owes a debt. 107. A Creditor is a person to whom a debt is owed. 108. A Bill is a written statement of a debtor's account, made by the creditor. 109. A Receipt is a creditor's written acknowledgment that he has received payment of part or all of a debt. 110. A bill is receipted when its payment is acknowl- edged in writing, by the creditor, or by some authorized person. NOTE. The sign @ is for at. Dr. is for debtor, and Cr. for creditor. JAMES P. BARNES, BILL FORMS. SYRACUSE, tf. Y., July 1, 1896. Bought of BEY BROS. & Co. 50 yd. Brussels Carpet @ $1 15 I 24 Oil Cloth 35 4 doz. pair Merino Hose " 3 50 2 Willow Chairs " 4 50 $ SENIOR ARITHMETIC. RECEIPTED BILL WITH CREDITS. ROCHESTER, N. Y., Jan. 2, 1896. MRS. JOHN F. WHITE, To BURKE & WHITE, Dr. Nov. 6 4 Ib. Coffee @ $ 27 $ 6 28 Ib. Sugar sy 2 18 5 gal. Molasses " 60 Dec. 11 18 Ib. Rice " 7X it 15 2 bbl. Potatoes " 1 80 tt 19 28 Ib. Butter " 21 CR. Nov. 18 Cash 3 50 Dec. 28 u 4 75 Balance due, Received payment, Jan. 15, 1896, BURKE & WHITE, By JOHN R. PIERCE. FORM OF A RECEIPTED BILL. NEW YORK, June 30, 1896. JEROME A. PHELPS, In account ivith D. 0. POTTER & Co. May 14 12 bbl. Flour @ $6 50 $ it 14 6 tubs Butter, 684 Ibs. " 24 June 10 5 bbl. Beef " 25 28 25 450 Ib. Ham 9% Received payment, . D. O. POTTER & Co. INDICATED OPERATIONS. 27 INDICATED OPERATIONS. 111. The Parenthesis, ( ), indicates that all the numbers contained therein are to be taken together. 112. Brackets, [ ], Braces, { }, and the Vinculum, ~~ ~~, have the same use as the parenthesis. 113. When the parenthesis is not used, operations indi- cated by X or -f- must be performed first. Thus, 1. 12 -.-4x2+ 36 -j-4-2x4 = ? SOLUTION. 12 -f 4 x 2 = 6. 36 -f- 4 = 9. 6+9-8 = 7. Ans. 2x4=8. 2. 4 + 3x2 = ? 5. 4 x (3 + 2) = ? 3. (4 + 3) X 2 = ? 6. 8 + 4 -r- 2 = ? 4. 4x3 + 2 = ? 7. (8 + 4) H- 2 = ? NOTE. When one parenthesis, brace, or vinculum includes another, first remove the inner one. 114. Find the value of : 1. 15 + 3 X 6 + 10 -*- 5. 2. (6 + 4) x (3 + 2) - (8 X 5). 3. 18 ^3x2 + 8x2-^4- 6. 4. 2 + 12 -^ 4 - (10 + 6 -f- 4) -^ 3. 5. (11 + 4) -s- 3 + 6 X 4. 6. 3 + 4 x 6 -f- (15 + 9 + 3). 7. 164 + 16 - 250 -r- 10 + 16 X 3. 8. 17 + 3x4x6 + 3-f-3 + 3. 9. [39 + 8 ^- 2 + 7] X 6. 10. [6 + 15 x 3 - (6 + 16 -v- 8 + 4)] -f- 8 + 5. 28 SENIOR ARITHMETIC. MISCELLANEOUS. 115. 1. I have four pieces of broadcloth. The first con- tains 13.7642 yd. ; the second, 22.008 yd. ; the third, 15.027 yd. ; and the fourth, 19.255 yd. How many yards in all ? vX& From a piece of ribbon containing 103f yd., 73| yd. were sold. How many yards were left ? 3. How many yards of -muslin at $.12^ a yard will it take for 4 pair of curtains, if each curtain contains 3.375 yd.? 4. I have 14.735 yd. of lace, and desire to cut it into seven equal strips ; how much will there be in each strip ? 5. What will be the cost of a hat at $7.50, a pair of gloves at $1.13, a veil at $1.25, and a parasol at $3.375 ? 6. Arrange the following articles in the form of a bill : 7 qt. of molasses at $.15 a qt., 3 pk. of apples at $1.28 a bushel, 30 Ib. of sugar at $.08^ a pound, and 12 bu. of potatoes at $.29 a bushel. y 7. A grocer bought three bunches of bananas at $1.54 a bunch. The first bunch contained 73 bananas, the second 54, and the third 97. He sold them all at 30/ a dozen ; did he gain or lose, and how much ? 8. The first year in business a grocer made $2374.68, the second $1529.47, and in the third year he lost $300. His expense each year averaged $928.45 ; how much money had he gained at the end of three years ? 9. What will 9 barrels of flour cost, if 28 barrels cost $173.60 ? 10. I bought 437 heads of lettuce at $5 a hundred, and sold them at $.08 a head ; what was my gain ? MISCELLANEOUS. 29 Find the cost of : 11. 6824 Ib. of coal at $4.68 a ton. 12. 2384 Ib. of coal at $5.67 a ton. i/is! 8972 ft. of lumber at $35.40 a thousand. 14. 6854 Ib. of hay at $16.50 a ton. 15. 4836 bricks at $9.45 per M. 16. 895 ft. of lumber at $19.75 per M. 17. What part of 4.50 is 3.33^ ? 18. What part of 3.625 is 1.5 ? /i9. What part of 6.2 is 3.25 ? 20. 1.1 is what part of 7.4 ? 21. A father left his son $24000, which was .375 of his estate. What was the value of the estate ? 22. Divide 26 by 2J, and multiply the result by 17.345. 23. Divide f of .375 by f of f of .298. 24. The product of three numbers is 167.7. Two of the numbers are 3.25 and 5.16. What is the other ? 25. What number divided by 2.86 equals .34 ? 26. What number diminished by 38.64 leaves .356 ? 27. A man bought 8.5 yd. of cloth at $3.33^ a yard, 12.4 yd. at $2.75, 18J yd. at $4.375, and 24f yd. at $2.875. How many bushels of corn at 43f cents a bushel will pay for the cloth ? 28. .5 of a number exceeds .45 of it by 20. What is the number ? SOLUTION: .5 .45 =.05. Now the question is, 20 is .05 of what ? 20 -f .05 = 400. 29. At 85^ a yard, how many yards of cloth can be pur- chased for $29.75. 30 SENIOR ARITHMETIC. 30. Divide $785 among A, B, and C, so that C will have $185 more than each of the others. 31. JL__ _^?046_ _ =? .05 .4 x .005 + .002 x .125 32. What part of .876 is .31536 ? 33. If .375 of a ton of coal cost $1.25, what will 7.125 tons cost ? 34. What is .3 of a number when .8 of it is 80 ? 35. How many thousandths in 3 units ? 36. How many thousandths in .1 ? 37. Express * of one hundredth as a decimal. ^38. The salary of the President of the U. S. is $50,000 a year. How much does he receive per day ? 116. Review of Factors, Multiples, Divisors, and Cancellation. 1. Define factor, composite number, prime number, and prime factor. 2. Find the prime factors of 5075; of 9576; of 3150; of 6006. 3. Find the sum of the prime factors of 34650. 4. Find the prime factors of 2310 ; of 17199 ; of 6840. 5. 81158 is the product of what prime factors ? 6. Find the largest prime factor of 12600. 7. What is a common divisor of two or more numbers ? 8. What is the greatest common divisor of two or more numbers ? 9. When are numbers prime to each other. Find the greatest common divisor of : /ID. 672 and 960. 13. 1650 and 1920. 11. 616 and 1012. 14. 696, 1218, and 1160. 12. 272 and 428. 15. 450, 720, and 810. MISCELLANEOUS. 31 16. What is the greatest prime factor common to 4242 and 2626. 17. A grocer had 84 bananas and 126 lemons, which he wished to put into bags, each bag containing the largest, number possible, and each containing the same number. How many could be put into each bag ? 18. A man has three fields containing respectively 14, 18, and 22 acres. He wishes to cut them into the largest possible lots of equal size. How much land will each lot contain ? How many lots will each field contain ? 19. What is a multiple of a number ? A common multi- ple ? The least common multiple ? Find the least common multiple of : ^. 96, 196, 42, and 54. 23. 252, 462, and 1092. 21. 45, 36, 70, and 90. 24. 120, 280, and 308. 22. 36, 40, 42, and 48. 25. 36, 110, 98, and 66. 26. Find the least common multiple of the even num- bers to and including 20. 27. What is the least sum with which I can buy an exact number of chairs at $6, $8, or $5 each ? 28. What is the smallest sum of money that may be expended by using an exact number of nickels, dimes, quarters, or 3-cent pieces ? How many pieces of each kind will the sum contain ? r 29. John can run around a block in 6 minutes, James in 8 minutes, and Henry in 9 minutes. If they start to- gether, how long before they will all be together again at the starting-point ? 30. What is the shortest piece of rope that can be cut into pieces 32, 36, and 44 feet long ? 31. W T hat is cancellation ? 32 SENIOR ARITHMETIC. 32. Of what use is cancellation ? Find results of the following by cancellation : 33 18 X 36 X 48 36 28 x 56 x 30 24 X 6 X 12 14 x 3 X 5 10 X 6 X 4 37 28 X 32 x 7 4x6 14 X 35 X 2 35. 18 X 9 X 10 X 5 3g 34 x 9 X 5 6x8x2 25x17x3 39. 240 X 48 X 70 X 18 -v- 42 x 15 X 54 x 7 = ? 40. Divide the product of 25, 14, and 11 by the product of 15, 7, and 22. 41. How many bushels of wheat at $1.10 a bushel must be given for 6 pieces of cloth each containing 33 yards at 50 cents a yard ? 42. How many cords of wood at $3 a cord will pay for 30 Ib. of sugar at 5 cents a pound ? 43. If 8 men can do a piece of work in 6 days, in how many days can 12 men do it ? Y/ 44. How many pounds of sugar can be bought for $7 if 21 Ib. cost $1.05 ? 45. How many pounds of maple sugar at 12 cents a pound must a farmer exchange for 15 pounds of coffee at 24/ a pound ? 46. A milkman exchanges 8 cans of milk, 30 quarts in a can, at 4 cents a quart, for 3 pieces of sheeting, 40 yards in a piece. What is the price of the sheeting per yard ? MISCELLANEOUS RE VIE W OF COMMON FRA CTIONS. 117. Oral. Y 1. Add \ and | ; ^ and 1 ; f and f ; f and | ; and $ ; | and 1^. 2- i-i = ? i-J = MISCELLANEOUS KEVIEW OF COMMON FRACTIONS. 33 3. Reduce to improper fractions 3i, T|, 8S|, 5*, 7f, 8|, 16|, lof. 4. Reduce to integers or mixed numbers J ? 5 -, - 1 /-, -y 5 -, -y, -> , ft, -W-, 2- u/-l Multiply 16 by ; 45 by f ; 18 by | ; 45 by ft 5 by 9; | by 32; * by 16 ; J by 27. 6. Find product of : f X f ; T 9 T X ^ ; 3J X U ; J X i?5 f X -V; 41X6J. 7. Find | of 24 ; f of 12 ; of 30 ; | of 27 ; f of 45 ; I of 40. 8. Find l of i ; i of f ; f of j ; ^ of f ; | of j ; J of 2J.. 9. Divide f by 3; f by 4; ^ by 12; T e T by 11; 4J by 3; i by 6; 4by6; 10 by $ ; 8 by ; by 8 ; t \ by 22 ; f by. 9. \xm Divide 4 by J ; 8 by J ; 9 by -| ; 16 by f ; 24 by f ; 13 by |; 11 by f; 12 by If. 11. Divide i by | ; by | ; J by J ; f by | ; ^ by f ; f by f ; 5i by 2J. 12. Divide 1 by : ) 1) 8) T^J T> NOTE. 1 divided by a fraction equals that fraction inverted. 13. | of 12 = ? 9 = what part of 12 ? 9 is | of what ? 14. 3X? = ; f *?=* iJ iX? = J; (f + r-ij ) 2 = i ; | - ? = f . 6. What part of 6 is 4 ? J is J ? | is ? 5 is 2\ ? 11 is 5? j is K j is J? | is i? 17. 9 is | of what ? 5 is f of what ? 6 is f of what ? 34 SENIOR ARITHMETIC. 18. Change | to 24ths ; f to loths ; f to 32nds ; to 20ths. 19. A man owning | of a farm, sold ^ of his share. What part did he sell ? How much remains ? 20. At 12i/ a dozen, how many dozen of eggs can I buy for $3 ? ^r * 21. /At 6J/ a box, how much will 8 boxes of berries cost ? v/22. John has 56 cents, and James \ as much. How much have both? 23. A can do a piece of work in 4 days ; B can do the same piece of work in 2 days. What part of the work can each do in a day ? 24. A can mow a field in 3 days, and B in 4 days. What part of the field can they mow in a day if both work together ? A can mow \ of it in 1 day, and B can mow \ of it in 1 day. Both working together can mow the sum of ^ and = ^ of it in 1 day. 25. C can do a piece of work in 2 days and D can do it in 4 days. In what time can they both do it, working together ? C does \ of it in 1 day, and D i of it in 1 day. Therefore both can do i + | = of it in 1 day. Since both can do f of it in 1 day, it will take as many days to do | or the whole of it, as f is contained times in -|, or \\ days. .4ns. NOTE. f divided by f gives the same result as 4 divided by 3. 26. \ of my money is gold, and \ as much is silver. What part of my money is silver ? /27. If a boy can earn $2^ in 1 week, how much can 3 boys earn in 4 weeks ? 28. James sold a book for 28 cents, which was of what it cost him. What did it cost him ? ' MISCELLANEOUS REVIEW OF COMM.6X FRACTIONS. 35 . . 29. The difference between of a number and J of it is 6. What is the number ? SOLUTION. The difference between \ and \ is i. Now the ques- tion is 6 is | of what ? 30. A boy 12 years of age is \ as old as his father. How old is his father ? Written. i/%\.. A farmer having 1200 bushels of potatoes, sold \ of them at one time, \ at another, and 350 bushels at another. How many bushels had he left ? 32. A mechanic whose wages are $5 per day uses -fV of his weekly earnings for board, arid f for clothing and other expenses. How many dollars does he save weekly ? 33. Which is greater and how much, \\ or \\ ? 34. If it takes 27 days to do a piece of work, how long will it take to do f of it ? 35. If a horse is worth $100, and a cow is worth f as much as the horse, what is the cow worth ? 36. John has in the bank $45 and draws out f of it. How much remains in the bank ? 37/ What will 16 pair of shoes cost at f 3| a pair ? v'aS. If a farmer has 23 sheep and sells them at $3 T 9 150| for oxen, and $407f for sheep. How much money had he left ? 54. Mr. Blanchard paid $8 T 9 Q- for shovelling his walk, $5f for trimming his grape-vines, and $6f for sifting his ashes. He gave the man a 20-dollar bill and a c ^lar bill. How much money should Mr. B. receive in retun MISCELLANEOUS REVIEW OF COMMON FRACTIONS. 37 55. If I add 2 to each term of the fraction J, will its value be increased or diminished, and how much ? 56. Mr. Homer has 10i acres of wheat, 6| acres of corn, 20| acres of barley, and 16| acres of rye. How many acres of grain has he ? ^57. What is the quotient of 389 divided by 1556, ex- pressed in its simplest form ? 58. -JiJ) -M = ? f of 13 of ft of 3 59. 816 is I of what number ? 60. From of } take f . 61. The product of two factors is 10^ ; one factor is 3|. What is the other ? 62. | + 6f + 9| + A + ^ = ? 63. ( T \ of 2i of T 3 7 ) x (f of 3i of 8 of J) = ? 64. The sum of two numbers is 19|^. One of the num- bers is 12f. What is the other ? 65. Reduce - to its lowest terms. 66- (I - I) X (| + |) = ? 67. Change to simple fractions : il -H i_ofj 7^ |_ofj 9 ' 61' 16 ' Jof2i' Jof A can do a piece of work in 4 days, B can do the same work in 5 days, and C in 6 days. In what time can all do it together ? 69. A tank has 3 supply pipes. It can be filled in 6 hours by the first pipe, in 7 hours by the second, and in 8 hours by the third. In how many hours can the tank be filled by the three pipes together ? 38 SENIOR ARITHMETIC. 70. A and B can do a piece of work in 3 days. A can do it alone in 5| days. In what time can B do it alone ? SOLUTION. Both can do | of it in 1 day. A, alone, can do fi of it in 1 day. 3 i 2 r / 35 the part A can do in 1 day. Since he can do 3 5 j of it in 1 day, he can do -jj-jj, or the whole of it, in as many days as /3 is contained times in ||, or 33 -=- 5 = 6| days. 71. of my property is invested in land, f of the re- mainder in business, and f of the remainder, which is $2400, is in the bank. How much property have I ? 72. What is the value of f 1 -j- 6 of -f i / 73. A farmer sold 11 doz. eggs at 14^/ a dozen, and took his pay in sugar at 5/ a pound. How much did he receive ? 74. Find the value of $. -- L i of 1 -s- 1. 1 f t 75. A boy having spent 1 of f of his money for a knife, had $2.25 left. How much did he pay for the knife ? 76. A father left $39000 to his two children, dividing it so that the daughter received | as much as the son. What was the share of each ? 77. A person owning f of a steamboat, sold j of iiis share for $17360. What was the value of the boat ? 78. After spending ^ of my money and ^ of the remain- der, I had $300 left. How much had I at first ? 79. If i of | of a bushel of apples cost f of T 9 ^ of a dol- lar, what will of | of a bushel cost ? 80. How many pounds of honey at \ of ^ of a dollar a pound can be bought for f of 2| dollars ? 81. Simplify * J of 82 ? of > + * f 12 ' 2 x 3 QUESTIONS. 39 83. Divide the product of 5 times ?1 plus L^Lt by 5i . 4$ * 15 J 4i 84. Divide f of -5 by f of -i . Find the cost of the following : 85. 315i Ib. of tea at $.37 a pound. 34| Ib. of coffee at $.18f a pound. 3105 j Ib. of pork at $.121 a pound. 30691 bu. of wheat at $1.12i a bushel. 36J doz. of eggs at $.12J a dozen. 26| yd. of sheeting at $.07f a yard. 86. A owns f of a farm and B owns the remainder ; f of the difference of their shares is worth $2400. What is the value of the farm ? 87. Divide $3 J among some poor children, giving each i of a dollar. What will be the number of children ? y/88. Two men hire a pasture for $25. A puts in 8 horses and B 12 horses. How much should each pay ? 89. Add 8 to both terms of the fraction , and find how muptf you have increased or diminished it. r 90. Subtract 4 from each term of the fraction f and find how much it has been increased or diminished. QUESTIONS. 118. 1. What is a decimal ? How are decimals writ- ten ? Why are they called decimals ? 2. How many decimal places are needed to write ten- thousandths ? Millionths ? Hundredths ? 3. What is the first place at the right of the decimal point ? What is the first period called ? The second place ? The second period ? 40 SENIOR ARITHMETIC. 4. What is a mixed decimal ? 5. What must the denominator of a decimal be ? 6. What is the effect of removing the decimal point one place to the right ? To the left ? Two places to the right ? Three places to the left ? 7. What is the effect of annexing a cipher to an inte- ger ? To a decimal ? Of prefixing a cipher to an integer ? To a decimal ? 8. How do we reduce decimals to common fractions ? Common fractions to decimals ? 9. Give rules for adding, subtracting, multiplying, and dividing decimals. 10. How do we locate the decimal point in the sum ? In the remainder ? In the product ? In the quotient ? 11. What are coins ? 12. What are the gold, silver, bronze, and nickel coins used in the U. S. ? 13. What are the aliquot parts of a number ? What are the aliquot parts of $1 ? Of 100 ? Of 1,000 ? 14. What is a bill ? An account ? A creditor ? A debtor ? Tell how to receipt a bill. 119. 1. Define unit, number, the unit of a number, ab- stract number, concrete number, like numbers. 2. Define notation, numeration, Arabic notation. 3. What is the value of the unit figure of a number ? The tens ? The hundreds ? 4. What is the largest number which can be expressed by four figures ? 5. What is the simple value of a figure ? The local value ? QUESTIONS. 6. What name is given to the first period at the right of the decimal point ? The second ? The third ? 7. What is addition? What kind of numbers can be added ? 8. Define subtraction, minuend, subtrahend, remainder. What is a proof of subtraction ? What is the sign of sub- traction, and where placed ? 9. What is a parenthesis ? A vinculum ? For what are they used ? 10. What is multiplication ? The multiplier? The mul- tiplicand ? The product ? 11. The multiplier and the multiplicand are what of the product ? 12. What is the sign of multiplication and how is it used ? Define division, divisor, dividend, quotient, re- mainder. 13. What is the sign of division, and how is it used ? 14. Express the division of 12 by 8 in as many ways as you can. 15. To what terms in multiplication do the divisor, quo- tient, and dividend correspond ? 16. How do you find the dividend when the divisor, quo- tient, and remainder are given ? 17. When is the quotient an abstract number ? 18. When the quotient and dividend are like numbers, what kind of a number is the divisor? 19. How can we divide when the divisor is 10 ? 100 ? 1000? When the divisor is 20? 50? 300? 20. Multiplying both dividend and divisor by the same number affects the quotient how ? 42 SENIOR ARITHMETIC. 21. Dividing both divisor and dividend by the same number affects the quotient how ? 22. Multiplying the dividend affects the quotient how ? The divisor ? Dividing the dividend ? The divisor ? 23. Define exact divisor, factor, prime factor, factoring. 24. How can you find the prime factors of a number? 25. Define divisor. Common divisor. The greatest com- mon divisor. Give rule to find greatest common divisor. 26. Define multiple, common multiple, least common multiple. Give rule for finding the least common multiple. 120, Define fraction, fractional unit, unit of a fraction, denominator, numerator, terms of a fraction, common frac- tion, integer, proper fraction, improper fraction, mixed num- ber, simple fraction, compound fraction, complex fraction. What is the value of a fraction ? State the principles of fractions. What is it to reduce a fraction ? How are fractions reduced to lowest terms ? To highest terms ? How can an improper fraction be reduced to a whole or a mixed number ? A whole or a mixed number to an improper fraction ? What are like fractions ? Unlike fractions ? How can fractions be reduced to others having a common denominator ? A least common denominator ? How can two or more fractions be added ? How can the sum of fractions be found ? Mixed num- bers ? How can the difference of fractions be found ? Mixed numbers ? How can a fraction be multiplied by a fraction ? A frac- tion by an integer ? COMPOUND NUMBERS. 43 How can an integer be multiplied by a fraction ? By a mixed number ? How can a fraction be divided by a fraction ? How do you reduce a complex fraction to a simple frac- tion ? COMPOUND NUMBERS. 121. A number composed of only one kind of unit is a Simple Number ; as, 5 pk., 4 apples, 6. 122. A Denomination is a name given to a unit of measure or of weight. 123. A number composed of different kinds of units is a Compound Number ; as, 3 bu. 2 pk. 1 qt. A number having one or more denominations is also called a Denominate Number. 124. Reduction is the process of changing a number from one denomination to another without changing its value. 125. Changing to a lower denomination is called Reduction Descending ; as, 2 bu. 3 pk. = 88 qt. 126. Changing to a higher denomination is called Reduc- tion Ascending ; 88 qt. = 2 bu. 3 pk. 127. Linear Measure is used in measuring lines or dis- tances. TABLE. 12 inches (in.) = 1 foot, ft. 3 feet = 1 yard, yd. 5^ yards, or 16 J- feet = 1 rod, rd. 40 rods = 1 furlong, fur 320 rods, or 5280 feet = 1 mile, mi. 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63360 in. 44 SENIOR ARITHMETIC. 128. Surveyors' Measure is used in measuring 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. A chain, or steel measuring tape, 100 feet long, is sometimes used by engineers. 129. Square Measure is used in measuring surfaces. TABLE. 144 square inches = 1 square foot, sq. ft. 9 square feet = 1 square yard, sq. yd. 301 square yards j = 272i square feet > 160 square rods = 1 acre, A. 640 acres = 1 square mile, sq. mi. 1 sq. mi. = 640 A. = 102400 sq. rd. = 3097600 sq. yd. 130. A square mile of land is called a Section. A square rod is sometimes called a perch (P.). A rood (K.) is 40 sq. rods. NOTE. 1 acre = 43560 sq. ft. There are 10 square chains^ in an acre. Roofing, paving, etc., are often estimated by the Square, which is 100 square feet. 131. Cubic Measure is used in measuring volumes or solids. TABLE. 1728 cubic inches = 1 cubic foot, cu. ft. 27 cubic feet = 1 cubic yard, cu. yd. 16 cubic feet = 1 cord foot, cd. ft. 8 cord feet, or 128 cu. ft. = 1 cord, C. 1 cu. yd. = 27 cu. ft. = 46656 cu. in. COMPOUND NUMBERS. 45 132. Liquid Measure is used in measuring liquids. TABLE. 4 gills (gi.) = 1 pint, pt. 2 pints = 1 quart, qt. 4 quarts = 1 gallon, gal. 1 gal. = 4 qt. = 8 pt. = 32 gi. A gallon contains 231 cubic inches. The standard barrel is 31 \ gal., and the hogshead 63 gal 129. Apothecaries' Fluid Measure is used in mixing medi- cines in liquid form. TABLE. 60 minims (TO) = 1 fluid dram, f. 3. 8 fluid drams = 1 fluid ounce, f . . 16 fluid ounces = 1 pint (0). 133. Dry Measure is used in measuring roots grain, vege- tables, etc. IABI ^ 2 pints = 1 quart, qt. 8 quarts = 1 peck, pk. 4 pecks = 1 bushel, bu. 1 bu. = 4 pk. = 32 qt. = 64 pt. The bushel contains 2150.42 cubic inches. 134. Avoirdupois Weight is used in weighing all common articles ; as, coal, groceries, hay, etc. TABLE. 16 ounces = 1 pound, Ib. ( 1 hundred- weight, cwt. 100 pounds = < ( or cental, ctl. 20 cwt., or 2000 Ib. = 1 ton, T. 1 T. = 20 cwt. = 2000 Ib. = 32000 oz. The Long Ton of 2240 pounds is used at the U. S. Cus- tom-House and in weighing coal at the mines. 46 SENIOR ARITHMETIC. The ounce is considered as 16 drams. The Avoirdupois pound contains 7000 grains. A hundred-weight is sometimes called a Cental. 135. Troy Weight is used in weighing gold, silver, and jewels. TABLE. 24 grains (gr.) = 1 pennyweight, pwt. 20 pennyweights = 1 ounce, oz. 12 ounces = 1 pound, Ib. 1 Ib. = 12 oz. = 240 pwt. = 5760 grains. 136. Apothecaries' Weight is used by druggists and physi- cians in weighing medicines that are not liquid. TABLE. 20 grains (gr.) = 1 scruple, sc. or 9. 3 scruples = 1 dram, dr. or 3. 8 drams = 1 ounce, oz. or %. 12 ounces = 1 pound, Ib. or R>. 1 Ib. = 12 oz. = 96 dr. = 288 sc. = 5760 gr. Dry medicines are bought and sold in large quantities by avoirdupois weight. Comparison of Weights. 1 Ib. Avoirdupois = 7000 gr. 1 oz. Avoirdupois = 437^ gr. 1 Ib. Troy or Apothecary = 5760 gr. 1 oz. Troy or Apothecary = 480 gr. 137. Measure of Time. TABLE. 60 seconds (sec.) = 1 minute, min. 60 minutes = 1 hour, hr. 24 hours = 1 day, da. 7 days = 1 week, wk. 365 days = 1 year, yr. 366 days = 1 leap year. 100 years = 1 century. COMPOUND NUMBERS. 47 The Civil Day begins and ends at midnight. The exact time in which the earth makes one revolution of the sun is 365 da. 5 hr. 48 min. 49.7 sec., or 365 days, nearly. For convenience the common year is regarded as 365 days; the fraction being disregarded until it amounts to a full day, which is in four years, nearly. Accordingly every fourth year contains 366 days. This day is added to the shortest month, February, and the year in which it is added is called Leap Year. But 365i days is a little more than the exact year, and we have added a little too much, when we have added 1 day to every fourth year, therefore only every fourth cen- tennial year is considered as leap year. This nearly cor- rects the excess, so that the error is less than 1 day in about 3600 years. Every year divisible by 4, and every centennial year divisible by 400, is a Leap Year. Circular or Angular Measure. 138. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from the centre. 139. The bounding line of a circle is the Circumference. 140. Any part of a circum- ference is an Arc. A to C and B to D are arcs of a circle. 141. A straight line through the centre of a circle termi- nating at the circumference is A circle. the Diameter. 142. A straight line from the centre to the circumference is the radius ; as, E to 'D, or E to C. 48 SENIOJi ARITHMETIC. 143. The circumference of every circle is divided into 360 equal parts called Degrees, each, degree into 60 parts called Minutes, and each minute into 60 parts called Seconds. 144. An Angle is the difference in direction between two straight lines. The point of meeting is the Vertex. The vertex is at the centre of a circle, and the angle is measured in degrees by the arc between its sides. Thus B D is the measure of the angle BED. TABLE OF CIRCULAR MEASURE. 60 seconds (") = 1 minute, ' 60 minutes = 1 degree, 360 degrees = 1 circumference, Cir. NOTE. An arc of 90 degrees or of a circumference is called a quadrant. A degree upon a great circle of the earth is 69.16 statute miles, or 60 geographical miles. A sign is an arc of 30 degrees. 146. Federal Money is the currency of the United States. TABLE. 10 mills = 1 cent, ct. 10 dimes = 1 dollar, $ 10 cents = 1 dime, d. 10 dollars = 1 eagle, E. The gold coins of the United States are the double-eagle, eagle, half-eagle, quarter-eagle, and one-dollar piece. The silver coins are the dollar, half-dollar, quarter-dollar, and the ten-cent piece. The five-cent piece is nickel, and the one-cent piece bronze. 147. English or Sterling Money. TABLE, 4 farthings = 1 penny, d. 12 pence = 1 shilling, s. 20 shillings = 1 pound, , or 1 sovereign. The coin which represents the Pound Sterling is the Sovereign, equal in value to $4.8665. COMPOUND NUMBERS. 49 148. Counting. TABLE. 12 things = 1 dozen, doz. 12 dozen = 1 gross, gr. 12 gross = 1 great gross, G. gr. 149. Paper. TABLE. 24 sheets = 1 quire. 2 reams = 1 bundle. 20 quires = 1 ream. 5 bundles = 1 bale. REDUCTION DESCENDING. 150. 1. Reduce 5 Ib. 6 oz. 12 pwt. 6 gr. to grains. 5 Ib. 6 oz. 12 pwt. 6 gr. 12 Since there are 12 oz. in 1 Ib., in 5 Ib. there are 5 times 12 oz. = 60 oz. (add 6 oz.) = 66 oz. _T Since there are 20 pwt. in 1 oz., in 66 oz. there 66 oz. are 66 times 20 pwt. = 1320 pwt. (add 12 pwt.) = 20 1332 pwt. 1320 Since there are 24 gr. in 1 pwt., in 1332 pwt. 12 there are 1332 times 24 gr. = 31968 gr. (add 6 gr.) 1332 pwt. " 319 ? 4 &- 24 Reduce to lower denominations : 2QQ 2 - If 3^ 2 ft. 9 in. to inches. 31968 3 - ^G r d. 4 yd. 2 ft. to feet. 6 4. 3 mi. 75 rd. 4 ft. to inches. 31974 gr. 5< 16 A 140 sq rd 26 S q. yd. to square yards. 6. 4 A. 15 sq. rd. 4 sq. ft. to square inches. 7. 50 ch. 45 li. to links. 8. 16 cu. yd. 25 cu. ft. 900 cu. in. to cubic inches. 9. 8 cd. 12 cu. ft. to cubic feet. 10. 15 gal. 3 qt. 1 pt. to pints. SENIOR ARITHMETIC. 11. 4 0. 6 f. I 3 f. 3 25 u to minims. 12. 7 bu. 3 pk. 5 qt. 1 pt. to pints. 13. 16^ bu. to quarts. 14. 25 Ib. 5 oz. 16 pwt. 10 gr. to grains. 15. 2 T. 6 cwt. 10 Ib. 14 oz. to ounces. 16. 16 Ib. 5 % 4 3 2 9 11 gr. to grains. 17. 28 14' 18" to seconds. 18. 18 155. Sd. 3 far. to farthings. 19. 27 da. 18 h. 49 min. to seconds* 3 wk. 48 min. 52 sec. to seconds. 21. How many quires in a bundle of paper ? 22. How many pints in a cask of molasses holding 84 gallons ? 23. How many articles in 7 G. gr. 5 gr. ? 24. How many hours in 10 years, allowing for two leap years ? ^S&. How many inches in 4^ rods ? 26. What is the cost of 10 miles of telephone wire at 28 cents a pound, if a pound measures 75 ft. ? 27. Find the number of square inches in a square yard ; square feet in a square chain ; cubic inches in a cubic yard. 28y How many hours in the month of February, 1896 ? ^/^b. How many cubic inches in 5 gallons. 30. How many square yards in 4 sq. miles ? 31. How many square feet in 1\ acres ? 32. How many ounces in 3 Ib. of silver ? 3 Ib. of iron ? 33. If I buy 3 bu. of nuts at $4 a bushel, and sell them at 5/ a pint, how much shall I gain ? 34. How many ounces in a long ton ? COMPOUND KUMBKKS. 51 35. At $12 a ton, what will f of a ton of hay cost ? In 1800 years how many centuries ? 37. If you can count sixty a minute, how long will it take to count 180000. 38. Through how many degrees does the hour-hand of a clock pass in 6 hours ? 39. Through how many degrees does the minute-hand pass in 6 hours ? 40. What will 3 reams of paper cost at 40/ a quire ? 41. Eeduce 3 mi. 4 fur. 20 rd. 5 yd. 2 ft. 8 in. to inches ? 42. Eeduce 6 mi. 240 rd. to feet. 43. Reduce 3 A. 8 sq. rd. 5 sq. yd. 3 sq. ft. to sq. inches. 44. Reduce 16 cu. yd. 9 cu. ft. 3 cu. in. to cu. inches. 45. Reduce 58 cd. to cu. feet. 46. Reduce 2 T. 3 ctl. 16 Ib. to ounces. 47. Reduce 3 Ib. 9 oz. 15 pwt. 12 gr. to grains. \^&T Reduce 60 gal. 3 qt. 3 gi. to gills. 49. How maiiy sheets in 5 bales of paper ? 50. Reduce 3 wk. 6 da. 5 hr. to minutes. REDUCTION ASCENDING. 151. 1. Reduce 1306 gills to higher denominations. 4 j 1306 gi. Since in 1 pt. there are 4 gi., in 1306 gi. 2 / 3^6 pt -4- 2 gi there are as many pints as 4 gi. is con- A I TAQ tained times in 1306 gi., or 326 pt. and 2 gi. 4 / JLuo Qt. . _ * * remainder. gal. + 3 qt. Since in j qt> there are 2 pt ., i n 326 pt. 40 gal. 3 qt. 2 gi. there are as many quarts as 2 pt. is con- tained times in 326 pt., or 163 qt. Since in 1 gal. there are 4 qt., in 163 qt. there are as many gal- lons as 4 qt. is contained times in 163 qt., or 40 gal., and 3 qt. remainder. Therefore, in 1306 gills there are 40 gal. 3 qt. pt. 2 gi. 52 SENIOR ARITHMETIC. 2. How many rods in 334 yd. ? 5J yd. 334 yd. Since in 1 rd. there are 22 5 yd., in 334 yd. there 11 half yd./ 668 half yd. are as man y rods as 5 * ? d - 5 o T. in 7 is contained times in 334 60 rd. + 8 half yd. yd ., or 60 rd ., and 4 yd . remainder. 334 yd. -^ 5i yd. = 668 half yd. -f- 11 half yd. 8 half yd. = 4 yd. 334 yd. = 60 rd. 4 yd. 3. Reduce 225932 in. to miles, etc. 4. How many miles and rods are there in 35640 ft. ? 5. Reduce 19922544 sq. in. to higher denominations. 6. Reduce 762051 cu. in. to cu. yards, etc. 7. How many cords in 7424 cu. ft. 8. Reduce 69056 oz. to tons, etc. 9. Reduce 21076 gr. to higher denominations. 10. Reduce 1947 gi. to gallons, etc. 11. How many bales in 24000 sheets of paper ? ^}ti Reduce 39180 min. to weeks, etc. 13. Reduce 5762 far. to higher denominations. 14. Reduce 84623'' to higher denominations. 15. Reduce 62341 M. to higher denominations. 16. How many chains, etc., in 13025 li. ? 17. How many bushels, etc., in 35842 pints ? 18. How many pounds, etc., (Troy) in 32563 gr. ? 19. Reduce 39632 gr. to lb., etc. (Apoth.). 20. How many tons, etc., in 35682 lb. ? 21. A box contains 75832 pens. How many Gr. gross, etc., in the box ? 22. Change 1384 dry pints to higher denominations. 23. In 139843 sq. in. how many miles, rods, etc. ? 24. How many cords of wood in 3692 cu. feet ? COMPOUND NUMBERS. 53 REVIEW PROBLEMS. 152. 1. Bought 2 gal. 8 oz. of fluid extract at 20/ an ounce, and sold it at 15/ an ounce. What was lost ? 2. How many minims are there in 10 fluid ounces (f. 5), 7 fluid drachms (f. 3) ? 3. Find the difference in value between 4 gal. of am- monia water at 10 cents a pint and 8 ounces of cinnamon water at 5 cents an ounce. 4. The pendulum of a certain clock beats seconds. How many times will it tick in 1 day, 9 hours, 25 minutes ? 5. How many degrees in 3492.58 statute miles, measured on the equator, a degree being equal to 69.16 statute miles ? 6. How many degrees of longitude will a steamship pass through, sailing due west on the equator, at the rate of 15 knots an hour for 5 days ? NOTE. A knot = 1 geographic mile or minute. 7. Find cost of each of following : (a) 5 gallons, 3 qt. 1 pt. of molasses at 20/ a gallon ; (I) 10 acres, 50 sq. rd. of land at $50 an A. 8. What will it cost to build 112 rd. 3 yd. of fence at 48/ a yard ? 9. If a man steps 2^ ft. at each step, how many miles will he travel in stepping 4820 times ? 10. If 17 ft. is | of the height of a tree, how high is the tree? 11. Reduce 6f mi. 317 rd. 4 yd. 2 ft. to feet. 12. Change 16571 ft. to miles. 13. At $3.20 a bu. how many quarts cvf nuts can be bought for $4.80 ? 14. How many pint bottles of camphor may be filled from 96 fluid ounces (f. 3) ? 54 SENIOR ARITHMETIC. 15. Find the cost of the following : 4 oz. iodine at 10/, 8 oz. spts. camphor at 5/, 10 oz. aqua ammonia at 10^, 14 oz. cinnamon water at 5/. 16. Reduce 4 bu. 3 pk. 7qt. 1 pt. to pints. 17. How many quart boxes will hold 2 bu. 3 pk. 5 qt. of berries ? 18. If 4 bu. of berries are bought for $.70 per bushel and sold for $.05 per quart, what is the gain or loss ? 19. Both sides of a railroad track are fenced with wire for 40 yards. What is the cost of the fence at 4/ a foot ? v2tT What will 8 Ib. 6 oz. of sugar cost at 8/ a pound ? 21. When pens are bought at 75/ a gross, and sold at 2 for 3/, what is the gain ? 22. If a man can walk 10 miles in 2 hours, how far can he walk in 6 hours ? in 30 minutes ? in 50 minutes ? 23. What will \ bu. berries bring at 8/ a quart ? 24. A silver chain weighs 18 pwt. ; what is its value, when silver is worth $.65 an ounce ? 25. What will 24 qt. of milk cost at 20/ a gallon ? 26. If I buy peanuts at 5/ a quart, and retail them so as to gain $6.40 on 4 bushels, what do I sell them for ? 27. At 4 pens for 3 cents what will 1 great gross cost ? 28. How many table-forks, each weighing 2^ oz., can be made from 4 Ib. 4 oz. 10 pwt. of silver. 29. In | of a gallon how many pints. 30. How many rods of fence will enclose a mile square of land ? 31. What is. the cost of 1 yd. and 27 inches of fringe at 60 cents a yard. 32. How many rods of fence are required to enclose a lot that is 20 rods wide and three times as long ? COMPOUND NUMBERS. 55 33. Kequired the distance around a room that is 13 feet long and 15 feet wide. 34. A shoe-box is 4 in. deep, 6 in. wide, and 12 in. long. How much twine will it take to wind twice around the box each way to hold on the cover, allowing 6 inches for tying. 35. I have a lawn that is 30 ft. by 70 ft., and wish to lay a board walk around it that is 3 ft. 6 in. in width. What is the distance around the walk, outside measurement ? 153. A Denominate Fraction is a fraction having a de- nomination. 154. To reduce Denominate Fractions to Integers of Lower Denominations. 1. Reduce f of a mile to rods, yards, feet, etc. SOLUTION. f of 320 rd. = JL 7 . How much did he earn ? 64 SENIOR ARITHMETIC. Fred was born Dec. 20, 1875 ; how old is he now ? 11. How much time has elapsed since George Washing- ton was 15 1 years old ? 12. Gen. Grant was born April 27, 1822. How old would he be if he were alive to-day ? 13. How long since Lee surrendered to Gen. Grant ? 14. Find the exact number of days between Jan. 10, 1896, and May 5, 1896. 15. When can a boy who was born May 5, 1882, cele- brate his 25th birthday ? 16. John goes to bed at 9.15 P.M. and gets up at 7.10 A.M. How many minutes does he spend in bed ? MULTIPLICATION OF COMPOUND NUMBERS 159. 1. Multiply 4 yd. 2 ft. 8 in. by 8. yd. ft. in. 8 times 8 in = 64 in = 5 ft 4 in Place the 4 428 in. under the inches column, and reserve the 5 ft., 8 to be added to the product of 2 ft. by 8, which 39 4 equals 16 ft. (add 5 ft.) = 21 ft. 21 ft. -j- 3 = 7 yd., with no remainder. Add 7 yd. to the product of 4 yd. by 8 = 32 yd. (add 7 yd.) = 39 yd. 2. gal. qt. pt. gi. bu. pk. qt. pt. 31 3 2 3 12 3 2 1 5 8 3. If a man travel at the rate of 60 mi. 240 rd. 16 ft. in one day, how far will he travel in ten days ? 4. A man owns 6 farms, each containing 75 A. 49 sq. rd. 25 sq. yd. of land. How much land in all the farms ? 5. If 6 loads of hay weigh 6 T. 18 cwt. 75 lb., how much will 48 loads weigh ? NOTE. 48 loads will weigh 8 times as much as fi loads. COMPOUND NUMBERS. 65 6. If 12 spoons weigh 3 Ib. 8 oz. 15 pwt., how much will one gross of spoons weigh ? 7. How much oil will 7 barrels hold if each barrel con- tains 35 gal. 2 qt. ? 8. What is the value at $4 per cord of 10 piles of wood, each containing 5 cd. 5 cd. ft. 12 cu. ft. ? 9. What is the weight of 15 packages, each weighing 1 Ib. 4 oz. (Avoir.) ? 10. If a bicyclist travels 75 mi. 140 rd. in one day, how far can he travel in ten days ? 11. In a watch-chain there are 2 oz. 12 pwt. 15 gr. of gold. How much gold is required for 25 such chains ? 12. A farmer has six bins, each containing 60 bu. 2 pk. of wheat. How much wheat has he ? 13. If a train is run for 8 hours at the average rate of 50 mi. 30 rd. 10 ft. per hour, how great is the distance covered ? 14. It takes John Smith 5 hr. 20 min. 11 sec. to plough one acre of ground. At the same rate, how long will it take him to plough 4 acres ? 15. 4 gal. 3 qt. l.pt. X 11 = ? 16. 2 A. 40 sq. rd. 16 sq. yd. X 20 = ? DIVISION OF COMPOUND NUMBERS. 160. 1. Divide 16 Ib. 9 oz. 17 pwt. 8 gr. by 10. Ib. oz. pwt. gr. SOLUTION. - ^ of 16 Ib. = 1, and 10 /16 9 17 8 6 Ib. remaining. 6 Ib. = 72 oz. 72 oz. 18 3 17^ +9 oz. = 81 oz. T V of 81 oz. = 8 oz., with 1 oz. remaining, = 20 pwt., to -which add 17 pwt., = 37 pwt. T l ff of 37 pwt. = 3 pwt., with 7 pwt. remaining, == 168 gr., to which add 8 gr. ; and taking J ff of the sum, we have 17 T 6 o gr. When the divisor is large, employ long division. 66 SENIOR ARITHMETIC. 2. Find ^ of 42 rd. 4 yd. 2 ft. 8 in. rd. yd. ft. in. 35/42 4 2 8 ( 1 rd. -J :> of 42 rd. = 1 rd. ; remainder, 35 7 rd. = 38^ yd. ; add 4 yd. = 42i yd. 3^ of 42^ yd. = 1 yd.; remainder, 7i yd., = 22^ 5 L ft.; add 2 f t. = 24i ft. 3 V, of 24i ft. - ft. 24^ ft. = 294 in. ; add 8 in. = 302 in. ^ of 302 Og- . 022 ' 35 38^ NOTE. When both dividend and divisor are -|- 4 compound, reduce them to the same de- 35/42^.yd. (1 yd. nomination, and divide. The quotient 35" will be abstract. 3. Divide 169 bu. 3 pk. 5 qt. by 7. 4. If a man travelled 607 mi. 169 rd. 11 ft. 6 in. in 10 days, what average distance did he travel in 1 day ? 5. If one gross of spools weighs 44 Ib. 9 oz., how much will one dozen weigh ? . 6. If one bottle holds 1 pt. 3 gi., i rci. i VQ. oj^ in. * . how many dozen bottles will be re- quired to hold 65 gal. 2 qt. 1 pt. ? 7. A man has 451 A. 138 sq. rd. 29 sq. yd. of land, which he wishes to divide equally among his six children. How much land will each child receive ? 8. If 12 persons share equally in the contents of a bin containing 20 bu. 2 pk. 4 qt. of apples, what is the share of each ? 9. If the entire area of 24 equal fields is 242 A. 20 sq. rd. 15 sq. yd., what is the size of each field ? COMPOUND NUMBERS. 67 10. A man walked 50 mi. 71 rd. 2 yd. in 15 hours. What was his rate per hour ? 11. If it takes a man 12 hr. 35 min. 15 sec. to walk 45 miles, what is the average time taken for each mile ? (Divide by the factors of 45.) 12. When $12 will buy 11 gal. 2 qt. 1 pt. of maple sirup, how much will $1 buy ? 13. A man travelled 100 miles in 9 hours. What was the average rate per hour ? 14. If a horse eats 12 qt. of oats per day, how long will 10 bu. 1 pk. 4 qt. last him ? 15. If a package weighs 4 cwt. 15 lb., how many such packages will it take to weigh 3 T. 2 cwt. 25 lb. ? 16. A man had 5 acres of land which he divided into 12 equal parts. How much land did each part contain ? 17. Divide 102 T. 15 cwt. 27 lb. 9 oz. by 8. 18. Divide 16 bu. 3 pk. 6 qt. by 2 bu. 1 pk. 19. I have 84 lb. 14 oz. of salt which I wish to put into packages of 2 lb. 6 oz. each. How many packages will there be? 20. If a horse eats 1 pk. 2 qt. of oats a day, how many days will 16 bu. 3 pk. 6 qt. last him ? 21. How many sacks, each containing 2 bu. 3 pk. 2 qt., will be needed to hold 165 bu. 2 pk. of meal ? 22. 16 cwt. 75 lb. 9 oz. of butter are to be put -into jars each containing 9| lb. How many jars will be needed ? To multiply or divide a compound number by a fraction. NOTE. To multiply by a fraction, multiply by the numerator, and divide the product by the denominator. To divide by a fraction, multiply by the denominator, and divide the product by the numerator. 68 SENIOR ARITHMETIC. 23. How much is f of 16 hr. 17 min. 14 sec. ? 24. How much is of 30| 5^ 19 8 gr. ? 25. Divide 120 cd. 50 cd. ft. 34 cti. in. by |. 26. How many times is ji contained in 840 T. 15 cwt. 98 Ib. 3 oz. ? 27. A man sold 4 bu. 3 pk. 2 qt. of potatoes, which was f of what he raised ; how much did he raise ? 28. A butcher sells 120 tons, 9 cwt. 75 Ib. of beef every month. How much does he sell in of a month ? 29. If 6 bottles hold 5 gal. 2 qt. of milk, how much milk will 3 such bottles hold ? 30. A field contains 10 acres 12 sq. rd. of land, which is | the size of the whole farm. Find the size of the farm. fl. A railroad track extends 144 miles, 40 rd. 3 yd. How far has a train of cars gone which has travelled T 9 ^ of this distance ? 32. If a pipe discharges 25 gal. 3 qt. 1 pt. of water in 1 hr., how much will it discharge in 5| hr., if the water flows with the same velocity ? NOTE. When the multiplier or divisor is a mixed number, reduce to an improper fraction, and proceed as above. 33. Divide 8 Ib. 11 oz. 15 pwt. 18 gr. by 2. 34. If a railroad train runs 60 mi. 35 rd. 16 ft. in one hour, how far will it run in 12f hr. at the same rate of speed ? 35. Divide 14 bu. 3 pk. 6 qt. 1 pt. by . 36. Divide 5 yr. 1 mo. 1 wk. 1 da. 1 hr. 1 min. 1 sec. by 3*. MISCELLANEOUS PROBLEMS. 69 MIS CELL A NE O US PR OBLE ^ I S. 161. 1. Xame two numbers which multiplied together make 14. 2. Write three sets of factors for 24. 3. Find the prime factors of 2205. 4 Ux$X3 X2X$ _.> 5 X 6 X 2 X 9 X 24 5. How many yards of silk, 24 inches wide, will it take to line a skirt containing six yards of cloth 28 inches wide ? 6. Find the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9. 7. Find the greatest common divisor of 285, 465. 8. Find the smallest number that will exactly contain 9, 15, 18, 20. 9. Find the length of the longest stick that will exactly measure the sides of a room 216 yd. by 111 yd. 10. What is the smallest sum of money with which you can buy pears at 75/ a basket, peaches at 90/, and grapes at 50/, using the same amount of money for each kind ? 11. How many times is 1 contained in 1 ? 12. How many times is 1 contained in 1 ? 13. A man bought a horse for $240, which is f of what he sold it for. What did it sell for ? 14. A man bought a horse for $240, and sold it for f of what he paid for it. What did it sell for ? 15. A pole stands ^ in the mud, 1 in the water, and the remaining 10 feet are above the water. How long is the pole? 16. A man owns 4 farms containing 365^, 375, 284f. and 2544 acres respectively. How many acres in all ? 70 SIvNIOK AK1THMET1C. 17. The man owning the above farms sells A 234^ acres, and B 366^\ acres. How many acres has he left ? 18. What is the value of 3f X W X f X 14 X 3 T 9 o X ^ X T 6 2 X | X T 2 3 X J. 19. Find the least common multiple of 273, 462, 1785, and 399. 20. A man owns a farm containing 400 acres. He sells J of the farm, and divides the remainder among his six children. How many acres does each child receive ? 21. Find the sum of 3 bu. 6 pk. 2 qt. 1 pt., 3 pk. 1 qt. 1 pt., 7 bu. 3 pk., 4 bu. 7 qt. 1 pt., and 19 bu. 2 pk. 2 qt. 1 pt. 22. Find the sum of 4 Ib. 6 oz. 21 pwt. 9 gr., 5 oz. 11 gr., 3 Ib. 9 oz. 18 pwt., 11 oz. 17 pwt. 5 gr., 16 Ib. 4 oz. 11 pwt., 18 Ib. 17 gr., 21 Ib. 15 pwt. 11 gr., 23 Ib. 10 oz. 21 pwt. 23 gr. 23. What part of a mile is 214 rd. 2 yd. 2 ft. 3 in. ? 24. Reduce 1 pk. 4 qt. If pt. to the fraction of a bushel. 25. What is the quotient of 184 bales, 4 bundles, 1 ream, 13 quires, 20 sheets, divided by ? 26. A farm is 60 ch. 25 1. long. How many rods long is it? 27. A surveyor measured my farm, and found that it is 80 ch. long and 60 ch. broad. How many acres does it contain ? 28. In 133128 in. how many miles ? 29. How many times will a wheel 12 ft. 3 in. in circum- ference turn round in going 15 mi. 20 rd. 12 ft. 2 in. ? 30. How many steel rails 30 ft. long are needed in the construction of 7 mi. 305 rd. 7 ft. 6 in. of double-track rail- road ? MISCELLANEOUS PKOBLEMS. 71 31. What fraction of the year is contained in the months of July and August, 1896 ? 32. The greatest depth of the Atlantic telegraph cable is 2 mi. 250 rd. 5 yd. 1 ft. How many feet is it ? 33. How many statute miles in 45 22' 30", measured on the equator ? 34. On an average when walking. Isaac steps 24 inches twice every second. How many minutes will it take him to Avalk li miles ? 35. Reduce ^ of an acre to square rods and decimals of a square rod. 36. A silversmith in making spoons uses 2 Ib. 3 oz. 19 pwt. of silver in one day, 3 Ib. 18 pwt. 20 grs. on the sec- ond day, and 11 oz. 19 pwt. 23 gr. on the third day. How much silver does he use altogether ? 37. What decimal of a pound Troy are 4 oz. 14 pwt. ? 38. Reduce 18 pwt. 164 gr. to the fraction of a pound. 39. Find the difference' between f Ib. and 4 Ib. 7.84 oz. Troy. 40. Find the cost of 2^ doz. spoons, each weighing 9 oz. 8.76 pwt., at $.045 a pennyweight. 41. What decimal part of a grain is 75 * 15 - of a pound? 42. What part of an acre is f rd. ? 43. Multiply 16 bu. 9 pk. 8 qt. by f . 44. What is the product of 20 bu. 9 oz. 11 pwt. 15. gr. multiplied by f ? 72 SEN10K ARITHMETIC. ME A S UREMENTS. 162. An Angle is the difference in direction of two lilies. 163. A Right Angle is the angle of a square. 164. Anything that has length and breadth, but not thickness, is a Surface. 165. A surface that does not change its direction is a Plane Surface. 166. A figure having four straight sides and four right angles is a Rect- angle. 167. A Square is a rectangle having equal sides. A Right Angle. A Rectangle. 168. A Square Foot is a square 1 foot long and 1 foot wide. 169. A Square Yard is a square 1 yd. long and 1 yd. wide. 170. The Area of a surface is the number of square units that it con- tains. A Square. NOTE. There are 6 sq. in. in a row, and in 4 rows there are 4 times 6 sq. in.= 24 sq. in. The multiplier is abstract, and the unit of the product must be the same as the unit of the multiplicand. MEASUREMENTS. 73 171. The length and breadth of a rectangle are called its Dimensions. Length X Breadth == Area. Area -r- Length = Breadth. Area -7- Breadth = Length. 172. A figure having three straight sides and three angles is a Triangle. The Base of a triangle is the line upon which it stands, and the Altitude is its height above the base, or the base extended. Thus, AC is the base, and BD the altitude, of the triangle shown below. D r / \ 173. It is evident from the accompanying figure that the area of a triangle is equal to one-half the area of a rec- tangle of the same base and altitude. Every circle may be regarded as composed of many equal triangles, the radius of the circle forming the alti- tudes, and the circumference forming the sum of the bases. Therefore, the area of a circle is equal to 1 the prod- uct of the circumference and radius. PRINCIPLE. The circumference of a circle is 3.1416 ' times the diameter, or about 3^. 174. Circumference -?- 3.1416 = Diameter. Diameter x 3.1416 = Circumference. 74 SENIOR ARITHMETIC. Oral. Find the areas of rectangles as follows : 1. 10ft. by 8 ft. 4. 50ft. by 20 ft. 2. 16 ft. by 4i ft. 5. 6 ch. by 8 ch. . 3. 14 rods by 10 rd. 6. 9 yd. by 6 yd. Find the other dimension. 7. Area 24 sq. ft., length 8 ft. 8. Area 72 sq. yd., length 8 yd. 9. Area 100 sq. in., breadth 5 in. 1 10. Length 16 yd., area 64 sq. yd 11. Breadth 4 ft., area 84 sq. ft. 12. Area 56 sq. ch., length 8 ch. Find the areas of the following triangles : 13. Base 10 ft., alt, 12 ft. 16. Base 5 ft., alt. 10 ft. y/14. Base 9 yd., alt. 6 yd. 17. Base 12 in., alt. 8 in. 15. Base 15 in., alt. 6 in. 18. Base 10 rd., alt. 5 rd. Find circumferences of circles having the following di- ameters : NOTE. Indicate the operation only. 19. 12 ft. 21. 16 in. 23. 16 rd. 25. 62 yd. 20. 18 ft. 22. 14 yd. 24. 25 ch. 26. 84 ft. Find the diameters having the following circumferences : NOTE. Indicate only. 27. 78 ft. 28. 19 ft. 29. 316.14 rd. 30. 189.68 ch. Written. Find areas. 31. Circumference 37.6992 ft., radius 6 ft. 32. Circumference 47.124 ft., diameter 15 ft. MEASUREMENTS. 75 33. Circumference 62.832 ft., diameter 20 ft. 34. Circumference 94.248 ft., diameter 30 ft. 35. Diameter 24 ft. 38. Diameter 160 ft. 36. Radius 16 ft. 39. Radius 62 ft. 37. Circumference 50 ft. 40. Circumference 314.16 ft. 41. How many square yards are there in a floor 24 ft. long and 15 ft. wide ? 42. The base of a triangle is 20 ft. and the altitude 18 ft. What is the area ? 43. The circumference of a circle is 31.416 ft. and its radius is 5 ft. What is its area ? 44. When the diameter of a circle is 50 ft., what is the circumference ? 45. When the radius is 6 ft., what is the circumference ? 46. When the circumference is 78.54 ft., what is the radius ? ^47. Mr. Clark's farm is 35 ch. long and 25 ch. wide. How many acres does it contain ? 48. A certain field is 70 rd. long and 65 rd. wide. How many acres are there in the field ? 49. A piece of land is 65 rods wide. How long must it be to contain 56 acres ? o 50. How many sods 10 inches square will be required to turf a lawn 100 ft. long and 50 ft, 6 in. wide ? 51. A building lot measures 60 feet in front. What must be its depth to contain 1 of an acre ? 52. How many tiles, each 8 inches square, will be required for the floor of a room 24 ft. by 30 ft. ? 53. How many shingles will be required for a roof 45 ft. long, and each of its two sides 20 ft. wide, allowing 8 shingles to the square foot ? 76 SENIOR ARITHMETIC. 54. How many square yards of oil-cloth are needed to cover a floor 18 ft. by 24 ft. 6 in. ? 55. A owns a city lot 168 ft. long and 42 ft. wide. He uses | of it for a lawn. How many square yards does the lawn contain ? 56. How long will it take a man to mow the above lot, if it takes him a minute to run a 2-foot lawn-mower length- wise of the lot ? 57. A pony can reach 40 feet in any direction from the stake to which he is picketed. Over how many square rods of surface can he graze ? 58. What is the diameter of a tree that is 10 feet in circumference ? 59. A basin measures 9 inches across the top and 6 inches across the bottom. How much farther around the top than around the bottom ? 60. How many acres of land are enclosed by a circular mile track ? 61. A landscape gardener lays off a circular grass-plot whose radius is one rod, and near it a semicircular plot having a radius two rods in length. Compare their areas. 62. If the area of a triangle is 9 acres and the base is 80 rods, what is the altitude ? 63. Find the area of one gable end of a building 40 ft. wide, the ridge being 15 ft. above the eaves. 64. Find area of a triangle whose altitude is 14 ft. and its base 12 ft. 65. Find the area of triangle whose altitude is 4 in. and base 12 in, MEASUREMENTS. 77 CARPETING ROOMS. 175. In making a carpet, the carpeting is cut from a roll into strips which are usually laid from end to end of the floor, or lengthwise. Sometimes the strips are laid across the room. 1. How much carpeting must I purchase to cover a room 6 yd. long and 4| yd. wide, strips running length- wise ? SOLUTION. It will be necessary to purchase as much carpeting as if the room were 5 yd. wide, the excess of yd. being turned under in the last strip. 1 strip contains 6 yd. 5 strips = 5 times 6 yd. = 30 yd. Ans. 2. How many yards must t i I purchase, if the strips are laid across the room ? SOLUTION. 1 strip contains 4| yd. 6 strips = 6 times 4f yd. = 28^ yd. Ans. Carpeting is commonly 1 yd. or | yd. wide. NOTE. It is often necessary ^ ^ to purchase more than enough carpeting to cover a room, on account of the waste in matching patterns. This diagram represents the floor in Ex. 1. in which the strips are laid lengthwise. Pupils should draw a similar diagram for each floor. NOTE. Carpeting is sold by lineal yards or meters, not by square measure. 3. A merchant bought a roll of carpet containing 74 yd. at 85/ a yard, and sold it at $1.15 a yard. What was his profit ? 78 SENIOR ARITHMETIC. 4. A roll of carpet | yd. wide contains 60 yards ; how many square yards of surface will it cover ? 5. How many strips of carpet 3 ft. wide will cover a floor 15 ft. wide ? 17 ft wide ? 18 ft. wide ? 6. How many strips 27 in. wide are required for a floor 12 ft. wide ? 14 ft. wide ? 16 ft. wide ? 7. If, in Example 5, the room is 16 ft. long, how many linear yards of carpet will be needed to cover the floor ? 8. If, in Example 6, the room is 19 ft. long, how many yards will be required to cover the floor ? 9. How many yards of ingrain carpet f yd. wide will be required for a floor 17 ft. wide and 20 ft. long, strips run- ning across the room ? 10. How much will a carpet cost at f .90 a yard to cover a floor 22 ft. long and 15 ft. wide, if the strips run crosswise, and no allowance is made for matching ? 11. How many yards of carpet f yd. wide will be re- quired for a floor 20 ft. long and 15 ft. wide, if the strips run across the room ? 12. How many yards of carpet 1 yd. wide will be re- quired for a floor 18 ft. long and 14 ft. wide, strips running across the room ? 13. How many yards of carpeting are needed to cover a floor 24 *ft. long and 17 ft. wide, strips running length- wise and | yd. wide ? 14. If my room is 16^ ft. long and 12 ft. wide, how many yards of carpeting 24 inches wide must I buy, if in cutting 6 inches is allowed on each strip for matching ? 15. I wish to have a carpet woven. My room is 21 ft. long and 17 ft. wide ; how much carpeting, 34 inches wide, must I order to exactly cover the room, no allowance being made for matching ? MEASUREMENTS. 79 16. How many yards of carpet 2^ ft. wide will cover a floor 7 yards long and 14 ft. wide, if strips run length- wise, and it requires ^ yd. for matching ? 17. What will it cost to carpet a room 15 ft. by 17^ ft., with carpet 30 inches wide, at $1.20 per lineal yard, if the strips run lengthwise, and an allowance of 9 in. to each strip be made for matching ? 18. How much less would be the cost with no loss for matching ? 19. A room 31 ft. by 17 ft. is to be covered with carpet- ing 30 in. wide. How many yards must be purchased, and how wide a strip must be turned under ? 20. At $2.50 a yard, what will be the cost of a carpet to cover a parlor floor 6 yd. long and 5 yd. wide, if f yd. is wasted in matching ? 21. How many yards of matting 1^ yd. wide will be required for an assembly room 85 ft. 8 in. long and 64 ft. 6 in. wide, strips running across the room ? 22. A room 17 ft. 6 in. long, 14 ft. wide, is to be car- peted with carpet f yd. wide. A border f yd. wide goes around the outside. How many yards of border, and how many yards of carpet if strips run lengthwise, and there is a waste of one foot on each strip for matching ? PLASTERING AND PAINTING, ETC. 176. Plastering and Painting are usually done by the square yard. Allowance is sometimes made for doors and windows, which are called openings. Allowance is also sometimes made for base-boards and wainscoting. 1. A room is 18 ft. long, 12 ft. wide, and 10 ft. high. How many square yards in the walls and ceiling, making no allowance for openings ? 80 SENIOR ARITHMETIC. 12 ft. NOTE. Let the pupils draw a diagram for each room, represent- ing the four walls in a line. The entire length of the walls will be 2 x (18 ft. 4 12 ft.) = 60 ft. The area of the four walls, 60 ft. by 10 ft. = 600 sq. ft. Area of ceiling 18 ft by 12 ft. = 216 sq. ft. 600 sq. ft. + 216 sq. ft. = 816 sq. ft. = 77 - sq. yd., area of walls and ceiling. 18 it. 12 ft. 18 ft. 12ft. 2. Find the cost of plastering the walls and ceiling of a room 35 ft. long, 26 ft. 6 in. wide, and 15 feet high, at $ .45 a sq. yd., allowing 1024 sq. ft. for doors, windows, and base-board. 3. How many square yards of plaster in the sides and ceiling of a room 30 ft. long, 24 ft. wide, and 10 ft. high, allowing for a base-board 1 ft. high, 2 doors 3 ft. by 8 ft., and 4 windows 3 ft. by 6 f t. ? 4. Find the cost of plastering the ceiling of a room 18 ft. by 20 ft., at 10 cents a square yard. 5. A room 15 ft. by 18 ft., and 10 ft. high, has 4 doors each 3 ft. by 7 ft., and three windows each 3 ft. by 6 ft. Find the cost of plastering the walls and ceiling of the room at 30 cents a square yard, deducting one-half the surface for openings. 6. How many sq. yds. of plastering in the ceiling of a room 20 ft. long, 9 ft. high, and 15 ft. wide, no allowance for openings ? 7. At $.30 a sq. yd., how much will it cost to plaster a room 21 ft. 6 in. long, 16 ft. wide, and 9 ft. high, the base-board being 8 inches wide, and allowing for 3 windows MEASUREMENTS. 81 7 ft. by 2i ft., and six doors of the same dimensions as the windows ? 8. Find the cost of plastering the walls and ceiling of a room which is 36 ft. long, 27 ft. wide, and 9 ft. high, at 25^ per square yard. 9. My study is 18 ft. long, 16 ft. wide, 8^ ft. high, and contains 1 door 3 ft. by 7 ft., and 2 windows, each 3 ft. by 6 ft. The base-board is 9 in. high. What will it cost, at 36/ per sq. yd., to plaster it, making full deduction for openings ? 10. At 35 cents a sq. yard, what will be the cost of plastering the walls and ceiling of a room 6 yd. long, 5 yd. wide, and 3 yd. high, an allowance of 20 sq. yards being made for openings, etc. ? 11. Find the cost of plastering a room 18 ft. square and 10 ft. high, at 25 cents a sq. yard, being deducted for openings ? 12. A close fence 6^ ft. high surrounds a vacant lot 450 ft. by 380 ft. At 7 cents a sq. yard, what will be the cost of painting both sides of the fence ? 13. Find the cost at 18/ per sq. yard to plaster the sides and bottom of a cistern 8 ft. 6 in. square, and 9 ft. deep. 14. Find the square yards of plastering on a room 20 ft. long, 17 ft. 6 in. wide, 9 ft. high. Allow for 6 windows, each 7 ft. 6 in. high, 3 ft. wide, and 4 doors, each 7 ft. high and 3 ft. 9 in. wide. PAPERING WALLS. 177. Wall-paper is sold by the roll. A Single Eoll is 8 yd. long ; a Double Eoll, 16 yd. long. Borders are sold by the lineal yard. The number of rolls needed for a room is found by 82 SENIOR ARITHMETIC. dividing the area of the space to be papered by the area of one roll. The width of wall-paper is commonly 18 in. NOTES. Unless otherwise stated, a roll is considered as 8 yd. long and 18 in. wide. Dealers in wall-paper do not sell a part of a roll. If a part of a roll is needed, a whole roll must be purchased. 1. What would it cost to paper the walls of a room 18 ft. long, 12 ft. wide, and 9 ft. high, with paper 8 yd. to the roll and ^ yd. wide, at 45/ a roll ? 2. How many strips of paper, and how many double rolls, will paper the sides of a room 15 ft. long, 12 ft. wide, and 8 ft. high, each roll being 1 ft. wide, and 16 yd. long, no allowance being made for matching ? 3. How many double rolls of paper 16 yd. to the roll, J yd. wide, will be required to paper the walls and ceiling of a room 25 ft. long, 10 ft. wide, and 10 ft. high, 110 sq. ft. being deducted for doors, windows, etc ? 4. How many rolls of paper will be required for the walls of a room 16 ft. by 20 ft., and 9 ft. high above the base-board, allowing for 3 doors, each 3 ft. by 7 ft., and 3 windows, each 3 ft. by 6 ft. ? 5. What will be the cost of the paper and border for the above room at 30 cents a roll for the paper, and 15 cents a yard for the border ? 6. How many rolls of paper must be purchased to paper the walls and ceiling of a library, 12 ft. long, 10 ft. 6 in. wide, and 8 ft. high, the base-board being 6 in. wide, and the border 1-J- ft. wide, with paper \ yd. wide and 8 yd. long, the paper extending from the border to the base-board ? BOARD MEASURE. / 178. A Board Foot is a square foot of the surface of a board, 1 inch thick, or less. MEASUREMENTS. 83 To find the number of board feet in lumber that is more than one inch thick, we must multiply the num- ber of board feet in the surface by the number of inches in the thickness. A board 10 ft. long, 1 ft. wide, and 1 in. thick or less con- tains 10 board feet ; but a beam 10 ft. long, 1 ft. wide, and 8 in. thick contains 8 times 10 board feet = 80 board feet. To find the number of board feet in a tapering board, the average width must be found by taking the sum of the widths of the two ends. Thus a board 10 ft. long, and 12 in. wide at one end, and 6 in. wide at the other, con- tains as many board feet as if it had a uniform width of 9 inches. (12 -f- 6) -5- 2 = 9. 179. The number of board feet = Length (in feet) x Width (in feet) X Thickness (in inches). NOTE. When the thickness is one inch or less, the number of board feet is the product of the length and width in feet. 1. How many board feet in a board 15 ft. long, 15 in. wide, and 1 inch thick ? 2. How many board feet would there be in the board (Ex. 1) if it were f in. thick ? 2 in. thick ? 1 in. thick ? 3. How many feet of lumber one inch thick will be required for a tight board fence 6 ft. high around a yard 4 rods square ? 4. How much lumber (Ex. 3) will be required for an open board fence, 4 boards high, boards 8 inches wide, and 5 inches apart ? 5. I need 213 planks 4 ft. 8 in. long, 1 ft. wide, and 2 inches thick, to build a sidewalk. How much will they cost at $13 a thousand ? 6. How many feet of lumber will it take to build a line fence 168 ft. long, the fence being 5 boards high, and the boards 12 ft. long and 6 in. wide ? 84 SENIOR ARITHMETIC. 7. What will be the cost of 10 planks each 12 ft. long, 10 in. wide, and 3 in. thick at $16 per M. ? 8. Find the cost of a stick of timber 8 in. square, and 40 ft. long, at $18 per M. 9. What is the cost of 8 sticks of timber each 36 ft. long, 10 in. wide, 8 in. thick, at $12 per M. ? 10. How many board feet of 2-inch planking will it take to make a box 2 ft. 8 in. long, 2 ft. wide, 1 ft. 6 in. deep inside ? 11. Find the cost of 7 planks 12 ft. long, 16 in. wide at one end, and 12 in. at the other, at $.08 a board foot. 12. At $18 per M., find the cost of flooring a room 21 ft. by 16 ft., allowing 1 of the lumber for matching. NOTE. Find area of floor and add &. 13. Find the cost of a board 20 ft. long, 22 in. wide at one end, and tapering to 16 in. at the other, and l in. thick, at $30 per M. ? 14. At $12 per M., what will' be the cost of 2-inch plank for a 3 ft. 6 in. sidewalk on the street side of a rectangular corner lot 56 ft. by 106 ft. 6 in. ? MISCELLANEOUS. 180. 1. My dining-room is 15 ft. long and 12 ft. wide ; the walls are 10 ft. high. What will it cost to paper the walls and ceiling with paper l ft. wide, if there are 8 yd. in a roll, and each roll costs $.37. ( T L allowed for openings.) 2. What will a carpet for the dining-room (Ex. 1) cost me at $.75 a yd., carpet f yd. wide ? 3. There are three windows in the dining-room. What will it cost to furnish them with shades at $1.10 each and sash-curtains at $1.37^ each ? MEASUREMENTS. 85 4. I bought a table at $14.50, six chairs at $2.75 each, a sideboard for $30, and other furniture for $28.97. I also spent $20 for new table linen ; what did it all cost ? 5. What was the entire cost of refurnishing my dining- room ? 6. What will it cost to carpet a room which is 24 ft. long, and 18 ft. wide, with Brussels carpet 1 yd. wide; no waste in matching, at $1 per yard ? 7. Find the cost of plastering sides and ceiling of a room 26 ft. long, 13^ ft. wide, 13 ft. high, at 9 cents a square yard, allowing 25 sq. yd. for openings. 8. What will it cost to build a cement walk 40 ft. long, and 6 ft. wide, at $1.25 per sq. yd. ? 9. A field, containing 8 acres, is 60 rd. long. How wide is it ? 10. How many yards of carpet will cover a floor 18 ft. long, 16 ft. wide ? Carpet one yard wide, strips to run lengthwise of room. How many yards if the strips run crosswise of the room ? 11. Find the cost of fencing a rectangular corner lot 68 ft. by 130 ft., the street fence costing 54 cts. a yard, and the line fences 25 cents a yard, but only half of the cost of the latter to be charged to the lot. 12. Find the cost of a carpet f of a yd. wide, at $1.50 per lineal yard, for a room 20 ft. long and 18 ft. wide, strips running lengthwise, and allowing a waste of of a yd. on each strip for matching. 13. What is the breadth of a rectangular lot whose area is 75 sq. ch., and the length 9 ch. ? 14. What is the circumference of a circle whose radius is 9 ft. ? 00 SENIOR ARITHMETIC. 15. How many square yards in the above circle ? 16. How many revolutions does the 5-foot driving-wheel of a locomotive make in going 30 miles ? 17. Find the area of a triangle whose base is 5 feet and altitude 3 ft. 18. If the circumference of the earth is 25000 miles, what is the diameter ? 19. The circumference of a circle is 18 ft. What is its radius ? 20. How many sq. yards in a triangle whose base is 48 ft. and whose altitude is 24 feet ? 21. Find the area of the gable-end of a house whose width is 25 feet and whose ridge is 10 feet 6 inches higher than the base of the gable. 22. If the diameter of the earth is 8000 miles, what is the circumference ? 23. How many board feet in 10 planks 18 ft. long, 15 in. wide, and 2 in. thick ? and what will they cost at $40 per M. ? 24. Find the cost of 10 joists, 3 in. by 12 in., 16 ft. long, at $25 per M. VOLUMES. 181. Anything that has length, breadth, and thickness is called a Solid or Volume. 182. A Rectangular Volume is a solid having six rectan- gular faces. 183. A Cube is a solid having six equal square faces. 184. A Cubic Inch is a cube 1 inch long, 1 inch wide, and 1 inch thick. 185. The Volume or Solidity of a body is the number of cubic units that it contains. MEASUREMENTS. 87 3 dn. wide 1. How many cubic inches in a block 4 in. long, 3 in. wide, and 2 in. thick ? The block is made up of two layers, each \ in. thick. In the top layer there are 4 times 3 cu. in. In the two layers, therefore, there are 2 x (4 x 3 cu. in.) = 24 cu. inches. The multiplier must be considered as abstract. The three dimensions must have the same unit. The length, breadth, and thickness of a rec- tangular solid are its di- mensions. 3 cu. in. _4 12 cu. in. 2 24 cu. in. 186. Length x breadth x thickness = Solidity. Solidity - either dimension = the product of the other two. Solidity -f- the product of two dimensions = the other. 2. Find the number of cubic feet of air in a school- room 32 ft. square and 12 ft. high. 3. How high is a room that is 24 ft. long, 20 ft wide, and contains 4410 cu. ft. ? 4. A cu. foot of ice weighs 56J pounds. How much will a load of 22 cakes weigh, each cake measuring 2 ft. square, and 1 ft. thick ? 5. The capacity of a rectangular box is 480 cu. in The box is 8 in. wide, and 5 in. deep. How long is it 9 6. A schoolroom is 25 ft. long, 18 ft. wide, and 12 ft. high. If 60 pupils are seated in it, how many cu. ft. of air are allowed for each child ? 7. A man sold 3 blocks of Vermont marble, each 8 ft. long, and 6 in. x 6 in. at the ends. How much did he receive for the marble at $3.50 per -cu. ft. ? 88 SENIOR ARITHMETIC. 8. A hot-house bed is 3 ft. 9 in. long, and 3 ft. 4 in. wide, inside measure. How deep must it be to contain 25 cu. ft. of earth, and allow 6 in. for the growth of the plants ? 9. How many bricks 8 in. by 4 in. and 2 in. thick will be needed for a wall 60 ft. long, 20 ft. high, and 2 ft. thick, making no allowance for mortar ? 10. How many rectangular blocks 12 in. by 8 in. by in. can be packed into a wagon-box 10 ft. long, 4 ft. wide, and 2 ft. 6 in. deep ? 11. How many cubic yards of earth must be excavated from a cellar 30 ft. 10 in. long, 21 1 ft. wide, and 5 ft. 8 in. deep ? 12. How many square feet in the surface of a rectangu- lar box 3 ft. 4 in. long, 2 ft. 2 in. wide, and 1 ft. high ? 13. How many cubes 2^ inches on each edge can be sawed from a block of marble 10 ft. 2J in. long, 6 ft. 5 in. wide, and 3 ft. 4 in. thick ? 14. A box is 1.5 in. long, .85 in. wide, and .58 in. deep. What is its capacity in cubic inches ? 15. The altitude of a cylinder is 8 ft. and the circum- ference of the base is 3 ft. What are the cubic contents of the cylinder ? NOTES. Contents of a cyclinder = Area of Base x Altitude. Area of curved surface of a cylinder = circumference of Base x Altitude. This may be seen by cutting a piece of paper so that it will exactly cover the curved surface of a small cylinder. 16. What is the area of the curved surface in the cylin- der mentioned in Ex. 15 ? 17. How much tin will be required to make 2 doz. cylin- drical shaped cans, with a diameter of 4 in. and altitude MEASUREMENTS. 89 of 7 in., allowing tin for the curved surface and the two circular ends ? 18. How many cubic inches of water will the 2 doz. cans (Ex. 17) contain ? 19. What will it cost to dig a cellar 36 ft. long, 24 ft. wide, and 6 ft. deep, at 20 cts. a cubic yard ? WOOD MEASURE. 187. A pile of wood 8 feet long, 4 feet wide, and 4 feet high makes a Cord. One of the 8 feet in length of a cord of wood is a Cord Foot. NOTE. This may be illustrated by placing side by side 8 books of equal size. One of the books represents a cord foot. How many cords of wood in the following : 1. A pile 18 ft. long, 4 ft. wide, 8 ft. high ? 2. A pile 50 ft. long, 8 ft. wide, 6 ft. high ? 3. A pile 19 ft. long, 2 ft. wide, 5 ft. high ? 4. A pile 16 ft. long, 4-J ft. wide, 7 ft. high ? 5. What is the cost of a pile of wood 10 ft. long, 4 ft. wide, and 8 ft. high, at $4J a cord ? 6. How high must a pile of wood be piled to contain 10 cords, if the pile is 50 ft. long ? 7. How many cords of wood can be piled in a shed 24 ft. long, 18 ft. wide, and 12 ft. high ? 8. How many cords of building-stone in a pile 18 ft. long, 6 ft. wide, and 3 ft. high ? 9. At $3.50 a cord, what will be the cost of a pile of stone 15 ft. long, 4| ft. wide, and 5 ft. high ? 10. How many cubic feet in a cord of 2-foot wood ? 3-foot wood ? 18-inch wood ? 90 SENIOR ARITHMETIC. CAPACITY OF BINS. 188. 1. A bushel fills 2150.42 cubic inches of space. How many bushels of wheat can be contained in a bin 5 ft. X 5 ft. X 4 ft. ? 5 x 5 X 4 x 1728 -s- 2150.42. NOTE. A bushel fills 1 cu. ft. of space nearly. 2. A wine gallon fills 231 cubic inches of space. How many gallons of water can be contained in a rectangular tank 10 ft. by 8 ft. by 4 ft. ? NOTE. A cubic foot of space contains ^/f- gal. = 7& gal. nearly. Find the contents in bushels: 3. Of a bin 6 ft. long, 5 ft. wide, and 4 ft. high. 4. Of a wagon-box 10 ft. long, 42 in. wide, and 22 in. high. 5. Of a box 3 ft. by 2 ft. by 2 ft. 6. How high must a bin 8 ft. long and 5 ft. wide be built to contain 120 bushels ? Find the contents in gallons. 7. Of a tank 8 ft. by 6 ft. by 2 ft. 8. Of a cistern 6 ft. by 5 ft. by 4 ft. 9. Of a tank 5 ft. square and 6 ft. deep. 10. How many barrels of water will a cistern contain that is 6 ft. by 6 ft. by 7 ft. ? 11. A circular cistern is 5 ft. in diameter and 6 ft. deep. How many barrels of water will it hold ? NOTE. Area of base x altitude. 12. How deep must I build a bin that is 6 ft. square, to hold 90 bushels of wheat ? 13. How deep must I build a tank that is 5 ft. square to hold 40 barrels ? LONGITUDE AND TIME. V 91j LONGITUDE AND TIME. 189. A Meridian is an imaginary line running from the north pole to the south pole. All places on a meridian have the same time. NOTE. The meridians of Greenwich and Washington are the meridians that run through Greenwich and Washington. 190. Longitude is distance east or west from some stan- dard meridian, as Greenwich or Washington. When two places are on the same side of the standard meridian, their difference in longitude is found by subtraction. When 011 opposite sides, their difference in longitude is found by addition. 1. What is the difference in longitude between two cities, one of which is 20 west longitude, the other 30 east longitude 20 o + 30 o = 50 o. 2. What is the difference in longitude between two places, one of which is 40 E., the other 70 E. ? 70 _ 40 = 30. Ans. NOTE. No two places can have a difference in longitude exceed- ing 180. If, in finding difference in longitude by addition, the sum exceeds 180, subtract the sum from 360 to find the true difference. The earth turns upon its axis from west to east once in 24 hours, thus ^ of its entire circumference, 360, or 15 of longitude, passes under the sun in 1 hour. Since the earth turns at the rate of 15 every hour, in 1 minute it turns -^ of 15, or 15', and in 1 second ^ of 15', or 15". Hence, The earth rotates 15 in 1 hour, 15' in 1 minute, and 15" in 1 second. SS. The difference in longitude between two cities is 18, 30'. What is the difference in time ? 92 SENIOR ARITHMETIC. SOLUTION. Since the earth turns 15 15 /18 3CK in 1 hr., 15' in 1 min., and 15" in 1 sec., the 1 hr. 14 iniD. time can be found by dividing the number of degrees, minutes, and seconds by 15. 4. The difference in time between two cities is 54 inin. 19 sec. What is their difference in longitude ? 54 min. 19 sec. Since the earth turns 15 in 1 hr., 15' in 15 1 min., and 15" in 1 sec., the distance in 13 34' 45" degrees, minutes, and seconds may be found by multiplying the number of hours, min- utes, and seconds by 15. STANDARD OR RAILROAD TIME. 191. The railroad companies have divided the country into four time belts, extending north and south. All places in each belt take the time of the meridian which passes through or near the middle of the belt. The belts are as follows : Eastern, Central, Mountain, and Pacific. The standard meridian for the Eastern belt is the 75th, for the Central belt the 90th, for the Mountain belt the 105th, and for the Pacific belt the 120th. These standard meridians are 15 degrees apart. Therefore, when it is noon in the Eastern belt it is 11 A.M. in the Central belt, 10 A.M. in the Mountain belt, and 9 A.M. in the Pacific belt. In going westward into another time belt, the traveller sets his watch back one hour. In travelling eastward, he sets his watch ahead one hour. When it is noon on the standard meridian of each belt, it is called noon at all places in the belt. NOTE. Time reckoned by this method is not true solar time, but it secures a uniformity of time which is very desirable. The time in general use is Railroad or Standard Time. LONGITUDE AND TIME. 93 Oral. 5. When it is 5 P.M. Mountain time, what is the time in the Pacific belt ? 6. When it is 11 A.M. Pacific time, what is the Central time? 7. In travelling from San Francisco to New York, how many times do I change my watch ? and do I set it ahead or back ? 8. When it is 4 A.M. at Augusta, Me., what is the stan- dard time at St. Louis ? 9. When it is 1 P.M. Mountain time at Denver, what time is it at Washington, D.C. ? 10. AVhat is the Pacific time at San Francisco when it is 5 P.M. at Chicago ? 11. The longitude of St. Paul is 93 4' 55" west, of Philadelphia is 75 10' west. W T hat is the difference in longitude ? 12. The longitude of New York is 74 3" west, of Paris is 2 20' 12" east. What is the difference in longitude ? 13. New York City is 74 4" west from London. When it is noon at London, what is the true time at New York ? /' 14. The longitude of Boston is 71 4' west, and Chicago is 87 36' west. Chicago is how far due west from Boston, if there are 51.27 miles in one degree at their latitude ? 15. A person travelled until his watch was 3 hours too fast. In what direction and how far did he go ? 16. What is the difference in standard time between Boston and Chicago ? 17. If a person goes from New York to San Francisco, will his watch be too fast or too slow ? and how much ? 18. The difference in longitude between two places is 17 54' 55". What is the difference in time ? 94 SENIOR ARITHMETIC. 19. The longitude of San Francisco is 122 26' 15" west, and that of Cincinnati is 84 26' west. When it is 9 A.M. at San Francisco, what is the tL^e at Cincinnati ? 20. The longitude of Boston is 71 3' 30" west, and that of Paris is 2 20' 12" east. When it is 30 min. past 2 P.M. at Paris, what is the tinie at Boston ? 21. Chicago is 87 38' west. When it is 27 min. 36 sec. past 11 A.M. at Chicago, it is 10 min. past 12 M. at Wash- ington. What is the longitude of Washington ? 22. St. Louis is 90 15' 15" west longitude. A gentle- man arriving there from Boston, in 71 3' 30" west, finds that his watch, which was set in Boston, is not right. What change must he make ? 23. Mr. Jones started from Philadelphia, and travelled until his watch was 1 hour 30 min. slow. How many degrees did he travel ? and in what direction ? 24. When it is 12 o'clock noon at Chicago, what time is it in a place 60 30' 30" west of Chicago ? 25. Two men start from the same place, and travel in the same direction, one going 3 degrees and the other 5 degrees per day. They travel until their difference in time is 4 hours. How many days are they travelling? REVIEW OF DENOMINATE NUMBERS. 192. 1. Define a simple number ; denominate ; compound. 2. For what is linear measure used ? 3. For what is square measure used ? 4. For what is cubic measure used ? 5. For what is liquid measure used ? 6. For what is dry measure used ? Give tables. KEVIEW OF DENOMINATE NUMBERS. 95 7. For what is Troy weight used ? table. Avoirdupois weight ? table. Apothecaries' weight ? table. 8. How many grains in a pound Troy ? Avoirdupois ? 9. How many grains in an ounce Troy ? Avoirdupois ? 10. What is a long ton ? and how used ? 11. How many days in a common year? a leap year? AY hat is the solar year ? Explain leap year. When does the civil day begin and end ? 12. What is the use of circular measure ? Define circle, circumference, diameter, radius, arc. Give table. What is the measure of an angle ? What is a degree ? A quad- rant ? How do we find circumference ? How do we find diameter ? What is a right angle ? 13. What is a surface ? a square ? a rectangle ? a tri- angle ? 14. Give rule to find area of a square ; of a rectangle ; of a triangle; of a circle. 15. Define solid, rectangular solid, cube, cylinder. How do we find the volume of rectangular solid ? of a cylinder ? 16. W^hat is reduction ? Reduction ascending ? Reduc- tion descending ? 17. Define a denominate fraction. Give the different kinds of reduction of denominate fractions. 18. How do we add compound numbers ? subtract ? multiply ? divide ? 19. Give the common method of finding the difference between dates. How do we find the exact difference ? 20. What is longitude ? How do we find difference in longitude between two places on the same side of a prime meridian ? On opposite sides ? 96 SENIOR ARITHMETIC. * 21. How do we find difference in longitude when differ- ence of time is given ? How find difference of time when difference in longitude is given ? 22. What is standard time ? What are the names of the four time belts ? In passing west into a time belt, how does the traveller set his watch ? in travelling east ? 23. How do we find length when area and breadth are given ? 24. How do we find length when volume, thickness, and width are given ? 25. What are the dimensions of a rectangular solid ? 26. Define cancellation, even number, odd number, prime number, composite. 27. When are numbers prime to each other ? 28. How many cubic feet in a cord of wood or stone? How long, wide, and high is a cord of wood ? What is a cord foot ? 29. For what is board measure used ? What is a board foot ? Give rule for finding board feet. 30. How do we find the capacity of bins ? cisterns ? THE METRIC SYSTEM. LINEAR MEASURE. 193. The standard unit of Linear Measure in the Metric System is the Meter. It is determined by taking one ten- millionth part of the distance from the earth's equator to either of its poles, measured on a meridian. It is equal to 39.37 inches. QUESTIONS. 194. l. What denomination in the English linear meas- ure is most nearly like the meter ? 2. Draw a line one meter long. THE METRIC SYSTEM. 97 3. Hold your hands one meter apart. 4. A meter is about how many feet long ? 5. How many meters long is your schoolroom ? Wide ? High? 6. About how many meters in a rod ? HOW THE TABLE IS MADE. 195. Divide a meter into ten equal parts. One of these parts is a Decimeter. Dec is a Latin stem meaning tenth. About how many inches long is a decimeter ? Show with your hands the length of a decimeter. What part of a meter is a decimeter ? 196. Divide a decimeter into ten equal parts. One of these parts is a Centimeter. Cent is a Latin stem meaning hundredth. What part of an inch is a centimeter ? Show its length. How many centimeters in one meter ? What part of a meter is a centimeter ? 197. Divide a centimeter into ten equal parts. One of these parts is a Millimeter. Mill is a Latin stem meaning thousandth. What part of a meter is a millimeter ? How many millimeters in a meter ? What part of an inch is a millimeter ? 198. Ten meters make one Dekameter. Deka is a Greek steam meaning ten. How many rods in a deka- meter ? How many feet ? How many dekameters long is your schoolroom ? 199. Ten dekameters make one Hektometer. Hekto is a Greek stem meaning hundred. How many meters in one hektometer ? How many feet long is a hektometer ? 200. Ten hektometers make one Kilometer. Kilo is a Greek stem meaning thousand. How many meters in one kilometer ? How many feet ? What part of a mile ? 98 SENIOR ARITHMETIC. 201. Ten kilometers make one Myriameter. Myria is a Greek stem meaning ten-thousand. How many meters in one myrianieter ? How many feet ? How many miles ? 202. These statements may be combined in the follow- ing table : 10 Millimeters (mm.) = 1 Centimeter (cm.) = .3937+ in. 10 Centimeters = 1 Decimeter (dm.) = 3.937+ in. 10 Decimeters = 1 Meter (m.) =39.37+ in. 10 Meters = 1 Dekameter (Dm.) = 32.808+ ft. 10 Dekameters = 1 Hektometer (Hm.) = 19.927+ rd. 10 Hektometers = 1 Kilometer (Km.) = .621+ mi. 10 Kilometers = 1 Myriameter (Mm.) = 6.213+ mi. 203. REDUCTION. 1 Myriameter = 10 Kilometers = 100 Hektometers = 1000 Dekameters = 10000 Meters 100000 Decimeters = 1000000 Centimeters == 10000000 Millimeters. 1 Millimeter = .01 Centimeter = o .001 Decimeter = .0001 Meter .00001 Dekameter = .000001 Hektometer = 3 .0000001 Kilometer = .00000001 Myriameter. W o 204. The following series of numbers read from the top downward is reduction ascending ; read from the bottom upward is reduction descending. All metric numbers may be reduced in this way. 75689132. mm. 7568913.2 cm. 756891.32 dm. 75689.132 m. 7568.9132 Dm. 756.89132 Hm. 75.689132 Km. 7.5689132 Mm. All these numbers might be read thus : 75689132. THE METRIC SYSTEM. 99 QUESTIONS. 205. 1. How can a metric number be reduced to higher denominations ? To lower ? 2. Eeduce 12345678 mm. to cm. ; to dm. ; to m. ; to Dm. ; to Hm. ; to Km. ; to Mm. 3. Eeduce 9.6538714 Mm. to Km. ; to Hm. ; to Dm. ; to m. ; to dm. ; to cm. ; to mm. 4. Eeduce 7 Mm. to lower denominations. 5. Eeduce 7 mm. to higher denominations. 6. Eeduce 6307.1 m. to Km. ; to cm. 7. Eeduce 31 meters to inches. 8. Write 2 Mm. as meters ; 7 Km. ; 6 Hm. ; 8 Dm. ; 5 m. 3 dm. ; 2 cm. ; 9 mm. Write them all as one number. 9. Eeduce 1 Mm. to feet. 10. Write 7 Mm. and 6 mm. in one number, as meters. Eeduce it to higher denominations ; to lower. 11. Eeduce .075 Km. to cm. 12. Eeduce 8 Dm. and 6 m. to Mm. ; to mm. 13. Write 75 Km. and 62 dm. in one number as meters ; as cm. ; as Mm. 14. State the value of each figure in 30769.543 M. 15. A ship sails 100 Mm. in one day. How many miles does it sail ? 16. Give the table of Metric Linear Measure. 17. Name the standard unit. 18. How is it determined ? 19. What is the scale of the Metric system ? 20. Name in order the Latin and Greek stems used in the table. 100 SENIOR ARITHMETIC. SURFACE MEASURE. 206. The standard unit of surface measure is the Are (pronounced like the English air). The Are is a square whose side is one dekameter. It is therefore a Square Dekameter. QUESTIONS. 207. 1. An are is how many meters long ? Wide ? 2. How many square meters does the are contain ? 3. An are is how many inches long ? Feet ? 4. The are is about how many rods long ? 5. About how many square rods does it contain ? 6. About how many ares equal one acre ? 7. How many ares does the floor of your schoolroom contain ? 8. Name all the surfaces you can think of that con- tain about one are. TABLE. 208. The table of surface measure, like that of linear measure, is made by prefixing the Latin and Greek stems to the standard unit, thus : 10 Centares = 1 Deciare, da. 10 Declares = 1 Are, a. 10 Ares = 1 Dekare, Da. 10 Dekares = 1 Hektare, Ha. NOTE. The denominations of the above table are little used, except the are, the hektare, and the centare, which are employed chiefly in measurements of land. 209. Draw a square whose side is one meter. How many square meters does it contain? It is how many THE METRIC SYSTEM. 101 decimeters on a side ? How many square decimeters does it contain ? How many square decimeters make one square meter ? 210. Draw a square whose side is one decimeter. How many square decimeters does it contain? How many centimeters long and wide is it ? How many square centi- meters does it contain ? How many square centimeters in one square decimeter ? In the same way find how many square millimeters in one square decimeter. How many sq. Meters = 1 sq. Dekameter ? How many sq. Dekameters = 1 sq. Hektometer ? How many sq. Hektometers = 1 sq. Kilometer ? 211. The answers to the above questions form the fol- lowing table of surface measure, which is used for all ordinary surface measurements : 100 sq. Millimeters = 1 sq. Centimeter, sq. cm. 100 sq. Centimeters = 1 sq. Decimeter, sq. dm. 100 sq. Decimeters = 1 sq. Meter, sq. m. 100 sq. Meters = 1 sq. Dekameter, sq. Dm. 100 sq. Dekameters = 1 sq. Hektometer, sq. Hm. 100 sq. Hektometers = 1 sq. Kilometer, sq. Km. QUESTIONS. 212. 1. Which denomination of this table is like the are? 2. Like the centare ? 3. Like the hectare ? 4. How far to the right must the decimal point be moved to reduce sq. m. to sq. dm. ? 5. How many places to the left must the decimal point be moved to reduce sq. m. to sq. Dm. ? 102 SENIOR ARITHMETIC. 6. To reduce sq. mm. to sq. cm. ? 7. To reduce sq. mm. to sq. dm. ? 8. Reduce 5555 ca. to Ha. 9. Reduce 3333 Ha to ca. 10. A field 134 M. long and 7 Dm. wide, contains how many sq. m. of land ? 11. How many ares ? 12. How many Ha. ? 13. How many sq. Dm. ? 14. How many sq. Hm. ? 15. How many sq. cm. ? 16. How many sq. cm. in an oblong 643 cm. long and 2.5 m. wide ? 17. How many sq. mm. ? 18. How many sq. Km. ? 19. One hectare equals about how many acres ? VOLUME MEASURE. 213. The unit chiefly used in measuring wood and stone is the Stere (pronounced stair), which is a cube whose edge is one meter. What denomination in the English volume measure is most nearly like the stere ? How many cubic meters does the stere contain ? How many decisteres ? How many centisteres ? How many millisteres ? QUESTIONS. 214. A cube whose edge is one meter long contains how many cubic meters ? It is how many dm. long ? Wide ? High ? How many cu. dm. does it contain ? How many cu. dm. = 1 cu. m. ? A cube whose edge is 1 dm. contains how many cu. dm. ? How many cm. long is it? Wide? THE METRIC SYSTEM. 103 High ? How many cu. cm. does it contain ? How many cu. cm. = 1 cu. din. ? A cube whose edge is 1 CHI. con- tains how many cu. cm. ? How many mm. long is it ? Wide ? High ? How many cu. mm. does it contain ? How many cu. mm. = 1 cu. cm. ? 215. From the answers to the above questions make the following : TABLE OF VOLUME MEASURE. 1000 cu. Millimeters = 1 cu. Centimeter, cu. dm. 1000 cu. Centimeters = 1 cu. Decimeter, cu. m. 1000 cu. Decimeters = 1 cu. Meter, cu. cm. QUESTIONS. 216. 1. How may cubic millimeters be reduced to cubic centimeters ? To cubic dm. ? To cu. m. ? 2. How many places to the right must the decimal point be moved to reduce cu. meters to cu. millimeters ? 3. Reduce 7 cu. meters to cu. millimeters. 4. Reduce 5 cu. millimeters to cu. meters. 5. How many steres in one cu. meter ? 6. A pile of wood is 30 dm. long, 3 m. wide, and 18 dm. high. How many cu. meters does it contain ? 7. How many steres ? 8. How many cu. millimeters ? 9. How many cu. centimeters of air in an empty box 2 m. by 12 dm. by 75 cm. ? 10. How many cubic dm. ? 11. How many steres of stone in a wall 30 m. long, 5 dm. thick, and 250 cm. high ? 104 SENIOR ARITHMETIC. CAPACITY MEASURE. 217. The metric capacity measure takes the place of both the liquid and the dry measure of the English system. The standard unit of capacity measure is the Liter (pronounced teeter), which is a cube whose edge is one decimeter. QUESTIONS. 218. 1. The liter is what part of a meter wide ? High ? Long ? 2. What part of a cubic meter does it contain ? '3. About how many inches wide is it ? High ? Long ? About how many cu. in. does it contain ? 4. Show with your hands how wide, high, and long a liter is. 5. What denomination of English dry measure corre- sponds most nearly to the liter ? 6. Make a full-sized picture of a liter. 7. What object the size of a liter do you know ? TABLE. 219. The table of capacity is formed similarly to the other metric tables, and is as follows : 10 Milliliters (ml.) = 1 Centiliter, c. 10 Centiliters = 1 Deciliter, dl. 10 Deciliters = 1 Liter, 1. 10 Liters = 1 Dekaliter, Dl. 10 Dekaliters = 1 Hectoliter. HI. 10 Hektoliters = 1 Kiloliter, Kl. 10 Kiloliters = 1 Myrialiter, Ml. QUESTIONS. 220. 1. How many liters in 1 myrialiter ? In 1 ml. ? 2. How many milliliters in 1 ML? THE METRIC SYSTEM. 105 3. Reduce 12345678 ml. to higher denominations. 4. Eead the above number, giving each figure the name of the denomination it represents. 5. Eeduce 154.67 cl. to Kl. 6. Reduce .012346 Ml. to dl. 7. How many liters equal one cubic meter ? 8. A bin is 2.5 in. wide, 6.4 in. long, and 17 dm. deep. How many liters of oats will it hold ? How many HI. ? How many Kl. ? 9. A tank is 3 m. long, and 3 m. wide. How many dm. deep must it be to hold 50 HI. of water ? 10. A stone containing 1 stere, if dropped in a pond, would displace how many liters of water ? MEASURES OF WEIGHT. 221. The Gram is the unit of weight. It is equal to the weight of a cubic centimeter of distilled water at its greatest density. TABLE. 10 Milligrams (mg.) = 1 Centigram, eg. 10 Centigrams = 1 Decigram, dg. 10 Decigrams = 1 Gram, g. 10 Grams = 1 Dekagram, Dg. 10 Dekagrams = 1 Hektogram, Hg. 10 Hektograms = 1 Kilogram, Kg. 10 Kilograms = 1 Myriagram, Mg 10 Myriagrams = 1 Quintal, Q. 10 Quintals = 1 Tonneau, T. or Metric Ton. QUESTIONS. 222. 1. How many grams in 1 Metric Ton ? 2. How many mg. in 1 metric ton ? 106 SENIOR ARITHMETIC. 3. Keduce 1 mg. to T. 4. Reduce 1 T. to mg. 5. Keduce 9876543215 mg. to higher denominations. 6. Read the above number, giving each figure the name of the denomination it represents. 7. Recite the table of weight. 8. Spell the name of each denomination. 9. Reduce 7.42 quintals to centigrams. 10. Reduce 543 mg. to Mg. 11. One gram equals 15.432 grains. How many grains in 1 Kg. ? 12. One pound, Avoirdupois, contains 7000 gr. How many pounds are equivalent to one Kg. ? 13. Mr. Smith weighs 100 Kg. ; how many pounds does he weigh ? 14. How many grams does a cubic meter of distilled water weigh ? 15. Would a cubic meter of any other substance weigh the same ? State your reason. 16. How many kilograms of water will a tank 4 m. X 3 m. X 12 dm. hold ? REVIEW QUESTIONS. 223. l. How many tables in the Metric System ? 2. Name the standard units in the order in which they have been given. Repeat them until you can say them as rapidly as you can talk. 3. Name the prefixes in the same way. 4. Name and describe the unit of capacity measure ; of weight ; of length ; of volume ; of surface. 5. Repeat the tables. GENERAL REVIEW. 107 6. The stere is the unit of what measure ? The meter ? The are ? The gram ? The liter ? 7. How can metric numbers be reduced to higher de- nominations ? to lower ? 8. How many things are to be committed to memory in the Metric System ? 9. What is 39.37 ? 15.432 ? 10 ? These are the only numbers that need be remembered. GENERAL REVIEW, 224. l. Define fraction, numerator, fractional unit, terms, reduction of fractions. 2. Change 217 to 20ths. 3. Give the principle upon which reduction of fractions is based. Illustrate. ^x4. Add 25i, 14, 7f. 5. Give the rule for reducing fractions to their least common denominator. 6. A man owned of a foundry and sold \ of his share for $1200 ; what was the foundry worth ? 7. Reduce to simple form (15-| 3|) X (2 + 5|). 8 9. Reduce to least common denominator six thirty- fifths, nine twentieths, and five sixteenths, and arrange the results according to value. 10. A man having $130 used f of it ; how much of it remained ? 1. C and D can do a piece of work in 24 days, D can do it alone in 45 days ; how many days will C require to do it? 108 SENIOR ARITHMETIC. 12. The numerator of a fraction is 6510, the denomi- nator 66495; reduce the fraction to its lowest terms. 13. If 7 be added to each term of the fraction |, will its value be increased or diminished, and how much ? 14. Two men are 140 miles apart, and travel towards each other, one at the rate of 3| miles an hour, and the other at the rate of 41 miles an hour; in how many hours will they meet ? 15. Define decimal fraction ; an account ; currency. 16. What will 6827 feet of lumber cost at $10.50 per- M. ? 17. 8.7625 + 31.735 - 17.382569 = ? 18. Write in words 365. 8752. 19. Find the cost of 7896 pounds of hay at $16 a ton. 20. Express in figures two hundred sixty-five and five thousand one hundred ten millionths. 21. Change .875 to a common fraction in its lowest terms. 22. How is a bill receipted ? 23. Give a rule for dividing a decimal by 10, 100, 1000, etc. 24. Keduce 3.25, 12.364, and .56087 to a common de- nominator. 25. When will a fraction reduce to a perfect decimal ? 26. 7.6875 -f- 187.5 x (5| + 2|) = what ? 27. How is the place for the decimal point in the pro- duct determined ? 28. Give the abbreviations of Creditor and Merchandise. 29. Name the seventh decimal order. 30. James Harris of Syracuse, N.Y., sold for cash to Preston White, on Nov. 4, 1887, 42 Ib. of sugar at 10 GENERAL REVIEW. 109 cents ; 3 Ib. Y. H. tea at $.60 ; 4 gal. molasses at $.75 ; 48 yd. sheeting at $.14 ; 1 box starch 46 cents j and 8 doz. eggs at $.24. Make the bill in due form. 31. Define a square ; a circle. 32. Write the table of cubic measure. 33. For what purposes are the following used : Troy weight, dry measure ? 34. The last war with England commenced June 18, 1812, and ended Feb. 17, 1815. How long did it continue ? 35. A jeweller made 3 Ib. 2 pwt. 2 gr. of gold into rings weighing 5 pwt. 10 gr. each. How many rings were there ? 36. Reduce 2 mi. 6 ch. 3 rd. to links. 37. Reduce to integers of lower denominations |, and .25256 T. 38. Change 4 % 5 3 2 9 8 gr. to a decimal of a pound. 39. Find the result of 4.8 bu. -f 2f bu. -f .8125 pk. + 2| pk. + | bu. 40. A grocer bought 35 casks of molasses, eaoh contain- ing 44 gal. 2 qt. 1 pt. How much did they all contain ? 41. A ship in 8 north latitude sailed due south until it reached 12 south latitude ; find the distance it sailed in statute miles. 42. Find the value of T 7 5 of a ton. 43. Reduce ^ of a year to integers of lower denomi- nations. 44. Reduce f of a Ib. Troy to integers of lower denomi- nations. 45. Express 120 rd. 2 yd. 1 ft. 6 in. as the fraction of a mile. 110 SENIOR ARITHMETIC. 46. Eeduce 45 sq. rd. 2 sq. ft. 9 sq. in. to the fraction of an acre. 47. ^hat part of a day are 6 hr. 13 min. 20 sec. ? 48. What part of 4 gal. 2 qt. 1 pt. are 1 gal. 1 qt. 1 pt. ? 49. At 25 cents an ounce, what is the value of 18 oz. 10 pwt. 12 gr. of silver ? 50. How much will it cost to fill a bin with corn at $.45 a bushel, if the bin is 10 ft. square on the bottom and 4 ft. deep ? 51. A cistern measures inside the walls 8 by 6 by 9 ft., and lacks f^ ft. of being full. How many gallons does it hold ? 52. How many bushels will a box of the same dimen- sions hold ? 53. How many cords of wood in a pile of wood that is twice the length, height, and width of an established cord ? 54. Find the total weight of 5 car-loads of coal, weigh- ing respectively 14 T. 18 cwt. 63 lb., 17 T. 4 cwt. 85 lb., 13 T. 19 cwt. 26 lb., 15 T. 10 cwt. 43 lb., and 14 T. 7 cwt. 90 lb. 55. How many bricks 8 in. by 4 in. by 2 in. will it take to pave a street ^ mile long and ^ mile wide, laying the brick 011 the longest narrow face ? How many if they are placed on end ? 56. How much wood in three piles, the first of which contains 10 cd. 6 cd. ft. 4 cu. ft, the second 12 cd. 12 cu. ft., the third 17 cd. 1 cd. ft. ? 57. A family consumes daily 6 lb. 14 oz. of bread. If each loaf weighs 1 lb. 6 oz. and costs 7 cents, how much does bread cost the family for the month of August ? 58. Find the sum of g mi., fur., rd., and ft. GENERAL REVIEW. Ill 59. From 6J mi. take 4 mi. 140 rd. 4 yd. 60. A man has a bin 6 ft. long, 4 ft. wide, 3 ft. deep, filled with wheat. If he sells 10 sacks, each containing 2 bu. 1 pk. 5 qt., how much is left ? 61. From a piece of land 20 rods long, 180 ft. wide, were sold 4 lots, each 50 ft. wide, 150 ft. long. What part remained ? 62. A merchant bought two casks of wine, each contain- ing 41 gal. 3 qt., at $1.80 per gallon. One-seventh of it leaked away. He sold 9 kegs, each containing 5 gal 1 qt. at 30/ a pint, and the remainder for 40/ a pint. How much did he gain ? 63. A coal-dealer bought 34160 Ib. of coal at $2.50 per long ton. He sold 8 loads, each 1 T. 4 cwt. 60 Ib., at $3 per short ton, and the rest for $3.25 per short ton. How much did he gain ? 64. Change 3,895,504" to higher denominations. 65. Three quadrants of a circle are equal to how many seconds ? 66. Through how many degrees does the minute-hand of a clock pass in 1\ hours ? Through how many does the hour-hand pass in the same time ? 67. How many minutes elapse between four o'clock Fri- day afternoon and nine o'clock the following Monday morning ? 68. A boy was exactly 10 years old when the United States declared war against Mexico, May 13, 1846. How old was he at the time of the first bloodshed of the Civil War, April 19, 1861 ? 69. A train leaves New York at six o'clock Monday evening, and travels an average of of a mile a minute. When will it reach Buffalo, a distance of 410 miles ? 112 SENIOR ARITHMETIC. 70. A cistern that holds 50 bushels is 6 ft. square ; how deep is it ? 71. A pile of wood is 6 ft. high and 4 ft. wide. - How long must it be to contain 3 cords ? 72. A man has a circular garden with a diameter of 36 feet. How many rods of fencing will be required to enclose it ? 73. How many square yards in the above garden ? 74. A city lot is 35 ft. front and 125 ft. deep. Find the area. 75. A meadow contains 8| acres. Its width is 35 rods ; find the length of it. 76. How many yards of carpeting f of a yd. wide will be required for a room 18 ft. wide and 20 ft. long, if the strips run lengthwise, and there is a waste of 6 in. in each strip for matching patterns ? 77. The platform in a schoolroom is 30 ft. long and 11 ft. wide. What will be the cost of oil-cloth, at 85 cents a sq. yd., to cover it ? 78. How many feet, board measure, in 6 boards 16 ft. long, 10 in. wide, 1 in. thick ? 79. Find the cost of 10 Norway sidewalk planks 16 ft. long, 12 in. wide, 2 in. thick," at $18 per M. 80. A class-room is 15 ft. long, 12 ft. wide, 10 ft. high. Find the cost of plastering it at 20 cents a yard. 81. A room is 30 ft. long, 40 ft. wide, and 16 ft. high. Find the number of square yards of plastering in it after making allowance for wainscoting 3 ft. high, 8 windows, 4 ft. by 8 ft., and 6 doors, 3 ft. 6 in. by 7 ft. 6 in. 82. What will it cost to paper a kitchen 12 ft. by 11 ft. and 9 ft. high, with 10-cent paper, if each roll covers 4 sq. yd.? GENERAL REVIEW. 113 X83. Find the cost of papering a room 16 ft. long, 12 ft. wide, 9 ft. 6 in. high, with paper 18 in. wide, 8 yards in a roll, at 50 cents a roll, if 20 sq. yd. be allowed for doors, windows, and base-boards. 84. If a shingle is 4 inches wide, and lays 5i- inches to the weather, how many shingles will it take to shingle one side of a roof that is 32 ft. long by 22 ft. wide, allowing ail extra course at the eaves ? How many for both sides ? 85. What would be the cost of the shingles for both sides of the roof in No. 84 at $3.25 per M. ? 86. The product of two numbers is l^y ; one of the num- bers is ^. What is the other? 87. What fraction multiplied by f will equal T 3 T ? 88. How many square yards of carpet will be required to carpet a room that is 27 ft. by 33 ft. ? How many yards of carpet will be required if the carpet is 30 inches wide ? 89. If a hotel uses 3 pounds of coifee a week, what would be paid for coffee at 38 cents a pound for January, February, and March, 1896 ? 90. When it is noon at New York, 73 59' 9" W., what is the time at Chicago, 87 36' 42" W. ? What is the time at New York when it is noon at Chicago ? 91. When it is noon at Greenwich, what is the longi- tude of a place whose time is 8.30 A.M. s 92. A and B start at a given point, and travel in opposite directions. A travels until his longitude is 30 40' greater than it was, and B travels half as far as A. What is the difference in time between the places they are then in ? 93. What part of a pound Avoirdupois is a pound Troy? 94. What part of an ounce Troy is an ounce Avoirdu- pois ? 114 SENIOR ARITHMETIC. 95. A druggist bought opium at $8 a pound Avoirdupois, and sold it at 75/ an ounce Troy. What was his profit on 10 pounds ? 96. How much heavier is a pound of iron than a pound of gold ? 97. What is the difference in the areas of two fields, one being 5 Hm. long and 8 Dm. wide, the other 8 Hm. long and 14 Dm. wide ? 98. In a cubic dekameter how many cubic millimeters ? 99. A rectangular field is 5.4 Hm. long and 1.5 Hm. wide. How many hektares does it contain ? 100. Three fields have an area respectively of 19 A. 146 sq. rd., 12 A. 73 sq. rd. 15 sq. yd., and 9 A. 127 sq. rd. 26 sq. yd. What is the total area ? 101. Find the volume and the area of the curved surface of a cylinder whose diameter is 8 in., and whose altitude is 11 in. PERCENTAGE. 225. Oral. How much is 1 of 20 ? 5 is i of what ? 5 is what part of 20 ? 5 is how many hundredths of 20 ? Questions of Relation may be solved by means of hun- dredths ; thus, a. How much is T 2 ^ of 20 ? b. 5 is T 2 , Oral. 1. 5 is ^V of what ? 4. 12 is 8% of what ? 2. 5 is 25% of what? 5. 30 is 12i% of what? 3. 40 is 10% of what ? 6. 15 is 50% of what .' Written. 238. Percentage and rate given, to find base. Percentage -5- Eate = Base. 7. $125 is 12i% of what? 8. 150 bu. is 33i% of what? 9. 240 is 120% of what? 10. $1644 is 40% of what? 11. 75 is 3i% of what? 12. 289 is 50% of what? 13. 25% of my property is $5000. What is the value of my property ? PERCENTAGE. 121 14. I sold a horse for $81, which was 90% of what it cost me. What did the horse cost me ? Question <, Oral. 1. What part of 45 is 15 ? 2. What per cent of 45 is 15 ? 3. What per cent of 80 is 60 ? 4. AVhat per cent of 90 is 30 ? 5. $40 is what per cent of $60 ? 6. 12 yd. is what per cent of 36 yd. ? 7. 14 bu. is what per cent of 56 bu. '/ 8. f is what per cent of * ? Written. 239. Base and percentage given, to find rate. Percentage -j- Base = Kate. 9. What per cent of $240 is $80 ? 10. 150 is what per cent of 900 ? 11. What % of a long ton is a short ton ? 12. What % of 5 days is 6 hours ? 13. 5 cwt. is what % of 3 tons ? 14. $28.16 is what % of $7040 ? 15. What per cent is J of 2 ? of ? f of 7 ? 16. My salary is $1600 and my expenses $1200. What % of my salary are my expenses ? 240. The sum of the base and percentage is called the Amount. 241. The difference between the base and percentage is called the Difference. 1. Find the amount in the following : How much is 10% of 20 ? Find the difference. 122 SENIOR ARITHMETIC. 2. 20 is what per cent of itself ? 3. If 20 is increased by 10% of itself, the amount is 22. What per cent of 20 is 22 ? SOLUTION. The base . . . 20 is 100% of 20. The percentage . _2 is 10% of 20. Therefore the amount . 22 is 110% of 20. Ans. 4. 100% + 10% = ? 1 + 10% = ? 5. If 20 is diminished by 10% of itself, the difference is 18. What per cent of 20 is 18 ? SOLUTION. The base ... 20 is 100% of 20. The percentage . _2 is 10% of 20. The difference . "li is 90% of 20. An*. 6. 100% - 10% = ? 1 - 10% = ? 7. What number increased by 10% of itself equals 220 ? SOLUTION. Since 220 is 10% more than the required number, 220 is 110% of the required number. The amount, 220, is now treated as the percentage, and 110% as the rate; and the question becomes, 220 is 110% of what number ? (Question 6.) 220 -f 1.10 = 200. Ans. Amount -f (1 + rate) = Base. -8. What number diminished by 10% of itself equals 180? SOLUTION. Since 180 is 10% less than the required number, 180 is 90% of the required number. The difference, 180, is now treated as the percentage, and 90% as the rate ; and the question becomes, 180 is 90 % of what number ? (Question 6.) 180 -f .90 = 200. Ans. Difference -I- (1 rate) = Base. 9. What number increased by 25% of itself equals 290 ? 10. What number diminished by 25% of itself equals 243? PERCENTAGE. 123 11. After selling 20% of his sheep, a farmer had 400 sheep left. How many had he at first ? 12. The population of a certain city has increased 12% in two years. If it now numbers 56000, what was it at the beginning of the two years ? 13. A clerk's salary was increased 6|%. If he now receives $850, what was his original salary ? 14. By selling goods at $630 I lose 12i%. What did I pay for them ? 15. $580 is 10% less than what number? 16. I sold goods at $450, which was 120% of the cost. What was the cost ? 17. After withdrawing 45% of my money from the bank, I still have $1300 on deposit. How much had I in the bank at first ? 18. A farmer increased his flock of sheep by 12^%, and then had 900. How many had he at first ? 19. A man, after spending a month in the Adirondacks, finds that his weight is 210 pounds, which is an increase of 5%. What was his weight before he went to the Adiron- dacks ? ^ 20. A regiment lost 12i% of its men in an engagement, and had 560 left. How many men were there before the engagement ? 21. A owes C 33^% more than he owes B. If he owes C $800, how much does he owe B ? 22. 1227.83 is 1% less than what number? 23. | is 20% less than what number? 24. A city lot cost $3600, which is 55% less than the cost of the house. What was the cost of the house ? 124 SENIOR ARITHMETIC. 25. A farmer raised 1500 bu. of corn, which was 33*% less than the number of bushels of wheat raised. How many bushels of wheat had he ? 26. In the year 1896 a merchant's profits were $1836.24, which was 25% more than his profits of 1895. What were his profits in 1895 ? PROFIT AND LOSS. 242. Oral. State the question only. 1. How much is a 10% profit on goods that cost $200 ? 2. T bought goods for $400, and sold them at a loss of 5%. How much did I lose ? 3. If I buy goods at $400, and sell them at $600, what per cent profit do I make ? 4. If I buy at $400, and sell at $350, what % do I lose? 5. By selling a house for $1600 I gain 33 J%. What was the cost ? 6. John sold his skates for 64 cents, and thereby lost 5 % . What did he pay for them ? 243. All computations in Profit and Loss come under the rules of Percentage. The cost corresponds to the base, and the gain or loss is a percentage of the cost. The selling price is the amount when there is a profit, and the difference when there is a loss. 244. Written. 7. If I buy eggs for 10 cents a doz., and sell them for 12J cents, what per cent do I gain ? 8. A grocer bought tea at 18 cents per pound, and sold it at 30 cents per pound. What was the rate of gain ? PERCENTAGE. 125 9. Find the profit on a bicycle that cost $75, and was sold at an advance of 30%. 10. Find the selling price of a horse bought at $88.65, and sold at 3J% below cost. 11. Find the rate per cent of loss on a cow bought for $80, and sold for $60. 12. Find the rate per cent profit on a car-load of Cort- land wagons sold for $1090, and bought for $1000. 13. Find the cost of a herd of cattle sold at 12-% above cost at a profit of $240. 14. A man bought books for $194, and sold them at a gain of 32 % . What was the gain ? 15. I sold a house and lot that cost $11225 at a loss of 51%. What was the loss ? 16. Mr. A., by selling his horse at a profit of 14%, made $32.20. What did the horse cost ? 17. By selling sugar at one-half cent per pound profit, a grocer makes ten per cent. What does he get per pound for his sugar ? 18. An agent gained $.09 by selling twine 25% above cost. What did it cost him ? 19. Find the cost of cotton sold at 16-2% above cost at a profit of $211.25. 20. By selling flour at a loss of 14f %, a grocer loses $13.45. What was the cost? 21. A farm that cost $2675 was sold for $3745. What was the gain per cent ? 22. Hats that cost $43.50 a doz. are sold for $4.50 apiece. What is the rate of gain ? 23. By selling boots for $206.40 a merchant gained 20%. What did they cost him? 126 SENIOR ARITHMETIC. 24. By selling corn for $92.61, a man gained What did it cost him ? 25. I sell a horse for twenty per cent less than my ask- ing price, and yet make twenty-five per cent profit. I asked $200. What did the horse cost me ? 26. My height is 6 feet 1J inches, my neighbor is 5 feet 10 inches. What per cent am I taller than he is ? 27. A farmer sold 160 acres of land for $2944, which was 8% less than it cost. What did it cost an acre ? 28. By selling a horse for $160, I lose 20%. What would have been the selling price had I gained 20% ? 29. 14f% was gained by selling tea at $.45 a pound. What did it cost a pound ? 30. Mr. Brown sold a lot for $4300, and by so doing made 11^%. What did he gain? 31. If I buy oranges at the rate of 3 for 3 cents, and sell them at the rate of 2 for 5 cents, what per cent profit do I make ? 32. A jeweller sold two watches at $24 each. On one he gained 20%, and on the other lost 20%. What did both watches cost him ? COMMISSION. 245. Oral. 1. A certain agent receives for his services 2% of the value of the goods which he sells. How much will he receive for selling $1000 worth of goods ? 2. A purchasing agent receives for his services 3% of the value of goods purchased. How much will he receive for purchasing $2000 worth of goods ? 3. How much must I pay my agent for selling $3000 worth of potatoes if I pay him 5% ? PERCENTAGE. 127 4. At 5% how much will a collecting agent receive for collecting $800 ? 5. How much must I pay my broker for selling $1000 worth of stocks, if I pay him % of their value ? 246. An Agent is a person who transacts business for another. 247. Some agents are known as Brokers, or Commission Merchants, according to the kind of business transacted. 248. The compensation of an agent is called Commis- sion, or Brokerage. The commission of a purchasing agent is usually a cer- tain per cent of the value of his purchases. The commission of a sales agent, or of a collector, is usually a certain per cent of the amount collected. 249. The merchandise sent to a commission merchant to be sold is called a Consignment. 250. The sender is the Consignor, and the person to whom the goods are sent is the Consignee. 251. The commission is the percentage, and the amount collected or invested is the base. 252. Written. 6. An auctioneer charges 5% commission for selling $864 worth of goods. What is the amount of his com- mission ? 7. Sold 850 barrels of flour at $5.25 a barrel, and charged 2|% commission. Find my commission. 8. What is an agent's commission for selling 6840 Ib. of butter, at 19 cents a pound, commission \\% ? 9. A dealer sells real estate for a commission of 2%. How much must he sell during the year to secure an income of $75 per month? 128 SENIOR ARITHMETIC. 10. A broker in New York received ^ of one per cent commission for negotiating a sale of 500 one thousand dollar bonds. What was his commission ? 11. A real estate man made $50 by receiving 2^ per cent instead of his regular commission of 2 per cent. What did his sales amount to ? 12. An agent's fee for collecting bills is 3%. If he receives $86.25 as his commission, how much money has he collected ? 13. An agent collected $1864 from a sale of some pictures, and received $4.66 as his fee. What was the rate of commission ? 14. An agent having sold 1250 velocipedes at $8 apiece, invested his commission of If % in a new stock company. How many shares at $25 each did he take ? 15. What per cent does an agent charge who receives $223 for buying $5575 worth of produce ? 16. A man sejids his agent $6000 to invest in flour, after deducting his commission of 2%. How much money is spent for the flour, and how much for the agent's commission ? 17. A man is paid 5% for collecting $235.75. How much must he pay over to his employer ? 18. A merchant sent his agent $3150 with which to buy flour after deducting his commission of 5%. At $4 per barrel how many barrels did the agent buy ? 19. An agent sold iron for $9872. He received $163.70, which included a freight charge of $52.64. What rate of commission did he receive ? 20. Received as net proceeds from a sale of cotton $1025.70, after paying my agent 1\% for selling. What did the sale amount to ? PERCENTAGE. 129 21. An auctioneer sells 15 tables at $1.45 apiece, 22 chairs at $1.121 apiece, and some pictures for $8.70, on a commission of o\/ . What was his commission, and the net proceeds of the sale ? 22. My agent in Boston sold a number of bicycles at $85 each. After deducting his commission of 3%, he re- turned to me $5759.60. How many bicycles did he sell ? 23. An agent who sold 150 lots at $233^ each, charged $262.50 for his services. What rate of commission did he get? 24. A collector pays over to his principal $23358.39^ after deducting a commission of 4%. How much was the entire collection ? 25. If I send my agent $367.20, with instructions to buy tea at 30 ct. a pound, and he charges 2% for buying, how many pounds of tea should I receive ? 26. A real estate agent charges me two per cent for sell- ing my property in Boston. He remits me $5880. What was his commission ? 27. A commission merchant _in New York charged $36 for insuring my goods, $14 for cartage, and $50 commis- sion at 2i per cent for selling them. How much money should he remit to me ? -- 28. Sent my agent $2050 to invest in coal at $4 per ton, after deducting his commission of 2i per cent. How many tons of coal could he buy ? 29. A cotton broker received $2531.71 with which to buy cotton at $.12 a Ib. He charged 2-J% commission. How many pounds of cotton did he buy, and what was his com- mission ? 30. A real estate agent receives $162,193.50 from a com- pany to invest in land. If he charges 5% commission, how 130 SENIOR ARITHMETIC. many acres of land can he buy at $9 an acre ? What is his commission ? 31. An agent sold 12000 Ib. of cotton at 10 cents a pound. He invested the proceeds in lumber at $25 per M. If his commission for selling was 4%, and for buying 2%, how many feet of lumber did he purchase ? 32. A grain-dealer received $4820.40 with which to buy wheat at 60^ a bushel after deducting his commission of 3%. How much wheat did he purchase ? 33. How much stock can be bought for $10827, allowing li% brokerage ? INSURANCE 253. 1. I keep my house insured for $4000. I pay the insurance company 1% annually. How much do I pay annually ? 2. If my house (Ex. 1) burns down at the end of three years, how much shall I receive from the company more than I have paid them ? 3. If I pay $75 per annum for insuring my house at 1%, for how much is it insured ? 4. At 2% what will be the cost of insuring $10000 of merchandise at j value ? 254. Insurance is security against loss. 255. The payment made for insurance is called the Pre- mium. Some of the different kinds of insurance are, 256. Fire, Lightning, Tornado, Accident, Life, and Marine. 257. The contract between the insurer and the insured is called the Policy. 258. The amount insured is the base, and the premium is the percentage. PERCENTAGE. 131 - 5. What will it cost to insure a house worth $ 2500 at | of its value, for three years, at f % ? 6. Insured a country store for $5,000 and goods for $10,000, at 30X on $100. $1 is paid for the policy. What does the insurance cost ? 7. What will it cost to insure a mill for $5000, the rate being one and one-half per cent for 3 years ? 8. How much will I save by insuring my property for $5000 at | of one per cent for 3 years, rather than taking an annual policy for | of one per cent ? 9. It costs me to insure my house $22.50 when the rate is | of one per cent. What is the amount of my policy ? 10. A stock of goods is insured for one-half the value, the premium being $30, and the rate T % of one per cent. What is the value of the goods ? 11. The semi-annual premium per one thousand dollars on my $6000 life-insurance policy is $26. What does it cost me a year ? 12. A person who pays $12 semi-annually for accident insurance is disabled by an accident for 13 weeks, during which time he receives $10 a Aveek. If he has paid three premiums, how much more does he receive than he has paid out ? 13. Paid for insuring a house for of its value, $151. The rate being 75^ on $100, and the policy costing $1, what was the house worth ? 14. To insure a house at ^ of 1% cost me $20. What was the house worth ? 15. Paid $18 for insuring goods worth $9000. What was the rate ? 16. A merchant pays $75 a year insurance on his stock of goods at 1 \%. What is the value of his stock of goods ? 132 SENIOR ARITHMETIC. 17. A block worth $30000 is insured for of its value at 2%. How much does the owner lose in case of its total destruction by fire ? 18. For how much must a cargo of wheat worth $24000 be insured, at 2i%, so that the owner, in case of loss, may recover both the value of the cargo and the premium. NOTE. The value of the wheat = 97% of the amount insured. TRADE DISCOUNT. 259. The deduction of a percentage from the price of merchandise is called Commercial Discount. It is used largely by manufacturers and wholesale mer- chants. The greatest discounts are for large purchases and cash payment. 260. The List Price is the price given in the price-list. 261. The Net Price is the list price -less the discount. 1. If I can purchase books at 25% off for cash, what must I pay for books listed at $80 ? SOLUTION. 100% 25% = 75%. 75% of $80 = $60. Ans. NOTP:. When two or more discounts are allowed, the first is deducted, the second computed on the remainder, and deducted from it, etc. 2. At what per cent above cost must a merchant mark his goods so that he may allow a discount of 25% from the marked price, and still make a profit of 10% ? SOLUTION. Selling price = 110% of cost. This selling price is 75% of the marked price. The question is, 110% is 75% of what. 1.10 -7- 75 = 1.46|%, therefore the marked price is 46|% above cost. 3. Find the sum to be paid on a bill of $264 with 10% off for cash. 4. What is the net price of a bill of goods, the list prict 1 of which is $56, subject to discount of 25% ? PERCENTAGE. 133 5. What must be paid on $935, if 15% and 10% off are allowed ? SOLUTION. Deducting 15$ is the same as allowing 85$ of the bill. 85$ of 935 = 794.75. 90$ of 794.75 = 715.28. Ana. 6. Which is the better for the buyer, 40%, or 25% and 15% off? 7. Find a single discount on a bill of $300 equal to 20% and 5% off. 8. A discount of $4 was allowed on a bill, which was then paid with a check for $36. What rate per cent was taken off ? 9. Consulting niy price-list, I find I can buy goods which are marked $450 at a discount of 20% and 5% off for cash. How much will the goods cost me ? and how much discount do I receive ? 10. Bought furniture amounting to $520 on credit for 6 months, or 5% discount for cash. What ready money will pay the bill ? 11. What is the cash value of a bill of books amounting to $40, on the face of which a discount of 20% and 5% is made ? 12. The net amount of a bill of goods is $359.10. What is the gross amount, the rate of discount being 10% and 5% ? 13. A set of Encyclopaedias, whose catalogue price is $100, can be bought at a discount of 2 tens and 5% off for cash. How much less than the catalogue price will they cost? NOTE. The expression 2 tens and 5$ means 10$, 10$, and 5$. 14. B offers me some hammocks for $450 with a dis- count of 20%, and 4% off for cash; and A offers me the same goods at a discount of 2 tens and 4% off. Which is the better offer ? and how much ? 134 SENIOR ARITHMETIC. 15. A dealer sold goods at 10% below his asking price, but still made a profit of 20%. What per cent above cost had he marked the goods ? 16. A merchant marked carpeting that cost him 60 cents a yard so that he could allow a discount of 10% and still make a profit of 20 % . At what price did he mark it ? 17. A book-dealer sold a stock of books for $1140, at a discount of 10% from the marked price, and finds that he has made a profit of 14%. What did he pay for the books ? and what was their marked value ? 18. Find the net amount of a bill for $386 subject to the following discounts, 20%, 10%, and 5%. TAXES 262. A tax is a sum of money levied upon property and persons for public use. NOTE. A tax upon persons is called Capitation or Poll Tax. It is levied in some localities upon men of full age, without regard to their property. It is usually but a small amount upon each person. The practice is going out of use, 263. Property is of two kinds, Real and Personal. 264. Real Property is immovable property, as lands and buildings. 265. Personal Property is property that is movable, as money, securities, household goods, horses, cattle, etc. 266. A tax assessed upon property is a Property Tax. 267. Assessors are officers chosen to make a list of tax- able property, estimate its value, and apportion the tax. 268. A tax is a percentage upon the assessed valuation of property. The tax on $1 is the rate. 1. The valuation of property in a certain town is $1,500,000, and the rate is 11%. What is the tax? PERCENTAGE. 135 2. The tax to be raised in a certain village is $37500. The taxable property is $2,500,000. What is the rate? What will be A's tax on $15000 real estate, and $3000 personal ? 3. The property of a town is assessed at $1,250,000. The tax to be raised is $15975. There are 650 polls, assessed at $1.50 each. What is B's entire tax, if his property is assessed at $2500, and he pays the poll- tax ? Rule. Deduct the amount of poll-tax, If any, from the whole tax. Divide the remainder by the assessed val- uation. The quotient will be the rate. To find each person's tax, multiply the assessed valua- tion by the rate, and to the product add the poll-tax, If any. 4. The officers of a certain town find that all the town expenses for the year 1896 will amount to $46000. The tax-roll shows real estate valued at $2,000,000, and per- sonal property at $300000. What is the rate of taxa- tion ? -5. A certain town votes to raise a tax of $14250, be- sides the collector's commission of 5%. What is the rate of taxation if the property valuation is $1,000,000 ? What is the collector's commission, and what is A's tax. on property valued at $4,500 ? 6. If the assessed valuation of a village is $2,384,564, and there are 750 polls at $1.50 each, what must be the rate of taxation to meet an expense of $29807.05 ? What is B's entire tax, if his property is valued at $3875, and he pays for 1 poll ? 7. What is the valuation of my property, if my tax, 15 mills on a dollar, amounts to $30. 136 SENIOR ARITHMETIC. 8. What is my entire tax, if I pay a poll-tax of $1.68, and my property is valued at $24750, when the rate of taxation is $16.28 on $1000? 9. The annual tax-rate for the State of New York for the year 1896 was 2.69 mills on the dollar. The amounts to be raised by tax are as follows: $961116 for general expenses, $4,062,903 for free schools, $2,360,103 for the canals, and $4,368,712 for the State care of the insane. What was the assessed valuation of the property of the entire State ? The tax-rate for 1895 was 3.24 mills. What was the entire tax of 1895? DUTIES. 269. Duties are taxes on imported goods, levied by the government, and collected at custom-houses. A port con- taining a custom-house is called a Port of Entry. 270. An Ad Valorem Duty is a certain rate on the value of goods at the place from which they were shipped. 271. A Specific Duty is a fixed sum charged upon an im- ported article, without regard to its value. Allowances are made as follows, in collecting specific duties : for Tare, which is weight of box, cask, etc. ; for Leakage, which is loss of liquids in barrels or casks ; and for Breakage, which is loss of liquids in bottles. 272. The Gross Weight is the weight of articles before any allowances are made. 273. The Net Weight is the weight after the allowances are made. 1. At 30 per cent ad valorem, what is the duty on goods valued at $725 ? 2. What is the duty on 10 gross of silver spoons, val- ued at $4.50 a dozen, at 30% ad valorem ? PERCENTAGE. 137 3. A. Mark's Sons imported from Lyons 1560 yd. of silk invoiced at 87 \fl per yard. What was the duty at 25/ a yard, and 30^ ad valorem? 4. If the average rate of duty under the McKinley law was 49.58 per cent, and under the Wilson law it is 37 per cent, what is the difference in revenue on $1,000,000 worth of dutiable imports ? 5. A merchant bought goods in London invoiced at 450. At the custom-house in New York he paid an ad valorem duty of 18%, and a specific duty of $325. What was the entire cost of the goods in United States money ? 6. Imported from England 5 cases of cloths and cash- meres, net weight 95 Ib. ; value as per invoice 375 10s. What is the duty, the rate being 50^ per pound, and 35% ad valorem ? QUESTIONS. 274. 1. What is the meaning of the term per cent ? How is per cent written ? 2. Define base, percentage, rate per cent, amount, dif- ference. 3. Tell how to find percentage when base and rate are given. To find base when percentage and rate are given. To find rate when percentage and base are given. 4. Tell how to find base when amount and 'rate are given. When difference and rate are given. 5. Define Commission, Brokerage, Insurance, Premium, Policy, Taxes, Real Estate, Personal Property. 6. What is trade discount ? List price ? Net price ? Give rule for finding net price. MISCELLANEOUS REVIEW OF PERCENTAGE. 275. l. Find 8% of 750. 61% of $12.75. |%of912. of 2140. 138 SENIOK AKITHMETIC. 2. A man gave his son 42% of his money, his daughter 25% of it, and his wife 16% of the remainder. If the son received $9350 more than the daughter, what did each receive ? 3. A dealer sold a horse and carriage for $637, which was 40% more than cost. If the horse cost f as much as the carriage, what did each cost ? 4. What per cent is gained when one-half an article is sold for what the whole cost ? When f of an article is sold for what one half-cost ? 5. A merchant pays $35 for a suit of clothes. What must he ask for it, so that he may drop 16% from his asking price, and still make 20% on the cost ? 6. 14.35 is T 7 ff % of what number ? 7. A man spent 20% of his salary for board and 15% of what was left for clothes. If he spent $132 more for board than for clothes, how much did he spend for each ? 8. What number diminished by 16% is 605 ? 9. What number increased 35% is 382.5 ? 10. Sold a load of wheat weighing 3240 Ib. at 68^ a bushel of 60 Ib., thereby making a profit of 6| per cent. Required the cost of the wheat ? 11. On a certain day the sun rose at 5 o'clock and 43 minutes, and set at 6 o'clock and 25 minutes. What per cent of the day was in sunlight ? 12. The salary of a certain teacher of arithmetic is $1600. His real estate tax is $90 ; his water tax is $25 ; gas bill, $15 ; coal bill, $45 ; other expenses, $325. What per cent of his salary does he save in the bank ? 13. The Oswego Starch Factory employs 700 operatives. The population of Oswego numbers 22000. What per cent of the population is employed in the starch factory ? PERCENTAGE. 139 14. | is 25 % more than what fraction ? 15. A dealer lost 10% of his capital, then gained 20% of the remainder, when he had $2160. How much had he at first ? 16. Goods bought for $400 are marked to sell at anj advance of 40%, but are finally sold at a reduction of 25% from the marked price. What is the per cent of gain ? What is the gain ? 17. An article is sold for $2.80, this being an advance of 25%. Find the cost. 18. A merchant buys sugar at an average price of 4 cents a pound, and sells at a profit of 8%. How many pounds must he sell to clear $500 ? 19. If by selling an article for 59 cents a dealer gains 10% more than by selling for 55 cents, what is the original cost ? 20. 15% of an estate is invested in city bonds, 40% in real estate, 25% in railroad stock, and the remainder, $5000, is deposited in a bank. W T hat is the estate worth ? 21. Define percentage, base, profit and loss, commission. 22. Give the five formulas of percentage. 23. Express as a decimal | per cent. 24. A man having a yearly income of $1500 spends 80% of it the first year, 75% of it the second year, 62 % of it the third. How much does he save in 3 years ? 25. 25% of 200 bushels is 2% of how many bushels ? 26. A man sold 80 acres of land for $1472, which was 8% less than it cost. What did it cost an acre ? 27. What terms in Profit and Loss correspond to base and amount ? 28. Find the cost of fruit sold for $207.70, at a gain of 140 SENIOR ARITHMETIC. 29. At what price must hats that cost $1.12 each be marked in order to abate 5 % , and yet make 25 % profit ? 30. What is the base in commission ? 31. A commission merchant sells 225 bu. of corn at $.65 a bushel, and 360 bbl. of apples at $2.40 per barrel ; com- mission 5%. Find the commission and the net proceeds. 32. The net proceeds are $3800, the rate 10%. Find the amount of sales and the commission. 33. Find the rate, the commission being $125, and the sum invested $2500. 34. A merchant owning f of a cargo valued at $44000 in- sures | of his share at 2|%. What premium does he pay ? 35. A man having $400 paid 62i% of it for a carriage. How many dollars had he left ? 36. An agent charged $432.46 for selling goods at $49424. W T hat was his rate of commission ? 37. A man sold four horses for $100 each ; on two he gained 25%, and on the other two he lost 25%. Did he gain or lose on the transaction ? and how much ? 38. If f the number of girls in a certain school exceed the boys 10%, and the girls number 275, what is the num- ber of boys ? 39. A farmer's sheep increased 10% each year for 2 years, when he had 242. How many had he at first ? 40. My New York agent buys for me 40 pieces of silk, 32 yards in a piece, at $5 a yard. He charges li% com- mission. How much money will it require to purchase the silk and pay his commission ? 41. A commission merchant in Boston has sold goods for me to the amount of $6932. He has charged 1|% commission, $18.50 cartage, and $12.15 for storage. How much is due me ? PERCENTAGE. M-O 42. A boy bought oranges at the rate of 3 for 5 cents, and sold them at the rate of 2 for 5/. What was his rate of gain ? - 43. Ten per cent of a number is 32 less than eighteen per cent of the same number. What is the number ? 44. I paid $28.87^- for insuring my house for $3850 for three years. What was the rate of the yearly premium ? 45. A stock of goods valued at $6300 is insured for its value at f %. W T hat will be the owner's loss if ,the goods are totally destroyed by fire ? 46. A man's income is $1720, which is 16f % of the sum he has invested. What sum has he invested ? 47. From a cargo containing wheat, 1620 bu., or 7%, was washed overboard. What number of bushels remained ? 48. A stock-dealer sold 38 head of cattle, which was 4% of his entire herd. How many had he left ? 49. In an orchard containing 820 trees, 20% of them were pear-trees, and the remainder were plum-trees. How many plum-trees were there in the orchard ? 50. From a cask of wine containing 65 gal. all but 15% was sold. How many gallons were sold ? 51. In a school containing 875 pupils 32% of them are boys and the remainder girls. How many girls are there ? 52. There is a loss of $500 on a house and lot sold for $5000. What is the per cent of loss ? 53. An agent reports that he invested the money re- mitted him in wheat, which he sold at an advance of 15% ; then investing the proceeds in a second quantity, he was forced to sell at a loss of 12|%. He now deducts $100 for expenses and commission, and remits $5333.75 to his employer as the balance due him. Find the loss to the employer. 142 SENIOR ARITHMETIC. 54. $90 are paid as premium for insuring a block for three-fourths of its value. If the rate of insurance is |%, what is the value of the property ? 65. New York State has a population of 5,998,000, and New York City has 1,515,000. What per cent of the population of the State live in New York City ? 56. During the war of 1861-1865, the State of New York paid $40,000,000 in bounties to her volunteers. Her population at that time was (in round numbers) 4,000,000. What was the average cost to each inhabitant ? 57. If I sell 6 horses for what 8 horses cost, what is my rate of gain ? 58. Sold wheat for $73.541, by which a gain of 15% was made. What did the wheat cost ? and what sum was gained ? 59. In a school containing 1160 pupils, 638 are girls and the remainder are, boys. What per cent are boys ? 60. A hall is 42 ft. wide, and 294 ft. long. What per cent of the length is the width ? .... 61. In a certain battle 22f % more than ^ of the soldiers were killed. If the loss was 110 men, what was the original number ? 62. In an orange-grove 8j % of the trees were ruined by frost. If 1100 remained uninjured, how many were destroyed ? 63. A merchant sold a lot of goods for $550, thereby gaining 10%. Find the cost of the goods. 64. A man sold a watch for $32, thereby losing 20% on the cost. Find the cost. 65. If a man owns 66 per cent of a factory, and sells 33^ per cent of his share for $1800, what is the value of the factory ? SIMPLE INTEREST. 143 66. A sold 30% of his steamship to B~; B sold 60% of his purchase to C; C sold 75% of his share to D for $27000. What was the value of the vessel ? What was each one's share in dollars after the sales had been made ? 67. After a discount of 30% had been made upon the catalogue price of a book, it was sold for $1.75. What was the catalogue price ? 68. Bought a horse for $120, and sold him for $135.' What part of the cost was the gain ? What per cent ? 69. Bought tea at 60 ct. a pound. What must I ask per Ib. so as to abate 10% and still make a profit of 25% ? 70. A merchant's profits for 1895 were $3402.84. If they were 6|% less than in 1894, what were they in 1894 ? 71. In one week John solved 75 problems correctly. If he failed in 16% of the number attempted, how many were there in all ? SIMPLE INTEREST. 276. 1. I borrow $500 for 1 year, and at the end of the year I repay the money and 6% -for the use of it. How much do I pay for the use of $500 ? -2. How much must be paid for the use of $50 for 1 year at 5% ? at 7% ? 3. How much at 5 % per annum must I pay for the use of $1000 for 1 year ? For 3 years ? 4. I loan James Barnes $500 at 6%. At the end of 2 years he pays me in full. How much does he pay me ? Money that is paid for the use of money is called Inter- est. The money for the use of which interest is paid is called the Principal, and the sum of the Principal and interest is called the Amount. Interest at 6% means 6% of the principal for 1 year. 144 SENIOR ARITHMETIC. 12 months of 30 days each are usually regarded as a year in computing interest. Oral. 5. What is the interest of $100 for 3 years at 6% ? SOLUTION. $100 Principal. .06 Rate. $6.00 Interest for 1 year. 3 $18.00 Interest for 3 years. 6. What is the interest of $80 at 5% for 2\ years ? 7. What is the interest of $1000 at 5% for 2 yr. 6 mo. ? 8. What is the interest of $100 at 6% for 1 year? For 1^ ? For 2 yr. 6 mo? For 3 yr. 3 mo. ? For 1 yr. 6 mo. ? When the time does not include days, find interest as follows : Principal X Kate X Time = Interest. 9. What is the interest of $297.62 for 5 yr. 3 mo. at SOLUTION. $297.62 .06 NOTE. Final results should $17.8572^ not include millg< Mills are dig . 44^43 regarded if less than 5, and called 892860 another cent if 5 or more. $93.75 Ans. Find the interest of : 10. $384.62 at 6 % for 2 yr. 12. $250.50 at 8 % for 5 yr. 11. $463.75 at 7% for 3 yr. 13. $685.20 at 4% for 6 yr. 14. $596.15 at 5% for 2 yr. 3 mo. 15. $386.42 at 5% for 6 yr. 5 mo. 16. $950.16 at 10% for 4i yr. 17. $283.25 at 6% for 2 yr. 8 mo. SIMPLE INTEREST. 145 Find the amount of : 18. $284.10 for 3 yr. 2. mo. at 7%. 19. $364.24 for 1 yr. 1 mo. at 6%. 20. $282.50 for 5 yr. 9 mo. at 5J%. 21. $298 for 4 yr. 3 mo. at 6%. 22. $389 for 7 yr. 10 mo. at 5%. 23. $894 for 6j yr. at 5J%. 24. A man buys a house and lot for $2800. He pays f of the amount in cash, and the remainder after 1 yr. 4 mo. with 5% interest. Find the amount of the second payment. 25. Kequired the simple interest and amount of $787.875 for 7 yr. 7 mo. at 7%. 26. Find the interest on a note for $12500 for three months at 8%. 27. A man paid his city tax five months after it became due. His tax was $560. In accordance with city ordi- nance, 1% is added for each 1 month the taxes are over- due. He pays to the city collector of taxes, who adds 5% collection fee. How much did he have to pay ? THE SIX PER CENT METHOD. 277. By the 6% method it is convenient to find first the interest of $1, then multiply it by the principal. 1. If $.09 is the interest of $1 for a certain time, what is the interest of $2 for the same time ? of $10 ? of $25 ? 2. The interest of $1 at 6% for a certain time is $.034. What is the interest of $36.25 for the same time ? EXPLANATION. The interest of $36.25 is 36$fe times the inter- est of $1. At Qf the interest of $1 for 1 year = . . . . $.06 for 1 month = fa of $.06 = $.00^ for 1 day = & of $.00^ = -$.0004 146 SENIOIt ARITHMETIC. 3. What is the interest of $50.24 at 6% for 2 yr. 8 mo. 18 da. ? SOLUTION. The interest of $1 for 2 yr. = 2 x 8.06 = $.12 for 8 mo. = 8 x $.00^ = .04 for 18 da. = 18 x $.000 = .003 The interest of $1 for 2 yr. 8 mo. 18 da. = $.163 The interest of $50.24 is 50.24 times $.163 = $8.19 4. What is the interest of $1 for 2 months? For 6 days? Rule. Find the interest on $1 for the given time, and multiply it by the principal, considered as an abstract number. Or, multiply the number of dollars by the number of days, and divide by 6. The quotient will be the inter- est in mills. Find the interest at 6% of: 5. $382 for 6 mo. 24 da. 6. $58.63 for 1 yr. 5 mo. 17 da. 7. $256 for 3 yr. 5 mo. ^ 8. $249.83 for 1 yr. 2 mo. 15 da. 9. $51 for 236 da. 10. $74 for 2 mo. 19 da. 11. $1500 for 1 yr. 3 da. 12. $287.15 for 2 yr. 11 mo. 22 da. Interest at any rate per cent may be found as follows : At 7%, find interest at 6%, and add \ of itself. At 5%, find interest at 6%, and subtract \ of itself. At 8%, find interest at 6%, and add f or ^ of itself. At 4%, find interest at 6%, and subtract f or of itself. At 5^%, find interest at 6%, and subtract | of itself. SIMPLE INTEREST. 147 Find the interest and amount of the following : 13. $2350 for 1 yr. 3 mo. 6 da. at 5%. 14. $125.75 for 2 mo. 18 da. at 7%. 15. $950.63 for 3 yr. 17 da. at 4%. 16. $336.48 for 90 da. at 1\%. 17. $738.53 for 2 yr. 2 mo. 24 da. at 8%. 18. $5000 for 6 mo! 19 da. at 4%. 19. $867.35 for 1 yr. 3 mo. 27 da. at 9%. 20. $260.50 for 5 yr. 21 da. at 10%. 21. $3050 for 3 yr. 3 mo. 3 da. at 12%. 22. $625.57 for 1 yr. 2 mo. 15 da. at 3%. 23. A grocer's bill for $84.36 is paid 8 mo. 12 da. after it becomes due, with interest at 5%. How much is paid? 24. Find the interest at 7 % on $37200 for 5 days. 25. A note for $125 was dated March 1, 1894. What was due Aug. 5, 1895, int. at 6% ? 26. Find the amount of $460.50 for 2 yr. 7 mo. 15 da. at 5%. 27. What is the amount of a note for $360 due in 3 mo., interest at 5% ? 278. On short-time notes, it is customary to compute interest for the actual number of days, using the 6% method. Find the amount of : 28. $684.23 from June 5, 1895, to July 23, 1895, at 6%. 29. $846 from Jan. 6 to March 9, 1896, at 5%. 30. $2064.28 from April 13, 1894, to June 3, 1894, at 8%. 31. $1428 from May 12, 1892, to June 9, 1892, at 6%. 32. $324 from April 1, 1896, to June 4, 1896, at 7%. 148 SEN lOll ARITHMETIC. 33. $3500 from Feb. 9, 1895, to March 12, 1896, at 34. $862.15 from May 25, 1893, to July 22, 1893, at 6%. 35. What is the amount of a note of $384.16 at 6%, given June 11, 1896, and paid Aug. 12, 1896? 36. A note of $395.80 dated April 5, 1896, was paid Aug. 4, 1896. What was the amount? 37. On Dec. 9, 1894, John Smith borrowed $484, agree- ing to pay interest at 5%. He paid the debt in full on March 3, 1895. What did he pay ? 38. What is the amount of $58.24 at 7% from April 23, 1893, to July 22, 1893? 39. A bill of $312 with interest at 5% was paid at the end of 90 days. What was the amount ? 40. What is the interest of $30000 at 8% for 7 days ? 279. Find the interest, using the best method. PRINCIPAL. TIME. RATE. 41. $364, 3yr., 8%. 42. $692.15, 1 yr. 3 mo., 9%. 43. $342, 62 da., 6%. 44. $243.50, 2 yr. 5 mo. 18 da., 7%. 45. $.392, 1 yr. 3 mo. 15 da., 4%. 46. $150.16, 7 yr. 2 mo. 27 da., 4%. 47. $284.10, 1 yr. 8 mo. 18 da., $%. 8. $1400, 2 yr. 1 mo. 12 da., 7%. 49. $124, 5 yr. 3 ino. 29 da., 6%. 50. $48, 33 da., 6%. 51. $124, 112 da., 5%. 52. $315, 45 da., 4%. 53. $214, 93 da., 8%. SIMPLE INTEREST. 149 280. Find the amount of : 54. $365 from April 1, 1895, to July 5, 1897, at 6%. 55. $250 from July 3, 1891, to April 21, 1893, at 9%. 56. $582 from Sept. 4, 1896, to July 8, 1897, at 8%. 57. $346.18 from May 10, 1893, to March 10, 1895, at 6%. 58. $287 from Jan. 1, 1895, to July 1, 1897, at 4J%. 59. $1684 from July 17, 1896, to Sept. 5, 1898, at 7. 60. $2500 from April 16, 1873, to Oct. 11, 1881, at 5%. 61. $186 from Feb. 12, 1896, to March 4, 1896, at 6%. 62. $346 from March 11, 1895, to Feb. 11, 1896, at 6%. EXACT INTEREST. 281. When the time includes days, interest computed by the 6% method is not strictly exact, by reason of using only 30 days for a month, which makes the year only 360 days. The day is therefore reckoned as ^^ of a year, whereas it is ^5- of a year. Rule. To compute exact interest, find the exact time in days, and consider 1 day's interest as 5 g- of 1 years interest. . 1. Find the exact interest of $358 for 74 days at 7%. SOLUTION. $358 x .07 = $25.06, 1 year's interest. 74 days' interest is f& of 1 year's interest. ^ of 25.06 = $5.08. Ans. Find the exact interest of : 2. $324 for 15 d. at 9%. 3. $253 for 98 d. at 4%. 4. $624 for 117 d. at 7%. 5. $153.26 for 256 d. at 5%. 6. $620 from Au6^J&-hiterest exceeiis-Ea^ment. A >4 \ 1893 4 1892 5 7 25 $1613.68 .052 $420 10 .052 12 583.91 1613.68 $1697.59 Amount. 420.00 Sum of 2d and 3d Payments. 1893 7 1893 4 13 7 $750 $1277.59 New Principal. .016 3 .016 6 20.44 1277.59 $1298.03 Amount. 750.00 4th Payment. 1894 7 1893 7 13 13 Settled. $548.03 New Principal. .06 1 .06 $28.88 548.03 $576.91 Ans. NOTE. The $45 payment, being less than the interest (64.55), is not deducted from the amount of the second principal ($1678.23). If this were done, and the remainder treated as a new principal, a portion of it ($19.55), being interest, would draw interest, which is not legal. Therefore, interest must be taken on $1613.68 until the date of the next payment (10 mo. 12 da.). The sum of the two payments, being greater than the interest, is subtracted from the amount. Write in proper form on paper a note for each of the following, indorse the payments, and solve : 158 SENIOR ARITHMETIC. 2. Date, Jan. 1, 1874, at Syracuse, N.Y. Face, $1000. Interest at 6%. Indorsements: July 7, 1874, $400; Oct. 19, 1874, $300; Dec. 1, 1874, $100. What remains due Jan. 1, 1875 ? 3. Face, $900. Date, March 1, 1886. Interest at 9%. Indorsements: Aug. 10, 1886, $300; Sept. 1, 1886, $100; Jan. 1, 1887, $50. What was due March 1, 1887? 4. Face, $2000. Date, Jan. 20, 1892. Interest at 6%. Indorsements: May 20, 1892, $100; July 20, 1893, $100; Sept. 10, 1893, $700; Oct. 20, 1894, $75. Settled Oct. 20, 1895. What was due ? 5. 00. Binghampton, N. K, On demand, for value received, (Q/^ promise to pay or order, , with interest. The following payments were made on this note : June 27, 1891, one hundred fifty dollars; Dec. 9, 1892, one hundred fifty dollars. What was due Oct. 9, 1895 ? 6. On a note for $573.25, at 6%, dated June 10, 1888, were the following indorsements: May 20, 1889, $50; July 10, 18'90, $16.50; April 5, 1891, $14.30; July 14, 1892, $250. How much was due Sept. 20, 1893? 7. A note of $850 was dated June 21, 1892, bearing interest at 6%. On this note were the following indorse- ments: Sept, 15, 1892, $150.90; Nov. 21, 1893, $45; Jan. 15, 1894, $256.88. What remained due June 21, 1894 ? 8. Find what was due June 1, 1896, on a note for $1928, with 4%% interest, dated Jan. 1, 1891, and bearing the following indorsements : March 1, 1891, $300; Oct. 16, 1893, $40; Feb. 4, 1894, $800; Dec. 16, 1895, $500. SIMPLE INTEREST. 159 9. On a note for $832.26 dated Aug. 3, 1889, due in 6 months, the following payments were indorsed : $350, Oct. 5, 1890 ; and $468.37, May 15, 1892. How much was due Dec. 12, 1893, interest at 1% ? 10. Face, $2950. Date, July 1, 1885. Interest, 1%. Indorsements : Oct. 1, 1885, $750 ; Jan. 15, 1886, $600 ; July 1, 1886, $900; Dec. 1, 1886, $300; March 1, 1887, $450. What was due July 1, 1887? MERCHANTS' RULE. 297. When notes and accounts are settled within a year after interest begins, and upon which partial payments have been made, it is customary for business men to make use of the following rule : Find the amount of the entire debt at date of settlement. Find the amount of each payment at date of settlement. Subtract the amount of the payments from the amount of the debt. Syracuse, N. Y., tf?lfM'g(/ t'il'M / fo-ntn- -tinu ~ Dollars, Value received, at the City Bank. (f^.^'' Discounted May 4 at 6%. BANK DISCOUNT. 171 to pay to the order of^ Brooklyn, N. Y., &W4. 3, 189$. after date, for value received, (Qr promise ne weeMdfoet/ -and Stvtn^u Dollars, at the Merchant 'j Bank, with interest. Discounted at date at 6%, 10. JVJ/J. Boston, Mass., IL/ /7?-tl*^ r. after date, for value received, &/ promise 'Wl&tti *$f. ^/W^iZ 1 or order ^ frttdant/ Dollar s, at the Citizen 's Bank. Discounted July 10 at 8%. 13. Syracuse, JV. Y , after date, ^ promise to pay to the order ollars, at the Third National Bank. J^K 'Q. Discounted at 6% at date. 14. A note for $135 is given for 90 days, and discounted the day it is given at 6%. What are the proceeds ? 15. ' William Johnson gave John Doe a note payable to the Binghamton Trust Co., time 60 days, amount f 204.60. Write this note. After 20 days Doe put the note in the bank. What are the proceeds of the note ? In the following problems, write the notes in full, and properly indorse them, using any names for payer and payee. 16. Find the bank discount of $400 for 3 months at 8%. 17. What are the proceeds of $250, with interest at 6%, discounted at bank for 60 days at 6% ? DISCOUNT. 173 18. What will be the proceeds of a note for $175 drawn at 4 mo., with interest at 6%. if the bank discount is 10% per annum ? 19. On the first day of January, 1896, a farmer gave his note at 90 da. for $525, with interest at 6%. When did the note become due? and what were the proceeds of the note if discounted at a bank at 1% a month on the tenth day of February ? 308. To find the Face of a note, when the Proceeds, Time, and Rate are known. 1. What must be the face of a 60-day note, without grace, which after being discounted at 6% will give $500 as proceeds ? SOLUTION. The bank discount of $1 at $% for 60 da. = $.01. The proceeds of $1 = $1.00 - $.01 = $.99. Since $.99 is the proceeds of $1, $500 must be the proceeds of as many dollars as $.99 is contained times in $500 505.05+ Ans. Therefore, $505.05 must be the face of a 60-day note which will give $500 as proceeds after being discounted at Rule. Divide the proceeds by the proceeds of $1. 2. A person must use $250 to-day. For how much must he make a bank note for three months that will give $250 proceeds, without grace ? 3. What must be the face of a 60-day note, payable at a Boston bank, upon which I can realize $350 after it is discounted at 6% ? 4. If you buy goods for $1200 cash, how large a note payable in 90 days, at 6% bank discount, must you make that the proceeds shall pay for the goods ? Without grace. 5. Find the face of a 60-day note that will yield $800 when discounted at bank at 7%, with grace. 174 SENIOR ARITHMETIC. 6. How large a note must I make at a bank for 30 days to pay a debt of $475, without grace ? 7. Wishing to borrow $ 494.90 at a Syracuse bank, for what must I make my note at 60 da., with interest at 6%, in order to obtain this amount ? Discount at 1 % a month. 8. The proceeds of a Buffalo note at 60 da., when dis- counted at a bank at 6% per annum, is $742.50. What is the face ? REVIEW OF DISCOUNT. 309. 1. Define Discount ; True Discount ; Bank Dis- count ; Proceeds ; Present Worth. 2. How is the present worth found ? The true dis- count ? The bank discount ? The proceeds ? 3. What is the term of discount ? and how is it found ? The day of discount ? and how found ? 4. How is the bank discount of an interest-bearing note found ? 5. How do you find the face of a note when the pro- ceeds, time, and rate are given ? 6. When does a "Rochester, N.Y., note mature if given for 1 month from Jan. 31 ? 7. State a point of difference between true discount and bank discount. 8. What kind of notes only can be discounted at banks ? ^ 9. Bought a city lot, and agreed to pay $546.94 at the end of 2 yr. 6 mo., without interest. Receiving some money unexpectedly after 6 months, I wish to pay cash. How much ought I to pay, money being worth 6% ? 10. What is the present value and true discount of $973.52, due in 1 yr. 7 mo. 24 da. hence, without interest, money being worth 8% ? DISCOUNT. 175 11. A man has an offer of $2000 cash for his house, or $2100 payable in 8 months. If money is worth 8%, which is the better ? and how much ? 12. Find the discount and proceeds of a note for $13500 payable at a bank in 90 days after date without grace, dis- counted at 5%. 13. For what sum must C. F. Norton draw his note on a Binghamton bank, that when it is discounted at 4% for 60 days he will have $800 ? 14. A man owes me $2540, due in 2 years, 3 months, with- out interest. If he pays it at once, what discount should I allow him ? 15. Find the discount and proceeds of a note on a Brook- lyn bank for $350, given May 12, 1896, for 4 months, and discounted at 6%, July 15. 16. Syracuse, N. Y., %4 after date, for value received, order, Dollars, with interest, at the Salt ( Springs National Bank. Discounted June 11 at 6%. 17. For what sum must I draw a 4 months' note so that the proceeds will be $800, discounted without grace at 6%? 18. I sell my horse for $216, and take a note due in 6 months without interest. If money is worth 6^ per an- num, what is the present value of my note ? 176 SENIOR ARITHMETIC. 19. For what sum must I give my note for 60 days at a bank in order to receive $650 proceeds, money being worth 8% ? 20. Find the face of a note, discounted for $2558.40 at 8%, for a term of 72 days, without grace. STOCKS AND BONDS. 310. Many kinds of business require so much capital that several persons must unite to raise the necessary amount. 311. The Capital to be raised is divided into Shares, usually of $100 each. Shares are then sold until the required amount is raised. Each purchaser of shares is a Stockholder, and receives a Certificate of Stock, which shows the number of shares purchased, and their value. This value is called the Par or Nominal Value. 312. The Market Value of stocks is the price for which they are sold. 313. The value of stocks depends upon the profitable- ness of the business. When the business is very profitable, the shares are worth more than par; they are then abov^- par, or at a Premium. When the business is unprofitable, the shares are not worth their par value. They are then below par, or at a Discount. 314. 1. The capital of a company is $100000. Into how many shares of $100 each can this be divided ? 2. A stockholder owns 25 shares of stock. How many dollars of stock has he ? STOCKS AND BONDS. Ill 3. If at the end of a year there has been a net profit of $ 10,000, what per cent profit has been made ? $10000 is what % of $100000 ? The profits are divided among the stockholders, and are called Dividends. 1. Dividends are usually declared semi-annually or quarterly. NOTE 2. When a 10 % dividend is declared, each stockholder receives 10% of the par value of his shares. 4. What will be A's dividend if he owns 35 shares ? When there is a loss, each stockholder is required to pay his share of the loss. This is called an Assessment. 5. What would be A's assessment to meet a 2% loss ? A person who buys or sells stocks for others is called a Stock-broker, and his commission is called Brokerage. NOTE 1. Brokerage is usually \ % or | % of the par value. NOTE 2. In all stock transactions, dividends, assessments, brokerage, premium, and discount are computed on the par value. NOTE 3. Shares are sometimes issued at $200, $250, $50, $25, or $10 each, but unless otherwise stated $100 is considered the par value of a share. 6. What is the market value of 10 shares of bank stock, when sold at par ? 7. What is the market value of 50 shares of railroad stock, at 10% premium ? SOLUTION. The market value of 1 share is $100 + $10 = $110. The market value of 50 shares is 50 times 8110. 8. What is the market value of 18 shares of mining stock at 15% below par? SOLUTION. The value of 1 share is $100 15 = $85. The value of 18 shares is 18 times $85. 178 SENIOR ARITHMETIC. Stock Quotations are the published prices of stocks. When railroad stock is quoted at 108, it means that it sells for 8% above par in the stock-market. When it is quoted at 92, it is selling at 8^ below par. 9. If I buy stock at 98 and sell it at 101, what gain do I make on 10 shares ? NOTE. Stock at 98 means $98 for a $100 share, and stock at 101 means $101 for a $100 share. 10. When stock is quoted at 85, what is the value of a share ? What is the value of 1 dollar of stock ? 11. What must I pay for 10 shares of stock at 95, if I pay the broker \c/ for doing the business ? SOLUTION. Cost of 1 share = $95 + Brokerage $i = $95i or $95.25. $95.25 x 10 = $952. 50. Ans. 12. If I sell 10 shares of stock at 110, and pay the broker \/oi what do I receive? SOLUTION. 1 share brings $110 $| = $109f , or $109.75. 10 shares bring 10 times $109.75 = $1097.50. Ans. 13. A man invested $4500 in street railway stock at 10% discount. How many shares did he purchase? 14. If I invest $2100 in bank stock at 105, ho\v many shares do I purchase ? 15. A capitalist bought 80 shares railroad stock at 87^, and 60 shares mining stock at 114. Find the cost. 16. $18200 will purchase how many shares of stock selling at 140 ? 17. A stock company declares a dividend of 2%. What does A receive, who owns 1500 shares of $10 each ? 18. How much is gained on 50 shares of railroad stock purchased at 98 and sold at 102 ? STOCKS AM) BONDS. 179 19. Bought stock at a discount of 2%, and sold it at a discount of 3 % . Did I gain or lose ? and how much on 20 shares? BONDS. 315. To meet extraordinary expenses, governments, States, cities, villages, counties, towns, and incorporated companies sometimes borrow money. The securities given by such corporations are called Bonds. Bonds bear a fixed rate of interest, payable annually, semi-annually, or quarterly. They are bought and sold in the same manner as stocks. Bonds are known by the rate of interest they bear : Vir- ginia 6's are bonds of the State of Virginia, bearing 6% ; U. S. 4's of '97 are U. S. bonds bearing 4% interest, and maturing in 1897. 316. A Coupon is an interest certificate attached to a bond. At the expiration of any interest period, the coupon ' is cut off and used in collecting the interest, being worth the amount of interest due on the bond for a specified period. 317. 20. What will be the cost, including brokerage at 1%, of 200 shares of C., B., and Q. R.R. bought at 67| ? SOLUTION/. Cost of 1 share = $67| + $i = $68J. Cost of 200 shares = 200 x $68 J. 21. How much, including brokerage at |%, must be paid for $5000 of U. S. 4's at 110J ? SOLUTION. $1 of bonds costs $1.10| + .OOJ = $1.11. $5000 worth will cost 5000 times $1.11. 22. What must I pay for $8275 of stock at 10% dis- count ? 23. What is the cost, including broker's commission of %%, of 150 shares of railroad stock bought at 89J ? 180 SENIOR ARITHMETIC. 24. I buy stocks at 5% discount, and sell at 5% pre- ^nium ; what per cent profit do I make on the investment ? 25. March 10, 1896, Western Union Telegraph stock was quoted at 84. How many shares could be bought for $1020, brokerage | per cent ? SOLUTION. Cost of 1 share, $84| + g = $85. As many shares can be purchased as $85 is contained in $1020. 26. How many shares of stock at 10% premium can be purchased for $2200 ? 27. I invested $5100 in K.Y. and N.H. railroad stock at 170. How many shares did I purchase ? 28. If I invest $42400 in 5% bonds at 106, what is my yearly income ? SOLUTION. $42400 -f- $1.06 = $40000, par value. How much is 5^ of $40000? 29. If I invest $21008 in 5% bonds at 104, what will be my annual income ? 30. What will be my yearly income if I invest $11100 in 5% at 92, brokerage \% ? 31. A man invests $9500 in Virginia 6's at 94|, broker- age i efine dividend ; assessment. 7. Upon what are premium, brokerage, dividends, and assessments reckoned ? 8. What is the average of payments ? Equated time ? Average term of credit ? RATIO AND PROPORTION. 189 RATIO AND PROPORTION. 325. Oral. 1. 5 bears what relation to 10 ? Ans. 5 is ^ of 10. 2. 10 bears what relation to 5 ? Ans. 10 is 2 times 5. 3. What part of 16 is 4 ? 4. How does $7 compare with $14 ? 5. John has 20/ and Mary 5/. What is the relation of John's money to Mary's ? Of Mary's money to John's ? 6. What is the relation of 15 to 3 ? Of $8 to $16 ? Of 28 men to 7 men ? Of 2 bushels to 2 pecks ? 326. Ratio is the relation between two like numbers. It is found by dividing one by the other ; thus : The ratio of 4 to 8 is 4 -=- 8 = f The sign of ratio is (:). It is the division sign with the line omitted. The ratio of 6 to 3 is expressed thus, 6:3. It may also be expressed fractionally, thus, . 327. The Terms of a ratio are the two numbers com- pared. The first term of a ratio is the Antecedent, and the second the Consequent. In the ratio 6 : 12, 6 is the antecedent, and 12 the conse- quent. 328. A ratio formed by dividing the consequent by the antecedent is an Inverse ratio. 12 -=- 6 is the inverse ratio of 6 : 12. 329. The two terms of a ratio taken together form a Couplet. 330. Two or more couplets taken together form a Com- pound ratio. 190 SENIOR ARITHMETIC. 3 : 6 "l A compound ratio may be changed to a sim- o . K I Qfi 1 50 ple ratio ^ takin tne product of the antece- T ~ dents for a new antecedent, and the product of 4 : 5 J the consequent for a new consequent. Antecedent -=- Consequent = Ratio. Therefore, Ratio -f Antecedent = Consequent; and, Ratio x Consequent = Antecedent. Multiplying or dividing both terms of a ratio by the same number does not change the ratio. The ratio 12 : 6 = 2. The ratio 3x12:3x6 = 2. The ratio 12 -f 3 : 6 -f 3 = 2. Find the ratio of : 7. 56 : 7. 11. 3 bu. : 3 pk. 14. 2^ : 16. 8. 20 : 300. 12. J : 4. 15. 1 : . 9. $55: $330. 13. 12 : J. 16. 3f : 5f 10. What is the ratio of T 9 n to ^ ? NOTE. Fractions with a common denominator have the same ratio as their numerators. Prove this in Ex. 10, by multiplying both terms by 10. 17. A:i = ? ff:7 7 5 = ? H'=? 18. | : f = ? f : ! = ? i : f = ? . 5 = ? 19. Find the inverse ratio of 75 to 25. Of 15 to 225. 20. 16 : (?) = . 14 : (?) = 2. 21. (?):5 = 4. (?):8 = J. 22. Find the value of the compound ratio, 8 : 10 ") 5:6 ) PROPORTION. 331. Oral. 23. Give three fractions having the same value as . 24. Give two numbers that have the same ratio as 5 to 10. 25. Give a fraction equal to f . RATIO AND PROPORTION. 31 26. Give a ratio equal to 3 : 4. 27. How does the ratio of 5 men to 10 men compare with the ratio of $ 5 to f 10 ? 28. How does the ratio of 8 Ib. to 4 Ib. compare with the ratio of 40/ to 20/? 29. Name two numbers that have the same relation as 5 to 10. As 4 to 24. As 8 to 16. As J- to J. 30. What number has the same relation to 5 as 12 to 3 ? 31. Find a number whose ratio to 4 equals 3 : 6. 32. Give three ratios equal to $ 100 : $50. 33. Give any two ratios that equal each other, and express their equality. 332. An equality of ratios is a Proportion. Thus, 4: 2 = 12 : 6. The ratio of 4 to 2 equals the ratio of 12 to 6. A proportion is usually expressed with the sign (::) between the ratios ; thus, 4 : 2 : : 12 : 6. This is read 4 is to 2 as 12 is to 6. A proportion has four terms, of which two are antece- dents and two are consequents. Each term is a propor- tional. 333. The first and fourth terms are called Extremes, and the second and third terms are called Means. NOTE. In the proportion 2:6 :: 6 : 18, the two means are the same number, 6. The 6 is called a mean proportional. PRINCIPLE. The groduct of the extremes equals the product of the means. Rule. To find an extreme, divide the product of the means by the given extreme. To find a mean, divide the product of the extremes by the given mean. 192 SENIOR ARITHMETIC. Supply the missing term : 34. 1:836::25:( ). 39. K) yd. : 50 yd. :: $20 : ( ). 35. 6 : 24 :: ( ) : 40. 40. 15 Ib. : 60 lb. :: ($ ) : $12. 36. ( ) : 15 : : 60 : 6. 41. 1 da. : ( da.) :: 12 : 6. 37. 25 :( ) :: 4 : 8. 42. ( men) : 75 :: $50 : $150. 38. 6:4:: J: ( ). 43. $ f : $3f :: ( ) : 5. SIMPLE PROPORTION. 334. An equality of two simple ratios is a Simple Pro- portion. It is employed in solving questions having three given terms, two of which have the same relation to each other as the third to the required term. 44. If 12 bushels of oats cost $4,what will 60 bushels cost ? SOLUTION. There must be the same relation between the coxt of 12 bu. and the cost of 60 bu. as exists between 12 bu. and 60 bu. We place $4 for the third term. The 12 : 60 : : $4 : ($ ) answer w n] ^e the fourth. We must now ,.~ . form a ratio of 12 and 60 that shall equal - = $20. the ratio of $4 to the answer. Since the third term is less than the required answer, the first must be less than the second, and we have 12 : 60 for the first ratio. The product of the means divided by the given extreme will give the other extreme, or $20. Ans. By analysis, Since 12 bu. cost $4, 1 bu. will cost $J, and 60 bu. will cost $20. Ans. Rule. Consider the required answer as the fourth term, and place the number that is like it for the third term. Place the two remaining terms as follows : If the answer is to be larger than the third term, the second must be larger than the first. If smaller, the second must be smaller than the first. Divide the product of the means by the given extreme. Cancel when possible. 11ATIO AND PROPORTION. 198 45. If 10 sheep cost $35, what will 23 sheep cost? What will 6 sheep cost ? 46. If 5 men can do a piece of work in 9 days, how long will it take 15 men to do the same work ? SOLUTIOX. Place 9 days for the third term, because it is like the required answer, thus, ::9 da. : ( da.) Since 5 men can do it in 9 days, 15 men can do it in less time. Therefore, since the answer is to be smaller than the third term, place 5 men for the second, and 15 men for the first. Multiplying and dividing we have 3 days, Ans. f 47. If 14 horses eat 36 tons of hay in a certain time, how many tons will 13 horses eat in the same time ? 48. If it costs $400 to lay 80 rods of street-car track, how much will it cost to lay 3^ miles at the same rate ? 49. If a pole 8 ft. high casts a shadow 4^ ft. long, how high is a tree which casts a shadow 48 ft. long? 50. If a man walks 280 miles in 8 days, how many days ought it to take him to walk 420 miles ? 51. If it costs $13.20 to supply a new arithmetic to each of a class of 24 pupils, what will be the expense of furnish- ing one to each of a class of 19 ? 52. How far can a train run in 3 hours, if it can run 160 Km. in 4 hours? 53. How many men will be required to do in 10 days what 15 men can do in 30 days ? 54. What will 8 tons of coal cost, when 171 tons cost $78.75? 55. If a certain sum of money yields $360 interest in one year, what would the interest of the same sum be for 15 months ? 194 SENIOR ARITHMETIC. 56. If $800 yield $48 interest in a certain time, how large a sum will yield $216 in the same time ? 57. If the interest of $3600 for a certain time is $216, what will be the interest of $800 for the same time ? ,~- 58. If a garrison of 240 soldiers have a supply of food sufficient for 150 days, how long would the same food last if the garrison were increased to 600 men ? 59. In the above example, how long would the food last if 80 men were sent away ? 60. Find the cost of l|f bushels of wheat, if f bu. costs 61. If 120 shoemakers make 40 dozen pair of shoes in a certain time, how many shoemakers would it require to make the same number of shoes in one-half of the time? 62. If a train runs 140 miles in 4 hr. 30 min., what is the rate per hour ? 63. When 5 tons 1650 Ib. of coal cost $24.75, what will be the cost of 18 tons 675 Ib. ? 64. If a 16-foot board 9 inches wide contains 12 sq. ft., how wide must a board of the same length be to contain 20 sq. ft. ? 65. It takes 26 yards of carpet 1 yard wide to cover a floor. How many yards will it take if the carpet is but 27 inches wide ? 66. The ratio of Simon's pay to Matthew's is f . Simon earns $18 per week ; what does Matthew earn ? 67. 25 men can do a piece of work in 70 days ; but after 30 days 15 of them refuse to work. In how many days can the rest complete the work ? RATIO AND PROPORTION. 195 COMPOUND PROPORTION. 335. An equality between a compound and a simple ratio is a Compound Proportion; thus, O . A \ > : : 12 : 20 is a compound proportion, o : 10 ) Find the fourth term. SOLUTION. First changing to a simple pro- 3:6) portion, we have, 4 : g| ::3: ( ) 3x4:6x8::3:(). Then divide the product of the means by the given extreme, using cancellation. Thus, gxgxg-M. An*. 3 x * 1. If 5 men earn $72 in 8 days, how much can 10 men earn in 6 days ? SOLUTION. Since the answer is to be in dollars, place $72 for the third term, and arrange the 5 men : 10 men ) terms of each couplet accord- 8 days : 6 days ( ' ing as the answer should be greater or less than the third term if it depended on that couplet alone. Since 5 men earn $72, 10 men can earn more, so we place 10 men for the second and 5 men for the first ; and since they earn 72 in 8 days, they will earn less in 6 days, so we place 6 days for the second term, and 8 days for the first. Dividing the product of the means by the extremes, we have, 9 2 &v = 35. 355. Square root may be explained by the aid of dia- grams. The area of every square surface is the product of two equal factors, length and width. Finding the square root of a number, therefore, is equiv- alent to finding the width of a square surface, its area being given. 356. The following formulas illustrate the principles which underlie the operations of square root : 1. Length x Width = Area. 2. Area 4- Length = Width. 3. Area -i- Width = Length. 1. Find the width of a square whose area is 1296 sq- ft. Fig . 7 . AREA. 1296 900 396 396 30ft. 6ft. 30 ft. 30 ft. 6ft. 2 x 30 ft. = 60 ft. 2 x 30 ft. + 6 ft. = 66 ft. SOLUTION. The greatest square of even tens contained in ^_ ^_^ 1296 sq. ft. is 900 sq. ft. | b | d \ c \ (Squared). Its width is Fig. 2. 30 ft. 1296 sq. ft. - 900 sq. ft. = 396 sq. ft., the area of 6, c, and d, considered as one rec- tangle (Fig. 2), whose width we desire to find. The length of this rectangle is (2 x 30 ft.) 60 ft. + the length of c. But we cannot 208 SENIOR ARITHMETIC. know the length of c till we find its width. By trial (Formula 2), we divide the area, 396 sq. ft., by 60 ft., its approximate length. The quotient, if correct, is 6 ft., the width desired. To test the cor- rectness : Add the 6 ft. to the trial divisor, and we have 66 ft., the entire length of a, />, and c, which (Formula 1), multiplied by its width, 6 ft., gives its area, 396 sq. ft. There is no remainder, and the work is correct. Therefore, 30 ft., the width of A, + 6 ft., the width of a, 7>, and c, = 36 ft., the width of the original square. NOTES. All the numbers in the middle column denote area. 1296 sq. ft. area of the original square; 900 sq. ft. = the area of A ; and 396 sq. ft. the area of b, c, and d. The numbers in the left-hand column denote length. 60 ft. the approximate length (or the trial divisor) of b, c, and d ; and 66 ft. the exact length, or the complete divisor. The numbers in the right-hand column denote width. 30ft = the width of A and 6 ft. the width of b, c, and d ; 36 ft. = the width of the original square. In dividing, to find the width of b, c, and d, since the divisor is too small, care must be taken that the quotient figure be not too large. SHORT METHOD. 357. Ex. 2. Find the square root of 1,306.0996. i3W.09'96 (36.14 9 66j 406 396 721 ) 1009 721 7224) 28896 28896 Rule. Beginning at the decimal point, separate the num- ber into periods of two figures each, pointing whole numbers to the left and decimals to the right. Find the greatest square hi the left-hand period, and irrlte its SQUATJR ROOT V 209 root at the rigid. Subtract the square from the left-hand period, and briny down the next period for a dividend. Divide the dividend by twice the root already found, and annex the quotient to the root, and to the divisor. Multiply this complete divisor by the last root figure, and bring down the next period for a dividend, as before. Proceed in this manner till all the periods are exhausted. NOTE 1. When occurs in the root, annex to the trial divi- sor, bring down the next period, and divide as before. NOTE 2. If there is a remainder after all the periods are ex- hausted, annex decimal periods. NOTE 3. If, after multiplying by any root figure, the product is larger than the dividend, the root figure is too large and must be diminished. Also the last figure in the complete divisor must be diminished. NOTE 4. For every decimal period in the power, there must be a decimal figure in the root. NOTE 5. If the last decimal period does not contain three fig- ures, supply the deficiency by annexing one or more ciphers. Ex. 3. Find the square root of 253009. SOLUTION. 25'30'09(j> As occurs in the 25'30'09 (_503 An*. 25 root, we annex to the 25 10) 30 trial divisor, 10, and an- 1003) 3000 other period to the divi- 3009 dend, and divide as before. Thus, NOTE. To find the square root of a common fraction, extract the root of each term separately. If both terms are not squares, change the fraction to a decimal, and then extract the root. The result will be the approximate root. Change mixed numbers to improper fractions. 4. What is the square root of T 8 J ? ? = ^ V144 210 SENIOR ARITHMETIC. Find the square root of : 5. 8836. 14. .06432 23. 6. 15876. 15. .005625 24. f- 7. 370881 16. .913936 25. fj$f 8. 46656 17. 25.6036 26. 4i 9. 820836 18. 24.3049 27. ^ 10. 29.0521 19. .612089 28. ff 11. 9.2416 20. 329.7643217 29. 36.45? 12. 3180.96 21. 1684.298431 30. 2863y 13. .007921 22. 389765268 31. 189 Jfj Find the square root to four decimal places : 32. .15 36. 72.5 40. 963 33. .18 37. 119 41. 13.2f 34. 17 38. 3.67 42. .009 T \ 35. 4.7 39. .222 43. .003f 44.- What is the length of one side of a square field that has an area equal to a field 75 rd. long and 45 rd. wide ? 45. How wide is a field containing 7056 square rods ? Perform the indicated operations. NOTE. Carry decimals to the third place. 46. V3.26 X .0063. 51. V3.532 -4- 6.28. 47. .OSxVfTf 52. vT+~6*T2. 1 ... 53. v 5- + (I ) . 54. V625 + 1296. 55. V625~+ V1296. RIGHT-ANGLED TKI ANGLES. 211 RIGHT-ANGLED T1UA XGLES. 358. A triangle having one right angle is a Right-Angled Triangle. 359. The side opposite the right angle is the Hypothe- nuse, as AB. BC is the Perpendicular, and AC the Base. In the triangle ABC, the hypothenuse is 5 inches, the perpendicular 3 inches, and the base 4 inches. 360. It will be seen that the square of the hypothenuse is 25 sq. in., which is equal to the square of the base, 16 sq. in., plus the square of the perpendicular, 9 sq. in. PRINCIPLE. The square of the hypothenuse equals the sura of the squares of the two shorter sides. Therefore, to find the hypothenuse, take the square root of the sum of the squares of the base and perpendicular. -\/ Base'- 2 -f Perpendicular 2 = Hypothenuse. 361. To find the base or the perpendicular, take the square root of the difference between the squares of the hypothe- nuse and the other side. Hypothenuse 2 Base 2 = Perpendicular. \/ Hypothenuse 2 Perpendicular- 1 = Base. SENIOR ARITHMETIC. 1. The base of a right-angled triangle is 32 ft, and the perpendicular 24 ft. What is the hypothenuse ? SOLUTION. 32 2 + 24 2 = 1000. A/1600 = 40 ft. Ans. Or, v/82 2 + 24* - 40 ft. 2. The hypothenuse of a right-angled triangle is 40 ft., and the base 32 ft. What is the perpendicular ? 402 _ 322 = 576 V576 = 2 4 ft. Ana. Or, A/40* - 32* = 24 ft. 3. A 40-foot ladder placed 24 feet from a house will just reach to the top of it. How high is the house ? 4. What is the length of a ladder that will reach the top of a house 40 feet high, when the foot is placed 30 feet from the house ? 5. A rope 150 ft. long fastened to the top of a flag-pole reaches the ground 40 feet from the base. How high is the pole ? 6. What is the hypothenuse of a right-angled triangle whose perpendicular is 36 feet, and whose base is 27 feet? 7. A square farm contains 360 acres. What is the di- agonal distance between its opposite corners ? 8. A telegraph pole 32 feet high casts a shadow 28 feet in length. What is the distance from the top of the pole to the end of the shadow ? 9. The base of a right-angled triangle is 16 in., and the perpendicular is 12.8 m. What is the hypothenuse ? 10. A boy rides his wheel due north at the rate of 15 miles an hour, and another boy starting from the same place, rides (hie east at the rate of 18 miles an hour. How far are they apart at the end of 5 hours ? SIMILAR SURFACES. 213 11. What is the length of the diagonal of a room 10 ft. long and 12 ft. wide ? 12. A crayon box is 6 in. long, 4 in. wide, and 4 in. high. What is the diagonal distance across the. bottom, between the opposite corners ? 13. A street is 32 ft. wide from curb to curb. A tele- graph pole 40 ft. high stands upon one side of the street. How long must a wire be to reach from the top of the pole to the opposite side of the street at the curb ? SI MIL A R S URFA CES. 362. Surfaces having the same form without regard to size are Similar Surfaces. NOTE. Any two squares or any two circles of different size are Similar Figures. Rectangles, triangles, etc., are similar when their corresponding dimensions are proportional. Oral. 1. What is the area of a square whose side is 2 ft. ? 2. AVhat is the area of a square whose side is 3 ft. ? 3. AVhat is the ratio of the two sides ? 4. What is the ratio of the two areas ? 5. Are these ratios equal ? (2 ft, : 3 ft*) (4 sq. ft. : 9 sq. ft.) SOLUTION. From the illustration it will be seen that the areas are to each other as the squares of the sides ; not as 2 to 3, but as 4 to 9. PRINCIPLES. Similar surfaces are to each other as the squares of their corresponding dimensions. Corresponding dimensions are to each other as the square roots of their areas. 214 SENIOR ARITHMETIC. 6. A circle is 4 inches in diameter ; another is 8 inches in diameter. What is the ratio of their areas ? 7. A circle has an area of 16 square feet ; another has an area of 64 sguare feet. What is the ratio of their diameters ? 8. The area of a rectangle 12 ft. long is 84 square feet. What is the area of a similar rectangle 6 feet long ? 9. Two similar fields have areas of 12 acres and 8 acres respectively ; the larger is 32 rods wide ? How wide is the smaller ? 10. The altitudes of two similar triangles are 20 ft. and 10 ft. ; the area of the smaller is 80 square feet. What is the area of the larger ? CUBE ROOT. 363. The cube of a number is the product of that num- ber taken three times as a factor. Blackboard. 13 = l. 10 3 = 1000. 10Q 3 = 1000000. 9 3 = 729. 90 3 = 729000. 900 3 = 729000000. 364. Annexing one cipher to a number, annexes three ciphers to the cube of the number, as shown in I 3 , 10 3 , 100 s , etc. 365. Cubes of even tens. 1. 10 3 = ? 4. 40 3 = ? 7. 300 3 = ? 2. 30 3 = ? 5. 80 s = ? 8. 800 3 = ? 3. 50 3 = ? 6. 200 3 = ? 9. 900 3 = ? 366. The cube of a number composed of tens and units may be found as follows : CUBE BOOT. 215 24 = 20 -f- 4 = 2 tens + 4 units ; 24* = (20 + 4) x (20 + 4) x (20 + 4). 20 + 4 = 20 + 4 = 24 24 96 480 (20 X 4) + 4 2 - 20 2 + (20 x 4) = 20 2 + 2 x (20 x 4) + 4 2 = 20 + 4 = 576 24 (20 2 x 4) + 2 x (20 x 4 2 ) + 43 = 2304 20^ + 2 (20 2 X 4) + (20 x 4 2 ) = 1152 20 3 + 3 x (20 2 x 4) + 3 x (20 x 4 2 ) + 4 3 - 13824 From the operation we find that, The cube of the tens 203= 8000 3 times the square of tens by units . . . 3 (20 2 x 4) = 4800 3 times the tens by the square of the units, 3 (20 x 4 2 ) = 960 The cube of the units 4 3 = 64 8000 + 4800 + 960 + 64 = 13824 367. PRINCIPLE. The cube of a number composed of tens and units is equal to the cube of the tens plus 3 times the square of the tens by the units, plus 3 times the tens by the square of the units, plus the cube of the units. FORMULA. Tens 3 + 3 X tens 2 X units -f 3 X tens X units' 2 -f- units 3 . 10. Separate the following into tens and units, and find their cubes : 35, 54, 63. 368. By reversing the process, we may find the cube root. 11. What is the cube root of 13824 ? SOLUTION. Separating into periods of three figures each, begin- ning at units, we have 13'824. Since there are two periods in the power, there must be two figures in the root, tens and units. 210 SENIOR ARITHMETIC. The greatest cube of even tens contained in 13824 is 8000, and its cube root is 20 (2 tens). 13'824 I 20 + 4 Tens 3 = 20 5 = 8000 ~ :J x tens- = 3 x 20- = 1200 5824 3 X tens X units = 3 X (20 x 4) = 240 Units' 2 = 4- = 10 3 x tens' 2 + 3 tens x units + units' 2 = 1456 (3 X tens' 2 + 3 x tens x units + units' 2 ) x units = 5824 Subtracting the cube of the tens, 8000, the remainder, 5824, con- sists of 3 x (tens' 2 x units) + 3 x (tens x units' 2 ) + units 3 . 5824. therefore, is composed of two factors, units being one of them, and 3 x tens' 2 + 3 x tens x units + units 2 , being the other. But the greater part of this factor is 3 x tens' 2 . By trial \ve divide 5824 by 3 x tens 2 (1200) to find the other factor (units), which is 4 if correct. Completing the divisor, we have 1200- + 3 x (20 -f 4) + 4- = 1456, which, multiplied by the units, 4, gives the product, 5824 proving the correctness of the work. Therefore the cube root is 20 + 4 - 24. 369. To find the cube root by the aid of blocks. Finding the cube root of a number is equivalent to find- ing the thickness of a cube, its volume being given. The following formulas illustrate the principles that underlie operations in cube root. NOTE. _ For convenience Z, 6, t, and r will represent length, breadth, thickness, and volume, respectively. (1) I X b X t = v. (2) V-*-(lX &) - t. (3) v + (lXt) = b. (4) v ~ (b X t) = L 12. What is the thickness of a cube whose volume is 13824 cubic feet ? VOLCMES. THICKNESS. 3 x 20 2 =1200 13'824 20ft. SOLUTION. The greatest 3 x 20 x 4 = 240 8000 4ft. cube of even tens contained 4-2 = 16 5824 24 ft. in 13824 cu. ft. is 8000 cu. ft. 1456 5824 (Cube .4.) Its thickness, therefore, is 20 ft. Subtracting 8000 (.4) from 13824 leaves a re- mainder of 5824 cu. ft., which are added in solids of equal thickir - ess CUBU 1400T. 217 to three sides of ^4, as seen in Fig. 3. It now remains to find the thickness of the additions (6, c, d), O,/, r/), and h, which have a uniform thickness. As the solids b, c, d, form the greater part of the volume of the additions (5824 cu. ft.), and the length and breadth of each is 20 ft. (the length and breadth of A}, by trial, using For- mula 2, we find 5824 -f (3 x 20'- 2 ) = 4 ft., thickness of the additions, if correct. Knowing the thickness, which is also the breadth of e, /, r/, /<, we find the product of the length and breadth of e, /, g = 3 x g 20 x 4 = 240 sq. ft. ; and that of h 4 2 = 16 sq. ft. ; both of which added to 1200 sq. ft. = the product of the length and breadth of all the additions. This product, by Formula 1, multiplied by the thick- ness, 4 ft.,= 5824 cu. ft.; proving the correctness. Therefore, The thickness of a cube whose volume is 13824 cu. ft. is 20 + 4 ft. = 24 ft. The numbers in the middle column (Ex. 12; all indicate volume: 13824 = volume of original cube. 8000 = volume of Cube A. 5824 = volume of the additions (6, c, d), (e,/, gr), and //. 218 SENIOK ARITHMETIC . The numbers in the left-hand column indicate product of length and breadth : 1200 = I x b of solids 6, c, d. 240 = I x b of solids e,/, g. 16 = I x b of cube /. The numbers in the right-hand column indicate thickness: 20 ft. = thickness of A. 4 ft. = thickness of all the additions. 24 ft. = thickness of original cube. 370. Short method. Rule for finding the cube root: Beginning at the decimal point, separate the number into periods of three figures each ; thus: 16'581'.375. Find the greatest cube in the left-hand period, and write its root at the right. Subtract the cube from the left- hand period, and bring down the next period for a div- idend ; thus, 8581 To find the trial divisor, square the root already found with a cipher annexed, and multiply by 3 ; thus, 8 20 Trial divisor, 1200/8581 _20 400 3 1200 To find the trial figure, find how many times the trial divisor is contained in the dividend j thus, 16'581'.375|_25 8 20 Trial divisor, 1200/8581 _20 400 8 1200 CUBE KOOT. 219 To find the correction, multiply the former root by 3, an- nex the trial figure, and multiply by the trial figure ; thus, 16 / 581 / .375|25.5 8 2 3_ 65 1200 325 Complete divisor, 1525 187500 3775 191275 8581 7625 956375 956375 2 325 Continue thus, until all the periods are-ex- hausted. NOTE 1. When there is a remainder after all the periods are exhausted, annex decimal periods, and continue the process as far as desired. The result will he the approximate root. NOTE 2. When a cipher occurs in the root, we annex two ciphers to the trial divisor, and bring down the next period. NOTE 3. The right-hand decimal period must have three places. 13. What is the cube root of 8.414975304 ? OPERATION. 8.414 / 975 / 304 I 2.034 120000 1809 414975 365427 121809 -6362700 24376 49548304 49548304 12387076 Since occurs in the root, an- nex 00 to the trial divisor, mak- ing 120000 ; bring down the next period. NOTE. To find the cube root of a common fraction, extract the root of each term separately. If both terms are not cubes, reduce to a decimal and then extract the root. The result will be the approx- imate root. Find the cube root of 14. 42875. 15. 884736. 16. 4492125. 17. 77854483, 220 SENIOR ARITHMETIC. 18. 8.615125. 21. 1879.080904. 19. 17.373979. 22. 32.890033664. 20. 450827. 23. 10077696. 24. What is the cube root of JffSSi? A? itH? 39^? A? Extract the cube root to the third decimal place : 25. 14.323. 27. .06324. 29. 3. 26. 31982.4. 28. .0015. 30. 7. 31. What is the width of a cube whose solidity is 91125 cubic inches ? 32. A cubical cistern holds 50 barrels of water. How deep is it ? 33. What is the entire surface of a cube whose side is 9 ft.? 34. -\/S()6 X 32.5 = ? SIMILAR SOLIDS. 371. Solids having the same form without regard to size are Similar Solids. Any two cubes or any two spheres are similar solids. Solids are similar when their correspond- ing dimensions are proportional. PRINCIPLES. Similar solids are to each other as the cubes of their corresponding dimensions. The corresponding dimensions of similar solids are to each other as the cube roots of their volumes. 1. A globe is 3 inches in diameter, and another 6 inches in diameter. What is the ratio of their volumes ? EXPLANATION. They are to each other as 3 3 to 6 3 = 27 : 216. 2. There are 64 cubic inches in a 4-inch cube. How many in an 8-inch cube ? SIMILAR SOLIDS. 221 3. Two similar solids contain 386 and 284 cubic inches respectively. If the larger is 11 inches thick, how thick is the smaller ? 4. If a man 6 ft. 2 in. tall weighs 215 pounds, what should be the weight of a man 5 ft. 10 in. tall of the same proportions ? 5. The width of a bin is 4 ft. 6 in. How wide must a similar bin be to hold 4 times as much ? 6. If an orange 2^ inches in diameter costs 5 cents, what should an orange 3^ inches in diameter cost ? QUESTIONS. 372. 1. What is involution ? A power of a number ? The first power ? The second power ? The third power ? What are the second and third powers called ? What is the exponent of a power ? 2. What is evolution ? A root ? The square root of a number ? The cube root of a number ? The fourth root of a number ? How is a root indicated ? The square root ? The fourth root ? e^. 3. Tell how to find the side of a square when the area is given ? 4. Tell how to find the edge of a cube when its volume is given ? 5. What kind of measure is a cube ? 6. A cube contains how many times as many figures as its root ? What is shown when the number is separated into periods of three figures each ? 7. What is the cube root of a number ? Two answers. 8. Cube the numbers from 1 to 10. 222 SENIOR ARITHMETIC. 9. What is the first root figure ? What kind of meas- ure is it ? 10. How is the trial divisor found ? What is the trial divisor? 11. What kind of measure is it? Why is it a trial divisor ? 12. How is the correction found ? 13. What kind of measure is the correction ? 14. What is the complete divisor ? What kind of meas- ure is it ? 15. What is a right-angled triangle ? 16. What principles are true of all right-angled tri- angles ? 17. Tell how to find hypothenuse, base, perpendicular. 18. What are similar figures ? What principles are true of them ? 19. What are similar solids ? What principles are true of them ? REVIEW. 373. Oral. 1. What is the cost of 20 pounds of sugar at 6 cents a pound ? _ ~ 2. A man owning f of a farm sold ^ of his share. What part does he still own ? 3. A can do a piece of work in 2 hours, and B in 3 hours. In what time can both do it, working together ? 4. Two men receive $60 for painting a house. One worked for $2 a day, and the other $3 a day. How much money should each receive ? 5. What is the interest of $ 500 for 2 years at 6< ? REVIEW. 228 6. What is the cost of 64 straw hats at $1 each ? At $ .50 ? $.25 ? At $.12i ? At $1.25 ? At $2.50 ? 7. If 4 oranges cost 12 cents, what will 7 oranges cost ? 8. If | of a yd. of silk costs $1J, what will 1J yards cost? 9. If a man 6 feet tall casts a shadow 8 feet long, how long a shadow will a boy 4 feet tall cast ? 10. If f of my money is silver and the rest bills, and I have $180, how much of each kind have I ? 11. If | of a cord of wood costs $1.50, what will a cord cost ? 5 cords ? 12. A boy buys papers at the rate of 3 for 2 cents, and sells them at the rate of 2 for 5 cents. How much does he make on 30 papers ? 13. What is the value of 8 bushels of wheat, if 6 bushels cost $4.50 ? 14. What is the cost of 2 Ib. 8 oz. of butter at 16 cents a pound ? 15. What is the difference between 5 square feet and 5 feet square ? 16. When it is noon in Syracuse, what time is it 7 east of Syracuse ? 17. Two places are 37| degrees apart. When it is 5 P.M. at the eastern place, what is the time of the western ? 18. When it is noon in Syracuse, what is the time 60 west of Syracuse ? 19. What is the standard time of Denver when it is noon in Boston ? V 20. 58 is of what number ? 21. A boy sold a knife for 60 cents, which was f of its cost. What did it cost ? 224 SENIOR ARITHMETIC. 22. The sum of two numbers is 32 ; their difference is 10. What are the numbers ? NOTE. The half-sum + the half-difference = the greater. The half-sum the half-difference the less. 23. At a village election there were 1200 votes cast for two candidates ; the successful candidate had a majority of 200 votes. How many votes were cast for each ? 24. The sum of two numbers is 68 ; their difference is 26. What are the numbers ? 25. What is 33i% o f $900 ? 66f % of $1200 ? 12^% of $96? 25% of $600? 26. A merchant, by selling goods at $80, lost 20%. What was the cost? 27. A farmer had a flock of sheep, and purchased 25% more ; he then had 250 sheep. How many had he at first ? 28. A lad had 45 marbles, and lost 33^ % of them. How many had he left ? 29. What is an agent's commission for buying 96 head of cattle at $33 a head, at 6f % ? 30. 75 X 66f - 26 X 12J = ? 31. How much is 500% of $12 ? 32. A druggist expended $20 in opium, which he sold at a profit of 300%. What did he sell it for ? 33. $18 is 600% of what ? 34. What is the difference between .6% of $50 and 1% of $70 ? 35. What per cent of a number is J of it ? \ of it ? T V of it ? f of it ? f of it ? f of it ? 16 is \% of what ? 36. A lot containing 48 square rods is 3 times as long as it is wide. What are its dimensions ? REVIEW. 225 EXPLANATION. As the length is three times the breadth, we divide the area by :>; the result will be the area of each of 3 equal squares, the square root of which will be the width, which multiplied by 3 will give the length. V^ = 4 rd., the width. 37. A and B had the same income. A saved 1 of his and B |. A had $1600 at the end of 8 years ; how much had B? 38. Which is greater, the square root of ^L, or the cube of i? 39. A two-inch pipe can discharge the contents of a cask in 8 hours. How long will it take a four-inch pipe ? 40. How many rods of fence necessary to fence a square lot containing 144 sq. rd. ? 41. A lot containing 144 sq. rd. is four times as long as it is wide. How many rods of fence does it require ? (Compare with result in Ex. 40.) 42. How many inches in a hektometer ? 43. How many milliliters in 4 dekaliters ? 44. How many ares in 5 Hektares ? -45. John and George divide 150 marbles in proportion to their ages. John is 7, and George is 8. How many marbles do each receive ? 46. If a boy can ride a bicycle at the rate of 18 miles an hour, how long will it take him to ride twice around a sec- tion of land ? 47. What is the interest of $600 at 8% for 3 months? for 3 years ? 48. If I owe a debt of $60, and pay $40 two months before it is due, how long after it is due should the re- mainder be allowed to run ? 49. At what time between 3 and 4 o'clock are the hour and the minute hand of a watch together ? 226 SENIOR ARITHMETIC. EXPLANATION. Both hands are together at 12 o'clock, and be- fore it is 12 o'clock again they will have been together 11 times. They will be together between 1 and 2 in ^ T of 12 hours, and between 3 and 4 in T 3 r of 12 hours. 50. If I buy 8% stock so that it pays me 6% on my in- vestment, what per cent do I receive ? Written. 51. Frost injured 72 peach trees on M's farm, which number was 9% of all the trees he had. How many did he have in all ? 52. At 2% an agent received $125.50 commission on the sale of some real estate. What was it sold for ? 53. Amsterdam, N. Y., ^OH. /, after date, (3^ promise to pay order, Dollars, with mterest. Value re- Find the proceeds of the above note, discounted at the Farmer's National Bank, Amsterdam, N.Y., Feb. 16, 1896. 54. A gentleman insured his house for $1800, . which was f of its value, at 1J$>. In case f total destruction by fire, what is the entire loss to the owner ? 55. A bill of goods amounting to $287.60 is sold with discounts of 10^ and 5% for cash. How much cash will pay it ? KKVIKW. H( 56. If a piano that cost $360 is to be sold at a profit of 16|%, what price must be asked that 12-J-% may be abated from the asking price ? 57. "I sold two articles for $1.50 each, thereby realizing a profit of 25% on one and a loss of 25% on the other. Did I gain or lose on both transactions ? 58. A bought a carriage at 20% and 10% from list price, and sold it at 10% and 5% from list price. What per "cent profit did he make ? 59. A grocer bought a cask of molasses containing 40 gal. for 38 cents per gallon. Seven gallons having leaked out, for how much per gallon must he sell the remainder in order to gain 12|% on the investment ? 60. Suppose a grocer bought a 42-gallon cask of vinegar at 12/ per gallon, and put 12 gallons of water with it, and sold it for the same price. What would be his rate per cent gain ? 61. A meter stick is what per cent longer than a yard stick ? 62. Buffalo is the largest flour depot in the world. It received by lakes and rail in 1895, 8,971,740 bbls. of flour. If the K Y. C. & H. K.E. shipped 18.9%, the N. Y., L. E., & W. R.K 12.15%, the Pennsylvania R.R. 8.33%, the West Shore R.E. 10.97%-, the Lehigh Valley E.R. 8.12%, the other roads 6.5%, and the remainder by water, what per cent was shipped by water ? and how many barrels ? 63. What will be the cost of 6 loads of wood, each con- taining 1 C. 6 cd. ft. 10 cu. ft., at $2.50 a cord ? 64. How many yards of carpet 2 ft. wide will be re- quired for a room 12 ft. by 15 ft. 6 in., if the strips run lengthwise, and there is a waste of of a yard in each strip in matching ? 228 SENIOR, ARITHMETIC. 65. The width of a building is 36 ft., and the ridge of the roof is 10 ft. higher than the eaves. How many square feet of boards will it take to cover one of the gable ends ? 66. With how long a rope must a goat be fastened to a stake that it may feed on four square rods of land ? 67. A room 24 feet long and 15 feet wide is to be car- peted with carpet | yd. wide. How many yards will be required if a waste of of a yard is made on each strip in matching, the strips to run crosswise ? 68. Oswego, N.Y., is in latitude 43 28' N. How many degrees is it from the North Pole ? From the South Pole ? 69. How many gallons in 32J hektoliters of wine ? 70. If it takes 2 Ib. 7 oz. 4 pwt. of silver to make 12 spoons, what amount will be required for one spoon ? 71. If it is one-half of a mile from your home to the school building, how many steps of 1 ft. 6 in. each will you take in reaching it ? 72. What decimal part of a week is 4 da. 3 hr. 36 min. ? 73. What part of 2 reams are 10 quires, 20 sheets ? 74. How many times is 132 X 75 X 42 x 104 contained in 26 X 22 x 150 X 168 ? . 75. Bought six loads of oats, each containing 32 bags, each bag containing 2 bushels, worth $.56 a bushel, and gave in return 8 boxes of ten. each containing 24 pounds. What was the tea worth a pound ? 4 x 7 X 32 X 15 X 7b. 16 X 56 X 5 X 4 X 6 77. If f of a box of oranges cost $4.50, what part of a box can be bought for $5.25 ? 78. Simplify the following complex fraction : * X | . fi REVIEW. 229 79. f of 63 is T 7 2 of what number ? 80. A gentleman invested $215380 in a knitting-mill, which was f of the value of the plant. What was the Nvalue of f of the plant ? 81. A and B, being 150 miles apart, travel toward each other. They start at the same time, and meet at the end of eight hours, when they discover that A has travelled 1| miles each hour more than B. How many miles has each man travelled ? 82. For how long a time must $4560 be placed on inter- est at 6% to gain $353.40 ? 83. A man borrowed $250 March 3, 1896, and paid the note Sept. 21, 1896, with 5% interest. What was the amount of the note ? 84. A merchant borrowed $165 at 6%, and when he paid the debt it amounted to $168.96. How long did he have the use of the money ? 85. The interest on a certain sum is $27.40, the time 2 years, 3 months, 12 days, and the rate 6%. What is the principal ? 86. A note for $250 was given Sept. 5, 1895 ; a payment of $75 was made April 25, 1896. How much will settle the note Oct. 3, 1897 ? 87. A man bought a farm for $4000, April 1, 1889. He gave a mortgage at 5% for $3000, and paid as follows: Jan. 1, 1890, $700; Oct. 1, 1890,' $1000 ; April 1, 1891, $850; and the balance of the mortgage April 1, 1892. How much was due at settlement ? 88. What sum of money must I loan at 6 per cent inter- est, that it may bring rue in a quarterly income of $300 ? 89. Compute the interest on $3450 for 2 yr. 6 mo. 20 da. at 6%. 230 SENIOR ARITHMETIC. 90. William Johnson holds a note for $1250 against James W. Way, dated Jan. 10, 1893, payable on demand. This note bears the following indorsements : March 10, 1893, $200: May 10, 1893, $300: July 10, 1893, $50; Oct. 10, 1893, $400. What is due Dec. 10, 1893, interest at5%? 91. Find the simple interest of $382.94, one half to be paid in 5 yr. 5 mo. 20 days at 3%, the other half to be paid in 5 yr. 5 mo. 20 days at 5%. 92. A man borrows $2000 which belongs to a minor who is 18 yr. 2 mo. 10 days old, and he is to keep it until the owner is 21 years of age. What will then be due, money being worth 6% ? 93. Bought a house for $6000, and gave a mortgage for $4000, dated Jan. 1, 1892, interest at 6%. Made the fol- lowing payments : July 1, 1892, $520 ; Jan. 1, 1893, $708 ; Jan. 1, 1894, $680 ; July 1, 1895, $725. How much was due Jan. 1, 1896 ? 94. A man owes me $463.50, payable in 6 months with- out interest. What sum can I afford to take now for the debt, money being worth 6% ? 95. A man bought goods amounting to $2100 on 6 mo. credit, but was offered a discount of 3% cash payment. If money was worth % a month, what is the difference? 96. Which is the more profitable, to buy goods worth $500 at 90 days, 3% off for cash, or put the amount at interest at 7%, and let the bill run to maturity ? 97. Face of a debt, $1256.25. Date. July 1, 1886. Time, 1 yr 6 mo. Rate, 6 % . What is the present worth ? 98. Had a note of $2500 discounted at a Rochester bank for 2 months. What were the proceeds, rate of discount being 1% ? REVIEW. 231 99. Find the difference between the true discount and the bank discount of a debt of $550, due in 4 months without interest. 100. April 1, A gave B a 3-mo. note for $300, which B had discounted at a bank May 1. What did B receive ? and what amount could the bank collect on July 1, dis- count at 6 % , no grace ? 101. Bought an invoice of goods amounting to $1360.58. How much will I make by discounting my note at the bank for 90 days at 6%, and paying cash for goods at 5% off? 102. A New York note of $2000, bearirig date May 24, 1895, and payable in 60 days, was discounted at 6%. The discount was $15. When was it discounted ? 103. Sweet and Johonnot sold 20 bicycles to a dealer, taking his note at 60 days, which they discounted immedi- ately at the Merchant's Bank at 6% with grace, receiving $1485. What was the price of each bicycle ? 104. On the first day of January, 1890, a man gave three notes, the first for $500 payable in 30 days ; the second for $400 payable in 60 days; and the third for $600 payable in 90 days. W T hat was the average term of credit, and what the equated time of payment ? 105. I wish to use $560.88 immediately. For what sum must I draw a bank note, due in 96 days at 6%, that I may receive the required amount ? 106. How many $500 U. S. bonds can be bought for $6630 at 10L% premium? 107. A guardian invests $1000 at simple interest at 3%, $1000 in 4% bonds at 112i, and $1000 in 5% bonds at 125. The bonds are to run 10 years, and be redeemed at par. Compare the three investments at the end of the ten years. 232 SENIOR ARITHMETIC. 108. The city of Buffalo pays $12425.72 for rented school buildings. On what amount of 3% c / bonds would this pay the interest ? 109. If I buy bank stock at 20% discount, and sell it at 10% premium, what per cent do I gain ? 110. What is the rate of income on a 4% stock bought at 62J. 111. I have $5000 to invest, and can buy 5% stock at 110, or 6% stock at 125. Which will be the better invest- ment ? and how much annually ? 112. A gentleman owned a house which he rented for $375 above all expenses. He sold the house for $5000, and invested the money in a 5% stock at 80. Did he gain or lose by the transaction ? and how much per year ? 113. The ratio of A's weight to that of B is |. B weighs 120 Ib. 8 oz. What does A weigh ? 114. If John is 6 years old and Henry 15, what is the ratio of John's age to that of Henry ? What will it be when each is 5 years older ? 115. If 4 horses eat 4 bu. of oats in 2 days, how many horses will eat 48 bushels in 12 days ? (Solve by analysis.) 116. If the antecedent is of T 9 g x ? ^j, and the ratio is of 2 9 ?> what is the consequent? 117. How wide can 20 men, working 8 hours a day for 8 days, make a ditch which is 75 rods long and 10 ft. deep, if 25 men, working 10 hours a day for 7 days, can dig a ditch 80 rd. long, 8 ft,, deep, and 2 ft. wide ? 118. If a baker's loaf weighs 10 ounces when wheat is 60 cents a bushel, what should it weigh when wheat is 70 cents a bushel ? 119. If a train moves at the rate of 30 miles in 48 min- utes, in what time will it run 450 miles? REVIEW. 233 120. One side of a shed is 8 ft. high, the opposite side 13 ft. 6 in. What is the ratio between the sides ? 121. If it costs $30 to lay a cement sidewalk 4 ft. wide and 16 ft. long, how much will it cost to lay the same kind of walk 7 ft. wide and 96 ft. long at the same rate ? 122. Write and solve a problem in proportion, using the following numbers: 8 men, 9 Ib. of beef, 1 da. ; and 2 da., 12 Ib. of beef. 123. If T \ of a yard of cloth cost $1, what will 4 yd. cost? 124. A, B, and C, engaged in trade. A put in $400, B $250, C $600 j they gain $300. Find each man's share of the gain. 125. A merchant failing in trade has debts amounting to $34560 ; his assets are $30240. What can he pay on the dollar ? and how much will a creditor receive to whom he owes $3840 ? 126. A man willed his property, which was valued at $6000, to his four children in the following proportion, giving to each one , , , and respectively. How much did each one receive ? 127. Three families rent a cottage for the summer. The first family occupies it for 6 weeks, the second for 2, and the third for 3 weeks. The rent for the entire season of 11 weeks, is $440. How much should each family pay ? 128. Scrantom, Morris, and Jackson were associated in business for a period of 1 yr. 6 mo. Scrantom furnished $5000, Morris $3000, and Jackson $2000 of the original capital. When the partnership terminated, they divided $4000, the profits arising from the same. How much more did each make than he would have realized had his money been invested in a 6% mortgage ? 234 SENIOR ARITHMETIC. 129. Divide $60 among three boys so that one shall have as much as the other two, whose shares are as 3 to 7. 130. What is the distance between the diagonally oppo- site corners of a lot whose area is 16 sq. ft. ? 131. My dining-room is 16 ft. long, 14 ft. wide, 10 ft. high. Find diagonals of the shorter sides, of the longer sides, and of the room. 132. What is the length of one side of a cube, equal in volume to a solid that is 49 ft. long, 27 ft. wide, and 7 ft. high? 133. A ladder 25 ft. long, the bottom of which is 5 ft. from a building, reaches the base of a window. How many feet from the base of the window to the ground ? 134. At 40 cents a rod for fencing, which will cost the more, to enclose a square field containing 10 A., or a field of the same area whose length is twice its width ? 135. Find the cube root of 41781.923. 136. Find the square root of 41781923. 137. A cubical cistern contains 30 hhd. of water. How deep is it ? 138. The volume of a rectangular prism is 200 cu. ft., and its height is 8 ft. Find its surface contents, if its two other dimensions are equal. 139. The area of a right-angled triangle is 289 sq. ft., its base is of its altitude. What is the length of its alti- tude ? 140. If a railroad company pays 19/ per sq. yd. for ex- cavating, and 37/ per sq. yd. for drawing away the earth, what will it cost the company to remove a mound equal in volume to a cube whose side is 81 feet ? REVIEW. 235 141. Forty feet directly east from a column that is 75 ft. high, I measure clue north 30 ft., and find that I am in line with a stake and the column. If the stake is 25 ft. distant from my position, and 10 ft. high, what is the dis- tance from the top of the stake to the top of the column ? 142. How many rods of fence will enclose a rectangular field containing 20 acres, if the field is twice as long as it is wide ? and how much will it cost at $2.45 per rod ? 143. If a locomotive runs at the rate of 55 miles in 40 minutes, and its drive-wheels are 18 ft. in circumference, how many revolutions will the drive-wheel make in one hour ? 144. A insured his stock for $1200. He paid a premium of $24. What was the rate of insurance ? 145. Grant and Dunn bought a bill of glass amounting to $853.68, upon which they received a discount of 60%, 25%, 15%, and 2% off for cash. What was the net amount of bill ? 146. My agent in Chicago sold goods to the amount of $8640. He also purchased 6800 bu. of wheat at $1.10 a bushel, paid for expenses $10.40, and received a commis- sion of 2 ct. on every dollar. How much will he remit to me after paying all expenses ? 147. A merchant buys calico at 5| ct. per yard, and sells at 6. What is his rate per cent of gain ? 148. What is the rate of insurance when a $1000 policy for 3 years costs $7.50 ? 149. A man lost $13.45 on some flour by selling it at a loss of 14f %. What was the flour worth? 150. A farmer buys 4 tons of hay at $20 per ton, and 4 bbl. of flour at $5 per barrel. What is the cash value of the bill, if he is allowed a discount of 15%, and 5% deduction for cash ? 236 SENIOR ARITHMETIC. TEST QUESTIONS. 374. Arranged, by permission, from examinations given in various cities. 1. Define addition, sum, sign of equality, subtraction, remainder, subtrahend, minuend, parenthesis, multiplica- tion, factors, multiplicand. 2. Multiply 7258 by 395, and write each partial pro- duct in words. 3. Subtract 8969 from 9782, and prove the work. 4. Prove by an illustration that multiplication re- sembles addition. 5. Solve : $73.46 - ($.94 + $3.02) + $47 X 35. 6. Write in figures, XL VI I. 7. Write in figures, six hundred eight thousand seventy- two. 8. Multiply 6504 by 657. 9. I of 585 x 5 = ? 10. Divide 45897 by 490, and prove that your work is correct. 375. 1. Copy and find the sum: $23.17, $6043.05, $0.42, $208.97, $5486.04. 2. How many yards of linen, at 28 cents a yard, must be given for 35 bushels of potatoes, at 56 cents per bushel ? 3. A man paid $13,465 for a house and some land. The house alone was worth $8,978. What was the value of the land ? 4. Write this number in words, 3,782,013. 5. Write in words, XCV. ; 76508.904. TEST QUESTIONS. 237 6. How many bushels of potatoes at 50 cents a bushel will pay the entire cost of a hat at $7.50, a dress at $24, a cloak at $16.25, and gloves at $1.75 ? 7. 6460000 x 3000 -v- 25000000 = ? (Use shortest way.) 8. If the dividend is 1761184 and the quotient 4684, what is the divisor ? 9. What is the smallest number that will exactly con- tain 16, 20, 24, or 30 ? 10. Define dividend, remainder, product, the prime fac- tors of a number. How do you prove division ? 376. 1. Define multiplier ; concrete number. 2. How can you prove an example in subtraction ? 3. A merchant bought 375 bbl. of apples at $.95 a bbl. ; 43 bbl. rotted ; if he sells the rest at $1.10 per bbl., how much does he gain or lose on all ? 4. My salary is $2350 a year, and I spend $4 a day ; how much will t save in six years ? 5. If 23 men own 475 bbl. of apples each, and 4 of them divide theirs equally among the rest, how many will each have then ? 6. Find the prime factors of 1155. 7. If my salary is $1400 per year, and my expenses $90 per month, how long will it take me to save $4160 ? 8. What is the smallest quantity of grain that will fill an exact number of bins, whether they hold 312, 260, or 390 bushels ? 9. What are like numbers ? Give three. 10. From Albany to West Troy is 5 miles, from West Troy to Cohoes 2 miles, and from Cohoes to Saratoga is 30 miles. How far is it from Albany to Saratoga ? 238 SENIOR ARITHMETIC. 377. 1. Write in words 23456789. 2. Find the greatest common divisor of 75, 25, and 500, and their least common multiple. 3. If 7 tons of hay cost $105, what will be the cost of 289 tons ? 4. Write the number which is composed of 3 units of the eighth order, 6 of the fifth, 2 of the third, and 9 of the second. 5. Find the contents of the smallest measure that may be filled by using either a 4-quart, a 5-quart, or a 6-quart measure. 6. Find the prime factors of 1452. 7. Solve by cancellation : A man receives $21 for 15 days' work of 7 hours each. How much should he receive for 19 days' work of 5 hours each ? 8. The product of three numbers is 105840 ; one of the numbers is 42, the other 35. What is the third number ? 9. How many pounds of butter at 20/ a pound are worth as much as 1600 bushels of wheat at 75/ a bushel ? 10. What is the greatest common divisor of two or more numbers ? 378. 1. Two persons start from the same point and travel in opposite directions ; one at the rate" of 25 miles a day, and the other at the rate of 32 miles a day. How far apart will they be in 8 days ? 2. What is the product of 20202 x 10101 ? 3. What number multiplied by 1728 will produce 1705536 ? 4. A man has $8250 ; how much must he add to this to be able to pay for a farm worth $10000 ? TEST QUESTIONS. 239 5. (6070 - 1200) + (4680 -=- 15) = ? 6. Bought 144 acres of land at $41.25 an acre, and sold the whole for $7000. Did I gain or lose ? and how much ? 7. If 3 oranges are worth f of a melon, what part of the melon is 1 orange worth ? 8. Austin having 30 marbles, gave of them to one companion and I of them to another. How many had he left? 9. How many eggs in 12^ dozen ? 10. Write the present year in Roman numerals. 379. 1. George gave a beggar 9 cents, which was of all the money he had. How much money had he ? 2. Mary is 14 years old, and her sister is f as old. How old is her sister ? 3. What is a mixed number ? 4. How many ninths in 5 ? 5. At | of a dollar a pound, what will 8 pounds of butter cost ? 6. What will | of a pound of coffee cost at 28 cents a pound ? 7. If a man earns $15 a week and spends f of it, how much does he save ? 8. What do you understand by | of anything ? 9. Change 142 1 to an improper fraction. 10. A boy having 20 quarts of blueberries, sold f of them for $%. What was the price for a quart ? 380. 1. If I put of my money in one bank, \ in an- other, in another, and have $4,200 besides, how much have I? 240 SENIOR ARITHMETIC. 2. A can mow a field in 10 days, B in 8 days, and C in 5 days. When working together, how many days will they need? 3. If 6 is added to both terms of the fraction -J, how much is the fraction increased or diminished ? 4. The divisor is 46, the quotient 605, and the remain- der 23. What is the dividend ? 5. From the sum of | and | take the sum of T \ and . 6. How many cords of pine wood at $3.25 a cord must be given for 12 yards of broadcloth at $2.10 a yard ? Work and analyze. 7. Find the prime factors of 13860. 8. 13 x 16 X 42 x 51 _ 9 6 X 17 x 48 x 91 9. Find the sum of the prime numbers under 20. 10. Reduce |f|^ to lowest terms. 381. 1. Write in Roman notation 1894. 2. Write the prime numbers from 1 to 18 inclusive. 3. If 3 boxes of oranges cost $5f, how many boxes can be bought for $17 ? 4. A farmer sold 64 sheep, and had ^ of his flock left. How many had he left ? of 3' 6. How many barrels of flour at $6 a barrel must be given for 3 pieces of linen, each containing 36 yds., at 25 ct. a yard ? 7. A farmer sold at market 15 sheep at $-^ each, and bought 7 yards of cloth at $1|- per yard. How much money did he take home ? TEST QUESTIONS^pg^/ 8. From the sum of 5$, 9, 11$, take the difference between 32 and 13 1. 9. Reduce to their least common denominator Jj, ||, 21 24 2S> S* 10. Write a receipt for $20 paid you by Mr. John Dixon. 382. 1. Add $49.50, $43.62$, $75.05, $64.75, $35.09, $6.03$, $42, $73.98, $105.60. 2. How many sheep at $5 each must be given for 15 horses at $150 each ? 3. What is the sum of ^i -f ^ + t ? 6 o 4 4. From of J of 3 take of 1J. 5. At $39 1 apiece, how many cows can be bought for 6. How many times is T 3 of of 6| contained in | of 54 X -*- J ? 7. Define multiple and greatest common divisor. 8. Give and define proper fraction ; mixed number. 9. If a man spends of his money for a house, f for a farm, and has $3400 in cash left, what is the amount of his wealth ? 10. A grocer bought 3 barrels of apples of different qualities at $2.75, $3.12, and $3.25 a barrel. What was the average cost ? 383. 1. What is reduction of fractions ? 2. Reduce }J to 156ths. 3. Express T 2 7 \\ in its simplest form. 4. Change to fractions having the least common denom- inator, J, T *j, and T V 242 SENIOR ARITHMETIC. 5. If a merchant buys tea at $f a pound, and sells it at $| ? does he gain or lose ? and how much ? 6. Find the sum of f , 7, and 8j. 7- A + A + Wk + i-*? 8. A man engaged to labor 30 days, but was absent 5 T 7 ^ days ; how many days did he work ? 10. A young man received a salary of $60 f a month, and paid for his board $30, for washing $lj, and for other expenses $12 T 9 S . How many dollars had he left ? 384. 1. Define a proper fraction, and give an example of one. 2. A merchant bought three pieces of cloth containing 125$, 96|, and 48f yards. How many yards did he buy ? 3. What is the value of 2 times | of f of l ? 4. If 9 men consume f of 9| pounds of meat in a day, how much does one man consume ? 5. A farmer distributed 15 bushels of corn among sev- eral persons, giving them 1| bushels apiece. Among how many persons did he divide it ? 6. What is the value of lii ? 7. What number must be added to 22 that the sum may be 99^ ? 8. A can do a piece of work in 8 days, and B can do it in 6 days. In what time can they do it working together ? 9. A pole stands ^ in the mud, J in the water, and 21 feet above the water. What is its length ? 10. A man bequeathed to his son $3500, which was f of what he left his wife. How much did he leave his wife ? 385. l. If of a farm is valued at $1728, what is the value of the whole ? TEST QUESTIONS. 243 2. If 8 be added to both terms of the fraction f , will its value be increased or diminished ? and how much ? 3. If the sum of two fractions is f , and one of them is fjfj what is the other ? 4. Express in its simplest form the quotient of 2025 divided by 3645. 5. If the dividend is | and the quotient 3 \, what is the divisor ? 6. At $J a bushel, how many bushels of apples can be bought for $5 1 ? (Analysis.) 7. Define fraction, terms of a fraction, improper frac- tion, compound fraction, and complex fraction. 8. Change 5 T 2 g* to a whole or mixed number. 9. How many 8ths of a bushel in 9 bushels ? 10. Change g, |, T 7 5 , T 5 2 , f to equivalent fractions having a common denominator. 386. 1. A farmer sells 6 jars of butter holding 8 pounds, at 36/ a pound, and receives in payment 14 cans of coffee, each holding two pounds. What was the price of the cof- fee ? Work by using cancellation. 2. If a man walks 3^ miles in one hour, how far can he walk in 9 hours ? 3. Find the sum of 13, }, 6f, 20|f, and 1 7 H . 4. How many days' work at $lf a day will pay for 8\$ yards of cloth at $2^ a yard, and 56 Ib. of butter at 25 cents a pound ? 5. The product of two numbers is 41|, and one of them is 160 T f j ; what is the other ? 6. Find the sum of 93567 + 20754867 + 4756 + 925674 + 6543987 + 6579 + 98675 + 567923 + 645876 + 9346 + 878 -f 54562 + 888. 244 SENIOR ARITHMETIC. 7. Tell in words what these numbers are : 1950 ; 90 ; 4040; 73000007. 8. Find the difference between 76392 X 4506 and 985301 X 976. 9. What will 79 ten-ton cars of coal be worth at $5.50 a ton? 10. If you should buy 376 horses for $65123, how much would you sell them for apiece to gain $5189 ? DECIMALS. 387. 1. A merchant bought four pieces of cloth contain- ing 32f, 38 1, 40|, 45| yards, respectively. How many yards did he buy ? 2. Change to decimals and add : |, f , 4|. 3. From a farm containing 128 f acres, 84| acres were sold. How many were left ? 4. Find the prime factors of 1008. 5. 50 -5- .05 = ? 6. What will 2.47 pounds of coffee cost at $.48 per pound ? 7. If one yard of ribbon costs 34i cents, what will 6 pieces cost, each piece containing 13.12 yards? 8. Write in words 68.0642. 9. If 9 yards of cloth cost $1.17, what will 15 yards cost? 10. A lady went shopping with $45. She paid $4-^ for shoes, $5f for a hat, $12| for a dress. How much money had she left ? 388. 1. If a farm is worth $3200, how much is | of it worth ? 2. From one million take one millionth. TEST QUESTIONS. 245 3. What is the difference in cents between | of a dollar and | of a dollar ? 4. Point off into periods 96308796, and write over each period its name. 5. Express with figures the following numbers : Seven million ninety-five thousand, sixty-three and fifteen thou- sandths, and seven hundred and seven hundredths. 6. Head (write in words) the following : 642.0016 ; 100.01. 7. 353812416 -- 589 = ? 8. Find the sum of 684.8, 96.84, 6.0T5, .1906, 7508. 9. At $9 per M., what will 6728 feet of lumber cost ? 10. At $.65 per C., what will 1240 pens cost ? 389. 1. Eeduce to a simple fraction 2. What fraction of 11 f is 5| ? 3. What common fraction equals .0125. 4. Reduce $ T 3 ^ to a decimal. 5. A farmer sold 120 sheep, which were f of his flock. How many had he before the sale ? 6. A grocer sold ^ of a barrel of sugar to one man and i of it to another, and had 80 pounds left. How many pounds did the barrel contain at first ? 7. A and B can do a piece of work in 12 days, A can do it in 25 days ; in how many days can B do it ? 8. A man spent f of his money for a horse and f of the remainder for a buggy and harness, and had $37.50 left. How much money had he at first ? 9. In dividing by a decimal, how do you determine the proper place of the decimal point in the quotient ? 10. At $9.75 per thousand, what will 16544 bricks cost ? 240 SENIOR ARITHMETIC. 390. 1. Which is the greater, J or f g ? How much greater ? 2. Find the sum of 78f , 87 T \, 4f, and 79f . 3. A man has three lots, which are 120, 420, and 600 ft. wide respectively. He wishes to divide them into lots oi the greatest equal width possible. How wide will each lot be ? How many such lots can he make ? 4. If .375 of a ton of coal cost $2.40, what is the price of one ton ? How many tons can be bought for $80 ? 5. Reduce to decimals f , f, f, J, y 1 ^. 6. Write in figures thirteen thousandths, four hundred and five hundredths, five hundred fifteen millionths, and add the results. 7. Eeduce to a simple fraction Tg T? . f + 1 8. Sold a house for $4,797, which was two-sevenths more than it cost ; find the cost price. 9. Make a bill for the following articles, bought to-day of James Brown, No. 23 Warburton Avenue, Youkers, N. Y. : 30 oranges at 25 cents a dozen ; 7 Ib. of coffee at 28/ ; 3 1 Ib. prunes at 13/ ; 1 bag of sugar containing 28 Ib. at 5^/. Receipt the bill as though you were James Brown's clerk. 10. Bought three boxes of oranges containing 263, 220, and 156, at $3.50 per hundred, and sold them for 50^ per doz. Find the amount of profit. 391 i (7| - 2.05) -v- (5 X .23) .7| + 2.23 - .6 -j- .4 2. 1000 -^- .001 = ? 3. j + i + .75 + 1| + -330 = ? 4. Reduce 7 to a decimal. TEST QUESTIONS. 247 6- T 7 i x ay 9. A milkman's quart measure is too small by one gill. At 5 cents a quart, how much does he dishonestly make in the month of June, if he sells 500 false quarts daily ? 10. Find the length of the diagonal of an are of land in the form of a square. MENSURATION. 441. The process of measuring lines, surfaces, and solids is Mensuration. 442. A Line is that which has length, without breadth and thickness. 443. A Straight Line is the shortest distance between two points, or a line that does not change its direction at any point. 444. A Curved Line changes its direction at every point. 286 SENIOR AK1THMETIC. 445. A Plane Surface is a surface that does not change its direction. 446. A Quadrilateral is a plain figure having four straight sides. 447. Parallel Lines are lines having the same direction and equally distant from each other. 448. A Parallelogram is a quadrilateral whose opposite sides are parallel. 449. A Rhomboid is a parallelogram whose angles are not right angles. What is a parallelogram called whose angles are right angles ? 450. A Rhombus is a rhomboid whose sides are equal. A Rhomboid. A Rhombus. 451. The area of a rhomboid is found by multiplying the base by the altitude. NOTE. The altitude of a parallelogram is the perpendicular distance between the sides. 1. Find the area of a parallelogram whose base is 24 rods, and altitude 18 rods. 2. Find the area of a rhombus whose base is 15 ft. and altitude 8 ft. 3. Draw a rhomboid whose base is 15 ft. and altitude 10 ft. Find its area. Draw a rectangle having the same dimensions. MENSURATION. 287 A Trapezoid. 452. A Trapezoid is a quadrilateral having only two sides parallel. 453. To find the area of a trap- ezoid, multiply the sum of the parallel sides by the altitude. 4. Find the area of a trap- ezoid whose altitude is 10 ft., its longest side 20 ft., and shortest side 15 f t. ? 5. A board 20 inches wide at one end and 12 inches wide at the other is 16 feet long. How many board feet does it contain ? 454. A Trapezium is a quad- rilateral having no two sides parallel. NOTE. By drawing a diagonal between any two opposite sides of a trapezium, we have two triangles, the diagonal serving as the hase of each. The altitude of each is the perpendicular distance from its other angle to the diagonal. A Trapezium. 455. To find the area of a trapezium, multiply the diago- nal by half the sum of the altitudes of the two triangles. 6. The diagonal of a trapezium is 18 ft. ; the altitudes of its two triangles are 5 ft. and 3 ft. What is the area ? 7. A farm is in the form of a trapezium. The diagonal distance between the northern and southern corners is 108 rods, and the perpendicular distances from the east and west corners to the diagonal are 52 rods and 36 rods respectively. How many acres in the farm ? SOLIDS. 456. A solid whose two bases are equal and parallel, and its other faces parallelograms, is called a Prism. 288 SENIOR ARITHMETIC. A Triangular Prism. A Rectangular Prism. NOTE. Prisms take their names from the form of their bases, as triangular, quadrangular, pentagonal, hexagonal, etc., according as the bases have three, four, five, or six sides, etc. 457. To find the contents of a prism, multiply the area of the base by the altitude. 8. Find the contents of a tri- angular prism .whose altitude is 10 in., and the area of its base 7 sq. in. 9. What are the contents of a quadrangular prism whose base is 5 in. by 8 in., and whose altitude is 12 in. ? 10. What are the contents of a hexagonal prism, the area of whose base is 10 sq. ft., and whose altitude is 15 ft. ? PYRAMIDS AND CONES. 458. A solid whose base is a triangle, square, pentagon, etc., and whose sides are triangles meeting at a vertex, is called a Pyramid. NOTE. A pyramid takes its name from the form of its base. A solid whose base is a circle, and whose convex surface terminates in a point, is called a Cone. 459. The Altitude of a pyr- amid or cone is the perpendic- ular distance from its vertex to the centre of its base. The Slant Height is the shortest distance from the vertex to the perimeter of the base. 460. To find the contents of a pyramid or cone, multiply the area of the base by $ of the altitude. A Pyramid. A Cone. MENSURATION. 289 To find the convex surface, multiply the perimeter of the base by ^ the slant height. 11. Find the contents of a quadrangular pyramid whose altitude is 10 in., and whose sides of bases are 8 in. and 6 in. SOLUTION. 8 x 6 x V 1 = 16 cu in - ^ns. 12. Find the convex surface of a regular hexagonal pyramid whose slant height is 16 in., and whose side of base is 4 in. 13. Find the convex surface of a cone, when the circum- ference of its base equals 16 ft. and its slant height 18 ft. 14. Find the convex surface and volume of a cone whose radius is 4 in. and altitude 6 in. 461. The Frustum of a cone or pyramid is the part which is left after the top is cut off in a plane parallel to the base. 462. To find the contents of the frustum of a pyramid or cone, multiply i of the altitude by the sum of the areas of the two bases plus the square root of their pro- duct. 15. Find the COn- Frustum of a Pyramid. Frustum of a Cone. tents of the frus- tum of a quadrangular pyramid whose altitude is 15 ft., and whose ends are 6 ft. and 4 ft. square. 16. A log 16 ft. long is 30 in. in diameter at one end and 24 in. at the other. Find its cubical contents. 463. A Sphere is a solid bounded by a curved surface, all parts of which are equally distant from the centre. 290 SENIOR ARITHMETIC. 464. To find the surface of a sphere, multiply the circum- ference by the diameter. 465. To find the contents of a sphere, multiply the surface by of the diameter. 17. Find the surface of a sphere when the diameter is 16 inches. 18. Find the surface of a sphere when the radius is 3 yards. 19. Find the surface of a sphere when the radius is 5 cm. Find the volume when : 20. Diameter = 25 ft. 22. Kadius = 12 ft. 21. Kadius = 2 ft. 23. Circumference = 125664 in. 24. Radius = 3 dm. 25. Compare the volume of a 4-ft. cube and a 4-ft. sphere. 26. Compare the surfaces of a 4-ft. cube and a 4-ft. sphere. REVIEW OF MENSURATION. 466. 1. Find the cubic yards in a cone, the circumfer- ence of whose base is 20 ft., and whose altitude is 30 ft. 2. Find area of a semi-circle when its radius equals 14 ft. 3. Find area of a square inscribed in a circle of 4 ft. in diameter. 4. The circumference of a circle and the perimeter of a square are each 300 ft. Which has the greater area ? 5. A circle is inscribed in a 6-ft. square. Find the area of the circle. MENSURATION. 291 6. Find the value at $50 an acre of a farm in the form of a trapezoid, the parallel sides of which are 120 rd. and 160 rd. respectively, the distance between which is 80 rd. 7. How many miles does the earth travel in a revolu- tion around the sun, the distance between them being 95,000,000 miles? 8. If a bin is 8 feet square, how deep must it be to hold 100 bushels ? 9. Find the lateral surface of an equilateral triangular pyramid, the perimeter of the base being 12 m. and the slant height 14 m. 10. Find the volume of a square pyramid, the perimeter of whose base is 16 feet, and whose altitude is 9 ft. 11. What is the volume of the largest cone that can be cut from a pyramid whose base is 6 feet square, and whose slant height is 15 feet ? 12. A cylindrical tank is 14 ft. deep and 6 feet in diam- eter. Find the cost of cementing sides at 90/ a sq. yard. 13. Find the capacity in gallons of a cylindrical cistern whose inside diameter is 6 feet, and whose depth is 7 feet. 14. Find the capacity in Kl. of a cylindrical cistern whose inside diameter is 4 m., and whose altitude is 5 m. YB 17452 THE UNIVERSITY OF CALIFORNIA LIBRARY