yyVM^^/x^/s^jf/i^yi^^- UC-NRLF $B ETE Ebfl IN MEMORIAM FLORIAN CAJORl < Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/completearithmetOOmaglrich / V| r^n U-y,, H-- -f 7t'-^t^*f Greenleaf* s Mathematical Series. THE COMPLETE AEITHMETIC, ORAL AND WRITTEN, ON tritE, BASIS OF WOJlJiS" By BETJJAMm ^GEEEKLE LEACH, SHEWELL, AND SAKBOEK, bosto:n^ and new york. GREENLEAF'S MATHEMATICAL SERIES. INDUCTIVE COURSE. First Lessons in Numbers. A Brief Course in Arithmetic. The Complete Arithmetic. The Brief Course and the Complete Arithmetic are each published wiib,^^nd withoiit-f^Kswers. Key to T,HEJGoMl^Ll!r^¥'/A^•lTtIMteTIC, for Teachers ordy^ CAJORf Copyright, 18S1, By Henry B. Maglathlik. pBSttwoxK BY Berwick & Smith, Boston, U.S.A. PEEFACE. oi^o This Arithmetic, undertaken at the suggestion of many educators of distinction, has been prepared with special refer- ence to training for practical business, and to development of mind power through fixed habits of attention and lucid pro- cesses of reasoning. To secure skill, rapidity, and accuracy in the use of numbers, required in common transactions, a large number of compara- tively simple business questions has been provided, and promi- nence has been given to subjects of the most practical value. That useful mental discipline may be attained the theory and principles of numbers have, been clearly presented, and problems have been given requiring thought and discrimination. The inductive plan has been followed throughout, principles have been developed from methods, rules derived from analyses, and oral and written exercises combined in a rational manner. The greatest care has been observed to have the definitions brief, clear, and accurate, and the solutions simple, concise, and logical. The methods employed are those which business experience, or test in the school-room, has shown to be the best. Decimals to three places, and United States money, are simply treated at the beginning with integers. The problems are abundant and varied, based on recent and reliable data, and drawn from the acjiml experiences of life. In wn from the actual 9183iSr IV PREFACE. commercial arithmetic the usages of the best business houses have been followed. Several hundred examples which have been used in exami- nations by superintendents of public instruction in various cities and towns leading in educational matters, have been collected, and arranged as exercises for testing proficiency, and for supple- mentary practice, to be drawn from at the teacher's discretion. Much matter formerly considered necessary in an arithmetic, but which modern progress has rendered useless or antiquated, has been omitted. The Appendix contains tables for reference ; information of a somewhat technical nature for the business man, the mechanic, and the farmer; subjects of minor importance to the majority of pupils ; and rules and applications not needed in the body of the work. Indebtedness is gratefully acknowledged to school superinten- dents of vanous cities and towns for examination papers and valuable suggestions ; to the Metric Bureau for information and cuts ; to Boston Custom-House officials for copies of invoices ; to the Kegents of the University of the State of New York, and to various college authorities, for entrance test-papers to be found in the Appendix. Credit is due to all the able teachers who have aided, dur- ing the past three years, in the preparation of the Inductive Course, of which this book is a part. The larger share of this credit belongs to Mr. G. A. South worth, the successful and experienced Master of the Prescott Grammar School, Somer- ville, Mass., by whom much work has been done. His prac- tical knowledge of what both teachers and pupils need in a text-book has been of great advantage. The work is given to the public in the expectation that it will meet every reasonable requirement of our common schools and seminaries. CONTENTS. Notation and Numeration 1 Addition 11 Subtraction 20 Miscellaneous Exercises 26 Multiplication 27 Miscellaueous Exercises 36 Division 38 Miscellaneous Exercises 47 Review 49 Factors 54 Cancellation 56 Greatest Cornniou Divisor. . 57 Least Common Multiple 60 Miscellaneous Exercises 62 Common Fractious 63 Keduction of Fractions 65 Addition of Fractions 73 Subtraction of Fractions ... 74 Multiplication of Fractions. 76 Division of Fractions 81 Miscellaneous Exercises 89 Review 91 Decimal Fractions 95 Reduction of Decimals 97 Multiplication of Decimals. 100 Division of Decimals 101 Miscellaneous Exercises 104 United States Money 106 Aliquot Parts 109 Accounts and Bills 112 Weights and Measures 116 Length Measures 116 Surface Measures 117 Volume Measures 118 Capacity Measures 119 Weights 120 Time 121 Arc and Angle 123 Miscellaneous 124 Compound Numbers 126 Reduction 126 Addition 134 Subtraction 136 Multiplication 138 Division 138 Miscellaueous Exercises — 139 The Metric System 141 Length Measures 142 Surface Measures 143 Volume Measures 1 44 Capacity Measures 145 Weight Measures 146 Reduction of Units 148 Measurements 151 Surfaces 161 Volumes 154 Wood Measure 157 Board Measure 158 Miscellaneous Exercises... . 159 Review * 161 Percentage 167 Profit and Loss 173 Commission 175 Insurance 177 Miscellaneous Exercises — 178 Interest 181 Simple Interest 181 Exact Interest 189 VI CONTENTS. Problems in Interest 190 Partial Payments 194 Compound Interest 199 Discount 203 True Discount 203 Commercial Discount 205 Bank Discount 206 Miscellaneous Exercises 210 Stock Investments 212 Exchange 216 Domestic Exchange 217 Foreign Exchange 219 Average of Payments 222 Keview 225 Ratio and Proportion 228 Ratio 228 Proportion 230 Simple Proportion 232 Compound Proportion 234 Partnership 237 Involution and Evolution 241 Involution 241 Evolution 242 Square Root 243 Cube Root 249 Mensuration 256 Right-angled Triangles 256 Quadrilaterals 257 Prisms 260 Pyramids and Cones 260 Similar Surfaces 263 Similar Solids 264 Review 265 Examination Questions 273 Appendix. Roman Notation 308 Fundamental Processes 309 Prime Numbers 310 Circulating Decimals 310 Tables 312 Government Lands 313 Longitude and Time 314 Legal Interest 316 Twelve per cent Interest 317 Animal Interest 318 Aveiage of Accounts 321 Business Forms 323 Taxes 325 Duties 327 Measurement of Round Timber 329 Gauging 330 Tonnage of Vessels... 331 Farmers' Estimates 332 Stone and Brick Work 333 Builders' Estimates 335 Arithmetical Progression 337 Geometrical Progression 339 College Entrance-Examination Papers 341 Answers to Examples 347 THE COMPLETE ARITHMETIC. 1. A Unit is a single thing, or one ; as one hour, one dollar, one. 2. A Number is a unit, or a collection of units ; as one montli, four hours, seven. 3. Arithmetic treats of numbers and their use. NOTATION AND NUMERATION. 4. Notation is writing numbers in figures. 5. Numeration is reading numbers written in figures. 6. Figures are characters used to express numbers. Ten different figures are used ; they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Zero, One, Twc, Three, Four, Five, Six, Seven, Eight, Nine. 7. Zero, or cip/ieVy used alone expresses no units, oi nothing. The other nine figures express the number of units shown by their names. 8. To express numbers larger than nine, two or more figures are written side by side. 2 NOTATION AND NUMERATION. * 9. A figure written alone has only a simple name and value; but when used with other figures it has also a place-name and value. 10. The Place of a figure is its position with reference to another figure in a number. 11. When two figures are written side by side, the fig- ure at the right occupies the first jjlace, has the place-name ones, and expresses iinits of the first order. The figure at the left occupies the secoTul place, has the place-name tens, and expresses units of the second order, each of which is ten tinges larger than a unit of the first order. Thus, 80 is read 8 tens, ones, or briefly eighty. 75 " 7 tens, 5 ones, " seventy-five. By the use of the other possible combinations of two figures any number between nine and one hundred may be written. Eead the following : 1. 97 4. 84 7. 54 10. 48 2. 45 5. 77 8. 91 11. 89 3. 63 6. 38 9. 26 12. 98 Write in figures : 13. Eighty-six. 16. Thirty-six. 19. Eighty-one. 14. Sixty-four. 17. Four tens, eight ones. 20. Twenty-nine. 15. Seventy-nine. 18. Ninety-two. 21. Sixty-seven. 12. A figure at the left of the tens* figure occupies the third place, has the place-name hundreds, and ex- presses a third order of ttnits, each ten times as large as cue of those in the second place. Thus, 659 is read 6 hundreds, 5 ten^, 9 ones, or briefly, six hundred fifty-nine. 508 is read 5 hundreds, tens, 8 on^es, or briefly, five hundred eight. NOTATION AND NUMERATION. By using the different combinations of three figures any number between ninety-nine and one thousand may be expressed. Eead the following : 22. 789 25. 123 28. 656 31. 555 23. 456 26. 984 29. 700 32. 390 24. 804 27. 327 30. 608 33. 299 Write in figures : 34. Eight hundred forty-four. 38. Five hundred eighty-eight. 35. Two hundreds, seven ones. 39. Seven hundred. 36. Seven hundred fifty-three. 40. Six hundred sixty-six. 37. Nine hundred ninety. 41. Nine hundred four. 13. A figure at the left of the hundreds' figure occu- pies the fourth place, has the place-name thousands, and expresses the fourth order of units, each ten times as large as one in the third place. Thus, 8976 is read 8 thousands, 9 huridreds, 7 te7is, 6 ones, or briefly, eight thousand nine hundred seventy-six. In this way the various combinations of four figures may be used to express the numbers between nine hundred ninety-nine and te7i thousand, Eead the following : 42. 7382 45. 9308 48. 1333 43. 8641 46. 4629 49. 3004 44. 6047 47. 5432 50. 2908 Write in figures : 51. Six thousands, no hundreds, five te7is, seven ones. 52. Eight thousands, two hundreds, no tens, no ones. 53. How may the preceding number be read more briefly ? 54. Three thousand eight hundred forty-six. 55. Nine thousand two hundred seven. 4 NOTATION AND NUMEKATION. 56. Five thousand seven lumdred forty. 57. Two thousand eight. 58. Four thousands, four tens, four ones. 59. Eight ones, three tens, nine hundreds, seven thousands. 14. On the same principle larger numbers are writ- ^^en by using five, six, seven, eight or more figures, each additional figure filling a new place, named successively towards the left, ten-thousands, hundred-thousands, millions, ten-millions, and so on, and forming a new order of units, each of which is ten times as large as the next smaller. 15. The 2)lcice-names and values of figures are shown in the following TABLE. Place. Place-name. Order of Units. First, Ones, 1st, Second, Tens, 2d, Third, Hundreds, 3d, Fourth, Thousands, 4th, Fifth, Ten-thousands, 5th, Sixth, Hundred-thousands, 6th, Seventh, Mimons, 7th, Eighth, Ten-millions, 8th, Ninth, Hundred-millions, 9th, Tenth. Billions. lOLh. 16. For convenience in reading and writing numbers, each three places in a number, beginning with the ones, form a group. Each group is named from its right-hand order of units. The first group is the group of ones. " second " " " thousands, " third " " " niillions. " fotcrth " " " unions. The comma is used to mark oil' the groups. NOTATION AND NUMERATION. 5 Thus, in the number 109,876,543,210, the first group is 210 ones, the second group is 543 thousandSy the third group is 876 millions, the fourth group is 109 billions. 17. Each complete group contains hundreds, tens, and 07ies of its group-name. Thus, 847,298,341, instead of being read 8 hundred-millions, 4 ten-millions, 7 millions, etc., is more briefly read, eight hundred forty-seven millions, two hundred ninety-eight thousands, three hundred forty-one. EXERCISES. 60. In the number 321,654,987, what is the name of the first group ? Of the third ? Of the second ? 61. Give the place name of the 1. Of the 2. Of the 3. Of the 5. Of the 9. Of the 6. Of the 4. 62. What figure expresses the 4th order of units ? The 7th? The 5th? The 6th? The 8th? 18. The method of writing numbers in figures is based on the following Principles of Notation. 1. Ten units of any order equal one unit of the order next larger, 2. Each removal of a figure one place toward the left in- creases its value ten times, and each removal of a figure toward the right diminisftes its value ten times, 19. Our system of Numbers is called a Decimal System, from the Latin decem, ten, because of the uniform ten-fold increase of units from any order to the next larger. The successive orders of units in a number form a 6 NOTATION AND NUMERATION. Scale ; and, where ten units of any order always make a unit of the next larger, the scale is ten or decimal. 20. A period (.), called the decimal pointy may be writ- ten at the right of the ones figure to m'ark the ones' place. 21. A number at the left of the decimal point is an Integer, or whole number, because it is a collection of en- tire ones. 22. Figures may be written at the right of ones, if separated from them by the decimal point. 23. The first figure at the right of ones is in the first decimal place, has the place-name tenths, and expresses the first order of decimal units, each being one tenth of a one. A figure at the right of tenths is in the second decimal place, has the place-name hundredths, and expresses the second order of decimal units, each being one tenth of a tenth. A figure at the right of hundredths is in the third deci- mal place, has the place-name thousandths, and expresses the third order of decimal units, each being one tenth of a hundredth. Thus, 6.378 expresses six ones, and three tenths, seven hun- dredths, eight thousaTidtlis, or briefly six, and three hun- dred seventy-eight thousandtJis. Note. — In reading numbers use the conjunction and only after the ones to show the place of the decimal point 24. A number at the right of the decimal point is a Decimal, or a collection of tenths, hundredths, thousandths or other decimal parts of a one, I40TB. — In writing a decimal without an integer the ones' place may hA filled with a zero. Thus, Three hundred forty-one thousandths may be wntten 0.341. NOTATION AND NUMERATION. 7 25. The method of writing numbers, and the names of units, places, and groups are shown in the following TAJBLE. Integer. Decimal. ORDERS OF Units. 12th,Uth,10th, 9th, 8th, 7th. 6th, 5th, 4th, 3d, 2d, Ist, Ist, 2d. 3d. § Place-names. Figures. Groups. Group-names. c3 nfj ;^ ""g ■— { ^^ "T? rO ^ r^ 3 6 ,5 0,79 i 4th, Billious, 3d, 2d, Millions, Thousands, 5 < 1st, Ones, ^ 1 6 3 8 1st Decimal, Thousandths. The number in the table is, three hundred sixty-one billions, five hundred forty millions, seven hundred ninety- three thousands, one hundred fifty-four, and six hundred thirty-eight thousandths. Note 1. — Group-names above billions are trillions, quadrillions, quintillions, etc. ; below thousandths, millionths, billionths, etc. Note 2. — The left-hand group of the integer, or the right-hand group of the decimal may contain only one or two figures. The name of a partial decimal group is the same as the place-name of its right-hand figure. Thus, 16.84 is read sixteen and eighty-four hundredths. EXERCISES IN WRITING NUMBERS. 63. Write in figures, thirty-five million six hundred twenty-seven thousand two hundred three, and sixteen hun- dredths. Solution. — Writing 35 for the millions, 627 ot^ ^orr oAo 1 /» foi" tliB thousands, 203 for the ones, and .16 for the hundredtbs, we have as the result required, 35,627,203.16. 8 NOTATION AND NUMERATION. 64. Write in figures, three hundred twenty-nine thousand four hundred fifteen. 65. Write in figures, nine thousand seven hundred fifty- two, and one hundred one thousandths. 26. Rule for writing Numbers. Beginning at the lefty write the figures of each group in their order, filling vacant xjlaces and groups with ciphers. Write in figures: 66. Six thousand five hundred nine. 67. Eleven thousand nine hundred eleven. 68. Thirty-seven thousand four hundred eighty-nine. 69. Ninety thousand four hundred forty- four. 70. One hundred sixty-three thousand. 71. Two hundred twenty thousand two hundred sixty-two, 72. Five hundred seventy-four thousand three hundred thirty-five. 73. Seven hundred fifty-three thousand seven hundred fifty. 74. Five thousand four hundred eighty-nine. 75. Four hundred eighty thousand eight. 76. One million one hundred thirteen. 77. Three million three thousand thirty. 78. Nine hundred seven million eight hundred five thou- sand seventy-four. 79. Fifty-seven billion forty-four million ninety-three thou- sand eighty-three. 80. Thirteen million six hundred thousand one hundred seventy. 81. Five hundred six billion two million four thousand one. 82. One thousand seven hundred sixty-two, and five tenths, 83. Nine hundred ninety-four, and thirty-eight hundredths. NOTATION AND NUMERATION. 9 84. One hundred nineteen thousand, and one hundred nine- teen thousandths. 85. One million four hundred thousand three hundred, and thirty-three hundredths. 86. Sixteen thousand seven hundred forty-five, and eighty- six thousandths. 87. Three billions three millions two thousands seven hun- • dreds, and three thousandths. EXERCISES IN READING NUMBERS. 88. Write and r. ad 25362795. Solution. — 25362795 marked off into groups be- 25,362,795 comes 25,362,795. The first group from the right expresses 795 ones, or 795; the next group expresses 362 thousands ; and the third expresses twenty-five millions. The whole is read : twenty-five million three hundred sixty-two thou- sand seven hundred ninety-five. 89. Write and read 3257893. 90. Write and read 64058.109. Solution. — 64058.109 marked off into groups be- 64,058 109 comes 64,058.109, which is read sixty-four thousand fifty-eight, and one hundred nine thousandths. 91. Write and read 1673405.75. 92. Write and read 906005.805. 27. Rule for Reading Numbers. Beginning at the ones' place, mark off the numbers by com- mas, into as many groups as possible of three figures each. Begin at the left and read each group as if it stood alone, giving the group-name except to the ones' group. If there is a decimal, read it as if it were at the left of the 'point, and add the place-name of the last figure. 10 NOTATION AND NUMERATION. Copy and read : 93. 486 101. 6082 109. 0.31 94. 2385 102. 19009 110. 117.03 95. 7275 103. 163404 111. .064 96. 12361 104. 789685 112. 118646.5 97. 62004 105. 99999 113. 2567.02 9a 199 106. 1634562 114. 88.999 99. 78382 107. 25401300 115. 1158834.55 100. 160405 108. 312407981 116. 100600.789 QUESTIONS. N Art. 1. What is a unit ? 2. What is a number 1 3. What is arithmetic 1 4. What is notation ? 5. What is numeration ? 6. What are figures ? 7. What does zero, or cipher, express ? What do the other nine figures express ? 9. When has a figure a simple name and value ? When has it a place-name and value 1 10. What is the place of a figure? 11. If in the first place, what place-name has a figure? What units does it express? If in the second place, what place-name has a figure ? What units does it express ? 12. What place-name has a figure at the left of tens ? What units does it express ? 13. What place-name has a figure at the left of hundreds ? What units does it express ? 15. Give the place-names beginning with ones. Give the corre- sponding orders of units. 16. What form a group for convenience in reading and writing numbers ? How is each group named ? Name groups beginning with ones. What is used to mark off" the groups ? 17. What does each complete group contain ? 18. What are principles of notation ? 19. What is our system called ? What do the successive orders of units form ? 20. What is the decimal point ? For what is it written ? 21. What is the number at the left of the point ? 20. How arc numl)er8 written in figures? 27. How are numbers written in fi^cures read ? ADDITION. 11 ADDITION. 28. 1. James has 9 apples and Henry has 3. How many apples have both ? 2. Mary had 7 oranges and her brother gave her 5. How many had she then ? 3. A father gave to one of his children 8 cents, to another 5 cents, and to a third 6 cents. How many cents did he give them in all ? 4. How many are 9 and 3 ? 7 and 5 ? S, 5j and 6 ? 5. How many dollars are 8 dollars, 7 dollars, and 3 dol- lars ? 6 What is the unit of 8 dollars, 7 dollars, and 3 dollars ? 29. The Unit of a number is one of that number. Thus, One pound is the unit of 8 pounds, one quart is the unit of 27 quarts. 30. Like numbers are numbers having the same unit. Thus 3, 5, 7 ; and 6 cents, 4 cents and 5 cents, are like numbers. 31. Addition is finding a number equal to two or more given numbers. 32. The sum, or amount, is the result of an addition. 33. The sign of addition is +, named plus. It means more, and is generally read and. Thus, 4 + 5 + 6 is read four and five and six. 34. The sign of equality is = . It means equal, or equal to, and is often read are. Thus, 7 + 8 = 15 is read seven and eight are fifteen. 12 ADDITION. 35. The sign, $, written before a number, means dollars. Thus, $10 is read ten dollars. 36. CentSy expressed in figures, may occupy two deci- mal places, tenths and huThdredths, and mills one place, tJwusandths. Thus, $0.53, or $.53, is read fifty-three cents, and $28,005 is read twenty-eight dollars and five mills. ORAL EXERCISES. 7. How many dollars are $ 7, $ 6, and $ 2 ? 8. If you pay 10 cents for a slate, 9 cents for paper, and 3 cents for a pencil, how much do you pay for the whole ? 9. 12 boys are at play in one place, 6 in another, and 5 in another. How many boys are at plaj^ in all ? 10. John one day caught 8 trout, another day 9, and a third day 4. How many did he catch in the three days ? How many are : 11. 5 + 3 17. 2 + 3 + 7 23. 8 + 3 + 7 12. 8-1-5 18. 1 + 5 + 3 24. 9 -h 1 + 8 "13. 7 -h 2 19. 3 + 5 -I- 7 25. 7 + 6 + 5 14. 6 -h 6 20. 5 -f- 2 -f- 6 26. 8 + 2 + 3 15. 9 + 8 21. 4 + 1 + 8 27. 3 + 6 + 7 16. 11 + 7 22. 6 + + 9 28. 9 + 7 + 8 29. Add by 2's from to 24, naming only results. Solution. — 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24. Add: 30. By 2's from 1 to 25. 36. By 4's from 2 to 34. 31. By 3's from 2 to 29. 37. By 4's from 3 to 35. 32. By 3's from 3 to 30. 38. By 5's from 1 to 31. 33. By 3'8 from 4 to 31. 39. By 5's from 2 to 32. 34. By 4's from to 32. 40. By 5's from 3 to 33. 35. By 4'8 from 1 to 33. 41. By 5's from 4 to 34. ADDITION. 13 42. How many are 8 + 6 + 5? 5 + 6 + 8? 8 + 5 + 6? 6+8+5? 37. Principles of Addition. 1. Only like numbers^ and units of the same order can he added. 2. The sum is the same in whatever order the numbers are added. WRITTEN EXERCISES. 43. What is the sum of 142, 16, 201, and 410 ? ^ .^ Solution. — We write the numbers so that units of the same order may he in the same column. Beginning with ones, we add, naming results, ^^^ thus : 0, 1, 7, 9, and writing the 9 beneath in 410 ones' place. 769 Adding the tens, 1, 0, 2, 6, the sum, 6 tens, we write beneath in tens' place. Adding the hundreds, 4, 6, 7, the sum, 7 hundreds, we write be- neath in hundreds' place. The sum, then, is 7 hundreds 6 tens 9 ones, or 769. To test or prove the work we add the columns downward, and have, as before, the sum 769. 44. What is the sum of 121, 516, 361, ftnd 11 ? 45. What is the sum of 231 4- 114 + 324 ? 46. How many are 235 + 321 + 142 ? 47. Sold 416 bushels of corn to one man, 301 to another, and 42 to a third. How many bushels were sold in all ? 48. Paid L r labor $104, for boards $530, for timber $243, <»,nd for hardware $112. How much was paid for all ? 49. Bought a horse for $150, a carriage for $200, a harness for $45, and hay and grain for $104. What was the cost of the whole ? 50. Mr. Smith, in his will, gave to his son Arthur $500 ; to his son John $365; to his daughters $475, and to his brother $ 125. How much did he give them ill ? 14 ADDITION. ORAL EXERCISES. 51. In one basket there are 20 apples, in another 9, and in a third 7. How many are there in all ? Add: • 52. By 5's from 5 to 35. 58. By 7's from 1 to 43. 53. By 6's from to 36. 59. By 7's from 2 to 44. 54. By 6's from 1 to 37. 60. By 8's from 3 to 43. 55. By &s from 3 to 39. 61. By 8's from 4 to 44. 56. By 6's from 4 to 40. 62. By 9's from 1 to 46. 57. By 6's from 5 to 41. 63. By 9's from 2 to 47. 64. Add by 7's from 4 to 39; from 5 to 40 ; from 6 to 41. 65. Add by 8's from 1 to 41 ; from 2 to 42 ; from 5 to 45; from 6 to 46 ; from 7 to 47. 66. Add by 9's from 3 to 48 ; from 4 to 49 ; from 5 to 50; from 6 to 51 ; from 7 to 52. 67. Add by 7's from to 42 ; by 8's from to 48 ; by 9's from to 54. How many are : 68. 17 + 2 + 5 + 6 ? 70. 13 + 2 + 7 + 4 ? 69. 21 + 5 + 8 + 2 ? 71. 32 + 6 + 3 + 9 ? 72. What is the sum of 67 and 25 ? 73. If you should travel one day 20 miles and the next day 24 miles, how far would you travel in the two days ? 74. A man earned in one week $11, in another $16, and in a third $13. How much did he earn in all ? 75. A father gave to one of his sons 30 cents, to another 14 cents, and to his daughter 12 cents. How much did he give them in all ? 76. In one iield there are 55 acres, in another 40, and in the third 17. How many acres are there in the three fields ? ADDITION. 15 77. Johnson keeps on one farm 60 cows, on another 19, and on a third 21. How many does he keep on all ? WRITTEN EXERCISES. 78. What is the sum of 595, 961, and 23. 595 Solution. — We write the numbers so that units 961 of the same order may be in the same column. ^^ Beginning with ones, we add, thus : 3, 4, 9 ; 1579 sum, 9 ones, which we write beneath in ones* place. We add the tens, thus: 2, 8, 17; sum, 17 tens, which are 170, or 1 hundred 7 tens. We write the 7 tens beneath in tens' place, and add the 1 hundred with the hundreds in the next column. We add the hundreds, thus: 1, 10, 15 ; sum, 15 hundreds, which are 1500, or 1 thousand 5 hundreds. We write the 5 hundreds beneath in hundreds' place, and the 1 thousand in thousands' place. The sum is 1 thousand 5 hundred 7 tens 9 ones, or 1579. To prove the work, we add the columns downward, and have, as before, the sum 1579. 79. Add 626, 317, 529, and 12. 80. Add 368, 689, 73, and 19. 81. Add $ 21.642, $ 0.763, $ 3.05, and $ 5.90. $21,642 Solution. — Beginning with thousandths, we 0.763 ^^^> thus: 3, 6; sum, 5 thousandths, which we 2 QK write beneath in thousandths' place. K qrw We add the hundredths, thus: 5, 11, 15; sum, 15 hundredths, or 1 tenth 5 hundredths. We write $ ol.ooo ^jjQ 5 hundredths beneath in hundredths' place, and add the 1 tenth with the tenths of the next column. We add the tenths, thus: 1, 10, 17, 23; sum, 23 tenths, or 2 ones 3 tenths. We write the 3 tenths beneath in the place of tenths, and add the 2 ones with the ones of the next column. We add the ones, thus: 2, 7, 10, 11; the sum, 11 ones, or 1 ten 1 one. We write the 1 one beneath in the ones' place, and add the 1 ten with the tens of the next column. We add the tens, thus: 1, 3; sum, 3 tens, which we write beneath in tens' place. The sum is | 31.355, or 31 dollars and 35 cents 5 mills. 16 ADDITION. 82. What is the sum of $31,067, $8,091, $0.46, and $ 0.31 ? 83. What is the sum of $13,615, $24.25, $6.14, and $ 17.66 ? 84. What is the sum of $91.55, $82.35, $63, and $ 80.50 ? 38. Rule for Addition. Write the numbers so that units of the same order may be in the same column. Add the right-hand column, writing the units of the sum beneath, and adding the tens, if any, to the next column. So 2)roceed with all the columiis, writing the entire sum of the last column. 39. Proof. — Add the numbers a second time and in a different order (Art. 37). 85. 86. 87. 88. 346 800 1123 5050 275 455 678 7825 62 619 642 1367 6 104 53 1243 89. 90. 91. 92. 315 6541 841.5 1.3.14 640 1809 302.6 21.65 179 723 427.5 82.91 106 480 122.4 11.78 812 235 324.7 14.43 93. 94. 95. 96. % 53.19 $93.47 % 103.64 * 111.625 61.25 6.80 205.75 41.55 16.87 0.39 83.91 76.875 X* ADDITION. 17 97. Find the sum of 406, 781, 918, 846, and 67. . 98. Find the sum of 1081, 686, 423, 925, and 328. 99. Find the sum of 2642, 3417, 506, 689, and 482. 100. Find the sum of 391, 638, 402, 83, 493, and 16. 101. Find the sum of 555, 64, 380, 75, 878, and 74. 102. Find the sum of 159, 363, 4682, 8405, and 3377 How many are : 103. 62.48 + 31.41 + 8.95 + 73 + 0.56 ? 104. 8.15 + 31.68 + 9.60 + 18.53 + 85.93 ? 105. 106 + 57.25 + 46.17 + 9.36 + 62.94 ? 106. 3.19 + 11.93 + 61.85 + 376 + 4781 ? 107. $ 16.43 + $ 24.77 + $ 75.35 + $ 9.10 ? 108. $ 342 + $ 164 + $ 4.95 + $ 0.74 ? 109. $ 78.05 + $ 72.09 + $ 8.11 + $ 9.83 ? 110. $415 + $88.24 + $13.08 + $16.08? 111. On a farm 105 trees bear pears, 492 bear peaches, 85 bear cherries, and 316 bear apples. How many trees are there in all ? 112. An army consisted of 2358 infantry, 868 cavalry, and 1165 artillery. What was the number of the army ? 113. A man bought a house for $ 8750. He paid $36355 lov repairs, $95.63 for painting, and $ 106.50 for taxes. For how much must he sell it to gain $ 350 ? 114. A farmer has four fat oxen. The first weighs 1463 pounds, the second 1385 pounds, the third 1507 pounds, and the fourth 1264 pounds. What is the weight of them all ? 115. Bought a suit of clothes for $42.50, an overcoat for $ 22.75, a pair of boots for $ 5.25, and a hat for $ 4.63. What was the cost of the whole ? 116. The area of France is 204091 square miles and that of Italy 114290. What is the area of the two countries ? 117. What is the sum of $103, $0.06, $15.05, $19.75, and $7.31? 2 18 ADDITION. 118. Gave $ 73 for a watch, $ 15.50 for a carriage robe. $ 250 for a horse, and sold them so that I gained $ 21.50. What did I sell them for ? 119. A merchant began business with goods worth $ 6750 ; a store worth $ 5700 ; fixtures worth $ 555.25 j and gained 1 1165.45. What was he then worth ? 120. 1,21. 122. 123. 321 8106 31.46 93.045 406 7334 42.47 10.304 718 5570 60.05 7.105 304 2344 63.06 16.321 818 648 71.85 4.554 103 102 40.25 17.056 145 341 14.40 2.005 124. Mount Everest is 13270 feet higher than Mont Blanc, Mont Blanc 4832 feet higher than Etna, and Mount Etna rises 10900 feet above the level of the sea. What is the height of Mount Everest above the sea level ? 125. The area of the United States consists of territory ceded by Great Britain as the result of the Revolution, 815615 square miles ; acquired from France, 930928 square miles ; from Spain, 59268 square miles; by admission of Texas, 237504 square miles; Oregon by treaty, 280425 square miles ;- from Mexico, 677262 s<|uare miles ; and from Russia, 577390 square miles. What u the total area of the United States ? 126. In building a cottage, the excavating cost $ 34 ; the cellar walls, $ 110,50 ; chimneys, $ 18.46 ; the plastering, $73.42; the frame, $64.50; the boarding, $33.50; the sid- ing, «5 25; tli^ roof boards, $20.67; the shingling, $62.80; the gutters and hardware, $ 42 ; ti)e truss work, $13; the water tank, $ 15 ; the cornices, $ 46 ; the windows, $ 98 ; the doors, $126; tlie flooring, $45; the stairs, $20; the base, *34.40; the sink, $9.50; the cistern, $25; the painting, $80.12; and incidentals, $50. What was the entire cost? ADDITION. 19 X^127. Bought a house for $23650 and land for $73640. Paid $ 4500 for repairs and taxes. For house and land how- much must I receive to gain the' cost of the land ? 128. The first of four numbers is 8437, the second 9325, the third is the sum of the first two, and the fourth is the sum of the second and third. Find the result if the numbers are T)ut together. 129. A, B, C, and D go into business. A puts in $7430, B $ 3200 more, C as much as A and B together, and D as much as A and C together. What is the capital of the firm ? 130. In 1880 the Boston and Providence R. B. received from passengers $ 776362.87, from freight $486724.85, from rents $ 19395.08, from express companies $ 30202.34, and from mails $ 11240.49. What was the total income for the year ? 131. Find the sum of the numbers in Art. 26, Exercises 66 to 74 inclusive. 132. In Exercises 73 to 81 inclusive. 133. In Exercises 81 to 87 inclusive. 134. In Art. 27, Exercises 93 to 100 inclusive. 135. In Exercises 101 to 108 inclusive. 136. In Exercises 109 to 116 inclusive. 137. What are my sales for the week if my daily sales are as follows: Monday $347.19, Tuesday $847.62, Wednesday $9643.27, Thursday $9876.50, Friday $843.91, Saturday $10986.75? 138. What is the income of a gentleman who receives annually from rents $ 2465.29, from mining profits $ 3462. from other business $9478.50, and from interest of U. S. Bonds $8600? QUESTIONS. 29. What is the unit of a number ? 30. What are like numbers ? 31. What is addition ? 32. What is the sum ? 33. What is the sign of addition ? 34. The sign of equality 1 35. The dollar sign? 37. What are principles of addition ? How do you add ? What ts the proof ? 20 SUBTRACTION. SUBTRACTION. 40. 1. John has 8 marbles, and his brother 5. How many more marbles has John than his brother ? 2. Thomas had 9 apples, and gave 4 of them to Peter How many had he left ? 3. How much more are 11 cents than 6 cents ? 4. How many dollars are $ 13 less $ 7 ? 5. Ella is 14 years old, and Mary is 9 years old. How many years older is Ella than Mary ? 6. Sold a knife for 15 cents and a ball for 7 cents. How much more did I get for the knife than for the ball ? 41. Subtraction is taking one of two like numbers from the other. 42. The Difference is the result of a subtraction. 43. The Minuend is the number subtracted from. 44. The Subtrahend is the number subtracted. 45. The Sign pf Subtraction is — , named minus. It means less. Thus, 15 — 9 = 6 is read fifteen less nine are six. ORAL EXERCISES. 7. A boy raised 13 melons and sold 6. How many had he left? 8. Arthur had 11 cents and gave away 7. How many had he left? 9. In a nest there were 14 eggs, but 5 have been taken away. How many remain in the nest ? 10. In a pool were 13 lilies, and 8 have been carried away. How many remain ? 11. If of 16 peaches 8 should be eaten, how many would be I(;ft ? SUBTRACTION. 21 How many are : 12. 16-9 17. 27-> 9 22. 18-9 26. 21-5 13. 15-7 18. 19-10 23. 25-6 27 24-6 14. 11-5 19. 26-8 24. 19-9 28. 20-7 15. 13-8 20. 17-3 25. 23-8 29. 28-9 16. 14-6 21. 12—7 30. Subtract by 2's from 40 back to 20, naming only re- sults. 31. By 3's from 38 to 24. 37. By 5's from 58 to 28. 32. By 4's from 37 to 21. 38. By 4's from 56 to 40. 33. By 5's from 51 to 26, 39. By 7's from 57 to 36. 34. By 6's from 49 to 31. 40. By 6's from 54 to 30. 35. By 7's from 50 to 29. 41. By 8's from 53 to 31. 36. By 8's from 48 to 32. 42. By 9's from 49 to 22. 43. How many are $ 16 less $ 9 ? $ 7 and how many dol^ lars are $ 16 ? 46. Principles of Subtraction. 1. Only like numbers and units of the same order can he subtracted one from the other, 2. The difference and subtrahend together must equal the minuend. WRITTEN EXERCISES. 44. Find the difference between 865 and 242. Minuend 865 Solution. — For convenience, we Subtrahend 242 ^^^^^ *^^ subtrahend under the min- uend, so that units of the same order Difference 623 ^^^ ^^ ^^ ^^^ ^^^^^ ^^1^,^^^^ Proof 865 2 ones from 5 ones leave 3 ones, which we write beneath in ones' place ; 4 tens from 6 tens leave 2 tens, which we write beneath in tens' place; and 2 hundreds from 8 hundreds leave 6 hundreds, which we write beneath in hundreds' place. The difference is 6 hundreds 2 tens 3 ones, or 623. SUBTRACTION. 23 73. John travels 35 miles a day and Edwin 60 miles. How many more miles does the one travel than the other ? 74. A boy had 51 cents and spent 38. How many cents had he left ? WRITTEN EXERCISES. 75. What is the difference between 1624 and 342? Minuend 1624 Solution. — 2 ones from 4 ones Subtrahend 342 ^^^^^ ^ ones, which we write be^ neath in the place of ones. Difference 1282 ^ ^^^^ ^^^^^^^ ^^ ^^^^^ ^^^^ 2 Proof 1624 tens ; we therefore take 1 hundred, or 10 tens, from the 6 hundreds of the minuend, leaving 5 hundreds, and adding the 10 tens to the 2 tens we have 12 tens ; 4 tens from 12 tens leave 8 tens, which we write beneath in the place of tens. 3 hundreds from 5 hundreds leave 2 hundreds, which we write in hundreds' place ; no thousands from 1 thousand leaves 1 thousand, which we write in thousands' place. The difference is 1282. This we prove by adding the difference and subtrahend, and finding the sum to be equal to the minuend. 76. Find the difference between 167 and 476. 77. Find the difference between 389 and 581. 78. What number and 1543 make 1735 ? 79. Subtract 32.15 from 78.8. Minuend 78.80 Solution. — We make the deci- Subtrahend 32.15 ^^1 places the same in the two numbers by filling the place of Difference 46.65 hundredths in the minuend by 0. 5 hundredths cannot be taken from hundredths; we therefore take 1 tenth, or 10 hundredths, from the 8 tenths, leaving 7 tenths; 5 hundredths from the 10 hundredths leave 5 hundredths, which we write beneath in hundredths' place. 1 tenth from 7 tenths leaves 6 tenths, which we write beneath in tenths' place ; and subtracting the ones and tens, and writing the re- sult beneath, we have 46.65 as the result required. 24 SUBTRACTION. 80. Subtract 167.807 from 970.96. 81. Take $158.55 from $549.60. 82. How much more than $91.97 is $147.11 ? 83. How much and $ 6711.45 make $ 7050.25 ? 47. Rule for Subtraction, Write the subtrahend under the minuendj placing units of the same order in the same column. Begin with the units of the lowest order to subtract, and proceed to the highest, W7'iting the result beneath. If any order of the minuend has less units than the same order of the subtrahend, increase its units by ten, and subtract ; consider the units of the next minuend order one less, and pro- ceed as before. 48. Proof. Add the subtrahend and the difference together ; the sum should equal the minuend (Art. 46). 84. 85. 86. 87. From 8647 5375 6365 9406 Take 3451 _406 4506 8350 88. 89. 90. 91. From 31867 12805 148.48 63.859 Take 1905 9264 92.09 49.608 92. 93. 94. 95. From $63.95 $85.69 $96.70 $182.05 57.68 1.73 0.89 152.06 96. Find the difference between 34000 and 21345. (8X9X9X10) Solution. — There being ones, tens, 3 4 hundreds in the minuend, we take 1 of the 4 213 4 5 thousands (leaving 3 thousands), or 10 hun- dreds; then taking 1 of the 10 hundreds (leav- 12 6 5 5 iiig 9 hundreds), or 10 tens ; and 1 of the 10 tens (leaving 9 tens), or 10 ones; the minuend SUBTRACTION. 25 may then be considered 3 ten-thousands 3 thousands 9 hundreds 9 tens and 10 ones. Taking from the changed minuend the 2 ten-thousands 1 thousand 3 hundreds 4 tens 5 ones of the subtrahend, we have as the difference required, 12655. Subtract : 97. 5544 from 40000. 103. 67.055 from 73.607. 98. 180 from 98000. 104. 19.55 from 831.50. 99. 12453 from 35421. 105. 444.5 from 10060. 100. 97 from 10000. 106. 921.56 from 1000. 101. 58346 from 67500. 107. $13.63 from $500.20. 102. 9999 from 10000. 108. $ 127 from $ 1963.75. 109. What is the value of 83956 - 78415 ? 110. What is the value of 60440 - 33457 ? 111. What is the value of 109800 - 98799 ? 112. From one hundred nine take one, and nine hun- dredths ? 113. America was discovered in 1492 ; how many years from that date to 1881 ? 114. A man began business with $1760, and after two years had $ 2500.75. How much had he gained ? 115. The number of regular soldiers furnished by the sev- eral States in the war of the Revolution was 231771 ; of these a single State furnished 67907. How many were furnished by other States ? 116. A merchant bought goods to the amount of $ 7563.56;, and sold them for 1 11630.50. How much did he gain ? 117. The Atlantic slope contains 967576 square miles, and the Mississippi valley 1237111 square miles. How much does the latter exceed the former ? 118. The population of Kew York City in 1870 was 942192, and in 1880 was 1209561. How much was the gain ? 119. The product of gold in a certain year was from Nevada $19546516, and from California $17760679. How much greater was the product from Nevada than from California ? 26 SUBTRACTION. MISCELLANEOUS EXERCISES. 120. A man owing $ 767.50 paid one time $ 190, at another 8 131, and at a third $ 155.25. How much did he then owe ? 121. James Dow's real estate is valued at $ 3769, and his personal estate at $2648.75. He owes Job Smith $1728 and Abraham Tyler $ 1161.93. How much is he worth wher these debts are paid ? 122. Modern notation in arithmetic was introduced from Arabia into Europe in the year 991, algebra from the same country in 431 years later, and decimal fractions were invented in 1602. Required the number of years from each to 1882. 123. Sydney has $ 178.50, Albert $ 75.75 more than Sydney, and Charles has as much as Sydney and Albert less $ 80.93. How much more has Charles than Sydney ? 124. What is the value of 1645 + 635 + 416 - 1314 ? 125. Two vessels 4563 miles apart start to meet each other. When one of them has made 1575 miles of the distance and the other 1658 miles, how far are they apart ? 126. A man whose property was $ 50675 gave his son Edwin $ 8555.50, his son Eobert $7000, his daughter Mary $ 9563.75, his wife $ 20000, and a public library the remainder. How much was given the public library ? 127. The population, in 1880, of Philadelphia was 847542 ; of Boston, 362535; and of Providence, \04760. How much greater was the population of Philadelphia than that of Boston and Providence ? 128. A man who had $ 1250 in a savings bank took out $ 51.75 at one time, $ 84.93 at another, and $ 267 at another. He then put in $ 185. IJow much then had he in the bank ? QUESTIONS. 41. What is Bubtraction ? 42. What is the difference ? 43. The minuend? 44. The subtrahend ? 45. The sign of subtraction ? 46. What are principles of subtraction ? 47. How do you sub- tract ? If any order of the minuend has less units than the same order of the subtrahend, what is done ? 48. What is the proof of Bubtraction? MULTIPLICATION. 27 MULTIPLICATION. 49. 1. If a man can earn 8 dollars in a week, how many dollars can he earn in 4 weeks ? How many dollars are 4 times 8 dollars ? 2. What will 5 quarts of cherries cost at 10 cents a quart ? 3. William has 9 apples and Alfred 7 times as many. How many has Alfred ? 7 times 9 are how many ones ? 50. Multiplication is taking one number as many times as there are ones in another. 51. The Multiplicand is the number taken or multi- If^lied. 52. The Multiplier is the number that shows how many times the multiplicand is taken. 53. The Product is the result of a multiplication ; and the factors of a product are the numbers multiplied to- gether to produce it. 54. The Sign of Multiplication is x. It means multi- plied by, or tiincs. Thus, 7x5 may be read seven multiplied by five, or seven times five. 55. A Concrete Number is a number in which some kind of unit is named. Thus, 2 books, 3 days, $ 7, are concrete numbers. 56. An Abstract Number is a number in which no par- ticular kind of unit is named. Thus, 2, 5, 7 are abstract numbers. 28 57, 1X1 = 1 1X2 = 2 1X3 = 3 1X4 = 4 1X5 = 5 1X6 = 6 1X7 = 7 1X8 = 1X9 = 8 9 2 X 9= 18 3X9 = 27 4X9 = 36 1 X 10 = 10 2 X 10 = 20 3 X 10 = 30 4 X 10 = 40 1 X 11 = 11 2 X 11 = 22 3X 11 = 33 4 X 11 = 44 1 X 12 = 12 2 X 12= 24 3 X12 = 36 4 X 12 = 48 5X1 = 5 6X1= 6 7X 1 = 7 8X1 = 8 5X2 = 10 6 X 2= 12 7X2 = 14 8X2 = 16 5X3 = 15 6 X 3= 18 7X3 = 21 8X3 = 24 5X4 = 20 6 X 4= 24 7X 4 = 28 8X4 = 32 5X5 = 25 6 X 5 = 30 7X5 = 35 8X5 = 40 5X6 = 30 6 X 6 = 36 7X6 = 42 8X6 = 48 5X7 = 35 6 X 7 = 42 7X7 = 49 8X7 = 56 5X8 = 40 6 X 8 = 48 • 7X8 = 56 8X8 = 64 5X9 = 45 6 X 9 = 54 7X9 = 63 8X9 = 72 5 X 10 = 50 6 X 10= 60 7 X 10 = 70 8 X 10 = 80 5 Xll = 55 6 X 11 = 66 7 X 11 = 77 8 X 11 = 88 5 X12 = 60 6 X 12= -72 7 X 12 = 84 8 X 12 = 96 9X1 = 9 10 X 1 = 10 11 X 1 = 11 12X1 = 12 9X2 = 18 10 X 2 = 20 11 X 2 = 22 12X 2 = 24 9X3 = 27 10 X 3 = 30 11 X 3 = 33 12 X 3 = 36 9X4 = 36 10 X 4 = 40 11 X 4 = 44 12 X 4 = 48 9X5 = 45 10 X 5= 50 11 X 5 = 55 12 X 5 = 60 9X6 = 54 10 X 6= 60 11 X 6 = 66 12 X 6 = 72 9X7 = 63 10 X 7 = 70 11 X 7 = 77 12 X 7 = 84 9X8 = 72 10 X 8= 80 11 X 8 = 88 12 X 8 = 96 9X9 = 81 10 X 9= 90 11 X 9 = 99 12 X 9 = 108 9X10 = 90 10 X 10 = 100 11 xio = 110 12X 10 = 120 9 Xll = 99 10 X 11 = 110 11X11 = 121 12X 11 = 132 9 X12 = 108 10 X 12 = 120 11 X 12 = 132 12 X 12 = 144 MULTIPLICATION. 29 ORAL EXERCISES. 4. How many wings have 8 doves ? 5. How much can you earn in 3 days if you earn 10 cents a day ? 6. At 9 cents each what will a half-dozen writing-books cost ? 7. How many horns have 8 yoke of oxen ? 8. How many days in 12 weeks ? In 9 weeks ? 9. How far can you ride in 7 hours at the rate of 8 miles an hour ? 10. How many desks in a school-room having 8 rows with 7 desks in a row ? 11. How many legs have 6 span of horses ? 12. At 7 cents each what cost a dozen pencils ? 13. A ten-foot pole is how many inches long ? How many are: 14. 15. 16. 17. 7X9 6x12 2x 7 6x 7 8X8 8x10 10X12 12x12 9x6 6x31 8X7 11 X 8 7X7 9x12 9X 4 9x 7 8x3 8x10 18. 11X11 19. 4X 7 12 X 9 -8x12 6x 9- 7x 7 6x 6 -4x 7 8x 6- 4x10 7x 8 -5x 9 7X12- 9- 9 6x 7 -6x 6 12 X 11 - 10x10 9X X8x 4 12 X 6- 8x 8 20. Give the products by 2, from 2 X 2 to 2 X 12. Solution. — 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24. 2L Give the products by 4, from 4 X to 4 X 12. 22. Count by 7's from to 84. 23. Subtract by 9's from 108 to 0. 30 MULTIPLICATION. 24. Name all the products of which 12 is a factor to 144. 25. What cost 9 six-cent hooks ? 12 ? 8 ? 26. What will 12 dozen eggs cost at a cent apiece ? 27. How many days in a 9 weeks' vacation ? 28. How many corners have 11 squares ? 29. I pay 7 cents daily for milk. What is my milk bill for a week ? 30. How many inches in 8 feet ? In 9 ? In 11 ? 31. How many feet in 12 yards of ribbon ? 32. What cost 7 lbs. at 5 cents a pound, and 5 quarts at 4 cents a quart ? 33. What numbers multiplied make 36 ? 56? 81? 96 ? 34. What factors produce 48 ? 72 ? 121 ? 63 ? 49 ? 35. At $ 8 a barrel, how many times $ 8 will 9 barrels of flour cost ? 36. How many are9x$8? 8x$9? 3x2x2? 2 K3x2?2X2x3? 58. Principles of iVIultiplication. 1. The multiplier is always considered an abstract number. 2. The product and the multiplicand are always like num- bers, 3. The product is the same whatever the order of the factors, WRITTEN EXERCISES. 37. Find the product of 274 multiplied by 7. , Ti*- 1,. -1. 1 r^^A Solution. — We write T2, , ( Multiplicand 274 ^, ,,. ,. ^ Factors < ^ the multiplier, 7 ones, ( Multiplier 7 ^^^^^ ^^^ ^^^g, ^^^^.^ ^^ Product 1918 the multiplicand. Beginning with the onee, we multiply : 7 times 4 ones are 28 ones, or 2 tens 8 ones. We write the 8 ones beneath in the place of ones, and reserve the 2 tens. 7 times 7 tens are 49 tens, and 49 tens plus the 2 tens reserved are 51 tens, or 5 hundreds 1 ten. We write the 1 ten beneath in tens' place, and reserve the 5 hundreds. MULTIPLICATION. 31 7 times 2 hundreds are 14 hundreds, and 14 hundreds plus 5 hun- dreds reserved are 19 hundreds, or 1 thousand and 9 hundreds. We write the 9 hundreds beneath in hundreds' place, and the 1 thousand in thousands' place. The product is 1 thousand 9 hundreds 1 ten 8 ones, or 1918. 38. 39. 40. 41. Multiplicand 756 4567 1109 6201 Multiplier _J 3 _5_ 8 42. 43. 44. 45. Multiplicand 3416 2608 12345 24301 Multiplier 6 ^_2 7 9 46. What is the product of 64.25 by 5 ? ^^ - . .. T nA r,r \ SoluHon, — 5 tiuies 5 huu- Multipiicand d4.Jo t .^. or i. j i^i, ^ dredths are 25 hundredths, ^^l^iP^^^^ . ^ or two tenths and 5 hun- Product 321.25 dredths. We write the 5 hundredths beneath in hun- dredths' place and reserve the 2 tenths. 5 times 2 tenths are 10 tenths, and 10 tenths plus the 2 tenths re- served are 12 tenths, or 1 one and 2 tenths. We write the 2 tenths beneath in tenths' place, and reserve the 1 one. 5 times 4 ones are 20 ones, and 20 ones plus the 1 one reserved are 21 ones, or 2 tens and 1 one. We write the 1 one beneath in ones' place, and reserve the 2 tens ; and so on. The multiplier being an integer, the product has as many decimal places as the multiplicand. Multiply By 47. 35.07 8 48. 6.135 5 49. 1635.9 2 50. 7128.53 7 Multiply By 51. $ 124.35 4 52. $ 192.547 6 53. $ 823.50 8 54. $8537.64 9 32 MULTIPLICATION. ORAL EXERCISES. 55. How much will 12 coats cost at $ 9 each ? 56. When plows are $ 11 each, what will eight cost ? 57. If 7 men can do a piece of work in 9 days, in what time will 1 man do the same work ? 58. If a vessel sails at the rate of 12 miles an hour, how fai* will it sail in 12 hours ? 59. What will 11 barrels of flour cost at $ 10 each ? 60. How many horses will consume in one day as many bushels of oats as 11 horses consume in 12 days ? What is the value of 61. 4 X 3 X 6 ? 67. 12 X 10 + 9 ? 62. 8 X 7 + 9 ?, 68. 11 X 11 - 12 ? 63. 6 X 4 - 10 ? 69. 11 X 6 - 10 ? 64. 9 X 5 - 8 ? 70. 11 X 10 - 12 ? 65. 6 X 5 + 12 ? 71. 10 X 7 - 11 ? 66. 3 X 4 X 10 ? 72. 12 X 8 + 10 ? 73. Give the product of the following numbers multiplied by 12 ; by 11 ; by 5 ; by 4. 9, 11, 3, 5, 12, 8, 2, 7, 10, 6, 4. 74. Multiply the numbers of the preceding exercise by 8 ; by 9 ; by 7 ; by 6, and add 3 to each product. 75. How many are 6 times 24 ? Solution. — 24 is 20 + 4 ; 6 times 20 are 120; 6 times 4 are 24; hence 6 times 20 + 4, or 24, are 120 + 24, or 144. 76. At 26 cents a yard, how many cents will 7 yards of cloth cost ? 77. At the rate of 23 miles an hour, how many miles will a train of cars move in 8 hours ? 78. What will 9 suits of clothes cost at $ 30 a suit ? 79. What will 5 sewing-machines cost at $ 50 each ? MULTIPLICATION. 33 WRITTEN EXERCISES. 80. What is the product of 534 and 242 ? Solution. — We write the factors so 534 242 Multiplicand . . . 242 , Multiplier 534 1068 2136 1068 . 129228 Partial Products Product . 129228 that the right-hand figure of each shall stand in the same column. Multiplying by the 2 ones, we have 1068 ones as the first par- ^ tial product; multiplying by the 4 tens, we have 2136 tens for the second partial product, which we write so that its right-hand figure shall come in the tens* column ; multiplying by the 2 hundreds, we have 1068 hundreds for the third partial product, which we write so that units of the same order shall come in the same column ; adding the three partial products, we have 129228 as the product required. To prove the work, since the product is the same whatever the or- der of the factors, we multiply the 242by the 534, and have, as before, 129228. 81. 82. 83. 84. Multiply 763 1345 i06 1621 By 37 45 26 34 85. 86. 87. 88. Multiply 134.7 17.58 3.049 $ 25.75 By 86 285 329 703 Note. — In solution of Ex. 88, there being Otens in the multiplier, we pass to the hundreds of the nmltiplier. 59. Rule for Multiplication. Write the multiplier under the multiplicand, with a line beneath. Begin7ilnfj at the right, multiply each figure of the multipli- cand by each figure of the multiplier successively, placing the 34 MULTIPLICATION. right-hand figure of each partial product under the figure of the multiplier that produced it. Add the partial products, and from the right of the result 'point off as many decimal places as are found in both factors. 60. Proof. See if the same result is obtained by multiplying the multiplier by the multiplicand. Multiply : 89. 347 by 769. 104. $ 817.42 by 358. 90. 826 by 243. 105. 8.439 by 125. 91. 90.4 by 85. 106. 86491 by 688. 92. $ 32.13 by 91. 107. 49382 by 294. 93. 456.7 by (SS. 108. $ 887.95 by 761. 94. 8.901 by 542. 109. 4963 by 845. 95. $ 23.45 by 397. 110. $ 28.59 by 927. 96. 6789.0 by 645. 111. 938.42 by 347. 97. $ 198.06*by 805. 112. 61904 by 869. 98. 45.32 by 907. 113. $329.87 by 35. 99. 982.4 by 3004. 114. 42935 by 942. 100. $ 60.51 by 768. 115. $ 864.23 by 346. 101. 87.35 by 94. 116. 84917 by 809. 102. $ 80.42 by 832. 117. 173.24 by 935. 103. 30.69 by 907. 118. $ 98.983 by 871. 119. On board of a steamer there are 163 barrels of sugar, each weighing 295 pounds. What is the weight of the whole ? 120. The factors of a product are 1468 and 87 ; what is the product ? 121. The factors of a product are 681, 507, and 12 ; what is the product ? .122. The multiplicand is 804.51, and the multiplier 63; what is the product ? 123. What number = 914 08 X 64 ? 124. At $ 82.50 an acre, what is the value of 25 acres oi laud ? MULTIPLICATION. 36 125. What is the product of five thousand four hundred fourteen, and fifteen thousandths, multiplied by 38 ? 126. 8304.5 X 77 = what? 127. 7038.61 X 126 = what ? 128. 824.84 X 424 = what ? 129. $ 62.005 X 91 = what ? 130. $ 47.168 X 208 = what ? 131. $ 617.43 X 355 = what ? • 132. How many bushels of wheat can be raised on 5634 acres at the rate of 47 bushels per acre ? 133. Multiply nine hundred sixty-five, and thirteen hun- dredths, by three thousand seven hundred five ? The multiplier a number of tens, hundreds, etc. 134. Multiply 48 by 10 ; by 100. 4g 43 Solution. — 10 forty-eights is the j^Q ii^QQ same as 48 tens (Art. 58), or 480. —— : Also, 100 forty-eights is the same 480 4800 ^g ^g himdreds, or 4800. 135. Multiply 427 by 10, by 100, and by 1000, and add the results. 136. How many are a thousand times 7854 ? 138 ? "What is the product of $ 55.56 multiplied by 100 ? ^ Pjw ^r* Solution. — The removal of a figure one place to the left in a number increases the value expressed ten-fold (Art. 18). Hence, $ 5556.00 ^e multiply $ 55.56 by 10 by removing the decimal point one place to the right, and by 10 X 10, or by 100, by removing the point two places to the right. 137. Multiply 97 by 600. Solution. — 600 is 100 times 6 ; 600 times 97. is the same as 100 times 6 times 97. 6 __— times 97 are 582, and 100 times 6 times 97 58200 are 100 times 582, or 58200. That is, — 36 MULTIPLICATION. 61. To multiply by a number of tens, hundreds, etc. Midtiply without regard to the ciphers at the right of the multiplier, annex that number of ciph&rs to the product, and give it as many decimal figures as the multiplicand has. Multiply : 138. 814 by 1000. 143. 3450.9 by 800. 13a 3921 by 70. 144. $ 1604.05 by 2000. 140. 54.78 by 9000. 145. 446.008 by 4600. 141. 1342 by 450. 146. 33300 by 820. 142. $ 611.31 by 110. 147. $77880 by 300. 148. If the earth moves about the sun at tbe rate of 68000 miles an hour, how far does it move in 240 hours ? 149. A square mile is 640 acres. How many acres has a State whose area is 7800 square miles ? MISCELLANEOUS EXERCISES. 150. Bought 20 barrels of flour at $ 8.50 a barrel, 8 tons of fine feed at $19.50 a ton, and oats for $63.25, and gave in payment a $ 500 bank-bill. How much should be paid back ? 151. In a certain battle an army lost 416 killed and 3 times IS many wounded. The enemy's loss was 5 times as many. What was the entire loss of killed and wounded in the battle ? 152. A clerk had a salary of $ 1500 for a year of 52 weeks. His board cost him $4.25 a week ; he wasted for cigars and liquor $ 1.30 a week ; and his other expenses were for the year % 150. How much did he save ? How much could he have \j saved if he had avoided the cigars and liquor ? 153. Two trains of cars leave Boston at the same time on the same railroad for the West. One goes at the rate of 31.50 miles an hour, and the other at the rate of 16.25 miles an hour. How far apart will the two trains be at the end of 48 hours ? M MULTIPLICATION. 37 154. A drover has bought 120 fat oxen, each weighing on an average 1360 j)ounds. What is the weight of the whole ? ^ I 155. In a school of 56 pupils there are 29 boys whose aver- age weight is 85 pounds. The girls average 79 pounds each. What is the weight of the school ? 156. A merchant bought 130 yards of cloth at $ 3.75 a yard^ and sold the whole for $ 573.95. How much did he make ? 157. Multiply the sum of 842 and 796 by twice their differ- ence. 158. When boards are $31.50 a thousand feet, and shingles $ 4.25 a tlxousand, what will 40 thousand feet of boards and 22 thousand of shingles cost ? 159. James Hudson has a house worth $ 2500, another worth $ 1900, and- 60 acres of land worth $75 an acre. How much more is the land worth than the two houses ? 160. Bought 17 yards of silk at $ 2.75 a yard, 114 yards of carpeting at $ 1.80 a yard, and $ 17 worth of lining. What was the amount of the bill ? 161. A drover bought 60 oxen at $ 50 a head, 120 sheep at $ 4.25 a head, and 28 cows at $ 45.50 a head. He returned 20 of the oxen at cost and bought 5 horses at $125 each. What was the cost of the whole to him ? 1/162. Bought 24 car-loads of wheat, each car holding 325 bushels, at $ 1.48 a bushel. I sold the wheat at $ 1.65 a bushel. How much did I gain ? QUESTIONS. 50. What is multiplication 1 51. What is the multiplicand? 52. The multiplier ? 53. The product ? The factors of the product ? 54. What is the sign of multiplication ? 55. What is a concrete number ? 56. An abstract number ? " 58, What are principles of multiplication ? 59. How are num- bers written for multiplying 1 How do you multiply ? 60. What is the proof 1 61. How do you nuiltiply by a number of tens, hundreds, etc 1 38 DIVISION. DIVISION. 62. 1. James has 32 books. How many times 8 books has he ? 2. How many quarts of nuts, at 9 cents a quart, can be bought for 54 cents ? 3. How many times 9 cents are 54 cents ? How many times 9 cents in 54 cents ? 4. If 7 men share 42 pounds of tea, how many pounds will each man have ? 5. Ella has 27 cents. How many pencils at 5 cents each can she buy, and how many cents left ? 63. Division is finding how many times one number is contained in another; or, finding one of the equal parts of a number. 64. The Dividend is the number divided. 65. The Divisor is the number by which we divide. 66. The Quotient is the result of a division. 67. The Remainder is the part of the dividend left, when the latter does not contain the divisor an exact number of times. 68. The Sign of Division, -j- , or : , means divided by. Thus, 16 -T- 8, or 16 : 8, is read, sixteen divided by eight. Division is also indicated by writing the divisor at the left of the dividend, with a curve, ), between, or by writing the divisor under the dividend, with a horizontal line be- tween. Thus, 2) 6, or |, may be read, six divided by two. 6d. A Parenthesis, ( ) , or Vinculum, , is used to in- clude such numbers as are to be considered together. Thus, 6 + 8 ^ 2 + 5 = 15, but (6 + 8) -f- (2 + 5) = 2. Note. —The signs X and -j- have no force in either direction beyond a f or a — unless a parenthesis is used. Operations indicated by them must be performed first. 70. The Equal Parts into which a number may be di- vided are named, according to their size. Thus, — One of two equal parts, written |-, is called one half ; One of three equal parts, written ^, is called one third ; One of four equal parts, written \, is called one fourth; Two of three equal parts, written f, is called two thirds ; Three of four equal parts, written |, is called three fourths ; and so on. ORAL EXERCISES. 6. John has 84 cents. How many times 7 cents has he ? 7. How many times 12 cents in 84 cents ? 8. Ellen has 49 peaches. Should she divide them equally among 7 of her playmates, how many would each receive ? 9. What number is one of the 7 equal parts of 49 ? What is -V- ? 10. If $ 96 he equally divided among 12 men, what sum would each receive ? How much is ^^ of }> 96 ? 11. In 84 days how many weeks ? How many times 7 in 84 ? What is 4^ of 84 ? What is 12. 13. 14. 15. 16. 63-7? 8) 72? 72:9? w I of 36 ? 64-^ 8? 9) 81? 96 : 12 ? 1.0 8 ? i of 45 ? 54-^9? 5) 60? 99 : 11 ? A^-? 1 of 96 ? 49 -^ 7 ? 11) 66 ? 32 : 4 ? -V" ^ tV oi 110 ? 24 ~ 8 ? 10) 80 ? 63 : 9 ? i,^ ? ^ oi 132 ? 40 DIVISION. 17. Give the quotients of the numbers in the following line divided by 2 ; by 4 ; by 6 ; by 8 ; by 10. 8, 12, 16, 20, 24, 32, 36, 48, 60, 64. 18. Give the quotients of the numbers in the following line divided by 3 ; by 5 ; by 7 ; by 9 ; by 8. 28, 33, 35, 42, 49, 63, 81, 96, 100. Supply the necessary numbers in the following : 19. 20. 6 X ? == 42 11 X ? == 121 9 X ? == 108 ? X 8 = 64 ? X 12 = 72 7 X ? := 42 4x?=^32 ?x9r=72 21. 22. 3 X 4 X ? = 84 -8/- + 9 = ? 2 X ? X 6 = 144 132 : ? = 12 3 X 3 X ? = 81 144 : ? = 12 3 X 3 X ? = 27 ^-fii -{-5 = ? 2*3. In 24 how many times 2?6?8?4?12? 24. In 45 how many times 5?3?9?15? 25. In 60 how many times 5 ? 10 ? 6 ? 4 ? 12 ? 15 ? 20 ? 26. At 8 cents each, how many cocoa-nuts can be bought for 88 cents ? 27. If a dozen of eggs cost 48 cents, what does one cost ? 28. How many pounds of meat can be bought for 72 cents at 8 cents a pound ? 29. How many yards of cloth at $ 6 a yard will pay for 8 tons of coal at $ 9 a ton ? 30. If 4 men do a piece of work in 12 days, how long will it take 6 men ? 31. What number divided by 9 will give 6 ? 32. If 4 cords of wood cost $ 36, what will 7 cords cost ? 33. How many twelve-quart cans will hold 84 gallons of milk? 34. If a train of cars runs 108 miles in 9 hours, what is the rate per hour ? Divisioisr, ^1 71. Principles of Division. 1. The dividend is the product of the divisor and the quotient 2. When the divisor and the dividend are like numbers, the quotient will be an abstract number. 3. When the divisor is an abstract 7iumber, the dividend and the quotient will be like numbers. WRITTEN EXERCISES. 35. Divide 8574 by 6. Divisor 6) 8574 Dividend. Solution.— Wq write the 1429 Quotient. '^|^|^^^ ^^ *^^ ^^^'^ ^^' ^^^ P dividend, with a curve be- tween them, and begin at Proof 8574 the left to divide. 6 in 8 thousands, 1 thou- sand times, with 2 thousands, or 20 hundreds, remaining. We write the 1 thousand beneath in thousands' place. Uniting the 20 hundreds with the 5 hundreds of the dividend, we have 25 hundreds. 6 in 25 hundreds, 4 hundred times, with 1 hun- dred, or 10 tens, remaining. We write the 4 hundreds beneath in hundreds' place. Uniting the 10 tens with the 7 tens of the dividend, we have 17 tens. 6 in 17 tens, 2 tens times, with 5 tens, or 50 ones, remaining. We write the 2 tens beneath in tens' place. Uniting the 50 ones with the 4 ones of the dividend, we have 54 ones. 6 in 54 ones, 9 times. We write the 9 ones beneath in ones' place. The quotient is 1 thousand 4 hundreds 2 tens 9 ones, or 1429. To prove the work, we multiply the quotient by the divisor, and have as the product the dividend (Art. 71). 36. 37. 38. 39. 4) 1532 7) 5747 5) 4855 8) 9136 40. 41. 42. 43. 6) 351.6 8) 97.84 7) 83.30 9) 40.203 42 DIVISION. 44. f.. r^orro Solution. — Dividing, there is a final re- ^ mainder of 3. The division of the 3 we Quotient 1274§ indicate by -|, and writing this as a part 5 of the result, we have the quotient 1274|. Q'^jO To prove the work, we multiply the in- o teger of the quotient by the divisor, and adding the remainder, have the dividend. Proof 6373 45. 46. 47. 48. 5) 3576 12) 8143 8) 4899 7) 58965 49. 50. 51. 52. 4) 2693 3) 40.02 6) 364.27 9) 39.137 53. How many tons of coal at $ 5 a ton can be bought for $ 2025 ? 54. How far must a vessel sail each day to sail 1438 miles in 7 days ? 55. Paid for 8 yards of broadcloth $ 43.60. What was the cloth a yard ? 56. If the dividend is 20818, and the divisor 9, what is the quotient ? ORAL EXERCISES. 57. How many times is ^ of 32 contained in J of 60 ? 58. The product of two numbers is 56, and one of the num- bers 8. What is the other number ? 59. If the dividend is 72, and the divisor 9, what is the quotient ? 60. Mr. Jones can earn in a month $ 81, and his son one ninth as much. How much can his son earn ? 61. A farm of 120 acres has been divided into 12 lots. How many acres in each lot ? 62. What is i of 64 ? i of 81 ? -,V of JIO ? ^ij of 96 ? 63. One man can do a piece of work in 132 days. In what time can 11 men do it? ' DIVISION. 43 64. A man earned $ 144 in 12 weeks. What was that a week ? 65. How many lO's in 100 ? How many lOO's in 1000 ? 66. In 65 how many tens and what over ? In 235 how many hundreds and what over ? 67. How many times 20 in 100 ? How many timen (20 -^ 4) in (100 -^ 4) ? How many times (20 X 2) in (100 X 2) ? 72. A General Principle of Division. Dividing or Tnultiplying both dividend and divisor by the same number does not alter the quotient. WRITTEN EXERCISES. 68. Divide 1185 by 12. 98^9^ Quotient. Solution. — \2 is in 118 Divisor 12 )1185 Dividend. tens, 9 tens times. We "I^Qg . write the 9 tens over the "TT^p. tens of the dividend. 9 tens times 12 are 108 __ vtens, which taken from the 9 118 tens of the dividend leaves 10 tens, and uniting with these 10 tens the 5 ones of the dividend, we have 105 ones. 12 is in 105, 8 times. We write the 8 ones above the ones of the. dividend. 8 times 12 are 96, which taken from the 105 of the dividend leaves 9, and indicating its division by the divisor, we have ^\. Writing the y^Tj as a part of the quotient, we have as the quotient 98^^^. 69. Divide 9876 by 21. 72. Divide 106.97 by 19. 70. Divide 3276 by 14. 73. Divide 26.575 by 25. 71. Divide 31278 by 23. 74. Divide 1868.5 by 37. 73. Rule for Division. Write the divisor at the left of the dividend with a curve between them. Find how many tiiues the divisor is contained in the few* 44 DIVISION, est left-hand figures of the dividend that will contain it, and write the result under or over the right-hand figure of the div- idend used, as the first quotient figure. Multiply the divisor by this quotient, subtract the product from the partial dividend u^ed, and to the rew.ainder annex the next dividend figure for a second partial dividend. Divide and proceed as before, until all the dividend figures lave been used. If there is a final remainder, write it with the divisor under it, 05 a part of the quotient. Note 1. — The division is called Short Division when only tlie divisor, div- idend, and quotient are written; and Long Division when each process of the solution is written. In short division the quotient is usually written below the dividend, and in long division over, or at the right of, the dividend. Note 2. — When the divisor is an integer, and the quotient is written under or over the dividend, the decimal point in the quotient must be placed immedi- ately under or over the point in the dividend. 74. Proof. — Multiply the quotient by the divisor, and to the product add the remainder, if any. The result should be the dividend. 75. Divide 182.72 by 45, and prove the work. Solution. Proof. 4.06A Quotient. 4.06^ Quotient. Divisor 45)182.72 Dividend. 45 Divisor. 180 2030 2.72 1624 2.70 182.70 .02 Remainder. .02 Remainder. 182.72 Dividend. 76. Find the quotient of $ 114.48 divided by 36. 77. Find one twenty-eiglitli of $ 890.56. DIVISION. 45 73. Divide 347692 by 351. 85. 1298763 H- 873 := ? 79. Divide 8468.31 by 793. 86. -^Ij of 63450 = ? 80. Divide 947684 by 982. 87. 4127.098 -^- 2007 = ? 81. Divide 3287.64 by 735. 88. 8643.21 -f- 987 = ? 82 Divide 6798.341 by 1234. 89. (643 x 857) -^ 456 = ? 83. Divide $8392.476 by 987. 90. (984 X 895) -^ 359 :^ ? 84. Divide 3004760 by 5942. 91. ^^^J/-^^ = ? 92. Wbat is tbe price of one sleigh if 85 cost $10625 ? 93. How muny horses at $225 each will $5400 buy ? 94. If $671178.90 is the valuation of a hamlet having 98 inhabitants, what is the average valuation to an inhabitant ? 95. If the distance across the Atlantic Ocean is 3000 miles, in how many days will a ship, sailing 115 miles a day, make that distance ? 96. What number multiplied by 512 gives a product of 1763.68 ? 97. What number multiplied by 2135 gives a product of 6419945 ? 98. Divide 49300 by 432. 100. $5711.04-^108==? 99. Divide 700074 by 2047. 101. $ 50000 -^ 365 = ? 102. Divide 489 by 25 to hundredths. 19.56 25) 489.00 25_ 239 225 Solution. — A^^ =z 489.00. Dividing the 489.00 by 25, we obtain as theresidt required 19.56. If the division had not terminated, we could 14,0 have denoted this by placing the sign -\- after 12.5 the hundredths in the quotient to indicate the 7~^ incompleteness. 1.50 103. Divide 2722.5 by 44 to hundredths. 104. Divide 9164 by 86 to thousandths. 105. Divide $ 12625 by 404 to cents. 106. Divide 17552 by 128 to thousandths. n!> 46 DIVISION. 107. A gentleman dying left an estate valued at on<^ mit lion dollars to be equally divided among thirty-three heirs. What did each receive ? The divisor a number of tens, hundreds, etc. lOa Divide 1467 by 10 ; by 100 j by 1000. 1467. -^ 10 = 146.7 Solutio7i, — The removal of a 1467. -^ 100 = 14.67 figure one place to the right de- 1467. -L. 1000 == 1.467 creases its vakie ten times (Art. 18). To divide a number by 10, or to find 1 tenth of a number, it is only necessary to move each figure of it one place to the right. This is done by moving the deci- mal point one place to the left, giving 14:6^^ as the quotient. In the same way two removals of the point to the left divides by 100, three removals by 1000, and so on. That is, 75. To divide by 10, 100, 1000, etc., Move the decimal point of the dividend as many 2^l(^ces to the left as there are ciphers in the divisor. 109. Divide 5824 by 160. Solution. — \Wh\Q X 10. Dividing both divi- sor and dividend by 10, by removing the decimal point one place to the left, the division becomes the same as 582.4 by 16, which gives as the quo- tient .36.4. Or, if the exact remainder is required, we may indicate the division by 10, by marking off one place, and have as a quotient 582, and a re- mainder of 4 ones. Dividing,' 582 by 16, we have for a quotient 36, and a remainder 6 tens. Uniting the two remainders, we have as the true remainder 64. 36.4 16.0) 682.4 48 36|tVj 1610) 582|4 48 102 or 102 96 6.4 96 64 6.4 V DIVISION. 47 110. There are 1000 mills in a dollar ; how many dollars are there in 19650 mills ? 111. Divide 3962 by 100 ; by 1000. 112. Divide 9108 by 200 to hundredths. 113. Divide 12386 by 90, and find the true remainder. 114. Divide 5886990 by 5400 to hundredths. 115. In the State of Alabama 143727 acres have pro- duced 51000 bales of cotton. What was that for a bale ? 116. If light moves at the rate of 186000 miles in a second, how long is it in passing from the sun to the earth, a distance of 92000000 miles? Divide: 117. 8496453 by 8291. 125. 849 x 863 by 1252. 118. 2907654 by 3782. 126. 84 x 96 X 25 by 1189. 119.' 72659302 by 1234. 127. 694^87 + 956 by 609. 120. 67890123 by 5678. 128. 847 X 12 X 900 by 9. 121. 4567.890 by 2961. 129. 9008 x 7080 by 2090. 122. 184.837 by 349. 130. $ 945.65 by 850. 123. 6239076 by 6384. 131. 843 - 159 by 29 X 7. 124. 1689.783 by 945. 132. $ 849.625 by 975. MISCELLANEOUS EXERCISES. 133. A farmer obtained $7665 for the apples from his orchard at $ 3 a barrel. How many barrels did he have ? 134. Two men start from the same place and travel in opposite directions, one at the rate of 25 miles a day, and the other at the rate of 31 miles a day. When 5656 miles apart how many days had they traveled ? 135. A cargo of 25 tons of coal was bought by the long ton of 2240 pounds, and sold by the short ton of 2000 pounds. How many short tons was the gain ? 136. A farmer bought 75 acres of land at $ 34 an acre, and 85 acres at $ 20.40 an acre. What was the average cost of the whole an acre ? 48 DIVISION. \J 137. If a man can save $ 15 in each of the 12 months of a year, in how many years can he save enough to amount to 12520? 138. The product of two factors is 96000, and one of the factors is 150. What is the other factor ? 139. Bought a quantity of boards for $ 2650, and sold the same for $ 3286, and thereby gained $ 6 a thousand feet. How many thousand feet were there ? 140. The product of three factors is 33600. Two of the factors are 15 and 35. What is the third ? 141. Texas contains 274400 square miles and Massachu- setts 7800. How many States of the size of Massachusetts might be made out of Texas, and how many square miles over ? 142. If the divisor is 350 and the dividend 262500, what is the quotient ? 143. A man bought 164 acres of land for $ 80 an acre, and sold a part for $ 4480, at the same rate. How many acres had he then left ? 144. If the dividend is 71142 and the quotient 1002, what is the divisor ? 145. A man exchanged 159 cords of wood at $5 a cord, for a horse valued at $ 144, and the balance in sheep at $ 3 apiece. How many sheep did he receive ? QUESTIONS. 63. What is division ? 64. What is the dividend ? 65. The di- visor? 66. The quotient? 67. The remainder ? 68. What is the sign of division ? 69. What is a parenthesis, or vinculum, used to include ? 71. What are principles of division ? 72. What is a general prin- ciple of division ? 73. How are the numbers written for dividing? How do you di- vide ? How is a final remainder written ? 74. What is the proof of dtvision ? REVIEW. 49 REVIEW. ORAL EXERCISES. 76. 1. What number added to 18 will make 27 ? 2. John had 37 cents. He gave 7 to his brother, and shared the remainder equally with his two sisters. What was his share ? 3. A farmer had 18 chickens. The foxes killed 3 and he sold 7. How many has he left ? 4. Thomas gave 17 cents for a whip, 13 cents for a slate, 12 cents for a copy-book, and 10 cents for pencils. How much change did he receive from a dollar bill ? 5. (5x6 + 5) -^(14-7) = ? 6. John had 30 marbles. He lost half of them and then gave away 5. How many had he left ? 7. A man had $ 57 ; he lost at one time $ 5, and at another $ 10. How many had he left ? 8. Bought 7 tons of coal at $ 8 per ton, and gave in payment 2 twenty-dollar bills and 2 ten-dollar bills. How much change should be received back ? 9. I have 3 bags of nuts. There are 2 bushels in each bag. What is the whole worth at $ 3 a bushel ? 10. George had in a basket 41 apples. He gave 5 to Susan, 7 to Lucy, and 6 to Henry. What is the value of those left at 2 cents each ? Find the result of : 11. 31 _ 9 + (6 X 3). 15. (30 -f 25 - 23) X 3. 12. (63T9 ~ 12) X (14 - 8). 16. 46 - 16 + 10 -- 10. 13. (irx 8 - 16) ~ 9. 17. (81 -^ 9) X (75 -^ 25). 14. 65 ^ (60 -^ 5 4- 30). is. 100 - (15 X 5) -^ 5. 19. Two men start from the same place and go in the same direction ; one travels at the rate of 5 miles an hour, and the 50 REVIEW. other at the rate of 9 miles an hour. How far apart will they be in 12 hours ? 20. A steamboat can run 10 miles an hour down stream, and 8 miles up stream. After running down stream 4 hours, how long will it be in returning ? 21. If you should earn $ 60 in 10 weeks, and pay of your earn- ings $3 a week for board, how much will your net earn- ings be ? 22. William had 98 peaches. He gave 18 to his brother, and shared the remainder equally with 9 others. How many did he give away in all ? 23. If 6 men can do a piece of work in 8 days, in how many days can 4 men do it ? 24. In what time can 6 men excavate a cellar, which 14 men can excavate in 3 days ? 25. If 15 men can build a wall in 10 days, in what time can 25 men do it ? 26. If 9 barrels of flour are worth $ 81, what are 7 barrels worth ? 27. When 24 pounds of coffee can be bought for $ 6, how many pounds can be bought for $ 8 ? 28. If a man can earn $ 96 in 8 weeks, in what time can he earn $ 60 ? 29. A man having $ 95 bought 5 coats, and had $ 20 left. What did the coats cost apiece ? 30. Two men start 132 miles apart and travel toward each other, one at the rate of 6 miles an hour, and the other at the rate of 5 miles an hour. How many miles must each travel before meeting ? 31. A cistern can be emptied in 16 minutes by 5 pipes. In what time can it be emptied by only 2 pipes ? 32. A and B start together and travel in the same direc- tion, A traveling 21 miles a day and B 27 miles. When they have traveled 9 days, how much less than 60 miles are they apart ? REVIEW. 51 WRITTEN EXERCISES. 33. The area of Illinois is 55405 square miles, and of Ar- kansas 52198 square miles. How many square miles is Illi- nois larger than Arkansas ? 34. The minuend is 74760, and the subtrahend 34943. What is the difference? 35. The difference between two numbers is 4004, and the greater number is 5496. What is the smaller ? 36. A man had $ 5000. He expended for a stable $ 2560.75, and for a horse and carriage $ 375.87. What had he left ? 37. The multiplicand is 664 and the multiplier 19. What is the product ? NJ 38. Six men bought some property for $ 5670 and sold it for $ 7896.84. What was each man's share of the gain ? Sj 39. A merchant bought 3 casks of sugar, each weighing 255 ^pounds, at 9 cents a pound, and sold it at 11 cents a pound. How many dollars did he make ? 40. A man sets out to travel 223 miles at the rate of 27 miles a day. When he has gone 61 miles, in how many days can he finish the distance ? 41. What is the cost of 365 parlor organs at $ 97 each ? 42. The divisor is 48, the quotient 596, and the remainder 10. What is the dividend ? 43. Expended in goods $ A2.bl, buying 17 yards of cloth at 15 cents a yard, 46 pounds of coffee at 28 cents a pound, 16 gallons of molasses at 76 cents a gallon, and 107 pounds of confectionery. How much was the confectionery a pound ? I 44. Bought 310 tons of coal for $ 1472.50 and sold it foi \J 1 1549.60. How much did I make a ton ? 45. Bought two lots of land ; the first containing 144 acres at $ 12 an acre, and the second 108 acres at $ 15 an acre. I sold both lots at $ 18 an acre. What was the gain per acre ? 46. The multiplicand is 407, and the product 10989. What is the multiplier ? 52 EEVIEW. 47. Bought 288 barrels of flour for $ 1728, and sold it at a profit of $ 576. What did I get a barrel for it ? 48. A gentleman bought a house for $ 5100, and farm stock to the amount of $> 715.80. He paid at one time $ 2013, and at another $ 1981.95. How much remained to be paid ? 4 49. (194 + 65) X 7 + Il-^2-f^2 20 _ 952 ::::, ? 50. Bought 500 barrels of flour at 15.75 a barrel, 47 hun dred-weight of cheese at $ 9.25 a hundred, and 15 barrels ot pork at $ 21.50 a barrel. What was the amount of the whole ? 51. Bought molasses for $ 9212, and sold it at $ 67 a hogs- .head, and gained $ 20. How man}^ hogsheads were there ? L"^^. A cargo of 125 tons of coal was bought by the long ton of 2240 pounds, and sold by the short ton of 2000 pounds. How many long tons was the gain ? 53. I have $ 2973. I wish to invest it in as many horses as I can at $ 150 each, and the remainder in a carriage. How many horses can I buy, and what can I pay for the carriage ? 54. -7 5f J- + 549 - (2128 4- 7) X 2 = what ? y 55. (194 + 65) X 7 + (352 - 220) -f- 11 - 952 -^ (91 - 35) — ? 56. A man bought 360 acres of land at $ 45.50, and paid down $ 1368. He sold 125 acres at $ 63 an acre, and made another payment. How much did he then owe for the land ? 57. The product of three numbers is 40800. One of the numbers is 150, and another 16. What is the third number ? 58. The dividend is 18988 and the quotient 2bj\^^. What is the divisor ? 59. If I have a garden 320 feet long and half as wide^ how many times must I walk' around it to travel 100 miles of 5280 feet each? 60. A man has 11496, which he wishes to lay out in pur- chasing cows and oxen, an equal number of each. If he should pay $ 37 for each cow and $51 for each ox, how many of each can he buy ? REVIEW. 53 61. If 216 pianos cost $ 112320, what will 519 cost ? 62. How many pounds of coffee, at 38 cents a pound, will pay for 2 hogsheads of sugar containing 1160 pounds each. at 19 cents a pound ? 63. A farmer having $ 3038, bought 15 tons of hay at $ 11 per ton, 3 yoke of oxen at $ 155 each, 375 sheep at $ 5 each, and spent the rest for cows at $ 41 apiece. How many cows did he buy ? 64. A man in his will gave to each of his 2 sons $ 7600 ; to a third son $ 1500 ; to each of 3 daughters $ 3775, and the balance $6877 to his wife. His wife died, however, and the whole property was divided equally among his children ; what did each receive ? 65. The expenses of a picnic party of 9 gentlemen and 8 ladies were $ 2.40 each. The gentlemen paid all the expenses. What did each pay ? 66. Sold 160 tons of coal at $ 5 per ton, and a number of tons at $ 3 per ton. The value of all the coal sold was $ 965. How many tons were there ? 67. Bought 960 acres of land for $ 12000. Sold i of it at $ 12 per acre, J of it at $ 15 per acre, and the remainder for $ 20 per acre. Did I gain or lose, and how much ? REVIEW QUESTIONS. 2. What is a number? 21. What is an integer? 55. What is a, concrete number ? 56. An abstract number ? 4. What is notation ? 5. Numeration ? 18. What are principles of notation? 31. What is addition? 37. What are principles of addition? 41. What is subtraction? 46. What are principles of subtraction? , 50. What is multiplication ? 58. What are principles of multipli- cation? 63. What is division ? 71. What are principles of divi- sion ? 54 FACTORS. FACTOBS. 77. 1. What two numbers multiplied together produce 2 ? 3 ? 5 ? 2. What numbers, other than itself and 1, will divide 6 without a remainder ? 3. What number multiplied by 2 will produce 6 ? 4. Kame all the numbers, other than itself and 1, which will divide 21 without a remainder. 5. What number multiplied by 3 will produce 21 ? 6. Name the numbers between 1 and 15 which are the pro- duct of two or more numbers greater than 1. 78. The Factors of a number are the integers which produce the number when multiplied together. Thus, 3 and 5 are factors of 15, and 2, 3, and 3 are factors of 18. 79. A Composite Number is a number having other fac- tors than itself and one. Thus, 4, 6, 8, 9, and 10 are composite numbers. Note. — One number is said to be divisible by another wlien there is no re- mainder in the division. Thus, A composite number is divisible by any of its factors. 80. A Prime Number is a number having no other fac- tors than itself and one. Thus, 1, 2, 3, 5, 7, 11 are prime numbers, 81. A Prime Factor is a factor that is a prime number. Note. ~ As every integer is a factor of itself, and has one as the other fac- tor, in speaking of the factors of a number we usually exclude tlie number itself and one. FACTORS. 55 82. Every prime number, except 2 and 5, has 1, 3, 7, or 9 for its unit figure. Note. — 2 is a factor of any number ending in 0, or whose ones are divisible by 2. 3 or 9 is a factor of any number the sum of whose figures is divisible by 3 or 9. 5 is a factor of any number ending in or 6. WRITTEN EXERCISES. 7. Find the prime factors of 56, * 2) 56 Solution, — Dividing by the prime 2) 28 mimbers 2, 2, and 2, the last quotient -— obtained is 7, which is also a prime ^ -. number. The factors required are 7 2, 2, 2, 7 ; or 2^ 7. Ans. 2, 2, 2, 7 ; or 2\ 7. 8. What are the prime factors of 84 ? Of 144 ? Of 160 ? 9. What are the prime factors of 462 ? Of 576 ? Of 1008 ? 83. Rule for finding the Prime Factors of a Number. Divide the given number by any prime number above one that will exactly divide it Divide the quotient, if composite^ in like manner, and so continue until a prime quotient is found. The several divisors and the last quotient will be the prime factors. Find the prime factors of 10. 210 14. 1155 18. 16028 11. 2772 15. 2800 19. 17199 12. 426 16. 3420 20. 10323 13. 6105 17. 7800 21. 12496 22. What is the largest prime factor of 1184 ? 23. Find the prime factors of 4389. 24. Find the prime factors of 6300. 25. What prime numbers multiplied together produce 40579 ? 56 FACTORS. CANCELLATION. 26. What is the quotient of 63 divided by 21 ? 27. What is one seventh of 63 divided by ]~ of 21 ? 28. What is the quotient of J of 63, divided by J of 21 ? 29. What is the quotient of 5 X 3 X 2, divided by 2 X 3 *? Of 3 X 5, divided by 3 ? Of 2 X 5, divided by 2 ? 84. Cancellation is striking out- the same factor from both dividend and divisor. 85. Principles of Cancellation. 1. Striking out a factor of a number divides the number by the factor. 2. Striking out the same factors from dividend and divisor does not affect the quotient. WRITTEN EXERCISES. 30. Divide 3 times 80 by 48. 80 X3 ^0X5X^ f. Solution. — Indicating the To ^^ -*^ q " division, and canceUng, by striking out in both dividend and divisor the factors 16 and 3, common to both, there remains only !ihe factor 5 in the dividend, which is the required quotient. 31. Divide 11 times 1476 by 6 times 132. 41 Solution. — Striking out in dividend to^ and divisor all the factors common to 147^ y TT both, there remains in the dividend only — - = 20 J the factor 41, and in the divisor only the ^p^ X p factor 2. Completing, then, the division, Ar ^ we have for the quotient 20^. 32. Divide 45 X 20 X 7 by 49 X 4 X 9. 33. (54 X 3 X 4 X 15) -f- (18 X 12 X 10) = ? FACTORS. 57 86. Rule for Cancellation. Strike out all the factors common to both dividend and di* visor, and then divide as 7nay he required. 34. Divide 255 x 63 x 4 by 340 X 12 X 7. 35. Divide 18 X 16 X 10 X 5 by 12 X 8 X 6 X 2. 36. Divide 50 X 36 X 14 by 54 X 10 X 4 X 3. 37. Divide 545 X 105 X H by 35 X 33 X 5. 38. Multiply 64 by 63 and divide the product by 168. 39. What is the quotient of 36 X 21 X 14 divided by 2'^ X7x6? 40. How many tons of hay at $ 24 a ton must be given for 4 cows at $ 42 each ? 41. How many yards of cloth can be bought for $ 95, when 24 yards can be bought for $ 120 ? 42. If 100 men can perfprm a piece of work in 12 days, in how many days can 150 men perform it ? 43. A fort has pro^sions for 225 men 12 months. How long will it last 675 men ? 44. Exchanged 15 pieces of muslin, each containing 30 yards at 10 cents a yard, for 3 pieces of flannel, each contain- ing 50 yards. What was the flannel a yard ? GREATEST COMMON DIVISOR. 45. What number will divide both 21 and 35, without a remainder ? 46. What factor is common to 21 and 35 ? To 15 and 25 ? 47. What is the greatest number that will divide both 18 and 30, without a remainder ? 48. What is the greatest factor common to 18 and 30? To 16 and 24.^ 87. A Commoii Divisor of two or more numbers is any factor found in each of them. Thus, 7 is a common divisor of 14 and 21. 58 FACTORS. 88. The Greatest Common Divisor of two or more num- bers is the greatest factor found in each of them. Thus, 6 is the greatest common divisor of 18 and 24. 89. Numbers are Prime to each other when they have no common factor or divisor. Thus, 9 and 14 are prime to each other. 90. Principle. The greatest common divisor of two or more numbers is the product of all their conimon prime factors, WRITTEN EXERCISES. 49. What is the greatest common divisor of 84 and 132 ? 2) 84, 132 ^^ 84^=2x2x3x7 Solution.— 2)42, 66 "*' 132:^2x2x3x11 ^^. ^^^ *^^ ^ prime factors 3) 21, 33 common to the 7, 11 2 X 2 X 3 = 12, Ans. numbers are 2, 2, and 3. The pro- duct of these, 2 X 2 X 3 or 12, is the greatest common divisor required. 50. What is the greatest common divisor of 36, 81, 135 ? 51. What is the greatest common divisor of 24, 42, 54, and 60? 91. Rule for finding the Greatest Common Divisor. Separate the numbers into their prime factors, and find the product of all such as are common to the numbers. What is the greatest common divisor 32. Of 45 and 135 ? 56. Of 20, 26, and 38 ? 53. Of 90 and 105? 57. Of 32, 48, and 128 ? 54. Of 42 and 81 ? 58. Of 45, 72, and 81 ? 65. Of 132 and 156 ? 59. Of 24, 51, 105, and 729 ? FACTORS. 69 60. Having three rooms, the first 12 feet wide, the second 15 feet, and the third 18 feet, I wish to purchase a roll of the widest carpeting that will exactly fit each room without any cutting as to width. How wide must it be ? 92. When the numbers are large, or cannot readily be separated into factors, — Of two numbers divide, the larger hy the smaller, and the last divisor hy the last remainder, until nothing remains. The final divisor is the greatest common divisor. If more than two numhers are given, find the greatest com- mon divisor of two of them, then of this divisor and a third number, and so on, 61. What is the greatest common divisor of 247 and 323 ? Solution. — If 247) 323 (1 247 will exactly 247 divide 323, it will 76) 247 (3 ^^^ *^^ greatest QOQ common divisor of 247 and 323. It Greatest common divisor = 19) 76 (4 ^^^^ ^^^ ^^^^^^ ^i. '^ vide 323, for there is 76 remainder. If 76 will exactly divide 247, it will be the greatest common divisor of 76 and 247, and therefore of 247 and 323. It will not exactly di- vide 247, for there is 19 remainder. If 19 will exactly divide 76, it will be the greatest common divisor of 19 and 76, and of 76 and 247, and of 247 and 323. 19 will exactly divide 76, and therefore it is the greatest common divisor of 247 and 323. Find the greatest commou divisor of 62. 336 and 480. 67. 2145 and 3471. 63. 925 and 1475. 68. 582 and 3724. 64. 308 and 506. 69. 10353 and 14877. 65. 172 and 1118. 70. 3528 and 4424. 66. 275 and 440. 71. 1764 and 2660. 60 FACTORS. 72. 744 and 906. 75. 2883 and 3131. 73. 728 and 808. 76. 3178 and 3500. 74. 756 and 1140. 77. 4872 and 9048. LEAST COMMON MULTIPLE. 78. Name four numbers of which 4 is a factor. 79. 15 is an exact number of times what two numbers ? 80. What is the least number that will contain both 3 and 7 an exact number of times ? 81. What is the least number that both 4 and 9 will exactly divide ? 93. A Multiple of a number is any number divisible by it (Art. 79, note). Thus, 5, 10, 15, etc., are multiples of 5. 94. A Common Multiple of two or more numbers is any number divisible by each of them. Thus, 48 is a common multiple of 4, 8, and 12. 95. The Least Common Multiple of two or more numbers is the least number divisible by each of them. Thus, 24 is the least common multiple of 4, 8, and 12. 96. Principle. The least common multiple of two or more numbers contains each of the prime factors of those numbers^ but 7io others. WRITTEN EXERCISES. 82. What is the least common multiple of 21, 35, and 45 ? 21 = 3x7 „ 3)21,35,45 Jl^'T: 7.. !t!!: rating the numbers into their prime fac- 35 = 5x7 ^""^ 5) 7,35,15 45 = 3x3x5 7^ 3 tors, we find the pro- rr Qi r ^^^^ ^^ ^^^ different 3 X 3 X 5 X i = 31c» factors, using each the FACTORS. 61 greatest number of times it occurs in any number, to be 3 X 3 X 5 X 7, or 315. This is the least common multiple, because it contains each prime factor of the numbers, but no other (Art. 96). Or, the different prime factors may be found by the second process annexed, the two methods being alike in principle. 83. What is the least common multiple of 7, 14, 15, and 21? 84. What is the least common multiple of 18, 28, 30, and 42? 97. Rule for finding the Least Common Multiple. Separate the numbers into their prime factors. Take the 'product of all the different fax^tors, using each factor the greatest number of times it occurs in any number. Note 1. — The following rule is sometimes used. Strike out any of the given numbers that are factors of any of the others, and divide the remaining numbers by any prime factor common to two or more of them. Strike out from the resulting quotients and undivided numbers all that are factors of any of the rest^ and divide as before. Thus proceed until no tioo of the remaining numbers have a common factor. The product of the divisors and remaining numbers will be the least common multiple required. Note 2. — The least common multiple of numbers prime to each other is their product. Find the least common multiple 85. Of 21, 33, 6Q. 89. Of 8, 18, 24, 36. 86. Of 63, 72, 84. 90. Of 7, 25, 12, 41. 87. Of m, 88, 110. 91. Of 28, m, 100, 125. 88. Of 81, 63, 135. 92. Of 24, 42, 54, 180. 93. What is the least common multiple of 24, 96, 100, and 144? 94. What is the least common multiple of 4, 11, 18, 20, 36, and 48 ? 95. What is the least sum of money that can be exactly ex- pended for sheep, cows, or oxen, at $ 5, $ 35, and $ 50 each, respectively ? 62 FACTORS. MISCELLANEOUS EXERCISES. 96. What are the prime factors of 2520 ? 97. How many times are 2 and 3 respectively factors in 5760? 98. What is the greatest factor common to 689 and 1573 ? 99. rind the sum of all the prime numbers between 70 and 100. 100. Employ cancellation in dividing 14 X 15 X 16 X 24 X 48 X 60 by 7 X 30 X 8 X 8 X 6 X 12 X 3. 101. How many times is the greatest common divisor of 48, 36, 72, 24 contained in their least common multiple ? 102. Find the sum of the composite numbers between 100 and 120, inclusive. 103. What is the difference between the greatest common divisor and the least common multiple of 160, 352, and 992 ? 104. Divide 1008 by 168, using prime factors and cancella- tion. 105. When hay is $ 24 a ton, how many barrels of flour, at $ 8, will exactly pay for 35 tons of hay ? QUESTIONS. 78. What are the factors of a number ? 79. What is a composite number 1 80. What is a prime number 1 81. What is a prime factor 1 83. How do you find the prime fac- tors of a number ? 84. What is cancellation 1 85. What are principles of cancella- tion 1 86. How is cancellation performed ? 87. What is a common divisor of two or more numbers ? 88. The greatest common divisor of two or more numbers ? 90. What is a principle of common divisors? 91. How is the greatest common divisor found ? 93. What is a multiple of a number 1 94. A common multiple of two or more numbers ? 95. The least common multiple of two or more numbers ? 96. A principle of multiples ? 97. How is the least common multiple found ? COMMON FRACTIONS. . 63 COMMON FRACTIONS. 98. 1. If a unit, as an apple, is divided into two equal pieces, what part of the whole will one piece be ? 2. If an apple is divided into three equal pieces, what part of the whole will one piece be ? Two pieces ? 3. If a single thing is divided into four equal pieces, what part of the whole will one piece be ? Two pieces ? Three pieces ? 4. How many halves in an apple ? How many thirds ? How many fourths ? 5. What is meant by one half of a unit ? By one third ? By two thirds ? By one fourth ? By three fourths ? 6. Which are the larger parts of an apple, halves or thirds ? Thirds or fourths ? 99. A Fraction is one or more of the equal parts of a unit. The Unit of the Fraction is the unit divided, and a Fractional Unit is one of the equal parts into which it is divided. 100. The Denominator of a fraction is the number that shows into how many equal parts the unit is divided. Thus, Three is the denominator of two thirds. 101. The Numerator of a fraction is the number that shows how many of the equal parts of the unit are taken. Thus, Two is the numerator of two thirds. 102. The Terms of a fraction are its numerator and denominator. Thus, 2 and 3 are the terms of the fraction |. 64 . COMMON FEAGTIONS. 103. A Common Fraction is a fraction expressed by- writing the numerator above, and the denominator below, a dividing line. Thus, Three fourths of a dollar is written $f, 3 being the numerator, 4 the denominator, 1 dollar the unit of the fraction, and \ dollar t\\Q fractional unit 104. An Integer may be expressed in a fractional form, by writing 1 under it for a denominator. Thus, 2. may be written |, and read 2 ones ; 7 may be written ^, and read 7 ones ; etc. 105. A Proper Fraction is a fraction whose numerator is less than its denominator. Thus, ^ and Y^-j are proper fractions. 106. An Improper Fraction is a fraction whose numer- ator is not less than its denominator. Thus, 1^ and -l| are improper fractions. 107. A Mixed Number is an integer and a fraction united. Thus, 3J, read three and one fourth, is a mixed number. 108. A Fraction may be regarded as an indicated divis- ion (Art. 63), the numerator being the dividend, and the denominator the divisor. Thus, I of 1 inch is the same as ^ of 3 inches, or 3 inches divided by 4. 109. The Value of a fraction is the quotient of the nu- merator divided by the denominator. EXERCISES. Express in figures : 7. Three sevenths. 9. Nine sixteenths. 8. Seven elevenths. 10. Seventeen ones. COMMON FRACTIONS. 65 11. One twenty-first. 14. Three and three twenty-ninths. 12. Eleven thirty-seconds. 15. Twenty-three and three fifths. 13. Nineteen fortieths. 16. Eight and nine twelfths. REDUCTIOlSr OF FRACTIONS. ORAL EXERCISES. 17. In J of an apple how many fourths of an apple ? how many eighths ? 18. In § of an apple how many sixths ? how many ninths ? 19. Name a fraction equal to ^. Name a fraction equal to f. 20. Express f in terms 2 times as large. 3 times as large. 21. Change f to t%, f to j%, J to j%. 22. How is the fraction f changed to twelfths ? 23. How many halves are there in | ? How many thirds are there in j^^ ? 24. How many tenths of a melon in ^g ? how many fifths ? 25. Express f in larger terms ; f in larger terms ; -^ in smaller terms ; |f in smaller terms. 110. Reduction of Fractions is changing their form with- out changing their value. To Larger Terms. 26. Change | to sixteenths. Solu Hon. — The required den om- inator is 4 times the given denom- inator. Multiplying both terms of the fraction by 4 gives ^. 1" X 4 ^^ "if ^^ ^^^1 ^^ observed from the illustration, that f of the square equals }| of- it, the multiplication increasing the number of fractional uuits 4 times, and making each one fourth as large, so that the valuer of the fraction is not changed. 6 1 F ■ if : Si 66 COMMON FRACTIONS. 111. Principle. Multiplying both terms of a fraction by the same number does not change its value. 27. Change to 12ths ^ I, i, J, §, |, |. 28. Change to 18ths i, J, ^ ^, f, f, |, J, |, |. 29. Change to 20ths J, i, i, |, ^^, |, ^^, ^^, ^^. 30. Change to 24ths i, J, i J, i, ^i,, f, |, |, |, |, |, /^, |^. 31. Change to 36ths f, f , #, f , J, \^, i, f , ^^, /^. 32. Change to 48ths %, f, i, f, |, ^^^^ ^J, ,3^, ^^, i j, J|. WRITTEN EXERCISES. 33. Change | to twenty-fourths. Solution. — To change 8ths to 24ths, •| = 1^1 = 2^^ we must multiply both terms of the Qp fraction by 3. Doing this, we obtain 3 ■^. Or, "g" of 1 = 5; of 2^4 = 2"T Since 1 is 24 twenty-fourths, \ of 1 is ^ of 24 twenty-fourths, or 3 twenty-fourths, and f of 1 is 3 times 3 twenty-fourths, or ^. 34. Change f to seventy-fifths. 112. To change a fraction to larger terms : Rule. Divide the required denominator by the given denominator^ and multiply both terms of the fraction by the quotient. Change : 35. ^ to 84ths. 39; f to 168ths. 36. I to 32ds. 40. if to 189ths. 37. J to 54th s. 41. I to 48ths. 38. \l to 196ths. 42. f to 576ths. To Smaller Terms. 113. A fraction is in its Smallest Terms when its terms have no common factor. COMMON FRACTIONS. 67 ORAL EXERCISES. 43. Change |f to smallest terms. Solution. — Dividing both terms of ^f by 4, we have |, whose terms have no common factor ; hence, J[-| changed to smallest terms is f , It will be observed from the illustration that ^| of the square equals f of it, the division increasing the size of the fractional units 4 times, while their number is a fourth as large, so that the value of the fraction is not changed. 114. Principle. Dividing both terms of a fraction by the same number does not change its value. Change to smallest terms ; 44. i, h h I. T% A, A. \% A, t\. 45. f, f, S; A, Hv A, H, H, A; M. 46. A, A, \h Ih hh \h iT, ^T, hh «. 47. A. if; li Ih hh If, ih IS, if; if • 48. ft, ft, A; II; ft; f I; il; ft; A; %h 49. A; if; if; IS; If; M; If; ft; if; f«- 50. i«, fi, H, f §, ii f f, f J, Jt, A; f |. 51. A%; A%; A%; A^o; A%; A%; A%; A^o; A%; AV WRITTEN EXERCISES. 52. Change f § to smallest terms. Solution. — Dividing both terms of -f -J by 3, we have J-|, and dividing both terms of \^ by 5, we have -f, whose terms have no common factor ; hence, |f changed to smallest terms 3)3 0. _ 10. 3)75 ~ 2 5 5) 10. — 1 5)25 -" 5 68 COMMON FRACTIONS. 53. Change j^^ to its smallest terms. 54. Change Iff to its smallest terms. 115. To change a fraction to its smallest terms : Rule. Divide both terms of the fraction by any common divisor , treat the resulting fraction in the same way, and so continue until a fraction is found whose terms are jjrime to each other. Note. — The greater the common divisor used, the shorter will be the pro. cess. Change to their smallest terms : 55. m- 59. m- 63. m- 67. 1^7^- 56. i^- 60. m- 64. m- 68. im- 57. t'A- 61. m- 65. ^^Vn- 69. m- 58. Iff- 62. in- 66. VsV- 70. mh An Integer or Mixed Number to an Improper Fraction. ORAL EXERCISES. 71. In 2 apples how many fourths of an apple ? In 5 apples ? In 8 apples ? 72. Eeduce 6 to fourths ; 3 to fifths ; 7 to sixths. 73. How many fourths of an apple in 3J apples ? Iw 6J apples. Change to improper fractions : 74. 1§, If, li, 1^, 1|, 2f, 3^, 5J, 4i, 3§. 75. 2t, 25, 3i, 3^, 3f, 3f, 3J, 4^, 4§, 4f. 76. 4i, 4f, 4J, 51, 5/^, 5?, 5i, 5|, 5|, GJ. 77. 6§, 6f, 65, 6f, 65, 6|, 7i-, 8^, 8§, 8|-. 78. 81 9J, 85, 4^j, 6^„ 4f, 6/^, 8^, 9^, 5f COMMON FRACTIONS. 69 WRITTEN EXERCISES. 79. Change 19 to eighths. 19 8 eighths. Solution. — In 1 there are 8 eighths ; in 19 152 there are 19 times 8 eighths, or i|-^. 8 80. Change 197 to the form of a fraction. 81. Change 21| to ninths. 2lf Solution. — In 1 there are 9 ninths, and in 21 there zr-i are 21 times 9 ninths, or 189 ninths : 189 ninths and 194 "sT 5 ninths are 194 ninths, or J-|4-. 82. Change 27i^5 to thirteenths. 83. Change 37^^ to elevenths. 116. To change an integer or mixed number to an im- proper fraction : Rule. Multiply the integer by the denominatoi', and, if there is a fractional part, add its numerator to the product, and write the result over the denominator. 84. Eeduce 93 to fifteenths ; 107 to twentj^-firsts. 85. Reduce 115 to a fraction whose denominator is 13. Keduce to fractions in their smallest terms : 86. 9-^^. 90. 14|§. 94. 81^^. 87. 56t\. 91. 98|f. 95. 25/7J5. 88. 104^. 92. 3|f. 96. 153t^. 89. 4tJ^. 93. 142f. 97. ll^^U^, An Improper Fraction to an Integer or Mixed Number. ORAL EXERCISES. 98. How many dollars in 32 quarter-dollars ? In 40 quarter- dollars ? In $ -y ? In $ 4^- ? In $ ^^ ? 70 COMMON FRACTIONS, Change to integers or mixed numbers : 99. f , ¥, -¥-, ¥, h h ¥, -VS ¥. ¥• 100. -\% V-; f; h -¥-; V-, -VS -VS ¥; ¥-. lOL -V-, -2^s ¥; ¥, ¥. ¥-, ¥-, ¥. -¥-. -¥- 102. i^-, -3/-, -Y-, -% ¥; -¥; ¥; f f ; f i; -¥- 103. ^-i-, VS ¥; v. ¥; n, ¥. tf; tl; ¥- WRITTEN EXERCISES. 104. Change ^^^- to an integer or mixed number. 19) 793 Solution. — As 19 nineteenths are 1, 793 nine- -^ teenths will be as many ones as times 19 in 793, 19 or 41^|. Ans. 41i|. 14 105. Change -\-^ to an integer or mixed number. 106. Change ^yf ^ to an integer or mixed number. 117. To change an improper fraction to an integer or mixed number : Rule. Divide the numerator hy the denominator, Eeduce to integers or mixed numbers : 107. ff. 112. IJ, 117. ^V-. 108. ^J^. 113. -Vt^.. 118. ^^K 109. ^^^, 114. lf&. 119. m-' 110. fg-J. 115. ifp. 120. -^ili^. Ill le. 116. -Wjf-. 121. ^n^. Fractions to Fractions having a Common Denominator. ORAL EXERCISES. 122. Express J as tenths ; J as tenths. 123. Express J and \ each as twelfths. COMMON FRACTIONS. 71 124. Change f and | to fractions having the same denomi- nator. 125. Change -f and f to fractions having the same denom- inator. What is a multiple of 6 and 5 ? 126. Change ^, §, and J to fractions having the same de nominator. 127. Express f, |, and /^ each as fortieths. 128. Express §, f, and f each as eighteenths. 129. What is the least common multiple of 3, 6, and 9 ? 118. Fractions have a Common Denominator when their denominators are alike. 119. The smallest denominator common to two or more fractions is their Least Common Denominator. 120. A Common Denominator of several fractions must be some common multiple of their denominators. The Least Common Denominator of several fractions must be the Least Common Multi;ple of their denominators. WRITTEN EXERCISES. 130. Change f and f to fractions having a common denom- inator. 2X7 _ 14 Solution. — As the product of the denomina- tors, 5x7, or 35, is their common multiple, 35 5X7 — 3 5 3.x 5 __ 15. must he a common denominator of the fractions. Changing then the fractions to thirty-fifths, we have \\ and f|. 131. Change f and | to fractions having a common denom- inator. 132. Change §, |, and -g to fractions having the least com- mon denominator. 72 COMMON FRACTIONS. 4 ^ 01 12 — 12 Solution. — The least common multiple of the denominators is 12, which must be the least 3 „ ^ _9^ common denominator of the fractions. Chang- ^ ot 1 2 - 12 ij^g thentt- ---— - - « 2 5 ^ir >^ _ 1_0 T^^, and ^0. ^ ot 1 2 — 1 2 ^ ^ ^ ^ ^ ing then the fractions to tweKths, we have ^jy 133. Change |, f , and | to fractions having the least com-^ mon denominator. 134. Reduce |, |, and \^ to fractions having the least com- mon denominator. 121. To reduce to fractions having the least common denominator : Rule. Change each fraction to its smallest terms. Divide the least common multiple of the denominators hy the denomina- tor of each fraction, and multiply both terms of the fraction hy the quotients Note. — When the denominators are nmtnally prime, take their product for the common denominator, and multiply each numerator by all the denomina- tors except its own for the new numerator. Reduce to fractions having the least common denomina- tor: 135. f and ^^. 141. ^^, \% and \\. 136. f and -^. 142. ^-^ \, and -V-. 137. T^^andH. 143. ^, Jf, ^^, and ^^. 138. /^ and ^^, 144. ^7^, yj^, y^^^, and 7. 139- -ft, ifr; and ^^. 145. ^^, f , ^, and /^. 140. J, $, andif 146. |, -,i>^, and |. 147. Reduce ^, f , and -^ to fractions having the least com- mon denominator, and show by the numerators which of the fractions has the greatest value. COMMON FRACTIONS. 73 ADDITIOISr. ORAL EXERCISES. 148. Mary paid | of a dollar for a book, | of a dollar for a hat, and f of a dollar for a handkerchief. How many eighths of a dollar did she spend for all ? 149. Smith owns ^ of a vessel and Keen f . How many sixteenths does Keen own ? What part of the vessel do both own ? Give the sum of the following fractions : 150. 151. 152. 153. 154. h + i i + 1 f + i t^ + tV i+i+ i § + 1 t + § i + h 1 + i 4 + i+ i h + i i + f 1 + « tV+ f i + i + ^ 3 + i f + 4 tV+ i i + t § + 1 + 1^2 A+ 4 1 + i f + 4 A+ § l + f+ f 122. Like Fractions are like parts of the same unit. Thus, f ofa dollar and |- of a dollar • also, i\ and ^■"j- are like fractions 123. Principle. Only ^ like fradioTis can he added. WRITTEN EXERCISES. 155. What is the sum of f , %, and {^ ? 5 , 3 _^ 11 _ Solution. — Reducing the given frac^ ^ ^ ^^ tions to fractions having the least comnion If + Ti + If = denominator, we have |f, ^% and ||, ||. -^ 22 T = 2|- which added give f-J- = 2^, or 2-J. 156. What is the sum of f , H> ^^^ tS ^ 157. What is the sum of ^, \i, and ^ ? 74 COMMON FKACTIONS. 124. Rule for Addition of Fractions. Change the fractions^ if necessary, to fractions having a common denominator, add the numerators, write the sum over the common denominator, and simplify the result, if needfuL 158. Add I, -i^, and -i^, 162. Add -f^, /y, and ^^. 159. Add if, 11, and l{, 163. Add %S h and f J. 160. Add If, and f^. 164. Add J, ^^, IJ, and /^. 161. Add f , ^j, and^. 165. Add ^3_, ||, zj^ and AV- 166. What is the value of f + If + V + i ^ 125. When there are mixed numbers, the fractions and the integers may he added separately y and the results united, 167. What is the sum of 15| and 24f ? 168. What is the value of 37| -f 109f + 341^t- ? 169. A man paid $ 4| for a hat, $ 16 J for a coat, and $ 5^ for a pair of boots. How much did he pay for the whole ? 170. A has in his farm 160| acres, B has 67 1% acres, and C has 85^^ acres. What is the number of acres in the three farms ? SUBTRACTION. ORAL EXERCISES. 171. A man owned \^ of a ship and sold /^. What part of the ship had he left ? 172. \^ less -^^ is what part of 1 ? 3 73. How much is \l less |J ? ^f less \\ ? 174. How much is || less f ? Solution. — f is il ; and f| less ^f is ^. 175. 176. 177. § -i = ? * - A = ? 3- 1 =? 4 -i = ? i - f = ? ^-Tft=? T^ - i = ? 8- i =? i - i = ? # -4 = ? S- i =? S- f =? A-i = ? 1- § =? «- i=? COMMON FRACTIONS. 75 178. 179. f - T^ - ? # - A = ? I - A = ? f - t't = ? !--«-? il - « = ? 180. Jane had $ 2 J, and spent $ 1|. How much had she left ? 126. Principle. Only like fractions can he subtracted the one from the other. WRITTEN EXERCISES. 181. From if subtract /_. 13 ^7_ __ 3.1 _ 18. -^ 11 Solution. — Reducing the 16 12— 4848? or 48 . ^ . „ ^ . given iractions to iractiona having the least common denominator, we have ff and |f. |-| sub- tracted from II leaves W. 182. Find the difference between fj and f J. 183. From §g take §g. 184. From i J take ^\. 127. Rule for Subtraction of Fractions, Change the fractions, if necessary, to fractions having a common denominator, find the difference of the numerators, and write it over the common denominator. 185. From \l take ^^, 189. From f -f take f |. 186. From JJ take ■^^. 190. From ^V^ *ake yVo- 187. From §J- take ^^. 191. From ^^ take yi§^. 188. From \{\ take ^j^^. 192. From %\ take |f . 193. What is the difference between ^f and -^ ? 194. I copied by mistake Jf instead of ^^. What is the amount of error I made ? 195. How much is 49f less 31f ? 76 COMMON FRACTIONS. 49f = 49H 31| = 4811 = 3l|i 1711 Solution. — Changing the fractiuns to fractions having a common denominator, we have 49 Jf and 31f|-. As we cannot take 1^ from ^, we take 1, or |f, from 49, leaving 48 ; and adding the fl to |-|, we have 48|-|. Subtracting || from ||, and 31 from 48, we have, as the result required, 17|-|-. 196. 31| - 101 r= ? 199. 2911 -16'l = ? 197. 63 - 54/2 = ? 200. 311 - 30i| = ? 198. 73^^-67jIj, = ? 201. 103i-gf-99H=.? 202. A boy paid for a sled $ 7|, and for skates $ If. How much more did he pay for the sled than for the skates? > 203. The greater of two numbers is 150, and the smaller is 147iJ. What is their difference ? 204. Two men start from the same place and travel in the same direction. When one has gone 17 2^0 miles and the other 19 f miles, how far are they apart ? MULTIPLICATION. A Fraction multiplied by an Integer. ORAL EXERCISES. 205. Multiply j\ by 4. j — ■— E 3_ X 4 _ 12 ^^^, ;;^ _ a 16 — 16? ^^ 16:4— 'i Solution. — Multiplying the numerator of ^^ by 4 gives ^|, equal to I ; or, dividing the denominator of ^-^ by 4, gives |. In either case, ^ is multiplied by 4. For it will be seen by the illustration that multiplying the numerator by 4 increases the number of fractional units 4 times, while their size remains unchanged ; and that dividing the denominator hy 4 increases the size of the fractional units 4 times, wliile their number is unchanged. The eflfect of either process, therefore, is to increase the value of the fraction 4 times. COMMON FRACTIONS. 77 206. At $J each, how many dollars will 12 arithmetics cost ? 207. At the rate of f of a bushel of grain a week, how many bushels of grain will a horse consume in 6 weeks ? 208. If 1 hat costs $ 3 J, what will 8 hats cost ? 209. Multiply the fractions in Exercises 27 to 32 by 3 ; b}; 4; by 5; by 6; by 7; by 8; by 9; by 10; by 12. 210. Multiply the mixed numbers in Exercises 74 to 78 by 4; by 5; by 6; by 8; by 9; by 10; by 12. 128. Principle. A fraction is multiplied either by multiplying its numerator 9r by dividing its denominator, 7 / WRITTEN EXERCISES. 211. Multiply \\ by 7. Solution, Or, 212. Multiply ^y by 6. 216. Multiply ^ by 35._^ 213. Multiply ig- by 16. 217. Multiply ^^\ by 18. 214. Multiply §f by 40. 218. Multiply |J by 48. 215. Multiply II by 85. 219. Multiply -W- by 76. 220. What is the product of 63| by 5 ? 03 3 Solution. — 63f is 63 1X5== "V" = 3| 5 and |. I multiplied by 5 ^\~\§ gives 3|, and 63 multi- ^^' — ^ plied by 5 gives 315. 63 X 5 =315 ^1" These results added give, Q-|o3 as the product required, 318^ 31 8|. _ „ 78 COMMON FRACTIONS. 221. Multiply 40t7^ by 27. 223. Multiply 39^^^ by 75. 222. Multiply 81f by 63. 224. Multiply V25j\ by 80. 225. What will 72 barrels of St. Louis flour cost at $ 8f a barrel ? An Integer mulfiplied by a Fraction. ORAL EXERCISES. 226. If a bushel of corn costs 64 cents, what will J of a bushel cost ? J of a bushel ? 227. If a pound of tea costs 60 cents, what will J of a pound cost ? f of a pound ? 228. What is f of 4 ? Solution. — ^ of 4 is 4", and -f of 4 must be 5 times f , or ^', equal to 2f . 229. What is t3^ of 8 ? f of 3 ? f of 5 ? | of 10 ? 230. What is the cost of J of a cord of wood at $ 6 a cord ? What is $ 6 multiplied by J ? 129. Principle. Multiplying a number by a fraction is taking such a part yf the number as is denoted by the fraction. WRITTEN EXERCISES. 231. Multiply 180 by ■^, 15) 180 180 Solution. — 180 x j\ is the same "12 ^ _^ as ^4^ of 180. ^V 0^ 180 = 12, and 4 ^^' 15) 720 t\ of 180 = 4 times 12, or 48. 48 48 Or, as ^ is the same as -^ of 4, 180 X Y^ is the same as ^ of i times 180. 4 times 180 = 720, and ^ of 720 = 48. 232. Multiply 108 by J. 236. Multiply 144 by |J. 233. Multiply 89 by y\. 237. Multiply 239 by -^ir- 234. Multiply 375 by |J 238. Multiply 404 by y\. 235. Multiply 137 by f 239. Multiply 376 by |f COMMON FRACTIONS. 79 240. Multiply 33 by 3f. 33 X f - 13i 33 X 3 r= 99 Or 33 99 241. 242. 245. 112i 1121 Multiply 110 by 9|. Multiply 85 by 13j. Solution. — 3| is 3 and | ; 33 multiplied by | gives 13|, and 33 multiplied by 3 gives 99. These results added give, as the product required; 112f 243. Multiply 84 by 9t^3^. 244. Multiply 145 by llf What cost 25^ acres of land at $ 50 an acre ? A Fraction multiplied by a Fraction. ORAL EXERCISES. 246. What part of a dollar is J of J of a dollar ? J of J of a dollar ? 247. What part of a dollar is J of § of a dollar ? J of | of a dollar ? 248. What is i of i ? ^ of ^ ? i of ^ ? i of ^ ? 249. What is J of f ? i of f ? J of f ? I of f J ? ^ 250. How much is J of f ? 1 nf 1 1 6' 1 nf 3 _ 3 Solution. — ^ of f is ^^, and f of | is 3 times j\, equal to This, also, appears by the illustration. A- 251. § X| = ? i x| = ? i X + -? tVX| = ? Ax« = ? 252. i xf = ? ■A X f = ? A x f = ? Ax^ = ? 253. tV X i = ? f xi = ? I xf = ? ? x# = ? T^^X| = ? 254. Give the product of each pair of fractions in Exercises 150 to 153. 80 COMMON FRACTIONS. 130. A Comjpound Fraction is a fraction of a fraction, as I of |, and may be considered an expression of multi- plication. A Simple Fraction is a fraction not connected with another, and having both its terms integers. WRITTEN EXERCISES. 255. Multiply f f by f . g Solution. — f -J 3 v/ 5 _ 15 _ 2 5 oj. ,S^x 5 _ 2 5 multiplied by I SlXe — 186~3 1'"^' 31X^6-— 31 . .^ "^^ IS the same as f of f ? ; i ol' If is A'e . and | of f f is 5 times ^^^ equal to ^|i or f f . Note. — By canceUation we may shorten the process, as shown in the second form of the work. 256. Multiply If by \\ ; Jf by f f . 131. Rule for Multiplication of Fractions. Write the 'product of the numerators over the product of the denominators. Note. — The rule is general, since a mixed number may be changed to an improper fraction, and an integer may be expressed in a fractional form. 257. Multiply f f by 27. 260. Multiply J by f of ^. 258. Multiply tf by e. 261. Multiply -Y- by | of t%. 259. Multiply f gj by f . 262. Multiply f f by \ of f i. 263. What is the value of f of f of ^ ? 264. What is the value of|X|XtXf? 265. What cost llf cords of wood at $ 7| a cord ? 266 What is the value of V- of 3^- X f X § of 10 ? 267. What is the value of f of f of ^y multiplied by § of 18 ? 268. What is the product of ^ of 8^^ multiplied by f of 9i? 269. What will 5^^ tons of hay cost at $21^^ a ton ? 270. If a man walks 4-i^^ miles in an hour, how many miles can he walk in } of S-j^^ hours ? COMMON FRACTIONS. 81 DIVISION. A Fraction divided by an Integer. ORAL EXERCISES. 271. What is § divided by 2 ? Solution. — J divided by 2 is f-^^, equal to ^., Or, as f divided by 2 is the same as ^ of |. we have f^g? equal to It will be observed by the illustration that dividing the numerator by 2 halves the number of fractional units, while their size remains un- changed ; and that multiplying the denominator by 2 halves the size of the fractional units, while the number remains unchanged. Either process divides the fraction by 2. 272. What is ^^ divided by 3 ? \ihj5? 273. Divide |i by 7 ; Jf by 9 ; f | by 7 ; ^^^^ by 11. 274. Divide the fractions in Exercises 99 to 103 by 2 ; by 3 ; by 5 ; by 6 ; by 8 ; and simplify'- the result. 275. If 8 men can mow | of a field in a day, what part of it can one man mow ? 276. How is -^(j divided by 5 ? f by 8 ? f by 9 ? 277. Divide J by 6; H by 7; y^^ by 10; u'hy 6. 278. If 7 pounds of coffee cost $2y'^, how much will 1 pound cost ? Solution. — $ 2-^Q = $ ^1 ; if 7 pounds cost $ f|, 1 pound must cost|of$fi orf^V 279. How much is 2tV divided by 7 ? 11^ divided by 17 ? lOf divided by 25 ? 280. If 9 bushels of wheat cost $ 19|, what is the cost of 1 bushel ? a 82 COMMON FRACTIONS. 132. Principle. A fraction is divided either hy dividing its numerator or hj 7Rultiplying its denominator. WRITTEN EXERCISES. 281. Divide ^f by 9. 18 23 Q 18 -^ 9 = 23 2 Or, 1 8 . 0—11 _JL8_: 2'3 ~ ^— 23X9 — 2 OT Solution. — As a fraction is di- vided by dividing its numerator^ we divide the numerator by 9, and have 2^3-. Or, as a fraction is divided 2 by multiplying the denominator, 2^ 3" we multiply the denominator by 9, and have, as before, ■^^, 282. Divide tV ^y 12. 283. Divide hi ^y 49. 284. Divide ^^ybyll. 285. Divide m ^y 21. 290. Divide 17f by 6. 6)17f 943 ^48 Divide f J by 60. Divide f 4. by 16. 286. 287. 288. Divide Or 13 J.. 17 — XI -r- a — 13-^ —13 9 >43 T^T ^y 5- 289. Divide f f J by 75. Solution. — 6 in 17f , 2 times, and 5| remainder. 5| = ^/, and ^ divided by 6 gives ||, which, united with the partial quotient 2, gives 2|-|. Or, 17| zz: J^p, and ^3. di- vided by 6 gives, as before, m 291. When $ 28 is paid for 2-i^ tons of hay, what part of a ton will $ 1 purchase ? 292. When $ 107^^^ is shared equally by 7 men, how many dollars will each man receive ? 293. The product of two numbers is 30|^, and one of the numbers is 9. What is the other number ? 294. When 12 cords of wood cost $ 76^, what is the cost a cord ? COMMON FRACTIONS. 83 An Integer divided by a Fraction. ORAL EXERCISES. 295. How many fourths of a dollar in $ 1 ? In $ 3 ? 296. How many times is J in 1 ? In 2 ? In 3 ? 297. In 4 how many times i? J? §? J? f? 298. At $ J a pound, how many pounds of tea can be bought for I 6 ? Solution. — As many pounds as times f in 6. In 6 there are ^^, and I are contained in ^-^ 8 times. Ans. 8 pounds. 299. In how many days can a horse consume 4 bushels of oats, if he consume ^^3^ of a bushel a day ? 300. Divide any given integer by the fractions in Exercises 27 to 33. 301. At $1J a yard, how many yards of cloth can be bought for $ 10 ? 302. In how many weeks will a boy earn $ 8, if he is paid $2|aweek? WRITTEN EXERCISES. 303. Divide 65 by f Q^ _ A 5 A . JL 5 6. _^ 5 _ g-j^ Solution. — 65 r= ^^ ; and "I is contained in ^^ as many ^ 1 3 times as 5 in 455, or 91 times. 65 -^ f = ^^"^^ = 91 Or, I is contained in 65, 7 times 65 times, and f , ^ of 7 times 65 times, or 91 times. This process is the same as multiply- ing the dividend by the divisor inverted. 304. Divide 49 by Jf . 309. Divide 98 by H- 305. Divide 27 by |J. 310. Divide 128 by S^\. 306. Divide 84 by f |. 311. Divide 432 by 2§. 307. Divide 49 by +f . 312. Divide 100 by If 308. Divide 60 by j\. 313. Divide 118 by 5|. 314. How many yards of cloth at $ | a yard can be bought for $56? ,.^ 84 COMMON FRACTIONS. 315. If a family consumes 2| pounds of sugar a week, in what time will it consume 44 pounds ? 316. There is a board 17 feet long, which I wish to saw into pieces 2f feet long. What will be the number of pieces ? A Fraction divided by a Fraction. ORAL EXERCISES. 317. How many times f of a yard in f of a yard ? 318. At I of a dollar a pound, how many pounds of coffee can be bought for | of a dollar ? 319. How many times f in | ? | in | ? f in Y- ? 320. How many pounds of raisins at -^^ of a dollar a pound can be bought for f of a dollar ? Solution. — As many pounds as times y% in f . In f there are ^|-, and y\ are contained in ^|, 4 times. Ans. 4. 321. How many times ^V in f ? i in f ? | in ^3 ? 322. If it takes f of a yard of cloth for a vest, how many vests can be made from 2J yards ? 323. 324. 325. A- -i = ? i - -| = ? 1 - -4 = ? i - -f = ? « - -i = ? 2J- -t = ? H- -f = ? fT7- -f = ? 2t- -f = ? ^ - -§ = ? 5 - -§ = ? H- -§ = ? f - -i = ? A- -| = ? H- -i = ? WRITTEN EXERCISES 326. Di vide 1 by §. Solution. — t -^ I I - f = if - i^ = if = I = li •« equivalent to ^ di- vided by ^f or to {^, Or, ^ equal to |, or 1^. 1 5 ?A5 t r ^^1 is III, equal to i, or H- Thia is the same as multiplying tlic dividend by the divisoi inverted. COMMON FRACTIONS. 85 327. What is the quotient of J divided by f ? 328. What is the quotient of Jf divided by /^ ? 133. Rule for Division of Fractions. Change the fractions, if necessary, to fractions having a common denominator, and divide the numerator of the divi- dend by the numerator of the divisor. Or, Invert the divisor, and proceed as in multiplication of frac tions. Note 1. —The rule is general, since a mixed number may be changed to an improper fraction, and an integer may be expressed in a fractional form. Note 2. — A fraction having a fraction in one or both of its terms, as in ' 3 ' 7 ^^ called a Complex Fraction^ and may be considered an expression 3 £ s of division. 329. Divide f f by \%, 334. Divide f J by f f . 330. Divide if by f . 335. Divide 9| by 4J. 331. Divide \^ by If. 336. Divide f §| by iff 332. Divide iJ by f. 337. Divide 17 2^ by 5f. 333. Divide ^j% by iff. * 338. Divide lOOg by 8f. 339. How much is tea a pound when \^ of a pound cost if of a dollar ? 340. How much is hay a ton when | of a ton cost $ llj ? 341. If a man walks 3f miles in an hour, in what time will he walk 15^^^ miles ? 342. Simplify ||. Solution. — Consid' 3 1 _ JL5_ -^- 1 = IJl x^ — ^4 = 14-1- erinff the fraction as an ^ ^|— 4*3 4'^8 — 32 -*-32 ^ . o ^' - - expression oi division, ^^j we reduce its terms to 2"l^^'32"~l3"2" ' improper fractions, and, dividing, have f|, or 1 Jf . Or, multiplying both terms of the complex fraction by 12, the least common multiple of the denominators of the fractional parts, we have, as before, f|, or Ijf . 86 COMMON FRACTIONS. 343. Simplify S. 345. Simplify-^. 347. Simplify—-. 5 t ^A 12 5^ 6# 344. Simplify—. 346. Simplify j|. 348. Simplify -|. 104 349. Eeduce ^^ to its simplest form. 350. If a man can do J f of a piece of work in 23 f days, in what time can he do the whole ? 351. At the rate of 31 x\ miles an hour, in what time will a train of cars move 125^ miles ? To find the Fraction one Number is of another. 352. One is what part of 2 ? Of 3 ? Of 4 ? Of 5? 353. What part of 5 dollars is 1 dollar ? 2 dollars ? 3 dollars ? 4 dollars ? 354. What part of 7 miles is 3 miles ? 4 miles ? 5 miles ? 6 miles ? 355. What part of 5 is § ? Solution. — 1 is ^ of 5, and f of 1 is | of ^ of 5, or -^ of 5. 356. What part of 4 dollars is f of a dollar ? 357. What part of ^^ is ^\ ? What part of f is |, or | ? 358. What part of 8 dollars is 2J dollars ? 134. The Fraction one number is of another is found bj dividing the number denoting the jpart ly the number denot- ing the whole. WRITTEN EXERCISES. 359. What part of 125 is* 75 ? 363. What part of 20 is 3]^? 360. What part of 7 is G ? 364. What part of 90 is 6J? 361. What part of 12 is |f ? 365.' J J is what part of |§ ? 362. What part of 186 is 93? 366. ^J is what part of J J ?. COMMON FRACTIONS. 87 367. What fraction of 84 is 91 ? 368. What fraction of 3f is 2^ ? 369. A tree, whose height was 85 feet, was broken off in a gale, 55 feet from the top. What part of the tree was left standing ? 370. A sportsman started up a flock of 144 birds, and shot § of them. What number escaped ? 371. If 25 yards of carpeting cost $ 37.50, what should 35 yards cost at the same rate ? 372. John has 156 cents, and Peter 106 cents. How does Peter's money compare with John's ? 373. If a man can do a piece of work in 7f days, what part of it can he do in 6 J days ? 374. What will be the cost of ^ of a cord of wood, when 12 cords cost $68^? To find the Whole when a Fractional Part is given. 375. 4 is ^ of what number ? 6 is ^ of what number ? 8 is J of what number ? 3 is ^ of what number ? 376. I spent $ 6, which was ^ of all my money ; how much had I? 377. I lost f of my sheep, but had 8 left ; how large a flock had I at first ? 378. 18 years is f of my age ; how old am I ? Solution. — Since 18 years is 3 fourths of my age, 1 fourth of my age is J of 18 years, or 6 years; 4 fourths of my age is 4 times 6 years, or 24 years. Ans. 24 years. 379. -«- of what number is 24 ? 380. 32 is f of what number ? 48 is f of what number ? 381. 36 is ^^ of what number ? 54 is j% of what number ? 382. 42 is I of what number ? 84 is /^ of what number ? 383. 63 is ^j of what number ? 75 is f -f of what number ? 384. How far is it to Boston if f the distance is 21 miles ? 88 COMMON FRACTIONS. 385. My house is insured for f its value, or for $ 4000 •, what is it worth ? 386. The cost of my buggy is | the cost of my horse. The horse cost $ 360 ; what did the buggy cost ? WRITTEN EXERCISES. 387. 1728 is If of what number ? 1 Q g Solution. — Since 1728 is if of an —iA^ X 19 = 2052 unknown number, ^^ of the unknown number is ^ of 1728, or 108; || of the unknown number are 19 times 108, or 2052. Aus. 2052. 388. 351 is 2^5 of whaf number ? 135. To find the whole when a fractional part is given, divide the part hy the numerator of the fraction^ and multi- ply the quotient by the denominator, 389. 1368 is ^T of what number ? 390. 891 is 4 J of what number ? 391. 8000 is \U of what number ? 392. What is f of a number if 273 is f J of it ? 393. My income in 1881 was $ 1600, or ^ of my income in 1880 ; what was the falling off ? 394. jf of 340 is T^T of what number ? 395. f is ^^ of what number ? /T$96. 7 J is tV of what number ? 397. Sold goods for |462, and thereby lost ^ of the cost; what was the cost ? 398. A man failing in business could pay me only $ 3930, which was J of whftt he owed me. How much did he owe me ? 399. A has $ 2000, and B has l-,^ times as much. How much has B ? 400. My house and furniture cost % 10901, and the furniture cost f as much as the house. How much did the house cost ? COMMON FRACTIONS. 89 MISCELLANEOUS EXERCISES. 401. At $ f a yard^ how many yards of cloth can be bought for$43i? 402. If 2^ pounds of butter cost $^f, what is that a pound ? 403. How many yards of cloth f of a yard wide will equal 12 yards f of a yard wide ? 404. If I of a barrel of oatmeal is worth $ 5.60, what is a barrel worth ? j/^405. What is the value of -|- oi 12^ divided by i of 8| ? 406. What number must be multiplied by 7| that the pro- duct may be 20 ? 407. A can walk S^y miles in 60 minutes, and B can walk /t as fast as A. How long will it take B to walk the same distance ? 408. If $ 106^ is shared equally among 8 men, how much will each man's share be ? 409. If of $ 350 you should spend $ 125, what part of the money would remain ? 410. There is a board 19 feet in length, which I wish to saw into pieces 2f feet long. What will be the number af pieces, and how many feet will remain ? 411. How much is coal a ton when 10^ tons can be bought for $ 67^ ? 412. What is the relative value of || and {^ ? >^413. Eeduce -^3- — ^— — to its simplest form. 414. What number must 2^ be multiplied by to give 4J ? ' 415. Find the value of ^ of Ij X 4^ divided by /^ of IJ X3i. 416. Jones and Smith plowed for a certain sum of money ; Jones plowed 6 J acres, and Smith 9f acres. What should be Smith's money, if Jones's share is $ 20 ? 90 COMMON FRACTIONS. 417. What number must be taken from 12|, and the re- mainder multiplied by 10|, that the product shall be 50 ? 418. When $236 are paid for llf acres, what will 20/^ acres cost ? 419. In counting my money I found I had $969, which was just 2 1 times as much as my brother had. How many dollars had I more than my brother ? QUESTIONS. 99. What is a fraction ? 100. What ia the denominator of a frac- tion ? 101. What is the numerator ] 102. What are the terms of a fraction ? 105. What is a proper fraction 1 106. An improper fraction ? 107. A mixed number ] 108. How may a fraction be regarded 1 109. What is the value of a fraction ] 110. What is reduction of fractions? 112. How is a fraction reduced to larger terms? 113. When is a fraction reduced to its smallest tenns ? 114. What is a principle of fractions ? 115. How is a fraction reduced to smallest terms ? 116. How is an integer or mixed number reduced to an improper fraction? 117. How is an improper fraction reduced to an integer or mixed number ? 118. When have fractions a common denominator? 120. What must be a common denominator of two or more fractions? 121. How are fractions reduced to their least common denominator ? 122. What are like fractions? 123. What fractions only can be added? 124. How do you add fractions? 126. What fractions only can be subtracted? 127. How do you subtract one fraction from another ? 128. How is a fraction multiplied ? 129. How is a number mul- tiplied by a fraction ? 131. How is one fraction multiplied by an- other? 132. How is a fraction divided? 133. How is one fraction di* Vided by another ? KEVIEW. 91 REVIEW. ORAL EXERCISES. 136. 1. Kame the only even prime number. 2. Show that 56 is a composite number. 3. What is the greatest factor common to 16 and 24 ? 4. What is the least common multiple of 3, 4, and 6 ? 5. Eeduce 9| to fifths ; 7f to eighths ; lO^^^ to elevenths. 6. Eeduce to smallest terms |f ; /^ ; ^f ; ^^^ ; ^^. 7. A has worked f of a day, and B i of a day. What part of a day have both worked ? 8. I of a pole is in the water, | in the mud, and the rest in the air. What part is in the air ? 9. James is 18 years old, and his age is f of the age of his father. How old is his father ? 10. Damon sold a house lot for $ 30, which was | of what it cost him. What was the cost of the lot ? 11. Multiply J by 6; H by 7; 5 by f; I by f. 12. If a man can mow If acres of grass in a day, how much can he mow in 6 days ? 13. If cloth is worth -f^ of a dollar a yard, what is | of a yard worth ? 14. Divide ft l^y 3 ; ii by 4; 8 by § ; I by 1 ; 2i by f. 15. Bought 5 bushels of wheat for $ 7J. What was it a bushel ? 16. How much is J of 8| ; i of 5f ; i of 7f ? 17. There is ^ pole standing so that f of it is in the water, ^nd f as much in the mud. How much is in the mud ? 18. If f of a bushel of grain will serve a horse a week, how many weeks will 6| bushels serve him ? 19. If 1| tons of hay will keep a cow 8 weeks, how manj cows will 10 tons keep for the same time ? 20. What number is that, ^ of which exceeds J of it by 3 ? 21. The difference between f and -/^ of a number is 2. What is the number ? 92 REVIEW. 22. If to William's age there is added f of his age he will bt 25 years old. What is his age ? 23. A man bought a coat and a hat for $30. The hat cost f as much as the coat. What was the cost of each ? 24. A and B made an even exchange of horses. By the trade A lost $ 40, which was f of the value of the horse he had at first. What was the value of each horse ? 25. Joseph spent /^ of his money for clothes. He then paid away $ 6f j which was just ^ of all he had left. How many dollars had he at first ? 26. If 25 is I of some number, what is | of the same num- ber ? 27. A boy spent f of his money, and then had given him I as much as he had left. What part of his money did he then have ? 28. What cost 12 apples at the rate of 7 for 5J cents ? 29. If 6 men can do a piece of work in 4f days, in what time will 1 man do it ? In what time will 5 men do it ? 30. If you can travel 13 miles in 3 hours, how many miles can you travel in 8 hours ? 31. If 5 men can cut 6| cords of wood in a day, how many cords can 7 men cut in the same time ? 32. If A can do a piece of work in 7 days, and B in 5 days, what part of it can both do in a day by working together ? 33. If A can build a wall in 3 days, which B can build in 4 days, in what time can they build it by working together ? WRITTEN EXERCISES. 34. Change §j|^ to smallest terms. 35. Find the product of the common prime factors of 1728 and 2880. 36. What number contains each of the prime factors of 243y 972, 576, but no others ? 37. Show that J$f is greater than J and less than f . 38. Find the sum of Jg, f, and ^7^. KEVIEW. 93 39. rind the sum of f , Ij, V", and ISyV- 40. What number added to ^J^ will make 16 ? 41. If 9J yards cut from a roll of cloth leaves 24j yards, what is the length of the roll ? 42. What is the value of 2f times i oi ^^? 43. Find the number that must be added to f of | to make I off. 44. A man has 229J pounds of honey, which he wishes to pack in boxes containing S^ pounds each. How many boxes will he require ? 45. What fraction must %^ be divided by to make the quo- tient 31^ ? 46. Find the least number which, divided by 6, by 8, and by 9, gives in every case a remainder 4. 47. The tide rose f of a foot in one hour, *|| the next, and f the third hour. How much did it rise in the 3 hours ? -^48. Change ^ to an equivalent fraction, having 91 for its denominator. 49. How many cakes would be required for a school of 53 children, of whom 27 are boys, if each girl has ^ of a cake, and each boy ^ as much again as each girl ? ^ "So. How many times does t'^ + i + i contain J + i + i ? 51.. What is the product of § of V" multiplied 1^7 j| ? 52. What fraction of 3 bushels is -^^ of 2f bushels ? 53. The difference between f and | of a number is 10. What is the number ? 54. Eeduce f of ^^ of ^ of 8| X V ^^ ^ simple fraction in its smallest terms. 55. Sold a horse for $ 105|, which was | of its cost. What was its cost ? 56. What cost 13^^^ tons of coal at $ 7|- a ton ? 57. Bought IJ bushels of corn for $ 3f f ; what was it a bushel ? 94 KEVIEW. 58. How many acres of land at 1 17| an acre can be bought 59. When $ 2.13 are paid for | of a cord of wood, what cost 1 cord ? What cost lOj cords ? 60. A boy, after spending | of his money in oranges, finds that ^ of what he has left is 24 cents. How much money had he at first ? 61. If 28 men can build an embankment in 42 days, in how many dsijs can f as many men build it ? 62. How much cambric that is f of a yard wide will line 6| yards of cloth that is 1^ yards wide ? 2-^ 63. What is the value of (f of lj%) -^ ^ ? 64. A owns j^^ of a store, B ^7- and C the remainder. The profits were $ 960. What was the share of each ? 65. When $ 1 J will buy | of a gallon of oil, what part of a gallon will | ^j buy ? 66. If § of a store is worth $ 3300, what is the value of j^^ of the same ? 67. How many pounds of tea can be bought for $ 23f , if 7^ pounds cost $ 5/^ ? 68. If 4 bushels of corn cost $ 3^, how much will 7 bushels cost? 69. A will do § as much as B. The board of each is worth $ f a day. If B is paid $1 J a day and board, what should be paid to A in addition to his board ? 70. A saves I of his income, and B, having the same in- come, spends 1 J times as much as A, and finds himself $ 62^ in debt at the end of the year. What was the income of each ? 71. A cistern of 960 gallons is emptied by two pipes, A and B, in 5 and 7 minutes, respectively. How much water will pass through each if both are opened together ? 1/72. What is the value of ^ ^. + J of 1§ + (1^ -^ l^) DECIMAL FRACTIONS. 95 DECIMAL FRACTIONS. 137. A Decimal Fraction is a fraction whose unit is di- vided into tenths, hundredths, thousandths, etc. . 138. A Decimal is a decimal fraction expressed without its denominator by means of the decimal point (Art. 20)c Thus, 0.7, 0.05, and 0.168 are decimals. 139. A Mixed Decimal is an integer and a decimal Thus, 17.06 and 5.305 are mixed decimals. 140. The method of writing decimals, in continuation of the notation of integers (Art. 25), is shown in the fol- lowing TABLE. Integer. Decimal. Place-names. Figures. ■s § 0^ o §e S rf o 1 ^ F5 13 r^ ^ "^ -r^ 15 7 9 3,154.63 H ,4 Group-names. Thousands. Thousandths. Millionths. Billionths. 8 The number in the table is, seven hundred ninety-three thousand one hundred fifty-four, and six hundred thirty- eight million four hundred seventy-eight thousand one hundred thirty-nine billionths. Note 1. — The place-names above and below ones correspond to each other, except that the decimals liave the termination ths. Note 2. — A decimal with a common fraction is called a complex deciinal ; as, 0.16^, reatl sixteen and four-lifths Imndredths. 96 DECIMAL FRACTIONS. , 141. The Denominator of a decimal is 1, with as many ciphers annexed as there are places in the decimal. Thus, \j.\j xo Y0> v/.x«j xo YO^O' ^'^-^-^ ■'^'^ 1000^ ^^^ u. v.vuy -Lia Yo WRITTEN EXERCISES. Eead the following : 1. 0.35 8. 180.06i 15. 66766.71 2. 0.035 9. 4.7307f 16. 6676.671 3. 0.7077 10. 473.7f 17. 14.000014 4. 0.0095 11. 0.47378 18. 1.0000001 5. 0.0007 12. 0.000931 19. 0.0000077 6. 0.4003 13. 667.6671 20. 7000000.7 7. 5.55555 14. 6.676671 21. 96.00300315 Write decimally : 22. Ten, and eleven hundredths. 23. Sixty-seven, and seven tenths. 24. Six hundred seven, and nine hundred six thousandths. 25. One thousand five hundred forty-one, and one hundred seventy -eight ten-thousandths. 26. Sixteen, and one hundred twenty-two hundred-thou- sandths. 27. Thirty-seven thousand four hundred eighty-eight hun- dred-thousandths. 28. Nine hundred fifty-eight thousand four hundred thirty- three millionth s. 29. Four thousand four hundred four, and fifty-nine and three-fourths hundredths. 30. Sixty-eight, and three hundred three thousand eight hundred seven and two-sevenths miUionths. 31. Forty million three hundred three, and forty-three thousandths. 32. Forty-seven, and nine million nine hundred ninety-nine thousand nine hundj^.d ninety-nine ten-millionths. DECIMAL FRACTIONS. 97 Express in decimal form : 33. m- 38. 7aV 43. looo^ig-b. 34. TOIJ-G- 39. Th%Uh' 44. ^'^^t^^Atjd- 35. AVAV 40. 9060501f|. 45. 6958Tig§^. 36. ^^TxfoW 41. TOO%%(J^- 46. T^XroTJTTTT- 37. 945j15Vtt- 42. 4167^^. 47. T(J(TSuJ^T7ir' REDUCTION. A Decimal to Smaller or Larger Denominator. ORAL EXERCISES. 48. How many tenths in 1 ? How many hundredths ? How many thousandths ? 49. How many hundredths in 2 tenths ? In 3 tenths ? In 9 tenths ? 50. In 0.01 how many thousandths ? In 0.1 how many hundredths ? How many thousandths ? 51. How many tenths in 0.50 ? In 0.70 ? In 0.900 ? In 2.50? 52. How many hundredths in 0.600 ? In 0.850 ? In 6.500 ? 142. Principle. Annexing a cipher to a decimal, or removing a cipher from the right of a decimal, does not change its value* (See Arts. Ill, 114.) WRITTEN EXERCISES. 53. Write 0.16, 3, and 1.014 as decimals having the least common denominator. 0.16 = 0.160 Solution. — "YhQ smallest order of deci- 3. = 3.000 mals in the given numbers is thousandths. 1.014 0. 16 expressed as thousandths is 0.1 60, and 3. expressed as thousandths is 3.000. the requi'^d decimals are 0.160, 3.000, and 1.014. 7 98 DECIMAL FKACTIONS. 54. Eeduce 71.500 to tenths. 55. Change 19, 43.6, 0.64, and 53 to thousandths. 56. Express 15.600, 4.7, and 13 as hundredths. 57.- Eeduce 18.0156, 401.6, and 176.4700 to decimals having the least common denominator. A Decimal to a Common Fraction. 58. How many halves in .5 ? In .50 ? 59. How many fourths in .25 ? In .75 ? In .750 ? 60. How many fifths in 0.4 ? In 0.6 ? In 0.8 ? 61. How many twentieths in 0.05 ? In 0.15 ? In 0.45 ? WRITTEN EXERCISES. 62. Change .355 to a common fraction in its smallest terms. oF,K — _3_5_5_ _7_1_ Solution. — As .355 is 355 tbou- »ooo — 10 — 2 sandths, it may be written -f^-Q, which, changed to its smallest terms, is -jy^-. 63. Change .225 to a common fraction in its smallest terms. 64. Change .875 to a common fraction in its smallest termg. 143. To change a decimal to a common fraction : Rule. Omit the decimal point, write the denominator, and change the fraction to its smallest terms, Eeduce to common fractions in smallest terms : 65. .025 70. .375 75. 9.37i 66. .561 71. .368 76. 115.875 ' 67. .054 72. 11.75 77. .01375 68. M^ 73. 4.43| 78. 200.96 69. .625 74. .09375 79. .015625 A Common Fraction to a Decimal. 80. How many tenths in 1 ? In J ? In ^ ? In f ? 81. How many hundredths in 1 ? In J ? In f ? In g ? 82. How many thousandths in 1 ? In J ? In J ? In | ? DECIMAL FRACTIONS. 99 WRITTEN EXERCISES. 83. Change f to a decimal. 8) 5.000 Solution. — ^ is equal to| of 5. 5 equab 60 .625 tenths, or 5.0 ; -^ of 50 tenths is 6 tenths, with 2 tenths, equal to 20 hundredths, remaining. ^ of 20 hundredths is 2 hundredths, with 4 hundredths, equal to 40 thousandths, remain- ing. ^ of 40 thousandths is 5 thousandths. Ans. .625. 84. Change f to a decimal. 85. Change ^y to a decimal of four places. 11) 8.0000 Solution.— -f^ IS equal to Jy of 8. As .7272t\ there are to be four places in the decimal, we annex four decimal places of ciphers to the numerator, reducing it to 8.0000. j\ of 8.0000 is .7272-i\. 86. Change ^^ to a complex decimal of three places. 87. Change ;^ to a complex decimal of four places. 144. To reduce a common fraction to a decimal : Rule. Annex decimal ciphers to the numerator, divide hy the de- nominator, and point off as ma7iy decimal figures in the quo- tient as there are ciphers annexed. Note 1. — When the division does not terminate, or has been carried as far as is desirable, the remainder may be expressed as a common fraction and made a part of the result. Note 2. — When an approximate result is sufficient, a fraction of ^ or more than ^ in a result may be rejected, and the last fi^re of the decimal be made to express 1 more. Thus, .7272 ft: approximately expressed to the nearest ten-thousandth is .72X3. Note 3. — When the fraction as a part of a decimal is unimportant, it may be omitted, and the incompleteness of the result simply marked by +. Thus, .7272+ may be written instead of .7272A. Eeduce to decimals : 88. iJ. 91- U- 94. 4g^. 89. ^. 92. i^. 95. 2i^. 90. i^. 93. m- 96. Sxis- 100 DECIMAL FRACTIONS. 97. Reduce ^^ to a complex decimal of four places. 98. Reduce j\ to the nearest ten-thousandth. 99. Reduce |f to an approximate decimal of four places. 100. Reduce -^-^ to a complex decimal of four places. 101. Reduce /y to the nearest millionth. 102. Change to the nearest thousandth f , §^, and ^f ^y, and find the sum of the results. 103. Reduce to the nearest ten-thousandth, and add, 15f, 20|, 12|, and 2.68. 104. Reduce 3^^ to the nearest millionth, and subtract the result from 15.057. 145. For rules for Addition and Subtraction of Deci- mals, see Arts. 38 and 47. MULTIPLICATION. ORAL EXERCISES. 105. How much is 3 times 2 tenths ? 3 times 0.3 ? 106. How much is 7 times 1 hundredth ? 7 times -^^-^ ? 9 times 0.06 ? 107. How much is ^^^ of 1 ? Of 3 ? -^^ of ^Jq ? iV of .3 ? 108. t^^XtV? AXtSu? 3x.3? 4X.3? .04 x .3 ? 109. How many places of decimals in the product when tenths are multiplied by tenths ? When hundredths are mul- tiplied by tenths ? WRITTEN EXERCISES. lia Multiply 31.5 by .07. 31.5 Solution. — As .07 is the same as ^^ of 7, 31.5 ^07 multiplied by .07 is .the same as y^ of 7 times 31.5. 2 2Q5 7 times 31.5 == 220.5, and y^ of 220.5, which is found by removing the decimal point two places to the left (Art. 75), is 2.205. ill. What is the product of 671 by .305 ? 112. What is the product of 18.72 by 7.1 ? DECIMAL FRACTIONS. >^ ^ ],./. IPl, 146. Rul^ for Multiplication of Decimals. ' ' Multiply as integers j and point off as many figures for deci- mals in the product as there are decimal places in both factors. Note. — If there are ni)t jBgures enough in the product, supply the deficiency by prefixing ciphers. 113. 114. Multiply .126 3.18 By ^ ■ .00029 756 2862 126 636 Product .002016 .0009222 115. Multiply 5.64 by 45. 120. Multiply .563 by 47. 116. Multiply 96.5 by 100. 121. Multiply 19634 by .0073. 117. Multiply 6.34 by .0023. 122. Multiply .0703 by .0055. 118. Multiply 42.2 by 2.004. 123. Multiply .0505 by .001. 119. Multiply 1671 by .013. 124. Multiply .0076 by .017. 125. What is the product of one million by one millionth ? 126. What is the cost of 35.75 yards of cloth, % 4.50 a yard ? 127. How much must be paid for 13.375 cords of wood, at % 4.62 a cord ? 128. What is the product of one hundred one thousandths by ten thousand one hundred one hundred-thousandths ? DIVISION. ORAL EXERCISES. 129. How many times 2 tenths in 8 tenths ? 3 tenths in 9 tenths ? 130. How much is ^ of 8 tenths ? J of 9 tenths ? 131. How much is \ of 15 hundredths ? i of .24 ? J of .45? 132. Divide .8 by .2 ; .63 by 7 ; .63 by .07. 133. The product of two factors is .72, and one of the fac- tors is 9. What is the other factor ? 102 DECIMAL FRACTIONS. 134. Tlie product of two factors is .72, and one of the fac< tors is .9. What is the other factor,? 135. In .72 divided by 9, how many places of decimals are there in the quotient ? In .72 divided by .9 ? WRITTEN EXERCISES. 136. Divide 46.48 by .4. \A.) 46f4.8 Solution. — Both dividend and divisor may be 116.2 multiplied by 10 without changing the quotient (Art. 72). This is done by moving the point one place to the right (Art. 75). Dividing as in integers, and placing the quotient point under the new dividend point, we have the quo- tient 116.2. 137. Divide .7935 by .23. 3.45 ' X^o.) pv.oo Solution. — To make the divisor an integer, ^" we multiply both divisor and dividend by 100, 103 by moving the point in each two places to the 92 right. Dividing as in integers, and placing the quotient point over the new dividend point, we have 3.45 as the quotient required. 115 115 138. Divide 1.264 by 4. 139. Divide .00115 by .05. 147. Rule for Division of Decimals. If the divisor is an integer, divide as in integers, and point off as many decimal figures in the quotient as there are such places in the dividend* •*' v If the divisor is a decimal, make it an integer by moving the decimal point a sufficient number of places to the right ; move the decimal point in the dividend as many places to the right, annexing ciphers, if necessary, and then divide. / DECIMAL FRACTIONS. 103 Note 1. — When the division does not terminate, or has been carried as far as is desirable, the remainder may be expressed as a common fraction and made a part of the result ; or, the fraction may be rejected and an approximation ex- pressed to the nearest decimal ; or, the sign + may be nsed to mark the incom- pleteness of the division (Notes, Art. 144). Note 2. — When the divisor is an integer, the decimal point in the quotient must invariably be placed directly under or over that of the dividend. If it is desirable to divide without changing decimal points, it maybe done, care" being talj:en to give the quotient as many decimal places as those in the dividend ex ceed those in the divisor. 140. Divide 37.4 by 4.5 to 141. Divide 6.0512 by 3.7 to three decimal places. the nearest thousandth. 8.311+ 1.635 4i5.) 37^4.000 3i7.) 6^0.512 360 37 140 235 135 222 50 131 45 111 50 202 45 185 "6 17 142. Divide 783.5 by 6.25. 147. Divide .817 by .9147. -r 143. Divide .0189 by .025. 148. Divide 1.365 by 1000. 144. Divide .01001 by .001. 149. Divide 72 by .018. 145. Divide 1.0665 by .00135. 150. Divide 555 by .0037. 146. Divide .08748 by 1.08. 151. Divide .0016016 by .00143. ^152. Divide ninety-five and three tenths by two hundred Bixty-four thousandths, to the nearest thousandth. 153. Divide one hundred eighT: and twenty-nine thou- sandths by seven and two tenths, to^'^hfee decimal places. 154. If a man can walk three and seventy-five hundredths miles in an hour, in how many hours can he walk seven hun* dred eighty-seven and five-tenths miles ? 104 DECIMAL FRACTIONS. MISCELLANEOUS EXERCISES. 155. Write four hundred three ten-thousandths ; four nuiv dred, and three ten-thousandths. 156. Write and read : 0.00567 ; 2.13007 ; 1.00157. 157. Express as decimals : y^V^Ty ; t^o^tt 5 TTTxrWiy' '4. 158. rind the mixed decimal equal to 4| + ^J. 159. From 45 subtract 36.00073. 160. Find the least fraction which is to be added to the sum of 25.7, 8.389, and 23.056 to make the result an integer. ' 161. Find the difference between 6346 and .6346 ; 4.2 and .0042 ; .0000005 and .00005. 162. Simplify, and express the result in a decimal form, 31 + 17^ + 476 + 3.125. 163. Change .03125 to a common fraction in its smallest terms. 164. Change 4, 2f , 17, .136, and .0408 to equivalent decimals having a common denominator, and find their sum. f* 165. If the year is considered 365.25 days, instead of 365.242264, how great will be the error in 1880, years ? 166. The dividend is 7423.973, the quotient 12.130, and the remainder .413. What is the divisor ? .-^^167. Divide $ 7498.70 among A, B, and C, so that A shall have just $749.83 more than each of the others. 168. What is the value of 20004 + (20.104 X 5.07) - (6.44 ~ .005) ? - / .. y / v^ 9>7 ? r 169. Add 2§, 4U, and 51.652, expressing the sum to the nearest thousandth. 170. A man willed his property to his three sons, — to the youngest he gave $968.49; to the second, 3.4 times as much as to the youngest ; and to the eldest, 3.7 times as much as to the second. Kequired the value of his property. 171. If a train of cars moves at the rate of 30.25 miles an hour, find to the nearest ten-thousandth how many hours it will take to go 150.75 miles. DECIMAL FRACTIONS. 105 (l72. Beduce to its simplest form -1 - , ^ ^^ ^^,^ ^ ^^^ . j 173. Dividing a certain sum by .027, the quotient is 6116. and the remainder .003. What is the dividend ? 174. Bought wheat at $ 0.94 per bushel, to the amount of $ 59.22, and sold it for | 70.56. What was made on a bushel ? 175. What is the cost of 60.5 tons of coal when 0.9 of a tor costs $ 6M ? 176. Find .3 of .064f of $ 1728. 177 Divide (12^8 -^ .16) by (.128 ~ .0016). L f 178. What part of .84 is .012 ? y 179. If .6875 of a gallon of wine costs $3.75, what will 40.25 gallons cost ? 180. If I of an acre costs $ 15|, what will 46.78 acres cost ? 181. If one pound costs $ 0.18f , how many pounds can be bought for $ 14.40 ? 182. If 35.84 cubic feet of water weigh a ton, what will be the weight of 2458.6 cubic feet ? 183. If 48 is .08 of some number, what is .7 of it ? • 184. The product of three factors is 5.76 ; one of them is .024, another is .06 ; find the third. 185. How many times is the difference of (f -r- 0.33J) and (0.87 j^ -^ J) contained in their product ? QUESTIONS. 137. What is a decimal fraction? 138. What is a decimal ? 139 A mixed decimal ? 141. What is the denominator of a decimal ? 142. What is the prin«iple in relation to annexing or cutting off a cipher from the right of a decimal ? 143. How is a decimal reduced to a common fraction ? 144. How Is a common fraction reduced to a diecimal ? 146. How do you multiply in decimals ? 147. How do you di- nde in decimals 1 106 UNITED STATES MONEY. UNITED STATES MONEY. 10 mills (m.) are 1 cent, c. or /. 10 cents " 1 dime, d. 10 dimes " 1 dollar, $. 10 dollars " 1 eagle, e. 149. Coins are pieces of metal stamped by authority of the govern- ment to circulate as money. 150. The principal Coins of the United States are : — Bronze, — the cent ; Nickel, — the five-cent ; Silver, — the dime, quarter-dollar, half-dollar, and dollar; Gold, — the dollar, quarter-eagle, three-dollar, half-eagle, eagle, and double-eagle. UNITED STATES MONEY. 107 151. In ordinary business transactions, eagles, dimes^ and mills are rarely mentioned, eagles being expressed as dollars, dimes as cents, and mills as a fraction of a cent. Thus, 3 eagles, 2 dollars, 5 dimes, 8 cents, 5 mills are written, $32.58^-. 152. To change cents to millSy multiply by 10. To change dollars to centSy midtijply by 100. To change dollars to nfiills, multijply by 1000. Thus, 45 cents = 450 mills ; $ 5 = 500 cents, or 5000 mills. 153. To change mills to cents, divide by 10. To change cents to dollars, divide by 100. To change mills to dollars, divide by 1000. Thus, 850 mills = $0.85; 365 cents = $ 3.65 ; 47000 mms = $ 47. 154. The dollar being the unit, of which cents are hun- dredths and mills are thousandths, it follows that All the rules for the processes in decimals are applicable to processes in United States money, ORAL EXERCISES. 1. How many mills in 7 cents ? In 7 cents 4 'mills ? 2. How many cents in 70 mills ? In 75 mills ? 3. Change $ 5 to cents. 7. Change 500 cents to dollars. 4. Change $ 5 to mills. a Change 5000 mills to dollars. 5. Change $3.50 to cents. 9. Change 350 cents to dollars. 6. Change $ 6.37 to mills. 10. Change 6370 mills to dollars. 11. In changing cents to dollars, how many places to the left is the decimal point moved ? 108 UNITED STATES MONEY. 12. In changing mills to dollars, how many places to the left is the decimal point moved ? 13. Paid for a bag of flour $ 1.25, for coffee 40 cents, and for soap 20 cents. How much was paid for the whole ? 14. A boy earned one week $ 3.50, and the next 75 cents. How much did he earn in all ? 15. Jane has 60 cents, Alice 15 cents, and Henry $ 1.20. How much have they all ? 16. Bought an arithmetic for $ 1.25, and gave a two-dollai bill in payment. What change should I receive ? 17. Bought a plow for $ 12.50, and sold it for $ 15. How much did I make ? 18. How much is the profit on flour bought for $ 7.50, and 6old for $ 9.25 ? 19. If a man can earn $ 1.50 in one day, how much can he earn in 3 days ? 20. How much must be paid for 5 pairs of shoes at $ 2.25 a pair ? 21. At 25 cents a yard, how many yards of cloth can be bought for $ 1.75 ? 22. At 50 cents a bushel, how many bushels of apples can be bought for $ 4.75 ? WRITTEN EXERCISES. 23. Bought a coat for $13.50, a vest for $2.63, a hat for $ 5, and a pair of boots for $ 6.13. What was the amount ? 24. A man paid for tea 63 cents, for butter 80 cents, for flour $ 7.50, and for other articles $ 32. What did he pay for the whole ? 25. Mr. Avery bought a farm for $6500, and paid down $1356.85. How much more had he to pay ? 26. How much will be received for 56 pounds of crackers at 14 cents a pound, and 128 loaves of bread at 9 cents a loaf ? 27. How many yards of cloth at 19 cents a yard can bo bought for $47.50? UNITED STATES MONEY. 109 28. A man bought 491 bushels of corn at 81 cents a bushel. He used 29 bushels, and sold the rest at 95 cents a bushel. How much did he make ? 29. Bought a house and farm of 36 acres for $ 7975. The house is worth $ 4560. What is the value of the land an acre, k> the nearest cent ? 30. A farmer sold 187 acres of land at $ 37.50, and 131 acres at $ 63 an acre. At what rate an acre, to the nearest cent, did he sell the whole ? 31. At 67 cents a yard, how many yards of cloth can be bought for $ 29.70 ? 32. At 9 cents a pound, how many pounds of beef can be bought for $ 68.40 ? 33. When 7 barrels of onions are worth $ 26.25, what are 43.50 barrels worth ? 34. Bought land for $1040.98, and sold 35.45 acres at $ 20.40 an acre ; the rest was worth to me $ 20.50 an acre. How many acres did I buy ? ALIQUOT PARTS. 155. Aliquot Parts of a number are such parts of the number as will exactly divide it. Thus, 2, 2|-, 3J, and 5 are aliquot parts of 10. 156. Computations are often abridged by use of the following Aliquot Pai 50 cents = -J of f 1. ^ts of a Dollar. 12J cents = | of$l. 33J cents = l of $ 1. 10 cents = yVo^^ 1- 25 cents = 1 of $1. 8 J cents = ^i^ of $ 1. 20 cents = ^ of $ 1. 6^ cents = 3^^ of $1. 16f cents = 1 of $1. 6 cents = tjV 0^ ^ 1' 110 UNITED STATES MONEY. ORAL EXERCISES. 35. What will 42 yards of cloth cost at 16 § cents a yard ? Solution. — 16f cents is -^ of $ 1. Since one yard costs $ |, 42 yards will cost 42 times $ -J, or $ \% or $ 7. 36. At 121- cents a pound, what will 96 pounds of sugai cost? 37. At 25 cents a box, how much will 60 boxes of straw berries cost ? 38. At 12^ cents a pound, how many pounds of sugar can be bought for $ 12 ? Solution. — At 12^ cents a pound, $1 will buy 8 pounds, and $12 will buy 12 times 8 pounds, or 96 pounds. 39. At 25 cents a box, how many boxes of strawberries can be bought for $15? 40. At 33 J- cents a yard, how many yards of cloth can be bought for $ 9.33^ ? 41. At 50 cents a gallon, how many gallons of molasses can be bought for 1 11.50 ? 42. At 20 cents a dozen, how many dozen eggs can be bought for $ 10.80 ? WRITTEN EXERCISES. 43. What will 1832 pairs of shoes cost at $ 1.75 a pair ? Solution. — 1832 pairs at f 1 will cost $1832 ; at $^, will cost J of $ 1832, or $ 916; and at $ \, will cost J of $916, or $458, $3206= " $T75 $1832 + $916 + $458 =$3206. 44. What will 144 bushels of wheat cost at $ 1.37 J a bushel ? 45. What will 84 yards of cloth cost at 66J cents a yard ? 1 1832 = cost at $ 1 916== a .50 458 = cc .25 UNITED STATES MONEY. Ill 46. What will 60 pairs of boots cost at $ 3.87^ a pair ? 47. At $ 5.12^ a dozen; how many dozen pocket-knives can be bought for $502.25? 48. At $1.62J a pair, how many pairs of shoes can be bought for $29.25? 49. At $ 0.87^ each; how much must be paid for 56 caps ? 50. At $2.25 a day, how much will a man earn in 302 days ? 51. At 37^ cents a yard, how many yards of cloth can be bought for $ 793.87J ? 52. What will 575 njelons cost at $ 16 a hundred ? Solution. — 575 is equal to 5.75, or 5f hundreds. 5| hundreds, at y~' $ 16 a hundred, will cost 5f times •$ 16, or $ 92. 53. What will 1360 herrings cost at 37^ cents a hundred ? 54. What will 650 melons cost at $ 12.50 a hundred ? 55. What will 2250 feet of boards cost at $ 20.50 a thou- sand feet ? Solution. — 2250 is equal to 2.250, or 2 J thousands ; and 2^ times $20.50 = $46.12^. 56. How much will 5650 feet of plank cost at $ 44 a thou- sand feet ? 57. What is the cost of 4565 feet of joist at $ 23 a thousand feet ; 13640 feet of boards at $ 53.55 per thousand feet ; and 15250 shingles at $ 4.50 per thousand ? 5a If 18500 bricks are sold for $ 155.40, what is the cost per thousand ? Solution. — 18500 = 18.5 thousands. Since 18.5 thousands cost $ 155.40, 1 thousand will cost $ 155.40 -^ 18.5, or $ 8.40. 59. If 8375 feet of spruce scantling cost $ lOO.SO^ what is the cost of a thousand feet ? 112 UNITED STATES MONEY. 60. If 650 melons cost $ 81.25; what is the cost a hundred ? 61. Bought a farm, containing 75.8 acres, at $ 31.50 an acre, and sold it for $ 2274. What was the loss per acre ? ACCOUNTS AND BILLS. 157. An Account is a record of articles bought or sold, cash paid or received, or services rendered. 158. A Debtor is one who owes a debt, and a Creditor is one to whom a debt is owed. 159. A Bill is a written statement of an account made out by the creditor for the debtor. A bill is Receipted when its payment is acknowledged in writing by the creditor, or by some one authorized to sign for him. Note. — Dr. is for "debtor," Cr. for "creditor,' M. for "thousand," and (aX for " at." WRITTEN EXERCISES. Copy and find the amounts and balances of the following accounts and bills : — 62. Itt %amxd foitfe JAMES DELANY, gr. 1881. JOLTL. d 3-0 / Ul. moloM^ ff cfcLt. (0)56^ / // // " /OO 60.. &cvu>lina. RU& " 7^ S'd-. 8 " 50 0-. m-. " #J^ UNITED STATES MONEY. 63. Q/^. ^uTne^ ^oo^ie^. 113 New York, May 16, 1881, TERMS CASH. 1Sougl)t of ARTHUR OILMAN & CO. c^Ot^ ^/eui. M>il. ® ^^O.^O. SO/. Q4iiu^ &£e^ Mee/ss ^T^.^O... ^a/, ,^zU 4-^/^ M ^.//. Received Payment, 64. ©^. 'm^/^77z Ma/Li>. Providence, June 11, 1881. So JOHN O'BRIEN, Pt. (S/o^c/(zy^' o^^i @ ^S.^O n ^aUcTia n ^.yj y Received Payment, My M^ 0'Muem 114 UNITED STATES MONEY. 65. 1S81. |iOtt5l)t of ALBERT LANE & SONS. Burlington, July 19, 1881, (3^^ " ^ ;) // ?i /J' ^u?te 7 II // • I ^/ 66. 1882. I^ llCCOttttt XDltl) FELIX REMOND, pt. iV^ei(; Orleans, July 1, 1882. i^u?ie (^ /^ /// ©/f ^^ (^^^^ @ /c^. J'c^ / II J /^ ///(2% <^^^ (^^.^i .1 /4^ .. ^ c^ ^^^ ^u^€ei., /J^/ ^ II .c^^ It Ifj /J'^^ (M?zoiec^ (^dcc7?z II ./^:? .1 J^^ 4c^ #. ^/^^ II .// — — 1. ^ G//i^elcda?zc/ci>e J^^.<^0 M /<:^ ^a^^. OO.OO Ma^7zce c/ue S^. M, / Received Payment, July 3, 1882, UNITED STATES MONEY. 115 67. Trenton, Oct. 3, 1881. Andrew Shaw bought of In- gram, Smith, & Co., 25 pairs of Kip Boots at $ 2.50 a pair ; 20 pairs Calf Boots at $ 2.75 a pair ; 30 pairs of New Bruns- wick Kubber Boots at | 3 a pair ; 15 pairs of Kip Brogan Shoes at $ 1.25 a pair ; and there was charged for carting 25 cents. Make out Mr. Shaw's bill and find its amount. .68. St. Paul, Sept. 15, 1881. Hooper, Blake, & Dudley sold Jacob Van Husen 100 bbl. Best Test Patent Plour at $ 8.25 a barrel ; 50 bbl. Wilbur's Extra Elour at I 6.50 a barrel ; and charged for prepaid freight $ 6.25. Make out the bill and receipt it for Hooper, Blake, & Dudley. 69. Make out a bill, June 1, 1881, for Joseph Mclntire, for 3 months' services rendered him at $ 28 a month. 70. Chicago, Nov. 1, 1882. William Asbury owed George W. Ogden & Bro. for 50 tons of Franklin Coal at $ 5.25 a ton, bought Oct. 2 ; 75 tons of Cumberland Coal at $ 4.75 a ton, bought Oct. 10 ; 25 cords of Pine Wood at $ 4.75 a cord, bought Oct. 16 : and George W. Ogden & Brother owed him for cash paid Oct. 10, $ 262.50 ; and for bill of merchandise rendered Oct. 20, $ 31.65. Make out the account and receipt for the balance in your name for George W. Ogden & Brother. 71. Nov. 10, 1881. Arthur Koberts bought of Jordan, Marsh, & Co. 64 J yards of tapestry carpet at 87 J f ; 27 yards Brussels carpet at $1.85; 18| yards oilcloth at 37^/; the making and laying cost $13.22. What was the amount of the bill ? QUESTIONS. 148. Recite the table of United States money. 150. What are thb principal coins of the United States ? 154. Why are the rules for the processes in decimals applicable to processes in United States money ? 155. What are aliquot parts of a number ? 156. What are some of the aliquot parts of a dollar ? 157. What is an account? 158. A debtor? A creditor? 159. What is a bill ? When is a bill receipted ? 116 WEIGHTS AND MEASURES. WEIGHTS AND MEASURES. 160. 1. How can the distance between two places be found and expressed ? 2. How can you determine the space occupied by a floor ? The space enclosed by the walls of a room ? 3. How can you determine the amount of water in a pitcher ? The quantity of grain in a bin ? 4. How can you tell how much sugar you have bought ? LENGTH MEASURES. 161. A Line is that which has length only, as the dis- tance between two corners of this book. 162. Linear, or Length Measures are those used in meas- uring lines or distances. TABLE. 12 inches (in.) are 1 foot, ft. 3 feet " 1 yard, yd. 5i yd., or 16^ ft. " 1 rod, rd. 320 rd., or 5280 ft. " 1 mile, mi. 5. How many inches in 4 ft. ? In 5J ft. ? In ^ yd. ? In I yd.? 6. How many yards in 72 ft. ? In 90 in. ? In 16^ ft. ? In 38 ft.? In^rd.? 7. How many rods in J mi. ? In j^ mi. ? In ^ mi. ? In 33 yd. ? 8. At $ 2 a foot, what will it cost to build 24 rods of stone wall? 9. Required the distance in yards around a room 13 feet square. 10. How many rods of fence are required to enclose a lot 25 rods wide and twice as long ? WEIGHTS AND MEASURES. 117 11. In 2 yd. 2 ft. 2 in., how many inches ? 12. What cost 27 feet of picture-cord at 10 cents a yard ? 13. What cost 27 in. of ribbon at 60 cents a yard ? 14. How many inches in 0.33 J of 2 yards ? SURFACE MEASURES. 163. A Surface is that which has length and breadth only, as this page, or the outside of a block. A plane surface is one that does not change its direc- tion. 164. An Angle is the difference in direction of two straight lines which meet at a point, called the vertex of the angle ; as A D C or C J) B. ^ 165. Two lines are perpendicular to each other when they meet so as to form equal adjacent angles. The angles thus formed are called Bight Angles. Thus the lines A B,C D are perpendicular to each other, and form the right angles, A D C, C D B. 166. A Rectangle is a plane surface with four straight sides and four right angles. 167. A Square is a rectangle with equal sides. 168. Surface, or Square Meas- ures are those used in measuring surfaces. An Inch iS^Kore, or Square Inch. 118 WEIGHTS AND MEASURES. TABLE. are I square foot, sq. ft. 9 square feet " 1 square yard, sq. yd. 30^ square yards, or 2 72 J sq. ft. " 1 square rod, sq. rd. 160 square rods " 1 acre, A. 640 acres An acre contains 43,560 square 144 square inches (sq. in.) " 1 square mile, or section, sq. ml feet. A rood (R.) is 40 square rods. ORAL EXERCISES. 15. How many square inclies in ^ sq. ft. ? In f sq. ft. ? In 0.5 sq. ft. ? In \\ sq. ft. ? 16. How manj^ square feet in 3 sq. yd. ? In 5J sq. yd. ? In 200 sq. in. ? In 7 sq. yd. 8 sq. ft. ? 17. In 47 surface feet how many surface yards ? In 80 sq. ft. ? In 100 sq. ft. ? 18. How many square rods in i A. ? In y^^- A. ? In 0.75 A. ? 19. What part of an acre is 120 sq. rd. ? 50 sq. rd. ? VOLUME MEASURES. 169. A Solid, or Volume, is that which has length, breadth, and thickness ; as a brick or a box. 170. A Cube is a solid bounded by six equal squares. 171. Cubic Measures are those Au [nch c«^6e, 01 Cubic Inch. ^gg^j jj^ measuriuff solids or vol- umes. WEIGHTS AND MEASURES. 119 TABLE. 1728 cubic inches (cu. in.) are 1 cubic foot, cu. ft. 27 cubic feet " 1 cubic yard, cu. yd. A-lso, 16 cubic feet " 1 cord foot, cd. ft. 8 cord feet, or 128 cu. ft. " 1 cord, cd. ORAL EXERCISES. 20. In 3 cu. yd. how many cubic feet ? In 10 J- cu. yd. ? In 81 cu. ft. how many cubic yards ? In 108 cu. ft. ? 21. In 4 cd. how many cord feet ? In 5^ cd. ? 22. In 64 cu. ft. how many cord feet ? 23. How many cord feet equal a cubic yard ? CAPACITY MEASURES. 172. Liquid Measures are used in measuring liquids. 173. Dry Measures are used in measuring grain, roots, fruit, etc. , TABLES. Liquid. Dry. 4 gills (gi.) are 1 pint, pt. 2 pints " 1 quart, qt. 4 quarts " 1 gallon, gal. 2 pints (pt.) are 1 quart, qt. 8 quarts " 1 peck, pk. 4 pecks " 1 bushel, bu. The gallon contains 231 cubic inches ; 4 quarts dry measure, 2684 cubic inches ; and the bushel, 2150.42 cubic inches. A hogshead as a measure is 63 gallons ; and a barrel j 31^ gallons. ORAL EXERCISES. 24. How many gills in 5 pt. ? In 9J pt. ? In 2 qt. ? In 1 gal. ? In I qt. ? 25. How many quarts in 7 gal. ? In 8J gal. ? In 48 pt. ? In 100 gi. ? In 3 gal. 3 qt. ? 120 WEIGHTS AND MEASUKES. 26. How many gallons in 41 qt. ? In 53 qt. ? In 60 pt. ? In 120 gi. ? In 89 qt. ? 27. How many pecks in 48 quarts ? How many bushels ? 28. Change to larger measures, 100 pt. of milk ; 64 qt. of nuts ; 84 gi. of oil. 29. What is a six-gallon can of milk worth at 5 cents a quart ? 30. I take a pint of milk at night and a quart in the morn- ing. What is my weekly milk-bill at 7 cents a quart ? 31. What cost 6 1 bu. of potatoes at $ 4 a huohel ? 32. I burn a pint of kerosene every night. What will a 2 weeks' supply cost me at 20 cents a gallon ? 33. Bought nuts for $ 2 a bushel, and sold them for 10 cents a quart. How much did I gain on 5 bushels ? WEIGHTS. 174. Troy Weights are used in weighing gold, silver, and jewels. 175. Avoirdupois Weights are used in weighing all com- mon articles. , TABLES. Troy. Avoirdupois. 24 grains (gr.) are 1 pennyweight. 20 p^vt. " 1 ounce, oz. 12 ounces " 1 pound, lb. 16 ounces (oz.) are 1 pound, lb. 100 pounds " 1 cental, ctl. 2000 pounds " 1 ton, T. 1. The long ton of 2240 avoirdupois pounds is used at custom- houses, and in weighing coal at the mines. 2. A pound avoirdupois is equal to 7000 grains, and a pound troy to 5760 grains, so that 144 pounds avoirdupois are equal to 175 pounds troy. A hundred-weight (cwt.) is the same as the cental. WEIGHTS AND MEASURES. 121 ORAL EXERCISES. 34. In avoirdupois weight, 3 lb. are how many ounces ? 5^ lb. ? 21 lb. ? 3 lb. 7 oz. ? 2 lb. 9 oz. ? 3f lb. ? .75 lb. ? 35. In avoirdupois weight, 56 oz. are how many pounds ? 65 oz. ? 27 oz. ? 2500 lb. are how many tons ? 4000 lb. ? 7000 lb. ? 36. Change to the next larger troy weight, 35 oz. ; 75 oz. ; 60 pwt. ; 84 pwt. ; 48 gr. ; 30 gr. ; 100 oz. 37. Change to the next smaller troy weight, 3 lb. ; 2^ lb. ; 4 lb. 7 oz. ; 6 oz. ; 5^ oz. ; 2 pwt. ; 2J pwt. ; | lb. ; f oz. ; J pwt. ; 0.6 lb. ; 0.8 oz. 38. What cost 2 lb. 4 oz. of butter at 32 cents a pound ? 39. What can I get for 7 pwt. of old silver at $1.10 an oz. ? 40. How much hay at $ 25 a ton can be bought for $55? 41. If 200 lb. can be packed in a barrel, how many barrels \¥ill be needed to pack 2 T. 8 ctl. ? -K TIMB. 176. Time is a measured portion of duration. TABLE. 60 seconds (sec.) are 1 minute, min. Also, h The Civil Day begins and ends at 12 o'clock, midnight. 60 minutes u 1 hour, h. 24 hours a 1 day, d. 365 days u 1 common year, y. 366 days u 1 leap year, 1. y. 7 days are 1 week, wk. 12 calendar months (mo.) " 1 year, y. 100 years " 1 century. 122 WEIGHTS AND MEASURES. 2. The Calendar Months, and the number of days in each, are, - 1. January, 31 days. 7. July, 31 days. 2. February, 28 or 29 " 8. August, 31 " 3. March, 31 " 9. September, 30 " 4. April, 30 " 10. October, 31 " 5. May, 31 " 11. November, 30 " 6. June, 30 " 12. December, 31 " 3. The Solar Year is 365 d. 5 h. 48 min. 49.7 sec, or very nearly 365^ days. The fraction in 4 years amounts to nearly a day. Hence, When the number of any year is divisible by 4 and not by 100, and when it is divisible by 400, the month of February has 29 days, and the year is called a leap year. ORAL EXERCISES. 42. How many minutes in 2 h. ? In 3 J h. ? In f h. ? In 3 h. 20 min. ? 43. How many hours in 210 min. ? In 3 d. ? In ^ wk. ? 44. How many days in 96 h. ? In 60 h. ? 45. How many minutes from 10.25 A. m. to 11.45 A. m. ? From 11.60 A. m to 1.35 p. m. ? 46. What part of a minute is 42 sec. ? What part of an hour is 48 min. ? 47. How many days from Mar. 31 to May 31 ? From June 7 to July 15 ? From Oct. 19 to Nov. 12 ? 48. How old is a child on July 4, 1883, who was born May 31, 1880 ? 49. If a three weeks' vacation begins June 21, when does it end ? If it ends Oct. 9, when does it begin ? 50. What day was 25 days before Mar. 4, 1880 ? Sixty days after? 51. How long from Thursday noon to 4.30 p. m. of Friday ? 52. How long is the night when the sun sets at 7.35 and rises at 4.30 ? 53. Name the next six leap years. 54. What century is this? In wb^t century did 1620 come ? WEIGHTS AND MEASURES. 123 ARC AND ANGLE MEASURES. 177. A Circle is a plane surface bounded by a curve, all points of which are equally distant from a point within called the center. The circumference of a circle is its boundary line. An arc of a circle is any part of the circumference ; as A C oi I) B. The diameter of a circle is a straight line drawn through its center, and terminated by the circumference ; as ^ ^. The radius of a circle is half its diameter ; ^^AEoiE D, 178. A Degree is -^^ of any circumference. 179. An Angle whose vertex is the center of a circle is measured by the arc between its sides. Thus, the arc A C measures the angle AEG. TABLE. 60 seconds (f') are 1 minute, \ 60 minutes " 1 degree, ®. 360 degrees - " 1 circumference, C. 1. A degree of the earth's equator contains 69.16 miles, or about 691 miles. 2. A sign is an arc of 30° ; as D B. A sextant is an arc of 60° ; as DC. A quadrant is an arc of 90° ; as ^ C. A semi-circumference is an arc of 180° ; as ^ C^. A right angle is an angle of 90° ; as ^ ^ 0. Note. — The earth, by turning on its axis once in 24 hours, causes ^ of 360°, or 15°, of longitude to pass under the sun, from east to west, in 1 hour's time. ORAL EXERCISES. 55. How many degrees in 2 right angles ? In 4 right angles ? 56. How many minutes in 3^° ? In |° ? 124 WEIGHTS AND MEASURES. 57. In 195' how many degrees ? In 360' ? 58. How many signs in a quadrant? Draw an angle of 45° ; another of 75°. 59. How many degrees does the minute-hand of a clock move in 3 hours ? The hour-hand ? 60. How many degrees does the minute-hand of a watch move in 3 J- hours ? 61. How many degrees in half a meridian circle? How many miles in one such degree ? MISCELLANEOUS MEASURES. 180. Counting. 12 ones are 1 dozen. 12 dozen " 1 gross. 12 gross " 1 great gross. 20 ones " 1 score. 181. Paper. 24 sheets are 1 quire. 20 quires " 1 ream. 2 reams " 1 bundle. 5 bundles " 1 bale. ORAL EXERCISES. 62. How many pencils in a gross ? How many score ? 63. What cost 15 doz. pens at $ 1 a gross ? 64. How many sheets in a ream ? In J ream ? 65. What will 72 sheets cost at 15 cts. a quire ? 66. If I buy paper at 20 cents a quire, and sell for a cent a sheet, how much do I gain on a ream ? 67. Name some articles that are sold by the gross ? By the dozen ? 68. How much will a great gross of tacks cost at 10 cents a paper ? 69. Bought 3 gross of pens at $ 1.50 a gross, and sold them for 2 cents each j liow much did I gain ? WEIGHTS AND MEASURES. 125 70. How many dozen pint bottles will be needed to hold 5 gal. 3 qt. of currant wine ? 71. If I use 1 ream of paper in 10 weeks, how many sheets do I use per day ? 72. If you can count four score in a minute, how many can you count between 8.50 and 9.10, A. m. ? 73. How many gross of boxes at 2/ each can be bought for $5.76? 74. If the sun requires 24 hours to make his apparent journey around the earth, how many degrees does he travel in an hour ? 75. A druggist buys a cask of liquor containing 30 gallons for $ 30, and sells it at $ 0.75 a pint. What does he gain ? QUESTIONS. 161. What is a line ? 162. What are linear measures ? Eecite the table of linear measures. 163. What is a surface ? A plane surface? 164. What is an an- gle ? 165. When are two lines perpendicular to each other 1 166. What is a rectangle ? 167. A square ? 168. For what are surface, or square measures used ? Give the table. 169. What is a solid, or volume? 170. What is a cube? 171. What are cubic measures ? Give the table. 172. For what are liquid measures used ? Give the tables. 173. What articles are measured by the bushel ? 174. For what are troy weights used ? Give the table. 175. For what are avoirdupois weights used ? Give the table. What is the long ton? Which is the heavier, a pound of gold or a pound of sugar ? 176. What is time? Kecite the table. When does the civil day begin and end ? Name the calendar months. When has February 29 days ? 177. What is a circle ? The circumference of a circle ? The di- ameter ? 178. What is a degree ? 179. How is an angle measured ? Give the table. What is a degree at the equator ? What is a sign ? A right angle ? 180. Give the counting table. 181. Give the table for paper. 126 COMPOUND NUMBERS. COMPOUND NUMBERS. 182. 1. How many inches in 3 feet 6 inches ? 2. How many feet and inches in 42 inches ? 3. How many quarts in 6 gallons 3 quarts ? 4. How many gallons and quarts in 27 quarts ? 183. A Denomination is the name of a unit of measure oi weight. 184. A Denominate Number is a number composed of units of one or more denominations. Thus, 42 inches, 6 pounds 4 ounces, are denominate numbers. 185. A Simple Number is a number of a single kind or denomination. Thus, 2, $ 3, 5 months, 7 yards, are simple numbers. 186. A Compound Number is a number composed of two or more denominations of the same general kind. Thus, 3 ft. 6 in., 6 gal. 3 qt., are compound numbers. REDUCTION. To change Denominate Numbers to Smaller Denominations. ORAL EXERCISES. 5. How many inches in 7 feet ? In 5 feet 6 inches ? 6. How many gills in 5 gallons ? In 2 gal. 3 qt. ? 7. How many ounces in 3 lb. troy ? In 8 lb. 7 oz. ? 8. How many minutes in 4 h. 20 min. ? In 5 h. 15 min. ? 9. How many quires of paper in 10 reams ? How many sheets in 3 quires 16 sheets ? COMPOUND NUMBERS. 127 10. How many days in 12 weeks 6 days ? In 11 weeks 4 days ? 11. How many quarts in 3 pecks ? In 2 bu. 1 pk. ? 187. Reduction Descending is changing denominate num- bers to smaller denominations. WRITTEN EXERCISES. 12. How many inches in 124 rd. 4 yd. 2 ft. ? 124 (rd.) 4 yd. 2 ft. ^^^^^ .^^^ _ 1 rd. = 5^ yd. ; 124 ^ J^' rd. = 124 X H yd., or 682 yd., 62 which, with 4 yd. added, are 686 620 yd. ^ . . . 1 yd. 3= 3 ft. ; 686 yd. = 686 X DOD (^ya.; 3 ^^^ ^^^ ^058 ft., which, with 2 ft. _? ^*- added, are 2060 ft. 2060 (ft.) 1 ft. = 12 in. ; 2060 ft. 3= 2060 12 in. X 12 in., or 24720 in. 124 rd. 4 OATOf) * yd- ^ ^^' ^^^ 24720 inches. 13. How many pints in 19 bu. 3 pk. 7 qt. 1 pt. ? 14. How many cubic feet in 48 cu. yd. 15 cu. ft. ? 15. How many pounds in 17 T. 17 cwt. 90 lb. ? 188. Rule for Reduction Descending. Multiply the largest denomination by the number of units it takes of the next smaller denomination to equal one of that larger, and to the product add the given number, if any, of the smaller denomination. Multiply the sum in like manner, and so proceed until the given number is changed to units of the required denomina- tion. 16. Eeduce 12 A. 144 sq. rd. 144 sq. ft. to square feet. 17. Eeduce 5 cu. yd. 23 cu. ft. 725 cu. in. to cubic inches. 18. Reduce 60 gal. 3 qt. 1 pt. to pints. + 128 COMPOUND NUMBERS. 19. Eeduce 13 bii. 2 pk. 7 qt. 1 pt. to pints. 20. Reduce 47 miles to feet. 21. Eeduce 72 lb. 10 oz. 15 pwt. 7 gr. to grains. 22 Eeduce 43 T. 13 cwt. 20 lb. to pounds. 23. Eeduce 365 d. 5 h. 48 min. 50 sec. to seconds. 24. How many cubic inches in 8 cu. yd. 10 cu. ft. 728 cUo in.? 25. How many seconds in 45° 28' 54^' ? 26. How many cubic feet in 25 cd. ? 27. How many gills in 40 gal. 3 qt. pt. 2 gi. ? 28. How many minutes in 67 wk. 6 d. 9 h. 52 min. ? 29. How many sheets of paper in 4 bundles 1 ream of paper ? 30. What is the value of 1 A. 80 sq. rd. of land at 5 cents a square foot ? 31. What will be the cost of grading 3 m. 195 rd. of road at 1 6.50 a rod ? 32. Change ^ of a mile to feet. 660 Solution. — 1 mile = 5280 ft. ; J mi. = -J of 5280 ft., or | X ^^^jH ft. = 4620 ft. 33. Eeduce ^g- of a mile to yards. 34. What part of a grain is -g-^^jj of a pound troy ? 35. What part of a second is s^^jjjj of a day ? 36. What part of a gill is -z^^-g of a gallon ? 37. Eeduce ^ oi o, bushel to quarts. 3a Eeduce 1.375 gallons to pints. 1.375 (gal.) 4 q^^ Solution. — 1 gal. = 4qt. ; 1.375 gal. = 1.375 rKAA / i. \ X 4 qt., or 6.5 qt. ^^^'^ 1 qt. = 2 pt. ; 6.5 qt. = 5.5 X 2 pt., or 1 1 t.V^- pints. 11.000 pt. COMPOUND NUMBERS. 129 39. Reduce .024 of a- ton to pounds. 40. Reduce .0075 of an acre to square feet. 41. Eeduce .3945 of a day to minutes. 42. How many square yards in 1.364 acres ? To change Denominate Numbers to Larger Denominations. ORAL EXERCISES. 43. How many feet in 84 inches ? In 66 inches ? 44. How many gallons in 160 gills ? In 88 gills ? 45. How many pounds troy in 36 ounces ? In 103 ounces ? 46. How many hours in 260 minutes ? In 315 minutes ? 47. How many reams in 200 quires of paper ? In 76 sheets of paper how many quires ? 48. How many weeks in 90 days ? In 81 days ? 49. How many pecks in 24 quarts ? In 72 quarts ? 189. Reduction Ascending is changing denominate num- bers to larger denominations. WRITTEN EXERCISES. 50. Change 410 feet to larger denominations. 3 ft. 5iyd. 2 4^0 fi;^ Solution. — - As in 136^. +2 ft ■ 3 feet there is 1 yard, •^ there are as many 11 hf .-yd, _ yards in 410 feet as 272 hf.-yd. times 3 feet in 410 24 rd. + 8 hf.-yd., or 4 yd. feet, which are 136, and 2 feet remainder. As in 5^ yards there is 1 rod, there are as many rods in 136 yards as times 5^ yards in 136 yards, or times 11 half-yards in 272 half-yards, or 12, and 8 half-yards, or 4 yards, remainder. In 410 feet there are 24 rd. 4 yd. 2 ft. 51. Change 1279 dry pints to larger denominations. 52. Change 1311 cubic feet to larger denominations. 53. Change 35790 pounds to larger denominations. 9 ^ x 130 A COMPOUND numbehs. 190. Rule for Reduction Ascending. Divide the given number by the number of units it takes oj that denomination to equal one of the next larger, and reserve the remainder, if any* Divide the quotient in like manner, and so proceed until the required denomination is reached. The last quotient, and the several remainders, if any, will be the number sought. Note. — Reduction Ascending and Reduction Descending, being reverse processes, are proofs of each other. Change to larger denominations : 54. 562068 square feet. 61. 31556930 seconds. 55. 273749 cubic inches. 62. 391256 cubic inches. 56. 487 liquid pints. 63. 163734''. 57. 879 dry pints. 64. . 3200 cubic feet. 58. 248160 feet. 65. 1306 gills. 59. 419887 grains. 66. 684592 minutes. 60. 87320 pounds. 67. 4320 sheets of paper. 68. At 5 cents a square foot, how many acres and square rods of land can be bought for $ 3267 ? 69. At $ 5.50 a rod, how many miles and rods of road can be graded for % 6352.50 ? 70. Change 3080 ft. to the fraction of a mile. Solution. — 1 mi. = 5280 ft. ; 3080 ft. mi. = -j^j mi. are |§|^ mi., which, changed to sma^le^t terms, is ^ mi. ■f 71. Reduce 88 yards to a fraction of a mile. 72. What part of a pound troy are 3000 grains ? 73. What part of a day are 12600 seconds ? 74. What part of a gallon are 24 gills ? 75. Reduce 4^^ quarts to a fraction of a bushel. COMPOUND NUMBERS. 131 76. Reduce 11 pints to a decimal of a gallon. 2) 11.000 pt. Solution. — As in 1 quart there are 2 pints, 4") 5 500 at. there are in 11 pints ^ as many quarts, or ^ - — ~ 11 ~ 2 = 5.5 quarts. l.o / o ga . As in 1 gallon there are 4 quarts, there are in 5.5 quarts \ as many gallons, or 5.5 -~ 4 = 1.375 gallons. 77 Reduce 48 pounds to a decimal of a ton. 78. Reduce 326.7 square feet to a decimal of an acre. 79. What decimal of a day is 568.08 minutes ? 80. How many acres in 6601.76 square yards ? To change Denominate Fractions to Integers of Smaller Denominations. ORAL EXERCISES. 81. How many quarts in | of a peck ? 82. How many hours in ^ of a day ? 83. How many feet and inches in f of a yard ? Solution. — f of a yard = | of 3 feet = 2^ feet ; ^ of a foot = ^ of 12 inches = 6 inches ; f of a yard = 2 feet 6 inches. 84. How many days and hours in | of a week ? 85. How many ounces and pennyweights in .8 of a pound ? 86. How many quarts and pints in .6 of a gallon ? 191. A Denominate Fraction is a fraction of a denomi- nate number. WRITTEN EXERCISES. 87. Eeduce ^ of a bushel to units of smaller denominations. r^ bu. = I of 4 pk. = 3^ pk. Solution. K ^ pk. = I of 8 qt. = Of qt. (f qt. =f of2pt. =l^pt. Ana. 3pk. Oqt. l^^pt. 132 COMPOUND NUMBERS. 88. Keduce § of a mile to units of smaller denominations. 89. What is tlie value of .83 of a bushel in smaller denom- inations ? First Solution, Second Solution. .83 bu. = .83 of 4 pk. = 3.32 pk. .83 bii. .32 pk. = .32 of 8 qt. = 2.56 qt. f ^6 qt. = .56 of 2 pt. = 1.12 pt. 3.32 pk. 8 Ans. 3 pk. 2 qt. 1.12 pt. 2.56 qt. 2 1.12 pt. 90. Keduce .53 rods to units of smaller denominations. 192. Rule for Reduction of Denominate Fractions to Integers. Change the fraction^ as far as possible, to an integer of the denomination next smaller. If there is a fraction in the re- sult, proceed with it in like manner, and so continue as far as required. Eeduce to integers of smaller denominations : 91. f of an acre. J^ 95. .6725 of a cental. 92. I of a pound troy. 96. .282 of a ton. 93. -/y of a common year. 97. .875 of a rod. 94. ^ of a mile. 98. .761 of a day. To change Denominate Integers to Fractions of Larger D enominations. ORAL EXERCISES. 99. What part of a peck is 6 quarts ? 100. What part of a day is 21 hours ? 101. What fraction of a pound is 14 ounces ? 102. What fraction of a yard is 2 feet 6 inches ? 103. What fraction of a gallon is 2 quarts 1 pint ? COMPOUND NUMBEKS. 133 104. What decimal of a bushel is 3 pecks ? 105. What decimal of an ounce is 16 pennyweights ? WRITTEN EXERCISES. 106. What fraction of a bushel is 3 pk. qt. 1 J pt. ? rHpt. =:^/pt.=:V--2,orfqt. Solution. -( Of qt. = f -^ 8, or ^ pk. ( 3i pk. = ^- pk. = 2^- ^ 4, or ^ bu., Ans. 107. What part of a mile is 71 rd. 1 ft. 10 in. ? 108. What decimal of a bushel is 3 pk. 2 qt. 1.12 pt. ? r 1.12 pt. = 1.12 -^ 2 == .56 qt. ^ 2) 1.12 pt. Solution. J 2.56 qt. = 2.56 -^ 8 = .32 pk. I Or, 8) 2^ qt. ( 3.32 pk. = 3.32 -^ 4 =: .83 bu. J 4) 3^ pk. Ans. .83 bu. .83 bu. 109. Eeduce 2 yd. 2 ft. 8.94 in. to a decimal of a rod. 193. Rule for Reduction of Integers to a Denominate Fraction. Change the given number of the smallest denomination to a fraction of the next larger. Write this fraction as a part of the number of that larger denomination. Change, in like manner, the number thus formed, and so proceed as far as required. 110. Eeduce 68 sq. rd. 155 sq. ft. to a fraction of an acre. 111. Eeduce 10 oz. 13 pwt. 8 gr. to a fraction of a pound. 112. Eeduce 232 d. 10 h. 21 min. to a fraction of a year. 113. Eeduce 248 rd. 4 yd. 2 ft. 8 in. to a fraction of a mile. 114. Eeduce 67 lb. 4 oz. to a decimal of a cental. 115. Eeduce 5 cwt. 64 lb. to a decimal of a ton. 116. Eeduce 4 yd. 2 ft. 5.25 in. to a decimal of a rod. 117. Eeduce 18 h. 15 min. 50.4 sec. to a decimal of a day. 134 COMPOUND NUMBERS. 194. When it is required to find the part that one de- nominate number is of another, Reduce the numbers to the same denomination, and divide the result denoting the part hy that denoting the whole, . 118. What part of 2 A. 112 sq. rd. is 144 sq. rd ? 119. What part of 3 mi. 120 rd. 4 yd. is 2 mi. 120 rd. 3 yd. \ 120. What part of 1 lb. 4 oz. 12 pwt. 12 gr. is 5 oz. 10 pwt. ? 121. What decimal of 74 mi. 80 rd. is 9 mi. 90 rd. ? 122. What decimal of 7 bu. 1 pk. 5 qt. is 82 bu. 3 pk. 1 qt. ? V ADDITION. 195. The operations with compound numbers differ from those with simple numbers only in the fact that compound numbers have an irregular instead of a decimal scale. * The principles, however, being the same, no special rules are required for adding, subtracting, multiplying, or divid- ing compound numbers. WRITTEN EXERCISES. 123. Add 15 lb. 11 oz. 19 pwt. 22 gr., 71 lb. 10 oz. 13 pwt. 17 gr., and ^h lb. 9 oz. 17 pwt. 14 gr. 15 lb. 11 oz. 19 pwt. 22 gr. Solution. — The sum of the 71 10 13 17 grains is 53 gr. = 2 pwt. 5 gr. QK 9 17 14 ^^ write the 5 gr. beneath, r^jT— ; and add the 2 pwt. with the 153 lb. 8 oz. 11 pwt. 5 gr. . , ^ ^ ^ pennyweights. The sum of the pennyweights is 51 pwt. = 2 oz. 11 pwt. We write the 1 1 pwt. beneath, and add the 2 oz. with the ounces. The sum of the ounces is 32 oz. = 2 lb. 8 oz. We write the 8 oz. beneath, and add the 2 lb. with the poundrt. The sum of the pounds is 153 lb., wliich we write beneath, and have, as the entire sum, 153 lb. 8 oz. 11 pwt. 6 gr. COMPOUND NUMBERS. 135 124. 125. mi. rd. ft. in. A. sq. rd. sq. ft. sq. in, 21 296 11 1 65 169 272 143 46 279 10 11 80 134 260 116 35 214 9 9 14 110 166 135 68 276 16 10 66 68 177 131 64 70 16 1 60 161 69 117 216 177 I'ti 8 278 136 131i 66 Or, Or, 216 177 15 2 278 136 131 102 In Ex. 124, the ^ ft. in the sum, reduced to units of a lower denom- ination, gives 6 in., which added with the 8 in. of the sum, and re- duced, gives the second form of answer. In Ex. 125, the -^ sq. ft., in like manner, gives 36 sq. in., which added to the 66 sq. in. of the sum, and reduced, gives the second form of answer. 126. What is the sum of 14 cu. yd. 20 cu. ft. 1463 cu. in., 9 cu. yd. 20 cu. ft. 1463 cu. in., 11 cu. yd. 23 cu. ft. 67 cu. in., and 27 cu. yd. 1305 cu. in. ? 127. Find the sum of 18 gal. 3 qt., 60 gal. 3 qt. 1 pt., 61 gal. 3 qt., and 57 gal. 3 qt. 1 pt. 128. Find the sum of 15 d. 23 h. 55 min. 17 sec, 13 d. 15 h. 17 min. 38 sec, 10 d. 23 h. 42 min. 17 sec, 16 d. 16 h. 38 min. 47 sec, and 20 d. 52 min. 57 sec. 129. What is the sum of 28° 56' 58'', 21° 51' 37", and 13° 39' 57" ? 130. Add .761 d. 4-1- h. and 1 h. 8 min. 31 sec. 761 d. = 18 h. 15 min. 50.4 sec. Solution. — .761 d. 4j h. = 4 15 ^^^ ^i ^' ^^^ ^^^^ ^^' j[ g 3^ duced to units of small- ■ er denominations, and 23 h. 39 min. 21.4 sec ^jj^^ .^ith the 1 h. 18 min. 31 sec, give, as the answer, 23 h. 39 min. 21.4 sec. 136 COMPOUND NUMBERS. 131. Find tlie sum of ^^ of a ton, \\ of a cental, and 1 T. 2 cwt. 3 lb. 132. A man traveled one day 60j miles, the second day 50 mi. 120 rd., and the third day h^"^ miles. How far did he travel in the three days ? , 133. I have three lots, — the first contains f of an acre, the second § of an acre, and the third |J of an acre. How many acres have I ? SUBTRACTION. 134. Find the difference between 15 lb. 3 oz. 12 pwt., and 9 lb. 1 oz. 17 pwt. 15 lb. 3 oz. 12 pwt. Solution. — As 17 pwt. cannot be taken 9 1 17 from 12 pwt., we take 1 oz., = 20 pwt., ■"T; I ~ from the 3oz., leaving 2oz., and add 6 lb. 1 oz. 15 pwt. + ^1, in / I.' ^ • owT ^ to the 12 pwt., which gives 32 pwt. ; 17 pwt. taken from 32 pwt. leaves 15 pwt. We write the 15 pwt. beneath. 1 oz. from 2oz. leaves 1 oz., which we write beneath; 9 lb. from 15 lb. leaves 6 lb. Ans. 6 lb. 1 oz. 15 pwt. 135. 136. From 73 bu. 2 p. 5 qt. 17 mi. 311 rd. 1 yd. 1 ft. 3 in. Take 59 3 7 3 79 1 2 7 137. From 116 A. 53 sq. rd. 100 ft. 113 in. take 87 A. 137 &q. rd. 100 sq. ft. 113 sq. in. 138. The longitude of Boston is 71^ 4^ 9'^ W. and that of Chicago 87° 35' W. What is the difference in the longitude of the two places ? 139. From {^ of a pound troy take 2 oz. 19.2 gr. 140. From a hogshead of molasses J had leaked out. How much remained ? 141. From .367 of a year take .761 of a day. 142. The distance between two places is .7895 of a mila How much is that more than } of a mile ? COMPOUND NUMBEliSo 137 196. To find the Time between two Dates. 143. What is the time from May 16, 1819, to April 9, 1881 ? 1881 3 mo. 9 d. Solution. — Of the year 1881, 3 mo. 9 d 1819 4 16 and of the year 1819, 4 mo. 16 d. have passed at the time named. As 16 d. cannot be taken 61 10 24 from 9 d. we take the 3d month, March, hav- ing 31 d. and add it to the 9 d., making 40 d. and 40 d. less 16 d. = 24 d. 1 y., or 12 mo., added to the 2 mo. remaining in the minuend = 14 mo., and 14 mo. less 4 mo. =: 10 mo. 1880 y. less 1819 y. = 61 y. Hence the difference between the dates is 61 y. 10 mo. 24 d. Note. — The calendar month taken to add to the days is the month preceding the one named in the later date, and the number thus added must always be the number of days in the month taken — that is, 30 d. for Apr., June, Sept., or Nov., 28 or 29 d. for Feb., and 31 d. for any other month. 144. "Wliat is tlie time l)etween Oct. 16, 1876, and Aug. 9, 1882? 145. A note was given Kov. 15, 1879, and paid July 5, 1881. How long did it run ? 146. How long from the surrender of Cornwallis, Oct. 19, 1781, to the battle of New Orleans, Jan. 8, 1815 ? 147. What time elapsed from the declaration of American independence, July 4, 1776, to the emancipation proclamation, Jan. 1, 1863 ? 148. How many days are there from Dec. 19, 1879, to Mar. 16, 1880 ? 12 4- 31 4- 29 4- 16 = 88 Solution. — There are 12 days remaining in December, 31 in January, 29 in February, and 16 in March, or, in all, 88 days. 149. A note dated April 9 is to be paid June 8, 1881. How long has it to run ? 150. Find the exact time, in days, between May 25, 1880, 10 p. M., and March 4, 1881, 9 A. m. 138 COMPOUND NUMBERS. MULTIPLICATION. 151. Multiply 11 bu. 3 pk. 2 qt. by 7. 11}^ ^ Ir 9 f Solution. — 7 times 2 qt. are 14 qt. = 1 pk. and 6 qt. We write the 6 qt. uii- der the quarts, and reserve the 1 pk. to 82 bu. 2 pk. 6 qt. add to the 7 times 3 pk. 7 times 3 pk. =21 pk., which plus the I pk. reserved = 22 pk. = 5 bu. and 2 pk. We write the 2 pk. under the pecks, and reserve the 5 bu. to add to the 7 times 11 bu. 7 times 11 bu. = 77 bu., which plus the 5 bu. reserved = 82 bu. Ans. 82 bu. 2 pk. 6 qt. 152. Multiply 17 wk. 4 d. 23 h. 47 min. by 8. 153. Multiply 3 mi. 40 rd. 4 yd. 2 ft. by 12. 154. If 1 load of hay weighs 1 T. 3 cwt. 17 lb., what will 9 loads weigh ? 155. How much land in 14 farms of 25 A. 60 sq. rd. 21 sq. yd. each ? 156. If the moon moves in her orbit 13° 11' 35^' in 1 day, how far will she move in 20 days ? DIVISION. 157. Find 1 of 82 bu. 2 pk. 6 qt. 7\ 89 V. 9 V fi f Solution. — 4" ^^ 82bu. = 11 bu., ^ ! £_ i_l with a remainder of 5 bu. The 5 bu. 11 bu. 3 pk. 2 qt. = 20 pk., which added to the 2 pk. in the dividend = 22 pk. | of 22 pk. ^= 3 pk., with a remainder of 1 pk. The 1 pk. = 8 qt, which added to the 6 qt. = 14 qt. -f of 14 qt. = 2 qt. Ans. 11 bu. 3 pk. 2 qt. isa Divide 139 wk. 6 d. 10 h. 16 min. by 8. 159. Divide 40 mi. 210 rd. 5 yd. 2 ft. by 12. 160 If 12 spoons weigh 31b. 10 oz. 11 pwt, what is the weight of each spoon ? COMPOUND NUMBERS. 139 161. What is the daily motion of the moon, if it moves 197° 38' 4:5'' in 15 days ? 162. A planter has 1634 gal. 1 qt. 1 pt. of molasses, which he wishes to put into 25 equal casks. What must be the least capacity of each cask to exactly receive the molasses ? MISCELLANEOUS EXERCISES. 163. How many inches in 18 rd. 5 yd. 2 ft. 11 in. ? 164. What will 5 T. 17 cwt. 25 lb. of iron cost at 3 cents a pound ? 165. At 3 cents a pound, how many tons of iron can be bought for $396.18? 166. What fraction of a common year is 27 of a day ? 167. In one barge there are 50 T. 5 ctl. 75 lb., and in another 47 T. 17 ctl. 35 lb. How many tons in both barges ? 168. Two boats start in a race, and one of them gains 5 feet upon the other in every 55 yards. How many rods will it have gained at the end of 2 miles ? 169. If 2 A. 65 sq. rd. can be plowed in a day, how much can be plowed in 8J days ? 170. Find the exact number of days from June 11, 1879, to Aug. 5, 1881. 171. If 9 acres produce 21 T. 537 lb. of hay, what does one acre produce ? 172. If a pendulum vibrates 47 times in a minute, in what time will it vibrate 13267583 times ? 173. What decimal of 20 acres is 7 A. 148 sq. rd. ? 174. If a man can cut 24 cords 102 cubic feet of wood in 12 days, how many cord feet can he cut in one day ? 175. How many silver spoons, each weighing 2 oz. 10 pwt., ann be made from a bar of silver weighing 11 lb. 5 oz. 10 pwt. ? 140 COMPOUND NUMBERS. 176. A stationer buys 25 reams of commercial note paper at $ 1.75 a ream, and retails it at 12 cents a quire, with tlie exception of one outside quire of each ream, which he sells at 8 cents. How much does he make ? 177. How many years, months, and days did a man live who was born March 15, 1767, and died June 8, 1845 ? 178. Bought a cask of oil, containing 6S^ gallons, at 72 cents a gallon ; | having leaked out, the remainder was sold at 90 cents a gallon. Did I make or lose, and how much ? 179. Show that any article is worth as many five-cent pieces a cental as dollars a ton. 180. A carpenter sent two of his apprentices to ascertain the length of a certain fence. The first made it 17 rd. 16 ft. 11 in., and the second made it 18 rd. 5 in. The carpenter, fearing they might both be wrong, measured for himself, and found it to be 17 rd. 5 yd. 1 ft. 11 in. What was the differ- ence in their measurements ? 181. If A and B should commence, March 5, 1882, to go to bed at the same hour, and A should rise at ^ before 6 o'clock and B at f past 7, how much more time for labor would A have had than B, by March 5, 1900, paying attention to the leap years ? \ \ QUESTIONS. \J^ 183. What is a denomination 1 184. A denominate number 1 185. A simple number ? 186. A compound number ? 187. What is reduction descending ? 188. How is a denominate number reduced to smaller denominations ? 189. What is reduction ascending? 190. How is a denominate number reduced to larger denominations ? 191. What is a denominate fraction ? 192. How is a denominate fraction reduced to integers of smaller denominations ? 1 93. How are denominate integers reduced to fractions of larger denominations ? 196. How is the difference between dates, in yeais, months, and days, found ? THE METKIC SYSTEM. / 141 THE METRIC SYSTEM. 197. The Metric System of weights and measures, now coming into use in the United States, has for its base a unit called the meter. Note. — This system, in extensive use in the arts and sciences, adopted for the United States Coast Survey, and partially employed in the Mint and Gen- eral Post Office, was legalized for use in the United States by Congress in 1866. 198. The Meter, which was intended to be, and is very nearly, the ten-millionth part of the distance on a meridian from the equator to the 2)ole, is the principal unit of lengths, and the standard unit from which all metric measures are derived. 199. The Are,* the principal unit of the measures of land, is a square whose side is ten meters. 200. The Stere, the principal unit of the measures of wood and stone, is a cube whose edge is a meter. 201. The Liter, the principal unit of the measures of capacity, is a cube whose edge is the tenth of a meter. 202. The Gram, the principal unit of weight, is the weight of a cube of pure water at its greatest density, whose edge is a hundredth part of a meter. 203. The Names of the divisions of the unit are formed by prefixing to the name of the unit the Latin words, milli for 1000th, ce7iti for 100th, and deci for 10th; and the names of multiples, by prefixing the Greek, deka for 10, hekto for 100, kilo for 1000, and myria for 10000. * Are is pronounced air ; stere, stair ; and liter, lee'ter. All metric names have the accent on the first syllable. 142 THE METRIC SYSTEM. 204. In the metric system, as in United States money, only a few of the denominations are much used. These will be distinguished in the tables by the difference in type. The unit corresponds to the dollar, and dcci^ centi, milli to climes, cents, mills. 205. LENGTH MEASURES. 10 millimeters (^^) are 1 centimeter, « 10 centimeters 10 decimeters 10 meters 10 dekameters 10 hektometers 10 kilometers 1 decimeter, ^™. 1 METER,- '". 1 dekameter, ^"^. 1 hektometer, ""^. 1 kilometer,^'". 1 myriameter, ^^"^. Equivalents. 1 centimeter = 0.3937 inch. 1 decimeter = 3.937 inches. 1 meter = 39.37 inches. 1 kilometer = 0.6214 mile. 1. The meter is used in measuring woven fabrics and short lengths and dis- tances. 2. The kilometer is the unit in meas- uring roads and long distances. 3. The decimeter is nearly 4 inches ; the m^ter, about 3 feet 3| inches ; and the kilometer, about 200 rods, or | of a mile. 206. A Metric Number is writ- ten with the decimal point separat- ing the unit from its decimal parts. Thus, ;i^Km gHm yDm 3 m ^dm 2 cm ^^ittcu as mctcrs, is 1573.42' and, written as kilometers, is 1.57342 ^'". ^l . •"' ~: tD — E luches. 1 2 ill ill I 1 1 1 1 1 CO — UN INI II 1 1 Ml II 4 5 Centimeters. OJ - "* - CO QO — o - THE METRIC SYSTEM. 143 207. In reading metric numbers, the name of the unit may be applied to all on the left of the decimal point, and the name of the smallest denomination denoted to all on the right of the point. Thus, 42.73 ™ may be read forty-two, and seventy-three hun- dredths meters; or, forty-two meters, and seventy-three centimeters. 8.675 ^"^ may be read eight, and six hundred seventy-five thousandths kilometers ; or, eight kilometers, and six hun- dred seventy-five meters. 1. How many meters and hundredths of a meter are ex- pressed by 7.25 •" ? 2. How many kilometers and thousandths of a kilometer are expressed by 8.407 ^"^ ? 3. How many meters and centimeters does 7.25 " express ? 4. How many kilometers and meters does 8.407 ^™ ex- press ? 5. How many centimeters in 4.15 '" ? How many meters in 7.384 ^^" ? 6. Reduce 784 centimeters to meters ; 6453 meters to kilo- meters. 208. SURFACE MEASURES. 100 square millimeters (^^i ™™) are 1 square centimeter, ^^ ^"^. 100 square centimeters ** 1 square decimeter, sqdm^ 100 square decimeters " 1 square meter, s^'", or centare. 100 square meters ** 1 square dekameter, sqDm^ 100 square dekameters " 1 square hektometer, sqHm, • 100 square hektometers " 1 square kilometer, sq^m^ Also, 100 centares («»), or sq. meters, are 1 are, a. 100 ares •• 1 hektare, "*. 144 THE METRIC SYSTEM. Equivalents. 1 square centimeter = 0. 155 sq. inch. 1 are = 3. 954 sq. rods. 1 square decimeter = 0. 1076 sq. foot. 1 hektare = 2. 471 acres. 1 square meter = 1.196 sq. yards. 1 sq. kilometer =0.38 61 sq. mile. 1. The square meter is used in measuring ordinary surfaces ; the square kilometery in measuring the area of countries ; and the a/re and hektare^ in measuring land. 2. The square meter is about lOf square feet, or 1-|- square yards ; and the hektare, about 2-|^ acres. 3. As 100 units of a smaller denomination make a unit of a denomination next larger, the scale is 100, and two places of figures must be allowed for each denomination. Thus, ^1 Square ' Centimeter. 31 "^ 14 ^ 17 *=% written as ares, is 3114.17 % which may be read 3114 ares, and 17 centares; and, written as hektares, is 31.1417 "% which may be read 31 hektares, and 1417 centares. 7. In 4 square meters how many square decimeters ? In 4 square centimeters how many square millimeters ? 8. Express 65.41 ^ as centares ; as hektares. 9. How many ares in 5734 ""^ ? How many hektares in 6893 ^ ? 209. VOLUME MEASURES. 1000 cubic millimeters (cumm) ^pe 1 cubic centimeter, <="«=™. 1000 cubic centimeters " 1 cubic decimeter, cu^m^ 1000 cubic decimeters ** 1 cubic meter, '="™, or stere. Also, 10 decisteres (^^^) are 1 stere, ^t. ^ Equivalents. 1 cubic centimeter = 0.061 cu. inch. 1 cubic meter = 1.308 cu. yards, 1 cubic decimeter = 61.022 cu. inches. 1 stere = 0.2759 cord. 1. The cvhic meter, the unit of ordinary solids, takes the name of stere when applied to the measuring of wood and lumber. THE METRIC SYSTEM. 145 1 Cubic Centimeter. 2. The cubic decimeter is about 61 cubic inches ; the cubic meter, or stere, about 35j cubic feet. 3. Where, as in the table, 1000 units of a smaller denomination make a unit of a denomination next larger, the scale is 1000, and three places of figures must be allowed for each denomination. Thus, 13cum 4ogcudm 573 cu cm^ Written as cubic meters, is 13.406578 '^^ ™ which may be read 13 cubic meters, and 406578 cubic centimeters ; and, written as cubic decimeters, is 13406.578 ^"^'^ 10. In 6 "^""^ how many cubic decimeters ? In 8 ''"^"' how many cubic centimeters ? 11. In 7000 '^^ '"'^ how many cubic centimeters ? In 9000 cu dm j^Q^ many cubic meters ? 12. Express 76.006^"'^"' as cubic centimeters; as cubic meters. 13. Express 0.3125 ''""' as cubic decimeters ; as cubic centi- meters. 210. CAPACITY MEASUKES. 10 milliliters Q^^) are 1 centiliter, ^^ 10 centiliters " 1 deciliter, ^^ 10 deciliters " 1 liter, K 10 liters " 1 dekaliter, ^K 10 dekaliters " 1 hektoliter, "i. 10 hektoliters " 1 kiloliter, ^K Equivalents. 1 liter = 61.022 cu. inches. 1 hektoliter = 3.531 cu. feet. 1 liter = 1.0567 liquid quarts. 1 hektoliter = 26.417 gallons. I liter = 0.908 dry quart. 1 hektoliter = 2.837 bushels. 1 MiUiliter = 10 - 1 Cubic Centimeter. 146 THE METEIC SYSTEM. 1. The liter is used in measuring liq- uids ; and the hektoliter is used in measuring grains, roots, and liquids in casks. 2. The liter is about 1.06 liquid quarts, or 0.9 of a dry quart ; and the hektoliter is about 26^ gallons, or 2| bushels. 14. How many liters, and what decimal of a liter, are ex pressed by 6.45 * ? 15. How many kiloliters, and what decimal of a kiloliter, are expressed by 9.750 ^^ ? 16. How many centiliters are 6 liters ? 7.55 liters ? 17. How many liters in 600 centiliters ? How many kilo liters in 6000 liters ? 211. WEIGHT MEASURES. 10 milligrams (""s^) are 1 centigram, <=ff. 10 centigrams " 1 decigram, ^s, 10 decigrams " 1 gram, st. 10 grams " 1 dekagram, ^s, 10 dekagrams " 1 hektogram, "«, 10 hektograms " 1 kilogram, ^^, or ^^ 10 kilograms " 1 myriagram, ^s, 10 myriagrams " 1 quintal, Q. 10 quintals " 1 metric ton, '^. THE METRIC SYSTEM. 147 Equivalents. In weight 1 gram = 1 cu. centimeter, or 1 milliliter of water 1 kilogram = 1 cu. decimeter, or 1 liter of water. 1 metric ton = 1 cu. meter, or 1 kiloliter of water. 1 gram = 15.432 grains troy. 1 kilogram =: 2.2046 pounds av 1 gram = 0.03527 ounce av. 1 metric ton== 1.1023 tons. Note. — The weight of the gram is determined when the water is pure and at the temperature of its greatest density, which is 39. 2° Fahrenheit. 1. The gram is used in weighing letters, gold, and jewels, and in mixing medicines ; the hilogram, or, for brevity, kilo^ is used in weighing common articles ; and the metric ton, in weighing very heavy articles. 2. The kilo is about ^\ pounds ; and the metric ton, about 1^ common tons. 3. Of the United States coinage, the nickel five-cent piece weighs 5 grams ; two silver half-dollars, 25 grams ; and 80 silver half-dollars, a kilo. • 4. A letter sent for a single postage must not exceed the weight of six nickels, or 30 grams. 18. In 5.65 ^, how many decigrams ? How many centi- grams ? 19. In 5650 ""^j how many centigrams ? How many deci- grams ? 20. In 6.315 ^^, how many grams ? In 4.670 "^^ how many kilos ? 21. What decimal part of a kilo is a gram ? What decimal part of a metric ton is a kilo ? 22. At 20 cents a kilo, how many dollars will a metric ton of sugar cost ? 23. A tank has the capacity of 5.250 kiloliters. How many metric tons of pure water will it hold ? 24. I wish to weigh 8.75 ^^. How many silver half-dollars will serve for the weights ? 148 tHE METRIC SYSTEM. REDUCTION OF UNITS. 212. The Units shown in the metric tables form a deci- mal system (Art. 19), to which apply the folio whig Principles. 1. Ten units, or some multiple of ten units, of any de nomhiation make one of the next larger unit. 2. A metric number may he changed from one denomination to another next smaller, or larger, by moving the decimal point one or more places to the right, or left, as the case may he, 3. Any denomination may he taken as the unit, the num- ber at the right of the point being read as a decimal of the unit. 213. The units of the metric and the common system may be readily compared by means of the equivalents which have been given, and by means of the following COMPARATIVE TABLE. Length. 1 cu. yard = 0.7646 cu. meter. 1 inch = 2.54 centimeters. 1 ^'^^^ = 3.625 steres. 1 foot = 30.48 centimeters. 1 rod = 5.029 meters. Capacity. 1 mile = 1.6093 kilometers. 1 liq. qnart = 0.9465 liter. « - 1 gallon = 3.785 liters. 1 dry quart = 1.101 liters. 1 sq. inch = 6.452 sq. centim. ^ ^^^^^ ^ 3^24 hektoliters. 1 sq. foot = 9.2903 sq. decim. 1 sq. yard = 0.8361 sq. meter. 1 sq. rod = 0.2529 are. 1 acre = 0.4047 hektare. 1 grain troy = 0.648 centigram. 1 sq. mile = 2.59 sq. kilometers. 1 ounce troy = 31.1035 grams. Volwme. 1 o^^nco av. = 23.35 grams. 1 cu. inch = 16.387 cu. centimeters. 1 j)oiind av. = 0.4536 kilogram. 1 cu. foot = 28.317 cu. decimeters. 1 comnsonT. = 0.9072 mot. ton. Weight, THE METRIC SYSTEM. 149 WRITTEN EXERCISES. 25. Express as meters and add 1365 ™", 497 ^"'j and 145.51 '"c 26. Express as ares and add 15.16 "^ 111.55 % and 3615 '=\ 27. Erom a range of wood containing 45 steres, I have sold 276 decisteres. How many steres remain ? 28. How many liters in 6 casks^ each containing 3.40 ^ ? 29. Eight men shared equally 21.080 metric tons of sugar. What is each man's share worth at 20 cents a kilo ? 30. Erom a farm containing 365.50 "^ there have heen sold two small lots ; the one containing 8.42 ^^, and the other 87.25 ^. How much remains ? 31. Change 125 meters to feet. 39.37 in. Solution. — As 1 meter = 39.37 1^5 inches, 125 meters must equal 125 19685 times 39.37 inches, or 4921.25 7874 inches. As 1 foot = 12 inches, 3937 there will be as many feet as 12 12 in.) 4921 25 in. inches are contained times in 410 IOt^ ft 4921.25 inches, or 410.10i5^ ft. 32. If your weight is 55 kilos, what is it in pounds avoir- dupois ? 33. A garden plat contains 306 square meters. How many square yards are there in it ? 34. A farm is 450 hektares in extent. How many acres does it contain ? 35. A barrel of flour weighs 196 pounds. What is its weight in kilos ? 36. The produce of 7 acres was 210 bushels of wheat. What was it in hektoliters ? 37. When butter is 35 cents a pound, how much should it be a kilo ? Solution. — At 35 cents a pound, a kilo, which is 2.2046 pounds, must cost 2.2046 times 35 cents, or 77+ cents. 150 THE METRIC SYSTEM. 38. At 65 cents a bushel, what should be the price of corn a hektoliter ? 39. What is the value of an eighth of a plantation of 600.58 hektares at $ 25 an acre ? 40. The distance by railroad between Boston and New Orleans is 1607 miles. What is it in kilometers ? ► 41. The dome of the capitol at Washington is 287 ft. 6 in high, surmounted by a statue of Liberty 19 ft. 6 in. high. What is the whole height in meters ? 42. Bought a roll of carpeting of 65 yards at 1 1.20 a meter, and sold it at the same price a yard. How much did I make by the transaction ? 43. The capacity of a certain bin is 40.64 cubic meters. What is the value of the grain that can be put in it at 80 cents a bushel ? QUESTIONS. 197. What is the metric system? 198. What is a meter ? 199. An are ? 200. A stere ? 201. A liter ? 202. A gram ? 203. How are the names of the divisions of the metric units formed 1 205. Recite the table of metric measures of length. For what is the meter used 1 The kilometer ? 206. How is a metric number written 1 207. How are metric numbers read ? . 208. Recite the table of metric measures of surface. How is the square meter used ? The square kilometer ? The are and hektare 1 209. Name the measures of volume. How is the cubic meter used ? When does it take the name of the stere ? 210. Recite the table of measures of capacity. How is the liter used ? The hektoliter 1 211. Recite the table of measures of weight. For what is the giam used ? The kilogram ? The metric ton 1 212. How many units of one denomination make a unit of another in the metric system ? How may a metric number l)e changed from one denomination to another ? When any denomination is taken as the unit, how may the number at the right of the point be read ? MEASUKEMENTS. 151 MEASUREMENTS. SURFACES. 214i 1. How many square inches in a surface 8 inches long and 1 inch wide ? In a surface 8 inches long and 2 inches wide ? 2. A path 12 feet long and 2 feet wide has how many square feet of surface in it ? 3. A table is 6 feet long and 4 feet wide. How many square feet of surface has it ? 215. A Plane Figure is a portion of a plane surface (Art. 163) bounded by lines. 216. The Perimeter of a plane figure is the sum of its bounding lines. 217. The Area of a plane figure is the surface included within its perimeter. 218. The Dimensions of a rectangle are its length and breadth. A rectangle 3 inches long and 2 ^^H^^HK inches broad contains in one row 3 " squares of 1 square inch each ; and 2 such rows contain 2 times 3 square inches, or 6 square inches. That is, The area of a rectangle is equal to the product of its length and breadth, taken in the same denomination. Also, One of the dimensions of a rectangle is equal to the area divided hy the other dimension. 152 MEASUREMENTS. 219. A Triangle is a plane figure bounded by three straight lines. The Base of a triangle is the line upon which it stands ; and the Alti- tude is its height above the base, or ^ the base extended. Thus, ^ ^ is the base, and C B the altitude, of the triangle A BG, 220. Draw the lines A E, D B per- pendicular to the extremities of the base of the triangle A C B, and draw the line E D through C, parallel to A B, and it is evident that the triangle A Bis half the rectangle ABBE, of the same base and altitude. That is. The area of a triangle is half the area of a rectangle of the same base and altitude. 221. A Circle may be regarded as consisting of a great number of tri- angles, whose bases form the circum- ference of a circle, and whose altitude is the radius of the circle. Hence, The area of a circle is equal to half the product of the cir- cumference hy the radius, 222. The quotient of the circumference of a circle di- vided by the diameter, to the nearest ten-thousandth, is 3.1416. Hence, The circumference is equal to the diameter multiplied hy 3.1416 ; tlu diameter is equal to the circumference divided by 3.1416. 62.75 26 31660 10660 1371.60 sq. ft. 1371 sq. ft. 72 sq. in. MEASUKEMENTS. 153 WRITTEN EXERCISES. 4. What is the area of a floor which is 52 feet 9 inches long and 26 feet wide ? Solution. ^52 ft. 9 in. = 52.V5 feet. The product of the length by the width gives as the number of square feet of surface 1371.50 sq. ft., or 1371 sq. ft. 72 sq. in., which is the Or, area required. 5. The area of a floor is 1371 square feet 72 square inches, and its width is 26 feet. What is its length ? 52.75 ft. = 52 ft. 9 in. 26) 1371.50 Solution. — 1371 sq. ft. 72 sq. 130 in. = 1371.50 sq.ft. Asthearea 71 1371.50 sq. ft. is the product of g2 ^he length and width, the width YqF must equal the quotient of the ^^^ area 1371.50 divided by the i^ length 26, which is 52.75 ft. = ^^^ 52 ft. 9 in. 130 6. The base of a triangle is 46 ft. 3 in., and the altitude 35 ft. 6 in. What is its area ? 7. The circumference of a circle is 314.16 feet, and. its ra= dius 50 feet. What is its area ? 8. The diameter of a circle is 400 feet. What is its cir- cumference ? 9. The circumference of a circle is 1256.64 feet. What is its diameter ? 10. How many yards of carpeting 1 yard wide will be re- quired to carpet a room 18 ft. long and 15 ft. 6 in. wide ? 154 MEASUREMENTS. 11. A room that is 18 ft. 9 in. square requires 50 yards of carpeting. What is the width of the carpeting ? *r* 12. How many acres in a field 45 rods long and 48 rods wide ? 13. The diameter of a circle is 15 meters. What is its cir- cumference in feet ? 14. What is the area of the gable end of a house 32 ft wide, the ridge being 14 ft. 6 in. higher than the base of the gable ? . 15. How many hektares in a rectangular meadow 564.50 meters long and 260 meters wide ? 16. What will it cost at 60 cents a square yard to concrete a walk 288 ft. long and 12 ft. 6 in. wide ? 17. A horse is fastened to a stake by a chain 60 feet long. How many square rods of surface can the chain sweep over ? 18. How many square feet of sheet zinc will be required to line a cistern 6 ft. deep, having a square bottom, of which each side is 2 ft. 6 in. ? 19. The capitol at Washington is 751 feet long and 348 feet wide. How many acres does it cover ? 20. On laying the pavement of a court with stones 2 ft. 6 in. long by 9 in. wide, it is found that it requires 75 stones to form one strip extending the whole length of the court, and that 8 J strips will exactly cover it. What is the area in square yards, and what is the cost of the pavement at 20 cents a square foot ? VOLUMES. ^ 21. How many cubic feet in a beam 10 feet long, 1 foot wide, and 1 foot deep ? In a beam 10 feet long, 1 foot wide, and 2 feet deep ? 22. How many cubic feet in a block of marble 8 feet long, 2 feet wide,, and 1 foot thick ? In a block 8 feet long, 2 feet wide, and 2 feet thick ? MEASUREMENTS. 155 223. A Rectangular Volume is a body bounded by six rectangles. The Dimensions of a rectangular volume are its length, breadth, and tniCKneSS. a RectaDguliar Volume. 224. The Contents of a rectangular volume are the space contained within its bounding surfaces. The dimensions of a rectangular volume determine its contents. Thus, A rectangular volume 3 inches long, 3 inches wide, and 2 inches thick, contains in one layer 3 rows of 3 viubic inches, or 9 cubic inches, and 2 such layers, or 2 times 3 times 3 cubic inches, contain 18 cubic inches. That is. The contents of a rectangular volume are equal to the pro- duct of its length, breadth, and thickness^ taken in the same denomination. Also, One of the dimensions of a rectangular solid equals its contents divided hy the product of the other two dimensions. 225. A Cylinder is a round body of uni- form diameter, whose bases are equal and parallel circles. The altitude of a cylinder is the straight line joining the centers of the two bases ; as the line A B. 226. The Contents of a cylinder are equal to the product of the area of the base by the altitude. The Cu7^ed Surface of a cylinder is equal to the ^product of its circumference and altitude. ^ 156 MEASUREMENTS. WRITTEN EXERCISES. 23. How many cubic feet in a rectangular block 8 feet long, 4 feet 6 inches wide, and 2 feet 3 inches thick ? Solution. — 4 ft. 6 in. = | ft. ; 2 ft. 3 in. = | ft. The product of the three dimensions, 8 X f X | = 81 cu. ft. 24. The contents of a rectangular block 8 feet long and 4 J feet wide are 81 cubic feet. What is its thickness ? 8 X 4i := 36 Solution. — The contents of the block are the product of its three dimensions. 81 -^ 36 = 24- The product of the length and width is 36. The given product of the three 24 ft. = 2 ft. 3 in. dimensions divided by 36 gives for the thickness 2^ ft., or 2 ft. 3 in. 25. What are the contents of a trunk 4 feet long, 2^ feet wide, and 18 inches deep ? 26. How many cubic feet of space in a room whose dimen- sions are each 10 feet 6 inches ? 27. A rectangular cistern, whose length is 13| feet and breadth 6 feet, contains 294J cubic feet of water. What is the depth of the water ? 28. A box is 2 meters long, 15 decimeters wide, and 1 meter deep. What is its capacity ? 29. How many cubic yards of earth must be removed in ex- cavating a cellar 60 feet long, 42 feet wide, and 8 feet 6 inches deep ? 30. A block is 3 meters long, 2 meters wide, and 1.45 me- ters thick. What are its contents ? 31. A circular well is 32 feet deep and 3 feet in diameter. How many cubic yards does it contain ? 32. A roller is 4 feet 6 inches long, and 6 feet 4 inches in circumference. How much surface will it pass over in revolv- ing 36 times ? MEASUREMENTS. 157 WOOD MEASURE. -/? ,. .-. A COKD 227. A Range of wood 8 feet long, 4 feet wide, and 4 feet high, makes a cord of wood (Art. 171). A Cord Foot is 1 foot in length of this range, or 16 cubic feet. WRITTEN EXERCISES. 33. How many cords of wood in a range 32 feet long, 8 feet high, and 4 feet wide ? 34. How many cords of 4-foot wood in a range 64 feet long and 4 feet high ? 35. A range of 4-foot wood is 28 feet long and 6^ feet high. How many cords does it contain ? 36. A range of 4-foot wood is 56 feet long. How high must it be piled to contain 10 cords ? 37. Wood is loaded upon a cart in 2 piles, 3 feet 6 inches, wide and 4 feet 6 inches high. Allowing the wood to have been cut 4 feet long, how much wood is there on the cart ? 38. A shed contains a pile of wood 30 feet long, 16 feet wide, and 12 feet high. What is the value of the wood at $ 5.25 a cord ? 158 MEASUREMENTS. BOARD MEASURE. 228. A Board Foot is a square foot of board one inch thick; and 12 board feet make 1 cubic foot. 229. Lumber, or sawed timber, as boards, plank, joists, and the like, is estimated in board feet. Heivn Timber is estimated either in board feet or cubic feet. In the measurement of lumber, or hewn timber which tapers, half the width or half the thickness of the two ends is taken. 230. To find the contents of a piece of lumber in board feet, Multiply the product of the length and width taken in feet by the thickness in inches. Note. — Lumber less than an inch thick is measured as if it were an inch thick. In computations, therefore, thickness may be disregarded if an inch or less. In the measurement of box boards, however, the standard of thickness is § of an inch. WRITTEN EXERCISES. 39. How many board feet in a joist 21 feet 6 inches long, 4 inches wide, and 3 inches thick ? Solution. — Multiplying to- 21 ft. 6 in. = -^2^- ft. gether the length and width expressed in feet, we have, as 4 inches = i% = g- ft. surface feet, ^ x ^, and, mul- tiplying by the inches in thick- "V" X i X ^ = ^^i- ^^* ^^' ne^s, have as board feet, ^ x J X 3 = 21^. 40. How many board feet in a board 16 feet long, 1 foot 6 inches wide, and | inch thick ? MEASUREMENTS. 159 41. Required the contents in board feet of a plank 20 feet long, 16 inches wide, and 2 J inches thick? 42. How many board feet in a piece of hewn timber 20 feet long, 10 inches thick, and whose width tapers from 18 to 16 inches ? 43. What will be the cost of 44 spruce joists, each 18 feet long, 9 inches wide, and 3 inches thick, at $ 23 a thousand feet ? 44. What is the value of 18 planks 2 inches thick, 24 feet long, 21 inches wide, at $ 35 a thousand feet ? MISCELLANEOUS EXERCISES. 45. A rectangular lot containing 9432 square feet has a frontage of 131 feet. What is its other dimension ? 46. A wagon 8 feet long and 3 feet wide has wood piled on it 5^- feet high. How much is the wood worth at $ 5.50 a cord ? 47. How many board feet of plank 2 inches thick will be required to cover a platform 30 ft. 6 in. long and 20 ft. wide ? 48. What must be paid, at the rate of $ 40 a thousand feet, for 6 boards 16 feet long, and in width tapering from 2S to 20 inches ? 49. A cistern 2.50 meters long, 90 centimeters wide, and 110 centimeters deep, will hold how many hektoliters of water ? 50. How many rolls of paper 12 yards long and 1 foot 8 inches wide will be required to paper a room 18 feet long, 12 feet wide, and 9 feet high, no allowance being made for win- dows and doors ? 51. Allowing .8 of a bushel to a cubic foot, how many bushels of corn can be put in a bin whose inside measure is 4 feet long, 3 feet wide, and 4 feet deep ? 52. Allowing 7J- gallons to a cubic foot, how much will a tank hold whose inside measure is 2 ft. 6 in. long, 2 ft. wide, and 1 ft. 9 in. deep ? 160 MEASUREMENTS. 53. Allowing for each square foot of surface of a brick wall twice as many bricks in number as the wall is inches thick, how many bricks are there in a wall 40 feet long, 12 feet 6 inches high, and 8^ inches thick ? 54. If a ton of 2000 pounds of Lehigh coal fills 40 cubic feet, and of Lackawanna coal 45 cubic feet, how much of each kind will fill a bin 8 feet long, 5 feet wide, and 4 feet deep ? 55. Water is flowing into a cistern whose rectangular base is 4840 square inches. How many cubic feet will have been supplied when the depth of water is 3J feet, and what w^ill be its w^eight at the rate of 1000 ounces a cubic foot ? 56. How many cubic feet in the walls of a cellar, whose length is 33 feet, width 30 feet, and depth 9 feet, allowing the thickness of the walls to be 18 inches, and the lap of the one wall by the other at each corner of the cellar to be 18 inches ? -1- QUESTIONS. 215. What is a plane figure? 216. What is the perimeter of a plane figure ? 217. The area of a plane figure ? 218. To what is the area of a rectangle equal ? 219. What is a triangle ? The base of a triangle ? The altitude ? 220. How is the area of a triangle found 1 221. How may the circle be regarded ? To what is the area of a circle equal ? 222. To what is the circumference equal ? To what is the diameter equal ? 223. What is a rectangular volume ? What are the dimensions of a rectangular volume ? 224. The contents ? 225. What is a cylinder ? The altitude of a cylinder 1 226. What are the contents of a cylin- der ? To what is the curved surface of a cylinder equal ? 227. What range of wood makes a cord ? How much of such a range is a cord foot 1 228. What is a board foot 1 229. How is lumber estimated ? How is hewn timber estimated 1 230. How are the contents of a piece of lumber found ? 224. The contents of a room divided by the area of the floor will give what ? REVIEW. 161 REVIEW. ORAL EXERCISES. 231. 1. The top of your desk is 2 ft. long and 18 in. wide. How many yards around it ? 2. It is IJ miles to the post-office. How many miles do I travel in making three round trips to it ? 3. How many score in a gross ? 4. My hens lay an egg apiece daily. How many dozen do 3 of them lay in 3 weeks 3 days ? 5. If a ton of coal measures 40 cu. ft., how many tons will two 5-foot cubical bins hold ? 6. How many degrees does the minute-hand of a clock move over in 45 minutes ? 7. What is the perimeter of the largest surface measure ? 8. What is the area of the entire surface of a 4-foot cube ? 9. How wide must a yard of ribbon be to contain 144 sq. in. ?_/ 10. W^hat will it cost to carpet a room 18 ft. square at $ 1 ' per square yard ? 11. What costs a mile of wire at a cent a foot ? 12. At 30 cents a gallon, what is my milk-bill for April ? I take 2 quarts daily. 13. What will a 12-inch cube of marble cost at one cent per cubic inch ? 14. f of 48 is ^Y of what number ? 15. If ^ ounce of tea costs 5 cents, what will 2^ pounds cost ? 16. How many acres in my grandfather's farm, which is J of a mile square ? 17. If 3^- dozen cost I 3.50, what will y^^ 2^^ 79. A room 8 feet high is 16 feet long and 14 feet wide. How many yards of paper 2 feet wide will cover the walls ? 80. A steamer reaches Boston at 3.15 p. m., Aug. 3, 1881, by the steamer's time, after a voyage of 7 d. 15 h. 30 min. Find the exact time of her sailing. 81. How many square yards of silk in 300 feet of 3-inch ribbon ? 82. What cost 18 gal. 3 qt. 1 pt. at % 0.45 per gallon ? ' 83. Find the difference between 0.7| and 0.7 + |. 84. From an acre of land I sold 8 square rods, and also a piece 8 rods square. How much have I left ? "7^ 85. A pile of wood containing 4 J cords is 5 ft. 6 in. high and 4 J feet wide. How long is it ? 86. A room 30 feet long requires 80 yards of carpeting J yd. wide. How wide is the room ? 87. A field containing 5 acres is 600 feet long. Required the distance around it. 88. The ^^ desks in a scbool-room are 2 feet by 18 inches. How many yards of 40-ini'h cloth will cover them? 89. How many 2-inch cubes will a cubic yard make ? 90. How many feet of inch boards can be sawed from a stick of mahogany 15 feet long and 20 inches square, 0.05 be- ing wasted in sawing ? 166 REVIEW. 91. From a lot 90 rods square I sold 90 square rods. What is the value of the remainder at $ 240 per acre ? 92. From a piece of cloth containing 9| yards, 3f yards were cut. What part of the whole piece remained ? 93. A garden is 12 rods long and 10 rods wide. At 75 cents a square yard, what will a concrete walk 6 feet wide, and Bxtending around the garden inside the fence, cost ? 94. A garden 240 feet long and 160 feet wide is enclosed by a tight board fence 6 ft. high. What will it cost to paint both sides of the fence at 5 cents per square yard ? 95. How many days, hours, and minutes from 10.30 p. m., Feb. 4, to 3.40 a. m.. May 11 ? 96. A regiment of 900 soldiers is to be clothed ; each suit requires 9 yards of cloth 1^ yd. wide. How many yards of ^ flannel | yd. wide will be required to line the suits ? 97. I have a room to carpet that is 30 feet long and 27 feet wide. Which is cheaper, to buy yard-wide carpeting at $ 1.25, or carpeting f yd. wide at $ 1, and how much ? 98. If the weight of a hektoliter of wheat is 75 kilos, what weight of wheat in metric tons will fill a bin 2 meters long, 1.4 meters wide, and 1 meter deep ? 99. What will it cost to shingle a house 112 ft. long, each side of the pitch-roof being 25 ft. wide, 10 shingles covering a square foot, the shingles costing $ 6.50 per thousand. 100. Bought 7|J dozen at 18J cents a dozen. Required the cost of 12 dozen. 101. Sold 145 tons of coal at $ 5f per ton, and 12 J tons at $6J per ton, and charged 25 cents a ton for housing the coal. Find the amount of my bill. 102. From the product of 3J and 4J take half their dif- ference. 103. If 17^ gallons cost $ 19g, what will | of a gallon cost ? 104. From a piece containing J of a yard I sold J of a yard at $ I a yard, and gained $ i. What did the piece cost me ? H- PERCENTAGE. 167 PEBCENTAGE. 232. 1. What is tW of 100? ^g^ ? ji^? ^^^? ^^\? 2. What is T-3_o of 500 ? jU ? AV ? 1^0% ? AV ? 3. A farmer lost 15 sheep from a flock of 100. How many hundredths of the flock did he lose ? 4. How many hundredths of $ 1 are 17 cents ? 39 cents ? 50 cents ? ^ of a cent ? 5. How many hundredths of anything is I of it ? J of it ? 6. In a catch of 100 fish f were perch. How many hun- dredths of the whole were perch ? 233. Per cent means hy the hundred. Thus, 3 per cent means 3 of every 100, or 3 hundredths. 234. The Sign % is used for the words per cent. Thus, 3 % means 3 per cent, and 4J % means 4|- per cent. 235. Percentage treats of computing in hundredths. 236. The Rate per cent is the number of hundredths. 237. The Base is the number of which the hundredths are taken. 238. The Percentage of a number is the part of it denoted by the rate per cent. 239. The Rate per cent, being a number of hundredths, is a fraction, and may be expressed in the form of a deci- mal or of a common fraction. Thus, 5% =0.05 =rh = 2\ 12| %= 0.121 = f 21= ^ 25 /. = 0.25 =^io-i 621% = 0.621-= in = I 100 % = 1.00 = -i-ff = 1 1371/. = 1.371 = -\U^ = ¥ 225 % = 2.25 = If-I = I 168 PERCENTAGE. EXERCISES. Express decimally : 7. 5 % ; 6 /. ; 7 %. 10. ^^ % ; f % ; IJ % ; .07 %. a J % 5 2i % ; 8J %. 11. 50 % ; i % ; 5 % ; .05 %. 9. 12 % 5 125 % ; 200 %. 12. | % ; 60 % ; 6 % ; 600 %. Express as common fractions in smallest terms : la 10 % ; 12J % ; 25 %. 16. m^ % ; 75 % ; 80 %. 14. 33 J % 5 37| % ; 50 %. 17. 125 % ; 150 % ; 200 %. 15. 16| % ; 621 % . 831 %. is. 40 % ; 60 % ; 14f %. Change the following fractions to hundredths : 19- i; i; i; i; *; I; i; i- 22. f ; f ; f; y^oJ A; #tj; ?%; ^(7- 23. f; 2%; t; |; f; i; |; |; H- 24- if? J iftr 5 A" j "/r 5 T4 ? "sV- The Base and the Rate given, to find the Percentage. ORAL EXERCISES. 25. What is 5 % of $ 4 ? Solution, — As 5 % is -^^j 5 % of $ 4 is yIij-j ^^ ^o> ^^ ^ "^j which is $0.20. 26. What is 4 % of $40? 6 % of |44 ? 3 % of $400? 27. What is 8 % of 200 ? Of 500 ? Of 700 ? Of 600 ? 28. What is 10 % of 90 yards ? 12^ % of 72 miles ? 29. What is 20 % of a cubic yard ? 5 % of an acre ? 30. What is 16§ % of a day ? Of a foot ? Of a gross ? 31. What is I % of a ton ? 75 % of it ? | of it ? 32. How many square inches in 66^ fo oi sl square foot ? 33. What is the difference between ^ of a mile and i % of a mile ? PEKCENTAGE. 169 240. The Amount is the base plus the percentage. 241. The Difference is the base less the percentage. WRITTEN EXERCISES. 34. What is 37^ ^ of $ 8.24 ? $ 8.24 X 0.37i = $ 3.09 Solution, — As 37^ % is 0.37J, )j, or I, 37i% of $ 8.24 is 0.37^ times 1.0 3 $8.24, or I of $8.24, which is I^.^^X 1 = 13.09 $3.09. 35. A's money is $ 2575. If by trading he should increase it by 40 %, how much would he then have? 36. A's income is 11890. If he should spend 83 J % of it, how much would he have left ? 242. Rule for finding the Percentage. Multiply the base by the rate per cent. Let h represent the base, r the rate per cent, 'p the percentage, a the amount, and d the difference, we have the Formulas : p = b X r, a = b -\- p, d — b — p, 37. Eind 2^ % of 8500 tons. 41. Find 9 J % of $ 5000. 38. Find 3 % of 6840 gal. 42. Find 8 % of $ 645.50. 39. Find f % of 2584 miles. 43. Find 6 % of $ 13.56. 40. Find 35 % of 3460 men. 44. Find ^2\ % of $817.68. 45. Johnson bought a house for $4850, and paid down 15 % . How much did he then owe for it ? 46. A merchant sold goods which cost him $ 9675.75, at a profit of 16 %. How much did he get for the goods ? 47. Bought a bill of goods amounting to $ 186.80, and for cash payment obtained a deduction from it of 5%. How much was the deduction ? 48. How many persons are engaged in agriculture, when they constitute 24 % of a population of 61450 ? 170 PERCENTAGE. 49. How much bank currency could be bought for $ 4500, in coin, in 1864, when gold was at 285 % ? 50. A man having $ 8550 bequeathed 33 J % to his wife and the remainder to his children. How much did he give his children ? 51. A bought goods to the value of $ 345.75, and sold them to B at 15 % advance on his outlay, and B sold them to C at 15 % less than his outlay. How much did C give for them ? Base and Percentage given, to find the Rate per cent ORAL EXERCISES. 52. What per cent of 12 is 3 ? Solution. — 3 is y% of 12 ; and ^ = ^, or 0.25, or 25 %. 53. What per cent of $ 35 is $ 7 ? Is $ 14 ? Is $ 28 ? 54. What per cent of 24 is 6 ? Is 12 ? Is 18 ? What per cent 55. Of 42 miles is 21 miles ? 58. Of $ 6 is $ 4.50 ? 56. Of 75 tons is 45 tons ? 59. Of $ 64 is $ 40 ? 57. Of 48 pounds is 30 pounds ? 60. Of $ 96 is $ 36 ? 61. If 4 quarts of grain are given for grinding a bushel, what per cent is the cost of grinding ? 62. John earns each week $7.50. He spends for board 1 2.50, and as much more for other things. What per cent of his earnings are his spendings ? WRITTEN EXERCISES. 63. What per cent of 128 is 16 ? 16 -i- 128 = 0.12^, or 12^ % Solution,— 16 is ^ of 128. Or, As i^i= .12^, or 12^%, 16 is ^ = i = .12i, or 12i % 12i % of 128. 64. What per cent of 600 bushels is 57 bushels ? PERCENTAGE. 171 243. Rule for finding the Rate per cent. mtage by the base, extern Formula, t —^ ^h. Divide the percentage by the base, extending the division t:* hundredths. What per cent is 65. 182.75 of 2150 ? 69. 600 of 720 ? 66. $ 8 of $ 130 ? 70. 70 tons of 500 tons ? 67. 16 bu. of 62 J bu. ? 71. $ 85 of $ 1700 ? 68. $490 of $5000? 72. $ 57.375 of $ 765 ? 73. Of a farm containing 1640 acres there are 246 acres in meadow. What per cent is in meadow ? 74. If the income on $ 1346 is % 168.25, what is the rate ? Hate and Percentage given, to find the Base. ORAL EXERCISES. 75. $ 25 is 5 % of my money. How much have I ? Solution. — As $ 25 is 5 % of my money, 1 % of my money is \ of $ 25, or ^ 5 ; 100 % of my money must be 100 times $ 5, or $ 500. Or, As $25 is 5 %, or -^^ of my money, |^ of my money must be 20 times $ 25, or $ 500. 76. $ 6 is 20 % of what number ? 40 rods is 12 J % of what number ? 77. 1 pound 4 ounces is 25 % of how many pounds ? 78. A number increased by 25 % of itself is 60. What is the number ? Solution. — As a number increased by 25 % of itself must be 125 %, or I of itself, 60 must be | of the number, and f , or the number, must be 48. 79. Sold a watch for $ 55, which was 10 % above its cost. What was its cost ? 80. 42 is 33 J % less than what number? 33J% more than what number ? 172 PERCENTAGE. WRITTEN EXERCISES. 81. $ 3550 is 25 % of what number ? ^4 ^ ^ .^^^ Solution. — As $ 3550 is $ ^11^ X ;pp = 114200 25 f„ of an unknown num- ber, 1 % of the unknown Or, $3550 4 number is -^-^ of $ 3550, oi $ a|M. ; 100 % of the unknown $14200 number must be 100 times $ ^f ^ or $ 14200. Or, as $ 3550 is 25 %, or J of the number, the number must be four times $ 3550, or $ 14200. 82. $ 16.22 is 5 % of what number ? 83. Having sold 40 % of my sheep, I have 177 left. What number had I at first ? 244. Rule for finding the Base. Divide the percentage by the rate per cent, Formulas, h =p -^ r. & = a-f-(l4-r). b = d^ {1 — r), Find the number of which 84. $ 125 is 8 %. 87. 7.80 yd. is | %. 85. 0.108 of a ton is f %. 88. 21.6 rd. is f %. 86. 16 bu. is 24 %. 89. $ 14 is ^ %. 90. I burned 1750 lb. of coal, or 25 % of my supply, in a single week. How much had I ? 91. The expenses of a charity concert were 40 % of the receipts. The poor received $ 250. What were the expenses ? 92. An agent's salary having been decreased 33 J % is now $ 1600. What was it at first ? 93. A man owes $ 6750, which is 75 % as much as he is worth. How much is he worth ? 94. The voters of a certain city number 16386, wliich is 20 % more than the number 3 years ago. What was the number then ? PERCENTAGE. 173 PROFIT AND LOSS. / 95. At a gain of 20 % of the cost of an article, what part of the cost equals the gain ? 96. How much is gained by selling goods at 10 fo profit when the cost is $ 25 ? 97. How much is lost on goods which cost % 40 by selling them at a loss of V2i\ % ? 98. A grocer buys tea at 45 cents a pound, and sells it at 60 cents a pound. What is the gain per cent ? 99. Sold a cow which cost me % 40 for % 45. What was the gain per cent ? 100. Sold a horse for % 120, and gained 16§ %. What did it cost ? 101. Sold goods for % 108, and lost 10 %, What was their cost? 245. Profit and Loss, as commercial terms, express the gain or loss in business transactions. 246. The hase of computation is the cost, the gain or loss is the "percentage, ; and the cases which occur correspond to, and are solved like the preceding cases of percentage. WRITTEN EXERCISES. 102. How much is made by selling flour at 20 % profit which cost % 7.75 a barrel ? 103. A dealer sold a stock of goods, damaged by fire, at an average of 66f % less than cost. The cost being % 18240, what was the loss ? 104. Of a carload of fruit 25% is spoiled. I sell the re- mainder for 16f % advance on its cost. What is my profit or loss on the load if it cost % 400 ? 174 PERCENTAGE. 105. A merchant sold a cask of molasses whicli cost him $ 69.60 at a profit of 15 %. What did he gain ? 106. If I pay for goods $ 350.50, and sell them at 6 J % loss, how much shall I lose ? 107. Sold a house which cost me $ 3500 at a gain of 8 %. What did I receive for it ? 108. Sold a carriage for $ 210 at a loss of 16 %. What was the cost ? 109. What per cent was gained hy selling property that cost $2400 for $2550? 110. Bought hams for 8^ cents a pound. What per cent will be gained by selling them at 12 cents a pound ? 111. Sold goods for $ 3312.70, and made 5 J %. What was the cost ? 112. If I am compelled to lose 12j^% on damaged goods, how should I sell those that cost me $ 560 ? 113. When the cost of flour is $ 7.50 a barrel, and the ex- pense of selling 6 %, at what price must it be sold to gain 5 % ? \^ 114. Of goods worth $ 1600, one fourth is sold at a profit of 15 %. For how much must the remainder be sold to gain 20 % on the whole ? 115. Sold a watch for $28, and gained 12%. What per cent would I have gained or lost if I had sold it at $ 24 ? 116. Bought a cask of molasses containing 120 gallons for $ 50. But, a fifth of the molasses having leaked out, for what must the remainder be sold a gallon to gain 10 % on the pur- chase ? 117. By selling hay at $15 a ton I lose 10%. At what price must I sell it to gain 15 % ? 118. A merchant sold goods for $ 150 and lost 10 %, whereas he should have gained 30 % per cent. How much were they sold under their proper value ? 119. Bought goods for $ 14500. Half of them I am obliged to sell at a loss of 20%. If I sell the other half at a gain of 20 %, what shall I gain or lose on the whole ? PEKCENTAGE. 175 COMMISSION. 120. How much should be received for selling $ 500 worth of goods if 3 % is allowed ? 121. How much must be paid for selling $ 800 worth of goods at 5 % ? 122. A collector of $ 700 was paid 2 J % for his services. How much did he receive ? 247. Commission, or Brokerage, is the compensation or percentage allowed an agent for transacting business. The agent may be known as a factor, broker, or commis- sion merchant. 248. The commission is usually a certain per cent of the sum invested or collected by the agent, the investment or the collection being the hase, and the commission the percentage. WRITTEN EXERCISES. 123. What is the commission on the sale of $ 5678 worth of cotton cloth at2i%? 124. A broker negotiated the sale of $ 3500 United States securities at a brokerage of \ %. What was the brokerage ? 125. A commission merchant sold goods to the amount of $ 7896.50. What was his commission at 2 % ? 126. Pind the commission on the sale of 368 barrels of flour for $ 6.50 a barrel, the rate of commission being 2J%. 127. An agent invests $ 5000 for me in the purchase of land. His commission being \ %, how much shall I send him to cover all the cost ? 128. My agent has sold goods for me amounting to $ 13500. His charges are : commission, 2^ % f guarantee, 2 % ; cartage and storage $ 16.50. How much is due me ? 1 76 PERCENTAGE. 129. I sent my agent $ 1500 to invest after deducting his commission of 2J %. Wliat sum can he invest, and what will be his commission ? Solution. — The remittance includes the investment and the com- mission. The investment is 100 % of the investment, and the com- mission is 2-| % of the investment. The remittance, therefore, or $1500, must he 102^% of the investment. $1500 is 102^% of I J ^^^- X 1 , or $ 1463.41 +. This, subtracted from the remittance, $ 1500, gives as the commission $ 36.59. 130. A commission merchant receives $ 650 to invest in goods after deducting his commission of 3 %. What will be his commission ? 131. I have remitted to an agent $1426.80, with instruc- tions to lay it out for me in flour at $ 6.50 a barrel, after de- ducting his commission of 2^ %. How many barrels can he buy? 132. An agent received a sum of money to lay out after de- ducting his commission of 2J %. He laid out $ 1392. What was the sum he received ? 133. A commission merchant sold on commission goods for i 8134.75, and received $334.75, which included a charge for cartage, freight, and storage of $ 22.75. What was the rate of commission ? 134. A lawyer collected 65% of a note of $ 950, and charged 6J % commission. Find his commission. 135. A factor received 5 % for buying wool. His commis- sion amounted to $208.50. How much did he pay for the wool ? 136. Sent my agent $4100 for the purchase of iron after taking out his commission of 2 J %. After he had bought the iron, I changed my business and telegraphed him to sell the iron at cost. He did so, taking a commission of 2^ % on the sale, and sent me the balance. How much did I lose by the transa(;tion ? ■^ PERCENTAGE. 177 INSURANCE. 137. What is the cost of securing the payment to me of $ 1000 in case my house is destroyed by fire, when 2 % is paid to those taking the risk ? 138. What will be the annual charge at 1^ % for taking the risk of $ 3000 against loss or damage on merchandise ? 249. Insurance is security against loss. 250. The Premium is the sum paid for insurance. 251. The Policy is the writing containing the contract of the insurer with the insured. 252. The sum insured is the hase, and the premium is the percentage. WRITTEN EXERCISES. 139. What is the cost of insuring $ 3600 on a house for 5 years at 2 %, and $ 1 for the policy ? 140. What is the premium for insuring $ 5545 on merchan- dise for one year at 2 J % ? 141. What is the amount paid for insurance on | of a ship valued at $ 68000 at 3 %, and $ 1 for the policy ? 142. A factory is insured for $ 55000 at 2^ %. If the prop- erty should be burned, what loss would the insurer actually sustain ? 143. A house was insured for | of its value at 1^ %, and the premium was $ 27. What was the value of the house ? 144. Paid $ 73, including $ 1 for policy, for the insurance of $ 3600 in a house. What was the rate of insurance ? 145. A man 44 years of age takes out a life-policy for $ 15000 for the benefit of his wife, at the yearly rate of $ 26.50 per $1000. Should his death occur at the age of 74, how much more would his widow receive than had been paid in yearly premiums ? 178 PERCENTAGE. MISCELLANEOUS EXERCISES. 146. The sum of | %, 5 %, 24 % and 55 % of a number is 60.25. What is the number ? 147. An article is composed of 37 parts of pure silver and 3 parts of copper. What per cent of the whole is each of the components ? 148. A clerk receiving a salary of $ 950 pays $ 275 a year for board, $ 180 for clothing, and $ 150 for other expenses. What per cent of his salary is left ? 149. I bought 150 apples at 2 for a cent, and 150 at 3 for a cent. I sold them all at 5 for 2 cents. How much did I gain or lose ? 150. A man bought 8 books at the rate of $ 10 a dozen, and sold them for $ 1.75 each. What per cent was gained ? 151. Bought a bill of goods amounting to $ 1540, but a discount of 25 % was allowed with 5 % off for cash. How much did I pay ? 152. If I should forward % 603.75 to a broker in St. Louis for the purchase of flour, his brokerage being 5%, how many barrels of flour should he return at $ 5 a barrel ? 153. Lost $ 17.25 by selling a watch 15 % below cost. What was the cost ? 154. For what sum must a store and its goods, valued at $ 25640, be insured so as, in case of its destruction, to recover the entire value of the building and goods and the premium of 2 % ? 155. If I purchase 15 pieces of cloth, of 35 yards each, at % 4.25 a yard, for how much a yard must I sell the whole so as to gain 25 % ? 156. A man bought 80 tons of coal at $ 5 a ton, 10 % of which went down in a boat and was lost. For how much a ton must the remainder be sold that he may lose nothing ? 157. A man bought 4500 bushels of wheat at % 1.20 a bushel. He sold 10% of it at 3 % loss, 50% of it at 10 % gain, and the remainder at 5% gain. What was gained by the entire trans* action ? + PERCEITTAGE. 179 158. If you should sell a house for $ 6000, and lose 33 J %, for what should you sell another at the advance of 32 % for just enough to cover the loss upon the first house ? 159. In 1875 there were 104513 illiterate persons in Massa- chusetts out of a population of 1652000. What was the rate per cent of illiteracy ? 160. I receive a remittance of $ 13195 to be spent, after paying commission of 1^ %, in the purchase of coal. Eequired my commission. 161. My house cost me $ 12000. It is insured for | of its value at |% premium. What is my actual loss in case it burns ? 162. The premium on an insurance of I 930 was $ 23.25. What was the rate ? 163. An agent collects money at 2^%, and pays his prin- cipal $ 4387.50. What was the amount of the collection ? 164. Leonard & Co. sell a lot of goods for me at auction to the amount of $ 11500. Their charges are as follows : com- mission, 2J%; guarantee, 2^% ; advertising, $35; labor and storage, $ 17.25. How much is due me ? 165. Sold property for $1400, 25% of which is gain. I found myself, however, able to collect only 90 % of the pro- ceeds of the sale. What was my actual gain % ? 166. Giles lost 8| % of his money in speculation, but had $ 920 left. How much had he at first ? 167. My room is 24 feet long; its width is 50% of its length ; how many yards of carpeting, J yd. wide, will it re- quire ? 168. A grocer after losing 11 % of his apples has 133.5 bar- rels left ; if they cost him $ 2.50 per barrel, for what must they be sold that he may lose nothing on his purchase ? 169. My agent sells 500 barrels of flour at $ 10 per barrel and remits me $4750. What rate of commission did he charge for selling ? 180 PERCENTAGE. 170. My store is insured for $ 8000 at 1 J % premium, and mj stock for $ 15000 at f %. If both are entirely consumed, what is the underwriters' loss ? 171. The difference between 24 % and 55 % of a number is 60.45. "What is the number ? 172. Bought a range of wood 20 ft. long, 12 ft. high, and 4 ft. wide, at $ 5 per cord, and sold the whole for $ 50. Be- quired the per cent of gain. 173. Paid $30 for my winter's wood, which was to have been 4 feet in length. It averaged, however, but 44 inches. Out of how much money was I cheated ? 174. In a school of 400 scholars there were 120 absences in 4 weeks ; the school has 2 sessions 5 days in a week. What was the per cent of attendance ? 175. How must I mark cloth which cost $2.50 so as to gain 20 % and still fall 25 % from my marked price ? 176. A cubic foot of water weighs 62f pounds and a cubic foot of ice 57 1- pounds. Ice is what per cent lighter than an equal bulk of water ? 177. The population of Chicago in 1880 was 503620, an in- crease of 69 % in ten years ; what was the city's population in 1870 ? QUESTIONS. 233. What does per cent mean ? 235. Of what does percentage treat? 236. What is the rate per cent ? 237. The base? 238. The percentage of a number 1 239. How may the rate per cent be expressed ? 240. What is the amount ? 241. The difference ? 242. The base and rate being given, how is the percentage found ? 243. The base and percentage being given, how is the rate found? 244. The rate and percentage being given, how is the base found ? 245. Define profit and loss. 246. What is the base of computa- tion ? 247. What is commission, or brokerage? 248. What is the base ? The percentage ? 249. What is insurance ? 250. The premium ? 251. The policy 1 INTEKEST. 181 INTEREST. 253. 1. When money is loaned for a year at 7%, what part of the money is the per cent ? 2. How much must be paid for the use of $ 15 for 1 year at 57c? At6%? 3. How much must be paid for the use of $ 20 at 7 % for 1 year ? For 2 years ? 4. When $ 200 is borrowed for 2 years at 7 % a year, what amount should the borrower pay at the end of that time ? 254. Interest is the money paid for the use of money. 255. The Principal is the money for whose use interest is paid. 256. The Amount is the sum of the principal and the interest. 257. The Rate of interest is the number of hundredths of the principal taken as the interest for one year or other specified time. Note 1. — The rate for one year and at 6 % is to be understood in this book when no other time or rate is specified. Note 2. — The rate of interest is regulated by law. The legal rates in the diiferent States may be found in a table in the Appendix. SIMPLE INTEREST. 258. Simple Interest is interest on the principal alone. Interest is an application of percentage, the principal being the base, the annual rate multiplied by the time in years being the rate per cent, and the interest the per- centage. 259. In the computation of interest it is customary to consider a year as consisting of 12 months of 30 days each. 182 INTEREST. General Method. ORAL EXERCISES. 5. What is tlie interest of $ 50 for 1 year at 4 % ? Solution. — At 4 % 1 year's interest is .04 of the principal, and. 04. jt $50 is $2. 6. What is the interest of $ 60 for 1 year at 5 % ? 7. What is the interest of $ 200 for 1 year at 6 % ? For 3 years ? For 5 years ? 8. What is the interest of $ 200 for 2 years 6 months at 7%? Solution. — As at 7 % the interest of $ 200 for 1 year is $ 14, for 2 years 6 months, or 2J years, it must be 2^ times $ 14, or $35. 9. What is the interest of $ 100 for 2 years 3 months at 8 % ? For 3 years 1 month ? 10. What is the amount of $ 400 for 2 years 9 months at 6 % ? For 3 years 4 months ? WRITTEN EXERCISES. 11. What is the interest and what is the amount of $26.25 for 2 years 4 months at 7%? $ 26.25 = Principal .07 = Eate Solution. — One yearns in- 1 1.8375 = 1 year's interest terest is .07 of $26.25, or 21 $ 1.8375 ; 2^ years' interest gj25~ is 2 J X S 1.8375, or, to the oQjf^Q nearest cent, $4.29. Adding x-.-oo^K T i. i. the principal to the interest, 1 4.2875 = Interest , .! ^ dt on ^a r.^ r»^ -r. . . 1 we have the amount, $ 30.54. 26.25 = Principal $30.54 = Amount 12. What is the interest and what is the amount of $ 1728 for 3 years 9 months at 6 % ? INTEREST. 183 13. What is the interest of $ 144 for 1 year 8 months at 5%? 14. What is the interest of $556 for 3 years 5 months 7 days at 8 % ? $556 .08 $ 44.48 Solution, — One year's interest is .08 41.24 ^^ $^56, or 1 44.48 ; the interest for 3 TTooi years 5 months 7 days, or 41. 2 J months, 8896 or ^i|i years, is ^^ times $ 44.48, 4448 . 17792 12) $ 1834.0581 or, to the nearest cent, $ 152.84. $ 152.838+ 15. What is the interest of $ 720 at 5 % for 1 y. 7 mo. 18 d. ? $ 720 Solution. — One year's interest is .05 of .05 $ 720, or $ 36 ; the interest for 1 y. 7 mo. 18 d., 12) $36 00 ^^ ^^'^ ^^'' °^ ^^^^ ^^^^^' ^^ ^^^ ^ *^^' ^^ • '- — $58.80. In this example, 1 year's interest $ 3.00 being a multiple of 12, we divide by the de- 1^'^ nominator of the multiplier before multiplying $ 58.80 by the numerator. 16. Find the amount of $ 1500 for 2 y. 6 mo. 15 d. at 6 %. 260. General Rule for Interest. Multiply the principal by the rate, and this product by the time in years. To find the amount, add the principal and the interest. Let jp represent the principal, r the rate per cent, i the interest, t the time, and a the amount, and we have the Formulas, i =p xr x t a —p + %, 184 INTEREST. Note. — In interest partial results may be carried to four places of deci- mals. The answers, in business transactions, are deemed sufficiently exact if the mills are omitted, and when they are five or more, the cents are increased byl. 17. What is the interest of $ 2464 for 2 y. 9 mo. 15 d. at 18. What is the interest of $ 2503.75 for 3 y. 10 mo. 21 do at6%? 19. What is the interest of $ 560.50 for 4 y. 10 d. at 7 % ? 20. What is the interest of $ 97.16 for 1 y. 5 mo. at 6 % ? 21. What is the interest of $156.80 for 3y. 1 mo. 3d. at 4%? 22. What is the interest of $865 for 1 y. 9 mo. 24 d. at 8%? 23. What is the interest of $ 890 for 5 y. 7 mo. 8 d. at 6 % ? 24. What is the amount of $ 5000 for 3 y. 11 mo. 10 d. at 7%? Six per cent Method. 261. The Interest of any sum, at 6 per cent a year. For 12 months, or 1 year, is .06 of the principal. For 2 months, or ^ year, is .01 of the principal. For 1 month, or 30 days, is .00|- of the principal. For \ month, or 6 days, is .001 of the principal. For -^ month, or 1 day, is .000 1 of the principal. 262. Hence, as a convenient method of reckoning in- terest at 6 per cent. Multiply J of .01 of the pi'incijpal by the time in months. Or, Multiply .001 of the principal hy \ of the time in days. Or, Of the principal take 6 times as many hundredths as years, ^ as many hundredths as months, and J as many thoiosandths as days. INTEREST. 185 / 25. What is the interest of $ 926 for 3 years 11 months 15 days at 6 % ? 2) $926. = Principal $ 4.63 = 1 mo.'s interest 47| = Time in months 2311 3241 1852 $ 219.921 = Eequired interest « 219.92J. Solution. — Two months' interest is .01 of the prin- cipal ; 1 month's interest is ^ of .01 of the principal, $926, or $4.63; the interest for 3y. 11 mo. 15 d., or 47^ mo., is 47i X $4.63, or 26. What is the interest of $ 340 for 103 days at 6 % ? $340 = Principal $ 0.34 =z Six days' interest 17J = J- time in days 238 34 $ 5.83| = K-equired interest Solution. — Six days' inter- est is .001 of the principal, or $0.34 ; one day's interest is ^ of $0.34, and 103 days' interest is 103 X i of $0.34, or J^ of $0.34, or 17^ X $0.34, or $5.84, to the near- est cent. 27. What is the interest of $ 1650 for 2 y. 7 mo. 18 d. at 6%? Int. for 2j. = .12 of principal. $ 1650 " 7 mo. = .035 " 18 d. = .003 .158 .158 13200 8250 1650 Solution, — Taking .06 for each year's in- terest, .00 J for each month's interest, and .000 J for each day's interest, we find that the interest for the $260,700 given time is .158 of the principal, or .158 of $ 1650, or $260.70. What is the interest at 6 % of 28. $ 56.80 for 1 y. 8 mo. 17 d. ? 29. $ 6000 for 4 y. 2 mo. ? 30. $ 17.28 for 1 y. 11 mo. 3d.? -/- 186 INTEREST. 31. $ 1850.75 for 9 mo. 24 d. ? 32. $253.50 for 2 y. 4mo. 7d.? 33. $ 85.90 for 3 y. 6 mo. 27 d. ? 34. $ 1992.25 for 93 days ? 35. $ 15600 for 4 y. 7 mo. 19 d. ? 36. What is the amount of $ 1400 for 2 years 6 months ? 37. What is the amount of $ 7000 for 5 years 3 months ? 263. For any other rate than 6 per cent we may, when more convenient than to apply the general rule (Art. 260), Find the interest at 6 ^er cent, and increase or diminish this interest hy such part of itself as will give the interest at the required rate. That is, to find 4 % interest subtract from the 6 % inter- est J of itself ; 4 J % interest, subtract from the 6 % interest \ of itself ; 5 % interest, subtract from the 6 % interest \ of itself ; 7 % interest, add to the 6 % interest \ of itself ; and so on. 38. What is the interest of $ 545 for 8 mo. 24 d. at 4% ? 39. What is the interest of $ 78.50 for 123 days at 5 % ? 40. What is the interest of $ 64.70 for 2 y. 5 mo. at 7 % ? 41. What is the interest of $ 1440 for 11 mo. 23 d. at ^%? 42. What is the interest of $ 9500 for 3 y. 6 mo. 17 d. af 7%? 43. What is the interest of $ 600.80 for 2 y. 11 mo. 3 d at8%? 44. What is the interest of $ 20000 for 63 days at 5 % ? 45. What is the interest of $340.90 for 4y. 7 mo. lid. at7%? 46. What is the interest of $ 15420 for 9 mo. 24 d. at 6i % ? 47. What is the interest of $ 374.75 for 3 y. 9 rao. at 8 % ? INTEREST. 187 4a Ernest Williams borrowed, April 5, 1881, $525 at 7% interest, and kept it until May 16, 1882. What was the amount ? 49. A note for $ 450.60, dated March 5, 1880, was paid Dec. 31, 1881, with interest at 7 %. What was the amount ? Short Method. 264. The following method of computing 6% interest is often very convenient, especially for short times. Find 60 days' interest hy taking .01 of the principal. Then take such multiples or parts of this interest as the given time may require, 50. What is the interest of $ 2460 for 3 mo. 18 d. ? Solution. Time, 108 d. $ 2460 = Principal. Int. for 60 ^^ =$ 24.60; or .01 of principal. " 30 " = 12.30, or i of 60 days' int. " 15 " = 6.15, or I of 30 '' " " __§_!!__ = 1.23 , or tV of 30 " " " 3 mo. 18 d. = $ 44.28, or the sum of the ahove. 51. What is the interest of $480 for 84 days at 7 J % ? Solution, Time, 84 days $ 480 = Principal. Int. for 60 " = $4.80, or .01 of principal. " 20 " = 1.60, or J of 60 days' int. " _4 " =: .32, or i of 20 '' " 84 days at 6 % = $ 6.72, or the sum of the above. Int. at 1J% = 1.68 , or J of 6 % int. Int. at 7^ % = 1^40, or $ 6.72 + $ 1.68, 188 INTEREST. 52. What is the interest of % 66.42 for 4 mo. 12 d. at 5 % ? 53. What is the interest of $ 8000 for 3 mo. 3 d. at 7 % ? 54. What is the interest of 1 130.50 for 45 days at 8 % ? 55. What is the interest of $ 7500 for 2 mo. 21 d. at 6 % ? 56. What is the interest of 1 225 from Feb. 3, 1880, to May 9, 1881, at 6 % ? 57. What is the amount of $ 163.20 from Dec. 12, 1881, to March 27, 1882, at 10 % ? 58. What is the amount of $ 900.65 from Sept. 16, 1881, to Nov. 8, 1882 ? 59. What is the interest of $ 4000 from May 12, 1879, to June 24, 1880, at 4i- % ? 60. What is the amount of $653.63 from Feb. 11, 1880, to Nov. 9, 1882, at 7 % ? 61. A bill of goods amounting to $ 4498.25 was paid at the end of 60 days, with interest at 5 %. What was the amount ? ^ By any of the preced; Lng methods find the interest Of For At 62. $248 6 mo. 18 d. 3|%. 63. 1845 13 y. 2 mo. 13 d. 4%. 64. $245.80 * 2 y. 5 mo. 7 d. 41%. 65. $960 3 y. 7 mo. 9 d. 5%. 66. $849.50 8 y. 4 mo. 12 d. 6%. 67. $2846 3 y. 5 mo. 10 d. 6|%. 68. $180 1 y. 9 mo. 15 d. 7%. 69. $948.39 3 y. 11 mo. 6 d. n%- 70. $862 4 y. 7 mo. 22 d. 8%. 71. $1500 1 y. 3 mo. 27 d. 9%. 72. $8400 2 mo. 17 d. 7.3%. 73. $9398 1 mo. 18 d. 10%. 74. $479.85 106 days 5%. 75. $948.25 89 days 4i% 76. $84.32 45 days 4%. 77. * 961.18 111 days 6%. INTEKEST. 1 Find the amouni Of From To At 78. $549.82 Dec. 14, '80 May 5, '81 9%. 79. $856.84 Aug. 17, '81 Apr. 4, '82 8%. 80. $1248 Jan. 24, '81 Mar. 31, '81 6%. 81. $ 960.50 Mar. 5, '81 Sept. 9, '82 H %. 82. $ 849.25 May 5, '81 Aug. 11, '81 5%. 83. $ 562.15 Aug. 15, '80 Dec. 29, '82 7%. 84. $476.84 Sept. 30, '81 May 6, '82 7i %. 85. $942 Aug. 31, '81 Dec. 30, '81 3^^ %. 86. $1728 Jan. 16, '80 Oct. 11, '81 8%. 87. $ 945.96 June 4, '81 Sept. 10, '81 9%. 88. $200 May 9, '82 Aug. 5, '82 10%. 89. $ 816.42 ^ Nov. 4, '80 May 1, '82 7%. 90. $945.55 May 19, '80 Oct. 19, '84 7%. 91. $ 624.87 Sept. 4, '81 Dec. 15, '81 5%. 189 EXACT INTEREST. 265. In the computation of Uxad Interest, as by the United States on its securities, for parts of a year, the actual number of days in each calendar month included in the time is counted, and each day's interest made a 365th of a year's interest. That is, to compute exact interest. Multiply the interest of the jprincijpal for one year at the given rate hy the exact number of days in the time, and di- vide hy 365. 92. What is the interest on a United States Treasury cer- tificate for $500 from April 1, 1881, to July 15, 1881, at 4%? $ 500 X .04 = $ 20 Solution. — The exact time from 2 1 April 1, 1881, to July 15, 1881, is 105 ^ — 5^#^=^ = $5.75 days. The interest of $500 for 1 "^ 3 year at 4 % is $ 20 ; and the interest for 105 days must be ^ of $ 20, which is $ 5.75f|, or, to the nearest cent, $5.75. 190 INTEREST. 93. What is tlie exact interest on a note for $ 3000 from Feb. 15, 1880, to June 5, 1880, at 5% ? ** 94. What is the exact interest on a $ 1000 bond from Nov. 1, 1881, to March 15, 1882, at 4i% ? 95. A note for $ 225.50 was given March 16, 1881, bearing exact interest from date at 6 %. What sum should discharge the note Jan. 13, 1882 ? PROBLEMS IN INTEREST. Principal, Interest, and Time given, to find the Rate. 96. At what rate must $ 450 be on interest to yield $ 81 in 3 years ? $ 450 X .01 X 3 = $ 13.50 Solution. — As the interest of 1 81 — $ 13 50 = 6 ^ ^^^ ^°^ ^ ^^^^^ ^^ 1 % is ^ 13.50, $81 is the interest at as many per cent as there are times $13.50 in $81, or 6. 97. The interest of 1250 for 1 year 3 months is $28.12^. What is the rate per cent ? 266. Rule to find the Rate of Interest. Divide the given interest hy the interest of the principal for the given time at 1 per cent, and the quotient will he the rate. Representing the principal by ^, the interest by t, the time by i, and the rate by r, as in Art. 260, we ha\'e EORMULA. r = t -^ (^ X 0- 98. If $ 1400 yields $ 126 in 1 year 6 months, what is the rate? 99. At what rate must $ 1000 be on interest to yield $ 282 in 4 years 8 monthe 12 days ? INTEREST. 191 100. At what rate must $ 416 be on interest to yield $ 88.64 in 3 years 16 days ? 101. At what rate must $ 1600 be on interest to yield $46.20 in 66 days? 102. At what rate will $241.20 amount to $260.58 m 6 months 20 days ? 103. At what rate will $ 480 yield $ 52.20 in 2 y. 5 mo. ? 104. What is the rate of interest if $ 640 gains $ 10.56 from August 12 to October 18 ? 105. In 1 y. 3 mo. 15 d. $ 960 amounts to $ 1084. What is the rate ? 106. At what rate will $ 444 gain $ 156.695 in 6 y. 5 mo. ? 107. At what rate must I invest a trust fund of $ 25000 to secure a semi-annual income of $ 500 ? Principal, Interest, and Rate given, to find the Time. 108. In what time will $ 450 yield $ 94.50 at 6 % ? Solution. — As 1 27 is the interest $ 450 X .06 = $ 27 of $ 450 for 1 year at 6 %, $ 94.50 is $ 94.50 -f- $ 27 = 3 J- the interest for as many years as 31 y. = 3 y. 6 mo. t^^r® ^^^ times $ 27 in $ 94.50, or, 3-J years, equal to 3 years 6 months. 109. The interest of $ 140 is $ 49 at 7 %. How long has it been on interest ? 267. Rule to find the Time. Divide the given interest by the interest of the jprinciiml for one year, and the quotient will he the time. Formula, t ^ i -^ (p X t). 110. How long must $ 98 be on interest to gain $ 23.48 at 8%? 111. How long must $ 75 be on interest to gain $ 6.25 at 6%? 192 INTEREST. 112. How long must $ 3600 be on interest to gain $ 46.20 at7%? •*r^ 113. How many days must $ 875 bear interest to gain $ 7 at6%? 114. How long must 1 9080 be on interest to gain $ 794.50 at3j%? 115. In wliat time will $ 750, on interest at 6 % gain % 750^ or double itself ? ^ 116. I deposited $ 540 in a bank paying 4 % simple interest until it amounted to % 700. How long did it remain ? 117. Principal, % 892 ; rate, 10 % ; interest, % 187. Eequired the time. 118. In what time will $ 12000 yield $ 2500 at 4^ % ? 119. In what time will a trust fund of % 4500 amount to $6000 at 31 %? ' Interest, or Amount, Time, and Rate given, to find the Principal. 120. What principal at 6% will gain $94.50 in 3 years 6 months ? $ 1 X -06 X 3i = $ 0.21 Solution. — As the interest of $ 94.50 -^ $ 0.21 = 450 $ 1 at 6 % for 3 years 6 months is $ 0.21, $ 94.50 must be the in- terest of as many dollars as $ 0.21 is contained times in $ 94.50, or $450. 121. What sum of money at 7% interest will amount to $ 320 in 4 years ? Solution. — As the amount of $ 1 X .07 X 4 = $ 0.28 $ 1 at 7 % for 4 years is $ 1.28, $ 1 + $ 0.28 = $ 1.28 $ 320 must be the amount of as $320 -^ $ 1.28 = 250 many dollars as $320 is times $ 1.28, or $ 250. 122. What principal on interest at 6 % will gain $ 6.25 in 1 year 4 montlis 20 days ? INTEKEST. 193 268. Rule to find the Principal. If the interest is given, divide it by the interest 0/ $ 1 at the given rate for the given time, and the quotient will be the 'principal. Or, If the am^ount is given, divide it by the amount oj %1 at the given rate for the given time, and the quotient will be the 'principal. Formulas, p = i -^ (r x t) -, p = a -^ (1 -\- r X t). 123. What principal at 7 % interest will gain $ 46.20 in 66 days ? 124. What sum at 4 % interest will amount to $318 in 1 year 6 months ? 125. What sum at 5 % interest will amount to $ 734.20 in 5 months 10 days ? 126. What principal at 6 % interest will give a quarterly in- come of $ 210 ? 127. How large a sum in the savings-bank at 5 % interest will give a 3^early income of $ 1200 ? 128. The interest on a note for 2 y. 6 mo. at 7% was $ 118.23. What was the face of the note ? 129. What sum must be invested in stock paying 3| % semi- annually to yield $ 924 per year ? 130. How much must I invest in U. S. 4 % bonds to pay thr college expenses of my son, $ 560 per year, with the income ? 131. A gentleman owns stock in a manufactory which pays annually 9 %. He receives quarterly $ 324. What sum has he invested ? 132. Mr. y. is said to have an income of $ 5400 per day. If his income from railroad stock, paying 8 %, is equal to his income from government securities, paying 4%, what is he worth ? 194 INTEREST. PARTIAL PAYMENTS. 2oB. A Promissory Note. C/ne yecil a-J^e^ k/ci^ QJ^ ^laTTztae ^ ^lay (344^^'^ o/otane, 270. A Promissory Note is a written promise to pay absolutely a specified sum of money for value received. 271. The Promisor, or Maker, of the note is the person who makes the promise to pay. 272. The Promisee, or Payee, is the person to whom the maker of the note promises to pay the money. 273. The Face of a note is the sum named in it. 274. A Negotiable Note is one payable to the bearer, or to the payee's order. 275. The Indorser of a note is the person who WTites his name on its back as security for the payment of the note. 276. The Holder of a note is the person who owns it. 277. Partial Pajrments are payments in part of a note or debt. 278. Indorsements are records of the partial payments with their dates made on the back of the note. 279. A note matwres, or is legally payable, on the third day after the time named in the note has expired. INTEREST. 195 2B0. A note draws interest from maturity at the legal rate, unless it contains the words "with interest." In such case interest accrues from the date of the note. Thus, The preceding note draws 6 % interest from Dec. 14 1882, and the following note draws 7 % interest from Dea 28, 1881. 281. Form of a Demand Note. /^^^^. Q^/^?iy, Mec. ^§. ^§§f. ^a^c iececzf-ec/. Q^. Q/^ci-n/y ^ ^a. 282. The Supreme Court of the United States has adopted for finding the amount due on a note on which partial payments have been made the following, called The United States Rule. Find the amount of the principal to the tivie when the pay^ ment, or the sum of the payments, equals or exceeds the inter- est dice. Then subtract such payment or payments from the amount, and, with the remainder as a new principal, proceed as before to the time of settlement. 133. i 304y%%. Providence, Dec. 8, 1876. »0n demand, I promise to pay J. B. Anthony, or order, three hundred four y^^^ dollars, with interest at 6 %. Value received William C. Thomas. Indorsements : Sept. 2^, 1877, $ 60 ; July 4, 1878, $ 90 ; Aug. 1, 1879, $ 10 ; Dec. 2 1879, $ 100. What was due Jan. 7, 1881 ? 196 INTEREST. Solution. Principal $304.84 Int. from Dec. S, 1876, to SepL. 25, 1877, 9 mo. 17 d • 14.58 Amount * 319.42 1st payment 60.00 New principal % 259.42 Int. from Sept. 25, 1877, to July 4, 1878, 9 mo. 9 d 12.06 Amount $271.48 2d payment 90.00 New principal .- $181.48 Int. from July 4, 1878, to Aug. 1, 1879, 12 mo. 28 d 11.74 Int. from Aug. 1, 1879, to Dec. 2, 1879, 4 mo. 1 d. 3.66 Amount $ 196.88 3d payment, less than int. due $ 10 4tli payment 100 110.00 New principal $ 86.88 Int. from Dec. 2, 1879, to Jan. 7, 1881, 13 mo. 5d 5.72 Amount due Jan. 7, 1881 $ 92.60 134. $ 600. Springfield, Jan. 6, 1880. For value received, I promise to pay James Dennis & Co., or order, six hundred dollars, on demand, with interest at 7 pei cent. Benjamin Pool, Jr. Indorsements : April 6, 1 880, $ 50 ; Nov. 21, 1880, $ 60.50; March 31, 1881, $ 150. What was due June 30, 1881 ? INTEREST. i97 135. A note for $ 750, dated Oct. 12, 1880, had two indorse- ments, — Dec. 27, 1880, $325; Aug. 7, 1881, $25. What was due July 1, 1882, at 6 % ? 136. $ 1500. New Orleans, March 10, 1880;_^,^ Six months after date, we jointly and severally promise to pay John Hyde fifteen hundred dollars, with interest at 5 per cent. Value received. Joseph Eaymond. Louis Bernardin. Indorsements : ISTov. 25, 1880, $ 45 ; July 20, 1881, $ 500 ; Jan. 30, 1882, $ 600. What was due May 15, 1882 ? 137. May 16, 1881, I gave my note, on demand, with inter- est at 7 %, for $ 563.50 ; Sept. 26, 1881, I paid $ 250. What was due May 16, 1882 ? 138. You borrow, Feb. 9, 1880, of Charles E. Lowe, $ 3000 by note, with interest at 5 %, and pay $ 1000 March 9, 1881, and $ 800 Kov. 24, 1881 ; write the note and the indorse- ments on it in proper form, and find the balance due Mr. Lowe Jan. 3, 1882. 139. On a note for $ 1200, dated Aug. 7, 1880, drawing 4 % interest, there were paid. May 13, 1881, $ 300 ; Kov. 23, 1882, $ 275. What was due Jan. 1, 1883 ? 140. What is due May 15, 1881, on a $ 6000 note drawing 71% interest, on which $400 was paid Aug. 11, 1879, $ 700 Dec. 15, 1880, the note being dated Jan. 1, 1879 ? 141. A note for $ 500, given Jan. 1, 1879, at 10 % interest, has on it two indorsements of $ 100 each, paid on the first day of each year. What was due June 19, 1881 ? 142. The face of a note is $ 2400 ; its date, Aug. 12, 1880 ; rate of interest, 4^ % ; indorsements : Sept. 12, 1881, $ 25 ; Oct. 12, 1881, $ 700 ; settled, Feb. 15, 1882. What was due ? 143. What sum will discharge a note "Nov, 10, 1881, for $ 1728 at 9 %, dated Nov. 23, 1878, which is indorsed as fol- lows : May 15, 1879, $248; Aug. 28, 1880, $301; May 30, 1881, $ 300 ? 198 INTEREST. 283. Business men, when settlement is made within a year after interest begins, often make use of the following, called The Merchants' Rule. Find the amount of the note or debt from the time of itb beginning to draw interest to the time of settlement ; also, the amount of each payment froin its date to the settlement ; and then subtract the sum of the amounts of the payments from the amount of the note or debt. 144. $ 850. Philadelphia, Jan. 2, 1882. For value received, I promise to pay John S. Moreland, oi bearer, eight hundred fifty dollars, on demand, with interest at 6 per cent. Arthur Ayer. Indorsements: March 18, 1882, $200; May 2, 1882, $ 150 ; Aug. 18, 1882, $ 300. What was due Dec. 2, 1882 ? Solution, Amount of $ 850 for 11 mo $ 896.75 $ 200 for 8 mo. 14 d $208.47 " $150 for 7 mo 165.25 " $ 300 for 3 mo. 14 d 305.20 $668.92 $227.83 145. $ 1164^%. St. Louis, July 6, 1881. For value received, I promise to pay to the order of Simeon H. Wright eleven hundred sixty-four ^^ dollars, with inter- est at 7 %. Alfred Shaw. Indorsements : Sept. 21, 1881, $ 250 ; Nov. 22, 1881, $315; March 6, 1882, $100; May 17, 1882, $200. What was due on settlement, July 6, 1882 ? INTEREST. 199 COMPOUND INTEREST. 146. Alfred Nickerson deposits in a savings-bank $600, with the understanding that at the end of every 6 months he is to receive interest on his deposit at the yearly rate of 4 %. How much interest is due him at the end of 6 months ? 147. If he does not choose to draw this interest, it will be placed to his credit with his original deposit. Of what sum should he receive the interest for the next 6 months ? 148. How much interest will be due him from the bank for the second 6 months ? 149. Suppose that he allows all his money to remain in the bank, on how much will he receive interest for the third period of 6 months ? 284. Compoimd Interest is the interest on both the prin- cipal and its unpaid interest added to it at stated intervals. 285. Interest may become due, and made a part of the principal, or compounded, according to agreement, at the end of each year, half-year, or quarter, or any other period of time. 150. What is the compound interest and the amount of $500 for 2 y. 7 mo. 12 d. at 6% ? Solution, Principal for 1st year $500.00 Interest " " 30.00 Principal for 2d year $ 530.00 Interest " " 31.80 Principal for 7 mo. 12 d $ 561.80 Interest " " " . 20.79 Compound amount for 2 y. 7 mo. 12 d. . . $ 582.59 Given principal 500.00 Compound interest for 2 y. 7 mo. 12 d. . . $ 82.59 200 INTEREST. 151. What is the compound interest of $ 750 for 4 years at5%? 286. Ri'le for Compound Interest. Find the amount of the given principal for the first period of time. Using this amount as a principal, find its amount for the second period, and so on for the entire time. The last amount, less the given principal, will he the compound in- terest. Note. — When the interest is compounded half-yearly, the rate must be considered one half the yearly rate, and when quarterly, one fourth the yearly rate. Interest compounds annually if nothing is said to the contrary. 152. What is the compound interest of $ 600 for 3 years 6 months at 5 % ? 153. What is the compound interest of $ 320 for 2 y. 9 mo. at 7 % ? 154. What is the compound interest of $ 500 for 4 y. 4 mo. 15 d. at 4 % ? 155. AVhat is the amount of $ 1000 for 2 years at 6 %, com- pounded half-yearly ? 156. What is the amount of 1 1200 for 1 y. 6 mo. at 4 %, compounded quarterly ? 157. Willard Aldrich deposits $200 in a savings-hank pay- ing 4% interest, compounding half-yearly. He withdraws his money after three dividends have been declared. How much has he ? 158. Charles Underhill borrows $ 2000 for 1 y. 8 mo. 24 d., paying simple interest at 6%. He lends it immediately at 7 % compound interest. What does he gain ? 159. What is the compound interest of $ 300 for 4 y. 8 mo. 12 d. at 8 %, interest compounding semi-annually ? 287. The computation of compound interest may be abridged by means of the following INTEREST. 201 TABLE SHOWING THE AMOUNT OF $ 1 AT COMPOUND INTEREST, FROM 1 TO 20 YEARS, AT 1^, 2, 21, 3, 3^, 4, 5, 6, 7, 8, 9, and 10 per cent. Yrs. 1^ per cent. 2 per cent. 2^ per cent. 3 per cent. 3| per cent. 4 per cent. 1 1.015000 1.020000 1.025000 1.030000 1.035000 1.040000 2 1.030225 1.040400 1.050625 1.060900 1.071225 1.081600 3 1.045678 1.061208 1.076891 1.092727 1.108718 1.124864 4 1.061363 1.082432 1.103813 1.125509 1.147523 1.169859 5 1.077283 1.104081 1.131408 1.159274 1.187686 1.216653 6 1.093442 1.126162 1.159693 1.194052 1.229255 1.265319 7 1.100843 1.148686 1.188686 1.229874 1.272279 1.315932 8 1.126491 1.171660 1.218403 1.266770 1.316809 1.368569 9 1.143388 1.195093 1.248863 1.304773 1.362897 1.423312 10 1.160539 1.218994 1.280085 1.343916 1.410599 1.480244 11 1.177947 1.243374 1.312087 1.384234 1.459970 1.539454 12 1 195616 1.268242 1.344889 1.425761 1.511069 1.601032 13 1.213550 1.293607 1.378511 1.468534 1.563956 1.665074 14 1.231753 1.319479 1.412974 1.512590 1.618695 1.731676 15 1.250229 1.345868 1.448298 1.557967 1.675349 1.800944 IQ 1.268982 1.372786 1.484506 1.604706 1.733986 1.872981 17 1.288017 1.400241 1.521618 1.652848 1.794676 1.947901 18 1.307337 1.428246 1.559659 1.702433 1.857489 2.025817 19 1.326946 1.456811 1.598650 1.753506 1.922501 2.106849 20 1.346849 1.485947 1.638616 1.806111 1.989789 2.191123 Yrs. 1 5 per cent. 6 per cent. 7 per cent. 8 per cent. 9 per cent. 10 per sent. 1.050000 1.060000 1.070000 1.080000 1.090000 1.100000 2 1.102500 1.123600 1.144900 1.166400 1.188100 1.210000 3 1.157625 1.191016 1.225043 1.259712 1.295029 1.331000 4 1.215506 1.262477 1.310796 1.360489 1.411582 1.464100 5 1.276282 1.338226 1.402552 1.469328 1.538624 1.610510 6 1.340096 1.418519 1.500730 1.586874 1.677100 1.771561 7 1.407100 1.503630 1.605781 1.713824 1.828039 1.948717 8 1.477455 1.593848 1.718186 1.850930 1.992563 2.143589 9 1.551328 1.689479 1.838459 1.999005 2.171893 2.357948 10 1.628885 1.790848 1.967151 2.158925 2.367364 2.593742 11 1.710339 1.898299 2.104852 2.331639 2.580426 2.853117 12 1.795856 2.012197 2.252192 2.518170 2.812665 3.138428 13 1.885649 2.132928 2.409845 2.719624 3.065805 3.452271 14 1.979932 2.260904 2.578534 2.937194 3.341727 3.797498 15 2.078928 2.396558 2.759031 3.172169 3.642482 4.177248 16 2.182875 2.540352 2.952164 3.425943 3.970306 4.594973 17 2.292018 2.692773 3.158815 3.700018 4.327633 5.054470 18 2.406619 2.854339 3.379932 3.996019 4.717120 5.559917 19 2.526950 3.025600 3.616527 4.315701 5.141661 6.115909 20 2.653298 3.207136 3.869684 4.660957 5.604411 6.727500 202 INTEREST. Note 1. — When the time extends beyond the limits of the table, find the amount for a convenient length of time, and use this amount for a new principal. Note 2. — If the interest is compounded half-yearly, take one half the given rate and twice the number of years ; and if compounded quarterly, take one fourth the given rate and four times the number of years. 16a What is the compound interest of $ 400 for 15 years 6 months at 6%? Solution, Amt. of II for 15 y. at 6%, from table .... $2.396558 Amt. of $ 400 for 15 y. at 6 % = $ 2.396558 X 400 .= $ 958.623 Int. of $ 958.623 for 6 mo 28.759 Amount of I 400 for 15 y. 6 mo. at 6 % $ 987.382 Int. of $400 for 15 y. 6 mo. = $ 987.38 - $400 . . $ 587.38 161. What is the amount of $ 500 for 20 years at 7 % com- pound interest ? 162. What is the compound interest of $ 120 for 14 years at 8 % compound interest ? QUESTIONS. 254. What is interest ? 255. The principal ? 256. The amount ? 257. The rate? 258. What is simple interest ? 260. What is the general method of computing simple interest 1 262. What is the ^ix per cent method ? 264. How may the six per cent method be shortened ? 265. What is the method for finding the exact interest for parts of a year 1 266. How do you find the rate of interest, the principal, interest, and time being given 1 267. The time, the principal, interest, and rate being given ? 268. The principal, the interest or amount, time, and rate being given ? 270. What is a promissory note 1 273. The face of a note ? 277. What are partial payments ? Indorsements ? 282. What is the United States Rule for partial payments 1 284. What is compound interest ? 286. What is the process of •omputing compound interest 1 DISCOUNT. 203 DISCOUNT. 288. 1. What sum put at interest at 5 % will in 5 years amount to $ 40 ? Solution, — 5 years' interest is .25, or J of the principal. The amount is | of the principal. As $ 40, then, must be | of the prin- cipal, I of the principal is 4 X i of $ 40, or $ 32 2. What sum on interest at 6 % will become $ 36 in 3J years ? 3. When an article whose list price is $ 25 is sold for cash at 10 % off, how much is the deduction ? 4. If I borrow $ 200 for 4 months, and pay the interest at 6 % in advance, how much is deducted from the debt for the interest ? ^ 289. Discount is the sum deducted from a debt or price. TRUE DISCOUNT. 5. Fred Wood has this day bought a horse of. me, agreeing to pay me $ 525 for it in 1 year without interest. If I prefer, he will pay me cash, provided I make him an equitable dis- count from the price. I propose a discount of $ 15. He ob- jects, saying that I can then take the proposed price, $ 510, put it at interest at the current rate, 5 %, and at the end of the year I shall have more than the original $ 525 which he agreed to pay. What discount ought I in justice to make ? 290. The Present Worth of a debt, payable at a future time without interest, is the sum which will amount to the debt when it becomes due if put at interest at the current rate. 291. True Discount is the difference between the face of a debt and its present worth, and is equal to the interest of the present worth of the debt for the given time. 204 DISCOUNT. WRITTEN EXERCISES. 6. What is the present worth of a debt of 1 25.44 due one year hence, the current rate of interest being 6 % ? What is the true discount ? $1 X 1.06 = $1.06 Solution. — As the amount of $ 25.44 -^ 1.06 = $ 24 $ 1 for 1 year at 6% is $ 1.06, the $ 25.44 — $24: = $ 1.44 present worth of $ 1.06, due 1 year hence, is $ 1 ; and as the present worth of $ 1.06 is $ 1, the present worth of $25.44 must be as many dollars as $1.06 is contained times in $25.44, or $24. $25.44 — $ 24 = $ 1.44, the true discount. Note. — The process is the same as finding the principal (Art. 268), the debt being the amount, the present worth the 2^'^i'^cip(^l, and the true discount the interest. 7. What is* the true discount of $ 192 due 4 years hence, money being worth 7 % ? 292. Rule to find the Present Worth and True Discount. Divide the given debt by the amount of$l for the given time and rate, and the quotient will be the present worth. Subtract the present worth from the debt, and the difference will be the true discount. 8. What is the present worth of $ 3450, due in 1 year 6 months, without interest, the current rate being 7 % ? 9. What is the true discount of $ 172.86, due in 3 years 4 months, money being worth 6 % ? 10. What is the present worth of $ 360, due in 90 days, the current rate being 4 % ? 11. What is the difference between the true discount and the interest on % 5000 for 2 years 6 months at 7 % ? 12. What must be the face of a note, due in 2 years 7 months 15 days, with interest, to exactly cancel a debt of $ 347.25, due in the same time, without interest, money being worth 6%? X DISCOUNT. 205 COMMERCIAL DISCOUNT. 293. Commercial Discount is a certain percentage de- ducted from the price of an article, or from the face of a bill, for cash payment, without regard to time. 294. The Net Price of an article is the selling, or list price, less the discount. 13. When goods whose list price is $ 125 are sold at 5 % oft, what is the commercial discount ? 14. What is the net cash price of a carriage hilled at $ 350, on 30 days, or 6 % off for cash ? 15. What is the discount on a hill of goods invoiced at $ 1344.50, sold on 30 days, at 2 % off for cash ? 16. What is the net cash value of hooks amounting as per hill to $ 460.50, less a discount of 10 % and 5% off for cash ? 17. Paid $ 433 for goods after a discount of 6 % had been made from the list price. What was the list price ? 18. What is the difference in net cash value between a bill of $ 1600, less a discount of 25% and 5% off the remainder, and the bill less a discount of 30 % ? 19. Find the amount of the following bill : Boston, Oct. 19, 1884. Taylor & Ames, Bought op Leach, Shewell, & Sanborn. 200 Greenleaf's Algebra 95 f ... 10% off. 50 First Lessons in Numbers , . . ISJ;^^ ... 7^% off. 3 Cases School Slates $6.35 ... 20% off. 25 Webster's School Dictionary . . 1.12 . . . 12j% off. A discount of 2i % was allowed for cash payment. 20. Bought of A. L. Davenport 62 yards Brussels carpeting at $1.87J; 118^ yards 3-ply carpet at 90/ ; 1 set parlor fur- niture, $ 285 ; 2 sets black walnut chamber furniture at $ 125 and $ 140. Eeceived a discount of 5 7o. What was the amount of my bill ? 206 DISCOUNT. BANK DISCOUNT. Form of a Discountable Note. /^^4^. &iim 1 note for 93 days are $0.9845, to give $ 500 proceeds the face of the note must be as many dollars as ^500 is times $0.9845, or $507.87+. 45. What must be the face of a 60-day note which, dis- counted at 7 %, will give as proceeds $ 1500 ? 46. The proceeds of a 4-month note, discounted at 6 %,. are $ 293.85. What was its face ? Date Time to run. Day of Discount. Rate. Jan. 14 4 mo. Feb. 27 6% Feb. 12 3 mo. Mar. 8 7% Apr.l 2 mo. Apr. 15 n% May 5 60 d. May 27 8% July 10 90 d. Date 5% June 15 5 mo. Aug. 4 9% Sept. 20 6 mo. Nov. 27 4% Aug. 25 30 d. Sept. 1 4i% Oct. 31 1 mo. Nov. 1 3i% Sept. 7 3 mo. Oct. 10 6% Nov. 12 75 d. Dec. 15 7% Jan. 17 4 mo. Feb. 10 10% Dec. 27 90 d. Jan. 15 3% Mar. 5- 3 mo. Mayl 8% Dec. 11 6 mo. Mar. 18 5% Aug. 8 4 mo. Oct. 27 9% 210 DISCOUNT. 301. Rule for finding the Face of a Note. Divide the given proceeds by the proceeds of $ 1 for the given rate and term of discount, 47. The proceeds of a 60-day note, discounted at 7 %, are $ 444.48|. What was its face ? 48. A merchant discounted a bill payable in 6 months, by deducting the interest for the time without grace at 10 %, and received as the cash proceeds $ 1520. What was the face of the bill ? 49. Receiving a 90-day note, I had it discounted at once at 6 %, and received as proceeds $ 828.95. What was the face of the note ? 50. For what amount must a note be payable in 8 months, so that when discounted at 1\ % the proceeds may be $ 483.56 ? MISCELLANEOUS EXERCISES. 51. I owe a debt of $ 924, payable without interest April 18, 1882. What shall be discounted for payment to-day, Oct. 6, 1881, money being worth 5 % ? 52. Find a year's interest of the present worth of $ 540, dutf 12 months hence without interest, money being worth 8 % ? 53. At what date must a $ 1200 note have begun to draw interest which at 6 % amounted to $ 1380, Oct. 15, 1880 ? 54. Wood owes Davis $ 5811. He pays him with a 60-day note. For what sum should the note be written to pay the exact debt if discounted at 1^? a month ? 55. A 4-month note is dated August 22. On exactly what day must it be paid to save a protest ? Why ? 56. What is the present worth of $477.71, due 4 years hence, without interest, money being worth 6 % ? 57. What is the true discount on ^ 900, due in 72 days, the current rat*^ being 7 % ? K DISCOUNT. .211 58. What is the difference between the interest and the true discount of $ 576^ due 16 months hence, at 6 % ? 59. What are the proceeds of a note for $ 368 payable in 90 days, discounted at bank at 6 % ? 60. On what month and day will a note for 60 days, dated Jan. 31, 1882, become legally due ? 61. A man was offered $ 3675 in cash for his house, or $ 4235 in 3 years without interest. He accepted the latter offer. Did he gain or lose, and how much, money being worth 7%? 62. Wishing to borrow $ 500 at a bank, for what sum must my note be drawn at 30 days to obtain that amount, discount being 6 % ? 63. What must be the face of a note, due in 45 days, that, when discounted at a bank charging 7 % interest, will enable me to take up my note for $ 750, that has been on interest at 7^^ % for 3 months and 15 days ? 64. Pratt, Davis, & Co. sold an acre of land, which cost them $ 400, at 5 cents per square foot, taking in payment a 6 mo.- note which they immediately get discounted at the Maverick Bank, at 5 %. What were their profits ? QUESTIONS. 289. What is discount? 290. What is the present worth of a debt) 291. What is true discount ? 292. How is the present worth found 1 The true discount ? 293. What is commercial discount ? 294. What is the net price of an article ? 296. What is bank discount ? 297. What are the proceeds of & note? 298. When is a note said to mature ? 299. What is the term of discount ? 300. How are the proceeds of a note found ? 301. How do you find the face of a note to yield given proceeds ? 274. What is a negotiable note ? 275. What responsibility does a person incur by indorsing a note ? 212 STOCK INVESTMENTS. STOCK INVESTMENTS. 302. 1. A company start business with $ 10000. Into how many shares of $ 100 each can this be divided ? 2. How much do 10 shares of $ 100 each represent ? 3. What is the value of five shares of $ 100 each at a dis- 30unt of 20 % ? 4. When $ 100 shares sell at $ 120 each, how much is the advance on the original value ? 5. When a $ 100 share sells at $ 125, what per cent is the selling price above the original value ? 6. If you own 10 shares of $ 100 each, and receive as the profits $ 60, what per cent are the profits ? 303. A Share is one of the equal parts into which the capital of a corporation is divided. The share is usually of the original value of $ 100, and may be so considered unless otherwise denoted. 304. Bonds are the interest-bearing notes of govern- ments or corporations. The interest on bonds is usually paid quarterly or semi- annually. A Coupon is the interest certificate attached to a bond. 305. Bonds are commonly named according to their rate of interest and date of maturity. Thus, U. S. 4J's '91, means United States Bonds bearing 4^ per cent interest and payable in 1891. 306. Stocks are the shares of companies and the bonds of governments and corporations. 307. The Par of stocks is their face value. Thus, When a stock is quoted at 105, it is worth 105 %of its face value. STOCK INYESTMENTS. 213 308. The Market Value of stocks is the price at which they sell. Stocks are at a p''emium when the market value is above par, and at a discount when the market value is helow "par. 309. A Stock Certificate is a document signed by the officers of a corporation specifying the number of shares owned by the holder. 310. A Dividend is a sum divided among the stock- holders as the profits of the business. 311. An Assessment is a sum required of the stock- holders to meet the losses or expenses of the business. 312. The market value of leading stocks and bonds at commercial centers is given in the daily papers. The fol- lowing is an extract : 5 Union Pacific 59| 60 N. Y. Central 112f 200 Erie K. R 34|^ 114 Am. Bell Telephone . . 205 13 Continental Mills ... 90^ 2 Exchange Bank .... 145 50 Eastern R. R 123 Stock Quotations. U. S. 4's reg. U. S. 4's coup. $1000 So. Kan. 5's . . . $5000 Pacific 6's '95 . . $5000 Ogd's «& Lake C. 6's $10000 N. Y. and N. E'. 6's $5000 Wisconsin Cent. 2ds. 127 127 100 125 100 106J 57| 313. Brokerage is computed on the par value of stocks, and the usual rate is either | % or ^ %. 314. The rules of percentage and interest already given apply to stocks, the premium, discount, dividend, and assess- ment heing always a percentage of the par. 7. What must be paid for % 5000 Union Pacific Eailroad Bonds at 114, brokerage \%2 Solution.— 114% + 1% ^ 1141%, and 114|% of $5000 = 85712.50, Ans. 214 STOCK INVESTMENTS. 8. What will 20 shares of New Jersey Central Eailroad cost, at 102|, brokerage i % ? 9. How much, including brokerage at | %, must be paid for $ 30000 of U.S. 4's at 1131? 10. When gold is at lOlJ, what is the value in currency Df S1250 in gold? 11. When the cost of 20 shares of the Atlantic National Bank, including I % brokerage, is % 2200, what is the market value per share ? Solution. $ 100 X 20 = $ 2000, par value. ^ % of $ 2000 r== $ 5.00, the brokerage. % 2200 — $ 5 = $ 2195, the market value. % 2195 -^ 20 = $ 109.75, the market value of 1 share. 12. When the cost of 25 shares of the Adams Express Com- pany, including brokerage at ^%, is $3206.25, what is the market value per share ? t 13. When gold is at lOlf , what is the value in gold of % 126.75 in currency ? 14. How should Pacific Railroad bonds be quoted when 10 hundred-dollar bonds cost, including brokerage ^ %, $ 1302.50 ? 15. What income will be realized from investing $ 1905 in 6 % stock bought at 95, allowing \ % for brokerage ? Solution. % 1905 -f- .95i = $ 2000, par value. $ 2000 X .06 = $ 120, yearly income. • 16. What will be the income from investing $ 2650 in State 5's at 105|, brokerage \%? 17. How much can be realized yearly from an investment of $ 6900 in a ^ % stock, bought at S^, brokerage i % ? 18. Which will yield the greater income in amount, Citj' 6^8, at 1055, purchased for $2650, or State 5's at 104J, pui> chased for $ 3135, brokerage in both cases being J % ? STOCK INVESTMENTS. 215 19. How much must be invested in 5 % stock, purchased at 103, to afford an income of $ 800 ? Solution. % 800 ~ .05 = $ 16000, par value of stock. $ 16000 X 1.03 = $ 16480, amount to be invested. 20. How much must I invest in Government 4J's at 105|^ to secure annually $ 900 ? 21. How much must be invested in 7% railroad bonds at 108f , brokerage \ %, to afford an income of $ 1050 per annum ? 22. What sum must be invested in U. S. 5 % bonds of $ 500 each, at 108|, brokerage \ %, to secure a yearly income of $2500? 23. When a 6 % stock is at 95|, brokerage \ %, what rate of income on the investment v/ill the stock yield ? Solution. — The annual income of a share of 6 % stock is $ 6. If the cost is $ 96, the income is -^, or y\, or .6 J %, of the cost. 24. What is the rate of income on E-ailroad 5's at 110, no allowance for brokerage ? 25. Which will yield the greater per cent income, Railroad 6's bought at 120, or 5 % stock bought at 105 ? How much ? 26. What per cent income will land-grant 8's purchased at 125 yield ? 27. How much must be paid for a 5 % stock that the in- vestment shall yield 6 % ? Solution. — A 5 % stock yields f 5 on an investment of $ 100. But if the S 5 is 6 % of the investment, 1 % of it is ^ of $ 5, and 100 % of the investment 100 X i of $ 5, or $ 83J. 28. How much must I pay for Eailroad 6's, that my invest- ment shall yield 7 % ? 29. At what price must I purchase 8 % stock that the in- vestment shall pay 6 % ? r^ 216 EXCHANGE. EXCHANGE. 315. 1. How can you pay a creditor in ISTew Orleans $ 800 without actually sending him the money ? 2, What Vrill an order of John Hall of Chicago on Abram Brown of New York for $ 300 cost, payable at sight, ii ^% premium is charged for it ? 3. What will an order for $ 250, payable at sight, cost, if purchased at |^ % discount ? 316. A Draft is a written order for the payment of money, made in one place, and payable in another. 317. The Draivei' is the maker of a draft ; the Drawee, the party to whom it is addressed ; and the Payee, the party to whom it is payable. 318. A Sight Draft is a draft payable on presentation to the drawee. 319. A Time Draft is a draft payable at a time named after presentation, or after date. 320. Form of a Sight Draft. om^ ^n(^u4ci?ic( c/otuzid, anc/ cnai^ ^ ^de account oj- 321. If the drawee agrees to pay the money specified in the draft, on its presentation he writes his name under the word " Accepted " across its face. This act is called the Acceptance of the draft. EXCHAITGE. 217 322. Exchange is the method of making payments by- means of drafts. The exchange is at par when a draft sells for its face. Three days of grace are usually allowed on time drafts. DOMESTIC EXCHANGE. 323. Domestic Exchange is between persons in the same ' country. WRITTEN EXERCISES. 4. What is the cost of a sight draft for $452 at 1J% discount ? Solution. $ 452 X 0.015 = $ 6.78, discount. $ 452 - I 6.78 = $ 445.22, cost of the draft. 5. What is the cost of a draft on ]^ew York for $ 1164 at 1 % premium ? 6. How much must be paid for a draft on St. Louis for $4000 at 2^% discount? 7. What is the cost of a draft of the Girard National Bank, Philadelphia, on the National Bank of Commerce, Boston, for $ 2517.70 at \ % premium ? 8. Find the face of the draft at 1| % discount that can be bought for $ 445.22. Solution. $ 1 - $ 0.015 = $ 0.985, cost of $ 1 of draft. $ 445.22 -f- 0.985 = $ 452, face of the draft. 9. How large a draft at ^ % premium can be bought for $2520.84? 10. Bought a draft, at 21% discount, for $3900. What was its face ? 218 EXCHANGE. 11. A merchant in Mobile bought a draft on New York a* 1 % premium for $ 1175.64. What was the face of the draft ? 12. A man in Trenton bought a draft on Richmond at | % discount, for $ 447.18f . What was the face of the draft ? 13. What must be paid for a draft of $ 1000^ on Hartford, at 30 days, interest at 6 %, when exchange is at 2 % premium ? $ 1 X 1.02 = $ 1.02 Solution. — The cost of $ 1 at $ 1.02 X 1000 = $ 1020 sight at 2 % premium is $ 1.02, $1000 X 0.0055 = $5.50 ^'^^ ^^ * ^^^^ i« $1^20. As $ 1020 - $ 5.50 = $ 1014.50 ^^^ ^^''^^^ ^^^ ^^ ^^^^ ^^ ^^^» interest at the given rate for that time, amounting to $5.50, must be deducted ; ana the proceeds, $ 1014.50, wiU be the cost of the draft. 14. What must be paid for a draft of $ 1500, at 60 days, at 7 7oj exchange being at J % discount ? 15. I wish to obtain a draft on Boston for $ 3000, at 60 days, interest at 6 %, exchange being at 1 % premium. What must I pay for it ? ^ 16. What is the face of a 30 days' draft, at 2 % premium, which can be bought for $ 2029, interest at 6 % ? Solution. $ 1 + $ 0.02 = $ 1.02, cost of $ 1 at sight. $ 1 X 0.0055 = $ 0.0055, int. of $ 1 for 33 days. $ 1.02 — $ 0.0055 = $ 1.0145, cost of $ 1 of exchange. $ 2029 -^ 1.0145 = $ 2000, face of the draft. 17. How large a draft on Philadelphia, at par, at 30 days, can be bought for $ 3978, interest 6 % ? 18. What is the face of a 60 days' draft at | % discount, which can be bought for $ 491.37i, money being worth 7 % ? 19. How large a 30 days' draft can I buy for $ 2998.50, in- terest at 6 %, and exchange at 1 % premium ? EXCHANGE. 219 FOREIGN EXCHANGE. 324. Foreign Exchange is between persons in different countries. In foreign exchange drafts or bills are expressed in the money of the country in which they are payable. 325. English or Sterling Money is expressed in pounds, shillings, pence, and farthings. TABLE. 4 farthings (far.) are 1 penny, d, 12 pence " 1 shilling, s. 20 shillings " 1 pound, £, Also, 10 florins (fl.) are 1 pound, £. 326. The Pound Sterling is represented by a gold coin, the Sovereign (sov.), whose value is $4.8665. 327. French Money is expressed in francs and centimes ; a franc being 100 centimes. A franc has the value of $0,193, and about 5.18 francs are equivalent to a dollar. 328. The Money of the German Empire is expressed in raarJcs (reichmarken) and pe7inies (pfennige) ; a mark being 100 pennies. A mark is equivalent to $ 0.238, and 4 marks are about 95 cents. Note. — For the value of other foreign coins as fixed by the U. S. govern- ment refer to the table in the Appendix. 220 EXCHANGE. 329. sterling Bills, or drafts on England, Ireland, and Scotland, are quoted at the exchange value of a sovereign or pound sterling in United States dollars. Exchange on Paris, Antwerp, and Geneva is quoted at a certain number of francs per dollar ; and on Bremen, Ham- burg, Frankfort, and Berlin at a certain number of cents per 4 reichsmarks. Thus, Foreign exchange may .be quoted as follows : Sterling sight 4.861 @ 4.87, 60 days 4.85; Francs sight 5.18 @ 5.18|^, 60 days 5.15^ @ 5.15|; and Eeichsmarks sight 94-| @ 94f, 60 days 93-7 @ 941 330. Bills of exchange, or drafts, on foreign countries are usually made in sets of three of the same tenor and date, named first, second, and third of exchange. Any one of the set being paid, the others are void. 20. Find the cost in Boston of the following bill drawn on London, exchange at 4.85. of -^^ (U2>??ze cui^ C177.CO -^no^ tin/Kzcc/, /^^y ^ ^^^ oiaei^ o/^ (^nirnuet ^Jr. /raMel one duncAec/ t^ccc^y ^lounc/d ecaM (zccouTi^ 0/ '^cc^i, .£^&a^oc/y, £^ ^o. Solution. — £ 160 8.s. = £ 1G0.4 ; $4.85 X 160.4 = $777.94. 21. When exchange on London is at 4.85, what will be the face of a draft that can be bought for f 777.94 ? EXCHANGE. 221 22. Bought a set of exchange on England for £ 1320 10s, at 4.87^. What was the cost ? 23. rind the value in New York of a set of exchange on Paris for 2380 francs at 5.15. 24. Andrew Taylor, of Providence, wishes to remit 1500 francs to Antwerp. What will be the cost of a draft for that sum, exchange at 5.19 ? 25. When exchange on Paris is 5.20, how many francs of exchange will $ 3195 buy ? 26. How much must be paid for a set of exchange on Ham- burg for 1304 reichsmarks, exchange at 95 ? 27. When exchange on Berlin is 95^, what must be the face of a draft that $1420.20 will purchase ? 28. Find the cost of a set of exchange on London at 60 days for £ 1254 15s. 6d., exchange being quoted at 4.87J. 29. What will be the face of a draft in francs that can be bought for $ 1042.50, exchange being 5.21J ? QUESTIONS. 316. What is a draft? 317. Who is the drawer? The drawee? The payee ? 318. What is a sight draft ? 319. A time draft ? 321. How is a draft accepted? 322. How many days' grace are usually allowed on time drafts ? 322. What is exchange ? 323. Domestic exchange ? 324. What is foreign exchange ? 325. How is English money ex- pressed ? Recite the table. 326. What is the value of a pound or sovereign ? 327. How is French money expressed ? What is the value of a franc ? 328. What is the money of the German Empire ? What is the value of a mark ? 329. How is exchange on London quoted ? On Paris, Antwerp, and Switzerland ? On Bremen, Hamburg, Frankfort, and Berlin ? 330. How are drafts on foreign countries usually made ? i- 222 " / •" AVERAGE OF PAYMENTS. AVERAGE OF PAYMENTS. 331. 1. How long should $ 1 be kept to equal the use of $2 for 1 month? % 2. In how many months will the interest of $ 6 balance the interest of $ 18 for 4 months at the same rate per cent ? Solution. — $ 6 being but J- of $ 18 will require three times as many months to gaui as much interest as $ 18 at the same rate, or 3 X 4 months, or 12 months. 3. The interest of % 15 for 2 months is balanced by the in» terest of $ 1 in how many months ? Of ^3? Of $5? 4. If I should be allowed the use of $ 50 for 3 months, how long in return should I lend $ 2^ ? 332. Average, or Equation of Payments, is the process of finding when several debts, due at different times, may be paid at one time without loss to either debtor or creditor. 333. The Average, or Equated Time, is the date of pay- ment. WRITTEN EXERCISES. 5. July 1, A owes B $ 100 ; of which $ 20 is due in 2 months, $40 in 3 months, $30 in 4 months, and $10 in 5 months. When may the $ 100 be equitably discharged by a single payment ? Solution, A is entitled to 2 months' use of $ 20 = 40 months' use of $ 1 cc u 3 u u $40 = 120 " '' u u 4 u u $30 = 120 " " " " 5 " " $10= 50 " " A is entitled to 330 months' use of $1, which is equivalent to the use of $ 100 for yj^ of 330 mo., or 3^5^ mo., or 3 mo. 9 d., which, added to July 1, gives the equated time, Oct. 10. Or, briefly, AVERAGE OF PAYMENTS. 223 20 X 2 mo. := 40 mo. 40 X 3 " = 120 " 30 X 4 " = 120 " _10 X 5 " = _50 " 100 )330 ^^ 3^^ mo., or 3 mo. 9 d. July 1 + 3 mo. 9 d. = Oct. 10. 6. What is the average time of paying $ 200 due April 1, 1200 due May 11, and $ 400 due June 30 ? Solution. April 1, $ 200 is due. May 11, or 40 days after April 1, $ 200 is due. June 30, or 90 " " $400 " 200 200 X 40 days = 8000 days 400 X 90 days = 36000 " 800 ) 44000 " 55 days April 1 + 55d. = May 26, the average time. 334. Rule to find the Average Time of Payment. Multiply each debt by its term of credit, and divide the sum of the products by the sum of the debts. The quotient will be the average term, of credit. This added to the date from which the credits were reckoned luill give the average time of pay ments. Note. — When the cents are 50 ar more, reckon them as one dollar ; but if less than 50, disregard them. Also, when in any result the fraction of a day ia ^ or more, reckon it one day; otherwise, disregard it. 224 AVERAGE OF PAYMENTS. 7. I have purchased goods of A. B. Blake, as follows : Jan- uary 3, a bill of $ 150 on 30 days ; January 15, a bill of $ 125 on 3 months ; and February 1, a bill of $ 200 on 60 days. Find the average time of payment. 8. May 7, A owes B $ 100, of which $ 40 is to be paid in 3 months, and $ 60 in 5 months. Find the average time of pay ment. 9. John Oldham owes Henry Smith $ 1000, $ 250 of which is due now, $ 350 in 2 months, and the remainder in 6 months. What is the average time of payment ? 10. May 16, 1881, Joseph Milton owes $ 169.85, payable in 40 days, $ 200.15 in 60 days, and $ 150 in 90 days. Find the equated time. 11. Three bills are due as follows : April 15, $ 200 ; May 1, $ 310.50; June 1, 1 160.25. What is the average date of pay- ment ? 12. Find the average time for paying $ 800 due in 30 days, $ 500 in 60 days, and 1 120 in 90 days. 335. When there is a common term of credit, we may find the average time, withottt regard to that teriiiy and then add it to the result. 13. Albert Thayer bought merchandise as follows on 60 days : July 5, 1881, $ 600 ; July 15, 1 400 ; August 10, $ 500. Find the average time of payment. 14. Bought the following bills on 4 months : September 9, 1880, $ 140 ; October 9, $ 160 ; November 6, $ 200. What \^ the average time for payment ? 15. Three 60-day notes bear date as follows : April 11, 1881, 1450; April 30, $600; May 16, $400. What is the average date of maturity ? 16. Bought goods on 6 months' credit as follows : July 2, 1881, $225; August 4, 1360; September 10, 1500; Septem« ber 24, 1 320. When shall a note to settle for the whole be made payable ? X REVIEW. 225 REVIEW. ORAL EXERCISES. 336. 1. I have $ 120. If I pay away 25 % of it, what sum shall I have left ? 2 Sold a horse which cost me $ 225 at 10% profit. What did I get for him ? 3. Bought a watch for $ 75 and sold it for I 84. What was the gain per cent ? 4. When goods are sold for | of their cost, what is the gain per cent ? 5. If J of f of the cost of my horse is the cost of my chaise, what per cent is the cost ol the horse more than the cost of the chaise ? 6. A knife which cost me 31J cents was sold for 25 cents. What was the loss per cent ? 7. At 2^ %, what is the premium on an insurance of $ 3000 ? 8. An agent received $40.50 for selling goods at 5% com- mission. What was the amount of goods sold ? 9. What will be the interest of $ 550 at 6 % for 2 years 6 months ? 10. What principal at 6 % interest in 2 years 6 months will give I 66 interest ? 11. What principal in 5 years at 5 % will amount to $ 50 ? 12. The interest of $ 550 for 2 years 6 months is $ 66. What is the rate per cent ? 13. What is the market value of 6 shares of stock at 112 ? 14. What is the cost of a draft for $ 400 at 2-i % premium ? 15. At what price must a 4 % stock be bought for a 5 % in- vestment ? 16. How much must be paid for a 10 % stock that 8 % may be realized on the investment ? 15 226 I REVIEW. ^ WRITT EN EXERCISES. 17. 21 is what % of 3| ? 18. If 35 % of my money is $ 2359, what is my money ? 19. Of goods worth $ 1200, one fourth is sold at a profit <>i 15 %. For how much must the remainder be sold to gain 17 % ? 20. Bought a horse for $ 360. which was 20 % less than his real value, and sold him for 30 % more than his real value. Re- quired the selling price. 21. A class of 50 pupils miss 75 words in spelling 10 each. What per cent of words were correctly spelled ? 22. What per cent of the year 1880 expired at midnight, June 15 ? 23. Bought a bill of goods which, at 5 % off for cush, amounted to $ 232.75. How much was the discount ? 24. Bought 12 Webster's Unabridged Dictionaries at $9.50, 10% off, and 15 Longfellow's Poems at $ 1.50, 5 % off. Paid cash, and received an additional discount of 3 %. Required the cost. 25. At 7J % interest, how much is due, July 5, 1881, on a note for $325, dated Jan. 7, 1880 ? 26. How long must $922 be on interest at 5% to gain $53.78i? 27. For what sum must a note be made payable in 60 days, so that when discounted at 6% the proceeds may be $593.70? 28. At what rate of interest will $ 640 amount to $ 774.40 in 3 years 6 months ? 29. In what time will $ 6000 at 8 % gain an interest equal to S of itself ? 30. A merchant bought a bill of goods amounting to $ 1550, on 30 days' credit, but could have bought the same for cash at a discount of 5%. What was the difference ? 31. What is the difference between the true discount and the simple interest, both at 5 %, on $ 6415.50 for 3 years 6 mouths ? -/- REVIEW. 227 32. How much more is the compound interest than the simple interest of $ 1300 for 4 years at 7 % ? 33. On a note for $ 2000, dated Jan. 1, 1880, at 6 % inter^ est, there was paid, July 1, 1880, $ 600. Eequired the bal- ance due Jan. 1, 1882. 34. A man makes a difference in his income of $ 82.50 by transferring a 4 % stock at 92 to a 5 % stock at 110. What amount was transferred ? 35. Invested $ 26250 in bonds at 87j^, and sold the same at 91. What was the gain ? 36. Find the difference between the income derived from $9080 invested in 3% stock at 85 J, and that derived from $ 9800 invested in 5 % bonds at 122^. 37. A owes B $ 460, of which $ 100 is to be paid in 50 days, $ 130 in 40 days, and the remainder in 140 days. Find the average time. 38. A merchant bought a draft on St. Louis for $ 2660, at 60 days, paying $ 2570.89. What was the rate of exchange ? 39. A note for $ 500, dated Oct. 8, 1880, and bearing inter- est at 6 %, is indorsed as follows : Nov. 4, 1881, $ 30 ; Jan. 30, 1882, $ 250. What will be due July 1, 1882 ? 40. I have purchased goods to the amount of $ 800, on a credit of 4 months. At the end of 2 months I pay $ 100, and at the end of 3 months I pay $ 200. How long after the ex- piration of the 4 months ought the balance in equity to remain unpaid ? REVIEW QUESTIONS. 235. What is percentage 1 254. What is interest ? 296. What is bank discount ? 277. What are partial payments ? 284. What is compound interest ? 270. What is a promissory note ? '304. What are bonds 1 What is a coupon ? 306. What are stocks ? 309. A stock certificate ? 316. What is a draft ? 322. What is exchange ? 329. How are sterling bills, or drafts, quoted 1 332. What is average, or equation of payments 1 333. What is the average, or equated time ? 228 RATIO AND PROPORTION. RATIO AND PROPORTION. RATIO. 337. 1. Thomas has 20 books and John 5. Thomas has how many times as many as John ? 2. Peter is 16 years old and his brother 8. How do their ages compare ? 3. What part of 27 feet is 9 feet ? 4. How does 20 compare with 5 ? 24 with 6 ? 27 with 9 ? 5. What is the relation of $ 35 and $ 7 ? Of 51 miles and 17 miles ? Of 7 pounds and 42 pounds ? 338. Ratio is the relation of two like numbers shown by their quotient. It is determined by dividing the first by the second. Thus, The ratio of 15 to 5 is 15 ~ 5, or 3. 339. Eatio is usually indicated by :, which is an ab- breviated form of -^. Thus, 18 : 6 expresses the ratio of 18 to 6. 340. The Terms of a ratio are the two numbers com- pared. The Antecedent is the first term of a ratio, the Con- sequent is the second term, and the two terms together are called a Couplet. 341. An Inverse Ratio is a ratio formed by inverting the terms of a given ratio. Thus, 8 : 9 is the inverse of 9 : 8. 342. A Simple Ratio is the ratio of two numbers. Thua> 21 : 3 = 7 is a simple ratio. RATIO AND PROPOETION. 229 343. A Compound Ratio is the product of two or more simple ratios. It is usually indicated by means of the brace. Thus, 8 : 2) 9:3>: or 8x9X10: 2x3X5 is a compound ratio 10 : 5) aqual to the simple ratio 720 : 30. A compound ratio may be changed to a simple ratio by muUvplying antecedents together for a new antecedent, and consequents for a new consequent. A ratio, like a fraction, is simply an indicated division (Art. 108). The principles of common fractions are equally applicable to ratios, the antecedent being the numerator and the consequent the denominator (Arts. Ill, 114). 344. Principles of Ratio. 1. The ratio is equal to the antecedent divided by the con- sequent. 2. The consequent is equal to the antecedent divided by the ratio. 3. The antecedent is equal to the consequent multiplied by the ratio. 4. Multiplying or dividing both the antecedent and the con- sequent by the same number does not change the ratio. EXERCISES. ind 1 the ratios of 6. 65 : 15. 9. 5:i. 12. #:f 7. 25 : 625. 10. 2.25 : 0.75. 13. 63 : 72. a $ 256 : $ 228. 11. 3i:13. 14. 6J : 7|. 15. What is the inverse ratio of Q^ : 15 ? 16. What is the inverse ratio of 25 : 625 ? 17. Which is the greater ratio, 12 to 13 or 25 to 27 ? 230 RATIO AND PROPORTION. 18. If 6.25 is the antecedent and 5 the ratio, what is th^ consequent ? 19. If 27 is the consequent and ^ the ratio, what is the antecedent ? 20. What is the value of the compound ratio o ! ^ [■ ? 21. What is the ratio compounded of ( 6 ; 8) X (16 : 10) X (12 : 9) ? PROPORTION. 22. The ratio of 21 to 7 is what number ? Of 51 to 17 ? 23. Name two numbers having the same ratio as 51 to 17. 24. What number has the same relation to 17 as 21 has tK)7? 25. If 12 yards of cloth cost $ 40, what part of $ 40 will 3 yards cost ? 26. How does the ratio of 3 yards to 12 yards compare with the ratio of $ 10 to $ 40 ? 345. A Proportion is an equality of ratios. Thus, 12:3 = 40:10isa proportion. The equality of ratios may be indicated either by = or : :. Thus, 8 : 2 = 16 : 4," or 8 : 2 : : 16 : 4. means 8 to 2 equals 16 to 4, or 8 is to 2 as 16 is to 4. 346. Each term of a proportion is called a Proportional ; the first and fourth terms are called Extremes; and the second and third terms, Means. When the two means are the same number, that number is a Mean Proportional between the two extremes. Thus, In 12:6 = 6: 3, 6 isa mean proportional between 12 and 3. RATIO AND PROPORTION. 347. In the proportion 6 : 3 = = 4: 2, as ; the ratios are 1 equal, we have 6 3 ' 4 ~ 2' 231 Changing these fractions to a common denominator, we have 6x23X4 As these fractions are equal and their denominators alike, their numerators must be equal, or 6 X 2 = 3 X 4. But 6 and 2 are the extremes, and 3 and 4 the means. Hence the following 348. Principles of Proportion. 1. I?i a proportion the product of the means is equal to the product of the extremes, 2. Either extreme is equal to the product of the means di- vided hy the other extreme. 3. Either mean is equal to the product of the extremes di- vided hy the other mean. WRITTEN EXERCISES. Find the missing term represented by x in the following proportions : 27. 14 : 7 =: 18 : iK. 31. |^ : cc : : 4 : 8. 28. 5 : 20 = cc : 60. 32. $45 : $24 : : 15 yd. : x. 29. cc : 8 = 65 : 13. 33. ir : $ 9 : : 60 men : 18 men. 30. 648 : 243 == 24 : x. 34. 5 tons : J ton : : x : $7.50. 232 KATIO AND PROPORTION. SIMPLE PROPORTIOlSr. 349. A Simple Proportion is an equality between two simple ratios. It applies to the solution of questions in which three terms of a proportion are given to find the fourth. Note. — Of the given terms two must be of the same kind, and constitute a ratio ; and the other must be of the same kind as the required term, and con- stitute with it another ratio equal to the first. WRITTEN EXERCISES. 35. If 37 yards of cloth cost $ 111, what will 19 yards cost ? 37 : 19 = $ 111 : $ x Solution. — As 37 yards must evi- o dently have the same ratio to 19 19 X $111 yards that 111 1, the cost of 37 yards, — = $ 57 has to the cost of 19 yards, or the answer, we arrange the terms so as to express the equality of these ratios. Or, As the fourth term is to be dollars, we make $ 111 the third term. The fourth term is to be less than the third term, because 19 yards will cost less than 37 yards. Hence the second term must be smaller than the first, and the first ratio is 37 : 19. Dividing the product of the means by the given extreme, we have as the answer 1 57. Or, If 37 yards cost I 111, 1 yard costs ^ of $ 111, and 19 yards 19 times as much, or J-| of $ 111, or $ 57. 36. If 12 barrels of apples cost $ 51, what will 30 barrels cost? 37. If the rent of 183 acres of land is $ 273, what will be the rent of 61 acres ? 38. What number of men will be required to perform in 16 days a piece of work that would take 30 men 48 days ? 39. If 24 men can mow a field in 15 days, how many days will it take 20 men to dp it ? RATIO AND PROPORTION. 233 350. Rule for Simple Proportion. Make that number which is of the same kind as the answer the third term.. If from the nature of the question the answer is to he larger than the third term^ make the larger of the remaining numbers the second and the smaller the first term ; but if the answer is to be smaller than the third term, make the second term smaller than the first. Divide the product of the means by the given extreme, and the quotient is the foiorth term, or answer. 40. When $ 120 are paid for 15 barrels of flour, what will 79 barrels cost ? 41. If 7 gallons of molasses cost $ 5.88, what will 27 gal- lons cost ? 42. If a man travel 319 miles in 11 days, how far will he travel in 47 days ? 43. If 27 men can do a piece of work in 12 days, how long will it take 36 men to do it ? 44. Find the cost of 7 sheep when 98 cost $ 441. 45. What time should 24 men take to perform a piece of work which 18 men can perform in 15 days ? 46. A garrison of 2100 men has provisions for 9 months, but receives a reinforcement of 600 men. How long will the provisions last ? 47. If 4| bushels of oats cost $2^, what will 19J bushels cost ? 48. 74 men had provisions for 35 days, but after five days 20 men were sent away. How long will the provisions last the remaining 54 men ? 49. If 6336 stones of 3i feet in length will make a certain quantity of wall, how many similar stores of 2f feet in length will make a like quantity ? 234 RATIO AND PROPORTION. 50. If 3f tons of coal cost $ 27.50, what will 4f tons cost ? 51. A certain piece of work was to have been performed by 288 men in 72 dajs, but, a number of them having been sent away, it was performed in 108 days. What was the number of men sent away ? 52. If 3^ cords of wood cost $ 11.37|, what will 12| cords cost ? COMPOUND PROPORTION. 351. A Compound Proportion is an equality between a compound and a simple ratio. Thus, 3:4; ^ ^ J 45 : 96 is a compound proportion. It applies to the solution of questions which would re- quire several simple proportions. WRITTEN EXERCISES. 53. If 4 men can earn $ 64 in 8 days, how much can 12 men earn in 3 days ? 4 : 12 ) Solution. — As the answer sought 8 : 3| = ^^"^ • ^^ is in dollars, we make the $64 the Qj. third term, and cc, representing the o o answer, the fourth term. If the an- T9 V ^ V (M ^^^^ depended only on the number ^ A^ ~ ^^ ^^ "^^"' ^* would be larger than the ^ '^ ^ third term, as 12 men will earn more than 4 men ; hence the first ratio is 4 to 12. But if the answer de- pended only on the number of days worked it would be smaller than the third term, as less can be earned in 3 days than in 8 days ; hence the second ratio is 8 to 3. Dividing the product of the means by the product of the given ex- tremes, we have $ 72 as the answer. Or, if 4 men can earn $64 in 8 days, 12 men in the same time can earn y of 5^ 64, and in 3 days | of l^ of $ 64, or $ 72. RATIO AND PROPORTION. 235 352. Rule for Compound Proportion. Make that number which is like the answer the third term. Form a ratio of each joair of the remaining numbers of the same kind according to the rule for simple proportion^ as if the answer depended on them alone. Divide the product of the means by the product of the given extremes, and the quo- tient is the fourth term, or answer. 54. If 3 men can make 108 pairs of shoes in 2 days, how- many pairs can 2 men make in a week ? 55. If $ 250 yields $ 175 interest in 7 years, how long will it take $ 500 to yield $ 360 at the same rate ? 56. If 24 men can reap 76 acres in 6 days, how many men will reap 114 acres in 9 days ? 57. How many acres can 10 men plow in 14 hours, if 5 men plow 6 acres in lOJ hours ? 58. Two cogged wheels, one of which has 15 cogs and tht other 28, work in each other. If the first turns 16 times in 7^ seconds, how often will the other turn in 4 seconds ? 59. If 15 men are fed for 7 days when flour is $ 8 a barrel, what must be the price when '10 men are fed 8 days at the same cost ? 60. If a man travels 117 miles in 15 days, employing only 9 hours, how far would he go in 20 days, traveling 12 hours a day ? 61. If 96 horses eat 192 tons of hay in one winter, how many tons will 150 horses eat in 6 winters ? 62. If a man, walking 12 hours each day, travels 250 miles in 9 days, in how many days, walking 10 hours each, at the same rate, would he travel 400 miles ? 63. If the expenses of a family of 8 persons amount to $ 84 in 16 weeks, how long will $ 200 support a family of 6 persons ? j4. If a pasture of 16 acres will feed 6 horses for 4 months, how many acres will feed 12 horses for 9 months ? 236 RATIO AND PROPORTION. 65. If 1080 bricks, 8 inches long and 4 inches wide, are re- quired for a walk 20 feet long and 6 feet wide, how many bricks will be required for a walk 100 feet long and 4 feet wide ? 66. If 34 men can saw 90 cords of wood in 6 days, when the days are 9 hours long, how many cords can 8 men saw in 36 days when they are 12 hours long ? 67. If 12 men in 15 days can build a wall 30 feet long, 6 feet high, and 3 feet thick, working 12 hours a day, in what time will 30 men build a wall 300 feet long, 8 feet high, and 6 feet thick, working 8 hours a day ? 68. If a loaf which sells for 20 cents when wheat is $ 4 a bushel, weighs 3 pounds, what is the price of wheat when a 12-cent loaf weighs 2J pounds ? 69. If a bin 8 ft. long, 41 ft. wide, and 2| ft. deep, holds 67| bu., how deep must another bin be made, that is 18 ft. long , and 3f ft. wide, to hold 450 bu. ? 70. If the annual salary of a man who works 8 hours a day, 48 weeks in the year, is $ 1200, how much ought a conductor to receive per month who works 14 hours daily the year round ? QUESTIONS. 338. What is ratio 1 338. How is it determined ? 340. What are the terms of a ratio 1 341. What is an inverse ratio? 342. A simple ratio? 343. A compound ratio ? 344. What are the principles of ratio ? 345. What is a proportion ? 346. What is each term of a propor- tion called ? What is a mean proportional ? 348. What are the principles of proportion ? 349. What is a simple proportion 1 350. Which number is made the third term 1 How are the terms arranged 1 How, then, is the required extreme found ? 351. What is a compound proportion ? 352. Which term is made the third term ? How is each pair of the remaining numhers ar- ranged ? How, then, is the required term found ? PARTNERSHIP. 237 PARTNERSHIP. 353. 1. Two men share between them $35, the one re- ceiving $ 3 as often as the other $ 4. How much does each receive ? 2. What number is f of 35 ? f of 35 ? 3. A and B are in Business together. A put in $ 3000 and B $ 5000. They gain $ 800. What are their respective shares of it? 4. Divide $ 800 into two parts having the ratio of 3 to 5. 5. Divide 54 oranges between two boys in the ratio of 4 to 5. 354. Partnership is the association of two or more per- sons in business. 355. The Company, or Firm, is the association, and the Partners are the members. 356. The Capital, or Stock, is that which is invested in the business, and the Dividend is the profits shared by the partners. The profits or losses are usually shared according to the terms of the agreement, or contract, made when the part- nership is formed. In the absence of a special agreement, dividends are in proportion to the capital invested and the time during which it is invested. WRITTEN EXERCISES. 6. A, B, and C engage in trade. A furnishes $ 200, B $ 250, and C $350. They gain 1100.80. What is each partner's share of the gain ? 238 PARTNERSHIP. Solution, $200 + $250 + $350 :r= $800, the capital. A's stock = f g-g = 1 ; 1 of $ 100.80 =: $ 25.20, A's gain. B's " = li% = j^ ; j% of $ 100.80 = $ 31.50, B's gain. C's " = |5^ = tV ; i\ of $ 100.80 = $ 44.10, C's gain. As A's stock is ^ of the entire capital, B's stock y^g, and C's ^^^ A must have J, B ^^g, and C ^^ of the gain. Hence A's gain is $ 25.20, B's $ 31.50, and C's $ 44.10. 7. A, B, and C engage in trade. A puts in $ 6000, B $ 9000, and C $5000. They gain $1680. What is each partner's share of the gain ? 357. To find the gain or loss of partners, Rule. Take for each partner such a part of the gain or loss as his stock -is of the entire capital. Note. — The rule applies to the distribution of the assets of bankrupts and other like apportionments- 8. A, B, and C engage in trade, investing capital to the amount of $1280, $1760, and $1920, respectively. Their profits were $ 2790. How were they divided ? 9. A bankrupt owes three creditors. A, B, and C, $ 1750, $ 2100, and $ 2650, respectively. His assets are $4225. What should they each receive ? 10. Hall and Bishop gain by trade $ 728. Hall put in $ 1200, and Bishop $ 1600. What is the gain of each ? 13. B puts on board of a ship 400 barrels of flour, C 600, and D 400 ; but when at sea it was found necessary to throw 360 barrels overboard. How much of the loss should fall to each ? 12. A, B, and C hire a pasture for $ 300. A puts in 8 horses, B 6, and C 10. How many dollars should each pay ? PAKTNERSHIP. 239 13. A, B, and C engage in trade. A puts in $ 1400, B $ 600, and C 125 barrels of Hour. They gained $ 180 ; of which C took $60 as his part. What will A and B receive, and what was the value of C's flour a barrel ? 358. When the capital of the partners is employed for unequal times, Find the product of each partner's stock multiplied hj the time it was invested, and divide the gain or loss in propor- tion to the products. 14. A, B, and C engage in trade. A puts in $300 for 7 months, B $ 500 for 8 months, and C $ 200 for 12 months. They gain $ 170. What is each man's share of the gain ? Solution, A's $ 300 for 7 months = $ 2100 for 1 month. B's $ 500 for 8 months = 4000 for 1 month. C's $200 for 12 months = 2400 for 1 month. The entire stock is the same as $ 8500 for 1 month. m% = ti ; Si of $ 170 = $ 42, A's gain. m% = tS ; t» of * 170 = $ 80, B's gain. im - SI 5 If of $ 170 - $ 48, C's gain. As $300 for 7 months is the same as $ 2100 for 1 month, $500 for 8 months the same as $4000 for 1 month, and $200 for 12 months the same as $ 2400 for 1 month, the entire stock is $ 2100 + $ 4000 + % 2400 = $ 8500 for 1 month. Hence A's gain will be |^ of the whole gain, or $ 42 ; B's, |^, or $80; andC's, II, or $48. 15. A, B, and C had a joint stock of $ 2400. A's part was $ 750, and continued in trade 4 months ; B's was $ 850, and continued 8 months ; the remainder was C's, and continued in trade throughout the year. They lost $ 640. What was each man's share of it ? 240 PARTNERSHIP. 16. A, B, and C were in partnership. A liad in the busi- ness $ 5000 for 8 months, B $ 4000 for 12 months, and C $ 3000 for 15 months. The profits were $ 1330. How much is each partner's part of the profits ? 17. A and B rent a pasture for $46.80. A puts in 30 horses for 33 days, and B 21 horses for 42 days. How much ought each to pay of the rent ? 18. A, B, and C form a partnership. A furnishes $ 500 foi 9 months, B $ 700 for 1 year, and C $ 400 for 15 months. They lose $ 300. What is each man's share of the loss ? 19. A and B are in partnership. A put in $ 6000, and at the end of 6 months put in $4000 more; B put in $12000, and at the end of 8 months took out $ 6000. They trade 1 year, and gain $2160. What is each man's share of the gain ? 20. A and B enter into partnership for 1 year. A had $ 500 in the business during the first 4 months, and $ 300 more during the remainder of the year ; whereas B had only $400 during the first 6 months, but $900 during the last 6 months. They gained $ 2400. What was each man's share of the gain ? 21. Jan. 1, 1881, Wood goes into business with a capital of $ 6000. March 1, Furbush joins him witli $ 5000. eJuly 1, they take Davis into the partnership with $4000 capital, agreeing to pay him 6 % interest for his money, and give him an annual salary of $ 1200. The profits were $ 3160, out of which Davis was paid and the balance divided between the other partners. Find each man's share at the end of the year. 22. Jan. 1, 1882, A and B form a partnership for a year. A furnishes $2000, and B $3000. May 1, they take C into the firm with a capital of $ 5000. August 1, A receives a leg- acy of $4000, which he adds to the capital. Oct. 1, B with- draws $ 1000 of his capital. At the end of the year the firm's net gains were $ 5850. Divide it equitably among the part- INVOLUTION AND EVOLUTION. 241 INVOLUTION AND EVOLUTION. INVOLUTION. 359. 1. What is the product of 5 used twice as a factor ? 2. Of what number are 5 and 5 the factors ? 3. What is the product of 5 used three times as a factor ? 4. Of what number are 5, 5, and 5 the factors ? 5. What is the product of .3 used three times as a factor ? Of I used three times as a factor ? 360. A Power of a number is the product arising from taking the number a certain number of times as a factor. The First Power of a number is the number itself ; The Second Power of a number is. the product arising from taking the number twice as a factor ; The Third Power of a number is the product arising from taking the number three times as a factor ; and so on. Thus, The first power of 3 is 3. " second " 3 is 3 x 3, or 9. " third " 3 is 3 X 3 X 3, or 27. " fourth " 3 is 3 X 3 X 3 X 3, or 81. The second power is also called the Square of the num- ber, as the area of a square is the product of two equal fac- tors. The third power is called the Cuhe, as the volume of a cube is the product of three equal factors. 361. The Exponent of a power is a small figure placed at the right and above a number. Thus, 25^ means the second power or square, of 25 ; 3.1^ means the third power, or cube, of 3.1 ; (I)* means the fourth power of |. 242 INVOLUTION AND EVOLUTION. 362. Involution is the process of finding powers. WRITTEN EXERCISES. 6. What is the third power of 12 ? Solution. — 12^ =z 12 X 12 X 12 = 1728. 7. Find the squares and cubes of the first nine numbers. 363. Rule for Involution. Use the given number as many times as a factor as thero are units in the exponent of the required power. rind the powers indicated by the exponents : 8. 231 11. (1)2. 14. 25^ 17. 3.52. 9. 163. 12. .(2J)3. 15. Ill 18. (141)2 10. 13^ 13. 3.6". 16. 0.151 19. 0.073. EVOLUTION. 20. What are the factors of 9 ? Of 25 ? Of 49 ? 21. What are the two equal factors of 16 ? Of 64 ? Of 81 ? 22. What are the three equal factors of 27 ? Of 64 ? 364. A Root of a number is one oi the equal factors which produce it. The Second, or Square, Root of a number is one of the two equal factors which produce it ; The Third, or Cidje, Root of a number is one of the three equal factors which produce it ; The Fourth Root of a number is one of the four equal factors which produce it ; and so on. 365. A Perfect Power is a number whose exact root can be found, and an Imperfect Power is a number whose root cannot be exactly tound. INVOLUTION AND EVOLUTION. 243 366. The Radical Sign, y/, is used to indicate a root. Thus. V 16 means the second, or square, root of 16. Si V 25 means the third, or cube, root of 25. The number in the opening of the sign, called the Index of the root, denotes the name of the root. The index of the square root may be understood. Thus, V 81, or V 81, may indicate the second, or square, root of 81. 367. Evolution is the process of finding roots. It is the reverse of involution. SQUARE ROOT. 368. To extract the square root of a number is to find one of the two equal factors which produce it. 369. The square of a number contains twice as many figures as the root, or twice as mo.ny less one. Thus, 12 =1. 92 == 81. 102 ^ i/QO. 992 = 98^01. 1002 ^ i/oO'OO. 9992 == 99'80'01. 370. If a povMr is separated into periods of two figures each, beginning at the decinfial point, the numher of periods will show the number of figures in the root. Thus, The square root of 2'35.'92'96 contains two integral and two decimal figures. 371. The square of the highest order of units in the root is found in the highest period of the power. Thus, 9 tens2, or 902 = 81 hundreds, or 81'00. 9 hundreds2, or 9002 — 81 ten-thousands, or 81'00'OQ 9 thousands^, or 90002= 81 millions, or 81'00^00'00. 244 INVOLUTION AND EVOLUTION. 372. The parts which make up a second power may be learned by a careful inspection of the process of multipli- cation by which the power is produced. For example, let us square 36, keeping its tens and ones and their products distinct and separate. Thus, 36 = 30 + 6 = tens + onea 36 = 30 4- 6 = tens + ones. 2l6= (30x6) + 62= (tens X ones) + ones^ 108 = 30^ + (30 X 6) = tens^ + (tens X ones). 1296 = 802 -f- 2 (30 >< g) + 59 ^ tens^ + 2 tens X ones + ones^. That is, the square of any number composed of tens and ones equals 373. The square of tlie tens, plus two times the tens times the ones, plus the square of tlu ones, or t^ + 2t X + 0^. WRITTEN EXERCISES. 23. Find the square root of 1296. 12^96 (30 + 6 Solution. — This 302 — - goo := t2 power contains two 2t=:2x30:=:60"396 = 2tXo + o2 Periods ; hence its oor\ 9 f V o ^°°* ^^^ ^^^ figures, tens and ones (Art. 370). The square of the tens is in the 36 = o2 o2 = 6 X 6 == 36 highest period (Art. 371). Taking out of the 12 hundreds the largest "tens 2" possible, 900, and placing its root, 3 tens, or 30, at the right, there remains 396, which must be the " 2 tens X ones + ones*" (Art. 372). The "ones*" being but a small part of the 396, we may treat this number as the approximate product of the " 2 tens X ones." Dividing this product, 396, by one of its factors, " 2 tens," or 60, we have 6 as the other factor, the "ones." Taking from 396 the " 2 tens X ones," or 60 X 6, or 360, there remains 36, which con^ tains the " ones*." Taking the ones*, or 6*, from 36, nothing remain^ Hence we conclude that 30 + 6, or 36, is the root required. INVOLUTION AND EVOLUTION. 245 Geometrical Explanation of Square Root. 374. As the length of a square is the square root of its ftrea, the method of extracting the square root of a number may be illustrated by the process of finding the length of a square, its area being given. It is required to find the length of a square containing 1296 square inches. Solution, A square containing 1296 sq. in. cannot be 40 in. long, for a 40-inch square contains 1600 sq. in. It must be more than 30 in. long, for a 30-inch square contains but 900 sq. in. The length of the given square must therefore be between 30 and 40 inches. Removing from the given square. A, a 30-inch square, B, contain- ing 900 sq. in., there remains a surface containing 396 sq. in., largely made up of the rectangles G and Z), whose length is evidently that of the square, B. It is obvious that the width of these rectangles added to the length of the square, B, will give the required length of the given square, A. Now the width of a rectangle is found by di- viding its area by its length (Art. 218). The length of each of these rectangles, G and D, is 30 in., and their united length 2 X 30 in., or 60 inches. Dividing their approximate area, 396 sq. in., by their length, 60 in., we have as their probable width 6 inches. 246 INVOLUTION AND EVOLUTION. Removing the rectangles C and D, there remains the little square, E, whose length is evidently the width of the rectangles removed. Combining the two rectangles and the little square, we find their united length to be 60 -{- 6, or 66 in. Multiplying their length and width together, we find their area to be Q^ X 6, or 396 sq. in., the exact area of that portion of the given square. A, remaining after the removal of the square, B. We therefore conclude that the length of the given square is 36 inches. The work may be expressed thus : Length. 30 in. X 2 = 60 in 6 " Area. Width. 12'96 sq. in. (30 in. + 6 in. : . 900 " : 36 in. 66 396 396 The process may be shortened by the omission of the ciphers, and proved by involution. 24. Extract the square root of 3998.64. 39'98.'64 (63.234+ 36 123 1262 12643 398 369 2964 2524 126464 44000 37929 _ 607100 505856 101244 INVOLUTION AND EVOLUTION. 247 375. Rule for finding the Square Root of a Number. Beginning at the decimal point, sejparate the given number into periods of two figures each. Find the greatest square in the left period, and place its root at the right ; subtract the square of this root from the first period, and to the remainder annex the next period for a dividend. Divide this dividend, omitting the last figure, by double the root already found, and annex the quotient to ' the root and also to the divisor. Multiply the divisor as it now stands by the last root figure, and subtract the product from the diaidend. If there are more pjeriods to be brought down, proceed in the same manner as before. Note 1. — If occurs in the root, annex to the divisor and another period to the dividend, and proceed as before. Note 2. — If there is "a remainder after using all the periods, w^e can only ap- proximate to the root. But nearer and nearer approximations can be obtained by annexing and using successive periods of decimal ciphers. 25. What is the square root of 49.434961 ? ' 49.'43'49'61 (7.031 ^" Here in the process, as occurs in 4349 the root, we annex to the divisor, 4209 14, and annex the next period to the 1403 14061 14061 corresponding dividend. 14061 Find the square root 26. Of 9216. 31. Of 6.7081. 36. Of 0.9409. 27. Of 27225. 32. Of 4.2025. 37. Of 77841. 28. Of 182329. 33. Of 1866.24. 38. Of 16.2409. 29. Of 717409. 34. Of 0.009409. 39. Of 14.8996. 30. Of 948676. 35. Of 0.05625. 40. Of 39.0625. 248 INVOLUTION AND EVOLUTION. 41. What is the square root of 538 to the nearest hun* dredth ? 42. What is the value of V^ to the nearest thousandth ? 43. What is the value of ^0.002 to the nearest ten-thou- sandth ? 376. To extract the root of a fraction, First change it to smallest terms ; then, if both terms are perfect squares, take the root of each ; otherwise change the fraction to a decimal, and find the root. Mixed numbers may he clianged either to improper frac- tions or to mioced decimals. 44. What is the square root of 5*^^ ? Solution. — t/-__ — ' ^:3 — V529 v/629 23 Find the value of the following : *^- ^HU- *^- s/49f. 53. v/| + J + |. *6- ^lim- 50. y/^. 54. v/98l|. 47. y/go^. 51. y'72i. 55' vTJef. *8. V371I- 52. v/|i. 56. V8T^. 57. What is the square root of J to the nearest thou- sandth ? 58. What is the square root of W to the nearest thou- sandth ? 59. A general has an army of 226576 men. How many must he place rank and file to form them into a square ? 60. A gentleman has 3 fields, one containing 3 acres 1 square -rod, another 5 acres 69 square rods, and the third 6 acres 91 square rods. He wishes to excliange them for a square field of equal area. Find the side of the square field ? INVOLUTION AND EVOLUTION. 249 CUBE ROOT. 377. To extract the Cube Root of a number is to find one of the three equal factors which produce it. 378. The cube of a number contains three times as many figures as the root, or three times as many less one or tvjo. Thus, 13 =: 1. 103 = l/QOO. 1003 :^ I'OOO'OOO. 43 = 64. 253 .= 15^625. 2433 = 14'348'907. 93 = 729. 993 = 970^299. 9993 = 997'002'999. 379. If a third power is separated into periods of three figures each, beginning with the decimal point, the number of periods will show the number of figures in the root. Thus, The cube root of 4'080'659.a92 contains one decimal and three integral figures. 380. The cube of the highest order of units in the root is found in the highest period of the poiver. Thus, 9 tens3, or 993 = 729 thousands, or 729^000. 9 hundreds3, or 9003 ^ 729 millions, or 729'000'000. 381. Taking any number composed of tens and ones, as 36, separate it into its tens and ones, cube it, keeping the products distinct, and we have 36 =: 30+6 _36= 30+6 216 = (30 X 6) +62 108 ^ 802 _^ (30 X 6) 1296 = 302 + 2 (80 x 6) + 6^ _36= 30+6 7776 = (302 X 6) + 2 (30 X 62) + 63 3888 =303 + 2 (302 x 6) + (80 x 62) 46656 = 303 ^ 3 (302 x 6) + 8 (30 x 62) + 63 250 INVOLUTION AND EVOLUTION. That is, the cube of any number that can be separated into tens and ones equals 382. The cube of the tens, plus three times the product of the square of the tens and the ones, plus three times the pro- duct of the tens and the square of the ones, plus the cute of the ones. This statement may be represented by the following formula in which initial letters are used, and the sign of multiplication is the period. t3 + 3 t2 . + 3 t . o2 + o3. WRITTEN EXERCISES. 61. Eind the cube root of 405224. t3 + 3 t2. o + 3 t . o2 + o3 = 405^224 (70 + 4 t3 = 343 000 3t2=:702x3 ==14700 3 t . o = 70 X 3 X 4 = 840 o2 = 42 =. 16 3 t2 + 3 1 . o + o2 =: 15556 62224 =: 3 t2 . o + 3t . o2 + o3 62224 z= (3 12 4- 3 1 . o + o2) X o Solution. — This power has two periods ; hence its root has two figures, tens and ones (Art. 379). The cube of the tens is in the highest period (Art. 380). The greatest tens^ in 405 thousands is 343000, which, subtracted, and its cube root placed at the right, leaves 62224, which equals " 3 t^ X o + 3 t X o* + o^." 62224 consists principally of 3 t* X o ; hence, if we divide it by 3 t*, or 70' X 3 = 14700, we shall have the ones, which we find to be 4. Finding 3 t X o, or 70 X 3 X 4 = 840, and o^, or 4^ = 16, and adding them to 14700, we have 14700 + 840 + 16 = 15556, which ecpials 3 t' + 3 t X o + o*. Multiplying this by the ones, 4, we have 62224, or 3 t* X o + 3 t X o* + o'. Hence we conclude that 74 is the cube root required. INVOLUTION AND EVOLUTION. 251 Geometrical Explanation of Cube Root 383. As the length of a cube is the cube root of its con- tents, the method of extracting the cube root of a number may be illustrated by the process of ascertaining the length of a cube, its contents being given. 62. It is required to find the length of a cube containing 13824 cu. in. The length of a cube containing 13824 cu. in. cannot be as much as 30 in., for a 30-inch cube contains 27000 cu. in. It must be more than 20 in., for a 20-inch cube contains only 8000 cu. in. The length of the given cube, then, must be between 20 and 30 inches. Removing from the given cube, A, a 20-inch cube, 5, containing 8000 cu. in., there remains a solid containing 5824 cu. in. An in- spection of this solid shows that it is largely made up of three rectangular solids, C, D, E, having square faces corresponding in area to the faces of the cube removed. It is evident that the thickness of these solids added to the length of the cube removed will be the length of the given cube. Now the thickness of a rectangular solid is found by dividing its contents by the area of its face (Art. 224). We find the area of one of the square faces of 252 INVOLUTION AND EVOLUTION. each of these rectangular solids to be 20*, or 400 sq. in., and the area of their combined square faces to be 20^ X 3, or 1200 sq. in. Dividing the approximate contents of these solids, 5824 cu. in., by this area of their square faces, 1200 sq. in., we have as their probable thickness 4 inches. Removing the three rectangular solids, C, Dy E, there remain three smaller rectangular solids, Fy G, H, whose length is evidently that of the cube B removed, and whose width is the thickness of the larger rectangular solids. The area of one face of one of these solids is 20 X 4, and the area of one face of all of them is 20 X 4 X 3, or 240 square inches. Eemoving these three smaller solids, there remains the little cube J, whose face corresponds in area to the end of one of the smaller rectangular solids re- moved, or 4*, or 16 square inches. Combining the area of the faces of the six rectangular solids with that of a face of the little cube, we have a total area of 1200 + 240 + 16, or 1456 sq. in. Multiplying the area of one face of these seven F ---'" -'-' i >< > F GH 4.00 80 80 80 solids by their thickness, 4, we have 4 X 1456, or 5824 cu. in., which is the number of cubic inches remaining of the original cube, yl, after the removal of the cube B. the given cube is 24 inches. We therefore conclude that the length of INVOLUTION AND EVOLUTION. The work is expressed thus : 253 Contents. Length. ' 13^824 cu. in. (20 in. + 4 in. = Area. 8000 " [24 in. 202 X 3 = 1200 sq. in. 20 X 4 X 3 = 240 42= 16 1456 5824 5824 63. What is the cube root of 14348907 ? 14^348^907 (248 23r=8 202 X 3 = 1200 20 X 4 X 3 = 240 42=: 16 1456 2402 X 3 = 172800 240 X 3 X 3 = 2160 32= 9 174969 6348 5824 524907 524907 384. Rule for finding the Cube Root of a Number. Beginning at the decimal point, separate the given power into periods of three figures each. Find the greatest cube in the left period, and place its root at the right. Subtract the cube of this root from the left jyeriod, and to the remainder annex the next period for a div- idend. Annex a cipher to the root already found, and take three times its square for a trial divisor. Divide the dividend by this trial divisor, and place the quotient as the next root figure. 254 INVOLUTION AND EVOLUTION. Multiply the number last squared by the last root figure, and add three times the product and the square of the last root figure to the trial divisor for a complete divisor. Multiply the complete divisor by the last root figure, sub- tract the product from the dividend, and to the remainder annex a new period. Form a second trial divisor, and proceed as before until all tlie periods have been used. Note. — Note 2, under the rule for square root, applies likewise to cube root. Extract the cube root of the following numbers : 64. 91125. 69. 12977875. 70. 60236.288. 65. 421875. 66. 571787. 67. 912.673. 68. 3796416. 71. 101847563. 72. 258474853. 73. 6372.783864. 74. What is the cube root of 8.144865728 ? 8.a44'865'728' (2.012 6 ! 120000 600 1 144865 120601 120601 12120300 12060 4 24264728 1213236 4 24264728 Here, as occurs in the root, we annex 00 to the trial divi- sor, 1200, and bring down to the corresponding dividend another period. 75. Find the cube root of 64481.201. 76. Find the cube root of 37259704. 77. What is the cube root of 0.000001728 ? 78. What is the cube root of 1860867 ? 79. What is the value of V^8l44865728 ? INVOLUTION AND EVOLUTION. 255 80. What is the value of v/0.075686967 ? 81. Find the value of v^O.008649 to the nearest thousandth. 82. Find the cube root of 0.000007 to the nearest thou- sandth. 83. What is the cube root of 25 to the nearest hundredth ? 385. When both terms of a fraction are perfect cubes, the cube root may be found by taking the cube root of each term ; but, if not, reduce the fraction to a decimal, and then find the root. Mixed numbers may be changed either to improper fractions or to mixed decimals. 84. What is the cube root of 4^o¥^? .W"729 V^729" Solution. 1 / JTTTTTT = ^, = V 4096 ^4096 85. What is the cube root of UMh ^ 86. What is the cube root of 49/y ? 87. What is the cube root of J ? 88. What is the cube root of -Sj to the nearest hundredth ? 89. Find the cube root of 81 /y to the nearest hundredth. 90. What is the cube root of 166| ? 91. A bushel contains 2150.42 cubic inches. What is the depth in inches, to the nearest hundredth, of a cubical bin which shall contain 8 bushels ?. QUESTIONS. 360. What is a power 'I The first power ? The second power ? The third power? 361. The exponent of a power? 362. What is involution ? 364. What is a root ? The second, or square, root ? The third, or cube, root ? The fourth root ? 365. A perfect power ? 367. What is evolution ? 369. The square of a number contains how many times as many figures as its root ? 378. The cube of a number contains how many times as many figures as its root ? 256 MENSURATION. MENSUBATION. 386. Mensuration treats of the measurement of lines, surfaces, and solids or volumes. RIGHT-ANGLED TRIANGLES. 387. A Right-angled Triangle has one right angle. The Hypothenuse is the side opposite the right angle, and the Perpendicular is the side perpendicular to the base. A Right-angled Triangle. 388. It will be seen from the diagram that The square of the hypothenuse is equal to the sum of the squares of the other two sides. . Hence, 389. To find the hypothenuse, Take the square root of the sum of tlie squares of the other tioo sides. 390. To find the base or perpendicular, Take the square root of the difference of the squares of lln hypotlienuse and the other side. 1. The base of a right-angled triangle is 8, and the pcrpen Jicular 6. What is the liypothenuse ? Solution, — S' + 6^ = 64 + 30 = 100 ; V^lOO = 10. MENSURATION. 257 2. The hypothenuse is 30, and one of the sides is 18. What is the other side ? Solution. — 302 _ ;^32 ^ 900 _ 324 ^ 57(5 . ^^^57^ ^ 24 3. What must be the height of a ladder to reach to the top of a house 20 feet high, the bottom of the ladder being placed 15 feet from the base of the house ? 4. The hypothenuse is 157 feet, and the perpendicular 132. What is the other side ? 5. Two vessels sail from the same port ; one sails due south 48 miles and the other due west 36 miles. What is then their distance from each other ? 6. A tree stands upon the edge of a river 100 feet wide, and a line extending from the opposite shore of the river to the top of the tree is 400 feet. What is the height of the tree, to the nearest hundredth of a foot ? 7. A park in the form of a rectangle is 40 rods long and 36 rods wide. What is the length in rods of a walk between its opposite corners ? 8. A ladder 32 feet long was so placed in a street as to reach a window 25 feet from the ground, and when it was turned to the other side, without changing the position of its foot, it reached a window 20 feet from the ground. How wide was the street ? QUADRILATERALS. 391. A ftuadrilateral is a plane figure bounded by four straight lines. 392. Parallel Lines are lines in the same plane having the same direction. Parallel Lines. 393. A Parallelogram is a quadrilateral having its op- posite sides parallel. 17 258 MENSURATION. ■liiii^ A Rhomboid. A A Rhombus. A Eedangle (Art. 166) is a right-angled parallelogram ; a RJwvihoid is a parallelogram having no right angles ; and a Rhombus is a rhomboid having equal sides. 394. It will be seen by the dia- gram that the rhombus A B C D is equal to the rectangle EBC F of the same base and altitude (Art. 218). Hence, S A F D The area of a parallelogram is equal to the product of the base and altitude. 9. What is the area of a parallelogram whose base is 36 feet and altitude 15 feet ? 10. The base of a rhombus is 16 feet and its height 12 feet. What is its area ? 11. What is the difference in the area of two floors, the one being 37 feet long and 27 feet wide and the other 40 feet long and 20 feet wide ? 395. A Trapezoid is a quadrilateral aving only two of its sides parallel. 396. The area of a trapezoid is equal A Trapezoid. Iq //^^ pvoduct of half the sum of tlie par- allel sides and the altit^cde. 12. What is tlie area of a trapezoid, the longer of the two parallel sides being 120 feet, the shorter 100 feet, and the altitude 85 feet ? MENSURATION. 259 13. What is the area of a plank whose length is 6 meters, the width of one of the parallel ends being 60 centimeters and the other 40 centimeters ? 14. The parallel sides of a field are 131 and 243 yards, and the breadth 220 yards. How many acres does it contain ? 397. A Trapezium is a quadri- lateral having no two of- its sides parallel. A Diagonal is a straight line join- ing any two angles of a plane figure not adjacent, as the line A C. A Trapezium. 398. It will be seen from the above diagram that a diagonal divides a trapezium into two triangles. Hence, The area of a trapezium is equal to the product of the di- agonal and half the sum of the perpendiculars drawn to the diagonal from the vertices of opposite angles. Note 1. — Any plane figure bounded by straight lines is called a Polygon, and may be divided into triangles ; and the sum of the areas of the triangles will be the area of the figure. Note 2. — For the Circle see Art. 221. 15. The diagonal of a trapezium is 16 feet, and the perpen- diculars upon it from the opposite angles are 7 feet and 5 feet. Find the area. 7 ft. + 5 ft. Solution. — ' X 16 = 96 sq. ft. 16. What is the area of a trapezium whose diagonal is Q>b feet, and the length of the perpendiculars let fall upon it from opposite angles is 14 feet and 18 feet ? 17. How many square yards of paving are there in a trape- zium whose diagonal is found to measure 126 feet 3 inches, and the perpendiculars upon it 58 feet 6 inches and 0^^ feet 9 inches ? 260 MENSURATION. PRISMS. 399. A Prism is a body having two equal parallej poly- gons as bases and the other faces parallelograms. A prism is triangular, qiiadrangidar, pentagonal, etc., according as its bases have three sides, four sides, Jive sides, etc. 400. The contents of a- jprism are equal to the product of the area of the hase hi/ the altitude A Quadrangular '' *^ Prism. Qrp length. Note. — For the Cylinder see Art. 225. 18. What are the contents of a triangular prism whose length is 15 feet and the area of its triangular base is 6 square feet ? 19. What are the contents of a quadrangular prism whose length is 6 meters, and wbose base is 18 by 20 centimeters ? 20. The altitude of a pentagonal prism is 20 feet 6 inclies, and the area of its base 1075.30 square inches. What are its contents in cubic feet ? PYRAMIDS AND CONES. 401. A Pyramid is a body whose base is any polygon, and whose sides are trian- gles meeting at a point called the vertex of the pyramid. A pyramid, like a prism, is triangular, quadrangular, ^pentagonal, etc., to the form of its base. according A Pyramid. MENSUKATION. 26] 402. A Cone is a body whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex of the cone. The altitude of a pyramid or cone is the shortest distance from the vertex to the center of the base ; as ^ ^. The slant height is the shortest dis- tance from the vertex to the perimeter of the base ; as ^ (7. A Cone. 403. The contents of a pyramid or cone are equal to the product of its base by one third of the altitude. Its convex surface equals the 'product of the perimeter of the base by half the slant height, 21. What are the contents of a pyramid whose altitude is 14 feet 3 inches, and the area of whose base is 14.70 square feet ? 22. What are the contents of a cone whose altitude is 15.00 meters, the circumference of the base being 12.5 meters ? 23. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. Suppose its other sides to meet at a point 500 feet above the base. What are the contents of the pyramid in cubic feet ? 404. A Frustum of a pyramid or a cone is the part which remains after cutting off the top by a plane parallel to the base. 405. The contents of the frustum of a pyramid or cone are equal to the sum of the areas of the two bases plus the square root of their product, multiplied by a third of the altitude. 262 MENSUKATION. 24. What are the contents of the frustum of a square pyra- mid whose altitude is 30 feet, and whose side at the base is 20 feet and at the top 10 feet ? Solution, 20 X 20 := 400 ; 10 X 10 = 100 j 400 X 100 = 40000. V40000 = 200 ; 200 + 400 + 100 = 700. 30 4- 3 == 10 ; 700 X 10 = 7000 cu. ft. 25. What are the contents of a column whose altitude is 2S feet 6 inches, and whose diameter at the larger end is 3 feet and at the other 2 feet 6 inches ? 26. How many cubic feet in a cquare stick of timber whose length is 18 feet 8 inches, and whose side at the larger end is 27 inches and at the smaller is 16 inches ? 406. A Sphere is a body bound- ed by a curved surface, all parts of which are equally distant from a point within called the center. The Diameter of a sphere is any straight line drawn through its center, and terminating both ways in the surface; and the Circum- ference is the greatest distance around the sphere. A Sphere. 407. The surface of a sphere is equal to the product of 3.1416 hy the square of the diameter ; and The contents of a sphere are equal to the product of ^ of 3.1416 hy the cidte of the diameter. 27. What is the surface of a sphere whose diameter is 26 inches ? 28. What number of square meters of gold-leaf will gild a globe 18 centimeters in diameter ? MENSURATION. 263 29. How many cubic feet of gas will be required to fill a spherical balloon whose diameter is 50 feet ? 30. How many cubic miles of volume has the moon, allowing its diameter to be 2100 miles ? SIMILAR SURFACES. 408. Similar Surfaces are surfaces having the same form, without regard to size. 409. Similar surfaces are to each other as the squares of their correspoiiding dimensions. Hence, The corresponding dimensions of similar surfaces are to each other as the square roots of their areas. 31. A triangle whose base is 12 feet has an area of 72 square feet. What is the base of a similar triangle whose area is 32 square feet ? Solution, — . 72 : 32 =: 12^ : x% or 72 : 32 = 12*: 8.* Ans. 8ft. 32. A side of one of two similar triangles is 12 feet, and the corresponding side of the other is 8 feet. If the larger of the triangles has an area of 72 square feet, what is the area of the other ? 33. The areas of two similar rectangular fields are, respec- tively, 9 and 6.25 acres, and the first is 120 rods long. What is the length of the second ? 34. The diameters of two circles are, respectively, 100 feet and 50 feet. How much larger is the one than the other ? 35. If a pipe whose diameter is 1.5 inches fills a cistern in 5 hours, in what time will another whose diameter is 35 inches fill it ? 36. If it costs $ 30 to pave the bottom of a cellar whose width is 16 feet, what will it cost to pave a similar one whose width is 24 feet? 264 MENSURATION. 37. A gentleman has a park, in the form of a right-angled ijiangle, containing 105.55 square meters, the longest side of which is 15 meters. He wishes to lay out another in the same form, whose longest side shall be 3 times the length of the first. Required the area. SIMILAR SOLIDS. 410. Similar Solids are solids having the same form without regard to size. 411. Similar solids are to each other as the cubes of their corresponding dimensions. Hence, 412. The corresponding dimensions of similar solids are to each other as the cube root of their contents or volumes. 38. If a cone 3 feet in height contains 1539 cubic feet, what are the contents of a similar cone 2 feet in height ? Solution. — 33 : 23 = 1539 : a^, or 27 : 8 == 1539 : 456. 39. Two similar bins contain, respectively, 400 and 600 bushels. If the first is 4 feet deep, how deep is the second ? 40. A sugar loaf which is 12 inches high weighs 16 pounds. How many inches may be broken from the base, that the rest msij weigh 8 pounds ? 41. If a sphere 6 inches in diameter weighs 16.50 kilo- grams, what is the weight of a sphere of the same material 12 inches in diameter ? 42. Suppose the diameters of the earth and the moon to be, respectively, 8000 and 2000 miles, how many times larger is the earth than the moon ? 43. If the weight of a well-proportioned man 5 ft. 6 in. in height, is 140 pounds, what should be the weight of Bar- num's Chinese giant who is 8 ft. 3 in. in height ? REVIEW. 265 REVIEW. ORAL EXERCISES. 413. 1. If 6 barrels of flour cost $ 33, what will 11 bar- rels cost ? 2. If 9 tons of coal cost $ 54, bow many cords of wood at3 I 4 a cord would cost as much as 5 tons of coal ? 3. If 6 horses in a certain time consume 14 bushels ot oats, how many bushels will 10 horses consume in the same time? 4. If I of a barrel costs $ 4§, what will f of a barrel cost ? 5. If A can do a piece of work in 3 days, and B can do the same in 4 days, what part of the work can they together do ii* Iday? 6. If A and B can in 1 day perform ^ of a piece of work, how long will it take them to do the whole ? 7. Divide 75 into two numbers that shall be to each other as 7 to 8 ? 8. The first three terms of a proportion are 7, 8, and 9. What is the fourth term ? 9. How long will 40 bushels of oats last 3 horses, if they consume 8 bushels in 2 weeks ? 10. If 7 horses consume 3 tons of hay in 6 weeks, how many tons will 6 horses consume in 4 weeks ? 11. A and B run a race in which A gains 3 rods in running II rods. How far must he run to gain 7 rods ? 12. A box of tea lasted a man and his wife 9 months. When the man was absent it would last his wife 12 months. How long would it have lasted the man alone ? 13. Smith and Jones hire a pasture together. Smith pays $ 16 and Jones $ 20. Smith puts in 12 sheep. How many does Jones put in ? 14. A furnished 2 loaves for a lunch, and B furnished 3, while C contributed 30 cents to be divided between A and B. How much should each receive ? 266 REVIEW 15. Three men trade together. A puts in $ 3 as often as B puts in $ 4 and as often as C puts in $ 5. They gain 1 87. What was each man's share of the gain ? 16. A hare is 60 rods before a hound, and runs 3 rods while the hound runs 6. How many rods must the hound run to overtake the hare ? 17. What is the time of day when ^ of the time past mid aight equals the time to noon ? 18. At what time between 4 and 5 o'clock are the hour and the minute hand together ? 19. A can do a piece of work in 2 days, B can do it in 4 days, and C in 3 days. In what time can they do it together ? 20. If I pay $ 15 four months before it is due, how long after it is due may I keep $ 20 to balance it ? 21. If 24 men can mow 66 acres of grass in 2 days, how many acres can 14 men mow in 7 days ? 22. What is the interest of 1 800 for 36 days at 10 % ? 23. I bought 12 cords of wood that was to have be^n 4 ft. long. It proved, however, to be only 42 in. What was my per cent of loss ? 24. William Simonds receives a bequest of $840, with which he buys U. S. 4 per cents at 120. What is his annual income ? 25. In what time will $ 400 yield $ 72 interest at 4 % ? 26. What is the perimeter of a square 10-acre field ? 27. My semi-annual income from an 8 % stock is $ 84. How many shares do I own ? 28. What is the longest straight line that can be drawn on a sheet of 6 X 8 paper ? 29. A lawyer earns $ 275 collecting money at 5 % commis- sion. What amount does he collect ? 30. The diagonal of a garden 8 rods long is 10 rods. Ee- quired the area of the garden. REVIEW. 267 31. The interest of 1 400 is $ 70 for 2 y. 6 mo. What is the rate ? 32. What is the cube root of the square of | ? 33. Which is the larger, the square of J or its cube, and how much ? 34. Find a mean proportional between 3 and 48. 35. What is the inverse ratio of g to } ? 36. Received a discount for cash of 12^ % on a bill of goods amounting to $ 128. How much did I pay ? 37. Which is the better investment, 6 % stock at 120 or 8 % bonds at 144 ? 38. Find the missing term in the proportion 2^ : x = 5 : 7|. 39. A note for $ 800, which matures May 31, is dis- counted April 1. Required its proceeds. 40. In selling cloth at $ 6 per yard, I gain 20 fo. What % should I gain if I sold it at $ 6.40 ? 41. A bank renewed $ 100000 worth of U. S. 6 per cents at 3 J %. What is the annual loss of interest ? 42. What is the interest of the present worth of a debt of $ 520 due in a year, money being worth 4 % ? 43. When exchange on London is quoted at | 4.80, what will a £ 200 draft cost ? 44. I lent Willard Aldrich $ 1500 for three months. How long should he lend me $ 1000 to requite the favor ? • 45. What is the ratio of the square of 4 to the cube root of 125? 46. What is the inverse ratio of a decigram to a kilogram ? 47. A party of pleasure-seekers hired a sailboat for $ 4, each paying as many cents as there were persons in the party. What did each pay ? 48. A man with more money than learning willed -J his property to his wife, J to his son, and | to his daughter. He left $ 13000 ; how ought it to be equitably divided ? 268 REVIEW. WRITTEN EXERCISES. 49. I am offered $ 7000 cash for my house or $ 7700 in a year without interest. I can get 10 % for my money. Which is the better offer ? 50. I buy butter for 32 cents, and am obliged to sell it for 28 cents. What % do I lose ? 51. A suit of clothes marked to sell at % 40 is sold at 10 % discount, and yet the seller makes 25%. What did it cost him? 52. What is 33^ % of the square of the cube root of f | ? 53. What is the entire surface of a 14-foot cube ? 54. When Washington was inaugurated there were 13 States in the Union, and, March 4, 1881, there were 38. Re- quired the rate per cent of increase. 55^ I lost 16§ % of my property by fire and 20 % by failures. How much had I originally if % 5700 remained ? 56. I buy books at 80 cents, 20 % off, and sell for $ 1.00, 15 % off, and 5 % off the remainder. How much do I gain ? 57. Received a consignment of 2000 barrels of flour, which I sell at *8.50, paying $74 storage and $27 cartage. My commission is 1^%. How much do I remit ? 58. My insurance at J % cost me a premium of % 62.40, and is I of the value of my house. What do I lose in case of its total destruction by fire ? 59. Received a remittance of $ 25375 for the purchase of cotton at 12^/ per pound after deducting my commission of 1| %. How many pounds do I purchase ? 60. I buy 6 % stock at 112. What per cent do I receive on my investment? 61. What sum will cancel a note of $892, Dec. 24, 1881, dated May 30, 1878, drawing 4i % interest ? 62. Sold two pianos for $600 each, gaining 20% on one and losing 20% on the other. Required my gain or loss. REVIEW. 269 63. What is the compound interest, payable Jan. 1 and July 1 of each year, of $ 1200 at a yearly rate of 4 % for 1 y. 9 mo. 18 d. ? 64. Eobert Burns owes me $ 1500 due 9 mo. hence without interest. What will be a fair deduction for present payment if the current rate for money is 8 % ? «5. How long will $ 1000 be in amounting to $ 1500 at 7 % ? 66. A note for $ 1600 dated Jan. 1, 1879, has three indorse- ments of $ 200 each, — July 10, 1879, Aug. 15, 1880, and May 12, 1881. What was due at settlement, Jan. 1, 1882 ? 67. Bought 62| yards of carpet at $ 1.87|, receiving a dis- count of 15 %. What was my bill ? 68. Sold an estate which cost me $ 8000, | % for insurance, $ 64 taxes, and 1 1260 for repairs, for a 2-month note of $ 11000, which I had immediately discounted at a bank at 5 %. Did I gain or lose, and how much ? 69. What is the face of a 4-month note which will yield $ 800 when discounted at 4 % ? 70. What cost 125 shares of N. Y. Central E. R. stock at 143 i and brokerage ? 71. May 14, I get a 90-day note for $ 1292 discounted, which was dated April 10, 1882. What were the proceeds ? 72. Exchange on Paris is quoted at 5.14. What will a bill of exchange for 8500 francs cost me ? 73. Find the missing term : V^:0625 : (1)2 = X : y/ 15.625. 74. Find a mean proportional to 816 and 97 J. . 75. If J of a pound cost I f , what will f of an ounce cost ? 76. If an 8-cent loaf weighs 10 ounces with flour at $ 8.50 per barrel, what will a 12-cent loaf weigh with flour at $ 10 ? 77. If my gas-bill is $ 4 for the month of January, 1881, when I use 4 burners 3 J hours each evening, what will it be for the month of February, when I use 3 burners 4 h. 20 min. each evening ? 270 REVIEW. 78. Three children, aged 14, 12, and 7 respectively, share a bequest of $ 7000 in proportion to their ages. What is the share of the youngest ? 79. Ames and Stevens form a partnership January 1, Ames furnishing $ 8000 capital and Stevens $ 6000. May 1st, they take Conant into the firm with a capital of $ 5000, and Au- gust 1, Hubbell joins them with $ 3000. How shall the gain of $ 12000 be divided at the end of the year ? 80. If it costs $320 to fence a rectangular lot 120 rods long and 80 rods wide, what will it cost to fence an equal square lot at the same rate ? 81. If it is 90 feet between the bases of a ball ground, what is the shortest distance from the second base to the home plate ? 82. What is the diagonal of a section of land in rods ? 83. How long a cube contains 25934.336 cu. ft. ? 84 Grandmother Gray lives in the center of a square farm containing a quarter-section of land. How far must she walk by the shortest route to visit her four children, one of whom lives at each corner of the farm, and return home ? 85. How large a cubical pile will 32 cords of wood make ? 86. Eind the diagonal of a rectangular solid 60 inches long, 40 wide, 20 high. 87. How many feet long is a square containing an acre ? 88. What will it cost at $ 0.75 per rod to fence eight equal rectangular lots made from a square ten-acre field ? 89. The roof of how high a building can be reached by a 40-foot ladder, the bottom of which is 12 feet from the building ? 90. A certain cube contains 175616 cu. in. Find the diag- onal of one of its faces. 91. If 5184 is the square of a number, what is its cube ? 92. How long are the rafters of a house 40 feet wide, the ridge-pole being 12 feet above the attic floor ? 93. What is the ratio of V^ 2|" to V^ 4jf REVIEW. 271 94. What is the entire area of a cube containing 592704 cubic inches ? 95. If 8 men can make 300 pairs of shoes in 6 days, how many men must be added to their number in order that twice as much may be done in half the time ? 96. There are two numbers in the ratio of 6 to 11, and the ' larger is 167 J^. What is the smaller ? 97. Reduce the ratio 3^2 • f ^^ ^^^^ smallest integral terms. 98. If 4/g^ pounds of coffee cost $ 1.38, what will 11 1 pounds cost ? 99. If 24 men can build 405 yards of wall in 24 days, how many men will it require to build it in 8 days ? 100. A and B are in partnership for one year. January 1, A put in $ 2000, but B did not put in any until April 1. What did he then put in to have an equal share with A at the end of the year ? 101. A and B start from the same place, and travel the same road. A goes 5 days before B, at the rate of 20 miles a day. B follows at the rate of 25 miles a day. In what time and at what distance will he overtake A ? 102. A cistern has 3 pipes : the first will empty it in 2 hours, the second in 3 hours, and the third in 4 hours. In what time would they together empty the cistern ? 103. A certain piece of labor was to have been performed by 144 men in 36 days, but a number of them having been sent away, the work was performed in 48 days. What nun^* ber of men was se»t away ? 104. If 15 carpenters can build a bridge in 60 days by working 15 hours a day, how long will it take 20 men to build the bridge by working only 10 hours a day ? 105. Divide $ 2000 among A, B, and C, so that for every $ 3 given to A, B is to receive $ 5, and C$8. What sum did they each receive ? ^ 106. How many inches in the diagonal of a square whose side is 24 feet ? 272 KEVIEW. 107. What is the cube root of |, to the nearest hundredth ? 108. The side of a square is 8 feet 6 inches. What is the side of a square having 25 times the area ? 109. How many gallons per minute will a |-inch faucet discharge, if a ^inch faucet discharges 30 gallons ? 110. One person owes another $ 150 payable in six months, $ 180 payable in 8 months, and $ 270 payable in 4 months. Find the equated time of payment. 111. Three towns are to provide, according to their popula- tion, a contingent of 182 men. The population of the first town is 2456, of the second 735, and of the third 436. Find as exactly as possible the number of men to be provided by each town. 112. How many square yards in the area of the sides of a square pyramid, whose slant height is 100 feet and the perime- ter of whose base is 54 feet ? 113. How many globes 4 inches in diameter are equal in volume to one that is 12 inches in diameter ? REVIEW QUESTIONS. 134. How is the part that one number is of another found ? 338. What is ratio ? How is it determined ? 345. What is proportion 1 349. What is a simple proportion ? 351. A compound proportion ? 161. What is a hne? 164. What is an angle? 165. What is a right angle ? 179. How many degrees in a right angle ? 163. What is a surface? 166. What is a rectangle? 167. A square 1 391. A quadrilateral ? 219. What is a triangle? 387. A right-angled triangle? The hypothenuse ? The perpendicular ? 177. What is a circle? The circumference of a circle? The di- ameter ? 222. The ratio of the cicumference to the diameter ? 169. What is a solid, or volume ? 223. A rectangular solid ? 170. A cube? 399. ^ prism? 225. A cylinder? 401. A pyra- mid 1 402. A cone ? 406. A sphere ? EXAMINATION QUESTIONS. 273 EXAMINATION QUESTIONS. FOR TESTING PROFICIENCY, FOR PROMOTIONS, AND FOR S1DB PLEMENTARY PRACTICE. ARRANGED FROM PAPERS USED IN VARIOUS CITIES. FUNDAMENTAL RULES. 414. 1. Write in words the following: 4238 — 758 = 145 X 24. 2. What is the difference between 24 times 325 and 36 times 245 ? 3. How many more times is 16 contained in 192 than it is in 64? 4. Multiply 125 by 9, and write the product in figures and in words. 6. John has 20 marbles and James has 12. How many marbles must John give James that each may have the same number? 6. The salary of the President of the United States is $50000 a year. How much is that a month ? 7. What number must be added to 365 to make 730? 8. Henry is Id years old, and one half of his age is twice the age of his brother. What is his brother's age ? 9. (12 X 9) + 12 = 5 X? 10. How many hours are there in January ? 415. 1. The product of three factors is 56700; two of the factors are 42 and 75. What is the third factor ? 2. The dividend is 50000, the quotient is 136, and the re- mainder 360i What is the divisor? 3. Divide 149184 by 84; and write the quotient in figures and in words. 4. Two men, who are 1224 miles apart, travel towards each other ; one 32 miles a day and the other 36 miles a day. In how wiauy days will they meet ? 274 EXAMINATION QUESTIONS. 5. The miimend being 26402 and the remainder 18725, what is the subtraliend ? 6. The product is how many times the multiplicand ? When is it a concrete number ? 7. Multiply 40800 by 30600. Why is the product an ab^ fitract number ? Q. A farmer bought 8 horses for $ 75 each^ and 6 horses for $ 125 each, and sold them all for $ 120 each. How many dollars did he gain ? 9. Divide 381600 by 123, and prove your work. 10. A man bought 240 acres of land at $ 26 an acre, giving in payment a house valued at $ 2820 and horses valued at $ 180 each. How many horses did he give ? 416. 1. What is the difference between a figure and a number ? 2. Kead 40090.049. What name is given to the number at the left of the point ? 3. How may you prove subtraction ? Illustrate by an example. 4. Give the sum of all the numbers in the next four ex- amples. 5. If 3008.7 is the minuend and 299.99 is the subtrahend^ what is the remainder ? 6. If 8467 is the remainder and 44 is the subtrahend, what is the minuend ? 7. Multiply 387.5 by 6. Perform the same example by raddition. 8. Divide 86784 by 87, and prove the work. 9. A man bought a cow for $ 85, a horse for .$ 165, and a carriage for $ 276. How much more did he pay for the car- riage than for both horse and cow ? 10. A man sold 108 acres of land at $ 205 per acre, and svith the money purchased horses at $75 each ; how many did he ^(it ? EXAMINATIO]^ QUESTIONS. 275 417. 1. The multiplicand is 87040, the multiplier is 6080. What is the product ? 2. One cord of wood contains 128 cubic feet. How many cubic feet are there in 75 cords of wood ? 3. A farmer sold to a flour merchant 45 bbl. of apples at $ 3 per bbl., 65 bbl. of potatoes at $ 2 per bbl., and receiyed in payment 40 bbl. of flour at $ 6 per bbl., and the balance in money. How many dollars did he receive ? 4. If a silk dress containing 17 yards costs $ 38.25, what is the cost a yard ? 5. 84.61 X 27 = ? At the right of each term write its name. 6. How many pounds of sugar at 12 cents a pound can you get for 18 dozen eggs at 16 cents a dozen ? 7. Multiply 814 by 16 ; add 279 to the product ; subtract 384 from the sum, and divide the remainder by 18. 8. Show by an example that either of the factors in multi- plication maybe used as multiplier without changing the value of the product. 9. A man bought 8 cords of wood at $ 6.50 per cord, 18 tons of hay at $ 21 per ton, 7 bushels of potatoes at $ 0.90 per bushel. He paid $ 75 in cash. How much does he still owe ? 10. A man bought ten books ; for 4 of thorn he paid 1 1.50 each, for 3 of them he paid $ 1.80 each, and for the rest he paid 28 cents each. How much did he pay for all ? 418. 1. Divide the product of the sum and difference of 125 and 36 by 48. 2. Bought 360 acres of land for $ 32400, and sold it for $ 8400 more than cost. What was the selling price per acre ? 3. (i of 69543248) - (^ of 81369) = ? 4. I bought 1265 books, and sold ^ of them at $ 0.50 each and the remainder for $ 0.75 each. How much did I receive for them ? 5. Bought 18 lb. of steak at 24/, 41- doz. eggs at 36/, 3 qt. of molasses at 18/, and a bushel of potatoes for 75/. Required the amount of my bill. 276 EXAMINATION QUESTIONS. 6. In a certain church there are 40 pews that seat 6 people, 35 that seat 5, and 18 that seat 4. The gallery will accom- modate 115. At a lecture every seat is filled ; the price of admission being 25/, what were the proceeds, 62 compliment- ary tickets having been used ? 7. A farmer bought a cow and 254 sheep for $ 1134.50 The cow cost $ 55. What did 54 sheep cost ? 8. How many 5-cent pieces in $ 720 ? 9. Exchanged a farm worth $4278 for 75 sewing-machines worth $ 45 each, and $ 200 cash. Did I gain or lose, and how much ? 10. 6010 X 6020 X 9 = 18 X ? COMMOISr FRACTIONS. 419. 1. Subtract the sum of all the prime numbers from 1 to 37 from that of all the composite numbers from 4 to 40. 2. Name three composite numbers prime to each other. 3. Keduce 2% f ? i h ^^^ ^^ ^^ fractions having the least common denominator, and find their amount. 4. Reduce to their lowest terms ^^^j, }||, ||f, and x^Vt, using in each case the greatest common divisor. 5. What is 2i fractional unit, and what is the unit of a frac- tion ? Give an example of each. 6. Find the amount of the following mixed numbers : 4/^5, 6/2'V, and 12^*^^. Reduce to lowest terms and least common denominator before you add. 7. From 175 J take 95^, and from 45 g take 25f. 8. In what two ways can fractions be multiplied by an in- teger ? Which way is preferable ? Why ? 9. Multiply 175 J by 12, 124 by 6f , and ^ by 12|. 10. A man who owned 3 of a ship sold % of his interest for $30000. What part of the sliip did he sell, and what was the value of the wliole ship at that rate ? EXAMINATION QUESTIONS. 277 420. 1. Define a prime number ; a composite number ; a fraction. 2. Find the greatest common divisor of 182 and 196. 3. Find the least common multiple of 8, 7, 10, 14. 4. Change 13J, 61|, 15^, to improper fractions. 5. Keduce f f f and |ff to their smallest terms, using the greatest common divisors. 6. Reduce ^|§^, ^{j-y -?§-; ^^ whole or mixed numbers. 7. Eeduce |, f, -f^, f, to equivalent fractions having the least common denominator. 8. When is it necessary to reduce fractions having different denominators to equivalent fractions having a common de- nominator ? 9. A horse traveled 48^^ miles in one day, 56^ the next, 40 J I the third, and 45|J the fourth. How far did he travel in all ? 10. From a bin containing 25| bushels of grain there were taken out 5| bushels at one time and 6| bushels at another. How much remained ? 421. 1. Find the greatest common divisor of 36, 108, and 420. 2. Find the least common multiple of 24, 180, 45, and 60. 3. Required the amount of 12j, 16|, 24f, and 40f 4. Required the difference between 84J and 42j. 5. Multiply f of 12J by 36f. 6 Divide 27f by § of 8 J. 7. Reduce — and ^ to simple fractions. a If § of a farm is worth $ 7000, what is | of it worth ? 9. If a man travels 240 miles in 5 j days, how far will he travel in 3^ days ? 10. A coal-dealer sold f of what coal he had on hand for $ 120, at the rate of $ 6 a ton. How many tons had he ? 278 EXAMINATION QUESTIONS. 422. 1. Define factoring. Name and define tlie terms of a fraction. 2. When are two or more numbers said to be prime to each other ? 3. E,educe the following by cancellation : 800 X 378 X 44 X 15 160 X 63 X 11 X 4 4. E-educe f to a fraction having 135 for its denominator. 5. The sum of two numbers is 36f-, and one of them is 15f . What is the other ? 6. Divide 12 J by J of 17. 7. What is the value of an estate if § of f of it is worth $45000? 8. What is the value of | of a farm if f of it is worth $4000? 9. Eeduce j ^ to a simple fr.iction. XO. Is ^4f\-{^ a large or a small fraction, and why ? 423. 1. Change fgg-^, j%^^, 4f|, to lowest terms. 2. Add 4^y, 3|, 4-1, f . 3. Take 12f from 31|. 4. Having $ 4283}f, I gave $ 1597|| to my son. How much had I left ? 5. Multiply 641 by 5|. 6. Keduce ^ of 3} of 4] of 63§ of |^ to a simple fraction 7. — plus 71 equals what ? 8. If 3^ lb. cost $ 21, what will 3 lb. cost ? 9. 5^j is contained how man}' times in 2^ ? 10. If a bird can fly 10} miles in \ of an hour, how far can it fly in 2 J hours ? EXAMINATION QUESTIONS. 279 424. 1. Find the sum and the difference of 12|:J and 2. In what two ways can a fraction be divided by a whole number, and which is the better way ? 3. If a man can travel 32 1 miles in one day, how many miles can he travel in 5^ days ? 4. A man received $ 84| for laboring 18| days. How much did he receive each day ? 5 What is the value of a farm if f of it is worth $ 500 more than f of it ? 6. If a merchant who owns f of a store should sell J of his share for $ 12000, what is the value of the whole store at that rate? 7. A has $ 12000. If $ 600 be added to f of A's money, the sum will equal J of B's mone3^ How much money has B ? 8. Reduce pr^ -^ to a simple fraction. 9. A man who owned 100 acres of land sold 37i acres to one man and ^ of the remainder to another. How many acres had he left ? 10. A man who owned | of f of a ship sold J of his inter- est for $ 32000. What was the value of the whole ship at the same rate ? 425. 1. Three cheeses weigh, respectively, 46f, 49y^^, and 57| lb. What is their entire weight ? 2. What number is that from which if 28-f- is taken the re- mainder is 65r| J ? 3. 4H 4- 56- 24^ -41H = ? 4. Find the cost of | of 156§ acres of land at | of $54 an acre. 5. What number multiplied by 33f will produce 297 J ? 6. In 2i acres of land, how many building lots of | of an acre each ? 280 EXAMINATION QUESTIONS. 7. Find the value of f ^ ^t - tV X 5i 8. What part of 9 miles is f of 8 miles ? 9. § of a farm is worth $ 9000. Wliat is yV ^^ i^ worth ? 10. If 2 be added to both terms of the fraction J, will the value be increased or diminished, and how much ? 426. 1. (7i - 3i) - 145. 2. At $ 3 a day for work, what part of a day's work can be had for $2.50? 3. Multiply 94 by 26§, and from the product subtract the quotient of 12000 divided by 7|. 4. Give an example in solving which it is necessary to use the least common multiple of 63 and 84. 5. The denominator of a certain fraction is J of f -f J X 3J^, and the numerator is | of the denominator. What is the value of the fraction ? 6. Find the amount of the sum and the difference of 6§ X 2f and 5j -^ 3|. 7. A boy sold a book for $ 2 J, which was J of the cost. What did he lose by the bargain ? 8. Emma, having 2 quarts of berries, ate J of them, sold J of a quart, and divided the remainder equally among three friends. What did each friend receive ? 9. Multiply 24| by |. Divide 24^ by ^. How much does the quotient exceed tlie product ? 10. A bought f of a farm for $1760. B bought 1 the re- mainder at the same rate, and C took the remainder at an ad- vance of $ 375. What did C's share cost ? DECIMALS. 427. 1. Change ^\ to an equivalent decimal. 2. From eight hundred thousandths take eight hundred thousandths. 3. Write in words : 7.008 ; 9090.909 ; 0.00042. EXAMINATION QUESTIONS. 281 4. Change to an equivalent common fraction in its smallest terms 0.0025. 5. Why is it unnecessary to express the denominator of a decimal f I'action ? 6. From a piece of cloth containing 49 J yd. I sold ^ of a yd., 3f of a yd., and 21.125 yd. What was the length of the rem- nant ? 7. Eeduce |, f, ^, and ^^ to decimals, and add. 8. How much will 240 men earn at $ 1.37 J a day ? 9. If 4 bushels of beans cost $ 12.56, what will 9 bushels cost? 10. What part of the quotient of 24 thousandths divided by 25 ten-thousandths is the product of 16 thousandths by 3 hundred ? 428. 1. Change ^J§^ to a decimal. 2. Change 960 hundred-thousandths to a common fraction in smallest terms. 3. Multiply the quotient of 144 -^ 12000 by the quotient of .0144 -^ .00012. 4. Subtract 23 ten-millionths from .02 of .006. 5. Divide 17.28 by .0831 6. Multiply .027 J by .36|. 71 7. Change ^^r^ to a decimal. 8. Tell the shortest way of dividing a decimal by 1000. Of multiplying it by 100. 9. If .03 of a number is 90, what is .005 of it ? 10. Divide 1.2 by .0025, and subtract the divisor from the quotient. 429. 1. Change -^^ to a decimal, multiply by .0008, and divide the product by .02. 2. Divide one thousand by one thousandth, and from the quotient subtract the dividend, the divisor, and their product. 282 EXAMINATION QUESTIONS. 3. Multiply six hundred thousandths by six hundred-thou- sandths, and divide the product by .02j. 4. All of the six U. S. gold coins are equal in value to how many twentj^-five cent piece*? 5. Bought 15280 bricks at $ 40 per thousand. 350 were so broken that they were worthless. What was the actual cost per thousand of those used ? 6. If a man receives $ 1500 a year for labor, and his ex- penses are $ 968, in what time can he save enough to buy a farm worth $ 3724 ? 7. A man lost 0.60 of his money, and then earned $ 130, when he had 0.83J of the original amount. How much had he at first ? 8. What cost 19375 ft. of lumber at $ 17.25 per thousand ? 9. Find the amount of the following purchases : 21|- yd. carpeting at $1.75; 25 yd. lining, at 12^/: 2| yd. silk at $ 2.25, and f yd. velvet at $ 2.87^. 10. If my gas bill was 1 11 when I burned 4400 feet of gas, what will it be when gas costs I more and I burn 1500 feet less? COMPOUND NUMBERS. 430. 1. Find the difference between 25t\ and 15.064. 2. Define a complex fraction ; a mixed decimal ; a denom- inate number ; reduction. 3. Change ^ !} to a complex decimal of five places, and 0.0875 to a common fraction. 4. Eeduce 65 rd. 2 yd. 1 ft. to the decimal of a mile. 5. Reduce 0.5473 of a pound troy to lower denominations- 6. At $ 5.50 a cord, what is the value of a pile of wood 80 feet long, 12 feet wide, and 4 feet high ? 7. A man bought a farm containing 260 acres 45 square rods at $ 25.75 an acre. What was the cost of the farm? 8. In 16.45/> metric tons how many kilograms ? EXAMINATION QUESTIONS. 283 9. A man paid $ 3.46 for sugar and coffee ; he bought 6 lb. 5 oz. of coffee at 32 cents a pound, and paid 11 cents a pound for the sugar. How much sugar did he buy ? 10. If an average degree of latitude is 69/oV common miles, and a meter is one ten-millionth of the distance from the equator to the pole, measured on a meridian, what is the length of a meter in inches ? 431. 1. How many years, months, and days from the dis- covery of America, Oct. 11, 1492, to the Declaration of Inde- pendence, July 4, 1776 ? 2. A farmer had two farms, one of 104 A. 117 sq. rd., the other of 87 A. 78 sq. rd. He reserved 40 A. 40 sq. rd., and di- vided the remainder equally among his 3 sons. What was the share of each son ? 3. If a car runs 16 mi. 25 rd. 12 ft. in 40 minutes, how far will it run at the same rate in 24 hours ? 4. If the difference in the time of Greenwich and of St. Louis is 5 h. 55 min., what is the difference of longitude ? 5. The aggregate weight of 85 hogsheads of sugar is 39 T. 1625 lb. What is the average weight per hogshead ? 6. A ship sails east from Boston, longitude 71° west, 2° 30^ 20'^ a day. What is her longitude at the end of 5 days ? 7. Define a square ; a rectangle ; a cube ; a solid. 8. If a man wastes 4 minutes a day, how many days, hours, and minutes will he waste in the years 1880 and 1881 ? 9. There is a fence enclosing a circular field 32 feet in diameter. What will be the area of a square field which the same fence will exactly surround ? 10. How many cubic feet of water must be drawn from a reservoir 24 ft. 6 in. long and 20 It. 9 in. wide, to lower the gurface 3 inches? 284 EXAMINATION QUESTIONS. 432. 1. A room measures 14 ft. by 16 ft., and is 8 ft. high. How many rolls of paper 1 J ft. wide will cover the walls, there being 8 yd. in a roll and J allowed for openings ? 2. Divide 2| times | of 29^ by 4| times ^j of 8. a Change f of a great gross to integers of lower denomina- tions. 4. How many square inches in the entire surface of 15 bricks, each 8 inches lone, 4 inches wide, and 2 inches thick ? 5. What date comes 275 days after May 7 ? 6. What is the cost of 17 gal. 3 qt. 1 pt. at $ 0.45 per gallon. 7. 3.75 X 48.341 -^.5^ = ? 8. How many acres in ^ of a mile of street 60 ft. wide ? 9. A room 10 ft. high contains 30000 cu. ft. What will it cost to carpet it at $ .75 per sq. yd. ? 10. How many square yards of silk in 300 yd. of 3-inch ribbon ? 433. 1. A man paid $ 660 for a piece of land 8 rods long and 10 feet wide, and sold it at 60 cents per square foot. What did he gain ? 2. What will 7bu. 3pk. 4qt. of nuts cost at $4.80 per bushel ? 3. It takes 30 yards of carpeting | yd. wide to carpet a room 15 ft. long. How wide is the room ? 4. Bought an acre of land for $ 300, sold J of it at 30 f per square foot, and the remainder at cost. What did I gain ? 5. Add 0.24 lb. and 7 oz., avoirdupois. 6. What will 12750 feet of boards cost at $ 27.50 per thou- sand ? 7. What part of an acre is a lot of land 132 ft. long and 66 ft. wide ? 8. How many bags, each holding 2 bu. 1 pk. 3 qt., will it take to hold 124 bu. 0pk.'7 qt. ? 9. Find the exact number of days and hours from 9 o'clock K. M., Jan. 7, 1876, to 3 p. m. of March 1, 1881. EXAMINATION QUESTIONS. 285 10. How many building lots, each 75 ft. by 125 ft., can be made out of 1 A. 46 sq. rd. 18| sq. yd. of land ? 434. 1. Mr. Howes bought a farm, 198 rods long and 150 rods wide, and paid $ 32 an acre. What did it cost him ? 2. I bought a board 12 feet long, 16 inches wide at one end and 9 at the other, at $ 30 per M. What did it cost me ? 3. How many cubic feet of snow will there be on an acre of land, if it is uniformly 6 inches deep ? 4. In a piece of ribbon 2 J inches wide and 167 fset long, how many square yards ? 5. Of a street 50 feet wide, how many feet in length make one acre? 6. How many bricks (each 8 inches long, 4 inches wide, and 2 inches thick) will make a cubical pile 13 feet each way ? 7. If a ship sails at the rate of 2 miles in 11 minutes and 11 seconds, how many days will she require to cross the At- lantic where it is 2979 miles wide ? 8. I own five contiguous unfenced house-lots. Each lot is 50 ft. wide and 150 ft. deep. How many feet of boards will enclose said lots with a tight board fence 4 ft. high, and what will they cost at $ 16 per M. ? 9. A grocer uses a false gallon containing 3 qt. IJ pt. What is the actual worth of the liquor he sells for $ 240, and what does he make by the cheat ? 10. What costs a pile of wood 17 ft. 8 in. long, 8 ft. wide, and 8 ft. 3 in. high, at $ 8.32 per cord ? 435. 1. What part of an acre is a rectangular piece of land, 12 rods long and 110 feet wide ? 2. A meter is 39.37 inches. How many meters are there in a mile ? 3. Gold is 19.35 times as heavy as water. What is the weight in kilograms of a cubic meter of gold ? 286 EXAMINATION QUESTIONS. 4. What is the cost of 25 yards 2 feet 3 inches of tubing at $ 0.25 per yard ? 5. Paid $ 5 for constructing 2| rods of stone wall. What will a wall .875 of a mile in length cost at the same rate ? 6. Knowing the length of a meter, how can you find the length of a degree of latitude ? Write out the statement, but do not perform the work. 7. A pile of wood 8 feet wide and 4 feet high contains forty cords. How long is it ? 8. How many feet of boards will it take to cover the walls of a house 56 feet long, 25 feet wide, andv30 feet high ? How much will they cost at $ 10 per thousand feet ? 9. What will it cost to make a sidewalk 8 ft. wide and 225 ft. long, when it costs $ 30 to make one 10 ft. wide and 90 ft. long ? 10. What will be the cost of constructing a railroad 25 miles 145 rods long, at $ 700 per mile ? 436. 1. How many yards of carpeting | yd. wide will carpet a room 16^ feet long and 14 feet 9 inches wide? 2. The signal service reports 4^ inches of rain as falling in 24 hours. How many cubic yards fell on J of an acre ? 3. Two ships on the ocean, which are 120 miles apart, are sailing towards each other, one 8| miles an hour, the other lOf miles an hour. How far apart will they be in 4 hours and 40 minutes ? 4. What decimal of a mile is 575f feet ? 5. A cubic foot of water weighs 62^ pounds avoirdupois. The specific gravity of gold is 19.25. One pound avoirdupois equals 7000 grains troy. How many grains does a cubic inch of gold weigh ? 6. A farmer sold 71200 pounds of hay at $ 22 a ton, and purchased 19625 feet of boards at $ 15 a thousand. How much money had he remaining ? 7. If a car runs 16 miles 25 rods 12 feet in 40 minutes, how far will it run at the s^i?^e rate in 4 hours ? EXAMINATION QUESTIONS. 287 8. The aggregate weight of 67 hhd. of sugar is 39 T. 16 cwt. 35 lb. What is the average weight per hhd. ? 9. If the difference in the time of Greenwich and of Boston is 4 hours 44 minutes, what is the longitude of Boston ? 10. A room is 18 ft. 8 in. long and 10 ft. 6 in. wide. One kind of carpeting three-fourths of a yard wide can be obtained for $2 per yard ; another kind, a yard wide, can be obtained for $ 1.75 per yard. Which kind is the more expensive, and how much will it cost to carpet the room with each ? 437. 1. A field containing 24 acres is 80 rods long. What is its width ? 2. What is the difference between 25 rods square and 25 square rods ? 3. How many tons of ice can be packed in an ice-house 50 feet long, 20 feet wide, and 12 feet high, a cubic foot of ice weighing 58.5 lb. ? 4. What is the difference in time between two places whose difference in longitude is 4° 40' ? 5. If 1 ounce of sugar costs 1 cent, what will be the cost of 5 T. 9 cwt. 75 lb. ? 6. I have a piece of land 42 rods long and 6 rods wide. I wish to make 7 square lots of equal size. What will be the cost of boundary and cross fences at $ 2.37 J a rod ? 7. What is the value of a pile of wood 12 feet long, 8 feet wide, and 6 feet high, at $ 4.50 per cord ? 8. Mont Cenis Tunnel, which connects the railways of France and Italy, is 7 miles 190 rods long, and the Hoosac Tunnel is 25000 feet long. The latter is what decimal of the former ? 9. What is the cost of 4 loads of coal weighing 2436, 2150, 1735, and 3462 lb., respectively, at $5.25 per ton ? 10. My house is on a corner lot, 100 ft. on one street and 75 ft. on the other. The sidewalk is 8 ft. wide. How many cubic feet of snow do I shovel in clearing my walk after a 15-inch storm ? 288 EXAMINATION QUESTIONS. PERCENTAGE. 438. 1. Express the following as common fractions : 6 J %, 12i%, 8i%, 16f%/66f%. 2. What per cent is /^, ii f, ^, |, f, f^? 3. Define percentage ; base of percentage ; rate per cent. 4. A man owns 60 % of a ship and sells 75 % of his share. What part of the whole ship does he sell ? 5. What is the difference between 87 J% of $ 5000, and .87^% of the same sum ? 6. The difference in the time of two places is 37 J % of a day. How many degrees apart are the meridians of those places ? 7. How much money has a merchant on deposit if S3^7o of one third of it is $ 700 more than 25 % of one fourth of it ? 8. A man sold a house for $ 5000, which was 25 % more than it cost him. What would have been his gain per cent if he had sold it for $ 6000 ? 9. A broker received $ 12000 for the purchase of bank stock. The brokerage was J % on the purchase. What did he pay for the stock, and what was the brokerage ? 10. What is 62^ % of a sum of money if | of it is $ 1200 more than § of it ? 439. 1. A man had % 5420 in bank. He drew out 15 % of it, and afterward 37^ % of the remainder. How much money had he then in bank ? 2. A man sold 160 acres of land for % 4563.20, which was 8 % less than it cost. What did it cost an acre ? 3. A broker received % 45337.50 to invest in stocks after de- ducting a commission of 2\ %. What amount did he invest, and what was his commission ? 4. A man owns a boat-loa^l of corn valued at % 1800, and insures 87^% of its value at 1§ %. What premium does he pay? 5. \ of 75 per cent is what per cent of J of 90 per cent ? EXAMINATION QUESTIONS. 289 6. A man owning f of a ship sold 45% of his share for $ 36000. What part of the ship did he still own, and what was its value at the same rate ? 7. What per cent of a mile is 124 rods 2 yards 2 J feet ? 8. A merchant, after paying 40 % of his debts, found that $ 4800 would pay 75 % of the remainder. What was his whole indebtedness ? 9. An agent received $ 3675 to lay out, after deducting his commission of 2J%. What was the amount of his commis- sion ? 10. What was the value of the goods purchased, and what was the remittance, when the commission at 3 J % amounted to $ 92.80 ? 440. 1. What per cent of | is | ? 2. A bankrupt's assets are $ 45000, and his liabilities $67500. What per cent can he pay ? 3. Bought a horse for $ 200. What must I ask for him in order to gain 10 per cent and still fall 10 per cent on the ask- ing price ? 4. The premium for insuring a cargo at 2 J- per cent was $ 1000. What was the amount insured ? 5. If f of a barrel of flour is sold for what f of the barrel cost, what per cent is gained ? 6. A merchant paid $ 2500 for cotton, and sold it at 10 per cent advance, taking his pay in prints, which he sold at a loss of 10 per cent. Did he gain or lose, and how much ? 7. Sold 2 horses for $ 450 each, thereby gaining 25 per cent on the one, and losing 25 per cent on the other. What was the per cent of gain or loss on the investment ? 8. What per cent of $ 500 is 37^ % of $ 1200 ? 9. A merchant reduced the price of a piece of cloth 18 cents per yard, and thereby reduced his profit on the cloth from 12^% to 8 %. What was the cost of the cloth per yard ? 10. In 1864 the greenback dollar was worth only 35f cents in gold. What was the price of gold ? 19 290 EXAMINATION QUESTIONS. 441. 1. A owns 35 per cent of a steamboat that is valued at $ 125000, B owns 45 per cent of it, and C owns the re- mainder. What is the value of each man's share ? 2. A farm that cost % 4500 was sold for % 5400. What was the gain per cent ? 3. A merchant sold % 65000 worth of goods in a year ; 40 per cent of the receipts were sales at 25 per cent profit, and the rest at 30 per cent profit. What was the cost of the goods ? 4. An agent received % 4500 with which to purchase mer- chandise, after deducting his commission at 2 J per cent. How much did he expend, and what was his commission ? 5. A man had $ 5000 in bank. He drew out 15 per cent of it, then 20 per cent of the remainder, and afterward deposited 12J per cent of what he had drawn. How much had he then in the bank ? 6. A merchant owes % 15000, and his assets are % 9525. What can he pay on the dollar ? 7. If by selling land at $ 80 an acre I lose 25 per cent, how must I sell it to gain 40 per cent ? 8. What must be paid for insuring $4500 on a house, at | per cent ? 9. A man has secured a policy of % 6000 on his life, at the rate of % 26.30 a year on % 1000. The dividend this year will reduce his payment 35 per cent. What, therefore, will his payment be ? 10. A grain dealer bought corn at 55 cents a bushel, and sold it at 66 cents ; and wheat at $1.10, and sold it at % 1.37^. Upon which did he make the greater profit, and how much ? 442. 1. How many feet are there in 45 per cent of a mile? 2. A man whose income was % 2800 spent % 1600. What per cent of his income remained ? 3. A man invested in real estate % 7500, which was 37 J per cent of his property. What was the value of his property ? EXAMINATION QUESTIONS. 291 4. A has $ 1600. 75 per cent of his money is equal to 62 J per cent of B's money. How much have both together ? 5. Alpheus Cole bought 12500 pounds of coal, and received a discount of $ 16.25, or 20 %. What was the price per ton ? 6. What per cent of 1880 was the month of December ? 7. A merchant, after paying 40 % of his debts, found that $ 4800 would pay 25 % of the remainder. What was his whole indebtedness ? 8. In a town containing 2576 whites 77 % of the inhabitants wefe colored. Eequired the population. 9. What was the value of the goods purchased, and what, was the remittance, wdien the commission at 5^% amounted to 1 92.80 ? 10. A merchant owning 45 % of a ship and cargo valued at $ 125000, paid 4^% for insuring his share. Eequired the base and the premium. 443. 1. Sold tea at 115 per cent of its cost, and thereby gained 9 cents on a pound. What was the cost per pound ? 2. What sum invested at 4J- per cent will yield an annual income of $ 900 ? 3. A man owns a house worth $ 5000. Eepairs and insur- ance average 1 per cent yearly. Would it be more profitable for him to rent the house at $ 400 per year, or to sell it and invest the money in business which will return him 6 per cent ? How much ? 4. A merchant received for a lot of goods $ 874. He had deducted 5 % from the face of the bill, and yet found he had made 15% on his investment. What did he pay for the goods ? 5. A merchant sold goods for $ 5895, and gained as much as he would have lost had he sold them for 14585. What was the gain per cent ? 6. The premium for insuring a steamer at 4J per cent was $ 2925. What was the value of § of the steamer ? 292 EXAMINATION QUESTIONS. 7. A man bought a lot of apples, and sold them for 20 per cent more than they cost, by which he gained $24.80. How much did they cost him ? 8. By selling flour at $ 6.65 per barrel I shall lose 5 per cent of its cost. For how much must I sell it to gain 5 per cent ? 9. A farmer made 12 % by the purchase and sale of a horse> and thus gained as much as he had lost by selling a $ 500 lot of land for 7J^% less than its value. What did he pay and receive for the horse ? 10. By selling goods at $1537.90, a profit of 12§ per cent ' was made. What per cent would have been gained or lost if they had sold for $ 1651.65 ? INTEREST AND DISCOUNT. 444. 1. Find the amount of $ 125 for 3 years 3 months 3 days at 7 %. 2. In what time, at 8 %, will any principal double itself ? 3. What principal, at 7 %, will amount to $643.76 in 3 years 4 months 24 days ? 4. What is the interest of $ 475 from January 1, 1880, to July 4, 1882, at 7^ per cent ? 5. What is the difference between the interest and true discount of $ 450 for 1 year 4 months at 6 per cent ? 6. The difference between the interest of $450 and the interest of $ 300 for a certain time is $ 15.30. Rate, 6 per cent. Required the time. 7. A note for $ 875, dated Jan. 1, 1879, has the following indorsements : March 10, 1880, $ 225 ; April 1, 1881, $ 145. What was due Dec. 31, 1881 ? 8. Required the avails of a note for $ 450, due in 6 months, discounted at a bank at 7^ per cent. 9. The proceeds of a 6 months' note discounted at a bank at 6 per cent are $ 800. Required the face of the note. EXAMINATION QUESTIONS. 293 10. How much more is the interest of 15 cents for 15 years at 4 % than the interest of 15 dollars for 15 days at 8 % ? 445. 1. In what time will the interest of $ 480 at 7 % equal the interest of $ 356.50 for 3 years 9 months 25 days at 8%? 2. What sum of money will gain $ 153.75 in 3 months 24 days at 7 % ? 3. At what rate per cent will $ 500 gain $ 84 in 2 years 4 months 24 days ? 4. What is the accurate interest of $1525 from March 20th to October 20th, at 41 % ? 5. What is the compound interest of $ 1360 for 1 year 6 months at 8 %, interest compounding semi-annually? 6. What is the present worth and the true discount of $ 1275, due in 1 year 5 months 18 days, at 6 % ? 7. What is a negotiable note ? How may a note, payable to order, be made negotiable ? 8. When is a note due, if the time for payment is not specified ? If the words " with interest " are omitted, when will interest accrue ? 9. On a note for $ 750 at 6 %, dated Jan. 15, 1878, were the following indorsements : Sept. 20, 1879, $ 250 ; June 12, 1880, * 120. What was due May 25, 1881 ? 10. For what sum must a note be drawn, at 9 months 15 days, at 7 %, so that the proceeds, when discounted, may be 1 1240 ? 446. 1. What is the difference between the simple and the compound interest on $ 700 for 2 years 6 months at 7 per cent ? 2. What is the bank discount on a note of $ 700 for 3 months at the rate of 7| per cent ? 3. What principal, on interest at 6 per cent, will gain $ 27.47 in 1 year 3 months ? 294 EXAMINATION QUESTIONS. 4. At what rate per cent must $ 648 be on interest, to gain $ 81.873 in 2 years 3 months 17 days ? 5. For how much must a note, payable in 30 days, be given, that, when discounted at a bank, $ 900 may be received on it, money being 6 per cent ? 6. A note of $ 365 is dated July 1, 1878. Indorsements ; Jan. 1, 1879, $85; July 1, 1879, $125. What was the amount due Jan. 1, 1881 ? 7. Find the compound interest of $ 245 for 2 years 6 months at 4^ %. 8. What must be the face of a note that, when discounted at a bank for 5 months at 8 %, the proceeds may be $ 217.35 ? 9. Find the bank discount and proceeds of a note of $450, payable in 90 days, discounted at 8 %. 10. A note of $ 250, dated May 16, 1880, and payable in 4 months, with interest at 6 %, was discounted July 5, 1880, at 7 % . What were the proceeds ? 447. 1. What is the interest of $ 105.23 at 6 per cent, from May 6, 1879, to July 7, 1881 ? 2. In what time will $ 300 gain $ 47.25 at 6 per cent? 3. The interest of $ 560 for 2 yr. 4 mo. 15 d. was $ 106.40. What was the rate per cent? 4. The proceeds of a note discounted at a bank for 90 days, at 8 per cent, is $293.80. Kequired the face of the note. 5. What is the difference between the simple and the com- pound interest on $ 650 for 2 yr. 8 mo. at 6 per cent ? 6. What principal at 8 per cent will gain $78.08 in 4 months 24 days ? 7. If the interest on $ 500 from Jan. 6, 1880, to April 18, 1880, be $12.75, what will the amount of that principal be, Feb. 23, 1881 ? 8* Note given for $320, July 14, 1874, at 8 % ; payment, Dec. 24, 1874, $ 180 ; settlement, March 30, 1875. What was due ? 9. What are the avails of a note for $ 1728, due Nov. 18, 1881, and discounted July 6, 1881 ? EXAMINATION QUESTIONS. 295 10. $860 compounds semi-annually for eighteen months at 7 per cent. What is the amount ? 448. 1. At 6 % how long will it take $ 175 to amount to $275? 2. A note of $ 500 on interest from Jan. 10, 1881, to July 10, 1881, amounted to $ 529.16|. What was the rate ? 3. A note of $ 1250, dated July 5, 18G8, was paid June 1, i870, with interest at 8 per cent. What was paid ? 4. What is the difference hetween the bank discount and the true discount of $ 450 due in 60 days, discounted at 6 per cent ? 5. On a note for $ 425, at 8 per cent, dated March 25, 1880, were the following indorsements : June 1, 1881, $ 75 ; Dec. 30, 1881, $ 120. What was due Sept. 1, 1882 ? 6. If I pay a debt of $ 1410, 2 yr. 6 mo. before it is due, what discount should be made, money being worth 7 % ? 7. For what sum must a note, dated May 10, for 3 months, be drawn, to yield $ 395.80, if discounted at a bank June lOj money being worth 6 % ? 8. June 15, 1880, George Page borrowed of Henry Smith $ 2000j and gave his note for the same, with interest, at 8 per cent. Aug. 27, 1881, a payment of $ 1450 was made, and a new note given for the balance. For what sum was the new note given ? Write the note in its proper form. 9. A capitalist holds half a million of extended U. S. Government 5's, now paying 3 J % . What is his quarterly income ? 10. Sold an 8 % R. E. stock which cost me 160, and bought Government 4's at 120. Did I increase or lessen my annual income ? STOCKS AND AVERAGE PAYMENTS. 449. 1. A 6 % stock, bought at 120, pays what per cent on the investment ? 2. What sum invested at 4| % will yield an annual income of $ 1800 ? 296 EXAMINATION QUESTIONS. 3. Paid $ 9000 for stock at 10 % below par, and sold it at 112. What per cent did I gain ? 4. Bought 40 shares of railroad stock, par value $ 100 per share, at 33 % below par, and immediately sold them at 20 % below par. What per cent did I gain by the transaction ? 5. How many shares of B. & A. E-. R. stock can be bought for $ 25000 at 164| and brokerage at :J, and how much money will be left ? 6. If A lends B $ 300 for 4 months, how long ought B to lend A $ 800 to equal the favor ? 7. If a person, owing $ 700, payable in 10 months, pay $ 300 down and $ 200 at the end of 6 months, how long after the end of 10 months may he delay payment of the balance ? 8. A merchant owes in London £ 500 6 d. How much must he pay for a draft at $4.86^ a pound sterling, to remit in payment ? 9. March 11, 1880, a merchant sold goods to the amount of $ 1850, on a credit of 4 months. April 7, he received $ 400 ; May 15, $ 270 ; and June 20, $ 350. When in equity should the balance be paid ? 10. Mr. Adams bought goods Aug. 1, 1881, to the amount of $ 2400 ; for J of the bill he was to pay cash, J of it he was to pay in 6 months, and the balance in 10 months. On what day may he equitably pay the whole ? PROPORTION. 460. 1. The ratio is 2§, the antecedent J of f . What is the consequent ? 2. At the rate of 72 yards for £ 44 16 s., how many yards of cloth can be bought for £ 5 12 5. ? 3. If a bin 8 ft. long, 4^ ft. wide, and 2 J ft. deep, holds 67i bu., how wide, must another bin be made, that is 18 ft. long and 3g ft. deep to hold 450 bu. ? 4. If 7 men build 6^^ rods of wall in 15| days, in how many days can 12 men do as much ? EXAMINATION QUESTIONS. 297 5. If 10 cents will buy a 6-ounce loaf, when flour is $11 per bbl., how large a loaf will 11 cents buy, when flour is $10perbbl.? 6. If 9 men, by working 8 hours per day, can mow 30 acres of grass in 2| days, how many acres can 5 men mow in 3| days, by working 7| hours per day ? 7a A vessel has provisions for 50 men for 4J months, allow ing each man 1| lb. per day. How long would the same pro visions furnish 75 men, allowing 14 oz. per day to each ? 8. If 36 men can dig a trench 60 rods long in 48 days, working 8 hours a day, how many men will dig a trench 80 rods long in 32 days, working 6 hours a day ? 9. If 19 men build a wall 25 rods long, 4 feet thick, and 3 feet high, in 8 days, working 8 hours and 30 minutes each day, how many men will it take to build a wall 45 rods long, 7| feet thick, and 6 feet high, in 9 days, working 9 hours and 30 minutes each day ? 10. If 8 men can mow 36 acres of grass in 9 days of 9 hours each, how many men will be required to mow 48 acres in 12 days, when the days are 12 hours long ? 451. 1. A and B are partners in trade, B contributing f of the capital. What is A's share of the gain, the whole gain being $ 4500 ? 2. Three men rent a pasture for $ 55. The first puts in 3 cows 2 months, the second 2 cows 4 months, and the third 2 horses 3 months. Each horse eats ^ more than a cow. What part of the rent should each man pay ? 3. Ames and Howe entered into partnership the first of January, and each put in $ 2060. The first of May, Ames put in $ 1000 more. At the end of the year the profits proved to be 1 2800. What should each receive ? 4. Four men hired a pasture containing 280 acres at $ 1.25 per acre. A pastured 125 sheep ; B, 150 ; C, 200 ; D, 225. How much rent ought each to pay ? 298 EXAMINATION QUESTIONS. 5. A, B, and C gain $ 2250. A's gain is $ 800, and B's $ 1000. C's capital is $3000. What is the stock of each ? 6. Jan. 1, 1880, three persons began business with $ 1300 furnished by A. March 1, B put in $ 1000 ; August 1, C put in $900. The profits were, at the end of the year, $750, Find the gain of each partner. 7. Two partners engaged in business. One furnished f of the whole capital, and the other furnished $ 4000. They gained in trade 20 per cent of their capital, but lost $ 500 from bad debts. What was each partner's share of the net gain ? 8. Three persons engage in trade with a joint capital of $ 37680. A puts in $ 6 as often as B puts in $ 10, and as often as C puts in $ 14. Their annual gain is equal to C's stock. What is each partner's gain ? 9. A can do a piece of work in 6 days, B can do it in 8 days, and C in 10 days. In what time can they do it by working together ? 10. A man owes A $ 1800, B $ 750, and C $ 1950. His as- sets are : money on hand, $ 205 ; a horse and carriage valued at $ 260 ; and a stock of goods for which he is offered $ 1200. What per cent can he pay, and what will A, B, and C each receive ? ROOTS AND MENSURATION. 452. 1. Extract the square root of 925444. 2. va369 + Va296 = ? 3. If a line 160 feet long will reach from the top of a steeple 130 feet high to the opposite side of the street, what is the width of the street ? 4. If it cost $ 312 to enclose a field 216 rods long and 24 rods wide, what will it cost to enclose a square field of equal area with the same kind of fence ? 5. What is the difference between the cube and the cube root of 0.008 ? EXAMINATION QUESTIONS. 299 6. In the center of a square garden there is a pond covering an area of 810 square feet, which is a tenth of the whole gar- den. How many rods of fence will enclose the garden ? 7. Extract the cube root of 34012224. 8. A rectangular piece of land is 800 meters long and 600 meters wide. If you walk at the rate of a hektometer in IJ minutes, how long will it take you to walk from one corner to another, diagonally across the piece ? 9. What are the dimensions of a cube that has- the same volume as a box 12 feet 6 inches long, 10 feet wide, and 5 feet high?' 10. The altitude of a pyramid having a square base is 80 feet ; the length of each side of the base is 120 feet. Required its slant height, surface, and contents. 453. 1. If it costs $ 75 to paint a house 45 ft. long, what will it cost to paint a similar house 60 ft. long ? 2. The pedestal of a certain monument is a cubical block of granite containing 373248 cubic inches. What is the length of one of its sides ? 3. Two flagstaffs, one 80 feet high and the other 116 feet high, stand 160 feet apart on a horizontal plain. What is the distance between their tops ? 4. In a square field of one acre, if a man mow a space one rod in width around it next the inner border, how many square rods does he mow ? 5. A and B start from the same point, each walking 12 hours per day. A walks due north, at the rate of a mile in 15 minutes, while B walks due east, at the rate of a mile in 12 minutes. How far apart will they be at the end of the fourth day ? 6. Find the entire surface and the diagonal of a cube con- taining 262144 cubic inches. 7. How many globes 4 inches in diameter will it take to equal in volume a globe 12 inches in diameter ? 300 EXAMINATION QUESTIONS. 8. How much more ground will a board 20 ft. long, 2 ft. wide, protect from rain falling vertically, when it is flat on the ground, than when one end is raised 12 ft. ? 9. Two rafters, each 35 feet long, meet at the ridgepole of a roof 15 feet above the attic floor. What is the width of the house ? 10. I have a square garden, 100 feet on a side ; I wish to enclose the garden by a ditch 4 feet wide. How deep must the ditch be dug that the earth thrown out may raise the whole surface of the garden one foot ? MISCELLANEOUS. 454. 1, Eeduce to a simple fraction ^ ^ — TqT' Z4:^ — lof 2. What decimal of J of a ton is | of a cwt. ? 3. A man bought a house for $ 2000, and sold it for 25 % more than he paid for it and 12 J % less than he asked for it. What did he ask for it ? 4. On a note for % 400, at 7 %, there was paid % 100 annually for 3 years. How much remained due 3 years 4 months from the date of the note ? 5. Required the bank discount and the proceeds of a note for % 1250, due in 90 days at 7 %. 6. Eequired the time of day, provided the time past noon equals § of the time to midnight. 7. Find the cost of raising the surface of \ of an acre 9 inches, at 50 cents per cubic yard of earth. a It costs $202.80 to enclose a field 108 rods long and 48 rods wide, what will it cost to enclose a square field of equal area with the same kind of fence ? 9. What is the cube root of 74088 ? 10. How many days will 21 men require to dig a ditch 80 feet long, 8 feet wide, and 4 feet deep, if 7 men can dig a ditch 60 feet long, 6 feet wide, and 3 feet deep in 12 days ? EXAMINATION QUESTIONS. 301 455. 1. How much cloth J of a yard wide will cover 27 tables 6 ft. long and 3 ft. 2 in. wide ? 2. What is the amount at compound interest of $ 200, at 8 per cent, for 2 y. 6 mo. 6 d. ? 3. What is the length of one side of a square farm contain- ing 102 A. 64 sq. rd. ? 4. What is the length of a cubical block of granite contain- ing 47|f I cubic feet ? 5. Divide 600 by .012, multiply the .quotient by .05, and divide .005 by that product. 6. The first term of a proportion is (4f -^ .03), the second is 6 1, and the fourth is 8J-. E/Cquired the third term. 7. Bought a hogshead of wine for $ 420 ; but 10^ gallons having leaked out, how must I sell the remainder per gallon to gain 25 per cent ? 8. What is the interest of $ 1200, at 4J per cent, from Aug. 18, 1880, to April 30, 1882 ? 9. If 25 men, in 6 days of 10 hours each, build 200 rods of wall, how many rods will be built by 12 men in 5 days of 8 hours each ? 10. My friend lends me $ 7000 for 15 days. How long must I lend him $ 7500 to requite the favor ? 456. 1. In the number 78.342, the value of the 7 is how many times as great as the value of the 2 ? 31 .2. From -^ of 12i take 4f. 3. A merchant who owns .36 of a ship sells 85% of his share for $ 22950. What is the value of the whole ship at that rate ? 4. The capital stock of a company is $ 80000 ; its gross earn- ings annually are 1 8000, and its expenses $ 3200. What per cent on his investment does a stockholder receive who pur- chased at 20 % above par ? 5. How much more is the compound than the simple inter- est on $ 625 at 8 % for 2 years 6 months ? 302 EXAMINATION QUESTIONS. 6. Find the date when due, the bank discount, and the proceeds of a note of $ 1250, dated Feb. 12, 1881, payable in 90 days. 7. If 6 men can reap 14 acres of wheat in 2 days of 10 hours each, in how many days of 8 hours each can 5 men reap 25 acres ? 8. The length of a rectangular field containing an acre is twice its width. What is the length of its diagonal ? 9. Find the solid . contents of a cone whose height and diameter are each 12 inches. 10. The length of a meter being known, how can the length of the earth's diameter be ascertained ? 457. 1. I hire $ 200 for 1 y. 8 mo. 18 d. at 8 %, at simple interest, and loan it at compound interest. Required my gain. 2. If 8 men in 4 days of 12 hours each can mow 18 acres of grass, how many acres can 6 men mow in 10 days of 9 hours each ? 3. Bought a house for $ 10000 gold, and sold it for $ 15000 currency. If gold at time of selling is worth 125, what is my gain per cent ? 4. How many gross of tacks can be bought for $ 12, if each tack cost .02 of a mill ? 5. What will be the cost of a close fence, 5 J ft. high, around a field 14 rd. long, 12 J rd. wide, the boards costing $42 per thousand ? 6. Bought a piece of cloth containing 72 yd. for $ 280 less 10 %, and sold it at $ 4.50 per yd., receiving in payment a note on 90 days, which I immediately have discounted at a bank at 6 %. Do I gain or lose, and how much ? 7. If a bullet J inch in diameter weighs IJ oz., what will be the weight of a cannon-ball 7 inches in diameter ? 8. What is the area of one side of a cubical block contain- ing the same number of cubic feet as one tliat is 27 ft. by 8 ft. by 125 ft. ? EXAMINATION QUESTIONS. 303 9. Sold 20 loads of hay, each weighing 16 cwt. 40 lb., at $ 35 per ton. How long must what I receive be on interest, at 8 % simple interest, to amount to as much as would have been received for 20 tons ? 10. I paid $ 4.50 for a line that would just reach from the top of a spire to the ground, at a distance of 56 ft. from the foot of the spire ; price being $ 0.50 per lb., allowing 18 ft. to a pound, what was the height of the spire ? 458. 1. After ^ per cent of a flock of sheep had been killed by dogs, and 68 had been sold to a butcher, ^ of the original flock were left. Required the number of sheep in the flock. 2. A pile of 4-foot wood 256 ft. long and 5 ft. high was sold for $ 152M, What did a cord cost ? 3. The floor of a public house, 56 by 85 feet, is of boards 14 ft. long, 6 in. wide. There are 8 nails in each board. Allowing 68 nails to a pound, how many pounds are in the floor? 4. Divide $ 1000 between two children aged, respectively, 6 years and 8 years, in proportion to the square of their ages. 5. If you hire $1200 for 1 y. 8 mo. 19 d., at 8 per cent simple interest, and lend it for the same rate and time at compound interest, payable semi-annually, how much do you gain ? 6. The square root of a number is 2304. What is its cube root? 7. A |-in. faucet fills a cistern in 3 hours. How long will a Ij-in. faucet require ? 8. Exchange is 2 per cent below par. How large a 60-day draft can you buy for $ 1939, the rate of discount being 6 per cent? 9. At what rate will $ 508.50 earn $ 89.609 in 2 y. 2 mo. 13 d.? 10. Sent my agent $ 7315 with which to buy apples, after deducting 4J per cent commissioUc What per cent is his com- mission of $ 6300 ? 304 EXAMINATION QUESTIONS. 459. 1. What would be my annual income on $ 10000, in- vested in 5 per cent bonds at 5 per cent premium ? 2. What would be the cost of a section of U. S. land con- taining 640 acres at $ 1.50 per centare ? 3. Calling the earth's diameter 7912.5 miles, what is the 'ength in miles of a minute on the equator ? 4. Give two ways of changing the form of a fraction witb' out changing the value, and explain why the value is not changed. 5. What will it cost to carpet a room 18 by 15 ft. with car-^ peting 27 in. wide at $ 2.50 per yard ? 6. Two men starting at different points travel till they meet ; one finds his watch 70 minutes fast, the other 120 minutes slow. How far, and in what direction, does each go ? 7. Sold two farms for $ 3000 each ; on one I make 16 1 per cent, and on the other I lose 12 per cent. Find the net gain or loss. 8. How much must be invested in U. S. 4's at 120 to pro- duce a semi-annual income of $ 960 ? 9. A man having a field 40 rods square sold to A 100 square rods, to B 4 acres, and to C 20 rods square. How much re- mained unsold ? 10. What per cent of the square root of 21316 is the cube root of 150568768 ? 460. 1. What will be the cost of fencing a section of land at 12 4^ cents per foot ? 2. If 40 yards of tapestry will carpet a room 18 feet long,- what is its width, tapestry being | yd. wide ? 3. Reduce to a simple fraction .\ — . 18i — 12f 4. What is the length of a pile of wood containing 12 cords if it^ width is 9 feet and its height 6 feet ? EXAMINATION QUESTIONS. 305 5. A merchant bought 45 pieces of cloth, each piece con- taining 30 yards, at $ 3.75 per yard, on 9 months' credit, and sold the same immediately at $ 4 per yard, on 4 months' credit. What was his gain at date of purchase ? 6. Kequired the proceeds of a note for $ 4500, dated June 15, running 4 months, and discounted August 18 at 7J%. 7. What sum of money will amount to $520 in 3 years 9 months, at 4 % ? 8. If 6 men in 16 days of 12 hours each build a wall 15 feet long, 8 feet high, and 3 feet thick, how many men will be re- quired to build a wall 45 feet long, 9 feet high, and 6 feet thick, in 24 days of 9 hours each ? 9. What is the distance, in rods, from the center to each corner of a section of land ? 10. How many cubic feet of water must be drawn from a reservoir 24 feet 6 inches long, and 20 feet 9 inches wide, to lower the surface 8 inches ? 461. 1. Write a note due in 90 days, dated March 30, 1881, and signed by John Smith, on which Wm. White can obtain $ 850 at a bank, the bank discounting at 8 per cent. 2. Find the difference between 3J divided by .31, and .31 divided by 3^. 3. For what must hay be sold per ton to gain 18|: per cent, if by selling at $ 46 per ton 25 per cent be gained ? 4. At $ 50 an acre, what will be the value of a field which can be divided into 12 lots, each 80 feet square ? 5. How many reams of paper will be required to make 32 octavo books, each containing 348 pages ? 6. A commission merchant receives $ 720 to be expended in butter, reserving his commission of 2J per cent on the amount expended. He pays 37 cents per pound. How many pounds can he buy ? 7. A sells cloth at a profit of 9 per cent ; B sells at a profit of 7 per cent, but sells 5 yards while A is selling 3. Which will make a greater profit on a capital of $ 1000 ? Give the Teason for your answer. 306 EXAMINATION QUESTIONS. 8. There is a fence enclosing a circular field 48 feet in diam- eter. What will be tlie area of a square field which the same fence will exactly surround ? 9. A loans $ 325.50 for 3 y. 4 mo. 24 d., and recei\'es for in- terest $ 77.469. B loans $ 1.80, and his money amounts in the same time to $2,259. Which receives the greater per cent? 10. Find the difference between the square root and the cube root of .0064 462. 1. What will 37^ bushels of salt cost, if 13f bushels cost $ 11.75 ? 2. How many stones 10 inches long, 9 inches wide, and 4 inches thick, will it take to build a wall 80 feet long, 20 feet high, and 2^ feet thick ? 3. If you buy eggs for 12|^ cents, and sell them for 15 cents per dozen, for how much must wood be sold, that cost you $4.20 per cord, to gain in the same ratio ? 4. What must I pay for 25 shares of bank stock, at 7f per cent advance, the par value being $ 250 per share ? 5. For how much less than its face should a note of $ 840 be sold, if payable in 4 months ? 6. What is the value of a lot of land in the form of a right- angled triangle, the base being 15 yards and the hypothenuse 25 yards, at 8J cents per square foot ? 7. In a school-room 35 feet long, 30 feet wide, and 12 feet high, there are 40 pupils, each breathing 10 cubic feet of air per minute. In what time will the air be unfit for respii ation, if no pure air be admitted ? 8. A man bought 3 loads of hay. The first load weighed 4832 lb., tare 1124 lb. ; the second weighed 4628 lb., tare 1136 lb. ; the third weighed 4976 lb., tare 1142 lb. What did it cost him at $ 33.37 i per ton ? 9 A cargo of 4000 bushels of wheat, worth $ 1.20 per bushel, is insured at -J of 1 J per cent on § of its value. If the cargo be lost, how much will the owner of the wheat lose ? EXAMINATION QUESTIONS. 307 10. Mr. B. mortgaged his farm for $ 6000, Oct. 1, 1879, to be paid in 6 years, with interest at 8 per cent. Three months from date he paid I 500 ; Sept. 10, 1880, $ 1126 ; March 31, 1881, $ 2000; and Aug. 10, 1881, $876.50. How much was due at the expiration of the time ? 463. 1. What cost a piece of land 70 rods 5| feet long, 52 / rods 8 J feet wide, at $ 256 per acre ? 2. What is the interest of $ 725 for 9 months 4 days, at 5 per cent ? 3. Bought a farm for $ 4200, agreeing to pay $ 600 down and the rest in six equal semi-annual installments. When could I justly make one payment for the whole ? 4. A chimney 5 ft. square and 50 ft. high has two flues, each 1 ft. square. How many bricks, 8 in. by 4 in. by 2 in., were used in its construction ? 5. Having used my carriage three years, I am willing to sell it at a loss of 20 per cent. If I receive $ 250 for it, what was the cost ? 6. What is the circumference of the largest wheel that can be got through a door 5 ft. wide and 12 ft. high ? 7. A, B, C, and D hired a pasture for $ 75. A put in 7 cows, 6 weeks ; B 3 cows, 13 weeks ; C 4 cows and 9 sheep, 7 weeks ; D 42 sheep, 5 weeks. A cow would eat as much as 3 sheep. What should each pay ? 8. Divide $ 3000 among A, B, and C, giving A $ 50 more than B, and B $ 250 more than C. 9. If a cistern 6 ft. in diameter hold 80 barrels of water, what must be the diameter of a cistern of the same depth to hold 1280 barrels ? 10. A can do a piece of work in | of an hour ; B can do | of it in one hour. In what time can both do it ? 308 APPENDIX. APPENDIX. ROMAN NOTATION. 464. Roman Notation uses seven capital letters in ex- pressing numbers : I, V, X, L, C, D, M. 1, 5, 10, 50, 100, 500, 1000. All other numbers are expressed by repeating or combining these letters, according to the following 465. Principles. 1. Repeating I, X, C, or M, repeats its value. Thus, I stands for one ; II, two ; III, three ; X, ten ; XX, twenty ; XXX, thirty, etc. 2. When I is placed before V or X, X before L or C, or C before M, the difference of their values is expressed. Thus, IV stands for four ; IX, nine ; XL, forty ; XC, ninety. 3. When a letter is placed after another of greater vc^lue^ the sum of their values is expressed. Thus, VI stands for six; XI, eleven; XV, fifteen; XXV twenty-five, etc. APPENDIX. 309 4. A dash ( — ) over V, X, L, C, D, or M, increases the value of the letter a thousand fold. Thus, V stands for five thousand ; X, ten thousand ; L, fifty thousand. 466. TABLE. I 1 XIV 14 xo 90 II 2 XV 15 c 100 III 3 XVI 16 cc 200 IV 4 XVII 17 ccc 300 V 5 XVIII 18 CD 400 VI 6 XIX 19 D 500 VII 7 XX 20 DC 600 VIII 8 XXX 30 M 1000 IX 9 XL 40 MD 1500 X 10 L 50 C 100000 XI 11 LX 60 M 1000000 XII 12 LXX 70 MDCXX 1620 XIII 13 LXXX 80 MDCCCLXXXI 1881 Express infi gures : 1. XX] ax. 4. XMCCXXII. 2. LX} CXIII. 5. MMDXLIV. 3. CD] ax. 6. MDCCCXCVIII. Express in h itters : 7. Twenty-i line. 10. One hundred sixty-one. 8. Seventy- three. 11. Fifteen hundred eighty. 9. Ninety-e ight. 12. Two thousand eight hundred. 13. One thousand seven hundred seventy-six. FUNDAMENTAL. PROCESSES, 467. The Fundamental Processes of arithmetic, or those upon which all others depend, are addition, subtraction, mul- tiplication, and division. These are also sometimes called the ground rules of arith- metic. 310 APPENDIX. PRIME NUMBERS. 468. Ko direct method of detecting prime numbers has been discovered. The prime numbers to 1009 are included in the following TABLE. 1 59 139 233 337 439 657 653 769 883 2 61 149 239 347 443 663 669 773 887 3 67 151 241 349 449 569 661 787 907 6 71 157 251 363 457 571 673 797 911 7 73 163 257 359 461 677 677 809 919 11 79 167 263 367 463 587 683 811 929 13 83 173 269 373 467 693 691 821 937 17 89 179 271 379 479 599 701 823 941 19 97 181 277 383 487 601 709 827 947 23 101 191 281 389 491 607 719 829 953 29 103 198 283 397 499 613 727 839 967 31 107 197 293 401 603 617 733 863 971 37 109 199 307 409 609 619 739 857 977 41 113 211 311 419 521 631 743 859 983 43 127 223 313 421 623 641 761 863 991 47 131 227 317 431 641 643 757 877 997 53 137 229 331 433 547 647 761 881 1009 CIRCULATING- DECIMALS. 469. A Circulating Decimal is a decimal in which a figure or a set of figures continually repeats. Thus, § = 0.666 ... and T^j. = 0.727272 . . . 470. A Repetend is the part of the decimal that continu- ally repeats. It may be marked, when a single figure repeats, by a dot (.) over it, and when a set of figures, by a dot over the first and the last of the set. Thus, APPENDIX. 311 0.666 . . . = 0.6, read repetend 6 j and 0.727272 ... = 0.72, read repetend 72. 471. A Pure Circulate has in the decimal no figure but the repetend. Thus, 0.93 is a pure circulate. 472. A Mixed Circulate has in the" decimal a part before the circulate. Thus, 0.936 is a mixed circulate. 473. A Circulating Decimal arises from the reduction to a decimal of a common fraction whose denominator contains other prime factors than 2 or 5. 14. Express the circulate 0.93 as a common fraction in its lowest terms. 0.93 X 100 = 93.93 . . . Solution. — As the repetend has 0.93 X 1 = 0.93 . . . ^wo places of figures, we multiply JTaXT; ^ _ ^ it by 100, and have 93.93. Sub- tracting from this once 0.93 gives 0.93 = _- = — . ^ number with the same figures as 99 oo the circulate, but which do not re- peat. Thus, 99 times the circulate = 93, and once the circulate must beff, orfj. That is, 474. A repetend is equal to a common fraction whose nu- merator is the figures of the repetend, and the denominator as many 9'5 as there are figures in the repetend. 15. Express the circulate 0.093 as a common fraction in its simple form. Solution. — 0.093 = 0.09|- =.^ = — , ^ 100 75 Beduce to common fractions in their simplest form : 16. 0.753. 18. 0.425. 20. 0.135. 17. 0.594. 19. 7.345. 21. 53,00243. 312 APPENDIX. 22. What decimal will express the difference between 29.259 and 25.047 ? 23. What decimal will express the product of 5.9 by 0.08 ? 24. What decimal will express the quotient of 4.27 divided by 0.42 ? . TABLES. 475. Surveyors' Measures. 7.92 inches are 1 link, 1. 25 links " 1 rod, rd. 4 rods " 1 chain, ch. 80 chains " 1 mile, mi. 10 square chains are 1 acre. Note. — A Gunter's Chain is the unit of measure, and is 4 rods, or 66 feet long, and consists of 100 links. 476. Mariners' Measures. 6 feet are 1 fathom. 120 fathoms " 1 cahle-length. 1\ cable-lengths " 1 mile. 1.15 common miles are 1 nautical mile. 3 nautical miles " 1 marine league. 477. Apothecaries' Measures. Weight. Liquid. 20 grains are 1 scruple, 9. 3 scruples " 1 dram, 3 8 drams " 1 ounce, S . 12 ounces " 1 pound, ft. 60 minims ("it\^) are 1 fluid drachm, f. 5 8 fluid drachms '* 1 fluid ounce, f § . 16 fluid ounces " 1 pint, O. 8 pints " 1 gallon, Cong. APPENDIX. 313 478. M^'iceUaneous. Blue grass seed Timothy seed . Clover seed Wheat bran . 14 lb. = 1 bu. 45 lb. = 1 bu. 60 lb. = 1 bu. 20 lb. = 1 bu. Com or rye meal . 50 lb. = 1 bu. Corn or rye . . 56 lb. = 1 bu. Oats . . . Barley . Wheat . . Beans . . Castor beans Potatoes 32 lb. = 1 bu. 48 lb. = 1 bu. 60 lb. = 1 bu. 60 lb. = 1 bu. 46 lb. = 1 bu, 60 lb» = 1 bu. 100 pounds dry fish are 1 quintal, 196 pounds flour " 1 barrel, 200 pounds beef or pork " 1 barrel. 25. The distance between two places, as measured by a sur* reyor, is 120 chains 75 links. How far are they apart in miles ? 26. A rectangular field is 30 chains 25 links long, and 25 chains 40 links wide. How many acres are there in the field ? 27. A piece of land is in the form of a right-angled triangle. The sides forming the right angle measure 14 chains 50 links and 24 chains 20 links. How many acres in the piece ? 28. When a vessel at sea has made 500 nautical miles, how many common miles has she gone over ? 29. How many prescriptions of 20 grains each can be put up from 1 lb 8 i 2 3? 30. How many minims are there in 12 fluid ounces ? 31. What is the value of a car-load of corn weighing 20000 pounds at 63 cents a bushel ? 32. How much is niade by buying 4 barrels of clear pork at $ 19.50 a barrel, and retailing it at 14 cents a pound ? GOVERNMENT LANDS. 479. Government Lands are divided by parallels and me- ridians into Townships, 6 miles square, each containing 36 Sections, or square miles. Each section is subdivided into half-sections and quarter-sections. 314 APPENDIX. 480. The Townshijos are numbered witli reference to two special lines, the one running east and west, called the base line, and the other running north and south, called the prin- Clival meridian. Thus, A township in the fourth tier of townships north of the base line, and the second in that tier west of the principal meridian, would be designated Township 4 iV*., Range 2 W. 481. The Sections into which a township is divided are designated by numbers beginning with the northeast corner section, and running westward with the north tier of sections, eastward with the second tier, and so on. Thus, A section in the third tier from the north in a township and the fourth toward the west in the tier would be designated section 16. 33. A section is in the fourth tier from the north in a town- ship and the third toward the west in the tier. What is its number ? 34. Draw a plan of a township laid off into 36 sections, and designate by number each section ? LONGITUDE AND TIME. 482. Longitude is distance east or west from a given meridian, measured in degrees, minutes, and seconds. The Prime Meridian is the meridian from which the longi- tude is measured. 483* When two places are on the same side of the prime meridian, their difference of longitude is found hy subtracting the lesser from the greater longitude ; when on opposite sides, by adding the longitudes. Note. — If the sum of the longitudes exceeds 180°, it must be sub- tracted from 360°, since it is impossible for the difference in the longitude of two places to be more than 180°. 484* The earth, by turning upon its axis once in 24 hours, causes ^ of 360"^, or 15° of longitude, to pass under the sun in 1 hour (Art. 179;, and ^ of 15°, or 15', in 1 minute of time, and ^ of 15', or 1§", in 1 s^^op^ 9^ ^^™®* APPENDIX. 315 t 485. TABLE. A difference of 15° in long, gives a difference of 1 h. in time. " 15' in long. " " 1 min. in time. " 15" in long. ** " 1 sec. in time. Note. — The New Standard Time, adopted in the United States, since 1883, by general agreement, is based upon four standard meridians, 15° apart, giving a difference of just one hour in time from one meridian to the next, and the time of each meridian extending to 7|° on each side of it. The Eastern Standard Time is based upon meridian 75° west from Greenwich, England, which passes near Philadelphia; the Central Standard Time, upon meridian 90° west, which passes nearly through New Orleans and St. Louis; the Mountain Standard Titne, upon meridian 105° west, which passes near Pike's Peak and Denver ; and tlie Pacijic Standard Time, upon meridian 120° west, which passes 9^' east of San Francisco. 486. From the table is derived the Rule. Divide the difference of longitude, in degrees, minutes, and seconds, hy 15, and the quotient will give the difference o/ time in hours, minutes, and seconds. Multiply the difference in time, in hours, minutes, and sec- onds, hy 15, and the product will give the difference in degrees^ minutes, and seconds. 35. When it is noon at New York, 74° 0' 3^' W., what time is it at San Francisco 122° W 16" W. ? 122° 26' 15'' Solution. — Both cities being ^4° Qf ^if west of Greenwich, their dif- ■iK\~A^ ^f ^, ference of longitude is found ^^^ — — — . by subtraction. As 15°, 15', 15" 3 h. 13 min. 45 sec. • ^^ake a difference of 1 h., 1 min., 12 h. min. sec. i sec. of time, respectively, 3^ 8 h. 46 min. 15 sec. of the difference in longitude considered as hours, minutes, and seconds, or 3h. 13 min. 45 sec, will be the difference in time. San Francisco, being the more westerly, has earlier time than New York. Hence, when it is 12 M. at the latter city, it is 3 h. 13 min. 45 sec, before 12 M. at San Francisco, or 8 o'clock 46 min. 15 sec. a, M, 316 APPENDIX. 36. What is the difference of time between Greenwich and Washington, 77° 2' 48^' west ? 37. Paris is in 2° 20' 15^' east longitude, and Boston Tl"" 4' 9'' west. When it is 6 o'clock in the morning in Boston, what time is it in Paris ? 38. How many degrees east of Greenwich is E-ome, whose time is 49 min. 48}f sec. later than that of Greenwich ? 39. The difference of time between Baltimore and New Orleans is 53 min. 30 sec. What is the difference in longitude ? 40. A gentleman traveling west from Augusta, Maine, in longitude 69° 50' west, found, on arriving at Des Moines, that his watch, which was right when he left Augusta, was 1 h. 35 min. 20 sec. faster than the time at Des Moines. What is the longitude of Des Moines ? LEG-AL INTEREST. 487. The Legal Rate of interest is the rate established by law. 488. When no rate is mentioned the legal rate is that given in the left-hand column ; and if specified in writing, any rate not exceeding that in the right-hand column is legal. TABLE. state. Rate. Alabama... 8 8 Arkansas 6 10 California 7 Any Colorado... 10 Any Conn 6 6 N. Dakota 7 12 Delaware 6 6 Florida ... 8 Any Georgia ... 7 8 Illinois ... 6 8 Indiana ... 6 8 Iowa 6 10 Kansas . . . 7 12 Kentucky 6 8 State. Rate. Louisiana 5 8 Maine 6 Any Maryland 6 6 Mass 6 Any Michigan 7 10 Minnesota 7 10 Mississippi Missouri... 6 10 6 10 Montana 10 Any Nebraska 7 10 Nevada ... 10 Any N. H 6 6 N.J 6 6 N. Mexico 6 12 State. N. Y N. C Ohio Penn R. I S. C Tennessee Texas S. Dakota Vermont Virginia .. Washington W. Va. .. Wisconsin Rate. Any 10 6 12 12 6 8 Any 6 10 APPENDIX. 317 TWELVE PER CENT INTEREST. 489. The Twelve per cent method of computing interest is often the most convenient. At 12 % 1 year's interest = .12 of the principal. " " T^ y-5 or 1 month's, interest = .01 " " *^ " ^0 ^-y or 3 days', interest = .001 " " 490. Hence, to find interest at 12 %, Multiply .01 of the jpTincij^al by the time in months. Or, Multiply .001 of the principal by J of the time in days, 41. What is the interest of $ 825 for 2 y. 7 mo. 7 d. at 4 % ? $825 =: Principal. ) 8.25 = 1 mo's interest. 31.2J^ = Time in months. Solution. — 1 month's in- terest at 12% is yf^ of the ^'^^ principal, or $8.25; 31.2J 1650 months' interest is 31.2 J X 825 $8.25, or $257,675. The 2475 interest at 4% = ^ of 12% 3) $ 257.675 = 12 % interest. interest, or $ 85.89. $85.89 ==4% interest. 42. Find the interest of $ 840 for 73 days at 10 %. $ 840 = Principal. $ 0.84 = 3 days' interest. 24^ = \ time in days. 336 168 12) $ 20.44 = 12% interest. $ 1.703 = 1 % interest. $ 17.03 = 10 % interest. 318 APPENDIX. 491. Connecticut Rule for Partial Payments. TVhen at least a yeai^^s interest has accrued at the time of a payment, and in the case of the last payment, follow the Uiiited States Rule. When less than a year's interest has accrued at the time of a payment, except the last, find the difference between the amount of the principal for an entire year, and the amount of the pay^ TTient for the remainder of the year after it is vfiade, for a new ^principal. When the interest which has accrued at the time of a pay- ment exceeds the payment^ find the interest upon the principal only. Note. — At the option of the teacher, the exercises under the United Sf^ates Rule (Art. 282), may be performed by this rule. ANNUAL INTEREST. 492. Annual Interest is interest payable annually. The annual interest not paid when due draws simple intei est. 43. What is the amount due on a note of % 500, interest payable annually, on which no payments have been made, at the end of 3 years 6 months and 12 days ? Solution. Principal $500.00 Ist annual interest $30.00 Int. on 1st annual int. 2 y. 6 mo. 12 d. . $ 4.56 2d annual interest 30.00 [nt. on 2d annual int. 1 y. 6 mo. 12 d. . 2.76 3d annual interest 30.00 Int. on 3d annual int. 6 mo. 12 d. . . .96 4th annual interest 16.00 S 500.00 $106.00 $8.28 Total interest, $ 106 + $ 8.28 = 114.28 Amount due $ 614.28 APPENDIX. 319 493. Rule for Annual Interest. Compute the interest annually, and simple interest on each annual interest for the time it shall remain unpaid, 44. A debt of I 600 was contracted Jan. 1, 1880 ; allowing interest annually at 6%, and if no payments he made, what will be due April 1, 1884 ? 45. A note was given May 16, 1881, for $ 1250, interest an- nually at 5 % ; if no payment is made, what will be due March ^.6, 1884 ? 46. A note was given March 14, 1880, for $ 576, interest annually at 6 %. What will be due Sept. 26, 1883, no pay- ments having been made ? 494. When Partial Payments have been made on a note, or other obligation drawing annual interest, the following is The New Hampshire Rule. If in any year, reckoning from the time the annual interest began to accrue^ payments have been made, compute interest upon them to the end of the year in which they are made. The amount of payments is to be then applied, first, to cancel interest upon annual interest ; second, to cancel annual interest ; and thirdly, to the extinguishment of the principal. If however, at the date of any payment there is no interest except the accruing annual interest, and the payment, or pay- ments, do not exceed the annual interest at the end of the year, deduct the payment, or payments, without interest on the same. Omit the last paragraph of this rule and it is the Vermont Rule. 47. What was due, July 1, 1881, on a note dated July 1, 1878, for % 1000, with 6 % annual interest, and on which was paid, Dec. 1,1879, $400? 320 APPENDIX. Solution. Principal $1000 1st annual interest $60.00 Int. on 1st annual int. 2 y $3.60 2d annual interest 60.00 $1000 $120.00 $3.60 Payment Dec. 1, 1879 . .$400.00 Int. on paym't, July 1, 188 14.00 AnVt of payin't July 1, 1880 $414.00 = 290.40 + 120.00 + 3.60 Principal, July 1, 1880, . . $ 709.60 3d annual interest . . . $42.58 Due July 1, 1881 . . $709.60 + $42.58 = $ 752.18 48. $2000. Manchester, N. H., April 1, 1873. On demand, I promise to pay Charles West & Son, two thousand dollars, value received, with interest annually. James Goddard. Payments: Sept. 19, 1875, $500; Dec. 3, 1879, $600; and Aug. 9, 1880, $ 775. What will be due May 19, 1883 ? 49. $5000. Plymouth, N. H., Jan. 13, 1874. On demand, we promise to pay to the order of John M. Monroe & Co., five thousand dollars, value received, with in- terest annually. Presby & Lord. Payments : Sept. 23, 1878, $ 2000 ; Feb. 19, 1880, $ 1500 ; May 29, 1881, $ 125 ; and June 11, 1883, $ 20. What will be due at settlement, Aug. 30, 1885 ? Note. — At the option of the teacher, each of the above exercises may be i^erfonned by the Vermont Rule. APPENDIX. &21 AVERAGE OF ACCOUNTS. 495. The Balance of an account is the difference between its debtor and creditor sides. 496. The Average of an account is the equitable time of the balance becoming due, or being entitled to interest. 50. What is the balance of the following account, and at what date should the balance begin to draw interest ? Pr. JOHN L. MARTIN. (Er. 1880. 1880. May 16 Mdse. on 60 d. 1300 May 20 Mdse. on 30 d. $200 June 3 u u 60 d. 50 July 19 u a 60 d. 200 July 1 u a 30 d. 150 DeMts. July 15, 300 X 26 d. = 7800 d. Aug. 2, 50 X 44 d. = 2200 d. July 31, 150 X 42 d. — 6300 d. $500 16300 d. 400 $ 100 balance. Solution. Credits. June 19, 200 X d. : Sept. 17, 200 X 90 d. : $400 Od 18000 d. 18000 d. 16300 d. 1700 -f- 100 = 17 d. 1700 d- June 19 — 17 days = June 2, average time. We select for convenience June 19, the earliest date at which any of the items of account become due, as the point of reckoning, and find the aggregate of the terms of credit of the credit items, with ref- erence to the selected date, to be equal to the credit of $ 1 for 18000 days, and the aggregate of the terms of credit of the debit items to be equal to the credit of $ 1 for 16300 days. Striking the balance, it appears, at the selected date, $ 100 subject to a credit equal to the credit of f 1 for 1700 days is in favor of John L. 322 APPENDIX. Martin. But the credit of $ 1 for 1700 days is equal to that of $ 100 for Yoo of 1700 days, or 17 days; hence the $ 100 was due in equity 17 days before June 19, or June 2. If, however, the balance of items and terms of credit had been both on the same side of the account, the balance would have been due af- ter, instead of before, the selected date. 497. Rule for Averaging Accounts. Select the earliest date at which any of the items of account becomes due, and therefrom reckon the terms of credit. Multiply each term of credit hy the number denoting the cor- responding item, and divide the balance of the sums of the 'products by the balance of the sums of the items of the account, and the quotient will be the average term of credit. When the balances are both on the same side of the account, the time must be added to the selected date, but subtracted from that date when the balances are on different sides. The same result may be reached by what is called the Interest Method. Thus, Compute interest upon each item for the days intervening^ between its becoming due and the earliest date at which any item becomes due. Di- vide then the balance of interest by the interest of the balance of items for QTie day, and the quotient will be the average term of credit. Note 1. — A convenient rate of interest in averaging accounts is 12 %, which is .01 of the principal for 30 days and .001 for 3 days. Note 2. — The note under the rule Art. 334 applies in averaging accounts. 51. Find when the balance of the following account averages due. Pr. WILLIAM HOLT. dr. 1881. June 20 July 10 Mdse. on 30d. " net $600.60 149.40 1881. July 15 Cash $650 APPENDIX. 323 52. Find the balance of the following account, and, allow- ing each item to be on 30 days, the time the balance becomes due. Pr. BRYANT AND ROBERTS. &X. 1881. Sept. 30 Oct. 15 Mdse. 11 $550 850 1881. Oct. 1 " 5 Mdse. $400 30 53. Find the time when a note for the balance of the follow- ing equated account should begin to draw interest. Due March 2 Dr. $600 Due March 14 Cr. $400 54. Find the face of a note which must be given for the bal- ance of the following account, and the date at which it should begin to draw interest. pr. JAMES GRIMSHAW. (Ir. 1881. 1881. Aug. 10 " 28 Mdse. on 4 mo. " " 6 mo. $200 200 Sept. 11 Dec. 10 1882. Cash. it $60 140 Sept. 1 " net 150 Jan. 29 . u 100 BUSINESS FORMS. RECEIPTS. 498. A Receipt is a written acknowledgment that money or other property of value has been received. Receipt for Payment on Account ■ $ 250. Boston, Oct. 19, 1881. Received from Henry F. Jordan Two hundred fifty dol- lars on account. Smith & Watson, 324 APPENDIX. Receipt for Rent. Received, Portland , May 5, 1882, from James Johnson^ Thirty-one dollars for rent of hoicse, No. 14 Austin Street, for month ending April 30, 1882. $ 31. Catharine A, Pettes. Receipt in Full. $ 26-Mr. Philadelphia, Aug. 1, 1882. Received from John Randall Twenty-six -^^q Dollars in full of all demands to date. S. A. Caswell & Co. ORDERS. 499. An Order is a written request to deliver money or goods to some person mentioned, or to his order, or to the bearer, on account of the person signing the request. Order for Money. Burlington, Dec. 12, 1881. Mess7's. Simmons & Son, Gentlemen : Please jpay S. P. Wright, or order, Seventy- five Dollars, and charge to our account. Reed, Pratt, & Co. Order for Goods. Manchester, Nov. 4, 1881. Mr. W. N. Goddard, Please pay to James Brewer, or order, Sixty Dollars in Goods from your store, and charge to the account of Charles Dole. DUE-BILLS. 500. A Due-Bill is a simple acknowledgment of a debt in writing. APPENDIX. 325 Due-Bill for Goods. Due, New York, Aug. 9, 1882, to H. L. Chase, Twenty- three -^^Q Dollars in goods from my store. $ 23^^. Henry (7. Carter, CHECKS. 501. A Check is a written order addressed to a bank by a person having money deposited, requesting the payment on presentation of a certain sum of money to a person named therein, or to his order. Bank Check. The National Bank of Commerce. S^. ^lOT^nc'na, Oi Oidci, Scad^n-aezf^en . /^ ^o//aU. TAXES. 502. A Tax is a sum of money assessed upon the person property, income, or business of individuals for public use. 503. A Poll Tax is a tax upon the person, a Property Tax is a tax on property, and an Income Tax is a tax on income. 504. Assessors are officers appointed to take an inventory of taxables, and to apportion the tax to be raised among the tax-payers. 326 APPENDIX. 55. A town is to be taxed $ 14562. The taxable property is $1146000. There are 540 polls, each assessed $1.50. What will be A's tax, whose property is assessed at $ 8500, and who pays one poll-tax ? Solution. $ 1.50 X 540 = $ 810, sum to be assessed on the polls. $ 14562 — $ 810 = $ 13752, sum to be assessed on the property. $ 13752 -f- ^ 1146000 = 0.012, or 12 mills on $ 1 of valuation. $ 8500 X 0.012 = $ 102, As tax on property. a 102 + $ 1.50 = $ 103.50, As entire tax. 505. Rule for Assessment of Taxes. Deduct the amount of the poll-taxes, if any, from the entire tax to he raised, and the remainder, divided by the value of the taxable property, will give the rate. Multiply the value of each individuaVs taxable property by the rate, and to the product add the poll-tax, if any, and the sum will be the individuaVs entire tax. Note. — In Massachusetts, the assessors "in each year assess upon the polls the state and county taxes authorized or required by law ; provided, however, that in case either of said taxes shall exceed in amount the sum of one dollar upon each poll, the excess above said amount and in every case the whole amount assessed for other purposes shall be apportioned upon property. . . . The state tax assessed upon poll and property and the county tax assessed upon poll and property shall each constitute an entire and indivisible tax." — CAop. 299 o/ the Acts of 1879. Computation of taxes may be facilitated by the construction of a table. Thus, if the rate on $ 1 is 12 mills, we can have the following TABLE. Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. $1 $0,012 $8 $0,096 $60 $0.72 $400 $4.80 $2000 $24.00 2 0.024 9 0.108 70 0.84 500 6.00 3000 36.00 3 O.O.'iG 10 0.12 80 0.96 600 7.20 4000 48.00 4 0.048 20 0.24 90 1.08 700 8.40 5000 60.00 5 0.060 30 0.36 100 1.20 800 9.60 6000 72.00 6 0.072 40 0.48 200 2.40 900 10.80 7000 84.00 7 0.084 50 0.60 300 3.60 1000 12.00 8000 96.00 APPENDIX. 327 56. What is B's tax, by the table, his valuation being $ 2545, and he paying one poll-tax of $ 1.75 ? 57. Find, by the table, C's tax on $ 9565, D's on $ 1764, and E's on $ 5630, and each paying a poll-tax of $ 1.50. 58. The taxable property of a certain town is $ 1000000. The number of polls, 600. The tax to be raised is. State $348, County $1500, and Town $12100. The state and county tax are each to be assessed upon the polls, but so much as either of these taxes shall exceed $ 1 on a poll, is, with the town tax, to be assessed upon the property. What will be each poll-tax ? What will be the rate of county tax ? What will be the rate of town tax ? What will be A's county tax, his valuation being $ 5000, and he paying for one poll ? What will be his entire tax ? DUTIES, OR CUSTOMS. 506. Duties, or Customs, are taxes levied on imported goods and the tonnage of vessels. They are collected at custom-houses by the government officers in charge. 507. A Specific Duty is a fixed tax upon an article with- out regard to its value. 508. An Ad Valorem Duty is a tax at a certain rate on the cost of the goods in the country from which they are imported. 509. Tare is an allowance made for the weight of the box, cask, etc., containing the goods. It may be estimated by actual weighing, or by a schedule furnished by the government. Leakage is an allowance for waste of liquors in casks, and Breakage is an allowance on liquors in bottles. Gross Weight is the weight before any allowances are made, and Net Weight is the weight after the allowances are made. 328 APPENDIX. 610. Values of Foreign Coins, in United States money, to be followed in estimating values of foreign merchandise at custom houses, as proclaimed by the Secretary of the Treasury, January 1, 1890. TABLE. Country. Standard. Monetary Unit. Value in terms of U. S. Gold dollar. Argentine Repub. Austria Brazil British Poss. N. A. Chili Cuba Egypt France German Empire. . Great Britain .... India Japan Mexico Netherlands Norway Portngal Russia Tripoli Turkey Colombia Venezuela Gold and silver Silver Gold Gold Gold and silver Gold and silver Gold Gold and silver Gold Gold Silver Gold and silver Silver Gold and silver Gold Gold Silver Silver Gold Silver Silver Peso Florin Milreis of 1000 reis... Dollar Peso Peso Pound (100 piasters). Franc Mark Pound Sterling Rupee of 16 annas. . . (Gold $0,965 0.345 0.546 1.00 0.912 0.926 4.943 0.193 0.238 4.866^ 0.332 0.997 0.752 0.758 0.402 0.268 1.08 0.558 0.629 0.044 0.698 0.14 Y^^ ] Silver. .... .... Dollar Florin Crown Milreis of 1000 reis. .. Rouble of 100 copecks Mahbub of 20 piasters Piaster Peso Bolivar Note. The par of exchange of the monetary unit of a country with a gold, and gold and silver, standard is fixed at the value of the gold unit as compared with the United State? gold unit; and, in case of a single silver standard, the par is computed at the mean price of silver in the London market, from October 1, to December 24, 1889. The boliviano of Bolivia, the Sucre of Ecuador, the peso of Guatemala, Honduras, Nicaragua, Salvador, and Port a Rico, and sol of Peru^ silver standard, are each of the same value as the peso oiColombia. The franc of Belgium, the franc of Switzerland, the pesata of Spain^ the lira of Italy, and the drachma of Greece, gold and silver, are each of the same value as the franc of France. The crown of Sweden, and the crown of Denmark, gold, are each of the same value as the crown of Norway. 59. The Bay State Iron Co. imported from England 200 tons pig-iron, invoiced at £725 35. 4cZ., and paid a duty of $7 per ton. Find the duty and the cost of the importation. APPENDIX. 329 60. Jordan, Marsh, & Co. import from Paris 3 cases silk goods containing 2664.5 meters, invoiced at 6 francs per meter. They pay a duty of 60 %. What does the government receive ? 61. What is the duty on an importation from Russia of 516 bales of flax, weighing 7880 poods, at $20 per ton, a pood being a Russian weight of about 36 lb. ? 62. Spaulding, Nash, S^Co. receive from Cuba an invoice of 50381 gallons of molasses, valued at 11102.7 pesos. They pay a duty of 5/ per gallon and 25% ad valorem. How many dollars do they pay the collector of customs ? 63. R. H. White & Co. import from Germany a case of dress-goods, containing 640.4 meters, costing in Hamburg 2.05 marks per meter less a discount of 8 %. The width of the goods is 43| inches. The duty paid is 8 / per square yard and 40 % ad valorem. Find it. MEASUREMENT OF ROUND TIMBER. 511. Bound Timber (or logs) is usually estimated in cubic feet. 512. Spars from 10 to 4 inches in diameter, inclusive, are estimated by the inch diameter, taken clear of bark, at one third of their length from the larger end. Spars above 7 inches should have 4 feet of length, and below 7 inches should have 5 feet of length, to every inch of diameter. 513. The Mean Girt of a tapering piece of round timber is the girt, clear of bark, at one third its length from the larger end. 514. To find the contents of round timber in cubic feet. Multiply the length in feet hy the square of one fourth of the mean girt in inches^ and divide the product hy 144. Note. — The rule gives about a fifth less than the exact quantity, so much l»eing allowed for crooks and waste. 330 APPENDIX. 64. How many cubic feet of timber in a log whose length is 30 feet, and whose mean girt is 42 inches ? 65. The mean girt of a piece of round timber is 60 inches, and its length 24 feet. Required its contents in cubic feet. 515. To find the side of squared timber that may be hewr Dr sawed from a log, Multiply the diameter of the smaller end of the log by 0.707. 66. The smaller end of a log is 21 inches in diameter. What is the side of the squared beam that may be sawed from it ? 67. How many board feet in a log when squared, the length of the log being 18 feet, and its diameter at the smaller end 24 inches ? 68. A log is in the form of a cylinder, 6 feet in circumfer- ence and 20 feet long. What is the value of the largest squared timber that can be hewn from it, at $ 30 a thousand feet, board measure ? GAUGING. 516. Gauging is finding the capacity of casks. 517. The Mean Diameter of a cask is very nearly equal to the head diameter increased by 0.55 to 0.70 of the difference between the bung and head diameters, acrording as the staves are curved little or much. 518. To find the capacity of casks, Multiply the product of the square of the mean diameter and the lengtli, expressed in inches^ by 0.0034 for gallons^ or by 0.0129 for liters. As a cubic foot is about 1\ gallons, in finding the capacity of a cis- tern it is sufficiently accurate to estimate 7^ gallons to a cubic foot. APPENDIX. 331 69. A. cask whose mean diameter is 22 inches and length 30 inches will contain how many gallons ? 70. What is the capacity of a cask in gallons whose mean diameter is 30 inches and length 38 inches ? 71. What is the capacity in liters of a cask, staves much curved, whose head diameter is 24, bung diameter 30, and length 36 inches ? 72. How many gallons in capacity is a rectangular cistern whose inside dimensions are 4 feet 3 inches, 3 feet 6 inches, and 4 feet ? TONNAGE OF VESSELS. 519. The Tonnage of a vessel is the number of tons' burdeD it will carry. 520. Shipwrights generally make their estimates of ton- nage by the following Rule. For a single-deck vessel^ take the length in feet above the deck from the forepart of the mainstem to the after-part of the sternpost, the breadth at the widest part above the main wales on the outside, and the depth from the under side of the deck plank to the ceiling of the hold. From the length subtract three fifths of the breadth, multiply the remainder, breadth and depth, together, and the product divided-by 95 will give the tonnage. For a double-deck vessel, take the length above the upper deck, for the depth half the breadth, and proceed as before, 73. What is the tonnage of a single-decked vessel whose length is 75 feet, breadth 20 feet, and depth 9 feet ? 74. What is the tonnage of a double-decked vessel whose length is 160 feet and breadth 30 feet ? 332 APPENDIX. FARMERS' ESTIMATES. 521. Gram in a bin or granary occupies nearly 1:^ as many cubic feet as there are bushels. 522. Corn on the ear will yield about half its bulk in shelled corn. 523. Wheat, according to quality, less a sixth for toll, will yield from 26 to 33 pounds of flour per bushel. 524. Mixed Hay, in large mow, is estimated at 500 cubic feet, and Clover, at 550 cubic feet to a ton of 2000 pounds. 525. Horses, young cattle, and sheep are estimated to con- sume daily, for each 100 pounds of weight, about 3 pounds of hay ; and oxen and cows, about 2 J pounds. As food for stock, 100 pounds of average meadow hay is equal to about 56 pounds of corn, 56 pounds of wheat middlings, 60 pounds of oats, or 32 pounds of cotton seed meal. 526. Net Weight of fat beeves is about f of the live weight ; of fat swine, J ; of fat sheep, J ; and of fat fowl, /^. Average beeves, net weight, will cut up : rump and sirloin, ^ ; thigh and round, \ ; forequarter and rattlerand, f ; hide, ^^^ ; and tallow, ^^. An average swine, net weight, will cut up : hams and shoulders, ^^ and sides and clear pork, ^. 75. I have a bin 8 feet long, 4 feet wide, and 3 feet deep. How many bushels will it contain ? 76. A wagon 8 feet long, 3|- feet wide, and 2 feet deep, is filled with corn in the ear. How many bushels of shelled corn will it yield ? 77. How much average meadow hay will suffice to keep 3 horses 120 days, whose weight is 1200 pounds each, provided grain is allowed to replace one third of the hay ? APPENDIX. 333 78. How many bushels of best wheat must be carried to mill to get back, after allowing a sixth for toll, a barrel of flour ? 79. When corn is 75 cents a bushel, what is the correspond- ing value of average meadow hay as food for stock ? 80. The live weight of a fat ox is 1550 pounds. If slaugh fcered, how many pounds of his net weight will cut up into rump and sirloin, and how many into round ? 81. The live weight of 5 fat swine is 2250 pounds. How many pounds of the net weight will cut up into hams and shoulders, and how many into sides, or clear pork ? STONE AND BRICK WORK. 527. A Perch of stone or masonry is 16J feet long, 1^ feet thick, and 1 foot high, or 24 J cubic feet. 528. In Rubble masonry, a cubic yard laid in the wall re- quires IJ cubic yards of undressed stone, and ;|^ of a cubic yard of mortar. In Ashlar work, about ^ of the volume of the stone is al- lowed for mortar. A mason, with a helper, can in a day lay in courses 4 cubic yards of rubble stone dry, or 3 cubic yards in mortar. 529. Bricks when laid will average for each square foot of surface on the face of the wall about twice as many bricks ip number as the wall is inches thick. 530. A Cask of Lime is about 2 J bushels, or 240 pounds, and absorbs about 2 J times its bulk, or 2| times its weight, of water in slacking. A cask of lime, with about 10 bushels of sharp sand, will make mortar for laying about 1000 bricks, or, with the addi- tion of 5 pounds of hair, mortar for 35 square yards of plaster- ing, one coat work, or 30 square yards, two coat work slipped. 334 APPENDIX. 531. A Cask of Cement of 300 pounds, with twice its bulk of sharp sand, will make mortar for laying 650 bricks ; or, with four times its bulk, or about 12 bushels, of clean gravel, concrete for 9 square yards of flooring surface. 532. In Paving, about 40 bricks laid flatwise, or 75 bricks laid edgewise, are allowed for one square yard. A mason, with a helper, in a day can lay in mortar, 8-inch work, 1400 bricks, or 12-inch work, 2000 bricks ; and bricks flat in sand 20 Bquare yards, or in cement, 12 square yards. 82. What will be the cost of the material for mortar for plastering 900 square feet, the price of lime being $ 0.90 per cask, sand 8 cents per bushel, and hair 6 cents per pound ? 83. I have a walk 4 feet wide and 224 feet long. What will it cost to pave it with brick, laid flat in sand, brick $ 7.50 a thousand, wages of the mason $ 2.75 per day, and of a helper 1 1.50 per day ? 84. What will it cost to concrete the bottom of a cellar 40 feet long and 24 feet wide, cement being $ 2 a barrel and gravel 8 cents a bushel ? 85. What will it cost to build a 12-inch brick wall, 6 feet high and 100 feet long, laid in mortar, brick being $ 8 per thousand, lime $ 1.10 per cask, sand 10 cents per bushel, wages of the mason $ 3 per day, and of the helper $ 1.75 per day ? 86. A cellar is 34 feet long, 27 feet wide, and 9 feet deep. Its walls are IJ feet thick, made of rubble stones laid in mor- tar. The stone undressed cost $ 2.50 per perch, the lime cost $ 1 per cask, the sand used with it for mortar 10 cents a bushel, and a cask of lime and 10 bushels of sand made 15 cu- bic feet of mortar. The mason who built the walls was paid 1 3 a day, and his helper $ 2. What did the material and mason work of the cellar cost ? APPENDIX. * 335 BUILDERS' ESTIMATES. 533. Shingles are usually 16 inches long, and on an aver- age 4 inches wide, and are put up 4 bundles to the 1000. 1000 shingles, laid 4 inches to the weather, will cover 107 square feet ; laid 4J- inches to the weather, 120 square feet ; and laid 5 inches to the weather, 133 square feet. 534. Clapboards are usually 4 feet long, and put up 40 bundles to the 1000. 100 of 4-foot clapboards, laid 4 inches to the weather, will cover 130 square feet ; laid 4|- inches to the weather, 150 square feet ; and laid 5 inches to the weather, 165 feet. 535. Laths are usually 4 feet in length, and are put up 10 bundles to a 1000. loo laths, set J of an inch apart, will cover 6^ square yards. A workman in a day will set of laths about 100 square yards, lay of shingles on a roof about 2000, or put on of outside boards abou^ 1000 feet. 536. Nails are put up 100 pounds to a keg. 6 pounds of 4-penny, or 5 pounds of 3-penny, nails are allowed for laying 1000 shingles ; 3^ to 4^ pounds of 5-penny nails for laying 1000 clapboards ; 7 pounds of 3-penny nails for setting 1000 laths. 537. Paint for outside work may have for first coat 16J pounds of white lead, ground in oil, to a gallon of linseed oil ; and for second or third coat, 20 pounds of white lead to a gal- lon of linseed oil. For inside work, the spirits of turpentine may replace from one third to two thirds of the oil. 538. A Gallon of linseed oil weighs about 7| pounds, and a gallon of spirits of turpentine about 7 pounds. 336 APPENDIX. One pound of prepared white lead paint will cover of first coat about 4 square yards, and of subsequent coats from 4^ to 5^ square yards. A day's work for a painter is, of plain outside work, from 80 to 100 square j^ards, and of inside work, from 40 to 65 square yards. 87. Each of the two sides of the roof of a certain building is 34 feet long and 25 feet wide. How many shingles will be required for it if laid 4 inches to weather, and how many 4-penny nails must be allowed ? What will be the cost of laying the shingles, at $ 2.50 per day for labor ? 88. How many clapboards, laid 4 inches to the weather, will be required to cover the side pf a building 30 feet high and 63 feet long, no allowance being made for openings ? 89. A close board fence, 4 feet high and 64 feet long, is to be painted both sides with two coats of white lead paint. When the lead, ground in oil, is 9 cents a pound, and linseed oil 72 cents a gallon, what will the paint required cost ? How much must be paid a painter for putting it on, if he covers 80 square yards a day at $2.50 ? 90. A room is 20 feet long, 18 feet wide, and 10 feet high. Allowing 108 square feet for openings and mopboards, how many laths will be required for its ceiling and walls ? What will be the cost of nails for setting them at 4^^ cents a pound ? 91. A hipped-roof barn is 64 feet long, 40 feet wide, and 20 feet high to the roof. Allowing 360 square feet for open- ings, how much will rough boards for the sides cost at $ 20 a thousand feet, and how much should each of two men, at $ 2.25 a day, be paid for putting on the boards ? PBOGRESSIONS. 539. A Series of numbers is a succession of numbers, in- creasing or decreasing according to some fixed law. 540. The Terms of a series are the numbers forming the series. APPENDIX. 337 The first and last terms are called the extremes, and the in- tervening terms the means. Thus, 3, 6, 9, 12, is a series in which 3 and 12 are the extremes, and 6 and 9 the means. 541. A series is ascending or descending , according as the series increases or decreases from the first term. ARITHMETICAL PROGRESSION. 542. An Arithmetical Progression is a series of numbers which increase or decrease by a common difference. Thus, 2, 4, 6, 8, 10, 12, is an ascending series, 12, 10, 8, 6, 4, 2, is a descending series, in each of which 2 is the common difference. To find either Extreme. 92. The first term of an arithmetical progression is 3, and the common difference 2. What is the fifth term ? Solution. 1st term = 3. 3d term = 3 + (2 X 2). 2d term =3 + 2. 4th term = 3 + (2 X 3). 5th term = 3 + (2 X 4) = 11. 93. The fifth term of an arithmetical progression is 11, and the common difference 2. What is the first term ? Solution. 6th term = 11. 3d term = 11 — (2 X 2). 4th term =11—2. 2d term = 11 — (2 X 3). 1st term = 11 — - (2 X 4) = 3. Hence, 338 APPENDIX. 543. To find an extreme, Multiply the common difference by the number of terms les$ one, and the product plus the smaller extreme will be the larger ; or, the larger extreme minus the product will he the smaller, 94. The number of terms of an arithmetical progression is 100, the common difference 3, and the first term 5. What is the last term ? 95. A man bought 34 yards of cloth, and agreed to give 12 cents for the first yard, Vl\ cents for the second yard, 12 § cents for the third yard, and so on. What did the last yard cost him ? 96. A man travels 10 days, increasing each day's travel by I of a mile. If he goes the last day 17 miles, how many miles did he start with ? 97. If 16 persons give in charity, and the first gives 5 cents, the second 9 cents, and so on in arithmetical progres- sion, how much does the last person give ? To find the Sum of the Terms. 98. The first term of an arithmetical progression is 2, the last term 12, and the number of terms 6. What is the sum of all the terms ? Solution. Let 2, 4, 6, 8, 10, 12, be an arithmetical series, and 12, 10, 8, 6, 4, 2, be the series reversed. 14 -j- 14 _|- 14 _[- 14 -j- 14 _[_ 14 = 84, twice the sum of the series. But 84 = (2 -}- 12) X 6, or the sum of the extremes multiplied by 6 ; and half of 84, or (^ + 1^) X 6 __ ^^^ ^^^^ ^^ ^^^ ^^■^\^^. Hence, 544. To find the sum of the terms, Multiply the sum of the extremes by the number of terms, and take half the product. APPENDIX. 339 99. The first term of a series is 2, the last term 478, and the number of terms S6, What is the sum of the series ? 100. A man agreed to labor 12 months. For the first month he was to be paid $ 7, and for the last $ 51. If he was to receive the same addition to his wages each successive month, what sum would he receive for his year's labor ? GEOMETRICAL PROGRESSION. 545. A Geometrical Progression is a series of numbers which increase, or decrease, by a common rate or ratio. Thus, 3, 9, 27, 81, 243, is an ascending series ; 243, 81, 27, 9, 3, is a descending series. In the first series the rate, or ratio, is 3, and in the last J. To find any Term. 101. The first term of a geometrical progression is 4, the rate 2, and the number of terms 5. What is the last term ? Solution. 1st term = 4. 3d term = 4 X 2^. 2d term =4X2. 4th term = 4 X 21 5th term = 4 X 2* = 64. 102. The first term of a geometrical progression is 1458, and the rate J. What is the seventh term ? Solution. 1458 729 "729" Hence, & = W9-' ''''>< i-. 546. To find any term, Multiply the first term by that power of the ratio whose ex- 'ponent is equal to the number of terms less one. 103. The first term of a series is 10, the rate 20, and the number of terms 5. What is the last term ? 340 APPENDIX. - 104. When the first term is $ 120, the ratio 1.06, and the number of terms 4, what is the last term ? 105. What will $ 50 amount to in 4 years at 6 % compound interest ? To find the Sum of the Series. 106. A geometrical progression consists of 2, 6, 18, 54, the ratio being 3. What is the sum of all the terms ? Solution. 6 + 18 + 54 + 162 = 3 times the series. 2 + 6 + 18-1-54 = once the series. 162 — 2 =2 times the series. 162 — 2 en ^i, • = 80 = the series. 2 Subtracting like terms of once the series from three times the series, there remains 162 — 2, as two times the series, or 80 as the sum of the series. Hence, 547. To find the sum of the series, Multiply the last term by the ratio, subtract the first term, and the remainder divided by the ratio less one will give the 107. What is the sum of a geometrical progression whose extremes are 6 and 768, and ratio 2 ? 108. The first term of a geometrical progression is 10, the ratio J, and the number of terms 5. What is the sum of the series to the nearest hundredth ? 109. The first term of a geometrical series is $ 100, the rate 1.06, and the number of terms 4. What is the sum of the series ? 110. A lady, wishing to purchase 10 yards of velvet, thought $4 a yard too high a price. She, however, agreed to give 1 cent for the first yard, 4 cents for the second, 16 cents for the third, and so on. What was the cost of the velvet ? APPENDIX. 341 COLLEGE ENTRANCE-EXAMINATION PAPERS. 548. Brown University, 1. Divide fifty millionths by six hundred twenty-five ten- thousandths, and express the quotient in words. 2. A merchant owned \\ of a stock of goods ; | of the whole stock were destroyed by fire, and -^^ ^^ ^^^ remainder damaged by water. How much did the merchant lose, provided the uninjured goods were sold at cost for $ 4200, and the damaged at half the cost ? 3. How many hektol iters of oats can be put into a bin that is 2"^ long, 1.3'" wide, and 1.5'" deep ? 4. Sold a village lot for 1 230, which was 8 per cent less than cost. Had it been sold for $ 300, what would have been the gain per cent ? 549. Dartmouth College, 2f -^ -^ X 2 1. Y = ^ 2-1-5 2. Find the sum and product of J, J, f . 3. Find the cube root of 3845672000. 4. Find the square root of 3534400.5. 5. A platform bears a weight of 100 lb. per square foot. What is the weight in kilograms per square meter ? 6. A horse that cost 6 J- per cent of % 25000 was sold for % 1000. What was the loss per cent ? 550. Trinity College, 1. Subtract thirty million twenty-six thousand three from 45007021. Find what number must be added to the difference to make one hundred million, and write the answer in words. 342 APPENDIX. 2. The sum of f and ^ is diminished by ■^. How many times does the difference contain f^j of the sum of J, ^, and ^ ? 3. Divide 375 by .75, and .75 by 375, and find the sum and the difference of the quotients. 4. A loaded wagon weighs 2 T. 3 cwt. 48 lb. ; the wagon it- self weighs 18 cwt. 75 lb. The load consists of 215 packages, each of the same weight. Find the weight of each, and re- duce it to grams and kilograms. 5. Define interest, and give and explain the rule for com- puting the interest on any sum of money for any time and at any rate per cent. 6. Extract the square root of 184.2 to three places of deci- mals. 551. Harvard University. i. Find the greatest common divisor of 315, 504, and 441. 2. Express as a decimal f X ^ ,, , ^ o ^ • 3. How many hektoliters are there in 57 gallons 3 J pints ? 1 liter z= 0.2642 gal. ; 1 gallon = 8 pt. 4. How much paper, 1\ yd. wide, will be needed to paper the walls of a room 10 feet high, 18 feet long, 12 feet broad? 5. I sold a lot of sugar for $ 230, and thereby lost 8 per cent of the cost. What per cent should I have gained if I had sold the sugar for $ 300 ? 552. Yale College. 1. Find the value of ( 5-?- of — ) divided by ^, and exv tract the square root of the quotient to two decimal places. 2. Find a fourth proportional to .37, 8.9, 4.3, and extract the cube root of it to two decimal places. APPENDIX. 343 3. Eeduce 16 rods 2 feet and 8 inches to the decimal of a mile. 4. What is the length in meters and decimeters of a side of a square which contains .1335 are ? 553. Dartmouth College, ^ 41 + 2^ -^ I ^ , ' 61 - If X t • 2. Find the least common multiple and the greatest com- mon divisor of 6, 8, 20, and 36. 3. How many meters in 25 feet ? 4. Find the square root of 3530641. 5. Gold was quoted at $1.12|. What was a one-dollar greenback worth ? 6. 1 1200 includes a sum to be invested, and a commission of five per cent of the sum to be invested. What is the sum to be invested ? 554. Cornell University. 1. Define a composite number, a factor, an abstract number, the cube root of a number, equation of payments. 2. What is the value of 50 lb. 8 oz. of gold at 1 20.59J per ounce ? 3. Given the meter equal to 39.37 inches, reduce one mile to kilometers. Give the metric table of weights. 4. Divide f of 7f by f of 12^^J. Prove the result by re- ducing the fractions to decimals, and working the example anew. 5. A man said, " I will spend half my income, save a third of it, and devpte a fourth to business.'^ His income was $ 780 a year. Point out his blunder, and divide his income rightly in the proportion intended by him. 6. How long must $ 125 be on interest at 7J per cent to gain $ 15 ? 344 APPENDIX. 7. Eeceived 6 per cent dividend on stock bought at 25 per cent below par. What rate of interest did the investment pay? a Find the cube root of .726572699. 555. Yale College, 1. Eeduce to a common denominator, and add: .3 9 ' 15' 4' ^"""^ 10- /3 4 5\ ^ \^^ 5^ 9/ 11 2. Divide (1-1) by 1. 3. Find, to three decimal places, the value of -— 3 • V3 4. Find the fourth term of a proportion of which the first, second, and third terms are, respectively, 3.81, 0.056, 1.67. 5. Eeduce 3 K 13 sq. rd. 8 sq. ft. to the decimal of an acre. 6. {a) In a board 4 "" long and 0.4 "" wide, how many square decimeters ? {h) Divide 2700 "'by 90 ^^ 556. Harvard University. 1. Simplify 1 + 3 + (S) t)-^ 2. Find the length in dekameters of the side of a square, the area of which equals the area of a rectangle which is 1 kilo- meter 8 meters long, and 4jf hektometers wide. 3. Find the least common multiple of the even numbers from 10 to 20 inclusive. APPENDIX. 345 4. The interest on $2500 for 2 months 12 days is $45. Find the rate. 5. If 25 men, working 8 hours a day, do | of a piece of work in 24 days, in how many days of 10 hours each will 30 men finish the piece of work ? 557. University of the State of New York. 1. Copy and add : 20570, 6206, 98.007, 63000, 426.000626, 4287, 63.961, 102030, 405.0607, 8090, 543.21, 1028848.414995. 2. Express by Arabic notation, MDXCYDCCCLXIV. 3. Express by Roman notation, 84796. 4. Numerate 20567189.004321098. 5. Divide 31984875832 by 96813. 6. Find the value of (28 - 7) X 6 + (92 4- 7) -^ 9 - (86 + 10) -^ 12. 7. Divide, using cancellation, 15 X 80 X 27 X 28 by 7 X 20 X 8. 8. Change /y, Jf , t^%, and J, to similar fractions having their least common denominator, and 9. Reduce their sum to decimal form. 10. Find the greatest common divisor of 7955, 8769, 6401. 11. How much must be paid for making 52 rd. 14 ft. 8 in. of fence, at $ 0.75 per foot ? 12. A traveler, on reaching a certain place, found that his watch, which kept correct time for the place he left, was 2 h. 22 min. slower than the local time. Had he traveled east- ward or westward, and how far, in circular measure, had he come ? 13. What per cent (expressed in words) of 30000 bushels are 50 bushels ? 14. What number diminished by 36% of itself = 336 ? 346 APPENDIX. 15. What is the value of a lot 70 rd. long and 20 rd. wide, at $ 47.25 per acre ? 16. A cistern has 3 pipes. The first will fill it in 12 hours, the second in 16, and the third in 18 hours. If all run to- gether, in what time will they fill it ? (State this example as a proportion, if you can.) 17. What is the difference between the simple interest on $328 for 2 y. 7 mo. at 7%, and the compound interest on the same sum for the same time at 6 % ? 18. Find the balance due March 4, 1881, on a note dated Jan. 1, 1879, for $ 580, at 5 %, on which a payment of I 85 has been made every 6 months, using the United States rule. 19. How much should be discounted on a bill of $3725.87, due in 8 mo. 10 d., if paid immediately, money being worth 5%? 20. Bought bonds at 115, and sold at 110, losing $ 300. How many bonds of $ 1000 each did I buy ? 21. If A puts in $ 4000 capital for 8 months, B $ 6000 for 7 months, and C $ 3500 for 1 year, and they gain $ 2320, what is each partner's share of the gain ? 22. If 5 horses eat as much as 6 oxen, and 8 horses and 12 oxen eat 12 tons of hay in 40 days, how much hay will 7 horses and 15 oxen eat in 65 days ? 23. Find the value of V'0.000238328. 24. A steamer goes due north at the rate of 15 miles an hour, and another due west 18 miles an hour. How far apart will they be in 6 hours ? 25. Find the cost, at 30 cents per sq. yd., of cementing the bottom and sides of a cubical cistern that will hold 300 barrels. 26. What is the area of a circle 5 ft. in diameter ? 27. What is the difference between 5 square feet and 5 feet square ? Illustrate by a diagram. ANSWERS. Art. 87. 44. 1009 45. 669. J^6. 698. 47. 759. 4S. $989. Ji9. J499. 50. $1465. 79. 1484. ^(?. 1149. 81. $31,355. 82. $39,928. 83. $61,665. ^4. $317.40. Art. 39. 85. 689. 86. 1978. 87. 2396. ^5. 15485. ^P. 2052. 90. 9788. 9i. 2018.7. 92. 143.91 P5. $131.31. 94. $100.66. 95. $393.30. 96. $230.05. .97. 3018. 9^. 3443. 99. 7736. 100. 2023. i9i. 2026. 10^. 16986. i^^. 176.40. Art. 46. i94. 153.89. 4S. 332. 295. 281.72. 4^. 223. 106. 5233.97. 47. 205. 107. $125.65. 48. 222. 108. $511.69. 49. 1114 109. $168.08. 50. 3212. 110. $532.40. 51. 2213. 111. 998. 52. 1118. 112. 4391. 53. 1221. 113. $9665.68.^: 54. $325. in. 5619. 76. 309. 115. $75.13. 77. 192. 116. 318381. 78. 192. 117. $145.17. 79. 46.65. 118. $360. 80. 803.153. 119. $14170.70. 81. $391.05. 120. 2815. 82. $55.14. 121. 24445. 83. $338.80. 122. 313.54. 123. 150.390. Art. 48. 124. 29002 ft. 84. 5196. 125. 3578^92. 85. 4969. 126. $1046.87. 86. 1859. 427. $175i30 87. 1056. 128. 62611. 88. 29962. 129. $39320. 89. 3541. 130. 1323925.63. 90. 56.39. 131. 1863189. 9i. 14.251. 132. 563972744718. 509006545503.418. 92. $6.27. 133. 95. $83.96. 134. 323497. 94. $95.81. 135. 340522022. 95. $29.99. 136. 1380855.262. 97. 34456. 137. $32545.24. 98. 97820. 138. $24005.79. 99. 22968. us ANSWEES. 100. 9903. 101. 9154. 102. 1. 103. 6.552. 104. 811.95. 105. 9615.5 i{?^, 78.44. 107. $486.57. 108. $1836.75. 109. 5541. 110. 26983. iii. 11001. 112. 107.91. ii,5. 389. 114, $740.75.' il5. 163864. ii6. $4066.94. 117. 269535. ii,5. 267369. 119. 1785837. i^a $291.25. i^i. $3527.82. 122. 891; 460; 280. 123. $173.32. 124. 1382. i^5. 1330 miles. 126. $5555.75. 127. 380247. 128. $1031.32. Art. 68. 38. 3024. S9. 13701. ^(?. 5545. 41. 49608. ^. 20496. 4^. 5216. .^4. 86415. .^. 218709. 47. 2S0.56. i?. 30.675. 4.^ 3271.8. 50. 49899.71. 5i. $497.40. 52. $1155.282. 53. $6588.00. 54. $76838.70. 81. 2S2;U. 82. 60525. 83. 10150. 7. 30303gV. Art. 76. 110. $19.65. 111. 39.62; 3.962. ii^. 45.54. 113. 137ff. ii^. 1090.18iW. ^^^- mm- 116. 494ii|-|^sec. 117. 1024||||.' 118. 768f4|f. 119. 5888ij^%V i;^o. 11956mi 121. 1.542Aff i^^. .529fU. 123. 977iMf. ^^^. 1.788i||. i^5. 585,W^. 126. im^s%\. 127. 100|f|. 128. 1016400. 129. 30515^^. i^o. $l.ll|ii 131. 3^^. i^-^. .871|^. 349 138. 2555. , 135 3. '^ i<^6. |26.77ftV 137. 14. i^.? 640. > i55. 106. 7.^5. 64. I4I' 35; and 1400 1 mi. over, i^. 750. i4^. 108. lU 71. i-^. 217. Art. 76. .?5. 3207. ^.f 39817. 35. 36. 40. 41. 42. 4S. 44- 45. 46. 47. 48. 4^. 50. 51. 52. 53. 54. 55.^ 56. 57. 58. 59. 60. 61. 62. 63. 64 65. 66. 67. 1492. $ 2063.38. 12616. $371.14. $ 15.30. 6. $35405. 28618. $0.14. $0.24fio $4.71^11. 27. $8. $1820.85. 873. $3632.25. 137ff hhds. $123. 192. 1808. $7137. 17» 759. 550. 17. $ 269880. 1160. 13. $5817. $4.53f. 215. $4064, gain. Art. 82. 8. 22, 3, 7; 2^3-^ 2S, 5. 9. 2, 3, 7, 11; 2«, 32; 2^ 32, 7. Art. 83. 10. 2, 3, 5, 7. 11. 22, 32, 7, 11. 12. 2, 3, 71. 18. 3, 5, 11, 37. 14 3, 5, 7, 11. 15. 24, 52, 7. 350 ANSWERS. 16. 22, 32, 5, 19. 17. 23, 3, 52, 13. 18. 22, 4007. 19. 38, 72, 13. ^0. 32, 31, 37. 21. 2^ 11, 71. ^2. 37. ^«5. 3, 7, 11, 19. ;^^. 22, 32, 52, 7. ;^5. 7, 11, 17, 31. Art. 86. 35. 12^. 36. 3|. 57. 109. 5. 56. 71. 28. 7^. 6. 73. 8. 7.4. 12. 75. 31. 76. 14. 77. 696. Art. 96. 83. 210. ^4. 1260. Art. 97. 85. 1848. 86. 504. ^7. 1320. ^^. 2835. 89. 72. 90. 86100. 91. 7000. 9^. 7560. P«5. 7200. 9J^. 7920. 95. 350. 96. 23, 32, 5, 7. 97. 27, 32. 9f. ANSWERS. 351 Art. 117. 107. 9. 108. 27f 109. 169iV 110. 1. Ill- 1H|. 112. l^V 113. 7H- li^. 128. 115. llOff. 118. 90|f. 119. 91|f. 120. 100 iW. 1^1. 95fff Art. 120. isi- il 133. ^^, to"' tA* Art. 121. 135. 136. 138. 139. ii m- fA. 1%, T^v 6 6 14(10 J^^ M 36 M. Art. 123. 156. 2H. Art. 124. 158. If. 159. 2^\. 160. !«. lei. 2ff 165. 8f|. 16^. 2|||. 165. 2|4|. 166. 9^1^. Art. 125. 167. 40|. 16^. 488^. 169. 27i. 17(?. 313|f. Art. 126. 182. 183. m. ^. Art. 127. 185. 186. 187. 188. Uf 189. ^. 190. \. 191. T*k. 19^. X. 193. -,y m. eV- i^5- 17ff. 196. 21f|. 197. SX. 198. 6^. 199. 13^. ^9(?. 280|f. 201. 4^. ^(?^. ^5^. 203. 2ff . Art. 128. 212. 1|. ;^15. 154 214. 37| ^15. 49. ;^16. lOH- 217. 2A\. ;^1^. 43^. 219. 144. ^^1. 1090^. ^^^. 5131. 223. 2927f ^^.^. 10025. 225. $621. Art. 120. ^5^. 84. 233. 241}. ^5-4. 325. 235. 97f. ^.?6. 88. ;^,S7. 165iV. 238. 17A. ^59. 243A. 241. 10724. ;^4^. 1119|. ^.4<^. 791. 2U' 1719f. ^-^. 1281J. Art, 131. 257. 134. 259. l| ^66. ;^61. 2f. ■5. 2| ;^64. f 265. Jt589A' ^66. 82f^. ^67. Iff ;^6^. 25^. ^69. $122||. 270. U^. Art. 132. 285. ^4^. ;^^7. ■ \288. 352 ANSWERS. ^89. '^91. A ^92. 15fJ. 293. 3H. 29 J,. «6f. SOJf,. mi 305. 306. 29A. 874. 52|. 307. 308. 170. 309. 151iV 310. 40. 311. 162. 812. ml 313. su. 64. 815. 16. 316. 7. 827. 141. 828. iV Art. 133. 829. |. 830. ^. S31. WA. 832. lA- 833. 1 334. AV S35. 2tV 336. HH- 837. 2||. 838. iilj. 339. ^M- &40. u^. 341. 41 hours. 34s. ill- 344. m. 345. 16|f. 34B. i." 347. ^TS' 848. - |. 349. ■r^. 350. 45 days. 851. 4 hours. Art. 134, S59 f 360 4. 367. {%. 368. if. ^ra 112. .^7i. $52.50. 372. 4|. ^7^*. |'4iVo- Art. 136. ,^^9. 1672. 390. 1863. ^9i. 6400. «^9^. 520. 393. 11400. P. .9474. 100. .0933J. i(?i. .135135. iO^. 1.232. 103. 51.9383. i(?.^. 11.933877. Art. 145. 111. 204.655. 112. 132.912. Art. 146. 115. 253.80. 116. 9650. ii7. 0.014582. im 84.5688. 119. 21.723. i^O. 26.461. 121. 143.3282 122. 0.00038665. 123. .0000505. 124. .0001292. 125. 1.000000. 126. $160,875. 127. $61.7925. 128. 0.01020201. 139. 0.023. Art. 147. IJ,^. 125.36.^ U3. 0.756. lU. 10.01. 11^5. 790. 146. 0.081. 147. 0.893+ i4^. 0.001365. 4000. .^ 150000. 1.12. 360.984. 15.004-f-. 210. 11,9. 150. 151. 152. 153. 154. 155. 157. 158. 159. 5 0.0403. I 400.0003. ' 0.0087. 0.00162. 0.00Q009. 5.325. 8.9999.7. 161. 162. 163. I64. 167, 160. 0.855. ( 6345.3654. { 4.1958. (0.0000495. 500. A- 23.9268 sum. 165. 14^54368 days. 166. 612. $ 2999. 45J A's. $2249.62jB's. $ 2249. 62J C's. 18817.92728. 58.872. $ 16444.9602. 4.9835. ._.. 96. 165.135.-- $0.18.— $447.70. $33|fi. 1. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. $219.54t\. 180. $5941.06. 181. 7e>j\\. 182. 68|||tons. 183. 420. 184. 4000. 1^5. 3f. Art. 154, 28. $27.26. 24. $40.93. 25. $5143.15. 26. $19.36. 27. 250. 28. $41.19. $94.86+. $48. 44ff. 760. $ 163.12f 50.95^^. 156, J 29. 30. 31. 34 Art, 44. $198. 45. $56. 46. $232.50. 554 ANSWERS. ^7. 98. JiS. 18. 49. ^49. 50. $679.50. 51. 2117. 53. $5.10. 54. $81.25. 56. $248.60. 57. $ 904.042. 59. $12. 60. $12.50. 61. $1.50. Art. 159. 62. $90.47. 63. $222.71. 6J^. $58.55. 65. $56.23. 66. $104.78. 67. $ 226.50. 68. $1156.25. 69. $84. 70. $443.35. 71. $126.64. Art. 187. 13. 1279 pt. u. 1311 cu. ft. 15. 35790 lb. Art. 188. 16. 562068 sq. ft. 17. 273749 cu. in. 18. 487 pt. 19. 879 pt. 20. 248160 ft. 21. 419887 gr. 22. 87320 lb. 23. 31556930 sec. 2^. 391256 cu. in. 25. 163734 sec. 26. 3200 cu. ft. 27. 1306 ?i. 28. 684592 min. 29. 4320 sheets. 30. $3267. SI. $6352.50. 3S. 88 yd. 35. 86. q>>y is sec. 39. 48 lb. J^0. 326.7 sq. ft. Ji.1. 568.08 min. Ji2. 6601.76 sq. yd. Art. 189. 51. 19 bu. 3 pk. 7 qt. 1 pt. 52. 48cu.yd.l5cu.ft. 53. 17T.17cwt.901b. Art. 190. rd. 5Jt. 12 A. 144 sq, 144 sq. ft. 55. 5 cu. yd. 23 cu. ft. 725 cu. in. 56. 60 gal. 3 qt. 1 pt. 57. 13 bu. 2pk. 7qt. Ipt. 58. 4^7 xrl. 59. 72 lb. 10 oz. 15 pwt. 7 gr. 60. 43T.13cwt.201b. 61. 365 d. 5 h. 48 m. 50 sec. 62. 8 cu. yd. 10 cu. ft. 728 cu. in. 63. 45^ 28' 54''. 64. 25 cd. 65. 40 gal. 3 qt. pt. 2gi. 66. 67 wk. 6 d. 9 h. 52 min. 67. 4 bundles 1 ream. 68. 1 A. 80 sq. rd. 69. 3 mi. 195 rd. mi. lb. r^ day. gal. ■ bu. ).'024 T. 0.0075 A. 79. 0.3945 day. 80. 1.364 A. Art. 191. 88. 71 rd. 1 ft. 10 in. 90. 2 yd. 2 ft. 8.94 m. Art. 192. 91. 68 sq. rd. 155 sq. ft. 82f sq. in. 92. 10 oz. 13 pwt. 8gr. 93. 232d. 6h. 32min. 43 j\ sec. 94. 248 rd. 4 yd. 2 ft. 8 in. 95. 671b. 4oz. 96. 5 cwt. 64 lb. 97. 4yd. 2 ft. 5.25 in. 98. 18 h. 15 m. 50.4s. 107. f mi. 108. U.53rd. Art. 193. 110. AWoA. 111. 1 lb. 112. fiiffy. 9^ ini. 113. 114. 0.6725 ctl. 115. 0.282 T. 116. 0.875 rd. 117. 0.761 d. Art. 194. 118. ;.. 119. Im- 120. tVi- 121. O.J 25. 122. 11.17H- Art. 195. 126. 63 cu.yd. 11 cu. ft. 842 cu. in. 127. 199 gal. 1 qt. 12^. 77 d. 8I1. 26 m. 56 sec. 129. 64^ 28' 32". ANSWERS. X 131. IT. 15cwt. 531b. 182. 1()7 mi. 240 rd. 133. 2 A. 49f^sq. rd. 135. 13 bu. 2 pk. 6 qt. 136. 14 mi. 231 rd. 5 yd. ft. 2 in. 137. 28 A. 76 sq. rd. 138. 16° 30' 51". 189. 5oz.5pwt.20.8gr. 140. 12 gal. 2 qt. pt. 141. 133 d. 4h. 39 mill. 21.6 sec. 14^. ISlrd. 2Yd. 2 ft. 8.72 in. Art. 196. lU- 5 y. 9 mo. 24 d. 145. 1 y. 7 mo. 20 d. 146. 33 y. 2 mo. 20 d. 147. 86 y. 5 mo. 28 d. W. 60 days. 150. 282d. llh. 152. 141wk.4d. 22I1. 16 min. 153. 37mi. 170rd. 1yd. 154 lOT. 8cwt. 531b. 155. 355 A. 49 sq. rd. 21f sq. yd. 156. 203° 51' 40". 158. 17 wk. 3d. 10 h. 17 min. 159. 3 mi. 124 rd. 1yd. 11 ft. 160. 3oz. 17pwt. 14gr. 161. 13° 10' 35". 162. 61 gal. 1 qt. 1 pt. 163. 3779 in. 164. S 351.75. 165. 6 T. 12 cwt. 6 lb. 167. 98 T. 3 ctl. 10 lb. 168. 19 rd. 6 ft. 6 in. 169. 19 A. 136sq.rd. 68 sq. ft. 9 sq. in. 170. 786 days. 171 2T. 7% J lb. 172. 196 d. Oh. 49m. 173. 0.39625. 174. ] GU cd. ft. 175. 55. 176. $15.25. 177. 78 y. 2 mo. 24 d. 178. Lose$10.78f 180. None. 181. 547d. 20h. Art. 213. 25. 151.845"^. 26. 1663.70 ^ 27. 17.4 ^^ 28. 2040 \ 29. $527. 30. 356.2075 "^ 32. 121.2531b. 33. 365.976 sq. yd. 34. 1111.95 acres. 35. 88.9+^. 36. 74+ ^^ 38. $1.84+. 39. $4637.60+. 40. 2586.145+. 41. 93.57"^. 42. $6.68+. 43. $922.36. Art. 222. 6. 820 9375 sq. ft. 7. 7854 sq. ft. 8. 1256.64 ft. 9. 400 ft. 10. 31yd. 11. 0.78125 yd. 12. 13i A. 13. 154.56+ ft. 14. 232 sq. ft. 15. 14.6770 «\ 16. $240. 17. 41 sq. rd. 147+ sq. ft. 18. 66.25 sq. ft. 19. 5 A. 159 sq. rd. 260i sq. ft. Area,128Msq. ^0. { yd. Cost $232.03+ Art. 226. 25. 15 cu. ft. 26. 11571 cu. ft. 27. 3f ft. 28. 3 cu. m, 29. 793 J cu. yd. 30. 8.700 cu. m. 31. 8.3776 cu. yd. 32. 1026 sq.ft.' Art. 8c(i. 8cd, ^ ft. ^4rd 227. 34. 35. 36. 37. 38. $236.25. Art. 230. 40. U bd. ft. 41. 66§ bd. ft. 4^. 283ibd. ft. 43. $40,986. 44' $52.92. 45. 72 ft. 46. $5.67+. 47. 1220 bd. ft. $7.68. 24.75 "1. 9 rolls. 38.4 bu. 65f gal. 8500 bricks. 4 T. and 31 T, 7002 \\ lb. 1782 cu. ft. 4S. 49. 50. 51. 52. 53. 54. 55. 56. Art. 231. 50. 99. 51. 0.003. 52. 0.01875. N 56 356 53. 0.42f 54. 6400. 55. $132.37i ( $ 10000 land. \ $ 4000 house. 57. 124 acres. 58. $2195. 59. 42i|. 60. 15 sq. yd. 3 sq. ft. 128 sq. in. 61. 62f yd. e^. 1613^ cu. yd. 63. 47520 bricks. 64. 95040 bricks. 65. 85.84 sq. rd. 66. 316i|^tons. 67. 37i sq. yd. 68. $32.08^. 69. $11.50. 70. 8739yV7. 71. 32^1 cd. 7^. 303 ^3_ sq. ft. 73. 74|i yd. 74. 5/y acres. 75. $20.14ff. 76. $15.40. 77. 0.5184. 78. 0.625. 7a 80 yards. 5(9. July 26, 11 h. 45 min. P. M. 81. ^ sq. yd. 82. $8.49f. 5.5. Ans. If 54. 88 sq. rd. 85. 23y\ft. 5^. 18 ft. 87. 1926 ft. 55. 16|yd. 5a 5832. 90. 475 ft. Pi. $12015 92. .Vir- 95. $351. P^. $53.33f 95. 95 d. 5 h. 10 min. 96. 13500 yd. 97. yd. wide; $7.50. 9d. il T. ANSWERS. 99. $364. 57. 1040 yd. 100. $2.25. 55. 3240 rd. 101. $r73.45yV 89. $ 3500. 102. im. 90. 7000 lb. 103. $0.88f 91. $ 166f. 104. $ 0.52^. 92. $ 2400. 93. $ 9000. Art. 241. 94. 13655. 35. $3605. 5^. $315. Art. 246 102. $1.55. Art. 242. 103. $12160. 37. 2380 tons. 38. 205.20. 39. 22.61 mi. 104. 105. 106. 107. $ 50 loss. $10.44. $22.78^. $3780. 40. 1211 men. 108. $250. 41. $487.50. 4^. $51.64. 109. 110. 6i%. 41 A %. 4S. $0.8136. 111. 17 ''' $3140. 44. $511.05. 112. S490. 45. $4122.50 113. $ 8.34f . $1460. 4 % loss. $0.57^. $19 lOf. 46. $11223.87. 47. $9.34. 114. 115. 48. 14748. 49. $ 12825. 116. 117. 50. $5700. 118. $66.66|. 51. $337.97. 119. Neither. ^i 91 %. Art. 248 Art. 243. 123. $141.95. ^5. %\%. 124. $8.75. ^^- Mil 125. $157.93. 67. 25f %. 126. $59.80. 68. 9|%. 127. $5012.50. ea 83j%. 128. $12876. 7a 14%. 130. $18.94. 71. 5%. 131. 214Abbl. 7^. 7i%. 132. $1426.80. 73. 15%. 133. 3.83+%. 7^. m%. 134. $38.59. 5^. $324.40. 135. $4170. 83. 295 sheep. 136. $200. Art. 244. Art. 252 84. $1562.50. 139. $73. 85. 14.4 tons. 140. $124.76 86. 66f bu. 141. $1531. ANSWERS. 357 IJ^. $53625. 1J,S. $2400. lU' 2%. 145. $3075. 14^- m\i' 147. 921%; 7^%. 1J^8 36^V%- i^. Lose $ 0.05. 150. 110%. iJi. $1097.25. 152. 115 bbl. $115. $ 26163.26. $5.31. $5f. $ 361.80. $12375. $195. $ 3033.75. 153. 154 155. 156. 157. 158. 159. 160. 161. 162. 163. $4600. 164. $ 10872.75. 165. 20%. 1008.77H- 42| yd. $2.80ff perbbl. 5%. $ 22790. 195. 33i%. $2.50. 99J%. $4. 298000. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. Art. 259. $2116.80. $12. $1728.75. $ 343.931 $584.6256. 19. $158.0298. 20. $8.2586. n. $19.3909. 22. $125.71. 12, 13 16 17 18 34. 35. 36. 37. 23. $299.33. 24, $6380.55f Art. 263. 28. $5.8409J. 29. $1500. 30. $ 1.9958f . 31. $90.686f. 32. $35.7857i. "" $18.4255| $ 30.87987^ $4339.40. $1610. $9205. 38. $15.986f. 39. $ 1.3410^^. 40. $10.9450|. 41. $63.54. 42. $2358.90iV 4^. $140.5872. 44' $175. 45. $110.1012. 46. $818,545. 47. $ 112.425. 48. $565,935. 49. $508.0765. Art. 264. 52. $1.2177. 53. $144.66|. 54. $1.30^. 55. $ 101.25. 56. $ 17.10. 57. 1 167.96. 58. $962.6446. 59. $201. 60. $779.3264. 61. $4535 7355. 62. $4,774. 63. $446.25. 64. $26.9458. 65. $173.20. $ 426.449. $637.19. $22,575. $ 279.77505. $320.28. 66, 67 68, 69 70 71. $178.87i. 74 75. 76. 77. 78 79, 82. 85. 72. $ 131.15f . 73. $125,306. $ 7.06446. $10.54928. $0.4216. $17.7818. $ 569.201. $ 900.2532. 80. $ 1261.936. 81. $1025.814. $ 860.57i $656.19776. $498.1983. $952,899. 86. $1968. 87. $968.66304. 88. $204.83^ 89. $901.6677. 90. $1237.88254. 91. $633.63554. Art. 265. 93. $45.62. 94. $16.52. 95. $236.73. 97. ^%. Art. 266. 98. 6%. 99. 6%. 100. 61^^ %. 101. 15f i 102. 7%. 103. ^\%. 104. 9 %. 105. 10%. 106. h\%. 107. 4%. 109. 6 years. Art. 267. 110. 2 y. 11 mo. 28:^ d. 111. ly. 4 mo. 20(1. 112. 2 mo. 6 d. 113. 1 mo. 18 d. or 48 d. 114. 2 y. 6 mo. 358 ANSWERS. 115. 16 y. 8 mo. 116. 7 y. 4 mo. 26| d. m. 3y. lmo.4^fd. 118. 4y. 7 mo. 16fd. 119. 9 y. 6 mo. 84 d. 122. $ 75. Art. 268. 123. \ 3600. 124. 1300. i^5. |718.23fi-. i-^6. ^14000. i^r. $24000. 128. $675.60. 129. $14215.38yV 130. $14000. 131. $14400. 182. $36956250. Art. 282. 134. $394.57. 135. $448.35. 136. $492.06. 137. $342.40. 138. $1447.08. 139. $722.17. UO. $5947.63. Ul- $413.43. IJi^. $1828.69. lJi3. $1280.20. Art. 283. 145. $349.21. Art. 285. Ul. $161.63. Art. 286. 152, $111.94. 153. $65.60. WJi,. $93.70. 155. $1125.51. 156. $1273.8?. 157. %n%M. t.5$. Sil.8$ gajij. Art. 287. 161. $1934.84. 162. $232.46. Art. 291. 7. $42. Art. 292. 8. $3122.17. 9. $28.81. 10. $356.44. 11. $130.32. 12. $300. Art. 294. 13. $6.25. U $329. 15. $26.89. 16. $393.73. 17. $460.64. 18. $ 20. 19. $213.82. 20. $734.25. Art. 300. ( Bank discount \ $17.94. I Proceeds (. $857.06. i Bank discount 1 $3.05. ) Proceeds i $82.55. Bank discount $6.30. Proceeds $593.70. 4880. $10635.40. $479.56. $953.23. 30. $1013.33. 31. $284.39. 32. $640.12. 33. $957.93. 3J^. $323,77. 35. $693.74. 23^ 25. 27. 28, 29. 37. 38 39. 40. $ 839.88. $494.92. $8567.37. $235.50. $795.07. 41. $555.27. 42. $568.96. 43. $888.85. 45. $1518.60. 46. $300. Art. 301. 47. $450. 43. $1600. 4^. $842.00+. 50. $509.34. 51. $24 discount. 52. $40. 53. April 15, 1878. 54. $6000. 55. Dec. 24. 56. $385.25. 57. $12.43. 58. $3.41. 59. $362.30. 60. April 4, 1882. 61. $175 loss. 62. $502.77. 63. $773.63. 64. $1722.64. • Art. 314. 8. $2050. 9. $34087.50. 10. $1268.75. 12. $128.12f 13. $124,571^7. 14. $130. 16. $125. 17. $360. 18. Neither. 20. $21100. 21. $ 16350. 22. $54500. ^4- 4A%. 25. 6 s, .24% greater. 26. 6| %. ANSWERS. 359 les, $ 85f ^9. At 133^. Art. 323. 5 $1175.64. 6. $3900. 7. $2520.84|J. 9. $2517.69+. 10. $4000. 11. $ 1164. 12. $450. U- $ 1474.12i. i5. $2998.50. 17. $4000. i5. 36 men. 1. 5^9203200. 64. $84,906. 104. 67^ days. 2. 9600 cu. ft. 362 ANSWERS. 3. 825. 4. $2.25. 5. 2284.47. 6. 24 lb. 7. 717H. 9. $361. 30. 10. $12.24. Art. 418. i. 29S||. ;^. $113f .5. 8656742. 4. $885.50. 5. $ 7.23. 6. $135. 7. $229.50. ^. 14400. 9, Lose $ 703. 10, 18090100. Art. 419. 1. 424. 6. 23,; 7. 79ii; 19||. P. 210«f; 857; 85. 10. j\; $100000. Art. 420. ^. 14. 3. 280. -a 9 14' fV 6. (37. ha*- 9. i\)\}^ miles. -/(?. 13J bushels. Art. 421. 1. 12. ^. 360. 5. 95J^V ^- 41^. 5. 306H. ^. H; iVe- ^. $65621 9. 146-^^ miles. 10. 46f tons. Art. 422. ^. 450. ^- m- 5. 20if. ^- m- 7. $ 94500. 5. $4800. Art. 423. ■'■• 5 J 5 > T' 2. 16ij%. 5. isH- .4. 2685f|. 5. 3445f. ^. Hf^* 7. 8^Vo- ^. $ff ^- II*- 10. 80| miles. Art. 424. ^- 32H!; 8AV .^. 187| miles. ^ 4fff. 5. $14000. 6'. $57600. 7. $12000. ^- iff- 9. 50 acres. !(?. $ 109714f . Art. 425. 1. 153^V 2. 94jf. 5 31-i+. .4. $5640. 5. 8ff . ^. 6 lots. ^. t\. 5. f. 9. 15)5625. i6>. Diminished ^. Art. 426. if- «^- y58M. 5. |. ^. 35. 7. $lf. ^- 21fi;28f;6i|J. i^. $ I841f . ^^ Art. 427. i- 0.1875. ^. 0.79992, (5. 24.5 yd. 7. 1.6075. 8. $330. P. S 28.26. 10. 0.5. Art. 428. i. 0.009125. .^. 1.44. .4 0.0001177. 5. 207.36. e. 0.01010625. 7. 0.00125. P. 15. 10. 479.9975, Art. 429. 1. 0.003. 2. 998998.999. S. 0.00144. Jf,. 166. 5. $40,937. 6. 7 years. 7. $300. ^. $334.21875. ANSWERS. 363 9. $49.81f 10. $9.06^. Art. 430. 1. 10.2485. 3. 0.72727 j\; ^V 4. 0.20445+. 5. 6 oz. 11 pwt. 8.448 gr. 6. $165. 7. $6702.24^. 8. 16455 ^ P. 13 lb. lA oz. ia 39.37+ m. Art. 431. 1. 283 y. 8 mo. 23 d. 2. 50 A. 105 sq. rd. S. 578 mi. 286 rd. 3 ft. 4. 88° 45'. 5. 936if lb. 6. 58° 28'20''W.. ^. 2 d. h. 44 min. 9. 2.32+sq. rd. 10, 338 cu. ft. 1584 Art. 432. 1. llf rolls. 2. BiH- , 5. 7 gro. 2f doz. 4. 1680 sq. in. Feb. 6. $ 8.04f . 353.7if. 3AA. 25 $250. 10. 25 sq. yd. Art. 433, 1. $132. 2. $37.80. 3. 13ift. ^. $6384. 5. 10.84 oz. 6. $350.62i. . Lessen If %. Art. 449. 1. 5 7'. 2. $40000. ^. 24f %. ■4. 19ff % gain. /- (151 shares. ■ ($122f left. 6. 1\ mo. 7. 19 mo. ^. $2432.62+. 9. Sept. 22, 1880. ia Feb. 6, 1882. Art. 460. 2. 9 yds. ^. m ft. ^. 9X days. 5. T\^ oz. 6. 22^^ acres. 7. h\ mo. . 460 sq. rd. 10. 364f| %. Art. 460. 1. $2640. ^. 15 ft. ^- It^A; 4. 28f ft. 5. $449fiHgain. 6. $4442.81 J. 7. $452.17^- 5. 36 men. a 226.27 rd. 10. 338ii cu. ft. 366 ANSWERS. Art. 461. 1. $867.94. 2. 12|i S. $i:3.516. 5. l^V 6. 1898^ViVlb. 7. B. 8. 1421.2296 sq. ft. 9. B ^% greater. 10. 0.105+. Art. 462. 1. $32.56J^J. 2. 17280. S. $5.04. 4. $6726.56i. 5. $17.22. 6. $112.50. 7. 31f 8. $184.129|. 9. $1636. 10. $2969.30. Art. 463. 1. $5908. 2. $27.59. 5. At 18 months. 4. 31050. 5. $312.50 6. 40.84+ ft. r$ 15.75 A's share. $14.62i B's share. ' $18.37 J C's share. $26.25 D's sliare. $816|C'ssliare. $ 1066f B's share. $1116§ A's share. 24 ft. 29f^ minutes. 7. S. < 9. 10. 16. 17. 18. 19. 20. Art. Iff If- -W(7^-. 474. :^^. 4.21295. ;^.^. 0.476. 24. 10.083. Art. 478. 25. 1.509375 mile. 26. 76.835 A. 27. 17.545 A. 28. 575 miles. 29. 486. .^^. 5760. 31. $225. <^^. $34. Art. 486. 36. 5h.8min.llfsec. 37. 10 h. 53 min. 37| sec. 38. 1*2° 27' 14''. ^P. 13° 22' 30". 40. 93° 40' W. Art. 493. 44. $768.12. 45. 1435.41. 4<5. $707.54. Art. 494. 48. $1179.04. 49. $4584.33. Art. 497. 51. Aug. 7, 1881. 52. Nov. 12, 1881. 53. Feb. 6 5^. Dec. 16, 1881. Art. 505. 56. $32.29. 57. 58. -{ C's$ 116.28. D's $22.69. E's$ 69.06. fEachpoll$1.58. County rate 0.0009. Town rate 0.0121. A's county tax $5.50. A's entire tax L $66.58. Art. 510. ( $ 1400 duty. 59. I $ 4929.02+ ( cost. 60. $1851.29+. 61. $2532f 62. $5105.979+ 63. $182.68+. Art. 614. 64. 22.96+ cu. ft. 65. 37i cu. ft. Art. 515. 66. 14.847 in. 67. 431.869+ bd. ft 68. $13.12+ Art. 518. 69. 49.364 gal. 70. 116.28 gal. 71. 369.309+ L 72. 446.25 gal. Art. 520. 73. 119Xtons. 74. 672lf tons. Art. 526. 75. 764 bu. 76. 22| bu. 77. 8640 lb. 78. 5JJbu. 79. $ 15. ANSWEKS. 367 80. 81. '186 Ib.^ rump and sirloin. ' 186 lb. round. '4501b. hams & shoulders. ' 600 lb. sides. Art. 532. 82. ^5.71f 83. $51.02f. 84. $35.0S^V 85. $179.64. 86. $362.20+. Art. 538. 87. 88. 89. 90. 91. 15.888 M. shin- gles. 95.3281b. nails. $19.86 cost of labor. 14.53 hundred. !$2.47 cost of paint. $ 1.78 paid painter. ( 20|- hundred 1 laths. ^$0.64-|- cost { of nails. ( $ 76 cost of 1 boards. )$4.27ito each ( man. Art. 543. Pf 302. 95. 23. 96. 9 miles. 97. 65 cents. Art. 544. 99. 20640. 100. $348. Art. 546. 103. 1600000. 10^. $142.92+. 105. $63.12+ Art. 547. 107. 1530. 108. 30^^. 109. 437.46+. 110. $3495.25. Art. 548. 1. Eight ten-thou- sandths. 2. $29741.25. 3. 39"'. 4- 20%. Art. 649. 2 jSA^^sum. ( Ti pi'oduct. 3. 1566.7+. 4. 1880.0001+. 5. 488.25 + K. e. 36if %. Art. 550. Eighty-five mil- lions eighteen thousands nine hundred eighty- two. 9^. 5 500.002. (499.998. 5 Kg 217 s^. 13.572-f-. 1. Art. 551. 63. 2. 0.555f 3. 2.17+ "^ 4 59^Vyd. 5. 20 %. Art. 552. 1. 0.64+. 2. 4.69+ 3. O.OSO^J mile. 4 36,5+^'". Art. 553. 1. 2. L. C. M. 360. G. C. D. 2. 3. 7.62 '». 4 1879. 5. $0.88f 6. $1142f Art. 554. 2. $12520.24. 3. 1.6093+ Km. 4- i, or .5. i Spent $360. 5. I Saved $240. (Tobusiness$180. ^. If years. 7. 8 %. 8. 0.899. Art. 555. 3. 0.577+. 4 0.024||f. 5. 0.8314^^5^ acre. neo^^dm. • [300000. Art. 556. 1. 7. 2. 67.2^™. 3. 5040. 4. 9 %. 5. 26f days. Art. 557. 1. 1234567.654321. 2. 1 595864. JLXXXIVDCG- XCVI. Twenty millions five hundred sixty- seven thousands one hundred eighty-nine, and four million three - hundred twenty- one thousand nine- ty-eight billionths. 368 ANSWERS. 6. 330377111! J. 6. 129. 7. 810. 10. 87. il. $654.50. 12. 35° 30' eastward. 13. iofl%. i^. 525. i5. $413.43f. 16. 4|f hours. i7. $5.87. 18. $289.76. iP. $125.03. 20. 6. ;^^. 24. 25. 26. A's$640. B's $ 840. C's $ 840. 211 tons. 0.062. 140.5 miles. % 19.44. 19.635 sq. ft. 27. 20 sq.ft. TTSE I ^% ^r- ss / ? 2^ -^i ^ ■« 918313 THE UNIVERSITY OF CALIFORNIA UBRARY