LIBRARY OF THE UNIVERSITY OF CALIFORNIA Accession o b 1 9 8 Class COMPLIMENTS AMERICAN BOOK CO, A. F. GUNN, Gen'l Ag't, 204 PINE STRKET, SAN FRANCISCO. GRAMMAR SCHOOL ARITHMETIC BY A. R. HORNBROOK, A.M. NEW YORK : CINCINNATI : CHICAGO AMERICAN BOOK COMPANY HORNBROOK'S MATHEMATICS. HORNBROOK'S PRIMARY ARITHMETIC. Number Studies for the Second, Third, and Fourth Years. HORNBROOK'S GRAMMAR SCHOOL ARITHMETIC. A Course for the Last Four Years. HORNBROOK'S CONCRETE GEOMETRY. An Introduction to Geometry. COPYRIGHT, 1900, BY A. R. HORNBROOK. GUAM. SCU. ABITH. W. P. I DISTINCTIVE FEATURES THIS Arithmetic is designed for use in the last four years of the grammar schools. The method of presentation is the result of long and close observation in the schoolroom, and conforms to the order and manner in which mathematical concepts are most naturally developed in children. Practical work has been so combined with work of a purely disciplinary character that each reenforces and enhances the value of the other. In business arithmetic, where the practi- cal demands the greater emphasis, the most simple and direct methods of computation are presented. Applications of per- centage which are little used in business, but which have a value as a stimulus to thought, are introduced at the point where they will afford the best discipline. Some subjects that have neither a practical nor a high disciplinary value, though found in many text-books, are designedly omitted from this book. The time saved by the omission of such matter is devoted to more fruitful drill on practical exercises. A carefully planned and continuous system of reviews runs through the book. These reviews take the form, first, of an excursion at the end of each chapter over all the ground thus far traversed, and, second, of a constant correlation of acquired knowledge with concepts about to be developed. No hard and fast line is drawn between mental and written work. Economy of time and effort is the sole basis of distinc- tion, and this is a self-regulating principle. Rules and definitions are given as guides in the preliminary stages of acquirement. They are not to be formally memo- 3 86198 4 DISTINCTIVE FEATURES rized; and, when clear ideas of their contents have been gained, they are to be superseded by rules and definitions of the pupil's own framing. Constructive work with simple geometrical forms is intro- duced at intervals whenever the numerical relations of those forms offer valuable material illustrative of arithmetical princi- ples. Exercises to test and develop the pupil's power of visualiz- ing are inserted in every chapter. The pupil's activity is further brought into play by a series of exercises in which he is called upon to supply the conditions for the problems as well as their solution. Problems involving unknown quantities, which are solved arithmetically only by most complicated processes, are deferred until familiarity with some of the principles governing the use of literal quantities may suggest simpler methods of procedure. The' aim throughout has been to secure a ready skill in deal- ing with numbers and to develop thought power adequate to the attack of any arithmetical problem that may arise in practical life. CONTENTS CHAPTER p AGE I. INTEGERS AND DECIMALS ....... 7 Fundamental Operations and Proofs 16 Addition of Decimals .27 Subtraction of Decimals 30 Multiplication of Decimals ...... 39 Division of Decimals ........ 47 Miscellaneous Exercises 51 II. PROPERTIES OP NUMBERS ....... 62 Multiples and Factors . ... . .62 Composite Numbers ........ 64 Prime Numbers ......... 65 Prime Factors 68 Least Common Multiple 69 Divisibility of Numbers . 74 Common Divisors .77 Powers and Roots 79 Miscellaneous Exercises 84 III. RATIO 95 Miscellaneous Exercises . . . . . . .102 IV. FRACTIONS 108 Addition and Subtraction of Fractions . . . .116 Multiplication of Fractions 126 Division of Fractions 134 Miscellaneous Exercises 142 V. DENOMINATE NUMBERS 150 Miscellaneous Exercises ....... 185 5 6 CONTENTS CHAPTER PAGE VI. ALIQUOT PARTS . . . 199 Miscellaneous Exercises 210 VII. PERCENTAGE 216 Merchandising 228 Commission 230 Trade Discount 233 Interest 236 Promissory Notes . . . . . . . . 244 Partial Payments 247 Bank Discount 250 Insurance .......... 254 Taxes .259 Miscellaneous Exercises . . . ... . . 265 VIII. BONDS AND STOCKS 280 Bonds 280 Stocks 285 Miscellaneous Exercises 290 IX. LITERAL QUANTITIES ........ 296 Miscellaneous Exercises . 316 X. INVOLUTION AND EVOLUTION ...... 322 Miscellaneous Exercises 333 XI. PROPORTION 340 Proportional Parts 348 Miscellaneous Exercises 351 XII. MEASUREMENTS AND CONSTRUCTIONS ..... .366 Lines and Surfaces 366 Solids. . . . ' 383 Arcs and Angles 397 Longitude and Time 408 Miscellaneous Exercises , 412 GRAMMAR SCHOOL ARITHMETIC CHAPTER I INTEGERS AND DECIMALS 1. Write an integer of three places. 2. Read: 235 . 235 Read "two hundred, thirty five." 2/or Do n t use "and" in reading an 124,235 3. How many figures are used to express the last number in Ex. 2 ? 4. For what are figures used ? Explain. 5. Express a number of two places by the figures 5 and 3. Express another number by the same figures. Which is the greater, and how much ? 6. What is the largest integer that can be expressed by using once all the figures 3, 7, and 5 ? The smallest integer ? Find their difference. NOTE TO TEACHER. Strictly speaking, the largest integer would be 53", but in the exercises in this chapter powers of numbers are excepted. 7. Find the difference between the largest integer and the smallest integer that can be expressed by using once all the figures 5, 1, and 8. 8. There are six different integers that can be expressed by using once all the figures 1, 2, and 3. Write these numbers in the order of their size and find their sum. 7 8 INTEGERS AND DECIMALS 9. Can numbers be expressed without figures ? 10. Write in words the number represented by 105. 11. Express in good English the number represented by 228,427. By 699,108. 12. Name all the figures that are used to express number. 13. When the figure stands alone, does it express number? In the sentence " John has marbles," what does express ? 14. is called naught, zero, or cipher. The other nine figures used to express numbers in Arabic notation are called digits. What is the tens' digit of the number 75 ? Of 235 ? What is the thousands' digit of the number 8421 ? Of 29834 ? Of 127446 ? 15. In the number 815, which is greater, the hundreds' digit or the tens' digit ? How much ? What is the sum of all the digits of that number ? 16. Write a number the sum of whose digits is 10. 17. Write a number of four places the sum of whose digits is 12. 18. Bead: 3 30 300 What is the ratio of 30 to 3 ? Of o 00(] 300 to 30 ? Of each number in the ' list to the one j ust before it ? 300000 SUGGESTION TO TEACHER. If pupils are not familiar with the terra " ratio," substitute the question, " 30 is how many times 3 ? " 19. Write a digit and place at the right of it. The result equals how many times the original digit ? 20. Placing ciphers at the right of a digit is called annexing ciphers to the digit. Annex two ciphers to 5 and state how many times 5 the result equals. INTEGERS AND DECIMALS 9 21. The easiest way to multiply an integer by 10 is to annex one cipher to it. What is the easiest way to multiply an inte- ger by 100 ? By 1000 ? 22. Give at sight the following values : a 35 multiplied by 10 e 1000 times 16 e 10 times 30000 b 100 times 71 d 10 times 3000 / 1000 times 50 23. Give at sight the quotient of : a 40-hlO c 4370 --10 e 15000 -r- 1000 b 420 -T- 10 d 2500 H- 100 / 28000 -h 1000 24. Give at sight the following values : a T Vof520 c T i^ of 2300 e T ^ of 4000 b T ^ of 600 d T i of 3100 / T^TT of 18000 25. CLASS EXERCISE. may name a number ending in three ciphers, and the class may give ^ of it. y-^ of it. unr <7 of it- 26. Multiply 1000 by 1000. A thousand thousands equal a Million. How many figures are required to express a million ? 27. Kead 8,636,448. Bead, 8 million (not millions), 636 thousand (not thousands), 448. 28. Eead: 9,240,827. 31,676,201. 125,475,042. 29. Why is it useful to separate a number into periods of three figures each before reading it ? 30. Separate into periods and read: 8347621. 98470245. 616823146. 47825001. 31. CLASS EXERCISE. - may write 9 figures on the board in a horizontal line, and another pupil may tell what number they represent. 32. Write in figures, placing a comma after millions and also after thousands : 5 million, 323 thousand, 471. 81 mil- lion, 175 thousand, 241. 815 million, 278 thousand, 924. 10 INTEGERS AND DECIMALS Millions Thousands Units 00 a -a C 1 CO 00 'S 1 1 73 c o s a o ! T3 1 I 00 TJ a o3 CO j w a> C 3 g H 1 a s 1 O H 3 'S W H & 4 7 6 8 2 1 023 33. CLASS EXERCISE. Copy on the board the above dia- gram, placing different figures in the spaces and reading the. numbers thus expressed. 34. Write and read a number of seven places, having 3 in the millions' place, 8 in the thousands' place, 4 in the tens' place, and in all the other places. 35. Write and read a number of 8 places, having 2 in the ten-millions' place, 7 in the millions' place, 4 in the units' place, and in the other places. 36. When numbers are expressed in figures they are said to be written in Arabic Notation. Write in Arabic notation : a 323 million, 224 thousand, 24 b 27 million, 960 thousand, 7 c 169 million, 201 thousand, 25 d 41 million, 41 thousand, 41 e 75 million, 75 thousand, 76 / 121 million, 3 thousand, 3 37. Write a number of 7 places whose units' figure is 5. Find $ of it. | of it. -|- of it. INTEGERS AND DECIMALS 11 38. Write the largest number that can be written with 4 places. With 6 places. With 9 places. Give the sum of the digits of each of them. 39. How many can you count in a minute ? SUGGESTION TO TEACHER. Find by trial the rate of speed at which different pupils count, timing them by the watch. 40. At your rate of counting, how many could you count in an hour ? In a day of 10 hours ? 41. Mary Wallace, a little girl living in Philadelphia, counted 75 in a minute. At that rate, how many whole minutes would it take her to count a million? How many whole hours ? How many days if she counted 10 hours a day ? 42. CLASS EXERCISE. may report the number which he can count in a minute. The class may find how many minutes would be required for him to count a million at that rate. How many whole hours. How' many days of 10 hours each. 43. Write in Arabic notation : 1st. 435 million, 347 thousand, 526. 2d. The number that is 2 million greater than the 1st. 3d. The number that is 3 thousand less than the 2d. 4th. The number that is 300 thousand more than the 3d. 5th. The number that is 3 more than the 4th. 6th. The number that is 20 thousand less than the 5th. 7th. The number that is 30 million more than the 6th. 8th. The number that is 200 million more than the 7th. 9th. The number that is 40 more than the 8th. 44. Find difference between 1st and 9th number in Ex. 43. SUGGESTION FOR CLASS EXERCISE. A pupil may write on the board a number containing millions, and the other members of the class may direct modifications as in the previous examples. When the pupil at the board blunders, another pupil may take up his work. 12 INTEGERS AND DECIMALS 45. Beginning at 2, count by twos to 10. How many num- bers did you name ? 46. What is the sum of 4 twos ? 6 twos ? Numbers which are the sum of a number of twos are called Even Numbers. 47. What is the first even number after 20? How many twos does it equal ? 48. Write all the even numbers that can be expressed by one digit. 49. What is the 8th even number ? The 12th even number ? 50. Divide 1,735,328 by the 7th even number. 51. Can you write an even number which does not end with 0, or 2, or 4, or 6, or 8 ? 52. Write an even number the sum of whose digits is 9. Find 1 of it. Find of it. Find Jg- of it. s 53. Write an even number consisting of millions, thousands, and units. Divide that number by 32. By 102. By 104. 54. In 1895 the expenses of the United States government were $ 356,195,298. The revenues of the government for that year were $ 313,390,075. How much did the amount spent exceed the amount received ? 55. Mention some of the things for which the United States government spends money, and make an example in addition. 56. The cost of the United States army in the year 1895 was $51,804,759. The cost of the navy was $28,797,796. How much did they both cost ? 57. Africa contains 11,514,000 square miles, North America 6,446,000 square miles, South America 6,837,000 square miles, Asia 14,710,000 square miles, Australasia 3,228,000 square miles, Europe 3,555,000 square miles, the Polar Kegions 4,888,800 square miles. How many square miles of land does the whole world contain ? INTEGERS AND DECIMALS 13 58. The total exports of the United States in 1895 amounted to $ 807,538,165 ; the imports amounted to $ 731,969,965. How many more dollars' worth of goods were sold to foreign coun- tries than were bought from them ? 59. The earth is about 92,800,000 miles from the sun; the planet Mars about 140,000,000 miles from the sun. How much nearer to the sun is the earth than Mars ? 60. Multiply a million by a thousand by annexing ciphers. 61. A thousand millions equal a Billion. How many figures are required to express a billion ? 62. Point off and read : a 414141414141 c 232648648648 e 58914367281 b 673673673673 d 827345827345 / 42781632512 63. CLASS EXERCISE. may write twelve figures on the board in a horizontal line, and others may tell what number they represent. 64. Write in Arabic notation : 1st. 427 billion, 338 million, 484 thousand, 521. 2d. The number that is 4 billion less than the 1st. 3d. The number that is 2 billion, 7 million, 20 thousand less than the 2d. 4th. The number that is 1 billion, 1 million, and 1 thousand more than the 3d. 5th. The number that is 13,013,013,013 more than the 4th. 65. CLASS EXERCISE. may write on the board a num- ber containing billions, and the class may direct changes of it as in Ex. 64. 66. Write and read an even number consisting of billions, millions, thousands, and units. 14 INTEGERS AND DECIMALS 67. Write: a 98 billion, 348 million, 693 thousand, 207 b 15 billion, 279 million, 427 thousand, 48 c 216 billion, 849 million, 348 thousand, 7 d 821 billion, 326 million, 475 thousand, 75 e 2 billion, 2 million, 2 thousand, 2 / 21 billion, 21 million, 21 thousand, 21 g 78 billion, 78 million, 78 thousand, 78 68. CLASS EXERCISE. may write on the board num- bers consisting of billions, millions, thousands, and units which are given to him by the class. 69. To count a billion takes how many times as long as to count a million ? 70. From the time of the establishment of our govern- ment in 1789 till 1896 there -had been spent for pensions $ 1,950,403,063 and for interest on public debts $2,791,537,714. How much more had been spent for interest than for pensions ? To whom are pensions given ? Why ? 71. In 1881, the public debt of the United States was $ 2,077,389,253 and in 1882 it was $ 1,926,688,678. How much was the debt decreased during the year ? 72. In 1894, Europe produced 897,231,061 Ib. of wool, North America 342,210,712 Ib., South America 397,970,000 Ib., Cen- tral America 2,000,000 Ib., Australia 663,600,000 Ib., Asia 258,000,000 Ib., Africa 131,925,000 Ib. How many pounds of wool were produced that year ? 73. Africa has about 127,000,000 inhabitants, North America 89,250,000, South America 36,420,000, Asia 850,000,000, Aus- tralasia 4,730,000, Europe 380,200,000, Polar Regions 300,000. What is the entire population of the world ? 74. Write the largest number that can be written with 12 figures. INTEGERS AND DECIMALS 15 75. Write a number of 15 places and find from the follow- ing note how to read it. The period of figures next higher than billions is called trillions, the next quadrillions, then come quintillions, sextillions, septillions, octillions, nonillions, decillions. 76. Write a number larger than 999 trillions and read it. Why do we seldom use such large numbers ? 77. Write the largest number that can be written with 7 places. Find 1 of it. f of it. ^ T of it. 78. What people in ancient times used letters to express numbers ? . 79. Copy the Roman numerals and write under each the corresponding Arabic numeral. I V X L C D M 80. In Eoman notation when a letter is repeated its value is repeated. Eead : XX. COG. MMMM. Write in Eoman notation: 3. 30. 300. 3000. 50. 500. 81. V, L, and D are not repeated. Can you see why ? 82. When a letter of less value is placed after a letter of greater value the sum of their values is represented. Eead VIII. XVI. LXVI. CLV. MDCL. MDCCCC. Write in Eoman notation : 28. 36. 53. 75. 125. 381. 722. 1605. 1620. 1905. 83. When a letter of less value is placed before a letter of greater value, the difference of their values is represented. Eead : IV. IX. XL. XC. MXCIX. Write in Eoman nota- tion: 14. 49. 99. 144. 579. 714. 1239. 1569. 1889. 1909. 84. A line over a letter denotes that its value is multiplied by 1000. Eead: MVI. VDC. Write in Eoman notation: 10051. 5525. 10630. 4324. 8956. 5427. 6385. 85. Write the following numbers in Arabic notation and find their sum : MDIII. MDCCCIV. MDCXX. MDCCCCI. MMDLXV. MDLXII. MDCCCCIV. MDCCCCXIX. 16 INTEGERS AND DECIMALS 86. Write the following numbers in Arabic notation and find their difference : MCCXCIX and MDCCCXLV. 87. The poet Longfellow was born in MDCCCVII. How many years old was he at the breaking out of the Civil War in MDCCCLXI ? 88. How many years elapsed between July 4, MDCCLXXVI, the date of the Declaration of Independence, and July 4th of the present year ? 89. Write in Arabic notation, MDC and find y^- of it. 90. Write in Roman notation the following dates : a The present year. b 25 years hence. c 100 years before you were born. d The year in which our present president was elected. e The year in which your state was admitted to the Union. / The year of Dewey's victory in the Philippine Islands. SUGGESTION FOR CLASS EXERCISE. Let the pupils suggest important dates to be written in Roman notation by the class. FUNDAMENTAL OPERATIONS AND PROOFS 91. A statement that two quantities are equal is called an Equation, as 60 minutes = 1 hour, 14 days = 2 weeks, 8 -r- 4 = 2. Write an equation, using the numbers, 7, 5, and another number. 92. Numbers that are added are called Addends. In the equation 3 + 4 = 7, which numbers are addends ? 93. Fill out the following equations and name the addends. 5 + 7 = ? 6+2i = ? 1+3 = 9 94. Give two addends whose sum is 15. 27. T 8 T . 95. Give three addends whose sum is 14. 20. -f. 96. Give three equal addends whose sum is 27. 30, FUNDAMENTAL OPERATIONS AND PROOFS 17 97. When John has caught 5 more fish he will have caught 7 fish. How many has he caught ? 98. If Mr. Reed had $ 325 more, he could buy a farm cost- ing $ 2168. How much money has he ? 99. When the sum of two addends is 29, and one of them is 4, what is the other ? Give the missing addend when the sum is 29, and the known addend is 21. 100. CLASS EXERCISE. Think of two addends and their sum. Then give the sum and one of the addends to the class. The class may find the other addend. 101. In adding 25 and 18 James carelessly wrote 45 as the answer. If either of his addends were subtracted from the number he wrote, would the other addend be found ? Explain. 102. Add 13 and 26. If your work is correct, and if one addend is subtracted from your answer, what will be left ? 103. Complete these equations. Illustrate with small num- bers. Addend -f- Addend = Sum Addend = 104. Add 124 and 354 and prove your work. To prove the correctness of the addition of two numbers subtract one addend from their sum. If the work is right, the remaining number will equal the other addend. 105. Find sums and prove : abed e f 628 949 639 457 1639 1854 354 848 728 622 2136 237 106. Write an example in subtraction and show which number is the minuend. The subtrahend. The difference. 107. What is a minuend ? A subtrahend? A difference? 108. From 728 Add subtrahend and difference. If the take 516 work is correct, the result will equal the minuend. HORN. GRAM. SCH. AR. 2 18 INTEGERS AND DECIMALS 109. From 824 Subtract the difference from the minu- take 512 end. If the work is correct, the result will equal the subtrahend. 110. Complete these equations. Illustrate. Minuend Subtrahend = Mm. Dif. == Dif. + Sub. = 111. Find the number for which x stands in the following : Minuend Subtrahend Difference a 240 x 160 b x 16 30 c 40 x 10 d 60 50 x 112. CLASS EXERCISE. - may give a minuend and a difference. The class may find the subtrahend. SUGGESTION TO TEACHER. Every pupil should be required to bring to the class his contribution to the class exercise carefully prepared. 113. CLASS EXERCISE. - may give a subtrahend and a difference, and the class may find the minuend. 114. How can you prove an example in subtraction ? 115. Subtract and prove : a b c d e f g 849 623 814 338 599 451 2148 321 517 276 124 378 239 1939 116. Multiply 123 by 3. By 30. By 300. Find the sum of the products. Compare the sum of the products with the product of 123 multiplied by 333. 117. Multiply 275 by 137 I n this example of what two num- bers is 1925 the product ? 8250 ? 275QO 27500 ? How is 37675 obtained ? 37675 FUNDAMENTAL OPERATIONS AND PROOFS 19 SUGGESTION TO TEACHER. Show that in multiplying by a number of two or more places we are finding the sum of the several products of the multiplicand and the number expressed by each figure of the multiplier in its present position ; and that in practice the naughts are omitted for the sake of convenience. 118. Multiply a number of 3 places by a number of 4 places, writing out the full partial products. Why are the full prod- ucts not usually written out ? 119. What is a multiplicand ? A multiplier ? A product? Illustrate. 120. Use 25 as a multiplicand and 17 as a multiplier. 121. If the product of 25 and 17 is divided by 17, what result will be obtained ? If the product of 25 and 17 is divided by 25, what result will be' obtained ? 122. When the product of two numbers is divided by one of the numbers, what result is obtained ? Illustrate. 123. The product of two numbers is 35, the multiplicand is 7. Find the multiplier. 124. Take Ex. 123, substituting in turn for 35 the numbers 84, 42, 77, 91, 112. 125. Find the values of x. Product Multiplicand Multiplier a 75 x 15 b x 14 3 c 70 x 10 d 60 6 x 126. Complete the equations. Illustrate. Multiplicand x Multiplier = Product -r- Multiplier = Product -r- Multiplicand = 127. How can you prove an example in multiplication ? 128. Multiply and prove : a 18 x 20 b 13 x 14 c 15 x 16 d 14 x 25 20 INTEGERS AND DECIMALS 129. Multiply and prove : a b c d e f 836 457 791 625 927 654 125 243 348 244 238 289 130. What is a dividend? A divisor? A quotient? Illustrate. 131. Find quotients of 800 -f- 2, 60^2, and 8-5-2. Add the quotients and compare the sum with the quotient of 868 -r- 2. SUGGESTION TO TEACHER. Show that in dividing 868 by 2, we divide 800 by 2, then 60 by 2, and then 8 by 2. 132. Divide by 3 first 900, then 60, and then 6, and find the sum of the quotients. Show a shorter way of dividing the sum of those numbers by 3. 133. Divide 400 and 80 and 4 each by 4. Show the usual way of dividing the sum of these numbers by 4. 134. 764 -f- 2 = ? In dividing 764 by 2 we divide 600 by 2, then 160 by 2, and then 4 by 2. 135. 976 -i- 4 = ? (800 -v- 4).+ (160 - 4) + (16 -h 4) = ? 136. Divide 765 by 5, and show how many hundreds, how many tens, and how many units are used in the separate divisions. 137. Divide 5468 by 4, and show how many thousands, how many hundreds, how many tens, and how many units are used in the divisions. 138. Divide 3765 by 5, and show what parts of the number are used in each division. 139. What number divided by 8 will give 2 for a quotient ? 140. The quotient of a certain number divided by 8 is 7. What is the dividend ? FUNDAMENTAL OPERATIONS AND PROOFS 21 141. Complete the following equations. Illustrate. Dividend -+- Divisor = Dividend -*- Quotient = Divisor x Quotient = 142. Find the values of x. Dividend Divisor Quotient Dividend Divisor Quotient a 72 4 x c x 21 5 b 99 x 11 d 32 x 4 143. Of what number is 11 both divisor and quotient ? What number has for divisor and for quotient 7 ? 12 ? 13 ? 15? 17? 21? 144. Use 21 as a divisor and 378 as a dividend. Multi- ply divisor by quotient, and compare the result with the dividend. SUGGESTION TEACHER. Using small numbers, show that as multi- plication and division are reverse processes, they may be used to prove each other. 145. Make a rule for proving an example in division when there is no remainder. 146. Divide each of the following numbers by 23, and prove your work : 322 391 575 759 874 943 1288 147. Use 41 as a divisor and 618 as a dividend. Multiply divisor by quotient, add the remainder, and compare the result with the dividend. 148. Make a rule for proving an example in division when there is a remainder. 149. Divide each of the following numbers by 25, and prove your work : 625 879 579 824 386 758 1028 981 22 INTEGERS AND DECIMALS 150. Divide 26103 by 12 as follows : 12)26103(2175 24000 2103 1200 How many times is 12 contained in 24,000 ? How much, remains to be di- vided by 12 after 24,000 is subtracted from 26,103? 63 60 SUGGESTION TO TEACHER. Show the process of dividing each re- mainder after the successive subtractions, and call attention to the fact that it is more convenient to omit ciphers and that it gives the same result. 151. Divide some numbers by others by long division, writ- ing out all the work. Why is it better usually to omit some of the work ? 152. A schoolboy brought this example to his teacher and 121)87493(723 told her that he had discovered that . if the numbers here printed in heavy tvpe were added in the order in which 27Q ' * they stand, the result would equal the *^* dividend. He proved his problems in 373 long division in that way. Take an example in long division and prove it 10 in the same way. 153. Can you see why adding all the subtrahends and the remainder in long division will give a result equal to the dividend ? SUGGESTION TO TEACHER. Show that in this example 700 times 121 = 84,700, 20 times 121 = 2420, 3 times 121 = 363, and that the sum of these numbers, plus the remainder 10 must equal the dividend. 154. Divide a number by another number that it contains exactly 8 times. Double your dividend and see how the quo- tient is changed. FUNDAMENTAL OPERATIONS AND PROOFS 23 155. Work with small numbers and show the truth of the following principles : PRIN. 1. Increasing the dividend increases the quotient. PRIN. 2. Decreasing the dividend decreases the quotient. PRIN. 3. Increasing the divisor decreases the quotient. PRIN. 4. Decreasing the divisor increases the quotient. 156. Some of a milkman's customers buy a pint of milk at a time, and some buy a quart. How many of the customers who buy a pint will together dispose of a gallon ? How many who buy a quart ? 157. How many pints of water can be drawn from a 20- gallon tank ? How many quarts ? How many gallons ? 158. A 40-gallon tank contains how many times as many pints as a 20-gallon tank ? 159. Illustrate the following principles with small numbers : PRIN. 5. Multiplying both dividend and divisor by the same number does not change the quotient. PRIN. 6. Dividing both dividend and divisor by the same number does not change the quotient. 160. Find the product of 8 and 10 and divide that product by 4. If the following expression of the problem were used, 8 x 1Q , and if before multiplying, both 8 and 4 were divided 2 by 4 as follows, $ x 10 = 20, would the result be the same ? F SUGGESTION TO TEACHER. Let pupils prove by trial with many small numbers that canceling common factors in dividend and divisor does not change the quotient. 24 INTEGERS AND DECIMALS 161. Find the values of the following, canceling when you can: a b c 4 x 8 x 6 _ 6 x 9 x 18 21 x 4 x 6 2x4x2 3x3x6~ 7x2x3 d e / 40 x 8 x 10 3 x4 x 5 16 x = 9 ~ 20x4x5 7x8x10 8x8 21 x28x4 2 x 8 x 12 30 x 7 x 5 _ 9 _ 9 7x7x6 5x4x6 60 x 21 x 15 ^ k I 80 x 4 x 25 = ? 24 x 7 x 10 = 9 35 x 6 x 9 = 9 16 x 20 x 5 12 x 14 x 20 18 x 7 x 5 x 3 " Can you show how the process of cancellation depends upon Prin. 6 ? 162. Cancel and find values : a b c 5 x 8 x 10 x 27 7 x 9 x 56 x 65 21 x 84 x 6 x 8 54x4x20 26 x 49 x~32 x 5 12x49x18 d e f 25 x 6 x 7 x 30 16 x 9 x 28 42 x 15 x 10 x 6 50x9x5 8x36x2 25x14x5 163. CLASS EXERCISE. With 24 x 30 x 35 as the dividend, - may make a cancellation exercise, and the class may solve it. 164. CLASS EXERCISE. With 28 x 12 x 36 as a divisor, may give a cancellation exercise for the class to solve. 165. Copy upon the board and read 1,111,111. 166. Point out the figure that expresses a hundred thousand. How many hundred thousand make a million ? FUNDAMENTAL OPERATIONS AND PROOFS 25 167. What part of a million is a hundred thousand ? 168. How much is y 1 ^ of one hundred thousand? Show the figure that expresses it. 169. How much is y 1 ^- of ten thousand ? Show the figure that stands for it. 170. Show the figure that stands for y^- of a thousand, y 1 ^ of a hundred, y 1 ^ of ten. 171. Copy 111.1. The point after the units is called a Decimal Point. 172. The mtmber that is y L of 1 is written .1. Eead 111.1. This should be read 111 and 1 tenth. 173. 1 at the right of the tenths' place means y 1 ^ of yL or yi . How many tenths and hundredths in .11 ? 174. 1 at the right of the hundredths' place means y^- of j-^. or 10 1 00 . How many tenths, hundredths, and thousandths in .111 ? 175. What does 1 at the right of the thousandths' place mean? What does 1 in the next place to the right mean? In the next ? 176. Numbers written at the right of the decimal point are called Decimals or Decimal Fractions. They decrease in value at a tenfold rate from left to right just as integers decrease. A decimal of one place expresses tenths. What does a decimal of two places express ? Of three places ? 177.' Eead: .1 .2 .9 .07 .007 .09 .009 178. Eead .19. This is read 19 hundredths (^%). A decimal is read like an integer, and then the name of the last decimal place is added to show what kind of fractional parts it represents. 179. Eead: 12.1 4.3 8.01 7.02 9.021 .003 .009 .03 .006 26 INTEGERS AND DECIMALS 180. Write and read a decimal of two places. Of three places. Of four places. 181. Write in words : .123 4.5 .4 41.41 .103 19.2 .15 3.75 8.6 41.041 21.109 11.025 182. Eead: 576.137 432.25 57.41 32.06 75.37 45.81 183. Write in one number an integer of 4 places and a decimal of 3 places. Read it. 184. Write and read a number consisting of an integer of 6 places and a decimal of 2 places. 185. Write and read a number in which there is an integral part of 3 places and a decimal part of 2 places. 186. Write and read a number in which there is an integral part of 5 places and a decimal part of 3 places. 187. Which is greater and how much, the integer 1 or the fraction ^ ? Fractions whose denominators are 10, 100, 1000, or 1 with any number of ciphers annexed, may be written as decimals. This way of expressing such fractions is convenient because when thus expressed they may be added, subtracted, multiplied, and divided in the same way as integers. 188. Write as common fractions : .12 .029 .125 .17 .27 .013 .049 .019 Notice that the denominator of a decimal is not written, but is indicated by the number of places it occupies. It is always 1 with as many ciphers annexed as there are places in the decimal. Thus in .32 the denominator is 100. 189. Write as decimals: 190. Write as decimals : A TTnT A TTOF ifo TTfr T5~G ADDITION OF DECIMALS 27 191. Express T 9 ^, using only one figure. Express y^ by two figures, y-^ by three figures. 192. Write as common fractions : .6 .06 .027 .125 .0004 .00025 .000005 193. If you cut a string into ten equal parts, in how many places must you cut it ? What is each part called ? Express it as a decimal. 194. How ma^iy times must a string be cut to divide it into 100 equal parts ? Into 1000 equal parts ? 195. Show on a ruler .1 of 10 in. .1 of 5 in. 196. If 20 pupils were in a class and .1 of them were dis- missed, how many would remain ? 197. How much is .1 of 100 ? .01 of 100 ? Why can you not easily show on a ruler .001 of an inch ? ADDITION OF DECIMALS 198. Read and add: a be 462.001 321.12 725.375 25.01 56.37 409.003 63.475 81.07 361.1 181.312 73.22 448.0035 692.436 195.87 772.6 199. In adding decimals why is it best to arrange them so that the decimal points are in a vertical line ? 200. Write and add : a 57 and 123 thousandths, 181 and 28 hundredths, 49 and 3 tenths. 6 167 and 4 tenths, 2128 and 4 hundredths, 396 and 4 thousandths. 28 INTEGERS AND DECIMALS c 821 and 47 hundredths, 526 and 47 thousandths, 2936 and 1 tenth. d 674 and 37 hundredths, 25824 and 128 thousandths, 1948 and 4 tenths. e 666,666 and 6 tenths, 7,777,777 and 77 hundredths, 88,888,888 and 888 thousandths. 201. Add: a 7.3 in. and 4.7 in. b 8.4 sq. yd. and 5.6 sq. yd. c 3.35 sq. ft. and 4.95 sq. ft. 202. .1 is how many times as great as .01 ? 203. In the expression 333.333, which 3 expresses the great- est value ? The least value ? 204. 3 in the first integral place expresses how many times as much as 3 in the first decimal place ? 205. 5 in the first decimal place expresses how many times as much as 5 in the third decimal place ? 206. Which figures in a decimal stand for the greater value, those near the decimal point or those far away from it ? ' I TJ "O w ^ fl T3 13 3 3 73 1 . 1 f I ' I a. *- ' : .1 I I I I- I 1 I | a I I I : H S 1 a I I 1 207. GLASS EXERCISE. Copy the diagram, placing a figure in each decimal place, and read the numbers thus expressed. 208. How much is T J of .01 ? T V of .001 ? ^ of .0001 ? of .00001 ? ADDITION OF DECIMALS 29 209. Give values of x: a x = 1 in the 4th decimal place. b x = 1 in the 6th decimal place. ^ c x = 4 in the 5th decimal place. d x = 7 in the 4th decimal place. e q = 6 in the 6th integral place. / x = 2 in the 6th decimal place. 210. Add, and write the sum in words : a b c 1.235 24375 325685 123.5 2.4375 .325685 .1235 .24375 3.25685 211. Write and read a decimal of 4 places. Of 5 places. Of 6 places. 212. Which integral place is occupied by millions ? Which decimal place by millionths ? 213. Give the places of the following: Thousands, thou- sandths, ten-thousands, ten-thousandths, hundreds, hundredths, hundred-thousands, hundred- thousandths. 214. The expression .3 shows that some unit is divided into 10 equal parts, and that 3 of those parts are taken. What does .003 show ? .0003 ? .00003 ? .000003 ? 215. When we have .3 of an inch, what unit has been divided into 10 equal parts ? Explain the expression " .7 of a foot." .17 of a dollar." 216. If A and C were each 3 in. from It, 1 ' ' how far apart would they be ? 217. If A and Owere each 122.57 mi. from B, how far apart would they be ? 218. If A is 7.7 in. from B, and C is "J ? 3.8 in. from B, how far apart are J^and <7? UNIVERSITY 30 INTEGERS AND DECIMALS 219. CLASS EXERCISE. Let pupils give different lengths (with decimals) to AB and BC, and find distance from A to (7. 220. By rail the distance from Nashville, Tenn., to Evans- ville, Ind., is 155.07 mi., and from Evansville to Chicago, 111., 287.15 mi. Mary Allen lives in Nashville. How far will she travel in going from her home through Evansville to Chicago and returning by the same route ? 221. How many cents equal 1 hundredth of a dollar ? .17 ? 222. Add: a b c d $48.33 $ 75.25 $ 81.39 $ 813.45 76.48 38.60 47.50 425.15 13.15 49.76 86.72 327.40 SUGGESTION TO TEACHER. Call the attention of pupils to the fact that they have been using decimals of a dollar in their work with dollars, dimes, and cents. SUBTRACTION OF DECIMALS 223. Subtract: a b c d e 446.35 674.37 821.42 123,478.008 964,821.88 29.78 338.49 365.17 1,939.981 283,464.79 224. Which is greater and how much, .6 of a dollar or 225. Which is greater, .1 or .10? .4 or .40? .50 or .5? .7 or .700? 226. Write a decimal. Annex a cipher to it, and tell how the value of the decimal is affected. Without changing values, 227. Change to hundredths : .7. 2.1. 45.3. 228. Change to thousandths : .25. .4. 8.1. 2.56. 229. Change to ten-thousandths : .125. 2.4. .17. SUBTRACTION OF DECIMALS 31 230. Change to hundred-thousandths : .4758. 3.56. .417. .9. 231. Change to millionths: .85674. 18.35. 42.7. .489. .9249. 232. Write the expression .3. Place a cipher between the decimal point an0 24 P oint of the Q uotient directly under that of the dividend as ' 9 soon as it is reached. In this case the quotient of 9 units divided by 7 is 1 unit with a remainder of 2 units. The decimal point should be placed after the 1 unit before the division is continued. 420. By 7 divide: 2.583. 1.0332. 4.1328. 421. Find values of x: a b c d 19.64 38.82 5.76 _ 21.60 X = X = X = X 4686 e ' f 9 * x = 343.7 x = 13.25 g = 848.8 x = 1.989 48 INTEGERS AND DECIMALS 422. Find the length, of one side of an equilateral triangle whose perimeter is 7.5 in. 423. By 9 divide: 8.811. 34.569. 672.3. 4712.31 424. $ 12,384.75 were divided among 5 heirs. How much did each receive ? 425. In one week Mr. A. earned $ 123.66. What were his average earnings for each working day of the week ? 426. Find J of .0076. 4). 0076 Queries. How many tenths in \ of .0 ? How many .0019 hundredths in \ of .00 ? How many thousandths in \ of .007 ? How many ten-thousandths in \ of .0036 ? 427. Find 1 of .008. Of .016. Of .246. 428. By 8 divide : .01728. .002016. .12102. .025832. 429. Divide .12 by 9, carrying the division to three places of decimals. 9). 120 Annexing a cipher to .12, we have .120, which is .0131 equal in value to .12. .120 divided by 9 equals . 430. How many ten-thousandths in the quotients of the following ? 1.34 87.1 .128 .542 76.4 69783 431. Divide to three places of decimals: H. Sfi. 5if. 4 o 5 432. Divide 22.75 by 13. 1.75 13)22.75 In dividing a decimal by an integer by long division, 13 write the quotient above the dividend and place the ~~97 decimal point of the quotient above the decimal point ?J_ of the dividend as soon as it is reached. 65 433. By 21 divide: 8.82. 26.04. 10.353. 4.1349. 434. By 32 divide: 5.44. 1.632. .11424. 20.48. DIVISION OF DECIMALS 49 435. By 24 divide: .3456. .5184. .241584. .5544. 436. Divide 55.44 by 44. By 28. By 77. 437. If $ 3.15 were divided among 15 boys, how much would each receive ? 438. If hats are bought at $ 8.64 a dozen, how much does one hat cost ? 439. If $ 250 were divided equally among 3 men, how many dollars and cents would each man receive ? 440. If the following sums of money were divided equally among five persons, how many dollars and cents would each person receive ? $124 $661 $946 $12823 $67847 441. What is | of 18.24 ? Of 17.52? Of 86.25? 442. If 768.32 acres of land were divided equally among 16 men, how many acres would 3 men receive ? 443. If 11 doors cost $19.25, how much would 2 doors cost? 444. Multiply 549.36 by 3. By 4J. By 81 By 12J. 445. 46.125x21 = ? 46.125 x3fc=? 446. How much will one knife cost at $ 9.00 a dozen ? At 10.50 per dozen ? At $ 15.00 per dozen ? 447. How much is gained on each hat by buying hats at $ 20 a dozen, and selling them at $2.00 apiece ? 448. How much is gained on each quart of milk : a Bought at $ .28 a gallon, sold at $ .08 a quart ? b Bought at $ .25 a gallon, sold at $ .07 a quart ? c Bought at $ .30 a gallon, sold at $ .09 a quart ? 449. In buying milk at $ .20 a gallon and selling it at $ .06 a quart, how many quarts must a milk dealer sell to gain $ 1.00 ? HORN. GRAM. SCH. AR. 4 50 INTEGERS AND DECIMALS 450. In buying balls at $ 1.00 a dozen and selling them for $ .10 apiece, how much is gained on each ball ? 451. When 45 yards of calico are bought for $ 1.35, and sold at $ .05 a yard, how much is gained on each yard ? 452. Divide 8.64 by 2. If both dividend and divisor were ten times as large as they are, what would the quotient be ? SUGGESTION TO TEACHER. The principle, " Multiplying both dividend and divisor by the same number does not change the quotient," should be thoroughly reviewed and illustrated before the following work is done. 453. Divide 3.76 by .2. If both dividend and divisor are multiplied by 10, we have 37.6 ~ 2. This is similar to previous problems. 454. By the following rule perform this example in division of decimals, and give reason for the rule. 1.96 -r- .4. To divide by a decimal Move the decimal point of the divisor to the right until the divisor is an integer. Move the decimal point of the dividend an equal number of places to the right, annexing ciphers if necessary. Divide, and point off as many decimal places in the quotient as there are then in the dividend. 455. Divide each of the following by .09 : 1.125 12.33 43.119 62.91 4.815 456. Use .06 as a divisor with the following dividends : 221.4 13.2 54.6 91.2 .636 5940 2100 457. Find values of x : - a 78.3 / 168 .35 b x = Ws 9 x- 7S X ~T5 c *- 49 - 7 d x- 37 - 5 e x- SSA ' .14 k X - 65A "12 ".15 i ^_165 ~ .08 i a- 89 - 1 ~^06 MISCELLANEOUS EXERCISES 51 458. A music teacher earned $ 100 in a month, giving lessons at $ 1.25 each. How many lessons did she give ? 459. At 75^ a yard, how many yards of lace can be bought for $ 12.75 ? For $ 23.25 ? 460. Find quotients : Dividend Divisor Dividend Divisor a 2.25 1.5 / 2.057 12.1 b 2.75 2.5 g 3.144 1.31 c 137.5 1.25 h 539.6 14.2 d 396 1.2 i 114.92 .221 e 4.84 1.1 j 603.2 .232 461. How long is a rectangle which is a 5 ft. wide, and contains 35 sq. ft.? b .3 ft. wide, and contains .75 sq. ft.? c .7 ft. wide, and contains .77 sq. ft.? d .5 cm. long, and contains .125 sq. cm.? e .9 in. long, and contains .72 sq. in.? MISCELLANEOUS EXERCISES 1. Add, 1248.375, 115.67241, 3935.5428, and 138.463249. 2. From 13 thousand and 21 thousandths take 11 hundred and 4 hundredth s. 3. Multiply .246 by .89. 4. Divide 243.26647 by .98. 5. Write in Arabic notation and find the sum : MI, MV, MX, ML, MC, MD. 6. Find the sum of all the numbers less than 100 that are expressed in Eoman notation by 2 letters. By 3 letters. By 4 letters. By 5 letters. By 6 letters. By 7 letters. 7. Subtract from 100 the number less than 100 that is expressed in Eoman notation by 8 letters. 52 INTEGERS AND DECIMALS 8. Express decimally and add : 137 and 17 hundredths, 23 thousand 67 and 19 ten-thousandths, 38 thousand 5 and 11 millionths. 9. From 256 thousand 17 and 15 thousandths take 128 and 129 ten-thousandths. 10. How many square feet in a rectangle 1.75 ft. long and 1.25 ft. wide ? How long is its perimeter ? 11. How wide is a rectangle that contains 1.92 sq. in. and is 1.6 in. long? How long is its perimeter ? 12. 1.44 is how many times .0012 ? 13. A merchant bought $2125.75 worth of goods, and sold them so as to gain 12% of the cost. How much did he gain ? 14. Mr. Duncan bought goods that cost him $ 1226.35, and sold them so as to gain 16%. For how much did he sell them? 15. A man died, leaving $ 12,000. He willed 50% of it to his wife, 30% to his daughter, and the rest to a library. How much did each receive ? 16. Thomas bought a dime's worth of ice cream, which was only 50% of the amount he wanted. How many cents' worth of ice cream did he want ? 17. Jennie has 7 cents, which is 25% of her sister's money, and 50% of her brother's money. How many cents has each of them ? 18. $ 45.75 is 25% of how many dollars ? 19. How many pounds in 25% of a ton ? In 10% ? 20% ? 20. Mr. Wade invested $870, and gained 10% on it in a year. How much had he at the end of the year ? 21. Mr. Brooks invested $9000, gained 10% on it in the first year, and added the gain to his capital. He gained 10% MISCELLANEOUS EXERCISES 53 on. that amount in the second year, and added it to his capital. During the third year he increased his capital by 10%. Find how much he had at the end of each year. How much more than his original investment had he at the end of the third year? 22. How much is 100% of 2 watermelons ? Of 4 chairs ? 23. A chair that cost $ 3 was sold at a gain of 100%. For how much was it sold ? 24. What number is as much greater than 10 as 10 is greater than 8 ? 25. What number is as much more than 20 as 20 is more than 17 ? As much less than 20 as 20 is less than 21 ? 26. What is the average of 10 and 16, or what number is as much greater than 10 as it is less than 16 ? To find the average of two numbers, divide their sum by 2. To average three numbers, divide their sum by 3. To average four numbers, divide their sum by 4, etc. 27. What is the average of 18 and 20 ? Of 4 and 50 ? Of 9, 21, and 24 ? Of 8, 12, and 25 ? 28. If you stand 98% on an arithmetic test, 95% on a spelling test, and 92% on a geography test, what is your average per cent ? 29. Joseph worked 9 problems on Monday, 12 on Tuesday, and 12 on Wednesday. How many problems a day did he average ? 30. What is the average of 8.48, 10.24, and 4.96 ? 31. Mr. Harris earned $25.37 in one week, $38.75 the next week, $ 31.25 the next week, and $ 40.50 the next week. How much were his average earnings during the four weeks ? 32. High-water mark at a certain town on a large river was 48.3 ft. one year, 50.5 ft. the next, and 47.6 ft. the next. WJiat was the average ? 54 INTEGERS AND DECIMALS 33. What was the average height of a river for four suc- cessive days, if on the first day it was 33.9 ft. high, on the next 34.3 ft., on the next 34.9 ft., and on the next 35.1 ft. ? 34. Mr. Howe invested $ 36,000 in business. At the end of 8 years his capital was $ 64,000. What was his average gain per year ? NOTE TO TEACHER. The following work requires a Fahrenheit ther- mometer and an explanation of its use. 35. How many degrees are there between the freezing point and the boiling point on the Fahrenheit thermometer ? 36. The temperature at noon for four successive days in July was 90, 91, 92, and 84. Find the average temperature. 37. The temperature at noon for five successive days in January was 21, 19, 15, 18, and 27. Find the average temperature. 38. The temperature of our bodies is about 98. How much above freezing point is that ? Below the boiling point ? 39. How many degrees does the temperature fall when it changes from 57 to 3 below the freezing point ? * 40. What is the decrease in temperature from 20 to 1 above ? From 20 to 1 below ? From 15 to 5 below ? 41. The temperature at Minneapolis one winter day was 11. Before night it fell 20. What was the temperature then ? 42. The next day it rose 15. What was the temperature? 43. The next morning it was 4 below 0. How much had it fallen ? 44. Mrs. A. bought 19^ worth of groceries and offered $ 1.00 in payment. The clerk gave her 1^ and said, " Twenty." Then he gave her a nickel and said, " Twenty-five." He then gave her a quarter saying, " Fifty." He ended by giving her a half doliar and saying, " One dollar." In the same way find how that amount of change could be given with different coins. MISCELLANEOUS EXERCISES 55 45. CLASS EXERCISE. may tell a story of a purchase made and payment offered. Members of the class may show different ways of making change. 46. Draw two horizontal lines and two vertical lines. 47. Lines which lie in the same direction are called Parallel Lines. Find parallel lines in the room. In your book. Name some capital letters that have parallel lines when printed. 48. Think of your own name printed in capitals. Can you see any parallel lines in it ? 49. How many pairs of parallel lines has a rectangle ? Are there any parallel lines in a triangle ? A 4.5 B 50. A four-sided figure that has only 2 ^ ^\ parallel sides is called a Trapezoid. AB c and CD are parallel. How long is the perimeter of the trapezoid ABCD, the measurements representing inches ? 51. The sum of the parallel sides of Fig. 12 is how much more than the sum of the non-parallel sides ? 52. Draw trapezoids of different shapes. 53. Two lines meeting at a point form an Angle, Z. The point where the lines meet is called the Vertex of the angle. Draw an angle and mark its vertex A. A 54. When one straight line meets another straight line so as to make two equal angles, the angles are called Right Angles. What letter is at the vertex of each angle in Fig. 13? Right 55t Pl ace two pencils so as to show two Right Angle Angle right angles. 56. Fold a strip of paper so that the FIG. 13. line of the fold makes right angles with 1 the edge. 56 INTEGERS AND DECIMALS 57. Cut out a paper circle and fold it into fourths. What kind of angles are made by the folds ? 58. Find right angles made by lines in the surfaces of the room or of objects in it. 59. An angle less than a right angle is called an Acute Angle. Draw an acute angle. SUGGESTION TO TEACHER. As children naturally FIG. 14. Judge of the size of an angle by the length of the lines that form it, pupils should draw and cut out a right angle, and by applying it to given angles, find out whether they are acute, right, or obtuse. 60. Draw a trapezoid and mark the acute angles. 61. An angle greater than a right angle is called an Obtuse Angle. Draw an obtuse angle. FIG. 15. 62. Draw a trapezoid and mark obtuse angles and acute angles. 63. What kind of angles has a rectangle ? An equilateral triangle ? 64. Draw a trapezoid that has two right angles. Name the other two angles. 65. In the printed words "ADMIRAL DEWEY," how many right angles are there ? Acute angles ? Obtuse angles ? 66. If the name of the county in which you live were printed in Gothic type like the words " ADMIRAL DEWEY," how many right angles would there be in it? How many acute angles? How many obtuse angles? SUGGESTION TO TEACHER. Splints or toothpicks are useful in the following exercises. 67. With 3 lines make 2 right angles; 2 obtuse angles; 2 acute angles. Show the vertices of the angles. MISCELLANEOUS EXERCISES 57 68. With 2 lines make 4 angles, and tell of what kind they are. 69. With 3 lines make 12 angles, and tell their kinds. Make 10 angles. 9 angles. 70. With 4 lines make 16 angles. 20 angles. 24 angles. 71. With 5 lines make 4 angles. 5 angles. 20 angles. 72. CLASS EXERCISE. may tell how many angles he can make with a certain number of lines, and the class may make them. 73. A triangle that has a right angle is called a Right Triangle. Draw a right triangle. 74. Draw an isosceles triangle. The angles at the base are equal. What kind of angles are they ? 75. Draw an isosceles triangle on paper. Cut it out and fold it so that the equal sides coincide. Cut along the line of the fold, and you have two equal triangles. What kind of tri- angles are they ? 76. How long would the perimeter of one of these right tri- angles be if the base were 20 in., perpendic- ular 11 in. longer than the base, and hypot- enuse 8 in. longer than the perpendicular ? << Base FIG. 16. 77. Place together the two triangles you have made so that they form a rectangle. If the area of that figure were 28 sq. in., what would be the area of each right triangle ? 78. A figure drawn upon a flat surface is called a Plane Figure. Can you draw a plane figure on the surface of a ball ? Of a slate ? Of a piece of gas pipe ? 58 INTEGERS AND DECIMALS FIG. 17. 79. A plane figure bounded by five straight lines is called a Pentagon. When (as in Fig. 17) the lines are all equal and make equal angles, the figure is called a Regular Pentagon. What kind of angles has a regular pentagon? 80. Find the length of the perimeter of the pentagon represented by Fig. 17. 81. How long is one side of a regular pentagon whose perim- eter is 9.15 in. ? 82. Draw a pentagon that is not regular. 83. The pentagon in Fig. 18 is o divided into triangles by equal lines drawn from its center to the vertices of its angles. What kind of triangles are thus formed ? How many of them ? Each triangle is what part of the penta- gon? What %? FIG. 18. 84. What % of the pentagon is the figure ABCO ? AEDO ? BCDEO? CDEABO? 85. Figure 19 differs from Fig. 18 in having the lines Og, Oh, etc., drawn from the center of the pentagon to the middle point of each side. They are perpendicular to the sides. Each right triangle thus formed is what part of the pentagon ? What % ? E i D FIG. 19. 86. What % of the pentagon is AOB? EOi? BOhC? ABCh? 87. Give the outlines of a figure which is 70% of the pentagon. MISCELLANEOUS EXERCISES 59 88. A butcher bought a hog weighing 375 Ib. at $ .03 a Ib. How much did it cost ? 89. He sold 15 Ib. of it at 2^ per Ib., 50 Ib. at 5^ and the rest at 12^ per Ib. How much did he receive for it ? 90. A farmer sold 15 doz. eggs at 18^ a dozen, receiving for them sugar at 6^ a pound. How many pounds of sugar did he receive? 91. A fruit dealer buys 29 doz. oranges for $8.70. How much does he pay for each orange ? 92. If he sells the oranges at the rate of -5^ apiece, how much does he gain on each orange ? How much on all the oranges ? 93. What number multiplied by 9 will give the same prod- uct as 12 multiplied by 6 ? 94. Mr. Hale had $ 5728 and paid 25% of it for a farm. How much did the farm cost ? He sold the farm for $ 1200. How much did he lose ? ^ ; IVA A ' ^ 95. He left the other 75p of his money in the bank until it had gained $472 interest. How much money had he then, including the money he received from his farm ? 96. A grocer bought 185 barrels of flour at $ 3.75 a barrel, and sold it all for $ 740. How much did he gain ? 97. A miller bought 35 bu. of wheat for $ 22.75, and sold it at $ .61 a bushel. Row much did he lose ? 98. A farmer had an orchard of 276 trees. One year they averaged 13 bu. of apples to each tree. What was the value of that season's crop at $ .75 a bushel ? 99. The next year the trees averaged 9 bu. per tree, and the apples brought $ .80 a bushel. What was the value of the crop that year ? 60 INTEGERS AND DECIMALS 100. A merchant's profits in January, 1899, were $ 1428.75. In January, 1900, his profits were 20 % less. What were his profits in January, 1900 ? 101. Mr. Strong had $975.85 in a bank; he drew a check on the bank for $ 625.47. How much money had he remain- ing in the bank ? SUGGESTION TO TEACHER. Show bank checks. Explain their use and let pupils copy and fill them out for use in imagined transactions. 102. If you had $ 65.87 in a bank, and should draw a check for $ 38.45, how much of your money would be left in the bank ? 103. Mr. Gale had $1225 in a bank. He drew $12.25 every Saturday night for 10 weeks. How much had he left in the bank ? 104. Aline deposited $ 11.75 in a savings bank in February. She drew out $ 3.25 in March and $ 2.95 in April. She de- posited $ 14.45 in May. How much had she then in the bank ? 105. Mr. Davis bought a stove worth $ 18.75. The dealer allowed him 2^ a pound for an old stove, weighing 195 Ib. He gave a check on the bank for the balance. What was the amount of the check ? 106. There were 276 houses on the street. A postman delivered 3 letters at 28 of the houses, 2 letters at 41 of the houses, and 1 letter at 105 houses. At how many houses were no letters delivered ? 107. There were 559 books in a school library, which was an average of 13 to each pupil. How many pupils were there in the school ? 108. In February of a common year, Mr. Fisk's family burned a ton of coal in 14 days. At $ 8.50 per ton, what was the cost of the coal for that month ? 109. A lot is in the form of a trapezoid. One of the parallel sides is 16.8 rd. long, and the other is twice as long. Of the MISCELLANEOUS EXERCISES 61 sides that are not parallel, one is 19.7 rd. long, the other is 15.4 rd. long. How long a fence is required for the whole lot ? 110. Mr. Lee started to Denver with $ 300. He paid $ 47 for railroad fare, his hotel bill was $ 4 a day for a week, other expenses $ 7.50, and his return ticket was $ 47. How much did he spend ? How much had he left ? 111. Passengers were first carried on railroads in the United States in the year MDCCCXXVIIL How many years have we had railroads ? 112. Square: 13. 1.5. 1.7. 113. Ella had a flower bed a yard square. She divided it into square feet and placed a rosebush in the middle of each square foot. How many rosebushes had she ? Represent. 114. Draw a square decimeter and show into how many square centimeters it can be divided. Into how many square inches can a square foot be divided ? 115. How many square inches in 5 squares whose sides are each 11 in. long ? 116. How long is the perimeter of a square, a side of which is 3.1 in. long ? What is its area ? 117. Draw a right triangle. If its base were 7.5 in., its perpendicular 10 in., and its whole perimeter 30 in., how long would the hypotenuse be ? What would be its area ? 118. Find the length of the perpendicular of a right triangle whose perimeter is 90 in., its base 22.5 in., and hypotenuse 37.5 in. Find the area of the triangle. CHAPTER II PROPERTIES OF NUMBERS 1. When an integer can be divided by another number without a remainder, it is said to be divisible by that number. Is 9 divisible by 5 ? Give a reason for your answer. 2. Choose an even number and illustrate this statement : An even number is an integer that is divisible by 2. 3. Choose an odd number and illustrate this statement: An odd number is an integer that is not divisible by 2. 4. Name the first even number after 10. How many twos does it equal ? 5. Square the third odd number. Square the sixth even number. Multiply the seventh even number by the fifth odd number. Find the difference between the sixth odd number and the eighth even number. 6. Find | of the fifth even number. Find 7% of the fourth odd number. MULTIPLES AND FACTORS 7. A Multiple of a number is the product obtained by multiplying it by an integer. Thus 5 is the first multiple of 5, 10 is the second multiple of 5. Give quickly the first twelve multiples of 3 4 5 6 7 8 9 10 11 12 8. Figure 1 represents two lots of land owned by Mr. Smith and Mr. Brown. The fence between the lots is 150 ft. long MULTIPLES AND FACTORS 63 and cost 7^ a foot. How much of this expense should each man pay ? Give reasons for your answer. . , _ 9. Mr. Smith and Mr. Brown decided to take away the fence and leave a strip 10 ft. wide on each side of the line where it had stood. This strip is used as a com- mon playground by the children of both families. How many square feet in their common playground ? 10. A fine park in Boston is called Bos- ton Common. What does the word " com- mon " mean in this case ? " 20 is a com- What does that statement mean ? FIG. 1. mon multiple of 10 and 5." 11. A number which is a multiple of two or more numbers is called their Common Multiple. Give several numbers that are common multiples of 2 and 3. Of 3 and 7. Of 4 and 5. 12. Of what two numbers besides itself and 1 is 15 a mul- tiple? 10? 35? 21? 22? 33? 13. Write all the numbers of which 6 is a common multiple. 8. 14. 16. 12. 24. 40. 36. 14. Give two numbers which multiplied together make the product 18. 20. 27. 15. The numbers that make a product are called the Factors or Divisors of that product. 12 has 3 pairs of factors, 1 x 12, 2x6, and 3x4. Give all the pairs of factors of 24, except the pair of which 1 is the least factor. SUGGESTION TO TEACHER. Let pupils find factors of a number by using as a trial divisor each number in succession, beginning with 2. Lead them to see that as soon as the quotient which they obtain is less than the divisor they use, it is unnecessary to try" any more numbers, as they will merely get the same pairs of factors stated in* reverse order. 16. Give all the pairs of factors of 45. 28. 36. 60. 72. 17. Give all the numbers of which 30 is a multiple. 66. 64 PROPERTIES OF NUMBERS 18. CLASS EXERCISE. may give a number which is a multiple of some other numbers, and the class may find all its factors. COMPOSITE NUMBERS 19. A number that is the product of two or more integers is called a Composite Number. Give three, composite numbers and their factors. 20. What number is composed of the factors 2 and 11 ? 3 and 11 ? 2, 3, and 11 ? 7 and 7 ? 2, 3, and 7 ? 2, 2, and 3 ? 21 . What factors compose 77 ? 40 ? 18 ? 42 ? 22. Name an even composite number, and give factors that compose it. 23. Name an odd composite number, and give factors that compose it. 24. Name a composite number that is a multiple of 5, and give its other divisors. 25. Make and keep a list of all the composite numbers less than 41. 26. "Write a composite number whose tens' digit is 2, and give its factors. 27 . Write the following numbers and their factors : a A composite number whose units' figure is 5. I) The first composite number after 26. c A composite number between 30 and 40 that is not a mul- tiple of 5. d A composite number between 30 and 40 that is not a mul- tiple of 2 nor of 5. e A composite number between 20 and 30 that is not a mul- tiple of 2, 5, nor 7. 28. Divide .00168 by the 3d composite number. PRIME NUMBERS 65 29. Find 5% of the 9th composite number. 30. Multiply the 8th composite number by .009. PRIME NUMBERS 31. A number that has no integral factors except itself and 1 is called a Prime Number. Think of each of the numbers from 2 to 10 and tell which of them are prime. 2 is the first prime number, as 1 is considered neither prime nor composite. 32. No prime number of more than one place ends in 2, 4, 6, 8, 0, or 5. Can you tell why ? 33. Make a list of all the prime numbers less than 50 in their order. SUGGESTIONS TO TEACHER. Show pupils how to find prime numbers less than 50 by examining each number to see if it can be exactly divided by 2, 3, 5, or 7. Develop the fact that there is no need of dividing even numbers because they are all divisible by 2 ; nor numbers that end in 5, because they are divisible by 5. Lead pupils to see that if a number will not contain 2, it will not contain 4, 6, 8, or any number of twos ; that if it will not contain 3, it will not contain 9, 15, 21, or any number of threes, and so on. ' 34. Group the prime numbers less than 50 as they are found in each ten numbers, as : 1st ten 2dten 11 13 17 Keep the list. 19 35. Find the sum of all the prime numbers that are ex- pressed by one digit. 36. Divide 13.5 by the 2d prime number. 37. Divide the 3d prime number by .8. 38. Find the difference between 26.4 and the llth prime number. HORN. GRAM. SCH. AR. 5 66 PROPERTIES OF NUMBERS 39. Multiply the 8th prime number by .0004. 40. Find 6% of the 9th prime number. 41. Write the first 30 numbers in two lists, one of prime numbers, the other of composite numbers. Leave out the number 1. 42. Find the sum of all the composite numbers less than 15. 43. Find the sum of all the primes less than 20. 44. What prime number is nearest to 20 ? 45. What two prime numbers are near to 12? 46. 15 is half way between two prime numbers. What are they ? There are three pairs of these numbers. 47. What prime number is nearest to the 2d multiple of 5 ? To the 8th multiple of 5 ? SUGGESTION FOR CLASS EXERCISE. Let children select prime numbers and give a clew to them, and let the class identify them. 48. Find the difference between the prime number nearest to 20 and the prime number nearest to 8. 49. CLASS EXERCISE. may name a number larger than 50 which he thinks is prime, and the class may see if he is right. NOTE TO TEACHER. The following method of finding prime numbers less than 100 is very useful : Write the first hundred numbers as on p. 67, omitting 1 because it is con- sidered neither prime nor composite. Counting from 2, the first prime number, strike out as composite every second number because it is a multiple of 2 ; counting from 3, strike out as composite every third number. Lead the pupils to discover why it is unnecessary to strike out the multiples of 11 or of any larger primes in finding the prime numbers less than 100. This device, which is an adaptation of the well-known "sieve of Eratosthenes," may be used to any limit to separate prime from compos- ite numbers by writing additional columns of numbers and striking out all the multiples except the first multiple of those primes whose squares are included within the limit. t ' n f\ pr PRIME NUMBERS 67 .- 11 2; 31 41 $1 61 71 2J 91 2T9 99 319 AW 39 ft ' 79' S9 Q9 A^ ^ P^ rr Pr rr l*r Pr PP 3 13 23 33 43 53 03 73 83 ?3 7 17 27 37 .47 57 67 77 27 97 1% ?$ 32 2 ^2 02 72 22 R2 ? 19 29 3? 9 59 0? 79 89 99 50. A Greek mathematician named Eratosthenes, who was born 275 B.C., devised this plan of finding prime numbers. Instead of marking out the composites, he cut them out. Can you see why the table of primes that was left was called " Eratosthenes' sieve " ? 51. How many and what prime numbers in the 2d ten numbers ? In the 10th ten ? In the 5th ten ? 52. Give the primes less than 100 whose units' digit is 1. 3. 7. 9. 53. Find the sum of all the primes in the 3d ten. In the 6th ten. In the 4th ten. In the 7th ten. 54. Name all the prime numbers less than 100 whose tens' digit is 2. 4. 1. 3. 5. 7. 9. 6. 8. NOTE TO TEACHER. This adaptation of "Eratosthenes' sieve" may be made helpful in studying composite numbers, divisors, and multiples. Let the columns of numbers be written on the board in large figures. Instead of striking out multiples, draw a circle or square around each of them, using crayon of the same color to inclose the multiples of a partic- ular number. If, for instance, the multiples of 2 are inclosed in blue, those of 3 in red, those of 5 in green, those of 7 in yellow, 30 shows itself at once by its motley framing as a multiple of 2, 3, and 5 ; 42 by its slightly different framing as a multiple of 2, 3, and 7 j 66 as a multiple of 2, 3, and 11. 68 PROPERTIES OF NUMBERS The following questions are based upon a diagram of that kind and refer to numbers less than 101. 55. Point out the multiples of 3 whose units' digit is 2. 3. 5. 6. 7. 8. 9. 4. 56. Point out the multiples of 7 whose units' digits is 1. 2. 4. 9. 57. What figure ends the expression of all the multiples of 5 that are odd numbers ? Even numbers ? 58. Point out all the numbers that are multiples of 2 and also of 5, beginning with the least. 59. Show the common multiples of 2 and 3, beginning with the least common multiple. 60. Show the common multiples of 2 and 7, beginning with the least common multiple. 61. Beginning with the least common multiple, show all the common multiples of 3 and 7. Of 3 and 5. Of 5 and 7. Of 2, 3, and 5. Of 2, 3, and 7. Of 2, 5, and 7. 62. CLASS EXERCISE. The teacher or a pupil pointing to a number in the diagram, members of the class tell of what numbers it is a common multiple, and whether or not it is their least common multiple. PRIME FACTORS 63. Those factors of a number that are prime, are called Prime Factors. What are the prime factors of 4 ? Of 6? Of 8? 64. Find the prime factors of 42. 2 1 42 To find the prime factors of a number divide it by the smallest 3 "21 prime number of which it is a multiple. Then divide the quo- 4r tient by the smallest prime number of which it is a multiple. Continue dividing until the quotient is prime. In this case 42 divided by 2 gives a quotient of 21, 21 divided by 3 gives the prime quo- tient 7, Hence the prime factors of 42 are 2, 3, and 7. LEAST COMMON MULTIPLE 69 65. Find the prime factors of: 10 15 20 24 27 32 35 39 44 48 12 16 21 25 28 33 36 40 45 49 14 18 22 26 30 34 38 42 46 50 66. Find the prime factors of all the even numbers greater than 49 and less than 59. Of all the composite odd numbers between those limits. 67. Find the prime factors of all the even numbers between 59 and 69. Of all the composite odd numbers. 68. Find the prime factors of all the multiples of 5 between 69 and 91. Of all the multiples of 3 between those limits. Of all the composite numbers between 90 and 101. 69. Resolve into prime factors : a 336 c 1225 e 639 g 3105 t 1470 fc 1296 b 3456 d 2214 / 560 h 888 j 810 I 1488 70. CLASS EXERCISES. may give to the class a number that is the product of several small prime numbers, and the class may find its prime factors. 71. Separate the first 100 numbers into two lists, one of prime numbers, the other of composite numbers. Write oppo- site each composite number the prime factors of which it is composed. SUGGESTION TO TEACHER. Class drill upon these lists should be given frequently until their contents are learned. LEAST COMMON MULTIPLE 72. Forty contains how many more fives than eights ? 73. CLASS EXERCISE. '- may name a number which is a common multiple of two or more numbers. The class may give the numbers and tell how many times their common multiple contains each of them. 70 PROPERTIES OF NUMBERS 74. The Least Common Multiple of two or more prime numbers is their product ; the next common multiple is twice their product ; the next is three times their product. What is the next ? Write the first six common multiples of 2 and 5, and underscore the least. 75. Write the first four common multiples of 3 and 5. Give the least common multiple and find how many times 3 and 5 are each contained in it. 76. Write the first three common multiples of 2 and 7. Find how many times 2 and 7 are each contained in their least common multiple. 77. Find the least common multiple of 2, 3, and 5. How many threes does it contain ? How many twos ? How many fives? 78. Find the least common multiple of 2, 3, and 7. How many sevens in it ? Twos ? Threes ? 79. The abbreviation for least common multiple is 1 .c. m. Find the 1. c. m. of 2, 5, and 7. How many fives in it ? Sevens ? Twos ? 80. Find the 1. c. m. of 5 and 9. Use the following method of finding the 1. c. m. mentally : Think of the multiples of the larger number in order until one is found which is a multiple of the smaller. For instance, in finding the 1. c. m. of 5 and 9, think of the multiples 9, 18, 27, 36, until the first that contains 5 is reached. 81. Find the 1. c. m. of : a 10 and 3 10 and 5 10 and 8 10 and 12 10 and 15 b 6 and 9 4 and 9 7 and 9 11 and 9 8 and 9 c 8 and 3 8 and 5 8 and 6 8 and 10 8 and 11 82. Find the 1. c. m. of 6, 5, and 3. Can a number be a multiple of 6, without being also a multiple of 3 ? LEAST COMMON MULTIPLE 71 83. Find the 1. c. m. of : a 6, 2, and 3 10, 5, and 2 20, 10, and 5 7, 5, and 6 b 10, 5, and 6 8, 4, and 12 5, 10, and 15 8, 4, and 3 c 2, 3, 4, and 6 3, 4, 8, and 6 4, 8, and 7 3, 6, and 8 84. 12 is the 1. c. m. of 2, 3, 6, and 4. Make similar state- ments about the numbers 18, 20, 24, 25, 30, 36, 35, and 48. 85. Which of the first 12 multiples of 3 are common mul- tiples of 12 and 3 ? 86. Can you find the greatest common multiple of 3 and 4? Of any two other numbers ? Explain. SUGGESTION TO TEACHER. Some of the pupils inay discover that the search for the greatest common multiple leads into infinity. 87. Draw a line 18 in. long, and show how many times a 3-in. line can be laid off upon it. How many times can a 6-in. line be laid off upon it ? 88. How long is the shortest line that can be laid off into 2-in. lines or 7-in. lines? 3-in. lines or 7-in. lines? 7-in. lines or 5-in. lines ? 5-in. lines or 11-in. lines ? 89. How large is the smallest number that can be divided into groups of 2 and of 7 ? Into groups of 3 and of 7 ? Into groups of 7 and of 5 ? Into groups of 5 and of 11 ? 90. CLASS EXERCISE. may name a composite num- ber, and others may show the different groups into which it may be separated. 91. A company of children were playing games. At first they played games which required them to be divided into groups of three. Afterwards they played in groups of four. Every child played all the time. What is the least number of children there could have been in the company ? SUGGESTION TO TEACHER. Illustrate, by grouping children, for the benefit of those pupils who cannot imagine clearly. 72 PROPERTIES OF NUMBERS 92. How many roses must a girl have, to be able to divide them into bunches of 3 roses or bunches of 5 roses ? How many bunches may she have of 3 roses ? Of 5 roses ? 93. A teacher has just enough pupils to divide into groups of 7 pupils or groups of 4 pupils. How many pupils has she ? How many groups of each kind can she have ? 94. I have just enough books to be arrangedon a number of shelves, 12 books on a shelf, or by using more shelves, 9 books on each shelf. How many books have I ? How many shelves would be needed under the first arrangement ? Under the second ? t 95. What is the least number of gallons that can be exactly measured by either of two casks, one holding 6 gal., the other 8 gal.? How many times can the smaller cask be filled by them ? The larger cask ? 96. What is the smallest sum of money that can be en- tirely spent in buying books at 15 ^ apiece, or in buying books at 9^ apiece? How many of each kind of books could be bought? 97. Ho.w long is the shortest piece of ribbon that can -be cut without remainder into lengths of 2 yd., 3 yd., or 5 yd. each ? How many lengths of each kind could be made ? 98. What is the least number of bananas that a mother can exactly ivide between her 2 sons, or among her 4 daughters, or among all her children ? How many bananas would each child receive in each case? NOTE TO TEACHER. Written methods of finding the 1. c. in. and g. c. d. are useful, because convenient, but the reasons for the processes are beyond the comprehension of ordinary children in the grade for which this work is designed. Hence the methods should be presented as convenient rules that have been discovered by mathematicians. The reasons for these rules should be learned later. ' 9 LEAST COMMON MULTIPLE 73 99. By the following rule find the 1. c. m. of 8 and 10 4 To find the least common multiple of several numbers Arrange the numbers in a horizontal line, leaving out all numbers that are factors of any of the other numbers. Find the smallest prime number that will exactly divide anyjwo^ofjhejn^ and divide by it all the numbers of which it is a factor, placing the quotients and undivided numbers below. Continue this pro- cess until no prime number will divide two o the numbers in the last horizontal line. Find the product of the divisors, quotients, ojid undivided numbers. 100. Find the I.e. m. of 12, 16, and 18. SOLUTION. 2 12, 16, 18 2 x 2 x 3 x 4 x 3 = 144 1. c. m. 6, 8, 9 3, 4, 9 1, 4, 3 ' 101. Find by the written method the 1. c. m. of: a 25, .60, 72, 35 c 63, 12, 84, 72 e 54, 81, 14, 63 b 24, 16, 15, 20 d 16, 12, 24 / 15 9, 6, 5 102. Find 1. c. m. of 5, 6, 18, 15. 3)?, 0, 18, 15 6, 5 3 x 6 x 5 = 90. Ans. Why may the 5 and 6 be crossed out and not considered in finding the 1. c. m. of 5, 6, 18 and 15 ? 103. Find 1. c. m. of: a 1, 2, 3, 4, 5, 6, 7, 8, 9 d 4, 8, 12, 24, 48, 84 b 8, 12, 16, 24, 36, 48 e 5, 10, 15, 20, 30, 40 c 4,3,6,7,8,16,9, / 7,28,35,14,70 104. CLASS EXERCISE. may name four composite num- bers, and the class may find their 1. c. m. 105. CLASS EXERCISE. may name three prime num- bers, and the class may find their 1. c. m. 106. Numbers that are multiples of any given number are said to be divisible by that number. Is 7 divisible by 3 ? Name several numbers that are divisible by 10. 74 PROPERTIES OF NUMBERS DIVISIBILITY OF NUMBERS 107. Illustrate the following principle: PRINCIPLE 1. A number that ends in 2, 4, 6, 8, or is divisi- ble by 2. 108. Tell without dividing which of the following numbers are not divisible by 2, and what the remainder is in each case : 8906. 2127. 2139. 2111. 2145. 1898. 109. Name in order the first fourteen multiples of 5. 110. When a multiple of 5 is expressed in figures, what figures may represent the units' digit? PRINCIPLE 2. A number that ends in 5 or is divisible by 5. 111. Without dividing, select from the following the num- bers that are not divisible by 5, and tell what the remainder is in each case: 75. 120. 81. 22500. 393. 920. 112. When a number is divisible by 2 and by 5, with what figure must its expression end ? By what other number is it divisible ? 113. Among all the prime numbers less than 100, can you find any the sum of whose digits is 9 ? 114. Write a number of two places the sum of whose digits is 9. Find how many times 9 is contained in that number. 115. Write a number of three places the sum of whose digits is 9, and find how many times 9 is contained in that number. 116. Write a number of four places the sum of whose digits is 9, and find how many times that number contains 9. 117. Write a number the sum of whose digits is 18, and see whether it contains 9 exactly. 118. Write numbers the sum of whose digits is 9 or some multiple of 9, and divide those numbers by 9 until you see the truth of the following principle : DIVISIBILITY OF NUMBERS 75 PRINCIPLE 3. Any number is divisible by 9 if the sum of its digits is a multiple of 9. 119. Tell without dividing which of the following num- bers are not divisible by 9, and give the remainder in each case: 2025. 105. 117. 112. 2114. 189. 207. 1026. 4154. 120. Write several large numbers of which 9 is a divisor. 121. Write ten multiples of 3 no one of which is less than 36. Find the sum of the digits of each of them, and see if that sum is a multiple of 3. PRINCIPLE 4. A number is divisible by 3 if the sum of its digits is a multiple of 3. 122. Tell at sight which of the following numbers are not divisible by 3, and give the remainder in each case : 213. 411. 6951. 343. 1125. 123. Can you find a multiple of 9 that is not a multiple of 3 ? Name a multiple of 3 that is not a multiple of 9. 124. Would it be possible for a number to be a multiple of 10 and not a multiple of 2 and 5 ? Explain. 125. Choose numbers ending in 0, and show what factors they have besides 2, 5, and 10. 126. Can a number be divisible by 6 without being divisible by 3 and by 2 ? 127. Write an even number the sum of whose digits is divisible by 3. As the number is divisible by 3 and by 2, it is divisible by 6. Write three other numbers divisible by 6. 128. Write three numbers each of which is divisible by 3 and by 5. Find how many times each of them contains 15. 129. Write three numbers ending in the sum of whose digits is divisible by 3. Find how many times each of them contains 30. 76 PROPERTIES OF NUMBERS 130. Write three numbers divisible by 2 and by 9. Divide each of them by 18. 131. Write three numbers divisible by 5 and by 9. What number between 40 and 50 is a factor of each of them ? How can you tell ? 132. Count by 4's to 100. 133. Add some multiples of 4 to 100, and see if the num- bers thus obtained are divisible by 4. 134. Add to 100 some numbers that are not multiples of 4, and see if the resulting numbers are divisible by 4. Explain. PRINCIPLE 5. A number is divisible by 4 if the number expressed by its two right-hand figures is divisible by 4. 135. Tell without dividing the whole number which of the following numbers are not divisible by 4, and give the remain- der in each case: 127. 244. 365. 782. 728. 496. 338. 2672. SUGGESTION TO TEACHER. Lead pupils to see that as 100 is a multiple of 4, any number of hundreds is a multiple of 4, and if there is added to any number of hundreds a number which is an aggregation of fours, the result will be a still greater aggregation of fours. 136. Write a number of four places. Let the number expressed by the two right-hand digits be a multiple of 4. Let the sum of the digits of the whole number be a multiple of 3. Find how many times 12 is contained in the number. 137. Write three numbers divisible by 4 and by 5. What multiple of 10 besides 10 is a factor of each of them ? Explain. 138. Write three numbers divisible by 4 and by 9. What number between 30 and 40 is a factor of each of them? Explain. 139. How many eights in 1000 ? 140. To 1000 add 3 eights or 24. How many eights in the number thus formed ? COMMON DIVISORS 77 141. Add to 1000 a number which is not a multiple of 8. Is the sum divisible by 8 ? Explain. PRINCIPLE 6. A number is divisible by 8 if the number ex- pressed by its three right-hand figures is divisible by 8. 142. Tell without dividing the whole number which of the following numbers are not divisible by 8, and give remainders : 3640. 5728. 9076. 4126. 5345. 1724. 8638. 1124. 10008. 143. Make some numbers which are divisible by 8 and by 5, and tell how you make them. With what figure do they end ? What two multiples of 10 besides 10 are contained in each of them? 144. Tell how to compose numbers that are divisible by 8, and also by 9 and hence by 72. COMMON DIVISORS 145. A number which is a factor of each of two or more numbers is called their Common Divisor. Illustrate. 146. Turn to the diagram on page 67 and find all the num- bers in it of which 11 is a common divisor. 147. Name all the numbers less than 100 of which 8 is a common divisor. Give all the numbers less than 100 that have as a common divisor : 9. 10. 12. 6. 148. What common divisor have all even numbers ? 149. Give a common divisor of 14, 21, 28, 35, 42, 49, 56,. and 63. 150. Name three multiples of 11, and give a common divisor of them. 151. Make a list of sets of numbers that have one or more common divisors, and write the Greatest Common Divisor of each set. 78 PROPERTIES OF NUMBERS SUGGESTION FOR CLASS EXERCISE. Let a pupil name two or more numbers that have a common divisor, and let the class discover the divisor. 152. What divisor is common to the 7th even number and the llth odd number ? 153. 2 is a common divisor of 10 and of 20. Is it the greatest common divisor of these two numbers ? What is the greatest common divisor of 24 and 36 ? 154. Give at sight the greatest common divisor of : a 10, 20, and 40 g 70, 80, and 90 b 15, 30, and 45 h 60, 72, and 84 c 18, 27, and 45 i 63, 72, and 90 d 16 and 24 ./ 28, 32, 40, and 44 e 50, 75, and 100 k 15 and 25 / 25, 30, and 35 I 12, 18, and 30 155. A candy manufacturer filled some boxes with choco- lates, and some others of the same size with bonbons. There were 24 Ib. of chocolates and 28 Ib. of bonbons. What is the largest number of pounds each box can contain ? 156. A boy wishes to divide two ropes, one 42 ft. long, the other 56 ft., into pieces of equal length, each as long as possible. How long will each piece be after this division, and how many pieces will there be ? 157. Mr. Allen has three strips of land. The first contains 10 acres, the next 12 acres, the next 14 acres. He wishes to lay them off into the largest possible equal lots. How many acres will there be in each lot, and into how many lots can each piece be divided ? 158. The abbreviation for greatest common divisor is g. c. d. What is the g. c. d. of 35 and 65 ? POWERS AND ROOTS 79 The g. c. d. of two numbers may be easily found by the following process of continued division. It is to be used with numbers which are so large that their divisors cannot be readily found by inspection. 159. By the following rule find the g. c. d. of 8 and 10 : To find the greatest common divisor of two numbers Divide the greater number by the less. If there is a remainder, use it as a divisor of the preceding divisor, and continue until there is no remainder. The last divisor is the greatest common divisor. 160. Find by continued division the g. c. d. of 49 and 168. 49)168(3 Using 49 as a divisor of 168, the quotient is 3 and 147 remainder 21. Using 21 as a divisor of 49, the quo- 21,)49(2 tient is 2 and remainder 7. Using 7 as a divisor of 42 21, the division is exact, hence 7 is the last divisor, or 7)21(3 the g. c. d. of 49 and 168. 161. Find by continued division the g. c. d. of the following : a 24 and 132 i 77 and 847 q 198 and 252 b 36 and 120 ./ 18 and 243. r 176 and 242 c 35 and 105 k 96 and 224 s 361 and 431 d 49 and 140 I 85 and 187 t 288 and 536 e 64 and 480 m 125 and 175 u 84 and 154 / 72 and 252 n 105 and 195 v 189 and 405 g 30 and 735 o 135 and 245 w 960 and 204 h 44 and 242 p 795 and 1105 x 236 and 576 162. CLASS EXERCISE. Let compose two large num- bers having a common divisor, and let the class find this common divisor by continued division. POWERS AND ROOTS 163. The product obtained by multiplying a number by itself one or more times is called a Power of that number. Illustrate. 80 PROPERTIES OF NUMBERS 164. The product of two equal factors is the Square of each factor. 2 x 2 is expressed 2 2 , and read 2 square, or 2 to the second power. Find values of : I 2 . 30 2 . 50 2 . 120 2 . 15 2 . 20 2 . 165. Give quickly in order the first 12 numbers that are perfect squares. Learn them. 166. 9 2 equals how many times 3 2 ? 16 2 how many times 8 2 ? 8 2 equals how many times 2 2 ? 167. What two perfect squares less than 100 have 6 for their units' digit ? 9 ? 4 ? 1 ? 168. Can you find a perfect square less than 100 whose tens' digit is 9? 7? 5? 3? 169. How much is T V of 8 2 ? Of II 2 ? 170. Multiply 6 2 by the first prime number after 31. 171. The product of three equal factors is called the Cube of each factor. 2 x 2 x 2 is expressed 2 3 , and read 2 cube, or 2 to the third power. Find values of : 2 3 . 3 3 . 4 3 . 5 3 . I 3 . 172. Continue the following table through 12 3 . Learn the table. I 3 = 1 173. Name a perfect cube whose units' digit is : 1. 2. 3. 4. 5. 6. 7. 8. 9. 0. 174. Multiply the cube of 8 by .07. By .125. 175. Find 6% of :4 3 . 9 3 . 7 3 . II 3 . 50 3 . 12 3 . GO 3 . 176 *=* *=* ^=9 *=* i*=? ' 43 33 6 3 53 4 3 SUGGESTION TO TEACHER. By the following work lead pupils to dis- cover the relations between a solid that is a cube and the third power of the number that measures one of its dimensions. Inch cubes should be used in this work until pupils are able to image the solids clearly without them. POWERS AND ROOTS 81 177. How many cubic inches does a 2-inch cube contain ? A 3-inch cube ? A 4-inch cube ? A 5-inch cube ? A 6-inch cube ? A 7-inch cube ? An 8-inch cube ? A 9-inch cube ? 178. One of the boys may draw a square yard on the floor in one corner of the room. How many cubic blocks 1 ft. in dimensions would cover the square yard ? 179. One of the girls may show how high she thinks the blocks must be piled to make a cubic yard. Another member of the class may measure with the yard stick and see how nearly right she is. 180. How many cubic feet in the lowest layer of blocks? How many layers would it take to make a cubic yard ? . How many cubic feet in a cubic yard ? 181. How many layers of inch cubes would be required to cover a square foot ? How many layers of the inch cubes to make a cubic foot ? How many inch cubes in a cubic foot ? 182. Into how many 2-inch cubes can a 4-inch cube be divided ? 183. How many 6-inch cubes can be packed into a box whose inside dimensions are each 1ft.? 6 Square Centimeters FIG. 2. 184. Copy Fig. 2 on paper or pasteboard, making each square 1 sq. cm. Cut out the copy and fold and fasten it so that it will inclose a cubic centimeter. How many such cubes would be re- quired to cover a square deci- meter? How many layers of them would be required to make a cubic decimeter ? 185. A cubic decimeter is called a Liter, centimeters does it contain ? HORN. GRAM. SCH. AR. 6 How many cubic 82 PROPERTIES OF NUMBERS 186. Copy Fig. 2 on paper or pasteboard, making each square 1 sq. dm. Cut, fold, and fasten to make a cubic deci- meter or a liter. SUGGESTION TO TEACHER. The most perfect liters and cubic centi- meters made by the children should be kept as a part of the school apparatus. 187. How many square centimeters in all the surfaces of a liter ? 188. In the metric system the liter is the measure that cor- responds very nearly to the quart in the English liquid measure. How much will 12 liters of oil cost at 65 cents per liter? 189. Place a cubic centimeter upon a cube that holds a liter. How many cubic centimeters in the figure thus formed ? How many square centimeters in all its surfaces ? SUGGESTION TO TEACHER. It should be explained that a liter is a unit of measure, and not a fixed form. 190. How many liters will be contained in a box that is 3 dm. long, 2 dm. wide, and 5 dm. high? 191. About how many liters of wheat can be put into a peck measure ? 192. 2 x 2 x 2 x 2 is 2 4 , which is read 2 to the fourth power. Raise to the fourth power each of the first five numbers. 193. Kaise to the fifth power each of the first three odd numbers. 194. Raise to the sixth power each of the first two even numbers. 195. Raise 10 to the seventh power. 196. Give the number whose prime factors are 2, 2, and 3. 2, 3, and 3. 2, 5, and 5. POWERS AND ROOTS 83 197. Find values of x. a x = 3 2 x 5 e x = 2 s x 3 i x = 7 2 x 13 6 ^ = 2 2 x 7 / x = 2 6 x 5 j x = 2 3 x 5 x II 2 c a; = 2 3 x 3 4 g x = 3 3 x 11 A; a; = 2 s x 3 x 5 2 d = 2 4 x 3 3 h x = 5 3 x 11 J a> = 3 4 x 5 2 x 7 198. Each of the two equal factors that compose a perfect square is called a Square Root of that number. Give the square root of 9. 25. 16. 199. V is used as the sign of square root. Vl6 = ? = ? V144=? Vl21 = ? VI = ? V6l=? VlOO = ? 200. Divide .63 by Vl9. By V9. By V81. 201. Divide .36 by V9. By Vl6. By V4. By V8l. 202. Divide 5^44 by V1. By V36. By Vl21. By V49. By V64. By Vl6. By V9. By Vl44. 203. How long is one side of a square whose area is 9 sq.ft.? 49 sq. ft. ? 100 sq. ft. ? 81 sq. ft. ? 25 sq. ft. ? 204. How much is 50% of VlOO? 25% of V64? 75% ofV64? 25%ofVl44? 205. Multiply Vl44 by .3. By .05. By .007. 206. Find 6% of Vlll. Of V64. Of V8l. 207. How much is 3 times V9? 4 times V9? 208. "5V9" is read "5 times the square root of 9." Find the value of the expression. 209. 210. 3 V16 = ? 4 VlOO = ? 2 V81 = ? 6V64 = ? 3 Vlll = ? 211. How long is the perimeter of a square containing 49 sq. in. ? 84 PROPERTIES OF NUMBERS 212. At $ 1.25 per rod, how much will it cost to fence a square lot containing 25 sq. rd. ? Eepresent. SUGGESTION TO TEACHER. For oral "quick work" exercises similar to the following are useful : "Think of the 3d multiple of 6, subtract 2, take the square root, add 1, square, add 5, take ^, take j 1 ^, add 1, square, add 5, take square root." Allow children to lead the work, letting them prepare their numbers beforehand to read to the class, until they are able to extemporize. 213. Each of the three equal factors that compose a number that is a perfect cube is called a Cube Root of that number. Give the cube root of : 8. 512. 64. 1000. 729. 1331. 1728. 125. 216. 343. 214. Give the cube root of a perfect cube whose units' digit is: 1. 2. 3. 4. 5. 6. 7. 8. 9. 0. 215. How long is one side of a cube that contains 8 cu. in. ? 27cu.in.? 1728 cu. in.? 64cu.in.? 729cu.in.? 1000 cu. in. ? 512cu.in.? 216cu.in.? 125cu.in.? 343cu.in.? 216. ^/ is read "The cube root of." How much is V64? v/343? -v/729? -v/1728 ? 217. Multiply the -\/8 by the first prime number after 40. 218. Multiply -v/125 by .001. -^512 by .75. 219. Find 50% of A/64. Of ^/1728. 220. -\/729--v/2l6 = ? -^1000 -^- A/125 =? A/1331 X A/343 =? MISCELLANEOUS EXERCISES 1. Write in decimal form and add : 1 ten-thousandth, 1 hundred-thousandth, y^fo, T H^ 1 millionth, 1 00000* 2. If you have a string a foot long and cut one inch from each end, how long is the string that is left? MISCELLANEOUS EXERCISES 85 3. When a line 3.4 ft. long is cut from each end of a line that is 1 rd. or 16.5 ft. long, how long is the line that is left ? 4. The diagonal of a certain schoolroom is 35.1 ft. John makes a mark on the diagonal 7 ft. from one corner, and James makes a mark 9 ft. from the opposite corner. If each boy stands at the mark he has drawn, how far apart are they ? SUGGESTION TO TEACHER. In most classes there are some pupils who fail to visualize. Select two of these to take the parts of John and James in illustrating this and similar problems. 5. Find the sum of 81.375 and the prime number nearest to 24. 6. Find the difference between 21.84 and the largest prime that can be expressed by two digits. 7. When a decimal of 3 'places is multiplied by an integer, how many decimal places should be pointed off in the product? Illustrate. 8. When an integer is multiplied by a decimal of 2 places, how many decimal places should be pointed off in the product ? Illustrate. 9. How many decimal places in the square of .007 ? 10. 8.283-*- 3 = ? 45.6 --12=? .286-^22=? 11. Tell how you divide a decimal by a decimal. 12. .12-j-.4=? .15 -f-. 005=? .75 -=-.5=? .84 -.12=? 13. When one decimal is divided by another decimal of the same denomination, how many decimal places are there in the quotient ? 14. At $ .05 per pound, how many pounds of sugar can be bought for $ .45 ? For $ .75 ? $ 1.25 ? $ 2.50 ? $ 8 ? 15. Harriet has some money in the bank, the interest of which is $ 1.30 every year. How long must the money stay in the bank that the interest may be $ 5.85 ? 86 . PROPERTIES OF NUMBERS 16. 1.728 -s-. 0012=? 17. Give the prime factors of the first odd composite num- ber after 81. 18. The largest prime factor of 66 is how many times the smallest prime factor of 66 ? 19. Find the 1. c. m. of 3, 8, 4, 9, 6, 12. 20. Find the g. c. d. of 44 and 66. Of 128 and 144. 21. Divide 7235.2 by the 1. c. m. of 4 and 7. 22. Divide 4.725 by the g. c. d. of 45 and 105. 23. Find by cancellation the value of x : a 7x8x4 b 3x7x9 c 64x21 14x32 d 25x21 21 x 18 x 5 e 48x63 42 x 8 x 8 / 16 x 25 x 36 35x30 g 49 x 63 36 x 24 x 18 h 48 x 200 x 18 x 6 i 56 21x84 24x36 21 x 16 24. What is one of the two equal factors of 121 ? 25. Name a perfect square whose units' digit is 9, and give its square root. 26. Give quickly the first 12 numbers that are perfect cubes. 27. How long is one edge of a cube that contains 1000 cu. in. ? 1728 cu. in. ? 28. Give one of the three equal factors of 216. Of 729. 29. Find the difference between O 2 and I 2 . I 2 and 2 2 . 2 2 and3 2 . 3 2 and 4 2 . 4 2 and 5 2 . 5 2 and 6 2 . 6 2 and 7 2 . 7 2 and8 2 . 8 2 and 9 2 . MISCELLANEOUS EXERCISES 87 30. Write these differences in a column and tell whether they are even numbers or odd numbers. 31. Find the sum of the first 7 odd numbers. Compare that sum with the square of 7. 32. Compare the sum of the first 8 odd numbers with the square of 8. The sum of the first 5 odd numbers with the square of 5. Of the first 9 odd numbers with the square of 9. 33. Find the sum of the first 5 even numbers. Subtract the square of 5 from that sum. 34. Find how much the sum of the first 7 even numbers exceeds the square of 7. 35. Take Ex. 34, substituting other numbers for 7. 36. Ella's record on an arithmetic test was 75%. What fractional part of her work was right and what part wrong ? 37. Mr. Hudson had $ 8000 in bank and took out 20% of it. How much did he take out ? How much had he left ? 38. Edward buys oranges at the rate of 4 for 25 ^, which is just one half of what he receives for them. What is the selling price of each ? 39. A milkman's horse ran away with a wagon containing 4 gal. of milk, and 25% of it was spilled. How many quarts of milk were spilled ? If the milk was worth 6 ^ per quart, what money value was lost ? 40. From a liter of oil 13% was spilled. How many cubic centimeters of oil re- mained ? 41. Draw a circle. What is a radius ? Diameter ? Circumference ? The radius of a circle equals what part of the diameter ? 42. How long is the diameter of a circle FlG - 3 ' whose radius is 5 in. ? 3J in. ? 7.5 in. ? 88 PROPERTIES OF NUMBERS 43. What is the diameter of the largest circle that can be cut from a piece of paper 3 in. square ? 44. The surface passed over in 1 hr. by the minute hand of a clock is what figure ? The minute hand of a clock in a tower is 2-J- ft. long. How long is the diameter of the circle it passes over every hour ? 45. A plane figure bounded by six straight lines is called a Hexa- gon. When the sides are all equal, and the angles are all equal, as in Fig. 4, the hexagon is called a Regular Hexagon. What kind of angles has a regular hexagon ? 46. If each side of a regular hexagon is 6.75 in. long, how long is the perimeter of the hexagon ? FIG. 4. 47. Draw a hexagon that is not regular. SUGGESTION TO TEACHER. Show the following method of drawing a regular hexagon : Draw a circle with a radius of any convenient length. Beginning at any point of the circumference, lay off the radius as a chord six times consecutively. Erase the circle. 48. Draw a regular hexagon whose sides are each 3 in. long. How long is the perimeter ? How dees the side of a regular hexagon compare with the radius of the circle in which it is inscribed ? 49. By drawing diagonals the regular hexagon may be di- vided into 6 equilateral triangles. Draw them, and find how long each diagonal is. How long is the perimeter of each equilateral triangle ? 50. If the perimeter of the hexagon were 32.4 in., how long would the perimeter of each equilateral triangle be ? 51. How many triangles in 50% of the hexagon ? MISCELLANEOUS EXERCISES 89 FIG. 5. 52. A plane figure bounded by four equal straight lines, and having no right angles, is called a Rhombus. What kind of angles has a rhombus ? 53. Draw a rhombus by the following method : Draw the line AB of any convenient length. With AB as a base construct an isosceles triangle CAB, making AC greater than f of AB. With AB as a base construct an isosceles triangle ADB, making AD equal to AC. Erase the construction line AB. (A construction line is a line forming no part of a figure, but used simply to help in its con- struction.) 54. Construct a rhombus each of whose sides is 5 in. 55. Mr. Jones laid out a flower bed in the shape of a rhombus, each side of which was 4.75 ft. long. How long was the entire edge of the flower bed ? 56. If the entire edge had been 28.8 ft. long, how long would have been one side of the flower bed ? 57. Draw a rhombus and the long diagonal of the rhombus. Into what kind of triangles is the rhombus divided ? 58. If the side of a rhombus is 7.5 in., and its longer diago- nal is 10.875 in., how long is the perimeter of one of the tri- angles into which the long diagonal cuts the rhombus ? 59. Draw a circle and inscribe a hexagon. Join the vertex of each alternate angle with the center of the circle. Into what kind of figures is the hexagon divided ? How long would the perimeter of each of the figures be, if the radius of the circle were 8 cm. ? FIG. 7. 12 cm. ? 60. Each rhombus is what fractional part of the hexagon ? 90 PROPERTIES OF NUMBERS 61. Finish the following course of reasoning: Since the whole of anything equals 100% of it, .1 = 33J% of it. f = % of it. 62. Write in each rhombus the per cent which it is of the hexagon. Shade one rhombus and tell what per cent of the hexagon is unshaded. 63. How much is 331% of 12? 21? 24? 30? 45? 48? 64. How much is 66f% of 15? 27? 18? 36? 33? 6? 65. 331% of a school of 48 pupils are boys. How many girls are there ? 66. How much is 33^% more than $ 15 ? $ 300 ? $ 600 ? 67. How much is 66f % more than $ 900 ? $ 1200 ? $ 1800 ? 68. $ 3000 - 66f % of 3000 = ? 2100 - 66f % of 2100 = ? 69. Each of the equal sides of an isosceles triangle is 33^% longer than the base, which is 15 in. long. How long is the perimeter of the triangle ? 70. A merchant found that some of his goods were shopworn and marked them at a reduction of 25% of their cost. How were goods marked that cost 12^? 20^? 40^? $1.00? $1.60? $10.00? 71. Find the selling price of goods marked at the following prices, which are to be reduced in price 33^% on account of being out of style. Cloaks costing $ 7.50, bead trimming $ 1.50 per yard, lace ruffling $ .57 per yard. 72. A grocer bought goods at the following prises. For how much must they be sold to gain 33 J% ? 25% ? a Tomatoes @ 12^ per pound. d Oranges @ 24^ a dozen. b Raisins @ 6^ per pound. e Bananas @ 18^ a dozen, c Molasses @ 36^ a gallon. / Potatoes @ 30^ a bushel. MISCELLANEOUS EXERCISES 91 73. At the end of a season a merchant decided to reduce prices 33J% on all of the following goods whose prices were over $1.00 and to reduce them 66f % on all those whose prices were less than a dollar. Find the new selling prices. a Lace @ $ 1.80 per yard. d Silk @ $ 2.70 per yard. b Ribbon @ $ .75 per yard. e Velvet @ $ 1.68 per yard. c Calico @ $ .06 per yard. / Alpaca @ $ .60 per yard. 74. William is 15 years old. His age is 33|% of his father's age. How old is his father ? 75. Mr. Gage had $ 396.66 in a bank and took out 33^% of it. How much remained in the bank ? 76. Mrs. Wallace lent Mr. Brown $ 1200 until the interest amounted to 66f % of the principal. How much was the inter- est ? How much did Mr. Brown then owe, including principal and interest ? 77. Write the following fractions in a column and opposite to each its value in % : \. \. f . \. }. f f f . f . 78. How many minutes in 33^% of an hour? In 66f% ? 25%? 20%? 40%? 50%? 79. How many hours in 50% of the time from 9 A.M. Mon- day to 9 A.M. Tuesday ? In 331% o f it ? In 75% ? 20% ? 80. How many square centimeters in 80% of a square deci- meter ? In 25% ? In 331% ? 81. Draw the equilateral triangle ADC, one of whose sides represents 3 in. With DC as a base line construct another equilateral triangle ACB. Erase AC. What kind of a figure is ABCD ? How long is its perimeter ? 82. With either side of the rhombus as a base line, construct another equilateral triangle. Erase the base line. What kind of a figure have you drawn ? How long is its perimeter ? 92 PROPERTIES OF NUMBERS 83. Continue adding equilat- eral triangles until you have a regular hexagon. Complete the following reasoning. 84. Since the whole of any- thing = 100%, iofit= 16|% fofit=-%? FIG. 9. 85. On your copy v of Fig. 9 write in each equilateral triangle the % which it is of the hexagon. 86. If 16f % of the hexagon were shaded, what would be unshaded ? of it 87. What % of the hexagon is the figure AOCB? AODCB? ABCDEFO? CDEFAO? ' 88. How long is the perimeter of the six-pointed star repre- sented by Fig. 10 if each side is 3.5 in. ? What kind of angles are those whose ver- tices are at the points of the star ? 89. Make a six-pointed star. TJraw a regular hexagon, and construct an equilateral triangle upon each of its sides. Erase the sides of the original hexagon. The star may also be made by prolonging the sides of the hexagon until they meet. 90. Divide your star into 6 equal rhom- buses. Write in each rhombus the % which it is of the star. 91. Put a letter at the center and one at each angle of your copy of Fig. 11 and tell what figure is 33% of it. 831%. FIG. 11. 50%. 66|%. MISCELLANEOUS EXERCISES 93 92. What is 16f% of 12? Of 24? Of 72? Of 84? Of 120? Of 144? 93. What is 83% of 18 ? Of 30? Of 48? Of 66? Of 144? 94. To make a profit of 16f % for what price must goods be sold that cost 6^? 18^? 15^? 30^? 54^? 95. What must be the selling price of the same goods to allow a profit of 831% ? 96. Select from the following list the per cents which are most easily used by reducing them to common fractions in their lowest terms, and give the equivalent fractions : 33J% 11% 16f% 25% 831% 50% 17% 20% 3% 40% 75% 9% 66|% 97. James had a dollar and lost 17 cents. What per cent of his money was left ? 98. Mary has only a dollar. Can she lose 101% of it? Explain. 99. 20 equals what part of 30 ? Express it in per cent. 100. CLASS EXERCISE. may give a number, and the class may give 33|% of it. 16f %. 66J%. 831%. 101. CLASS EXERCISE. - may give a number, and the class may give the number of which his number is 16J%. 331%. 25%. 102. Draw a right triangle whose base is 3 in. and perpen- dicular 4 in. If your drawing is correct, the hypotenuse will be 5 in. Each side, of the triangle equals what part of its perimeter ? 103. Draw a right triangle whose base is 6 in. and perpen- dicular 8 in. Its hypotenuse is just twice as long as the hypot- enuse of the triangle given in Ex. 102. Each of its sides equals what part of its perimeter ? 94 PROPERTIES OF NUMBERS 104. Draw a square 3 in. in dimensions. If you drew a larger square, having each of its sides 1 in. from the corre- sponding side of the first square, how long would its per- imeter be? 105. Separate the following into two lists, one of odd numbers, the other of even numbers. How many are there of each ? 874; MDCCCLXXXVIII ; the square of 7 ; the fifth multiple of 4 ; the product of 7 and 8 ; the quotient of 84 divided by 2 ; the difference between 81 and 18 ; the sum of 85 and 37 ; the largest numbe*r that can be expressed by two figures ; the largest factor of 12 except itself ; the number that is 5 greater than 212 ; the largest number that can be expressed by three figures ; the smallest number that can be expressed by three figures ; the number that means a dozen ; the number that tells how many days in May; the integer between 17,345 and 17,347 ; one of the equal factors of 25 ; the factor that helps 7 to make 77 ; the square root of 100 ; the number that shows how many quarts in a peck; the denominator of the fraction T \; the greatest common divisor of 6 and 8 ; the remainder after dividing 25 by 11 ; the smallest multiple of 7 that will contain 5 ; the least common multiple of 4 and 7 ; the number that is just half way between 30 and 50; the smallest prime number greater than 25 ; the largest prime number less than 25; the numerator of the fraction |^; the number that tells how many square inches in a square foot ; the number that is just as much less than 15 as it is greater than 11; the average of 19, 20, and 21 ; the first composite number ; the number that shows how many pounds in a ton ; the number that shows how many cubic inches in a cubic foot ; the number that shows how many sides a pentagon has ; the largest prime number that can be written with two figures ; the smallest prime number that can be written with three figures ; the quotient of 13.14 divided by .06; 50% of 862; the largest prime factor of 102 j the number that shows how many millimeters in a meter. CHAPTER III RATIO SUGGESTION TO TEACHER. Review ratio as given in Hornbrook's "Primary Arithmetic." See notes on pp. 117 and 118 and tables on pp. 145, 160, 174, 183, of that book. 1. A 3-inch line equals what part of a 4-inch line, or what is the ratio of a 3-inch line to a 4-inch line ? 2. What is the ratio of a pint to a quart ? Of a quart to a gallon ? Of an inch to a foot ? Of a foot to a yard ? Of 2 ft. to a yard ? Of an ounce to a pound ? Of 8 oz. to a pound ? Of 15 oz. to a pound ? 3. CLASS EXERCISE. may name a number less than 100, and the class may give its ratio to 100. 4. 6 is how many times 3, or what is the ratio of 6 to 3 ? 5. What is the ratio of a yard to a foot? Of a foot to an inch ? Of a foot to 3 in. ? Of a foot to 6 in. ? Of a foot to 7 in. ? 9 in. ? 11 in. ? 6. CLASS EXERCISE. may name some number greater than 10, and the class may give its ratio to 10. 7. The ratio of two numbers is the quotient of the first of those numbers divided by the second. Thus the ratio of 10 to 5 is 10 -f- 5, or 2. The ratio of 7 to 5 is 7 -^ 5, or If. What is the ratio of 4 to 5 ? 8. Draw a rectangle 4 in. long and 1 in. wide. A rectangle 3 in. long and 1 in. wide equals how many fourths of the first rectangle ? A rectangle 8 in. long and 1 in. wide equals how many fourths of the first rectangle ? What do j equal ? 95 96 RATIO The ratios indicated by " parts " and " times " are really of the same kind. They both express the quotient of one quantity divided by another of the same kind. 9. What is the ratio of a second to a minute ? Of a year to a month ? 10. Build from inch cubes or draw right prisms like the following : 3x2x1 5x2x1 7 / / 2x2x2 3x3x1 a rn IIT I fir 5x1x1 7x2x1 I 1 1 1 1 . Find ratios of : a to b a to c b to c b to d c to b c to d FIG. 1. & to e e to d btof ftod c to/ c to e d to c /to a a to e 12. Mr. Jones works every day from 8 until 12 o'clock, and from 1 until 5 o'clock. At 9 o'clock in the morning, what is the ratio of the work he has done to the work he still has to RATIO 97 do that day ? What is the ratio of the work he has done to his whole day's work ? 13. At ten o'clock, what is the ratio of the work he has done to a day's work? At 12 o'clock? At 1 o'clock? At 3 o'clock? At 5 o'clock ? 14. What is the ratio of a rod to a mile ? 15. If your home is a mile from the schoolhouse, how many rods must you travel each school day of two sessions, if you go home at noon ? 16. Joseph rode a mile on his bicycle. When he had ridden a rod, what was the ratio of the distance he had ridden to that which he afterward rode ? 17. Ella walked to the home of her cousin, who lived a mile away. What was the ratio of the distance she had walked to the remaining distance after she had gone 16 rd. ? 32 rd. ? 80 rd. ? 120 rd. ? 18. What is the ratio of an ounce to a pound ? 19. Margaret had half a pound of candy and gave away all of it except one ounce. What was the ratio of what she had left to what she had at first ? 20. What is the ratio of a pound to a ton ? 21. Just after a ton of hay was weighed in market, a horse ate one pound of it. What was the ratio of what he ate to what was left ? 22. Ratio is expressed by a colon. Give ratios of : 15 : 3. 3:15. 16:2. 2:16. 3:18. 18:3. 5:20. 23. Give quickly the ratio of 2 to each of the first ten mul- tiples of 2. Give the ratio of the 2d multiple of 2 to each of the first ten multiples of 2. Do the same with the 3d multiple of 2. With the 4th, 5th, 6th, 7th, 8th, 9th, and 10th. 110KN. GRAM. SCH. AR. 7 98 RATIO 24. What is the ratio of the 2d multiple of any number to its 3d multiple ? Of its 2d multiple to its 4th ? Illustrate. 25. What is the ratio of 2471 to 17 ? 26. What is the ratio of 1.422 to 1.8 ? 27. The ratio of 3 to 6 is -J ; the ratio of 6 to 3 is 2. These two ratios between the numbers 3 and 6 are called reciprocal ratios. Give the reciprocal ratios between the following num- bers : 2 and 3. 3 and 5. 8 and 4. 9 and 12. 18 and 20. 28. 19 equals how many twentieths of 20 ? 20 equals how many nineteenths of 19 ? 29. John is 8 yr. old, and his sister Mary is 16 yr. old. What is the ratio of Mary's age to John's ? Of John's age to Mary's age ? 30. When is the ratio of one number to another number an integer ? Illustrate. 31. The line A B rep- ACT) JP f' B T I I I I 1 resents a distance of Fl - 2 - 35 mi. divided into 5 equal parts. How much is the distance AE ? CB ? AF ? EB ? 32. Find the ratio of AF to AB. CF to CB. AB to AD. AB to DB. AD to CF. AB to CF. 33. What is the ratio of the first composite number after 19 to the first composite number after 29 ? Of the first odd number after 20 to the first odd number after 5 ? Of the first prime number after 7 to the first even number after 20 ? 34. Draw a 2-inch square and a 4-inch square and find the ratio of each square to the other. 35. Find the reciprocal ratios of a 3-inch square and a 4-inch square. Of an 8-inch square and a 6-inch square. RATIO 99 36. What is the ratio of 3 V4 to 2 V25 ? 2V9:6Vl6 = ? 37. 2V49:3Vl21 = ? 3V36 : 2 VI44 = ? 38. 5Vl6:4V9 = ? 2 V81 : 3 V64 = ? 39. 3 VlOO : 5 Vl6 = ? 2\/25:5V36 = ? 40. Build with cubes and find the following ratios : An inch cube to a 3-inch cube. An inch cube to a 4-inch cube. A 2-inch cube to a 3-inch cube. 41. If a 5-inch cube is cut into inch cubes, what is the ratio of one of the small cubes to the large cube ? 42. What is the ratio of a cube an inch in dimensions to a 6-inch cube ? 43. 3 3 :4 3 =? 40 3 :60 3 = ? 11 3 :9 3 =? 6 3 :20 3 =? 44. Give quickly the cube root of : 27. 64. 8. 216. 512. 729. 343. 1000. 125. 45. A/125 : ^/lOOO = ? ^/729 : \/2l6 = ? -\/1728 : -J/512 = ? 46. Image the following figures and tell the ratio of one side of each figure to its perimeter. A square. A regular hexagon. A rhombus. A regular pentagon. Express the ratios in / . 47. What is the ratio of the perimeter of a square yard to the perimeter of a square foot ? Of the perimeter of a square centimeter to the perimeter of a square decimeter ? 48. What number is that whose ratio to 8 is f ? Or what is I of 8 ? 49. How much is | of 12 ? f of 49 ? f of 15 ? \ of 21 ? SUGGESTION TO TEACHER. In order to insure correct reasoning on the part of pupils, they should occasionally be required to explain the steps by which they arrive at results, as : since \ of 21 is 3, f of 21 are 4 times 3 or 12. After this is thoroughly understood, the habit of mental cancel- lation should be encouraged. For instance, in finding ^ of 21, children may be led to visualize the expression and mentally to cancel the terms. 100 RATIO | 50. Find values : | of 36 | of 56 of63 ^ of 77 J of 72 f of 36 51. If a boy earns $ 77 in 11 wk., how much would he earn in 3 wk. ? 5 wk. ? 9 wk. ? 52. When 9 yd. of calico cost 72^, what is the cost of 2 yd. ? 5 yd. ? 7 yd. ? 8 yd. ? 53. Goods that cost 8^ a yd. are sold for f of their cost. What is the selling price ? 54. Give quickly the selling price of goods : a Bought at $ 0.12 and sold at f of the cost. b Bought at f 0.18 and sold at more than cost, c Bought at $ 0. 20 and sold at \ more than cost. d Bought at $ 0.40 and sold at ^ more than cost. e Bought at $0.50 and sold at more than cost. / Bought at $ 0.60 and sold at ^ more than cost. g Bought at $ 0.80 and sold at \ less than cost. h Bought at $ 1.00 and sold at f of the cost. i Bought at $ 0.40 and sold at f of the cost. SUGGESTION TO TEACHER. Before the following work is taken up, pupils should be drilled in finding reciprocal ratios of pairs of numbers until they readily see the truth of the first statement in the solution of Ex. 55. 55. 6 is f of what number ? SOLUTION. 6 is f of the number that is f of 6. % of 6 is 2. f of 6 are 8. 56. 8 is | of? f of? of? I of? 57. 12 is | of? * of? fof? 58. Find the values of x. a 10 = of x. f of x. | of x. f of a. | of x. b 12 = | of x. | of x. f of x. |- of x. f of x. T 2 T of x. c 15 = | of x. f of x. f of x. f of x. | of x. f of x. KATIO 101 d 18 = f of ar. T % of x. T 6 T of . -& of x. f of a. f of a?, e 20 = J of a;. ^ of x. if of aj. -V- of x - V of ^ V of aj - / 5 = | of a;. 10 = fofx. 7 = |ofa?. 9 = $ of a;. 59. Anna's age is f of Mary's age. What is the ratio of Mary's age to Anna's age. If Anna is 12 years old, how old is Mary ? 60. James has 24 marbles. He has % as many as John. How many marbles has John ? 61. Land in one part of a certain county in Illinois is worth $20 an acre, which is only of the price of land in another part of the county. What is the price of the better land? What is the value of the land owned by Mr. Baxter, who has 40 acres of each kind ? 62. Mr. Walker sold gingham at 8^ a yard, which was f of what it cost him. How much did it cost ? 3. Find the cost of goods : a Sold at $ 0.09, which was f of the cost. b Sold at $0.20, which was f of the cost, c Sold at $ 1.98, which was f of the cost. d Sold at $ 2.97, which was f of the cost. e Sold at $ 1.47, which was -J of the cost. 64. 24 marbles will cost how many times as much as 12 marbles of the same kind ? 65. What is the cost of 24 marbles when 12 marbles cost $.25? $.08? $.60? 66. What is the ratio of the price of 10 hats to the price of 1 hat ? To the price of 2 hats ? 5 hats ? 7 hats ? 67. What will be the cost of 10 hats when 5 hats cost $3 ? $7? $9? $7.50? Use ratio. 10 hats cost how many times as much as 5 hats ? 102 RATIO 68. What will be the cost of 10 hats when 2 hats are worth $3? $4? $5.25? $7.65? 69. If 3 articles of the same kind cost $.17, how much will 12 such articles cost ? 18 articles ? 21 articles ? 30 articles ? 70. If 5 things cost $19, how much will 15 things of the same kind cost ? 71. Take Ex. 70, substituting another number for 5 and for 15 some multiple of the number that you have substituted. 72. If 10 acres of land are sold for $375, how much would 80 acres cost at the same rate ? 60 acres ? 100 acres ? 73. Find the cost of 48 oranges when 5^ are paid for 6 oranges. For 8 oranges. For 4 oranges. For 12 oranges. 74. If 15 marbles are sold for 9^, how much do 5 marbles cost ? 3 marbles ? 20 marbles ? MISCELLANEOUS EXERCISES 1. Add the square of 7.9 to 7.9. 2. Subtract the cube of 1.3 from 10. 3. Divide 14 2 by .007. 4. A regular hexagon is inscribed in a circle whose radius is 8 in. How long is the perimeter of the hexagon ? Eepresent. 5. How long is one side of a regular hexagon whose perim- eter is 5.4 cm. ? 6. How long is one side of a regular pentagon whose perim- eter is 8.45 cm. ? 7. Is it correct to say that an inch line is \ of a 4-inch line ? If the inch line were in Boston and the 4-inch line in New York, would the shorter line be a part of the longer ? What part of the longer line would the shorter line equal ? MISCELLANEOUS EXERCISES 103 8. What is the ratio of an hour to a day ? Of a week to a day? 9. Give the ratio of 21 to each of the first 12 multiples of 7. 10. What is the ratio of a square whose side is 3 ft. to a rectangle 9 ft. by 8 ft. ? 11. Find the prime factors of 546. Of 495. 12. Find the ratio of the largest prime factor of 35 to the largest prime factor of 39. Of the largest prime factor of 49 to the largest prime factor of 15. Of the smallest prime factor of 49 to the smallest prime factor of 15. 13. 60 3 : 30 3 = ? 6 3 : 5 3 = ? 7 3 : 5 3 = ? 8 3 : 12 3 = ? 14. 7V4:3V49 = ? 4V25 : 2 VTOO = ? 15. How long is the shortest line that can be divided into either 8-inch lines or 10-inch lines ? 16. Divide the 1. c. m. of 2 and 5 by the 1. c. in. of 3 and 5. 17. Divide the g. c. d. of 36 and 45 by the g. c. d. of 12 and 3. 18. Which power of 6 is 216 ? 19. Which power of 2 is 16 ? 64? 256? 20. Which power of 10 is the denominator of the decimal .11? .0125? .6? .345? .00^004? .000009? 21. AB represents a distance of 320 rd. J- 1 or 1 mi., which is just f of the distance from A to G. How far is it from A to C ? r IG. o. From B to C? 22. Mary had some money in a toy bank. She took out 35^ which was |- of it. How much was left ? 23. Lizzie spent 15^ for a singing book, which was J of the price of her arithmetic. The price of the arithmetic was -f of the price of her story book. How much did they all cost ? 104 RATIO 24. On Monday John rode on his bicycle 21 mi., which was f of the distance he rode during the rest of the week. How far did he 'ride during the whole week ? 25. A boy had 12 agates, for which he paid 60 He sold them for 6^ apiece. How much did he gain on each ? 26. Mr. Cooper paid $64, which was 50% of what he owed. How much does he still owe ? 27. When a man hires a house to live in, he is said to pay rent for it. When he hires money to use, he is said to pay in- terest for it. If you have deposited $ 100 in a bank that pays 4% interest, how much interest will you receive each year? 28. How much would you receive each year if you had $ 100 at6%? 3%? 7%? 29. If a man borrows $25 and pays .06 of $25 as interest for 1 yr., how much interest does he pay ? 30. At 6% what is the interest of $14 for a yr. ? Of $16? $30? $40? $60? 31. At 8% what is the interest each year of $ 7 ? $ 11 ? 32. At 1% what is the interest each year of $ 9 ? $12 ? 33. At 8% what is the interest of $12 for a year? For 2yr.? 6yr.? 8 yr. ? 12 yr.? 34. At 8% what is the interest of $ 7 for a year ? For | yr. ? Forl^yr.? 2 yr. ? 3 yr. ? 1J yr. ? 3J- yr. ? 1\ yr. ? 35. At 6% what is the interest of $8 for a yr. ? For \ yr. ? For3yr.? 4yr.? 5 yr. ? 6J yr. ? 36. CLASS EXERCISE. may tell how many dollars he would like to have at interest at 6%, and the class may tell how much interest he would have each year from it. How much in 2 yr. 3 yr. 4|- yr. 5 yr. 6 yr. MISCELLANEOUS EXERCISES 105 37. Mrs. Ware lent $800 @ 4% interest, $900 @ 3%, and $2500 @ 5%. How much interest did she receive each year from those loans ? 38. What is the ratio of a 5-inch square to an oblong 13 in. long and 5 in. wide ? 39. Draw on paper a rectangle 4 in. long and 3 in. wide. Draw a diagonal. Cut out the rectangle and divide it along the diagonal. Into what kind of triangles is a rectangle divided by a diagonal ? What is the area of the rectangle ? Of each triangle ? 40. Show the truth of the following statement : The area of a right triangle equals one half the area of a rectangle which has the same base and altitude. 41. What is the area of a right triangle whose base is 8 in. and altitude 5 in. ? 42. Give directions for finding the area of a right triangle. 43. Find the area of a right triangle whose base is 8 cm. and altitude 9 cm. Base 27 in., altitude 13 in. 44. Find the area of a right triangle whose base is 12 in. and whose altitude is 25% of the base. 45. Can you bisect a rectangle and place the two parts so as to form an isosceles triangle ? Kepresent. 46. Bisect a rectangle and place the parts as in Fig. 4. Show two horizontal parallel lines ; two oblique parallel lines. 47. A four-sided plane figure whose opposite sides are parallel, and whose angles are not right angles, is called a FIG. 4. * Rhomboid. Draw a rhomboid. 106 RATIO 48. How long is the perimeter of a rhomboid whose short sides are each 7 in. and long sides each 10 in. ? FIG. 5. 49. How long is the perimeter of a rhomboid two of whose sides are each 1.75 ft. long, and the other two each 2.5 ft. long ? 50. Represent and find perimeters of rhomboids having : A long side, 18 in. ; short side, 33^% of a long side. A short side, 12 in. ; long side, 25% longer than a short side. A short side, 1 ft. 4 in. ; long side, 50% longer than a short side. 51. What per cent of the angles of a rhomboid are obtuse ? 52. Find a rhombus in your book, and see whether it agrees with the definition of a rhomboid. A rhombus differs from other rhomboids in having all its sides equal. 53. Draw and cut out two equal equilateral triangles. Cut them in two, and arrange the right triangles thus formed as in Fig. 6. How long would the perimeter of Fig. 6 be if each side of the equilateral triangles were 6 cm. long? 16 cm.? 25 cm.? 54. Place the four triangles as in Fig. 7. What kind of a figure is formed ? How long is its perimeter if each side of the original triangles is 10 in. long ? 11 in. long ? 55. By changing the position of one triangle, change the figure into a rhomboid. Find length of perim- FIG. 6. FIG. 7. eter if each side of the original triangles is 7 in. long. MISCELLANEOUS EXERCISES 107 56. Place the four triangles as in Fig. 8, and name the figure. How long would the perimeter of Fig. 8 be if each side of the two equilateral triangles were 6 in. long? 1 ft. 3 FIG. 8. in - lon ? 57. Place the four triangles so as to form a rectangle. 58. Draw a 2-inch square. Draw its diagonals. With the point where the diagonals meet as a center and a radius of 1 in., draw a circle. At how many points do the sides of the square touch the circle ? At how many points do the diagonals cut the circumference ? Draw lines AB, BC, etc., between these points as in Fig. 9. 59. Erase all but the part shown in Fig. 10. Such a figure is called an Octagon. How many sides has an octagon? How long would the perimeter of your octagon be if each side were 7 in. ? 9 in. ? 60. Fold the octagon in various ways, and see whether the angles are all equal, and whether the sides are all equal. If they are, what kind of an octagon is it ? 61. What kind of angles has a regular octagon ? 62. Draw diagonals of the octagon as in Fig. 11. Into how many isosceles triangles is the octagon divided? Each triangle is what part of the octagon ? 63. If the area of the octagon were 60.48 sq. in., what would be the area of one of the FIG. 11. isosceles triangles ? FIG. 10. CHAPTER IV FRACTIONS 1. Draw a line an inch long and divide it into halves and quarters. How many halves of an inch are there in an inch ? How many fourths of an inch ? How many eighths ? How many thousandths of an inch ? How many millionths ? A Fraction is an expression of one or more of the equal parts into which a unit is divided. 2. In the expression J, 4, the denominator of the fraction, shows that some unit is considered as separated into 4 equal parts ; 3, the numerator, shows how many of those parts are expressed. What is meant by the expression of an inch ? 3. In the fraction |f which number is the denominator? What does the 16 show ? What name is given to the number above the line ? 4. What is meant by |- of an inch ? | of an apple ? 5. Make a mental picture of what each of the following expressions represents, and tell how much each lacks of a unit of its own kind : |- of a pie ; J of an apple ; T 7 ^ of a foot ; |- of a square yard ; f of a regular pentagon ; of a regular hexa- gon ; J of a 2-inch cube. 6. A fraction whose numerator is less than its denominator is called a Proper Fraction. Give some proper fractions and tell how much each lacks of being equal to the whole of which it is a part. 7. A fraction whose numerator is equal to or greater than its denominator is called an Improper Fraction. Give some improper fractions and tell how much each exceeds one unit. 108 FRACTIONS 109 8. Separate the following fractions into two lists, one of proper fractions, the other of improper fractions : f I * A H tt -3 V& 101% 9. Write a proper fraction whose terms are 5 and 7. Write an improper fraction with the same numbers as terms. 10. A number that consists of an integer and a fraction is called a Mixed Number ; as 3J. How many halves of a circle in 31 equal circles ? Illustrate. Does the following explana- tion seem to you to be true ? As there are 2 halves in 1 whole, in 3 wholes there are 3 times 2 halves, or 6 halves, 6 halves -f 1 half = 7 halves. 11. Change 2| to an improper fraction and explain the process. 12. Change to equivalent improper fractions : 7* Si 8i 16| Sf 7J of 20J 8* 15f 21f 6f 5* 1$ 6J llf 13. Tell how a mixed number is changed into an equivalent improper fraction. 14. Change to equivalent improper fractions : 7 5lf 12f 4f 6} 15. Write a mixed number whose fractional part is f. Change it to an equivalent improper fraction. 16. Write a mixed number whose integral part is 7. Change it to an equivalent improper fraction. 17. CLASS EXERCISE. - may give a mixed number, and the class may reduce it to an improper fraction. 110 FRACTIONS 18. How many wholes are there in f ? f ? SUGGESTION TO TEACHER. Lead pupils to express in their own way the evident fact that since it takes 2 halves to make a whole, there will be as many wholes in any number of halves as there are groups of 2 in that number. 19. To reduce a fraction is to change its form without changing its value. Reduce the following improper fractions to mixed numbers : f -- tt W W 20. Give directions for reducing N an improper fraction to a whole or a mixed number. 21. Reduce to integral or mixed numbers: m Hi W- 22. CLASS EXERCISE. may give an improper fraction, and the class may change it to a mixed number or to an integer. 23. A fraction is an expression of division, if- equals how many units ? In the expression -^ which number is the divi- dend ? Which is the divisor ? What is the quotient ? Show the same with regard to the expression - 1 - -. With -^k With - : 7 5 -. SUGGESTION TO TEACHER. Lead pupils to see that proper fractions also express division. indicates that one unit is divided by 3. Let lines be drawn and divided to illustrate such facts as that ^ of 2 yd. or of 6 ft. equals f of a yd. , that f of a ft. or 9 in. equals % of 3 ft. \ 24. Draw a line 3 in. long. Divide each inch into fourths and show that f of 1 in. equals of 3 in. 25. When the numerator and denominator of a fraction are made to change places, the process is called inverting the frac- tion, as f inverted is f. The fraction resulting from this inver- sion of a fraction is called the Reciprocal of the original fraction. Thus -f is the reciprocal of f . What is the reciprocal of the fractionf? ? T 9 T ? y~ ? 26. Which is greater, f or its reciprocal ? ^ or its reciprocal ? FRACTIONS 111 FIG. 1. 27. The ratio of Mr. A's money to Mr. B's money is -f . What is the ratio of Mr. B's money to Mr. A's money ? If Mr. A's money is $ 8, how much has Mr. B ? 28. Draw a regular octagon. Divide it by diagonals into 8 equal isosceles trian- gles. Divide each isosceles triangle into equal right triangles as in Fig. 1. Each isosceles triangle equals what part of the octagon ? Each right triangle equals what part of an isosceles triangle ? Of the octa- gon ? i of i = ? 4SuGGESTioN TO TEACHER. Let a large copy of Fig. 1 be drawn upon the board as a basis for the following exercises. 9. Find from the figure the values of the following: iofj i'afj ifj ioff ofi |off |of| JofJ 30. A fraction of a fraction is called a Compound Fraction. What is the value of the compound fraction 1 of ^ when expressed in simple form ? 31. How many inches equal J of \ of a foot? 32. See if the following reasoning is true : Since of => of = 2 tiroes or 33. The following rule is founded upon the same reasoning: To find the simple form of the value of a compound fraction Find the product of the numerators for the numerator of the simple fraction and the product of the denominators for its denominator. Cancel if possible. By the same reasoning find the value of % of |. 112 FRACTIONS 34. Simplify : a b c d f of | of f * of ^ of f| of | of 72 f of | of | of ^f e f g h |of of A<* foffiof* < j A; I f off of if foffofi^ fof2 3 5of2i foffiofS m no A of H of 7J fof^ofGi | of | of 8i 35. How many square centimeters in ^ of ^ of a square decimeter ? 36. Mr. King owned 1 of a farm and sold f of his share. What part of the farm did he sell ? If there were 200 acres in the farm, how many acres did he sell and how many had he left ? 37. Six boys divided a number of marbles equally among themselves. Edward Wells, one of the boys, gave ^ of his share to his younger brother. What part of the marbles did Edward keep ? If there were 54 marbles, how many did he keep? 38. Mr. Hubbard owned | of a mine and sold f of his share. If the mine was worth $80,000, how much did he receive ? 39. How many cubic feet in f of f of a cubic yard ? In f of f of a cubic yard ? 40. How many minutes in f of f of an hour ? In | of of an hour ? 41. -I of f of 4 of 4 is how much less than 1 unit ? Than 2 o 4 y / units? 42. f of f of 2f. is how much more than 1 unit ? How much less than 3 units ? Than 8 units ? FRACTIONS 113 43. Find, from the octagon on page 111, the values of x in the following equations : i=T Z 6 t = f i = f i = TF t = f f = T 104. Change ^ and ^ to equivalent fractions having their least common denominator, and explain your method of changing them. 105. Express with least common denominator and find values: +f. f + f + A- i + A- 106. Find values : i-i c t-i e *-* ? t-t b i-i-V d |-i /|-i h }-| Add: a 6 c d 107. i i i i, |, } i i, i i i, f 108. i, ^ | i, |, -f- ^ ^ * i, |, f 109. 1, ^ i i, |, f ^ i, T V i, f, H no. i, *, A i i A i i- i i i f in. f, i I i i, A f> i A i i f 112. i, |, i |, $, i i, i i> i t 113. i, |, A i f A i T> H i i A 114. f, f, A i, i, i I, f, f f, fc A SUGGESTION TO TEACHER. For additional practice let the class sub- tract the second fraction from the sum of the first and third in Examples 107-114. ADDITION AND SUBTRACTION OF FRACTIONS 119 116. How much greater than ^ is ^ ? ^ ? ? ? |? 117. & is how much more than -ft ? fa ? -fa ? fa ? fa? 118. CLASS EXERCISE. may give two fractions, and the class may find their sum or their difference as may be directed. 119. Draw a line AB and mark two points in it, C and D. If AC represents \ a mile, CD \ of a mile, and DB -fj- of a mile, what distance is represented by AB ? 120. John spent ^ of his money for a top, J of it for a ball, and of it for candy. What part of it had he left? 121. How much money did John spend for each article and how much had he left if he had at first 12^ ? 48^ ? 24^ ? 122. f + T \ is how much more than f T \ ? 123. The sum of -I and 4 is how much more than their o o difference ? 124. Express in lowest terms the ratio of 15 to 20 and of 16 to 30. Find their sum. Their difference. 125. Find the sum of -f- and its reciprocal. Of f and its reciprocal. 126. Find the sum of | and its reciprocal. Find the differ- ence between f and its reciprocal. 127. What is the square of 10? The 5th power? 3d power ? 128. Which power of 10 is the denominator of the decimal .01? .0015? .003? .00008? .000009? 129. Add -j 2 ^, -^ jfa>, TOTP Add nr

ii 130. Add: a 6 c rf .07 .165 .06 .485 .018 .2145 .016 .6 .5 .31 .07 .21 120 FRACTIONS 131. When several fractions have denominators that are powers of 10, is it easier to add them as common fractions or as decimals ? Why ? 132. Write in decimal form and add : jflT TflTFTF nfinnF T% 133. Write in decimal form and find values : TO" ~~ T^TJ" Tuinr ~~~ TTIF TTRT ~o ~~~ TTrluo" 134. Which is greater, 3 or 3.00 ? 7 or 7.000 ? 135. Change f to a decimal. 3 reduced to lOOths equals 3.00. As 3 equals 3.00, 3 -r- 4 equals 3.00 H- 4. 3.00 -r- 4 equals .75. Hence f = .75. 136. Change f to a decimal by the following rule. Explain. To reduce a common fraction to a decimal Annex ciphers to the numerator and divide by the denominator. 137. Change to decimals and add : t i * A it i A 138. Change -^ to a decimal, stopping at lOOOths. Common fractions cannot be reduced to exact decimals, if when re- duced to their lowest terms their denominators contain any factors other than 2 and 5. 139. Reduce the following fractions to decimals, not carry- ing the work beyond ten-thousandths : i r' . , jt i -.H ' .. ...t .. if - m H If A A A A A W 140. Reduce the following to decimals of not more than 3 places, and add them. If the division is not exact, make the remainder the numerator of a common fraction : f* TV I I A A f A ADDITION AND SUBTRACTION OF FRACTIONS 121 141. Tell at sight which of the following fractions can be reduced to exact decimals : A $ A I TT M 'fr T M Eeduce the fractions to decimals of not more than three places, and find their sum. 142. Divide 2 by each number larger than itself that is expressed by one digit. Express the quotient as a decimal of not more than three places. 143. Divide three by each number between 10 and 20, and express the quotient as a decimal of not more than three places. 144. Change to decimals, stopping at hundredths : * t i i * * * * t I 145. How many lOOths or per cent equal $ ? f? |? f ? |? 146. How many per cent equal ^ ? -f^ ? ft? |$ ? 147. How many per cent equal ft ? ft? ^? ft? ft ? | ? 148. What per cent of anything is \ of it ? % ? f I ? -fa? 149. Add 2| and 7$: 7$ f + = | or 1|, which added to the sum of 2 and 7 = 10. 10J 150. Add: a b c d e 7f lOf 8| 6f 2| 3f 6 3^ 8| 6ft 8f 5^ 4f 7j% 8J 151. Tell how the following mixed numbers are added: abed 8 16f 66| 18f 8J GJ: 12$ 16| 122 FRACTIONS 152. Add: 91| 41f 93f 68f 153. CLASS EXERCISE. may give three mixed num- bers, and the class may find their sum. 154. A farmer used 77^- acres of land for wheat, 40 J acres for corn, If acres for vegetables, 29^- acres for pasturage, and acres for an orchard. How many acres were in the farm ? 155. Mr. White has 3 fences on his farm; one is 168^- rd. long, another 456 T 4 T rd. long, and another 328 T 5 T rd. long. How many rods of fencing has he in all ? How much did his fence cost at 75^ a rod ? 156. After selling 3f acres, a farmer had left 123 j acres. How much land had he at first ? 157. How long is it from half past eight A.M. to noon? From quarter before nine to half past eleven ? From quarter past two to quarter to six? From half past ten to quarter past one ? From a quarter of eleven to half past three ? 158. From 14 We subtract \ from 1 of the units of the minu- take 7-i- en( ^' '^^ ie rema ' n der is \. As 1 unit has been ^ subtracted from the 4 units 3 units are left. 7 from 6 13 leaves 6. a b c d e f 159. From 8 9 18 25 16 20 take 6 5| 4f 3| 11 4 160. Mary's aunt sent her 6 yd. of cashmere for a dress. 5| yd. were used. How many yards were left ? 161. In a jumping match Thomas jumped 3 ft. and his brother jumped 2-jAj- ft. How much farther did Thomas jump than his brother ? 162. Make problems in which a mixed number is subtracted from an integer. ADDITION AND SUBTRACTION OF FRACTIONS 123 163. Subtract : a b c d e f g 2$ 3& 2| 2j_ 3| 2| 3i 164. A piece of string 3^- ft. long was cut from a piece ft. long. How much was left ? 5^ ft. were cut from the re- mainder. How much then remained ? 165. From 41 take -f. 4 i 2 As | cannot be subtracted from |, we subtract f from 1|. _Z This leaves only 3 units in the minuend. 3f 166. Find difference : a b c d e f g 8| 6i 8J 6i 8f 6$ 7i 1 J J J J J J 167. Find difference : a b c d e f g h i j k I m S$ 9i 7f 8i 4| 6i 3| 2| 4f 2| 2| 2| 168. Write two mixed numbers whose fractional parts have the same denominator. Let the mixed number whose integral part is the larger have the smaller fractional part. Find dif- ference between the mixed numbers. 169. Find difference: a b c d e f g 3J or -| 7J 8^ 2f 124 FRACTIONS 170. Write two mixed numbers the fractional parts of which have different denominators. Let the mixed number that has the larger integral part have also the larger fractional part. Find their difference. 171. CLASS EXERCISE. may give two mixed numbers like those described in Ex. 170, and the class may find their difference. 172. Find difference : a b c d e f g 91 97 "5" 9 I m n 6 7 4| 2J 2f 21 4^ 4 T \ C 2\ 2* 173. Write two mixed numbers whose fractional parts have different denominators. Let the mixed number whose integral part is the greater have the smaller fractional part. Find their difference. 174. If a line is 3J in. long, how many inches must be added to make it 5 J in. long ? Represent. 175. 3^ 176. Find values of x: a 10-7i = x. e 9 -3% = x. b 6 - 4 = x. f 10 - 5| - x. c 7 -3% = x. g 7$-2=x. 4 = z. h 12 -7=a. 177. A weighs 148 J lb., B 157f lb., C 1611 lb., D 175| Ib. How much do they all weigh ? 178. What is the difference between the weights of A and C? AandB? AandD? B and D ? B and C ? CandD? ADDITION AND SUBTRACTION OF FRACTIONS 125 179. Mr. Otis rode 23f miles on Monday, llf miles on Tues- day. On Wednesday he rode as far as on Monday and Tuesday. How far did he ride in the three days ? 180. Mr. Carr rode on his bicycle to a city 91f miles dis- tant. The first day he rode 16 miles ; the next day he rode 3^ miles more than on the first day. On the third day he rode 21 miles more than on the second day. How far did he ride in those three days ? How many miles more did he ride before he reached the city ? 181. Mr. Grey planted 75J acres in wheat, 45f acres in corn, and 7-J acres in oats. How many acres of grain did he cultivate ? 182. In a township containing 23,039 T ^ acres, the roads occupy 345f acres, and the rest is divided into farms. How many acres in the farms of that township? 183. A stove burned 180f Ib. of coal in one week, 175 J Ib. in another week, and 205^ Ib. in another week. How many Ib. did it burn in the three weeks ? 184. A, B, and C own a mine. A owns T 5 ^ of it, B owns f of it, and C owns the rest. How much does C own ? 185. If the mine is worth f 248,400, what is the value of each man's share ? 186. A has 75f acres of land, B has 13| more acres than A and 4| acres less than C. How many acres has B ? C ? A and B ? A and C ? B and C ? 187. A farmer has a field in the form of a trapezoid. One of the parallel sides is 71 T 4 T rd. long, the other is 68| rd. long. Of the non-parallel sides, one is 53f rd. and the other is 54^- rd. Represent and find length of perimeter of the trapezoid. 126 FRACTIONS MULTIPLICATION OF FRACTIONS 188. Multiply by . To multiply any number by ^ is to take of it. 189. By 1 multiply : f f ! T G T I I f 190. Make a rule for multiplying a fraction by a fraction. Multiply : abed 1 Q 1 4 \/ 7 .2.1 V 8 1 5 y 49 6 y 5 g 1 qo 4 5 v 1 5 v 1 1 1 8 v 7 27 v 8 IVA. -Q-f A YT 77 A TJ-J T5T A IF 32 * "9 194. 14 x 44 M x -rV M x -U *- x M 195. CLASS EXERCISE. may give two fractions and the class may find their product. 196. CLASS EXERCISE. may give a proper fraction and an improper fraction and the class may find their product. 197. CLASS EXERCISE. may give three fractions of such a kind that cancellation may be used in finding their product and the class may find the product, canceling wherever possible. 198. Multiply | by itself. 199. Square : 2 4 5 5 10 12 3 151917 25 41 3 o 7 9 11 la 7 16 20 18 30 53 200. What part of a square inch is a rectangle that is J- of an inch long and ^ of an inch wide ? Draw a figure and prove your work. 201. What part of a square yard is a rectangle ^ of a yard square ? How many square feet in it ? 202. Multiply f by the square of f . MULTIPLICATION OF FRACTIONS 127 203. Cube: t I * * I I I A T 4 T A 204. Draw a rectangle whose length is of a foot and width i of a foot. What fraction of a square foot is its area ? Prove by reducing the fractions of a foot to inches. 205. What fraction of a square foot is a rectangle f of a foot long and ^ of a foot wide ? How many square inches in it ? 206. How long is the perimeter of a rectangle of a foot long and ^ of a foot wide ? Give the area of the rectangle in fractions of a square foot, and also in square inches. 207. How many square feet in a square f of a foot in dimensions ? How many square inches ? 208. How many square feet in a square f of a foot in dimen- sions ? How many square inches? 209. Add the product of ^ x f to the product of f x 7. 210. Subtract the product of f x from the product of I x I- 211. The product of several numbers is called their con- tinued product. What is the continued product of 3, 5, and 7 ? Of i |, and f? Of , 2 4 T , and f ? Offhand}? 212. Multiply | xf Observe that the denominator of the fraction f may be omitted without changing the result. Multiply : a b c d e 213. f x 24 ^ x 35 5 x 27 J x 64 T 2 T x 33 214. ^ x 100 f x 18 f x 21 ^ x 18 | x 24 215. 36 x fV 48 X I 32 x A 39 x A 60 x ii 216. 77 x T 4 T 42 x f 63 x 2^ 54 x ^ 65 x fV 217. Multiplying the numerator of a fraction by an integer has what effect upon the value of the fraction ? Illustrate. 128 TRACTIONS 218. Dividing the denominator of a fraction by an integer has what effect upon the value of the fraction ? Illustrate. 219. Give either of two ways by which a fraction may be multiplied by an integer. 220. Multiply 12 by a proper fraction. Is the product greater or less than 12 ? 221. Multiply 12 by an improper fraction. Is the product greater or less than 12 ? 222. When will the product of an integer and a fraction be greater than the integer ? 223. Multiply 64 by 3J. 64 8* 192 = the product of 64 and 3. 16 = the product of 64 and . 208 = the product of 64 and 3J. Multiply : a b c d 224. 55 164 164 125 4 i 21 5 t 5 225. 72 81 208 144 226. 81 115 64 343 H 227. 66 201 192 512 51 11} 4f 7| 228. 172 981 111 169 21 101 MULTIPLICATION OF FRACTIONS 129 a b c d 229. 385 243 78 408 230. Multiply an integer by a mixed number and explain your method. 231. At 60^ a yard, what is the cost of 7 yd. of silk ? 8 yd.? lOfyd.? 12f yd. ? 14| yd. ? 15| yd. ? 18f yd. ? 232. 160 square rods equal 1 acre. How many square rods in | of an acre? In | A. ? ^ A. ? |f A. ? ^ A. ? A. ? 233. Mr. Hill has a lot 40 rd. long and 3 rd. wide. What fractional part of an acre is it ? 234. How many square rods in 2| A. ? 5| A. ? 7 T % A. ? A.? 235. Multiply 16 $ by 6. Multiply : a 16f 6 96 = the product of 16 and 6 4 = the product of f and 6 d 100 = the product of 16f and 6 b c 236. 15J 8 92J 39| 24 12 122| 22 237. 124f 16 72| 164| 15 4 109J 27 238. 24| 9 119f 81$ 15 16 98| 5 HORN. GRAM. 8CH. AR. 9 130 FRACTIONS 239. Tell how a mixed number is multiplied by an integer. 240. At 18f ^ per yard, what is the cost of 4 yd. of gingham ? 7yd.? 8yd.? 10yd.? 12yd.? 241. How long is the perimeter of an equilateral triangle, a side of which is 4 T 7 2 ft. long ? 242. If each side is 4 T 9 ^ ft. long, how long is the perimeter of a rhombus ? Of a regular octagon ? Of a regular hexa- gon ? Of a regular pentagon ? 243. At 331 ^ per yard, what is the cost of 8 yd. of dress goods ? 9 yd. ? 12 yd. ? 14 yd. ? 15 yd. ? 18 yd. ? 244. Multiply 2^ by 3^. Before small mixed numbers are multiplied together, they should be reduced to improper fractions. fx = Y = 8i Multiply : 245. a 21 by 3J d 5| by 12J b 3J by 6J e 18f by 11| c 4f by 6^ / 7i by 66| 246. Square: 2| H 3i 2} 6i If If 2f 47. Cube: 11 H 34 31 4i 6J 3f 2^ 248. CLASS EXEKCISE. may give two small mixed numbers, and the class may find their product. 249. CLASS EXERCISE. may give a small mixed num- ber, and the class may find its square. 250. Draw on the floor a square 5 yd. in dimensions. Each side is 1 rd. long. Find the number of square yards in a square rod. MULTIPLICATION OF FRACTIONS 131 251. How many feet long is a rod? Find the number of square feet in a square rod. 252. Each of the short sides of a rectangle is 7-f in. The long sides are 9| in. each. Find the area of the rectangle. 253. One side of a rectangle is llf in., and each of its adjacent sides is 3 in. shorter. Find the area of the rectangle. 254. Draw on paper or pasteboard a circle whose radius is 3 in. Cut it out. By measuring the circumference with a tape measure, it will be found to be nearly 22 in. %f- is considered the ratio of the circumference of a circle to its diameter. What is the ratio of a diameter to the circum- ference ? 255. Find the circumference of a circle whose diameter is 7 in. 21 in. 14 in. 256. How long is the circumference of a circle whose radius is 21 in. ? 3 in. ? 10 in. ? 17J- in. ? 257. How long is the circumference of the largest circle that can be cut from a piece of paper 4-| in. square ? 258. A round flower bed 14 ft. across has a border of pinks, set 6 in. apart. How many pinks in y^ of the border ? Represent. 259. Mrs. Smith's wash bench is 4 ft. long and If ft. wide. A tub is set upon it in such a way that the lowest hoop of the tub touches the front edge and also the back edge of the bench without extending over either edge. What is the circumference of the hoop ? 260. A round tin pail with straight sides is 8 in. across and 10 in. high. How long is the diameter of the largest plate that can be placed on the bottom of it ? The circumference ? 261. If a ball is cut into two equal parts by one cut, what is the shape of the flat surface of each part ? 132 FRACTIONS 262. A round apple 4 in. in diameter was cut into halves. One of the halves was laid with its flat side down upon a plate, in such a way that no part of the cut surface of the apple extended beyond the plate. Find the diameter and the circumference of the smallest plate that could be used for that purpose. 263. A part of a circumference is called an Arc. Draw a circle and divide its circumference into several arcs. 264. If the radius OA (Fig. 2) is 3 in., how long is the diameter AC? How long is the circumference ? . 265. What part of the circumference is the arc AC? How long is it ? 266. The arc AB is % of the circum- ference. How long is it? FIG. 2. 267. What part of the circumference is the arc BC ? How long is it ? 268. If the radius of a circle is 5 in., how long is the diameter? The circumference? An arc which is J T of the circumference ? of the circumference ? 269. Find the length of an arc which is -^ ference of a circle whose radius is 5 in. 12 in. of the circum- 37 in. 270. The circumference ABCD is divided into how many equal arcs? If the diameter of the circle ABCD is 7 in., how long is the arc AB? The arc ABC? The arc BCD? The arc ABCD? 271. An arc which is of a circum- ference is called a Quadrant. If the diameter of the circle ABCD were 28 cm., how long would a quadrant be? MULTIPLICATION OF FRACTIONS 133 272. Multiply 124f by 6J. 124f 744 = the product of 124 by 6. 62 = the product of 124 by J. 4 = the product of f by 6. = the product of f by . 810 = the product of 124| by 6. 273. This method of multiplying mixed numbers together is useful when the numbers are large. Can you see why ? Multiply : a b c d 274. 441 344f 288f 456f 16f abed 275. 819f 64| 816^ 9f 28^ 276. Square 64f. 32|. 24f 22 T 2 T . 36J. 277. Cube 12J. 16J. 16J. 14|. 278. At the rate of 17f mi. per hour, how far will a steam- boat go between nine o'clock Monday morning and half past ten on Tuesday morning ? Find the cost : 279. Of 51 yd. of cloth at $ 4| a yard. 280. Of 7-J- A. of land at $ 24^ an acre. 281. Of 7| T. of hay at $ 21 J a ton. 282. Of 10J yd. of ribbon at $ .23 a yard. 283. Multiply ^ by fa and write the product in decimal form. 284. Find the product of -}- and -f^ and express it as a mixed number. 134 FRACTIONS 285. Express fj and T 8 ^ in decimal form and find their product. 286. If asked for the product of two fractions having for denominators some power of 10, would you find it easier to multiply them as common fractions or as decimals ? Why ? 287. Give a rule for pointing off the product of two decimals. 288. Let a = .04, b = .02, c = .005, d = .0007, e = .00002, /= .3. Find the value of ab, ac, ad, ae, af, be, bd, be, bf, cd, cf. DIVISION OF FRACTIONS 289. How many times is 1 fourth of an inch contained in 3 fourths of an inch ? 1 of anything in of it ? |-^-i=? |-=-i=? 290. 6 sevenths -v- 2 sevenths =? f -j- -f = ? 10 elevenths ^- 5 elevenths = ? |f -*- ^ = ? 291. A-nA=? T 8 T contain ^ as many times as 8 units of any kind contains 3 units of the same kind. 8 -j- 3 = f or 2|. 292. What are similar fractions ? Illustrate. 293. Divide |f by a similar fraction. 294. CLASS EXERCISE. may give two similar fractions, and the class may divide the greater by the less. 295. How is a fraction divided by a similar fraction ? 296. Use |- as a dividend and % as a divisor. 297. Multiply | by its reciprocal. 298. Multiply several fractions by their reciprocals and compare the results. 299. How many times is f of anything contained in f of the same thing ? inj? |in|? 300. Which is greater, } -j- f or j x | ? }-5-'{orJx^? DIVISION OF FRACTIONS 135 301. $dividedbyf=? f multiplied by the reciprocal of f=? 302. Think of two similar fractions. Divide the larger frac- tion by the smaller. Compare the result with the result obtained by multiplying the larger fraction by the reciprocal of the smaller. Think of two fractions that are not similar. Reduce them to similar fractions and make the same com- parison. Continue this until you see the reason for the fol- lowing rule : To divide a fraction by a fraction Multiply the dividend by the reciprocal of the divisor. 303. By this method divide -^ by ^. Divide : a b c d 304. A by f & by \ &byf |by | 305. A b y T by* if by A iby | 306. A by | f by f ifby | Hby - 307. ft b y i 7 ft b> A ffby | Mby 308. by ff M by f 1 4 Vv V 7 "2TT J 10" Hby 309. 1 b y TT- Mby.fr M by A if by 310 Wby | tf by A ff by || liby - 311. How do you find the ratio of one number to another ? Find the ratio of f f to ^-. To ^-. To -fa. Find ratios : abed 312. A : f tt'f H : li tt : H 313. &* H : A A : f H^ 314. |f:A If^A *:* if I 315. Multiply the numerator of a fraction by an integer. Is the value of the fraction increased or decreased? 136 FRACTIONS 316. Find by trial how the value of a fraction is changed by multiplying its denominator. By dividing its numerator. By dividing its denominator. 317. Which gives the greater quotient, 16 divided by 8, or 16 multiplied by | ? 318. |f -5-8 = ? We may write 8 as f . The question is then f -4- f = what ? This is solved by the rule for dividing one fraction by another. Find values of x : a b c 319. f-=-2 = a; | T -j- 3 = a; T 6 7 -7- 12 = a 321. f -r- 4 = a; y 2 ^-8 = aj fj -j- 11 = a 322. Divide 28 by -f. Consider 28 as *. 323. Divide 24 by: f f T T . f. f |f f 324. Divide 41 by : f . f . f f . f . ||. |f Divide after reducing mixed numbers to improper fractions : a b c d 325. 3|-T-2i- 7J-*-18f 8J-f-lf 5f-r- 326. 2^-f-3J 13i-f-4| 6|- 327. 3-i-6J 12 ^-5f 37|- 328. CLASS EXERCISE. may give a small mixed number for a dividend and another small mixed number for a divisor, of such a kind that cancellation may be used in finding the quotient. The class may find their quotient. 329. At a picnic 5^- pies were divided equally among 44 persons. What part of a pie did each receive ? DIVISION OF FRACTIONS 137 330. Mr. Tod has 13^ acres devoted to celery, which is just four times as much as his brother has. How many acres of celery has his brother ? 331. The top of a newel post is an octagon whose perimeter is l|i ft. How long is one side of the octagon ? 332. The circumference of a circle is 9f in. How long is its diameter ? Its radius ? Eepresent. 333. Find the length of the diameter of a circle, the circum- ference of which is 11 in. 334. A quadrant of a circle is 4^ in. How long is the cir- cumference ? Diameter ? Eadius ? 335. For 75^, how many yards of lace' can be bought at 21 f per yard? At 3^? At 6^? At8i^? At 12^? At 16.}?? At 37J^? At 621^? At66|^? At 83^? At 336. If a coat costs 3 j dollars, how many coats may be bought for 62 dollars ? 337. If 1J yd. of cloth are required for a coat, how many coats can be made from 87-J- yd. ? 338. If Jerry walks 2^ mi. an hour, in how many hours will he walk 7-J- mi. ? 11 mi. ? 1 mi. ? | mi. ? J mi. ? 1 mi. ? 339. What number divided by 3 will give 5 for a quotient ? What fraction divided by 3 will give f for a quotient ? Prove. 340. What mixed number multiplied by 3 will give 181? 6f? 341. A fraction that has a fraction in one or both of its terms 1 2~ 7 is called a Complex Fraction ; as, |, =f> ^T. Write a complex fraction whose numerator is a mixed number, and denominator a whole number. | is read "| divided by i v 138 FRACTIONS 342. Write and read complex fractions as follows : a Numerator an integer, denominator a simple fraction. b Numerator a simple fraction, denominator a mixed num- ber. c Numerator an integer, denominator a mixed number. d Numerator and denominator both simple fractions. Both mixed numbers. 343. A complex fraction, like other fractions, is merely an expression of the division of the numerator by the denominator, and it is reduced to a simple fraction by performing that division; as, f=H r I x H- 3 Reduce to their simplest form the following complex fractions: a b c d e f 4} U 3 1} 3f 17 3J 4| T 5| 13 8j 344. Simplify: a b c d e f g h ^12J16J41|2|-3J6J7J 33! 83j 87j 91| 20 30 50 60 345. CLASS EXERCISE. may give the hardest complex fraction that he wrote in Ex. 342, and the class may reduce it to a simple fraction. 346. The product of two numbers is 15. One of them is 3. What is the other ? How is it found ? 347. The product of two fractions is -fa. One of them is f . What is the other ? 348. CLASS EXERCISE. may give the product of two fractions and one of the fractions. The class may find the other fraction. DIVISION OF FRACTIONS 139 349. Divide: liv ~^ iV T7TO"0~ ~*~ TO" 10000 "*" TOO" 1000 ~*~ TOTF 350. Reduce the same fractions to decimal form and find the quotients. Which is the easier way of dividing in this case ? Why ? 351. Leta=.4, 6 = .08, c=.032, d=.0016. Find values of : a a a b b 6 b c d a c d . - f? - - a b d a b c 352. Change f and ^f to decimals and divide the greater by the less. 353. Change to decimals and divide : f by J. f by |. 354. If 4.5 yd. of silk cost $ 6.75, how much will 1 yd. cost ? 3.7 yd. ? 6.75 yd. ? 355. Mr. K bought a lot in Washington, D.C., for $4500, paying $1.875 per square foot. How many square feet in the lot? 356. If 1.7 yd. of cloth is used to make -a coat, how many coats can be made from 81.6 yd. ? 357. How much cloth at $ .75 a yard can be bought for $ 45.75 ? 358. At the rate of 8.25 mi. an hour, in how many hours will a stage run 125 mi. ? 359. If a barrel of beef costs $ 14.25, how many barrels can be bought for $ 798 ? 360. Traveling 215.6 mi. a day, in how many days will a steamer go 1000 mi. ? 361. If a dollar gains 5 i interest each year, in how many years will it gain another dollar ? 140 FRACTIONS 362. In how many years will $1.00 double itself at 4% ? 363. One third of John's money is if. How much has he ? 364. Thomas spent f of his money and had 5 f left. How much had he at first ? Explain. 365. How much money has a boy who can spend J of his money and have left 70? 90? 20? 120? $1.00? $8.00? 366. How much money has a boy who after spending 30 will have left f of his money ? J ? T 9 7 ? ? f ? } ? 367. Three fourths of John's money is 150. How much is of it ? How much is the whole ? 368. 6= f of what number ? SOLUTION BY ANALYSIS. As 3 fifths of the number = 6 1 fifth of the number = 2 5 fifths of the number = 10 369. 8 is f of what number ? Analyze as above. 370. Analyze. 12 is f of what number ? 4. of what ? 371. Find values of x. Analyze. a -ff of x = 14 e f of x = 15 i f of x = 18 b | of a? = 10 / | of x = 16 J V of x = 20 c f of a = 25 g f ofx = 28 fc J of a; = 28 d f of a; = 24 h f of a? = 30 / | of a = 75 372. How long is a line f of which is 9 in. long ? 373. A man rode 16 mi. on Monday, which was f of the dis- tance he rode on Tuesday. How far did he ride on Tuesday ? 374. f of John's money is 210. How much will he have left if he gives away $ .05 ? 375. f of James's money is $20. How much will he have left if he gives away of his money ? DIVISION OF FRACTIONS 141 376. How much money must a boy have that he may lose | of it and have 12^ left? 30^? 42^? 60^? 377. How much money must a man have so that after gain- ing i as much, he may have $ 700 ? $ 210 ? $ 441 ? $ 7.70 ? 378. Mary has 12^. Her money equals y 3 ^ of Florence's money. How much has Florence ? 379. William has 8 marbles. He has f as many marbles as James has. How many marbles has James ? 380. Alice has 14^, which is just -f of the money she needs to buy her geography. What is the price of the geography ? 381. Make problems in which a certain number is a fractional part of the number which is to be found. 382. John gave away -- of his marbles and then had 30 marbles left. How many had he at first? 383. Susie gave -| of her money to her sister, and found that she had 16 cents left. How much had she at first ? 384. Harry gave away -| of his pigeons and sold f of them. He had 15 pigeons left. How many had he at first ? 385. In a storm, a ship's crew threw overboard 30 bbl. flour, which was y\ of the whole cargo. How much was the whole ? 386. A man owned f of a mine. He sold J of his share for $ 6000. How much was the whole mine worth ? 387. Mr. Buchanan sold J of his share of a store for $ 2000. What was his share worth? His share was f of the whole value. What was the whole value ? 388. Owning | of a quarry, Mr. Harris sold ^ of his share for $ 6000. What was the value of the quarry ? 142 FRACTIONS 389. Mr. Madison owned \ of an Indiana gas well. He sold f of his share for $ 1500. What was the value of the whole well? 390. yL of the pins in a cushion were crooked, and there were 66 straight pins. How many were there in all, and how many were crooked ? 391. If 8f yd. of tape cost $ .70, how much will 1 yd. cost? 9|yd.? 392. If 16f yd. of rope cost 100 cents, how much will 1 yd. cost ? 23f yd. ? 393. If 6 bu. of seed cost $15, how much will 19f bu. cost? 394. If | of a quart of seed cost $ .18, how much will 1 pk. cost? MISCELLANEOUS EXERCISES 1. Divide .0096 by .12. By .008. By 24. By 3.2. By .16. 2. Divide .000048 by .012. By .4. By .0024. By 2.4. 3. Divide .144 by .04. By 48. By 1.6. By .0003. 4. What is the ratio of 889.44 to .102 ? To .105? To .108? 5. Multiply 7* by the 4th prime. 9 2 by the 6th prime. 6. f of -f-g of y\ of 24 hr. is how much less than a day ? 7. 24 sheets of paper make a quire. How many sheets in | of | of | of a quire ? 8. How many sheets in y 7 ^ of ^ of -f of a quire ? In |- of of -fe of a quire ? 9. Mrs. Smith is f as old as Mr. Smith, who is 48 yr. old. Their daughter Alice is % as old as her mother. How old is Alice ? 10. Add yV, &, A, iV Add A A> A> A- 11. From L- take --. From - take . MISCELLANEOUS EXERCISES 143 12. Multiply the first prime number after 9 by -|. 13. Multiply the largest prime factor of 330 by 2^-. 14. Multiply 64 by the largest prime factor of 390. 15. Multiply 8f by the 1. c. m. of 5, 6, and 10. 16. Multiply the 1. c. m. of 8, 6, 9, and 12 by 3^. 17. Multiply the g. c. d. of 36, 48, and 60 by 3|. Leto = f; & = fi; c = 10; d = 5; e = f Find values of : 18. a x b or ab ac ad ae be bd cd ce de 19. a-f-c a + e a + d b + c b + a d + e 20. c a d a e a c b c d d b Let a = |; & = 3J; c = l^; d = 2f; e = f Find values of : ^ a a a a b b 6cd 6cdeeda6e 22. f of Anna's money is $ .50. How much will she have after giving away 7 f ? 23. Of what number is 21 three fourths ? f ? f ? ^ ? 24. Of what number is 16 four sevenths ? | ? f ? ^ ? 25. If ^ of the price of a house is $ 400, what is the price of the house ? How much will five such houses cost ? 26. If 1 apple costs |^, how much will 4 doz. apples cost ? 27. If f of the price of an orange is 3^, how much will a dozen oranges cost ? 6 doz. ? 28. At 16|^ a yard, how many yards of ribbon can be bought for$l? $2? $3? $5? 29. 12 doz. make a gross. When buttons are bought for 25 1 a gross, what is the cost of 1 button ? 30. If 500 pins cost 10 ^, how much will 1 pin cost ? 144 FRACTIONS 31. If a gross of pencils cost 50^, how much will 1 pencil, cost? 32. Mrs. Norton paid 5^ for a box of toothpicks, in which there were 2000 toothpicks. How many did she get for a cent ? What was the cost of 1 toothpick ? 33. She paid a nickel for a box of matches. What was the price of each match if there were 500 matches in a box ? 34. A gross of glass vials cost 48 ^. How much did 1 vial cost ? 35. Find the average receipts of a peanut stand for 6 days. Monday, $ 1.37 ; Tuesday, $ 2.11 ; Wednesday, $ 1.87 ; Thurs- day, $ 1.04 ; Friday, $ 1.75 ; Saturday, 9 3.10. 36. If the average cost of keeping up the stand was $ 1.25 per day, what were the owner's average gains per day ? 37. Suppose a pie to be exactly round, and 101 in. in diam- eter. If it were cut into 6 equal pieces, how long would the curved edge of each piece be ? 38. The surface which is bounded by an arc and two radii is called a Sector. Show five sectors in Fig. 4. You may remember the figure of a sector more easily if you recall the way in which pies, waffles, and round cakes are usually cut. 39. How long is the perimeter of a sec- tor of a circle whose radius is 4 in., if the arc of the sector is 5^ in. ? Represent. 40. Draw a circle whose radius is 3 in. Divide it into 4 equal sectors. Write the word " sector " in each. Write upon each line of the perimeter of a sector the length of the line and find the length of the perimeter of a sector. 41. What kind of an angle is the angle of a sector which is J of a circle ? Less than \ ? Greater than 1 ? MISCELLANEOUS EXERCISES 145 42. Draw a circle and apply its radius six times as a chord. What regular polygon have you drawn ? Each arc thus cut off is what part of the circumference? If the radius is 2f in., how long is the circumference ? Each arc ? 43. Draw lines from the ends of each arc to the center of the circle. What are these lines called ? What kind of angles do they make ? 44. Erase the chords. Find length of the perimeter of each sector, supposing the radius of the circle to be 2-f in. 5^ in. 10 in. 45. Erase radii so as to leave the circle divided into three equal sectors. Find length of perimeter of each sector, assum- ing the radius to be -j- in. T 9 T in. l T ^ r in. 46. If the radius of a circle is 2f in., how long is the perim- eter of a sector which is J of the circle ? 47. Find the perimeter of the sector which remains when a sector that is of a circle is subtracted from the circle. 48. If a strip of paper 5 in. long were curled around so that its edge inclosed a circle, how long would the circumference of that circle be ? 49. The circumference of the wheel of a toy wagon is 20 in. How far does the wagon run when the wheel turns around once ? 3 times ? SUGGESTION. Let pupils roll a coin, button, or other circular objects, as a help in realizing the conditions of these problems. 50. How far will a hoop 2^ ft. in circumference run in turning 7 times ? 9 times ? 51. How many times will a wheel 2 ft. in circumference revolve in running 8 ft.? 52. How many times will a wheel 6 ft. in circumference revolve in running 12 ft.? 8 yd.? HORN. GRAM. SCH. AR. 10 146 FRACTIONS 53. A mile is 5280 ft. If the front wheels of a wagon are each 6 ft. in circumference, and the hind wheels are 8 ft., how many times will each wheel revolve in running a mile ? 54. Draw a rhomboid whose long sides are each double a short side. How long would its perimeter be if each short side were 8 in.? 12 J in.? 55. How many square feet in a lot 30 ft. wide, and 150 ft. deep ? If the owner uses \ of the lot for a house, and 1 for a chicken yard, how many square feet remain ? 56. A house is 48 ft. long, and the distance from the ridge- pole to the eaves on each side is 23 ft. How many shingles will be required to cover it if 6 shingles are required to cover a square foot? 57. John is 8^ yr. old, and his sister is 6 T 7 T yr. old. What is their average age ? 58. A grocer bought 3 cheeses, one weighing 32|- lb., another 28J lb., another 41| lb. How many pounds were there in all ? 59. Which is greater, and how much, -f- x f , or -f- -=- J ? 60. What is the area of a rectangle 3f ft. long and If ft. wide ? 61. What is the area of a right triangle whose base is 4 in., and altitude 3 in.? 62. What is the use of reducing fractions to a least common denominator ? 63. Write the fraction that expresses the ratio of the first composite number after 18 to the first composite number after 30, and reduce the fraction to its lowest terms. 64. When Arthur was a year old his father placed $50 in the bank as money to be used in sending him to college. He put $ 50 in the bank on every birthday until, at the age MISCELLANEOUS EXERCISES 147 of 18, Arthur was ready for college. How much money had been placed in the bank for him ? 65. Arthur's expenses at college for the first year were $218.75; for the second year, $310.50; for the third year, $ 365.25. How much of the amount was left at the end of the third year ? 66. Arthur's expenses for the last year were $ 410.90. He received $465.67 as interest. How much was left of the money when he had finished his college course ? 67. A man bought a lot for $ 2000, built a house upon it for $ 2500, and sold the property so as to gain $ 100 on his invest- ment. For how much did he sell it ? 68. Charles bought a ball for $.08, and sold it for $.12. The gain equaled what part of the cost ? What per cent ? 69. A man's salary is $ 2400 a year. He saves J of it one year, 1 of it the next yea^ and -| of it the next year. How much has he saved at the end of the third year ? 70. A gentleman had 1200 books in his library, and gave away - of them. He lost -^ of the remainder. How many books were left ? 71. He added 200 more volumes to the library, and then gave away f of it. How many had he left ? 72. The base of an isosceles triangle is 3 ft. The ratio of one of the equal sides to the base is -J. How long is the perimeter of the triangle ? Kepresent. 73. How long is the perimeter of an isosceles triangle whose base is 14 in. and each of whose equal sides is 5 in. longer than the base ? 74. How long is the perimeter of an isosceles triangle whose base is 21 in. and each of whose equal sides is 33|% longer than the base ? 148 FRACTIONS 75. The perimeter of a certain isosceles triangle is 25 in. and one of the equal sides is 9 in. How long is the base ? 76. The base of an isosceles triangle is 11 in. and the perimeter 35 in. How long is each of the equal sides ? FIG. 5. 77. Draw a rectangle 4 in. long and 3 in. wide. Draw a diagonal of it. Into what kind of figures does a diagonal divide a rectangle? If the angles of your figure are exact right angles, and if your lines are exactly drawn, the diag- onal will be just 5 in. long. SUGGESTION TO TEACHER. Let pupils find by trial that if 3 in. be measured off upon one of the lines about a right angle and 4 in. upon the other line, the joining line will be 5 in. long. A 78. In the right triangle ABC how long is the hypotenuse AC if the numbers represent inches ? 79. If BC and AB were each twice as long as they are, AC would be twice as long as it is. If AB is 8 in. and BC 6 in., how long is AC? Prove by measuring. 80. If the perpendicular sides of a right triangle are in the ratio of 3 to 4, the ratio of the hypotenuse to the less side is f, and the ratio of the hypotenuse to the other side is f . In a right triangle whose base is 30 and altitude 40 how long is the hypotenuse? Represent. 81. How long is the hypotenuse of a -right triangle whose perpendicular sides are 9 in. and 12 in. ? 21 and 28 ? 15 and 20 ? 33 and 44 ? 82. The first steamship crossed the Atlantic Ocean in MDCCCXIX. Tor how many years has it been possible for Americans to go to Europe in a steamship ? 3 FIG. 6. MISCELLANEOUS EXERCISES 149 83. Imagine a block of ice 1 yd. in dimensions. How many square feet are there in all the surfaces ? 84. Imagine the same figure with one cubic foot cut out of one corner of it. How many square feet in all its surfaces ? 85. Imagine a cubic yard of ice, and suppose a cubic foot of it to be cut from the middle of one side. How many square feet in all the surfaces of the solid that is left ? 86. How many square feet in all the surfaces of the solid that would be left, if the cubic foot were put back in its place and the cubic foot above it were taken away ? 87. If a box 1 yd. in dimensions were packed f full of groceries, how many cubic feet of space would be left ? 88. What part of a cubic yard is a cube which is f of a yard in dimensions ? How many cubic feet are there in it ? 89. A coal dealer bought 1246 tons of coal at $ 4J- a ton, and sold it for $ 6 a ton. What was his gain on each ton ? On the whole ? 90. A man bought $ 88^- worth of furniture, paying in weekly installments of $ 14^ each. In how many weeks did he pay for the furniture ? 91. A grocer bought strawberries at the rate of 4 boxes for a quarter, and sold them, at the rate of 3 boxes for a quarter. How much did he gain on each box ? On a dozen boxes ? On a gross of boxes ? 92. Mr. Jones worked |- of a day on Monday, f of a day on Tuesday, and a whole day on Wednesday, on Thursday, and on Friday. On Saturday he worked % a day. At $ 3 per day, how much did he earn in the week ? CHAPTER V DENOMINATE NUMBERS 1. How many feet equal a yard? How many pints equal a quart? How many ounces equal a pound?. How was it decided in these cases how many units of a certain denomina- tion should make one of the next higher denomination? SUGGESTION TO TEACHER. Let facts concerning the origin of our systems of measuring be obtained from encyclopedias and other sources of informa- tion and brought to the class. Pupils should understand that the value of a unit in terms of lower denominations is an arbitrary value, varying in different kinds of measurements. 2. Numbers that show measurements whose values are set- tled by custom or law are called Denominate Numbers, as 5 bushels, 2 hours, 1 dollar. Denominate numbers that consist of more than one denomination are called Compound Denominate Numbers. Write a compound denominate number whose larg- est denomination is bushels. Hours. Tons. Miles. Acres. Gallons. Dollars. Meters. 3. The denominations of United States money are mills (m.), cents (), dimes (d.), dollars ($) and Eagles (E.). UNITED STATES MONEY One dollar is the standard 1 eagle = 10 dollars 1 dollar 1 dime = ^ of a dollar 1 cent = jfa of a dollar 1 mill = j^Vo of a dollar 150 DENOMINATE NUMBERS 151 4. Name each denomination of the following : $ 5875. $10,125. $20,705. 5. Express 3 dollars as cents. As dimes. As eagles. 6. A 10-dollar gold piece is called an Eagle. A 20-dollar gold piece is called a Double Eagle. A 5-dollar gold piece is called a Half Eagle. What is the value of a Quarter Eagle ? 7. Name the silver coins. What other coins are there? 8. How much money has a man who has 2 double eagles, an eagle, 3 half eagles, a quarter eagle, 2 dollars, 3 dimes, 2 nickels, and 3 cents ? 9. CLASS EXERCISE. may name a certain number of different kinds of coins, and the class may find the amount of money which their sum equals. 10. Treasury or bank notes are also used as money. If a man has eight $ 100 bills, seven $ 20 bills, a $ 10 bill, a $ 5 bill, a $ 2 bill, and three $ 1 bills, how much less than $ 1000 has he ? 11. How many mills in a dollar? In a half eagle? Why is there no coin to represent a mill ? 12. The denominations of liquid measure are gills (gi.), pints (pt.), quarts (qt.) and gallons (gal.). LIQUID MEASURE 424 gal. qt. pt. gi. Over the abbreviation of each denomination above you will find the number of units that equal a unit of tha.next higher denomination. Give the table of liquid measure, beginning with the units of the lowest denomination. 13. Fill the blank in the following table of equivalent values : 1 gal. = 4 qt. = 8 pt. = gi. SUGGESTION TO TEACHER. Pupils should make actual measurements, so far as is practicable, in connection with the study of each table, and should learn to change rapidly units of one denomination into units of another. 152 DENOMINATE NUMBERS 14. Illustrate each of the following statements : a As 4 gills equal a pint, any number of pints equals 4 times as many gills as pints. b As 2 pints equal a quart, any number of quarts equals twice as many pints as quarts. c As 4 quarts equal a gallon, any number of gallons equals 4 times as many quarts as gallons. 15. Express -f gal. as quarts, qt. as pints, f pt. as gills. 16. Express % gal. as quarts, -- qt. as pints. T 8 T pt. as gills. 17. Express .75 gal. as quarts. .5 qt. as pints. .625 pt. as gills. 18. Express .375 gal. as quarts. As pints. 19. Express 5|- gal. as quarts. As pints. As gills. 20. Express 1\ gal. as pints. 21. Express 2| gal. as gills. 22. Express 3^ gal. as quarts. 3 gal. and 2 qt. as quarts. 23. How many quarts in 5 gal. 2 qt. ? 7 gal. 1 qt. ? 24. How many pints in 7 qt. 3 pt. ? In 1 gal. 3 pt. ? 25. How many gills in 1 pt. 3 gi. ? 1 qt. 3 gi. ? 3 qt. 1 gi. ? 26. 1 gi. equals what part of 1 qt. 1 pt. ? 1 qt. 1 pt. = 5 pt., which equal 20 gi. 1 gi. = Js of 20 gi. 27. 1 gi. equals what part of 2 qt. 1 pt. ? Of 3 qt. 1 pt. ? Of 1 gal. 1 pt. ? 28. Illustrate each of the following statements : a As 4 gills make a pint, any number of gills equals \ as many pints as gills. DENOMINATE NUMBEKS 153 b As 2 pints equal 1 quart, any number of pints equals as many quarts as pints. c As 4 quarts equal 1 gallon, any number of quarts equals as many gallons as quarts. 29. Express 32 gi. as pints. As quarts. As gallons. 30. Express 40 pt. as quarts. As gallons. 31. Express 25 gi. as pints. Ans. 6| pt. 32. Express 7 pt. as quarts. 9 qt. as gallons. 33. Express 11 pt. as quarts. As gallons. 34. Express 13 pt. as quarts and pints. Ans. 6 qt. 1 pt. 35. Express 15^ pt. as quarts, pints, and gills. 36. Express 17^ qt. as gallons, quarts, and pints. 37. Express 19 pt. as gallons, quarts, pints, and gills. 38. 1 gi. equals what part of a pint ? Of a quart ? Of a gallon ? 39. 3 gi. equal what part of a quart ? Of a gallon ? 40. Express If pt. as quarts. As gallons. 41. Express 1 pt. 3 gi. as pints. 42. Express 2 qt. 1 pt. 2 gi. as pints. As quarts. As gallons. 43. Which is greater and how much, 2 gal. 1 qt. 3 pt. or 22 pt. ? 44. Express 1 pt. as a decimal of a quart. 3 qt. as a deci- mal of a gallon. 45. At 6 cents a quart, how much will a gallon of cider cost ? 31 gal. ? 4f gal. ? 1 gal. 3 qt. ? 1 pt. ? 3 qt. 1 pt. ? 46. Name several articles that are measured by liquid measure. 154 DENOMINATE NUMBERS 47. Add: gal qt pt i We find the sum of the gills to be 7 git 7 &* equal 1 pt. 3 gi. We place the 3 gi. under the col- 2312 urnn of gills and add the 1 pt. to the number of pints. 7103 The sum of the pints is 3 pt., equal to 1 qt. 1 pt. 4212 The * pt> is P^ aced un der the column of pints, and the 1 qt. is added to the number of quarts. The sum of 14 3 1 3 the quarts is 7 qt., equal to 1 gal. 3 qt. The 3 qt. are placed under the column of quarts, and the 1 gal. is added to the number of gallons, making 14 gal. 3 qt. 1 pt. 3 gi. Add: gal. qt. pt. gi. gal. qt. pt. gi. gal. qt. pt. gi. 48. 1 3 1 3 49. 5 2 1 1 50. 8 3 3 3302 7313 9302 gal. qt. pt. gi. gal. qt. pt. gi. gal. qt. pt. gi. 51. 6 3 1 2 52. 11 3 1 1 53. 15 2 1 1 7213 5213 11 3 1 3 54. A milkman leaves 25 gal. 3 qt. 1 pt. of milk at one hotel, and 33 gal. 2 qt. 1 pt. at another. How much does he leave at both hotels ? gal. qt. pt. gi. 55. From 7313 Subtract each number in the subtra- , -i 1 1 2 nen d from the corresponding number in . the minuend. 2201 56. CLASS EXERCISE. may write a compound denomi- nate number consisting of gal., qt., pt., and gi. The class may use it as a minuend, making every number in the subtrahend less than its corresponding term in the minuend. gal. qt. pt. gi. As 3 gi. cannot be taken from 1 gi., 57. From 7111 we reduce * P*- * S 1 - to S ills > which gives take 2303 5 gi. 5 gi. minus 3 gi. equal 2 gi. As the 1 pt. has been taken from the column 4202 of P ints and reduced to gills, there are no pints left in the minuend, from which pt. are to be taken. As 3 qt. cannot be taken from 1 qt., we reduce 1 gal. to quarts, which, with the 1 qt., equal 5 qt. 3 qt. from 5 qt. leave 2 qt. As 1 gal. has been taken from the column of gallons and reduced to quarts, only 6 gal. remain. 6 gal. minus 2 gal. equal 4 gal. Hence the difference is 4 gal. 2 qt. pt. 2 gi. DENOMINATE NUMBERS 155 Subtract : gal. qt. pt. gi. gal. qt. pt. gi. 58. 17 3 3 59. 16 3 1 1 11 111 8103 gal. qt. pt. gi. gal. qt. pt. gi. 60. 13 1 1 2 61. 15 2 1 6311 9303 62. 17 gal. 1 qt. 1 pt. of oil were in a tank. 11 gal. 2 qt. 1 pt. were drawn out. How much remained ? 63. Multiply 3 gal. 1 qt. 1 pt. 2 gi. by 9. gal. qt. pt. gi. 9 times 2 gi. equal 18 gi., which equal 4 pt. 2 gi. 3112 ^ e ^ &* are wr ^ten under the gills. 9 times 1 pt. p plus the 4 pt. already found equal 13 pt., which equal 6 qt. 1 pt. 9 times 1 qt. plus the 6 qt. already 30 3 J 2 found equal 15 qt. 15 qt. equal 3 gal. 3 qt. 9 times 3 gal. plus the 3 gal. already found equal 30 gal. Hence the product is 30 gal. 3 qt. 1 pt. 2 gi. gal. qt. pt. gi. gal. qt. pt. gi 64. Multiply 5312 65. 10 113 by 6 9_ 66. Multiply 7 gal. 2 qt. 1 pt. 3 gi. by 3. By 5. By 7. 67. A milkman sold 99 gal. 3 qt. of milk on Monday. If he were to sell the same amount every day for a week, how much milk would he sell ? 68. At 24^ a gallon, how much would he receive for the milk ? If the whole cost of the milk was $ 135, how much would he gain ? 69. How many times is 3 gal. 3 qt. 1 pt. contained in 19 gal. 1 qt. 1 pt. ? Express both dividend and divisor in the same denomination before dividing. 156 DENOMINATE NUMBERS 70. How many bottles each containing 1 pt. 2 gi. can be filled from a flask containing 3 gal. ? 71. How many bottles each containing 1 pt. 2 gi. can be filled from a 6-gallon tank ? 72. Divide 21 gal. 3 qt. 1 pt. 3 gi. by 6. In dividing denominate numbers, if there is a remainder after dividing, it is the custom to reduce that remainder to the next lower denomination instead of writing the quotient as a mixed number. In this way, frac- tions are avoided in all the denominations of the quotient except the lowest. Dividing 21 gal. by 6, we have 3 gal. for the quotient and 3 gal. for the remainder. 3 gal. or 12 qt. plus 3 qt. equal 15 qt. 15 qt. divided by 6 give 2 qt. for a quotient and 3 qt. for a remainder. gal. qt. pt. gi. 3 ctf" or 6 Pk P lus ! P fc - e( J ual 7 P*- 7 ?* divided 6 1 21 3 1 3 ky 6 give 1 pt ' for a ^ uotient and 1 P*- for a ' - - - remainder. 1 pt. or 4 gi. plus 3 gi. equal 7 gi. ^" 7 gi. divided by 6 equal 1 gi. Hence the quotient is 3 gal. 2 qt, 1 pt. 1 gi. gal. qt. pt. gi. gal. qt. pt. g 73. Divide 2)8 2 1 2 74. 3)6 3 1 2 75. Divide 9 gal. 3 qt. 1 pt. 1 gi. by 4. By 5. By 6. By 8. 76. How many half-pint bottles can be filled from a 10- gallon can of milk? 77- The denominations of dry measure are pints (pt.), quarts (qt.), pecks (pk.), and bushels (bu.). DRY MEASURE 482 bu. pk. qt. pt. Give the table of dry measure, beginning with the units of the lowest denomination. 78. Fill the blanks: 1 bu. = pk. = qt. = pt. 79. Give the ratio of 1 pt. to a unit of each denomination of dry measure. DENOMINATE NUMBERS 157 80. Express 3 qt. 1 pt. as pints. As quarts. As pecks. 81. Express 2 pk. 5 qt. 1 pt. as pt. As qt. As pk. As bu. 82. Name several articles that are measured by dry measure. 83. At 20^ a peck, how much does a bushel of tomatoes cost ? 3 qt. 1 pt. ? 5 qt. 1 pt. ? 84. At 25^ a quart, how much does a pint of strawberries cost ? 3 qt. 1 pt. ? 5 qt. 1 pt. ? 85. At 121 ^ a quart, how much does a peck of potatoes cost ? 1 pk. 2 qt. ? 1 bu. ? 86. Express -f bu. as pk. % pk. as qt. -J qt. as pt. f bu. as qt. 87. Express f pk. as bu. T 8 T qt. as pk. ^ pt. as qt. |- qt. as bu. 88. Express .125 pk. as qt. .875 bu. as pk. 55 qt. as pt. 89. Express .375 bu. as pk. As qt. As pt. 90. Express 4 qt. as a decimal of a peck. Of a bushel. 91. How many pints in -J of ^y of J of a bushel ? 92. By selling apples at f .40 a peck, Mr. Allen doubled his money. How much did they cost him per bushel ? 93. Express in pecks 7% of a bushel. Express in quarts 51 % of a peck. Add: bu. pk. qt. pt. bu. pk. qt. pt. 94. 20 3 5 1 95. 21 3 1 1 4111 33 201 5261 48 371 bu. pk. qt. pt. bu. pk. qt. pt. 96. 6 2 5 1 97. 10 3 6 1 3340 8151 6171 12 240 158 DENOMINATE NUMBEKS Subtract : bu. pk. qt. pt. bu. pk. qt. pt. 98. 18 3 2 1 99. 40 1 5 14 1 6 1 17 3 2 1 bu. pk. qt. pt. bu. pk. qt. pt. 100. 8360 101. 19 1 2 1 1241 14 361 Multiply : bu. pk. qt. pt. bu. pk. qt. pt. 102. 4261 103. 8371 5 7 bu. pk. qt. pt. bu. pk. qt. pt. 104. 7151 105. 8371 6 8 106. Multiply 6 bu. 3 pk. 3 qt. 1 pt. by 4. By 6. By 8. 107. A grocer has 3 bins, each holding 4 bu. 3 pk. 2 qt. of potatoes. How much do they all hold ? 108. How much wheat is there in 10 bins, if each bin con- tains 40 bu. 2 pk. 6 qt. ? 109. At 5 1 a quart, how much will a bushel of walnuts cost ? bu. pk. qt. pt. bu. pk. qt. pt. 110. Divide 5)6 1 7 1 111. 6)8 2 5 1 112. Divide 9 bu. 2 pk. 7 qt. 1 pt. by 2. By 3. By 4. 113. How many boxes containing 2 bu. 3 pk. of sawdust can be emptied into a bin which will hold 13 bu. 3 pk. ? 114. Place a cubic centimeter upon each of the corners of the upper surface of a cube that holds a liter, and find how many square centimeters there are in the surface of the figure thus formed. DENOMINATE NUMBERS 159 115. The standard unit of metric measure of capacity is the Liter, equal to about .9 of a quart dry measure and 1.05 quarts liquid measure. METRIC MEASURE OF CAPACITY 1 kiloliter (Kl.) = 1000 liters 1 hectoliter (HI.) = 100 liters 1 decaliter (Dl.) = 10 liters 1 liter (1.) 1 deciliter (dl.) = .1 of a liter 1 centiliter (cl.) = .01 of a liter 1 milliliter (ml.) = .001 of a liter Write table of equivalents. 1 Kl. = 10 HI. = Dl. = 1. = dl. = cl. = ml. 116. Learn to give quickly, forward and backward, the de- nominations of this table, and the meaning of each prefix to the word "liter." To help you remember the value of the units observe that D, H, and K, the abbreviations for Deca, Hecto, and Kilo, follow one another in alphabetical order. 117. Give the meaning of the following prefixes: Kilo, milli, Hecto, centi, Deca, deci. Of d, H, D, c, K. 118. Bead as a decimal 1235.576 1. Give the denomination of each figure. 119. How many centiliters in 5.37 1. ? In 8.4 1. ? 10.251.? 6.875 1. ? 120. How are liters reduced to centiliters ? 121. Express 7 kiloliters as liters. As hectoliters. As centiliters. As milliliters. 122. Express 2 HI. 5 Dl.'as 1. As cl. As Kl. As dl. 123. Reduce 2345.248 1. to units of each of the other de- nominations. 124. CLASS EXERCISE. may name a number of liters, and the class may reduce them to dl. To cl. To ml. 160 DENOMINATE NUMBERS 125. Write in one number, 5 Kl. 2 HI. 5 Dl. 2 1. dl. 7 ml. 126. Write and add : 3 Kl. HI. 2 Dl. 5 1. dl. 2 cl. 7 ml. 2 HI. 4 Dl. 1. 2 dl. 7 cl. 9 Kl. 3 HI. 2 Dl. 6 1. 5 dl. 2 cl. 8 ml. 127. From 8 Kl. 2 HI. 7 Dl. 4 1. 6 dl. 2 cl. 9 ml. take 6 HI. 4 Dl. 2 1. 3 dl. 8 cl. 5 ml. 128. 825.346 1. - 27.59 1. = ? 129. Multiply 125.275 1. by 5. By 8. By 12. 130. Multiply 341.626 1. by 10. By 100. By 1000. 131. Divide 239.268 1. by 4. By 6. By 12. 132. Under which system of denominate numbers is it easier to add, subtract, multiply, and divide, the metric system or the English system ? Why ? 133. How many liters in 7% of 132.5 1. ? Of 178.7 1. ? 134. 17 1. +3% of 17 1. = how many liters? How many dl.? How many cl.? 135. How much will 7.5 1. of wine cost at $ 1.25 a liter ? 136. What is the cost of a Dl. of wheat, at $ 7.25 per HI.? 137. What is the cost of a hogshead of wine containing 225 1. at $ .15 per liter ? 138. What is the cost of 25 1. of vinegar at $ 15 per HI.? 139. How much wheat is contained in 125 sacks, each hold- ing 1 HI. 2 Dl. ? 140. The denominations of avoirdupois weight are ounces (oz.), pounds (lb.), hundredweights (cwt.), and tons (T.). AVOIRDUPOIS WEIGHT 20 100 16 T. cwt. lb. oz. Fill out the following table of equivalents. 1 T. = cwt. = lb. = oz. DENOMINATE NUMBERS 161 141. 1 oz. is what part of a pound ? Of a hundredweight ? Of a ton ? 142. Express 5 Ib. 8 oz. as ounces. As pounds. As hun- dredweights. As tons. 143. Express 3 T. 10 cwt. 25 Ib. 12 oz. in each denomination of avoirdupois weight. 144. Express -fa T. as cwt. f cwt. as Ib. 145. Express f oz. as Ib. -f Ib. as cwt. f Ib. as cwt. 146. Express .7 T. as cwt. .17 cwt. as Ib. .125 Ib. as oz. .75 T. as Ib. 147. Express .625 T. as cwt. As Ib. As oz. 148. Express 15 Ib. as a decimal of a hundredweight. 149. At 20^ a pound, how much will 2 Ib. and 8 oz. of butter cost ? 3 Ib. 4 oz. ? 5 Ib. ? 12 oz. ? 7 Ib. 2 oz. ? 150. How many pounds in 21 T. ? In 3% of a ton ? 151. A farmer brought a ton of hay to market and sold 25% of it. How many pounds had he left ? 152. How many cwt. in 5% of a ton ? In 45% ? 153. 1 Ib. f of a pound = how many ounces ? Add: T. cwt. ib. oz. T. cwt. Ib. oz. 154. 7 15 75 8 155. 10 19 67 5 4 12 55 12 12 14 25 13 6 17 80 15 15 16 84 11 T. Ib. oz. T. Ib. oz. 156. 8 425 7 157. 16 875 10 9 375 8 14 985 11 7 425 5 6 435 13 HORN. GRAM. SCH. AR. 11 162 DENOMINATE NUMBERS T. cwt. Ib. oz. T. cwt. Ib. oz. 158. From 9 3 30 4 159. 10 5 take 27 41 7 6728 T. Ib. oz. T. Ib. oz. 160. From 8 201 8 161. 18 700 12 take 4 175 12 4 900 8 162. From 30 T. 800 Ib. of hay there were sold 7 T. and 900 Ib. How much was left ? 163. Mrs. Harris brought 20 Ib. 8 oz. of butter to market, and sold 18 Ib. and 12 oz. How much remained unsold ? 164. Name articles that are weighed by avoirdupois weight. 165. John weighs 87 Ib., Thomas 92 Ib., William 97f Ib. How much do they all weigh ? Give answer in pounds and ounces. SUGGESTION FOR CLASS EXERCISE. Find sums and differences of weights of pupils. T. cwt. Ib. oz. T. cwt. Ib. oz. 166. Multiply 5 10 40 10 167. 15 15 75 15 by 8 12 168. Multiply 15 T. 7 cwt. 25 Ib. 8 oz. by 3. By 5. By 8. By 9. Divide : T. cwt. Ib. oz. T. cwt. Ib. oz. 169. 4)9 15 20 8 170. 5)16 14 50 10 171. Divide 20 T. 12 cwt. 48 Ib. 12 oz. by 6. By 8. By 12. 172. One Christmas day 40 T. of coal were equally distrib- uted among 11 poor families. How many tons, hundred- weights, and pounds did each receive ? 173. Formerly 2240 Ib. were considered a ton, and that standard is sometimes used now. In that case the ton was called a "long ton." How many pounds in 4J long tons ? In of along ton? DENOMINATE NUMBERS 163 174. A coal dealer buys 150 T. of coal, 2240 Ib. each, at $ 4.50 per ton. He sells it at $ 4.75 per ton, giving 2000 Ib. per ton. How much does he gain ? 175. The standard unit of metric measure of weight is a Gram, equal to about 15^- Troy grains. METRIC MEASURE OF WEIGHT 1 kilogram (Kg.) = 1000 grams 1 hectogram (Hg.) = 100 grams 1 decagram (Dg.) = 10 grams 1 gram (g.) 1 decigram (dg.) = .1 of a gram 1 centigram (eg.) = .01 of a gram 1 milligram (mg.) = .001 of a gram Fill blanks : 1 Kg. = Hg. = Dg. = g. = dg. = eg. = mg. 176. The weight of a cubic centimeter of water is a gram. How many grams does a liter of water weigh? 177. What part of a kilogram is a decagram? Decigram? Milligram ? Hectogram ? Centigram ? 178. Name each denomination of the expression 1978.347 g. 179. Write in one number, making the gram the unit 4 Kg. 7 Hg. 6 Dg. 5 g. 2 dg. 6 eg. 3 mg. 180. Express 75 Kg. as grams. As eg. As mg. As Dg. As dg. As Hg. 181. Express 186 eg. as grams. As mg. As Dg. As dg. As Hg. As Kg. 182. Add 325 g., 55 Kg., 75 Dg. 183. How many grams in 20% of 425 g. ? 184. 126 g. - 4% of 126 g. equal how many Dg. ? dg. ? eg. ? mg. ? 185. A nickel weighs 5 g. $5.00 in nickels weigh how many kilograms? 164 DENOMINATE NUMBERS 186. A kilogram is equal to about 21 Ib. Find the approxi- mate cost of a kilogram of a drug that costs $ .60 a pound. 187. What is the cost of 2242 g. of coffee at $ .60 a kilogram ? 188. If a kilogram of wool costs $ 1.75, how much will 6.5 kilograms cost ? 189. The denominations of time measure are seconds (sec.), minutes (min.), hours (hr.), days (da.), and years (yr.). TIME MEASURE 100 365 24 60 60 century yr. da. hr. min. sec. Till out the following table of equivalents : 1 century = yr. = da. = hr. = min. = sec. 190. Find the ratio of one hour to a unit of each denomina- tion of time measure. 191. Express 2 hr. 10 min. as sec. As min. As hr. 192. Express f yr. as da. T 5 g- da. as hr. f hr. as min. 193. Express ^ of a year as da. As hr. As min. As sec. 194. Express |f min. as hr. 2| hr. as da. 100 sec. as min. 195. Express .12 yr. as da. .33^ da. as hr. .75 hr. as min. .66| min. as sec. .17 centuries as yr. 196. Express 108 sec. as a decimal of a minute. Of an hour. 197. How many minutes in 5% of an hour? In 65%? 95%? 198. 1 hr. + 15% of an hour = how many minutes ? 199. At 6% now much interest will $300 gain in 1 yr. ? In 1| yr. ? 3 yr. 6 mo. ? 2 yr. 9 mo. ? 200. Express in years, 1 yr. 1 mo. 15 da. Find the interest of 400 for that time at 6 % At 8 % At 5 % . DENOMINATE NUMBERS 165 201. Usually every fourth year has 366 da., and is called a leap year. In leap year the month of February, which usually has 28 da., has 29 da. How many hours in February of a leap year ? 202. Every year whose number is divisible by 4 is a leap year, excepting the years whose number ends in two ciphers, as 1500, 1700, 1800. Such years are leap years only if their number is divisible by 400, as 1200, 1600, 2000. Which of the following are leap years ? 1848 1862 1892 1900 1904 2000 2108 2200 1000 203. At a dollar a day, how much could a man earn, working 6 da. in a week, in the month of February, 1896, which began on Saturday ? How much in February, 1898 ? 204. A $ 1.50 per day, how much could a man earn in February of a leap year, in which the 1st of February fell on Sunday ? 205. A man's heart beats at the rate of about 72 beats in a minute. At that rate how many times does it beat in an hour ? In a day of 24 hr. ? In a common year ? 206. At that rate how many times would a man's heart beat in a lifetime of 80 yr., - of which are leap years ? 207. When a person born in 1883 is 20 yr. old, how many years of his life have been leap years ? 208. The time in which the earth passes once around the sun is 365 da. 5 hr. 48 min. 46 sec. That is how much more than 365 da. ? To how much will the extra time amount in 4 yr. ? How much does it lack of being a whole day ? 209. In adding the whole day to every fourth year or leap year, how many more minutes and seconds are added to the year than rightly belong to it ? 166 DENOMINATE NUMBERS 210. Find how nearly the error caused in a century by this arrangement is corrected by omitting the extra day in February at the end of the century. 211. Find the amount of error at the end of the fourth cen- tury, and see how nearly it is corrected by restoring the extra day in February of that year. 212. Henry rose at 5.50 A.M. and went to bed at 8.20 P.M. How long was his day ? 213. CLASS EXERCISE. may give a time for rising and a time for going to bed, and the class may find the length of the included day. 214. How many hours were there in the year 1800 ? " Thirty days hath September, April, June, and November. " 215. Learn the above rhyme and remember the fact that all the rest of the months except February have 31 days. 216. Write the names of the months in order, beginning with January, using abbreviations. Write opposite the name of each month the number of days it contains. 217. Find the number of days from May 1st to June 7th. There are 30 days after the 1st in May, which, with the 7 days in June, make 37 days. Find the number of days between the following dates : 218. May 7, 1896, July 4, 1896. 219. Jan. 1, 1900, Mar. 1, 1900. 220. Sept. 28, 1899, Nov. 5, 1899. 221. Dec. 15, 1899, Jan. 31, 1900. 222. How many days from to-day until next Christmas? Next 4th of July ? DENOMINATE NUMBERS 167 ' 223. CLASS EXERCISE. may give the date of his next birthday, and the class may find the number of days intervening. 224. Thirty days from the 4th of July, 1876, was what date ? 225. What was the date 60 days after Oct. 14th, 1492? Christmas, 1897 ? 226. On the 17th day of June, Mr. Herbert borrowed from a bank $ 100 to be paid in 60 days. When was it due ? 227. What will be the date 30 days after to-day ? 60 da. ? 228. What will be the date 33 days from next Monday ? 63 da. ? 93 da. ? 105 da. ? 229. What was the date 30 days before the first of May, 1891 ? 60 days before the 1st of March, 1892 ? 230. How many days since the 1st of January of this year ? 231. CLASS EXERCISE. - may give the date of his last birthday, and the class may find how many days have passed since then. Add da. hr. min. sec. da. hr. min. sec. 232. 17 20 30 40 233. 19 19 45 30 20 16 40 10 25 20 15 30 40 18 50 20 16 12 30 30 Add: da. hr. min. sec. da. hr. min. sec. 234. 19 14 30 45 235. 15 2 30 30 70 20 45 15 30 10 45 25 6 18 15 15 40 6 15 35 4 2 30 30 9 8 15 15 236. Mr. Cox earns $ 2.00 for each day of 10 hours that he works. On Monday he worked 8 hr. 30 min. ; on Tuesday, 9 hr. 10 min. ; Wednesday, 7 hr. 40 min. ; Thursday, 8 hr. 30 min. ; Friday, 7 hr. 50 min ; Saturday, 4 hr. 30 min. How much did he earn in that week ? 168 DENOMINATE NUMBERS 237. At $ 2.00 per day of 8 hours, how much would Mr. Cox have earned ? Find the differences : da. hr. min. sec. da. hr. min. sec. 238. 47 18 2 10 239. 62 10 30 15 25 20 1 40 48 20 19 45 240. Which is the 7th month ? 12th ? 3d ? 5th ? 241. Which month is February? September? June? November ? 242. Bead the following dates : 9/5/99. 10/5/98. 7/4/76. 243. Find the difference of time between March 5, 1898, and Oct. 21, 1902. Write as below and subtract : 1902 10 21 1898 3 5 In finding the difference between two dates in years, months, and days, we assume that 30 days = a month. American Authors Ralph Waldo Emerson, born May 25, 1803 died April 27, 1882. John G. Whittier, born Dec. 17, 1807 died April 27, 1892. Henry W. Longfellow, born Feb. 27, 1807 died March 24, 1882. James Russell Lowell, born Feb. 22, 1819 died Aug. 12, 1891. English Authors Alfred Tennyson, born Aug. 6, 1809 died Oct. 6, 1892. Charles Dickens, born Feb. 7, 1812 died June 9, 1870. 244. Find the number of years, months, and days since each of the authors mentioned above was born. Since each died. 245. Find the age of each author at his death. 246. Find the time between the birth of Emerson and that of each of the other authors. DENOMINATE NUMBERS 169 247. Find the time between the death of Dickens and that of each of the other authors. da. hr. min. sec. da. hr. min. sec. 248. Multiply 7 18 20 10 249. 11 15 8 30 by 7 9 250. Multiply 5 da. 10 hr. 20 min. 30 sec. by 5. By 6. By 8. 251. If you spend 4 hr. 30 min. in school every day, how many hours and minutes do you spend in a school week ? In a school month of 4 wk. ? In a school year of 10 mo. ? Divide : da. hr. min. sec. da. hr. min. sec. 252. 4)21 16 2 40 253. 6)13 13 13 30 254. Divide 15 da. 12 hr. 40 min. 30 sec. by 5. By 6. By 8. 255. The denominations of linear measure are inches (in.), feet (ft.), yards (yd.), rods (rd.), and miles (mi.). LINEAR OR LONG MEASURE 320 5| 3 12 mi. rd. yd. ft. in. Fill out the table of equivalent values. 1 mi. = rd. = yd. = ft. = in. 256. How many inches in 2 yd. 1 ft. 7 in. ? 5 yd. 3 ft. 7 in. ? 257. Express 2 yd. 1 ft. 6 in. as in. As ft. As yd. 258. Express 2 mi. 20 rd. as mi. As rd. As yd. As ft. 259. 1 ft. is what part of a yard ? Of a rod ? Of a mile ? Reduce complex fractions to simple fractions. 260. Express 1 ft. 6 in. as yd. As rd. As mi. 261. Express 1 yd. 1 ft. 9 in. as yd. As rd. As mi. 262. Express T 5 ^ mi. as rd. T 8 T rd. as yd. % yd. as ft. -| ft. as in. rd. as ft. 170 DENOMINATE NUMBERS 263. Express f in. as ft. ^ ft. as yd. | yd. as rd. 11$ rd. as mi. If ft. as rd. If in. as ft. 4 ft. as yd. 264. Express .875 mi. as rd. 3.6 rd. as yd. 5.5 yd. as ft. .9 ft. as in. 1.66| yd. as ft. .64 rd. as yd. 265. Express 7.2 in. as a decimal of a foot. As a decimal of a yard. 266. Express 115.5 ft. in yd. In rd. In mi. 267. How many rods in 5% of a mile ? In 15% ? 35% ? Add: yd. ft. in. yd. ft. in. 268. 2 1 Hi- 269. 6 2 7 5 1 2j 3 2 llf mi. rd. yd. ft. in. mi. rd. yd. ft. in. 270. 6 200 2 1 10 271. 10 20 1 2 10 8 120 1 2 6 7 300 1 1 8 mi. rd. yd. ft. in. mi. rd. yd. ft. in. 272. 8 150 4 2 11 273. 10 180 4 2 3 4 100 1 1 5 7 40 2 1 11 mi. rd. yd. ft. in. mi. rd. yd. ft. in. 274. 16 200 3 2 4 275. 18 2 4 2 10 14 150 1 1 10 5 319 2 1 7 mi. rd. yd. ft. in. 276. 15 3223 10 319 1 1 9 277. Mr. Smith's lot is 1 yd. 1 ft. shorter than Mr. Brown's lot, which is 30 yd. long. How long is Mr. Smith's lot ? 278. An elm tree is 32 ft. 9 in. high. How high is a fir tree that is 6 ft. 10 in. less in height ? DENOMINATE NUMBERS 171 279. Find the difference between the height of John, who is 5 ft. 3 in. tall, and his sister, who is 3 ft. 11 in. A B c 28 - ^ ^ were ^Of miles from A to C, and 5J- miles from A to J5, how far would it be from B to C ? 281. If it were 20 mi. 20 rd. from A to C, and 5 mi. 80 rd. from A to J3, how far would it be from BtoC? Multiply : yd. ft. in. mi. rd. yd. ft. in. 282. 312 283. 52109 3 4 mi. rd. yd. ft. in. mi. rd. yd. ft. in. 284. 7 80 2 6 285. 10 160 4 1 11 2 7 286. Multiply 2 mi. 240 rd. 3 yd. 5 ft. 6 in. by 2. By 3. 287. How long is the perimeter of a regular pentagon, each of whose sides is 3 yd. 1 ft. 9 in. long ? 288. A summer house is built in the shape of a regular hexagon, each side being 2 yd. 1 ft. 6 in. What is the entire distance around it ? 289. How long is the edge of the border of a flower bed in the shape of an octagon, if each side is 1 yd. 1 ft. 8 in. long ? 290. Divide 3 mi. 16 rd. 2 yd. 1 ft. 8 in. by 2. By 3. By 4. 291. Divide 19 mi. 10 rd. 5 yd. 2 ft. 6 in. by 5. By 6. By 7. 292. Divide 7 mi. 160 rd. 4 yd. 1 ft. 8 in. by 2. By 3. By 4. 293. Henry measured a cornstalk and found it to be 5 ft. 6 in. long. How many such stalks laid in a continuous line would it take to extend a mile ? 172 DENOMINATE NUMBERS 294. Harry has two dogs, Don Quixote and Sancho Panza. Don measures 4 ft. 3 in. from the end of his nose to the tip of his tail, weighs 77 Ib. 12 oz., and is 3 yr. 7 mo. 9 da. old. Sancho measures 3 ft. 8 in., weighs 53 Ib. 14 oz., and is 2 yr. 11 mo. 28 da. old. Find the difference of the lengths of the dogs. Of their weights. Of their ages. 295. Harry's dog, Sancho Panza, chased a rabbit 40 yd. and then gave up the chase. The rabbit had 10 yd. the start of the dog and ran twice as fast as he ran. How far apart were the animals when Sancho Panza gave up the chase ? Represent. 296. If from a string 2 yd. 2 ft. long, 2.5 ft. is broken off at one end, and 3.7 ft. at the other, how long a string is left ? 297. The denominations of square measure are square inches (sq. in.), square feet (sq. ft.), square yards (sq. yd.), square rods (sq. rd.), acres (A.), and square miles (sq. mi.). SURFACE OR SQUARE MEASURE 640 160 30 9 144 sq. mi. A. sq. rd. sq. yd. sq. ft. sq. in. Write a table of equivalent values. 1 sq. mi. = A. = sq. rd. = sq. yd. = sq. ft. = sq. in. SUGGESTION TO TEACHER. Let a diagram of the square rod be drawn on the floor, a square yard in one corner of the square rod being subdivided into square feet, and one of the square feet into inches, so that literally each square unit may be a part of the units of higher denominations. Let pupils make many practical problems upon the figures. 298. 1 sq. ft. equals what part of a sq. yd. ? Of a sq. rd. ? 299. How many square feet in 2 sq. rd. ? 3 sq. rd. ? 5 sq. rd. ? 7 sq. rd. ? 10 sq. rd. ? 12 sq. rd. ? 300. How many square inches in 5 sq. ft. 60 sq. in. ? In 4 sq. ft. 20 sq. in. ? In 2 sq. ft. 80 sq. in. ? 301. How many square feet in 4 sq. yd. 6 sq. ft.? In 20 sq. yd. 7 sq. ft. ? In 18 sq. yd. 5 sq. ft. ? DENOMINATE NUMBERS 173 302. How many square yards in 8 sq. rd. 15 sq. yd. ? In 10 sq. rd. 4f sq. yd. ? In 24 sq. rd. 19 sq. yd. ? 303. Express 1 sq. mi. 320 A. 80 sq. rd. in sq. rd. In A. In sq. mi. 304. Express 10 sq. yd. 7 sq. ft. 72 sq. in. in sq. in. In sq. ft. In sq. yd. 305. Express 2 sq. rd. 15 sq. yd. in sq. yd. In sq. rd. In sq. ft. In A. 306. Express ^ sq. mi. as A. -^ A. as sq. rd. 307. Express T 8 T sq. rd. as sq. yd. ^ sq. yd. as sq. ft. -ff sq. ft. as sq. in. - 1 / sq. in. as sq. ft. 308. Express .625 sq. mi. as A. .375 A. as sq. rd. .48 sq. rd. as sq. yd. .175 sq. yd. as sq. ft. .7 sq. ft. as sq. in. 4.32 sq. in. as sq. ft. .18 sq. ft. as sq. yd. 309. Express 345.6 sq. in. as a decimal of a square foot. Add: sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in. 310. 21 2 100 311. 36 3 70 786 15 7 60 16 5 40 20 7 20 Find difference : sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in 312. 17 8 85 313. 21 6 100 4 3 75 6 8 75 Multiply : sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in. 314. 24 3 140 315. 16 4 96 6 7 316. Multiply 2 A. 40 sq. rd. 10 sq. yd. 4 sq. ft. 20 sq. in. by 3. By 4. By 5. 174 DENOMINATE NUMBERS 317. How many acres in 5% of a square mile ? 7% ? 15% ? 318. 1 sq. ft. 37|% of a square foot = how many square inches ? 319. Divide by 3, 48 sq. yd. 7 sq. ft. 4 sq. in. 320. Divide by 5, 25 sq. yd. 6 sq. ft. 2 sq. in. 321. Divide 20 A. 80 sq. rd. 20 sq. yd. 4 sq. ft. 72 sq. in. by 4. By 8.. By 6. 322. How many square feet in a square f of a foot in dimensions ? How many square inches ? 323. How long is the perimeter of a square f of a foot in dimensions ? What fraction of a square foot is its area ? How many square inches in its area ? 324. Mr. Gilbert owns 400 A. 120 sq. rd. of land in Gibson County, 225 A. and 10 sq. rd. in Warrick County, and 14 A. 40 sq. rd. in Vanderburgh County. How much does he own in those counties ? 325. A farmer had 80 A. 50 sq. rd. of land. After selling 30 A. 10 sq. rd., how much had he left ? 326. Mr. Carter owns three times as much land as his cousin, who owns 120 A. 80 sq. rd. How much land do both own? 327. A garden 180 ft. long. 150 ft. wide is surrounded by a tight board fence 6 ft. high. How much will it cost to paint the fence on both sides @ 12 ^ per square yard ? SUGGESTION TO TEACHER. Pupils who cannot imagine the conditions of this problem may be required to inclose a surface on their desks by a strip of paper folded so that its divisions represent the parts of the fence. 328. From each corner of a square, a side of which is 2 ft. 5 in., a square measuring 5 in. on a side is cut out. Represent and find the area of the remainder of the figure. DENOMINATE NUMBERS 175 329. Find the area of the walls of a room 12 ft. long, 10 ft, wide, 8 ft. high. Find the areas of the walls and ceiling of rooms of the following dimensions, and the cost of plastering them at 20 cents a square yard, no allowance being made for openings. a b Length Width Height Length Width Height ft. ft. ft. ft. ft. ft. 330. 20 18 10 40 30 12 331. 30 25 9 21 20 9 332. 25 21 9 30 18 8 333. 18 15 8 4 20 9 334. 15 12 8 16 15 8 335. Which is greater, a rectangle 12 in. by 12 in. or a rectangle 16 in. by 9 in.? What is the difference in the length of their perimeters ? 336. Give dimensions of several rectangles, each equal to a square foot. Compare the lengths of their perimeters. 337. If two rectangles have equal areas but different shapes, which will have the longer perimeter, the one which is more nearly square or the other ? Illustrate. 338. Find the area of a square whose perimeter is 24 in. 28 in. 36 in. 40 in. 339. Find the perimeter of a square whose area is 9 sq. in. 340. The denominations of cubic measure are cubic inches (cu. in.), cubic feet (cu. ft.), and cubic yards (cu. yd.). CUBIC MEASURE 27 1728 cu. yd. cu. ft. cu. in. Write table of equivalent values. 1 cu. yd. = cu. ft. = cu. in. 176 DENOMINATE NUMBERS 341. How many cubic inches in 5 cu. ft. 192 cu. in. ? In 1 cu. yd. 624 cu. in. ? 342. How many cubic inches in .875 cu. ft.? In .625 cu. ft.? 343. John may draw a square yard on the floor in a corner of the room. How many blocks of ice 1 foot in dimensions would it take to cover that square yard ? 344. If another layer of cubic feet of ice were laid upon the first, how many cubic feet of ice would there be ? How high would the ice be piled ? 345. If a third layer of cubic feet of ice were placed upon the other two, how many cubic feet of ice would there be ? 346. What name is given to a solid which is 3 ft. long, 3 ft. wide, and 3 ft. high ? 347. In the square yard which John drew William may set a yard stick upright at that corner which is not against a wall. Two other boys may place sticks in such a position that a cubic yard is outlined in the corner of the room. SUGGESTIONS TO TEACHER. Devote a certain space in the room to the imaginary cubic yard. See that every member of the class images a cubic yard in that particular place. Let pupils show divisions of the cube by outlining them with their hands in the space devoted to it. Let pupils model before the class with inch cubes the figures given in the following exercises. 348. Image a cubic yard with one cubic foot cut from the upper layer at a corner that is not against a wall. Model the figure. Outline in the cubic yard in the corner the part taken out. What is the ratio of the part taken out to the whole cubic yard ? What is the ratio of the part taken out to the part left ? 349. Take two more cubic feet from the upper layer, one on each side of the vacant space. Model. Outline in the cubic yard. Tell the ratio of the vacant space to the whole cubic yard. Of the vacant space to the filled space. DENOMINATE NUMBERS 177 350. Take out the two cubic feet that were directly under the cubic foot first removed. Model. Outline. Find the ratio of the vacant space to the cubic yard. Of the filled space to the cubic yard. Of the vacant space to the filled space. 351. Restore the whole cubic yard. Take away the middle cubic foot on each side of the upper layer. Model. Outline. Tell the ratio of the vacant space to the cubic yard. Of the filled space to the cubic yard. Of the vacant space to the filled space. 352. CLASS EXERCISE; may give directions for taking away parts of the cubic yard. The class may tell the ratios of the spaces to one another. Some members may model the figures, and some one else may show the outline of the space imaged as vacant in the cubic yard. 353. How many feet in the sum of all the edges of a cube 1 yd. in dimensions ? 354. How much will a cubic yard of building stone cost at $ 2.50 a cubic foot ? 355. How many cubic feet in a right prism 6 ft. long, 2 ft. wide, and 1 ft. high ? Model the prism. 356. How many cubic feet in a right prism, a 4 ft. long, 2 ft. wide, 2 ft. high ? b 5 ft. long, 3 ft. wide, 1 ft. high ? c 8 ft. long, 2 ft. wide, 2 ft. high ? d 10 ft. long, 4 ft. wide, 2 ft. high ? 357. How many cubic feet in a tank 14 ft. long, 10 ft. wide ; 5 ft. high ? 358. How many cubic yards in a wall 81 ft. long, 3 ft. thick, and 9 ft. high ? In a wall 30 ft. long, 6 ft. high, and 3 ft. thick ? HORN. GRAM. SCH. AR. 12 178 DENOMINATE NUMBERS 359. A monument is in the shape of a right prism, 7 ft. long, 4 ft. wide, and 3 ft. high. How much will it cost at $ 3.50 per cubic foot ? 360. If the engraving on it costs $ 62.50, what will be the entire cost of the monument ? 361. If a cake 3 in. long, 3 in. wide, and 3 in. high, has icing all over it except on the under side, how many square inches of icing has it ? 362. To cut the cake into inch cubes, how many cuts would be necessary ? How many inch cubes would there be ? 363. How many of the cubes would have icing on three sides ? On two sides ? On one side ? On no side ? 364. How many inch cubes can be placed on a square foot ? How many layers of those cubes would it take to make a cubic foot ? How many cubic inches in a cubic foot ? What is the ratio of 1 cu. in. to a cubic foot ? 365. Imagine a cubic foot of marble with 1 cu. in. cut from each of the upper corners. What would be the ratio of the part cut out to the part left ? 366. Which is greater, a right prism 12 in. by 12 in. by 12 in., or one 24 in. by 12 in. by 6 in. ? Compare their surfaces. 367. Give dimensions of several right prisms, each of which equals a cubic foot. Compare their surfaces. 368. How many cubic feet in a stick of timber 12 in. wide, 9 in. thick, and 24 ft. long? 369. How many cubic feet in a cistern 5 ft. square and 6 ft. deep ? How many cubic inches ? How many gallons will the cistern hold ? (231 cu. in. = 1 gal.) Add: cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 370. 5 10 1700 371. 3 5 1400 4 8 129 10 24 300 DENOMINATE NUMBERS 179 Add: cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 372. 9 11 1720 373. 4 15 1600 6 18 10 8 20 200 Subtract : cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 374. 120 13 1700 375. 41 10 1634 65 15 1125 25 18 1507 Subtract : cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 376. 81 3 208 377. 16 4 800 40 20 125 4 21 525 Multiply : cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 378. 20" 5 1160 379. 15 10 989 3 4 380. Multiply 2 cu. yd. 20 cu. ft. 1000 cu. in. by 2. By 4. Divide : cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in. 381. 5) 6 20 72 382. 8) 10 4 36 383. Divide 10 cu. yd. 15 cu. ft. 180 cu. in. by 3. By 5. 384. Imagine a cubic rod of marble. Why do we have no such measurement as a cubic mile ? 385. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is a cord of wood. How many cubic feet in a cord of wood ? 386. Bepresent a cord of wood by drawing or by placing blocks or toothpicks. 387. How many cords in a wood pile 16 ft. long, 8 ft. wide, and 8 ft. high? 180 DENOMINATE NUMBERS 388. At $ 5.00 per cord, what is the value of a pile of wood 20 ft. long, 4 ft. wide, and 4 ft. high ? Of a pile 18 ft. long, 8 ft. wide, and 8 ft. high ? 389. The standard unit of metric linear measure is a Meter, which is 39.37 in. This length was obtained by calculating one ten-millionth of the distance from the equator to a pole of the earth. A kilometer is about f of a mile. METRIC LINEAR MEASURE 1 kilometer (Km.) = 1000 meters 1 hectometer (Hrn.) = 100 meters 1 decameter (Dm.) = 10 meters 1 meter (m.) 1 decimeter (dm.) = .1 of a meter 1 centimeter (cm.) = .01 of a meter 1 millimeter (mm.) = .001 of a meter "Write table of equivalents : 1 Km. = Hrn. = Dm. = m. = dm. = cm. = mm. 390. Express 42 m. as centimeters. As decimeters. As decameters. As hectometers. 391. Express 375 m. as kilometers. As decameters. As decimeters. As millimeters. 392. Express 4287 m. as kilometers. As decimeters. As hectometers. 393. How many centimeters in 11% of 12 m. ? Of 25 dm. ? 394. 7 % of 192 m. = how many meters ? Decameters ? SUGGESTION TO TEACHER. Let pupils find in meters and decimals the length and width of room. Length of blackboards. Length of diagonal of room or blackboard. Heights of pupils. 395. About how many inches in a decameter ? In a deci- meter ? 396. Find approximately the number of inches in a kilo- meter. In 4 Hm. In 12 Dm. In 7 dm. DENOMINATE NUMBERS 181 397. What is the cost of 12 m. of cloth at $.75 per meter ? Is the cloth cheaper or dearer than at $ .75 per yard ? Explain. 398. At the rate of 36 Km. per hour, how far will a train run in 3 hr. 30 min. ? 399. What is the value of a decameter of silk at $ 1.65 per meter ? 400. How many centimeters long is the perimeter of a regu- lar octagon, one side of which is 8 mm. ? 401. Find the length in decimeters of one side of a regular pentagon whose perimeter is 75 cm. 402. How long is the base of an isosceles triangle whose perimeter is 4 dm. and whose equal sides are each 12 cm. ? Represent. 403. How long is each of the equal sides of an isosceles triangle whose perimeter is 3 dm. and base 8 cm. Construct. 404. A kilometer is about what fraction of a mile ? 405. Find approximately the number of miles in 40 Km. In 72 Km. In 3.2 Km. In 6.72 Km. 406. Find the approximate number of miles in 9288 m. SUGGESTION. Express 9288 m. as kilometers before finding its equiva- lent in miles. 407. Find approximately the number of miles in 45864 m. In 63824 dm. In 59888 Dm. In 71848 Hm. 408. Find approximately the number of kilometers in 75 mi. In 235 mi. In 84.5 mi. 409. How many square millimeters in a rectangle 1 cm. long and 1 cm. wide ? 410. In a square decimeter, how many square centimeters ? Square millimeters ? 411. In a square meter how many square decimeters? Square centimeters ? Square millimeters ? 182 DENOMINATE NUMBERS 412. A square decameter equals how many square meters ? Square decimeters ? Square centimeters ? Square millimeters ? 413. A square hectometer equals how many square deca- meters ? Square meters ? Square decimeters ? 414. A square kilometer equals how many square hecto- meters ? Square decameters ? Square meters ? Square deci- meters ? Square centimeters ? Square millimeters ? 415. In long measure, under the metric system, what is the ratio of a unit of each denomination to a unit of the next higher denomination ? 416. In square measure, metric system, what is the ratio of a unit of each denomination to a unit of the next higher denomination ? 417. Write a table of square measure, metric system. Write a table of equivalents of units of square measure, metric system. 418. Express 3 sq. Km. 2 sq. Dm. 50 sq. m. in square meters. In square decameters. In square kilometers. 419. Express 7 sq. m. 20 sq. dm. 30 sq. cm. in square millimeters. In square centimeters. In square decimeters. 420. Express 1 sq. m. 2 sq. dm. 3 sq. cm. in sq. mm. In sq. cm. In sq. dm. In sq. m. In sq. Dm. In sq. Hm. In sq. Km. 421. What is the area of a square whose perimeter is 24 cm. ? 20 cm. ? 40 mm. ? 422. How long is the perimeter of a square whose area is 81 sq. cm.? 49 sq. cm.? 64 sq. dm.? 423. What is the area and the perimeter of a rectangle which is 35 cm. long and ^ as wide as long ? 424. A land measurement, 10 meters square, or its equivalent, is called an Are (a.). How many square meters in an are ? How long is the perimeter of an are in the form of a square ? DENOMINATE NUMBERS 183 425. What is the cost of f a. of land at $ 12.50 per are? 426. The standard unit of metric land measure is an Are, which is equal to a square decameter or approximately to ^ of an acre. METRIC LAND MEASURE 1 hectare (Ha.) = 100 ares 1 are (a.) 1 centare(ca.) =.01 are Notice that the final vowel of "hecto," and "cento" is dropped be- fore the word " are." 427. How many meters of fence would be required to inclose a hectare in the form of a square ? 428. Draw on the floor a square containing a centare. How long is its perimeter ? 429. How many square decimeters in a centare? In an are ? In a hectare ? 430. What is the cost of 24.7 Ha. at $425 a Ha.? Of 63.25 Ha. at $1032 a Ha.? 431. Approximately, how many acres in 280 a. ? In 160 a. ? In 240 a. ? In 120 a. ? 432. Find the approximate value in ares of 30 A. 75 A. 17| A. 8 A. 120 sq. rd. 12 A. 80 sq. rd. 6 A. 350 A. 500 A. 40 sq. rd. 433. Image a cubic centimeter and a cubic decimeter. How many cubic centimeters are equal to the cubic decimeter ? 434. Draw a square meter in one corner of the room. Imag- ine it covered with a layer of cubic decimeters or liters. How many are there ? 435. With a meter stick outline a cubic meter. How many layers of cubic decimeters are there in it ? How many cubic decimeters ? 184 DENOMINATE NUMBERS 436. In cubic measure, metric system, how many units of each denomination make one unit of the next higher denomina- tion ? 437. Write the table of cubic measure, metric system. 438. A cubic meter, or its equivalent, is called a Stere (s.). Image a stere of ice, of cubical form. How many square deci- meters in all its surfaces ? 439. Image a stere of marble, 2 m. long and 1 m. wide. How high is it? How many decimeters in all its edges? Represent with blocks. 440. A box which holds a stere is full of packages of Break- fast Food, each of which holds a liter. How many packages are there? 441. At 15^ a liter, what is the value of the contents of the box? 442. At 2^ a liter, what is the cost of 3 s. of wheat? 4726 s. ? 8347 s. ? 443. The standard unit of metric wood measure is a Stere, which is a little over ^ of a cord. METRIC WOOD MEASURE 1 decastere (Ds.) = 10 steres 1 stere (s.) 1 decistere (ds. ) = . 1 of a stere 444. A pile of wood 7 m. long, 6 m. wide, and 5 m. high contains how many steres ? How much is it worth at $ 1.50 a stere ? 445. At 70^ a stere, what is the value of a pile of wood 4.5 Dm. long, 3.5 m. wide, and 300 cm. high ? 446. About how many cords are there in 20 s. ? In 32 s. ? In 42 s. ? In 12.8 s. ? In 6.36 s. ? DENOMINATE NUMBERS 185 447. How many cubic feet in a cord? About how many cubic feet in a stere ? 448 v Approximately how many steres in 5 cd. ? In 7 cd. ? In 9J cd. ? In 3 cd. 64 cu. ft. ? In 6 cd. 32 cu. ft. ? In 12 cd. 16 cu. ft. ? In 4 cd. 8 cu. ft. ? In 11 cd. 4 cu. ft. ? In 24 cd. 8 cu. ft. ? 449. Which is greater and how much, a stere or a kiloliter? Explain. 450. About how many liquid quarts equal a liter? (See page 159, Ex. 115.) 451. About how many quarts are there in a decaliter? In a hectoliter ? In a kiloliter ? 452. About how many liters in 7.7 qt. ? In 132 qt. ? In 39 qt. ? In 17 qt. ? 453. A cask of oil containing 187 1. was bought at 20^ a liter and sold at 25^ a quart, a liter being counted as 1.1 qt. How much was gained ? 454. How much is gained by buying 209 1. of wine at 30^ a liter and selling them, at 40^ a quart, counting a liter as 1.1 qt. ? 455. Image a milliliter. What else is it called ? (See page 81.) 456. The weight of a cubic centimeter of pure water at its greatest density is called a Gram (g.). 457. Image a glass vessel of cubical shape containing a liter of pure water. How many grams would it contain ? 458. Approximately 1000 g. equal 2| Ib. What is the approximate equivalent of a gram? MISCELLANEOUS EXERCISES 1. Divide 62.5 by .0025. 2. Find the g. c. d. of 567 and 637. 186 DENOMINATE NUMBERS 3. Reduce to lowest terms : -J-JJ. 4. Find the 1. c. m. of 24 and 57. 5. Add -fa and T 4 T . 6. From T 7 -j take -fa- 7. Kesolve into prime factors 26,460 and 60,060. 8. Write the improper fraction that expresses the ratio of the first prime number after 40 to the first prime number after 20, and reduce it to a mixed number. 9. How many 6ths in J of 10 ? In J of 7 ? 10. One eighth of 88 is how many times 3 ? 4-J- ? 5 ? 11. If 13 is a divisor and 39 a dividend, what is the quotient ? If both divisor and dividend are multiplied by 4, what is the quotient ? 12. From 49J subtract a number which is ^ as large. 13. A minuend is 15, and a subtrahend 11. What is the difference ? If 3 is added to both minuend and subtrahend, what is the difference ? 14. If 3^ is added to both minuend and subtrahend in the preceding question, what is the difference ? 15. Find difference between % of ft of 40 and f of Jf of 3. 16. Square f f. f ft. 17. What fraction multiplied by itself will give f ? 18. What is the square root of ff ? T W ? ft ? ft ? yinr ? 19. CLASS EXERCISE. may give a fraction that is a perfect square, and the class may give its square root. 20. How many cubic yards of earth must be removed to make a reservoir 120 ft. long, 44 ft. wide, and 9 ft. deep ? 21. How much will it cost to dig a cellar 36 ft. long, 18 ft. wide, and 6 ft. deep, at $ 2.50 a cubic yard ? MISCELLANEOUS EXERCISES 187 22. How many cords of wood in a pile 36 ft. long, 4 ft. wide, and 8 ft. high ? At $ 3.50 a cord, how much would it cost ? Find the cost of plastering ceilings of the following rooms at 20 $ a square yard : 23. 18 ft. x 20 ft. 27. 10 ft. x 13| ft. 24. 16 ft. x 17 ft. 28. 15 ft. x 18 ft. 25. 141 ft. x 20 ft. 29. 27 ft. x 36 ft. 26. 9 ft. x 16J- ft. 30. 3 yd. x 16 ft. 31. Estimate the cost of plastering the ceiling of your schoolroom at 25^ a square yard. Find the cost of plastering the walls and ceiling of rooms of the following dimensions : length width height 32. 6m. 5m. 2.8m. 33. 8m. 7m. 3m. 34. 6.5m. 5m. 3m. 35. 7m. 6.5m. 3m. The price is 25 $ per square meter, and no allowance is made for openings. 36. An arc which is \ of a circumference is 1 yd. 1 ft. 3 in. long. How long is the circumference ? The diameter ? The radius ? 37. How many acres in a field 56 rd. long and 40 rd. wide ? 38. How much will it cost to pave a walk, 60 ft. long and 15 ft. wide, at $ 1.25 a square yard ? 39. How many trees can be planted on 3 A. of ground if only 1 tree is planted on each square rod ? 40. How many cubic feet in a pile of wood 24 ft. long, 3 ft. wide, and 8 ft. high ? How many cords ? 188 DENOMINATE NUMBERS 41. Express in grams the weight of the following measure- ments of pure water at its greatest density : 1 cu. dm. 3 1. 15 cu. cm. 1 ins. 1 ml. Is. 4 cu. m. 42. Express the measurements above in kilograms. 43. Taking 2^ Ib. as the equivalent of a kilogram, what is your weight in kilograms? 44. Express the following in avoirdupois on the basis of 2 Ib. to the kilogram : 75 Kg. 88 Hg. 15 Dg. 175 g. 395 eg. 45. Express the following in avoirdupois weight: 33 Kg. 275 g. 924 mg. 99 Hg. 16 Dg. 46. How much is gained by buying a barrel of flour (196 Ib.) for $ 6.00 and selling it at 7^ a kilogram ? 47. How much is gained by buying 99 Ib. of sugar at 5 f a pound and selling it at 13 ^ a kilogram ? 48. How much is gained or lost by buying 440 Ib. of dried fruit at 10 ^ a Ib. and selling it at 22 f a Kg. ? 49. How much is gained by buying 100 Kg. of coffee at 50^ a Kg. and selling it at 30^ a Ib. ? 50. How much is gained by buying 500 Kg. of raisins at 12 j a Kg. and selling them at 8 ^ a Ib. ? 51. Imagine a cubic decimeter cut from each corner of the upper layer of a cubic meter, and find the surface of the figure thus formed. 52. A sector whose arc is a quadrant was cut from a circle. If the area of the whole circle was 4| sq. in., what was the area of the part that was left ? 53. A California woman took 300 Ib. of honey from her hives in a month. What was its value at $ 5.00 per hundred- weight ? MISCELLANEOUS EXERCISES 189 54. Earning $ .75 per day, how long will it take a boy to earn enough to buy a $ 12.00 watch ? 55. Mr. Taylor bought 3 prize pigs whose respective weights were 3 cwt. 73 Ib. 12 oz., 4 cwt. 99 Ib. 15 oz., 5 cwt. 12 oz. How much did they all weigh ? 56. How many baskets, each holding 2J pk., can be filled with 10 bu. of apples ? 57. A garden containing 1089 sq. yd. is 49^- yd. long. How wide is it ? 58. A fisherman had a line 24 yd. 2 ft. long. A fish broke off 3 yd. 1 ft. 6 in. of it. How much was left ? 59. A dealer bought 2 T. 3 cwt. of carpet tacks in 8-oz. papers. How many papers of tacks were there ? 60. How long is one side of an equilateral triangle whose perimeter is 5 yd. 1 ft. 3 in. ? Of a regular pentagon having an equal perimeter ? Of a regular octagon of equal perimeter ? 61. A string, 4 yd. 2 ft. 6 in. long, was used to outline a regular hexagonal flower bed. How long was each side ? 62. A farmer sold 5 loads of hay, each Containing 17 cwt. 85 Ib. How much did he sell ? 63. How far will a man walk who begins walking at 9 A.M. and walks until 3.30 P.M., at the rate of 5 mi. an hour ? 64. A family started to go in a wagon to St. Louis from a town 132 miles away. They rode 24 miles a day for 5 days. On the morning of the sixth day, they started at 9 o'clock to ride the remaining distance at the rate of 6 miles an hour. At what time did they reach St. Louis ? 65. Mr. A has a lot 40 rd. square, and Mr. B has a lot con- taining 40 sq. rd. How many more square rods in Mr. A's lot than in Mr. B's ? 190 DENOMINATE NUMBERS 66. A flight of stairs in Mrs. Long's house consists of 18 steps, each 1 ft. wide and 8 in. high. How much will the stair carpet cost at $ .75 per yard, if 3 in. is allowed at each step for the turning in of the carpet ? 67. How much can be earned in two weeks by a person who earns $ 2.34 every working day ? 68. General McClellan was born Dec. 3, MDCCCXXVI, and died Oct. 29, MDCCCLXXXV. How old was he when he died? 69. April 4th, 1898, was Monday. At the close of that day, Kuth Mayo found that there were 8 weeks and 3 days left of the school term. On what day did the term close ? 70. She entered college Sept. 14, 1898. The first term closed Dec. 21. How long was it ? 71. Her expenses for the term were $ 95.75. What was the average per week ? 72. The Thanksgiving vacation began Nov. 24 and ended Nov. 28, and there was no other vacation in the term. She attended a Saturday class. How many working days had she in that term ? 73. Her second term began Jan. 3, 1899, and ended March 25, 1899. How long was it ? 74. Her expenses for that term averaged $ 7.50 per week, and she earned $ 25 during the term by outside work. Her expenses were how much more than her earnings ? 75. The floor of Mrs. Beed's dining room, which is 15 ft. long and 14 ft. wide, is laid with parquetry flooring. How much did it cost at $ .621 per square yard ? 76. The wainscoting is 3 ft. high. There are 4 doors, each 3 ft. wide. Two windows, each 3 ft. wide, extend down into the wainscoting 1 J ft. There is a fireplace 4. ft. wide. How many square yards in the wainscoting? MISCELLANEOUS EXERCISES 191 77. Her dining table is 6 ft. long and 4 ft. wide. How many square yards in the top of it ? 78. A rug under the table is 12 ft. long and covers 12 sq. yd. of the floor. How wide is it ? 79. Make a problem about the dimensions of a room. 80. Fourteen cords of wood are piled evenly on an open car 28 ft. long and 8 ft. wide. How high is the wood piled ? 81. If a leaf of a book is 12 cm. long and 9 cm. wide, how many square centimeters in the surfaces of both sides of the leaf ? 82. If your schoolroom were 36 ft. long and 30 ft. wide, how many square yards could be drawn on the floor, provided no two overlapped ? 83. In Pig. 1 the angles are all right angles. How long is the line repre- sented by GH? HA? 84. Copy Fig. 1, making the dimen- sions inches or centimeters. Draw a H G construction line BE. How long is it ? Find the area of Fig. 1 by finding the sum of the areas of the two rectangles that are thus formed. 85. Copy Fig. 2. Find its area by drawing a construction line from C per- pendicular to FE and finding the area of the two rectangles that are thus FIG. 2. formed. 86. Find the area of Fig. 2 by drawing a construction line A BE F from O perpendicular to AF and finding the area of two rectangles thus made. 87. Copy Fig. 3, making AB 6 in., EC 3 in., CD 3 in., DE 3 in., EF 3 in., FG 8 in. Find length of GH FIG. 3. and HA. FIG. 1. 10 E 192 DENOMINATE NUMBERS 88. Find area of Fig. 3 by drawing construction lines that will divide it into three rectangles and finding the area of the rectangles. Show different ways of dividing it. 89. Find area of Fig. 3 by finding area of a rectangle AFQH and subtracting the square BE DC. 90. Copy Fig. 4, making AB 5 in., BC 3 in., CD 4 in., DE 6 in., EF 2 in., FG 4 in. GH=? HA= ? 91. Show four different ways of di- viding Fig. 4 into three rectangles. Find its area. jT~ a 92. Beginning at a point marked A, FIG. 4. raw to t h e rig]^ 4 in 20, another $ 25. What per cent of the whole did each man give ? 137. Mr. Low had his money invested in three houses as follows : in the first $ 1620, in the second $ 8100, in the third $3240. What per cent of his money was invested in each house ? 138. Mr. Eves insured Mr. Croft's building for $ 1600, which is 75% of its value. What is its full value ? If Mr. Croft paid 1% upon the amount insured, how much did he pay ? 139. Mr. Jones's yearly income from a mine is $ 4000, which is 15% of the sum he invested in it. How much did he invest ? 140. A store is rented for $ 60 a month. The yearly rent is 8-J% of the value of the property. What is its value? 141. A lawyer collected some money for his client, receiving for his services $ 80, which was 5% of the sum collected. How much did he collect and how much did he pay over to his client ? 142. Mr. Eoy gained 8% by selling his cow for $20 more than it cost him. For how much did he sell it ? 143. Mr. Litch received $ 500 a year rent for one of his houses, which was 7 % of its value. What was its value ? 228 PERCENTAGE 144. After Mr. Lane had paid 37% of his debts, he found that $3568 would pay the remainder. What was his total indebtedness ? 145. Mr. Bingham sold two houses for $ 7000 each; for the first he received 12|% less than its value and for the second 16f % more than its value. What was the value of each ? 146. Mr. Allen bought for $ 3200 a store which had depre- ciated 40% in value. What was the original value of the store ? 147. When the Twentieth Century Novelty Company failed in business they had $ 3350, which paid 67 % of their debts. What was the amount of their debts ? 148. A grocer sold his stock for $ 2000, which was at a loss of 10%. How much did the stock cost ? 149. Mr. Drew bought 5 doz. oranges. Of these 30% were not good. How many oranges were good ? 150. A florist used in January 4000 bu. of coal. This quantity was 36% of the number of bushels used during the winter. How many bushels were used in all ? 151. Mr. M. receives $150 each ye.ar as the interest on a sum of money which pays 6%. What is the sum of money ? 152. How much money must one have at interest that he may receive from it $ 750 a year when the rate of interest is 5% ? 6% ? 8% ? 4% ? 3% ? MERCHANDISING 153. Buying and selling goods for profit is called Merchan- dising. Those who carry on merchandising are called mer- chants. Mention several lines of merchandising. 154. A merchant bought $ 6000 worth of dry goods, and in the first year gained 20% of his capital. How much did he gain that year ? MERCHANDISING 229 155. He used $ 900 that year for living expenses, and put the rest into his business. What was his capital at the begin- ning of the second year ? 156. In the second year he cleared 33-^% on his capital. What was his gain ? 157. The second year he used $ 1200 for his living expenses, and put the balance into his business. What was his capital at the beginning of the third year ? 158. In the third year he gained 37%% on his capital. His expenses outside of his business that year were $ 1800. The rest of his gain went into the business. What was his capital at the end of the third year ? 159. In the fourth year he gained 44|-% on his capital, and used $ 1700 for outside expenses. What was his capital at the end of the fourth year ? 160. January 1, 1899, Mr. A. went into business with a capi- tal of $10,000. At the end of the year he found that the amount of his sales for the year had been $ 18,448 ; that his business expenses, such as rent, clerk hire, advertising, etc., had been $ 6000 ; that he had spent $ 10,625 in replenishing his stock of goods. How much had he gained that year ? His living expenses were $1000. What was his capital Jan. 1, 1900, including the amount invested in goods ? 161. In 1900 he netted 10% on his capital, and his living expenses were $ 1500. What was his capital Jan. 1, 1901 ? 162. In 1901 his profits were 7% on his capital, and his living expenses were $ 1300. What was his capital Jan. 1, 1902? SUGGESTION TO TEACHER. Lead pupils to realize some of the condi- tions of merchandising, and let them make similar problems, tracing the course of simple business ventures. 163. A merchant buys broadcloth at $ 2.40 per yard. How shall he mark it that he may sell it at a gain of 40% ? 33-|% ? 35%? 50%? 230 PERCENTAGE 164. How much would he gain by selling 20 yd. at each of those rates of advance ? 165. Mr. B. bought 40 yd. of novelty goods at $ 1.60 per yard. He sold ^ of it at an advance of 75%, ^ of it at an advance of 50 /o i and the rest at 25 % advance. How much did he gain on the whole piece ? 166. Mr. C. bought 60 yd. of goods at $ 1.20 per yard. He sold i of the piece at a gain of 60%, -i- of the rest at a gain of 50%, 14 yd. at $ 1.30 per yard, and the rest at $.75 per yard. How much did he gain on that piece ? 167. Mr. Evans bought goods at $.75 a yard and marked them to sell at an advance of 33J%. Later in the season he sold them at a reduction of 10% from the marked price. What was the actual selling price ? Find actual selling price, and amount of gain or loss on goods : Bought Marked Sold 168. $.60 50% above cost 10% below marking 169. .80 25% above cost 30% below marking 170. .90 33^% above cost 16f % below marking 171. 1.75 $ 2.00 above cost 12J% below marking 172. 1.60 25% above cost 12^% below marking 173. CLASS EXERCISE. may mention prices at which goods might be bought, marked, and sold, and the class may find the amount of gain or loss. COMMISSION 174. Instead of buying and selling for themselves, some merchants buy or sell goods for others at a given per cent of their value. The money which they receive for buying or sell- ing is called their Commission, and they are called commission merchants, agents, or brokers. What is the commission on $ 900 at 10% ? COMMISSION 231 175. Mr. Ward sold a piano for $ 575, receiving a commis- sion of 10% on the selling price. How much was his com- mission, and how much money should he send to the owner of the piano ? 176. Mr. Clark is an agent for the Elliott Piano Co. which pays him a commission of 25% on all his sales. During the month of January, 1899, he sold one piano at $325, one at $ 400, two at $ 250, and three at $ 200. How much were his commissions, and how much should he return to the Elliott Piano Co. ? 177. The expenses of his store for rent, heating, lighting advertising, clerk hire, etc., for January, were $ 178.50. How much did Mr. Clark make that month above expenses ? 178. His living expenses for that month were $ 109.75. How much did he save ? Would it be safe to reckon his yearly income on this basis ? Why ? 179. A cotton broker receives a shipment of cotton from Alabama consisting of 400 bales. He sells it at $ 20 a bale, receiving 11% commission. How much is his commission on that sale, and how much should he return to the owners of the cotton ? 180. John sold his sister's sled for her, receiving $ 1 for it. When he gave his sister the dollar, she handed him a dime for selling the sled. What per cent was his commission ? 181. Mr. Adams sold a wagon for the Melvin Wagon Co. at $ 200, and after taking out his commission, sent the Wagon Co. -$180. How much was his commission? What per cent of -the value of the wagon ? 182. Mr. Gordon buys chickens for a Baltimore firm, re- ceiving a commission of 15% on all the money spent in buying them. If he buys $ 12,000 worth of chickens in a year, what is his income for that year ? 232 PERCENTAGE 183. Mr. Wilson travels in the South, selling stoves. He receives a commission of 10% on sales. When he has sold $ 18,000 worth of stoves, how much has he earned ? 184. Mr. Wood buys hogs for a pork-packing establishment, receiving a commission of 3% on the amount spent for the hogs. If he buys $ 7500 worth in a month, what are his earn- ings for that month ? 185. A commission merchant bought $ 5000 worth of pea- nuts in Tennessee, and shipped them to a Chicago firm. The cost of hauling them to the depot in drays was $ 2.75, the freight charges were $ 47.50. The buyer's commission was 4% on the amount paid for the peanuts. What was the amount of his commission ? How much did the peanuts cost the Chicago firm ? 186. A commission merchant in Kentucky bought for a New York firm 3000 Ib. of pecans at $ .06^ per pound. How much was his commission at 2% ? 187. CLASS EXERCISE. may fill out the following, and the class may find the amount received as commission: !1 worth of r sold . . ^ , , , I receiving % com' g J mission. 188. People are sometimes hired to collect money due to other people. These collectors receive a percentage on the amount collected. How much does Mr. Cox earn by collecting a bill of $ 600 at 2 % ? 189. Mr. Ho well gave Mr. Scott a list of bills to be col- lected, 5% of the amount of which Mr. Scott was to receive as commission. How much did he earn by collecting the follow- ing bills, and how much did he turn over to Mr. Howell ? John Andrews, Groceries, $ 28.75. Charles Stockton, 21 Ib. Sugar, @ $ .05. George Baldwin, 12 Ib. Turkey, @ $ .15. Peter Garrison, 10 gal. Molasses, @ $ .45. TRADE DISCOUNT 233 190. CLASS EXERCISE. - may fill out the following, and the class may find the collector's receipts : A collector collects , receiving for his work %. 191. A collector undertook to collect $2125 worth of bills due a physician. He collected 67% of them, receiving 10% as his fee. How much did he collect ? How much did he keep ? How much did he send to the physician ? 192. Mr. Blair sold a carriage for the Hale Carriage Co. at $450. His commission was 15%. The purchaser paid $300 cash, and Mr. Blair was obliged to pay a collector 10% on the balance for collecting it. How much did Mr. Blair gain on the sale ? TKADE DISCOUNT 193. A reduction in the selling price of goods is called a Discount. Mr. Reed found that he could buy the bicycle that he wanted for $ 70, on 3 months' time, or for $ 66.50 cash. How much was the discount ? 194. How much is paid for a bill of goods invoiced at $ 18.75, with a discount of 30%? 195. The employees of a certain large dry goods store are allowed to buy goods from it at a discount of 10%. What amount was paid for a bill of $ 17.25 bought by an employee ? 196. A man received a bill for $ 75. Printed on the bill head were the words, "5% discount if paid within 30 days." The bill was paid at the end of 4 weeks. What amount was paid? 197. Hale & Co., Springfield, 111., bought of D. W. Lamont & Co., St. Louis, Mo., 3 doz. plain gold rings, @ $ 20 per doz. ; 4 gold rings, diamond settings, @ $ 50 ; 6 gold watches, @ $ 15 ; 4 sets teaspoons, @ $ 6. Make out the bill, allowing a discount of 25%. 234 PERCENTAGE 198. Mr. K. sold Mr. D. furniture for his home to the amount of $ 75, on 60 days 7 time ; but Mr. D. accepted the offer of 3J% off for cash. How much did Mr. K. receive for the furniture ? Many manufacturers and wholesale dealers have a fixed price for their goods, called the list price. They sell to the retail dealers at a discount from the list price. For instance, a manufacturer of carriages sends out a catalogue containing a list of the different kinds of carriages he makes, and their prices, with the discounts he gives to retail dealers. 199. Howard, Cowperthwait, & Co., who sell carriages, select from a manufacturer's list a carriage whose list price is $ 750. The discount on this carriage is 55%. How much do Howard, Cowperthwait, & Co. pay for the carriage? If Mrs. Douglas buys the carriage for $ 700, how much do they make on it ? 200. At the same rate of discount, how much will be gained* by Howard, Cowperthwait, & Co. if they buy four carriages whose list price is $ 600, and sell them at the list price ? 201. Frequently more than one discount is given upon the same purchase. If Howard, Cowperthwait, & Co. buy a $ 1000 carriage at 55% off, its cost is $ 450. If they get an ad- ditional discount of 10%, it is reckoned on the $ 450 and deducted from it. In that case, how much will the carriage cost them ? Observe that when there is more than one discount, each successive discount is made upon a smaller sum of money than the preceding. 202. How much is left of $800 when it is discounted 25%, then 33|%, then 25%, then 33% ? Find cost of the following : List price Discount List price Discount 203. $ 175 45 and 10 off 206. $ 500 20, 10, and 5 off 204. $ 370 35 and 5 off 207. $ 600 40, 30, and 2 off 205. $ 350 15 and 10 off 208. $ 800 50, 10, and 5 off TRADE DISCOUNT 235 209. If you were buying goods should you prefer discounts of 15% and 10%, or a straight discount of 25% ? Find the discount on different sums at these rates, and state which terms are more advantageous to the buyer, and why. 210. A dealer buys wagons whose list price is $ 60, at a discount of 30%, 20%, and 5% off for cash. How much do these wagons cost him ? He sells them at the list price with 5% off for cash. How much does he make on a cash sale ? On a time sale ? If a time sale is made in which the time exceeds a certain limit, the buyer gives his note for the amount and pays the interest upon it. This time limit varies with different firms, being usually not less than 60 da. , nor more than 6 mo. 211. How much is paid for a bill of goods invoiced at $37.25, discounts 10, 15, and 5 off. How much is gained if the goods are sold 10 % above the list price ? 212. Ball & Co., piano dealers, select from the manufac- turer's catalogue, 3 pianos listed at $400, $450, and $700, and order 6 pianos of each kind. The trade discount is 60, 10, and 5. What is the cost of that shipment of pianos ? 213. Mrs. Fox pays cash for one of the $ 400 pianos sold at list price, and gets a discount of 5%. How much do Ball & Co. make on that sale ? 214. Mr. Shaw buys a $700 piano at list price, pays $75 cash, and gives his note for the balance, payable in 6 mo. At the end of that time, finding himself unable to pay for it, he returns the piano to Ball & Co., who spend $ 2 in polishing and tuning it, and then sell it for $ 650. How much do Ball & Co. make on that piano ? 215. Mr. King buys one of the pianos whose list price is $ 450, on the installment plan, paying $ 10 a month. Ball & Co. charge him $ 500 for the piano. How much do they make on that sale ? 236 PERCENTAGE 216. Mrs. Lee buys from Ball & Co. a piano listed at $ 700. She agrees to pay for it $750, in monthly payments of $15 each. After making 7 payments she returns the piano. Ball & Co. spend $ 5 for repairs and sell it for $ 625. What is their gain on it ? SUGGESTION TO TEACHER. Require the pupils to bring to class similar problems, describing business occurrences. Let them find out facts which will enable them to keep the conditions of their problems within the range of probabilities. Find gain on the following goods bought at discounts of 10, 20, and 5 off, and sold at an advance of 20% on list price : Items List price 217. 1300 yd. Carpet @ $ .75 218. 800 yd. Drapery Silk @ $ 3.50 219. 100 prs. Lace Curtains @ $6.75 220. On a bill of $ 675, what is the difference between discounts of 40% and 20%, and a straight discount of 60%? 221. Mr. Dow, a merchant in Kentucky, goes to New York twice a year to buy goods. He had been getting discounts of 30, 20. and 5% off for cash, but on his last trip he found that he could get discounts of 30, 40, and 5%. If he bought goods to the amount of $ 9000, how much was he benefited by the change of discounts ? 222. The list price of an article with three different houses is $400. One house offers discounts of 20, 10, and 5% ; the second 5, 10, and 20% ; the third 10, 5, and 20%. Which is the best offer, and why ? INTEREST 223. Money paid for the use of money is called Interest. The sum on which interest is paid is called the Principal. If $ 250 is loaned at 6 % for 1 yr., how much is the interest ? What sum is the principal ? INTEREST 237 224. When you know the interest of a sum of money for 1 yr., how may you find the interest of the same sum at the same rate for 2 yr. ? 7 yr. ? 31 yr. ? ^ of a year ? 2 yr. 1 mo. ? Find interest : Prin. Rate Time Prin. Rate Time 225. $ 200 6% 4yr. 232. $1800 5% 1 yr. 8 mo. 226. $ 700 8% 21 yr. 233. $ 400 7% 2 yr. 9 mo. 227. $ 300 7% 21 yr. 234. $ 600 4% 1 mo. 228. $ 800 8% 2 yr. 6 mo. 235. $ 800 6% 7 mo. 229. $ 900 5% 3yr. 4 mo. 236. $1200 3% 11 mo. 230. $ 600 4% 2yr. 1 mo. 237. $1800 9% 5 mo. 231. $5000 6% 3yr. 2 mo. 238. $1500 8% 10 mo. There are many ways of calculating interest, all depending upon this fact. Principal x Rate x Time = Interest. A 6% method and a cancellation method are given in this book. /Six Per Cent Method 239. From the first equation reason out the equations which follow it. Learn them. At 6% the interest of $ 1 for 12 mo. = $ .06 the interest of $ 1 for 2 mo. = $ k 01 the interest of $ 1 for 1 mo. = $ .005 the interest of $ 1 for 6 da. = $ .001 the interest of $ 1 for 1 da. = $ .0001 At 6 % what is the interest of $ 1.00 for : yr. mo. da. yr. mo. da. 240. 1 2 6 246. 8 10 18 241. 2 1 6 247. 9 3 1 242. 3 1 12 248. 7 5 5 243. 4 3 6 249. 8 9 7 244. 5 3 12 250. 6 11 8 245. 7 8 12 251. 7 7 9 238 PERCENTAGE 252. CLASS EXERCISE. may give a number of years, months, and days, and the class may find the interest on $ 1 for that time at 6%. 253. Find the interest of $ 1 at 6% for 2 yr. 6 mo. 24 da. and then find the interest of $ 2 for that time. $3. $7. $ 25. 254. Find the interest of $ 25.75 for 5 yr. 8 mo. 12 da. at 6%. 25.75 .342 The interest of $ 1 for 5 yr. 8 ino. 12 da., at 6%, is $.342. 5150 The interest of $25.75 is 25.75 times $.342. In practice it is 10300 more convenient to multiply 25.75 by .342. Of course the 7725 result is the same. $8.80650 Find the interest of $ 125.37 at 6% for: yr. mo. da. yr. mo. da. 255. 346 260. 297 256. 7 9 12 261. 3 10 8 257. 8 2 18 262. 659 258. 7 5 24 263. 5 5 10 259. 6 11 7 264. 11 2 11 Find interest at 6% : 265. $ 175.25 for 1 yr. 6 mo. 7 da. 266. $ 210.60 for 3 yr. 2 mo. 9 da. 267. $ 625.48 for 2 yr. 9 mo. 9 da. 268. $ 330.27 for 5 yr. 7 mo. 9 da. 269. $ 45.60 for 8 yr. 3 mo. 10 da. 270. $ 910.75 for 1 yr. 6 mo. 13 da. 271. $ 712.25 for 2 yr. 8 mo. 14 da. 272. $ 861.60 for 3 yr. 8 mo. 14 da. 273. $ 520.40 for 7 yr. 4 mo. 14 da. INTEREST 239 274. What is the interest of $100 for lyr. at 6%? At 3% ? 275. What is the ratio of the interest of a sum of money for a given time at 3%, to the ratio of the same sum of money for the same time at 6% ? 276. Find the interest of the following amounts for 1 yr. 8 mo. 24 da., first at 6% and then at 3% : $276 $24.76 $13.25 $417 $625 277. Find by the 6% method the interest of $318 at 5% for 2 yr. 7 mo. 12 da. SOLUTION. The interest of $ 318 for 2 yr. 7 mo. 12 da. at 6 % is $ 49.926. At 1 % the interest is - as much or $ 8.321. At 5 % the interest is 5 times as much as at 1 % or 41.605. Required interest : Prin. Rate Time 278. $ 28.35 6% 1 yr. 6 mo. 16 da. 279. $ 49.36 7% 2 yr. 9 mo. 12 da. 280. $ 30.75 8% 6 yr. 15 da. 281. $ 252.00 6% 4 yr. 7 mo. 27 da. 282. $ 160.00 7% 8 mo. 26 da. 283. $ 72.00 5% 6 yr. 8 mo. 13 da. 284. $ 75.00 4% 8 yr. 3 mo. 285. $ 112.00 5% 6 yr. 7 mo. 22 da. 286. $ 46.75 3% 2 yr. 11 mo. 20 da. 287. If you borrowed $ 100 and paid it back at the end of one year, with the interest on it at 6%, how much would you pay? 288. The sum of the principal and interest is called the Amount. What amount must be paid back when $ 200 is borrowed at 6% interest and kept 2 yr. 10 mo. 18 da. ? 240 PERCENTAGE Prin. Kate 289. $900 6% 290. $800 3% 291. $600 4% 292. $144 5% 293. $672 7% 294. $ 145.36 7% 295. $816.35 5% 296. $696 8% 297. $ 216.25 4% 298. $625 7% 299. CLASS EXERCISE. Find the amounts of the following: Time 7 yr. 4 mo. 6 da. 3 yr. 5 mo. 7 da. 11 yr. 2 mo. 8 mo. 12 da. 3 yr. 8 mo. 15 da. 2 yr. 2 mo. 1 da. 3 yr. 3 mo. 3 da. 5 yr. 7 mo. 13 da. 4 yr. 18 da. 9 yr. 9 mo. 9 da. may mention a sum of money, and the class may find the amount of it for any length of time and at any rate which he may decide. 300. Find the time from Jan. 1, 1898, to July 7, 1899. 301. Mr. Monroe borrowed $ 300 Jan. 1, 1897, at 6%. What was the interest March 1, 1898 ? Sept. 1, 1899 ? 302. On Nov. 7, 1896, what amount was due on $600 bor- rowed May 1, 1892, with interest at 6% ? 303. Find the amount of $376.25 borrowed July 1, 1883, and paid Nov. 13, 1887, with interest at 6%. At 6%, what is the amount of $ 700? 304. Borrowed Sept. 7, 1898 Paid April 19, 1899 305. Borrowed June 15, 1895 306. Borrowed Dec. 12, 1891 307. Borrowed Aug. 6, 1880 308. Borrowed Feb. 29, 1896 Paid Oct. 3, 1898 Paid May 15, 1894 Paid May 12, 1885 Paid June 30, 1897 INTEREST 241 Cancellation Method 309. If the interest of a sum of money for a certain time is $ 72, what will be the interest of that sum for J of that time ? f of that time ? 310. The interest of a certain sum of money for a certain time is $ 96. Find by cancellation the interest of that sum for | of that time, f 1 of it. of it. 311. Find the interest of $ 400 for 2 mo. 20 da. at 3%. SOLUTION. The interest of $ 400 for 1 yr. at 3 % is 400 x jfo. The interest of the same sum for 2 mo. 20 da., or 80 da., is -gfe as much. Hence the interest = ^ of $ 12. 4 2 Canceling we have x -|- x * = | = $2.66| J0J5 jJpjJ 6 m 3 Find the interest of the following by the cancellation method : Prin. Rate Time 312. $ 276 12% 3 mo. 9 da. 313. $ 184.50 4% 5 mo. 27 da. 314. $1200 6% 1 mo. 21 da. 315. $1400 5% 3 mo. 15 da. 316. $1800 7% 3 mo. 20 da. 317. $ 625 8% 90 da. 318. $ 800 1% 63 da. 319. $ 900 8% 100 da. 320. $1100 6% 33 da. 321. $2175 3% 93 da. 322. $4150 6% 63 da. 323. Find the interest of $840 at 5% for 1 yr. 3 mo. 840 x T ^ x |f. When no days are given, find the number ol months and express them as twelfths of a year. HORN. GRAM. SCH. AR. 16 242 PERCENTAGE Find interest : Prin. Rate Time 324. $560 3% lyr. 8 mo. 325. $218.64 6% 2 yr. 1 mo. 326. $175.25 8% 1 yr. 6 mo. 327. $165.36 4% 4 yr. 2 mo. 328. $500 5% 7 mo. 6 da. When days are given, reduce the whole time to days and express as 360ths of a year. Find interest of : Prin. Rate Time 329. $700 2% 1 yr. 1 mo. 6 da. 330. $750 6% 3 yr. 1 mo. 15 da. 331. $420 1% 2 yr. 1 mo. 10 da. 332. $800 9% 2 yr. 3 mo. 6 da. 333. $875 10% 1 yr. 8 mo. 20 da. 334. $630.25 7% 1 yr. 4 mo. 20 da. 335. There are some special rules for calculating interest which are derived from the principles of the cancellation method as, To compute interest at 8 % Multiply the principal by the number of days, move the decimal point of the product two places to the left, and divide the result by 4$- Find interest of $200 for 53 da. at 8%. By cancellation method By special rule 2 53 45)1060(2.35+ 90 It will be seen that dividing by 45 gives 160 the same result as multiplying by 8 and 135 dividing by 360. 225 INTEREST 243 336. Give the reason for the special rule for computing interest at 8%. 337. Show how the following rules are derived from the cancellation method : (a) To compute interest at 5 % Multiply the principal by the number of days, move the deci- mal point of the product two places to the left, and divide the result by 72. (b) To compute interest at 6% Multiply the principal by the number of days, move the deci- mal, point of the product three places to the left, and divide the result by 6. 338. Give a similar rule for computing interest at 9%. At 4%. At 12%. At 10%. At 3%. Exact Interest 339. Usually 360 da. are considered one year, but some- times calculations of interest are made, in which a year is considered as 365 da. This is called Exact Interest. To find the exact interest of a sum of money, use the cancellation method, expressing the exact number of days as 365ths of a year. Find the exact interest of $900 from Dec. 1, 1898, to Feb. 12, 1899, at 8%. SOLUTION. The exact number of days from Dec. 1, 1898, to Feb. 12, 1899, is 73 da., or ^ of a year. Find exact interest of : 340. $300, 1 yr. 1 mo. 1 da. at 9%. 341. $240, 11 mo. 25 da. at 4%. 342. $336, 1 yr. 2 mo. 10 da. at 6%. 343. $430 from Oct. 15, 1897, to Jan. 11, 1898, at 244 PERCENTAGE PROMISSORY NOTES 344. July 1, 1900, Mr. James Allen bought a horse of Mr. William Brown for $125, paying $25 cash and giving a promissory note, like the following, for the balance. The note was paid in full when due. What amount was paid ? ilntfi, neHUbi nT, A-i/x, njpn. n?/nl, jpn, n/n/ruj/m, REVENUE STAMP O.HW. 345. A Promissory Note is a written promise to pay money. Who is the maker of the above note? 346. Who should keep the note until it is paid? What should be done with it after the money is paid ? 347. Write a note promising to pay Robert Ruskin $300 with interest at 6% in one year from date. 348. The person to whom the note is to be paid is called the Payee. Who is the payee of the note you have just written ? 349. The sum mentioned in the note is called the Face of the note. What is the face of your note ? 350. The date at which the note becomes due is called the date of Maturity. What is the date of maturity of your note ? 351. Is it an interest-bearing note ? SUGGESTION TO TEACHER. Procure blank forms of promissory notes of different kinds, and let the differences between them be discussed in class. 352. What is the amount of a note for $425 that matures in 4 mo., interest being 5% ? PROMISSORY NOTES 245 353. In some states the law allows 3 days more than the specified time for the payment of a note, but interest is exacted for these 3 days, called Days of Grace. If a note is made payable Aug. 1, on what day is it really due, when grace is allowed ? In the problems of this book, days of grace are not to be considered unless mentioned. 354. A note for. $ 400 is dated March 1, 1896, and made payable May 1, 1896, with grace. It is said to mature May 1/4, and interest is computed to May 4. What is the interest at 6% ? 355. A note for $ 500, dated July 1, 1900, is made payable in 3 mo. with grace. When does it mature, and what is the interest at 6% ? At 5% ? At 7% ? 356. How much must be paid for the use of $ 625 from June 1, 1897, to July 1, 1898, with grace, at 6% ?. At 3% ? 357. How much must be paid for the use of f 500 from Dec. 1, 1899, to March 1, 1900, with grace, at 6% ? At 4% ? 358. What is the interest on a note for $ 300 at 6%, dated Aug. 31, 1895, and made payable in 30 da., with grace ? 359. June 17, 1897, Mr. Kent gave a note for $500 at 6%, payable in 60 da., with grace. When was the note due? What was its face ? Its amount at maturity ? 360. Notes, being promises, may be varied to suit the inten- tions of the parties concerned. Some notes draw interest from date, some after maturity, and some not at all. Some are made payable at a specified time, and are called Time Notes. Some are made payable upon the demand of the holder for payment, and are called Demand Notes. Some are made in such a way that they can be sold or transferred to other persons, and are called Negotiable Notes. Some are made payable only to a certain person, and are called Non-negotiable notes. What is the special advantage of a negotiable note? Of a non-negotiable note ? 246 PERCENTAGE Demand Note $300.75. BOSTON, MASS., Sept. 20, 1900. On demand, I promise to pay William D. Owen three hundred and -^-fa dollars, with interest at 6 %. Value received. EDWARD M. ARLINGTON. 361. What is due Jan. 19, 1901 ? 362. If the note above were not paid until May 20, 1902, how much would be due? 363. How much would be due on the above note if it were paid Sept. 25, 1902 ? January 11, 1903 ? March 1, 1903 ? 364. How does a demand note differ from a time note? 365. Write a time note for $ 500, due in 3 mo., at 4%. Negotiable Note $175.50. LOWELL, MASS., Sept. 15, 1899. One year after date, I promise to pay to Henry Scott, or bearer, one hundred and seventy-five and -ffa dollars, with interest at 6%. Value received. MARY GREEN 366. What two words in the above note make it a negotiable note? 367. If the negotiable note given above was paid when it was 6 mo. past due, how much was paid ? 368. Write a negotiable time note for $600, interest 6%, and find the amount of it when due. When 3 mo. past due. 11 mo. past due. $ 1000. SAN FRANCISCO, CAL., Nov. 12, 1901. One year after date, I promise to pay to the order of Ellen Eames, One Thousand Dollars, with interest. JAMES PORTER. 369. When a note includes the words "with interest/' but gives no specified rate, interest is computed at the rate legal in PARTIAL PAYMENTS 247 the state in which it is dated. Copy the above note, dating it at the place where you live, and find the amount of it at maturity under the laws of your state. 370. Write a note for $ 700 due in 3 mo., with interest after maturity. Find the amount due on it 8 mo. after its date under the laws of your state. PARTIAL PAYMENTS 371. May 7, 1896, Mr. James Smith gave Mr. John Brown a note for $ 700 payable on demand, with interest at 6%. How much was due May 7, 1897 ? At that time Mr. Smith made a partial payment of the amount due by giving Mr. Brown $ 442. On how much money ought Mr. Smith to continue to pay interest? How much was due May 7, 1898? At that time Mr. Smith made another partial payment, giving $ 218. How much was due on that note Nov. 7, 1898 ? SOLUTION May 7, '96, Mr. Smith owed Mr. Brown $ 700 Prin. Int. on prin. from May 7, '96, to May 7, '97 . . . . 42 Int. May 7, '97, Mr. Smith owed Mr. Brown $742 Am't. May 7, '97, Mr. Smith paid Mr. Brown 442 Pay't. Mr. Smith still owed Mr. Brown $ 300 New prin. Int. on new prin. from May 7, '97, to May 7, '98 . . 18 Int. May 7, '98, Mr. Smith owed Mr. Brown $318 Am't. May 7, '98, Mr. Smith paid Mr. Brown 218 Pay't. Mr. Smith still owed Mr. Brown $ 100 New prin. Int. on last prin. to Nov. 7, '98 3 Int. Nov. 7, '98, Mr. Smith owed Mr. Brown $103 Am't due. 372. When a partial payment is made, the holder of the note writes upon the back of it the amount of money paid and the date of payment. The writing is called an Indorsement, and serves as a receipt for the amount paid. What were the indorsements that Mr. Brown wrote ? 373. Aug. 1, '93, Mr. John Dow gave to Mr. Frank Rand his note for $800 at 6%. Aug. 1, '94, he paid $218. Feb. 1, '95, he paid $ 223.90. How much did he owe Aug. 1, '95 ? 248 PERCENTAGE SUGGESTION TO TEACHER. Let the pupils in one section of the class enact the part of Mr. Dow in writing the note, and those of another section take the part of Mr. Rand, making the indorsements upon the notes written by the others. Let class discuss justice of the settlements. $ 800. NEW ORLEANS, March 1, 1898. For value received, 60 da. after date, I promise to pay to the order of Amos Butler, Eight Hundred Dollars, with interest at 6%. HOWARD CURTIS. 374. On the back of this note these indorsements were written by Mr. Butler : Dec. 1, 1898, $ 300. June 1, 1899, $ 222.08. How much was due March 1, 1900 ? 375. Make a problem in which you suppose that you give a note for $ 900 due in 3 yr., with interest at 8%. What amount would you owe at the end of 3 yr. if you made no pay- ment before that time ? On what principal would the yearly interest be reckoned ? But suppose that instead of waiting until the end of the three years you made a payment of $ 12 at the end of the first year. The interest then due would be $ 72. If, now, in this case, as in the previous problems, the payment $ 12 were deducted from the amount $ 972, and if the difference, $ 960, were regarded as a new principal, observe that simply because you had made a payment on the note, you would be charged interest on a greater prin- cipal. Would that be just ? 376. To prevent injustice in such cases, the Supreme Court of the United States has adopted the following rule : UNITED STATES RULE. When the payment is less than the interest due at the time of payment, no change of principal shall be made at that time, but the interest shall be computed upon the same principal until the sum of the payments shall equal or exceed the interest due. Make a problem in which the first payment on a note is less than the interest due when the payment is made. $ 900. CINCINNATI, OHIO, Sept. 30, 1896. One year from date, I promise to pay Henry Moore, or order, Nine Hundred Dollars, with interest at 8%. Value received. MARTIN CAMPBELL. Indorsements : March 30, 1897, '9 16; Sept. 30, 1897, $ 56. PARTIAL PAYMENTS 249 377. How much was due June 1, 1898? SOLUTION. The first payment, $ 10, is less than the interest, $36, that has accrued at the time this payment is made (int. of $ 900 for 6 mo. at 8% = $36). Therefore, we compute interest to the time of the second payment. The interest of $900 for 1 year at 8% is $ 72, and the amount is $972. Subtracting from this amount the sum of the payments ($16 + $56 = $72), we find that the new principal on Sept. 30, 1897, is $900. The interest on $900 from Sept. 30, 1897, to June 1, 1898 (9 months), is $ 54. Therefore, the amount due June 1, 1898, is $ 954. 378. A note of $280 was dated June 25, '94, interest 6%, indorsed $ 20, Jan. 25, '95. How much was due June 25, '95 ? 379. Face of note, $ 700. Date, July 2, '95. Kate, 8%. In- dorsed, Jan. 2, '96, $ 225. Find the amount due Oct. 2, '96. 380. Face of note, $200. Date, Sept. 7, 1896. Rate, 7%. Indorsed, March 7, 1897, $30. June 7, 1897, $40. Sept. 7, 1897, $ 60. Find the amount due Dec. 7, 1897. 381. When settlement is made within a year, the following rule is generally used : MERCANTILE RULE. Find the amount of the principal from date to time of settlement. Find the amount of each payment from its date to the time of settlement. Subtract the amounts of the payments from the amount of the principal. Find by this rule the amount due at the end of a year on a note of $500 with interest at 6%, if a payment of $200 is made 4 months before settlement. 382. A note for $1000 was given Feb. 7, 1898. Kate, 6%. Settlement was made 63 da. later. A payment of $200 was made 30 da. before settlement, and 15 da. before settlement $ 300 was paid. How much was paid at settlement ? SOLUTION. The amount of the principal, $ 1000, from date of note to time of settlement (63 da.), is $1010.50. The amount of first payment, $200 (30 da.), is $201 ; the amount of second payment, $300 (15 da.), is $300.75. Subtracting the amounts of the payments, $501.75, from the amount of the principal, $1010.50, there remains to be paid $508.75. Find amount paid at settlement applying Mercantile Rule. 383. Face of note, $60. Date, June 20, '85. Rate, 8%. Indorsed, $ 20, July 6, '85. Settled, Aug. 23, '85. 250 PERCENTAGE 384. Face of note, $ 80. Date, Nov. 5, '91. Rate, 7%. In- dorsed, $ 30, Dec. 5, '91. Jan. 5, '92, $ 25. Settled, Feb. 5, '92. 385. Face of note, $ 120. Date, Aug. 9, '93. Bate, 6%. In- dorsed, Sept. 15, '93, $ 48. Oct. 1, '93, $ 45. Settled, Oct. 9, '93. BANK DISCOUNT 386. Mr. James Gage sold a carriage to Mr. John Lyman, price $ 600, terms $ 100 cash, and the balance by a note due in 6 mo. without interest. As Mr. Gage wished to use the money in his business, he took the $ 500 note immediately to Mr. Peter Reed, who discounted it at 8 % ; that is, in exchange for the note, he gave Mr. Gage what remained after the interest of the $ 500 at 8 % for 6 mo. had been deducted from the $ 500. How much did Mr. Gage receive for the note ? At the end of the 6 mo. to whom should Mr. Lyman pay the $ 500 ? How much did Mr. Reed make by the transaction ? 387. When a note is discounted, the payee indorses it, mak- ing it payable to the one who discounts it. The payee is then responsible with the maker of the note for its payment. Mr. Gage wrote on the back of the note when he transferred it to Mr. Reed, Pay to the order of Peter Reed. JAMES GAGE. If when the note became due Mr. Reed should be unable to collect the amount of it from Mr. Lyman, to whom could he look for payment ? SUGGESTION TO TEACHER. Let three pupils enact the parts of Mr. Gage, Mr. Lyman, and Mr. Reed, one making, signing, and giving the note ; another receiving, indorsing, and transferring it ; the third discounting it. Let the class discuss the purpose and the justice of each step in the transaction. 388. To discount a note is to take from its face the simple interest on it for the time between the date of discounting and the date of maturity. BANK DISCOUNT 251 At 5%, what is the discount on a non-interest-bearing note for $ 700 due in 60 da. ? 389. Discount that is found by computing interest for a certain time is called Bank Discount. How does it differ from trade discount ? 390. If you had a note which promised to pay you $ 300 at the end of a year's time without interest, would it be worth $300 now? If it were discounted at 8%, how much would the discount be, and how much would you receive for the note now? 391. The difference between the bank discount and the face of the note discounted is called the Proceeds of the note. What are the proceeds of a non-interest-bearing note for $400 due in 6 mo. discounted at 10%? 392. Hale & Co. sold a carriage for $ 400 ; terms $ 50 cash, balance by note, payable in 60 da., without interest. As they wished to use the money at once, they sent the note to a bank where it was discounted at 8%. What were the proceeds of the note ? How much did Hale & Co. really receive for the carriage ? Find bank discount and proceeds of non-interest-bearing notes for the following amounts: 393. $ 250, due in 90 da., discounted at 6%. 394. $450, due in 30 da., discounted at 9% 395. $900, due in 60 da., discounted at 8%. 396. $ 750, due in 60 da., discounted at 9%. 397. $900, due in 30 da., discounted at 6%. 398. $ 650, due in 100 da., discounted at 7 % - 399. A note for $800 due in 4 mo. was discounted at 6% 3 mo. before it was due. What were the proceeds ? As the note had only 3 mo. more to run, it was discounted for 3 mo. 252 PERCENTAGE Find proceeds of non-interest-bearing notes discounted at 8% 400. Am't $600 Date of note Feb. 1, '97 Date of discount Mar. 1, '97 Date of maturity Apr. 1, '97 401. $500 Feb. 15, '96 Mar. 31, '96 May 1, '96 402. $870 Sept. 1, '95 Oct. 10, '95 Dec. 1, '95 403. $660 June 4, '97 July I, '97 Aug. 4, '97 404. $745 Apr. 6, '84 June 3, '84 July 9, '84 405. Find proceeds of a note given April 15, 1894, due in 60 da., discounted May 15, 1894, at 7%. Find proceeds of the following non-interest-bearing notes discounted at 8% : Face Date of note Time Date of discount 406. $ 38 June 7, '95 60 da. July 1, '95 407. $ 900 May 10, '84 90 da. June 10, '84 408. $ 850 Sept. 7, '90 60 da. Sept. 21, '90 409. $ 750 Mar. 15, '87 4 mo. May 1, '87 410. $ 1500 Oct. 2, '93 60 da. Nov. 1, '93 411. If you had a note promising to pay you $ 400 in one year with interest at 6%, how much would it be worth at the end of the year ? If that amount were discounted at 8%, what would the proceeds be ? As national banks do not usually discount long-time notes, if you wished to obtain the money in advance on this note you might apply to a savings bank or to one of those persons who deal in money and are called capitalists, money lenders, brokers, or loan agents. Their rates of discount and their ways of computing it vary. In the case of interest- bearing notes, the discount is sometimes reckoned upon the face of the note, sometimes upon the amount due at maturity, and sometimes the face of the note is discounted at a rate per cent equal to the difference between the rate of interest and the rate of discount. 412. A note for $ 1500 due in one year with interest at 7% was taken to a money lender, who deducted 8% from the BANK DISCOUNT 253 face. What were the proceeds? At the end of the year, how much did the money lender receive ? How much did he gain? 413. A note for $ 1500 due in one year with interest at 7% was taken to another money lender, who calculated the amount due at maturity and discounted that amount at 8%. What were the proceeds ? 414. A note for $ 1500 due in one year with interest at 7% was taken to another money lender, who agreed to discount it at 8%. He found the difference between the rate of interest and the rate of discount, 1%, and took 1% of the face. What were the proceeds ? Find the proceeds of the following notes, discounted at 8% by the method explained in Ex. 414 : In these problems, the notes are supposed to be discounted on the days on which they are dated. Face of note Time Rate 415. $ 750 2 mo. 5% 416. 650 3 mo. 6% 417. 1700 1 mo. 6% 418. 1200 4 mo. 7% 419. 1000 60 da. 6% 420. Mr. Ashby bought an automobile for $ 1000 and sold it for $ 1250, receiving $ 1000 cash and the balance in a note due in 6 mo., with interest at 6%. On the day of the sale the note was discounted at 8 % by the plan given in Ex. 414. How much did Mr. Ashby gain by the sale ? 421. Mr. Day sold three bicycles, each of which cost him $75, 011 the following terms: For the first he received $50 cash and a note for $50 due in 4 mo., interest 6%. For the second he received $ 75 cash and a note for $ 25 due in 6 mo., interest 5%. For the third he received $ 99 cash. The notes were discounted at date at 8% by the method used in Ex. 414. Compare the profits on the three sales. 254 PERCENTAGE INSURANCE 422. Mr. Adams has a house worth $ 7000. He has made an agreement with the agent of an insurance company by which, if the house is destroyed by fire, he will receive from the company $ 5000, or, if it is injured but not destroyed, he will receive a sum in proportion to the damage done. For this insurance against loss by fire, he pays the company every year 1% of the sum for which the house is insured. How much does he pay for insurance ? 423. The money paid for insurance is called a Premium. Mr. Green, wishing to provide for his wife in case of his death, has taken, for her benefit, an insurance of $ 5000 upon his life. The company has agreed to pay her $ 5000 upon proof of his death. For this he pays a premium of $27.35 a year, for each thousand dollars. If he pays premiums for 25 yr., what amount will he pay to the company ? 424. Mr. Strong, wishing to provide for his future, has taken out what is called an endowment policy. This agreement provides that if he is alive at the end of ten years, he shall receive $3682.25, and in case of his death at any time during the ten years, his heirs shall receive $ 2500. For this he pays an annual premium of $ 347.47. To how much will his premi- ums amount in the ten years? 425. There are two kinds of insurance. Property Insurance and Personal Insurance. To which kind of insurance does each of the three preceding problems refer? There are many kinds of property insurance, as insurance against loss by fire, tornadoes, shipwreck, theft, unpaid debts, etc. 426. Mr. Campbell has a house worth $1800. If it were insured for f of its value at 1 % each year, what would be the annual premium? If the house were destroyed by fire, how much insurance would Mr. Campbell receive ? INSURANCE 255 427. What would be Mr. Campbell's annual premium if his house were insured for j of the value, and the rate of insurance were 1J%? How much would the premium be if the rate were 1^%, and the insurance covered of the value of the house? 428. A stock of goods invoiced at $ 10,500 was insured for f of its value at l-j-%. How much premium was paid ? 429. A ship worth $ 75,000 was insured for f of its value at If % . The cargo, valued at $ 7500, was insured for of its value at 24%. Find amount of premiums. 430. Insurance companies generally insure property for a period of years, as 1, 3, or 5 yr., charging a certain number of cents on each hundred dollars insured. They also charge a certain amount, usually $ 1, for the written agreement to pay the insurance in case of loss. This written agreement is called a Policy. An insurance company insured Mr. Allen's house, worth $ 1600, for f of its value, for a period of 3 yr., charging $ 1.30 for each $100, and $1 for the policy. How much did the insurance cost him? SUGGESTION TO TEACHER. Let an insurance policy be brought into the schoolroom to be discussed and examined by the pupils. After solv- ing the following problems let pupils compose similar ones. 431. Mr. Stevens takes out an insurance policy of $7000 for a period of 3 years. The 3 yr. rate is twice the annual rate, which is $ .65 for $ 100. Policy, $1. Find cost of insuring. Find cost of insuring the following property at the above rates ; the policy costing $ 1 in each case. 432. Barn, $ 600. Hay, $ 300. f value insured, 1 yr. 433. House, $ 2700. Stable, $ 600. value insured, for 3 yr. 434. Stock of goods, value $6000, |- insured, 1 yr. 435 . House, $ 5000, $ 4000 worth of insurance taken for 3 yr. 436. House, $ 9500, $ 7000 worth of insurance taken for 3 yr. 256 PERCENTAGE 437. Mr. Eice takes $3000 worth of insurance from the Helena Fire Insurance Co. for a period of 3 yr. at the rates given above. Policy, $1. Six months later his house is damaged by fire to the extent of $ 1000, for which amount he receives a check from the company. How much better off is Mr. Eice than he would have been had he taken no insurance ? 438. Mr. Wood insured his house five times successively for 3 yr. periods at $1.25 a hundred, each policy costing $1. During the first period the house was insured for $ 8000, and during the next period for $ 7500. He continued to reduce the amount of insurance $ 500 each time he renewed it. The house was never injured by fire. How much did he pay out for in- surance on the house during those 15 yr. ? What did he receive in return for his payments? 439. Mr. Charles Olney insures his furniture for $500 for 3 yr. The annual rate is $ .55 per hundred. The rate for 3 yr. is twice the annual rate. What is the premium ? 440. John Gibson took out two policies: $3000 on dwell- ing, and $12oO on furniture. Term, 5 yr. Annual rate, 40^. 5 yr. rate three times the annual rates. What was the amount of both premiums ? 441. The trustees of Perry Township hold a policy on a school building for $3250.00. Term, 5 yr. What premiums have they paid, the rate being as in the previous problem ? 442. West & Co. insure their stock against wind storms, for 3 yr., for $20,000. Eate, 40^ a hundred for 3 yr. What is the amount of premiums? 443. George Brown takes out a policy for 3 yr. : $3000 on his dwelling and $500 on his furniture. Eate, $.90 for 3 yr. Policy fee, $1. What is the cost of his insurance? 444. A house valued at $ 3000 and insured for f its value was struck by lightning. The adjuster for the insurance com- INSURANCE f TJ pany estimated that it was damaged 37% of its value, and paid that per cent of the amount insured. How much did the owner of the house receive from the insurance company ? There are several kinds of personal insurance, as Life Insurance, Acci- dent Insurance, Endowments, Annuities, etc. 445. Mr. Blake took out a life insurance policy of $5000 for the benefit of his wife, upon which he paid $33.30 per thousand, yearly premium. He lived 20 yr. How much more was paid to his widow than he had paid to the insurance company ? What would have been the difference between the amount paid to the company by Mr. Blake and the amount received from it by Mrs. Blake, if he had taken an insurance of $ 10,000 ? ' Successful insurance companies take the small savings of those who are not able to invest them to advantage, and, massing them, invest them profit- ably. The insured loses the interest of the money which he pays to the company, but receives an assurance that those for whose benefit he is insured will receive the full amount in case of his death at any time. How much more or less is received from the insurance com- pany than is paid to it in the following cases ? Amount insured Yearly premium per $ 1000 Years 446. $7,000 $30.70 20 447. 6,000 44.84 40 448. 10,000 28.28 21 449. 30,000 27.38 50 450. 50,000 35.65 5 451. 200,000 39.00 15 452. Mr. Bland takes out a life insurance policy for $ 4000, paying $ 24 per thousand. What is the annual premium ? If he pays it for 50 years, how much more does he pay than his heirs receive ? 453. Mr. Corlen insured his life for $ 13,000 paying his first premium of $ 57.50 per thousand on Jan. 1, 1901. He died Mar. 1, 1901. If the company paid the agent a commission of HORN. GRAM. SCH. AR. 17 258 PERCENTAGE 15% on the first premium for insuring Mr. Corlen's life, how much did it lose by the insurance ? 454. Mr. Hill took out a life insurance policy for $ 25,000 at the rate of $32.75 per thousand. He died three months after paying the first premium. How much more did his heirs receive than he paid ? 455. Many insurance companies divide a part of their earn- ings among those who are paying premiums, giving each one a certain per cent on the premium he pays. This amount is called a Dividend. Find the value of a dividend of 5% on a premium of $ 250. 456. For 25 yr. Mr. Field paid an insurance premium on a life policy for $ 7000 at the rate of $ 27.50 per thousand. Ten per cent of the amount of the premiums was returned to him in dividends. How much did his insurance cost him ? 457. Mr. A., who is a traveling salesman, carries an accident policy. When he had paid $ 205.75 in premiums, he was acci- dentally injured and received an allowance of $25 per week for 7 weeks. How much more or less did he receive from the company than he had paid to it ? 458. At 25 yr. of age Mr. B. took out an endowment policy by which he will receive $ 5000 when he is 45 yr. old. An- nual premium, $ 240.38. How much more will he receive from the company than he pays to it ? How can the company afford to do business in that way ? SUGGESTION TO TEACHER. Let pupils make problems under various imaginary conditions, getting facts about insurance from agents or circu- lars in order that their problems may approximate to the actual. 459. Neil & Co., agents for the Westchester Insurance Co., insured the following risks for periods of 3 yr. at $ 1.30 per hundred. Their commission was 15% on the premiums, and they received from the insured a policy fee of $ 1 in each case, which they retained. How much did Neil & Co. earn and how much did they send to the insurance company ? TAXES 259 Dwelling, value f 1,800, f value taken Store, value 15,000, -| value taken Stock of goods, value 17,000, % value taken Opera house, value 75,000, f value taken 460. Miss Otis bought from an insurance and annuity com- pany a yearly annuity of $ 100, paying for it $ 1382.50. How much more or less would she receive from it than she paid for it if she lives 20 yr. ? 7 yr. ? 461. Mrs. Green owns a house from which she receives a monthly rental of $ 25. The insurance on it for the year 1898 was $ 8 and the taxes were $ 36.75. It was vacant three months. How much was her net income from it? TAXES 462. It is necessary for all governments to tax the people to pay public expenses. Taxes upon property are calculated at a certain per cent of the assessed value. "At 1% how much are the taxes upon a piece of property worth $ 9000 ? SUGGESTION TO TEACHER. Explain the duties of the assessor and tax gatherer. Procure copies of a part of an assessor's list of taxables and let pupils compute the taxes. 463. How much are the taxes upon Mr. Hudson's property which is assessed at $ 8000, where the rate of taxation is 464. A fixed sum assessed, without regard to their property, upon male citizens who are at least 21 yr. of age, is called a Poll Tax. If Mr. Hudson lived where the poll tax was $1.50, how much would all his taxes be ? 465. Mr. Howe, who pays a poll tax of $ 2, owns property assessed at $ 6000, in a city where the rate of taxation is $ .75 per $ 100, and $ 2500 in another city where the rate of taxation is 2^%. What are his taxes ? 260 PERCENTAGE 466. What are the taxes of Mr. Hearn who owns real estate assessed at $ 11,375, and other property valued at $ 2500, the rate of taxation being l-g-%, and his poll tax $ 1 ? 467. Property is considered to be of two kinds : Real Estate, as lands, houses, stores, factories, mines, and other immovable property ; and Personal Property, such as money, notes, furni- ture, and other property which can be carried from place to place. Name a piece of real estate. Name different kinds of per- sonal property. 468. Mrs. Kent owns real estate assessed at $ 5600, and per- sonal property assessed at $ 1000. The rate of taxation is $ 1.20 on $ 100. What is the amount of her tax bill ? 469. A penalty of a certain per cent of the amount of the tax is sometimes enforced if the taxes become delinquent ; that is, if they are not paid at the time required. If Mrs. Kent allows her taxes to become delinquent, and the penalty is 10%, what will be the amount of her tax bill ? 470. In a certain city the rate of taxation is $ 1.35 per $ 100, and the poll tax is $ 1. How much are the taxes of an adult male citizen whose real estate is valued by the assessor at $ 1540, and personal property at $ 300 ? 471. What is the tax of an adult female citizen whose real estate is valued at $ 1500, and personal property at $ 650 ? 472. Mention several things which are paid for with the money raised by taxation. 473. Some property, as churches, government bonds, and public property of all kinds, is exempt from taxation. Why are public school buildings not taxed ? 474. A certain town raised $ 21,845 by taxation, a part of which was the assessment of 721 polls at $ 2 each. How much tax was raised from property ? TAXES 261 475. Taxes to the amount of $ 24,704.35 were raised in the town of Nalasco, of which $ 23,456.35 was raised from property, and the rest from polls at $ 2 each. How many citizens paid poll taxes in the town ? 476. In a town where there are 1236 polls assessed at $ 1.50 each, it was decided to raise $ 46,854 by taxation. How much must be raised from the property ? If the property valuation of the whole town was $ 3,600,000, how much must each dollar's worth of property yield ? What would be the tax of each of the following residents of that town ? 477. Mr. A., 31 yr. old, whose realty is $ 7000, personals $ 1375. 478. Mr. B., 19 yr. old, whose realty is $ 938, and who has $ 4500 worth of personal property, of which $ 3000 is in government bonds. 479. Mrs. C., whose realty is $ 7500, personal property $ 635. 480. Mr. D., 43 yr. old, who has no property. 481. Miss E., who has no real estate and $500 worth of personal property. 482. The money which defrays the public expenses of cities, counties, and states is raised by direct taxation upon property or person. Money for the expenses of the national government is raised by indirect taxation, of which there are two kinds, Internal Revenue and Duties or Customs. The internal revenue is mostly derived from taxes on the manufacture of liquors and tobacco products and from the sale of stamps which the government requires to be placed upon certain legal documents and articles sold. To defray the expenses of the war with Spain in 1898, a law was passed by Congress requiring among other provisions that a one-cent stamp should be affixed to every telegraphic message or express receipt, a two-cent stamp to every bank check, UNIVERSITY 262 PERCENTAGE sight draft, etc., two cents for every hundred dollars, or frac- tional part thereof, named in the face of time drafts, prom- issory notes, etc., and stamps of different values upon patent medicines, proprietary articles, insurance policies, contracts, leases, etc. What should be the value of a stamp affixed to a promissory note for $ 2500 ? For 9 275 ? For $ 39.50 ? 483. A drug firm sold in one week 1216 bottles of patent medicines, each requiring a stamp whose value is f of a cent, 1172 packages each requiring a stamp costing 1^ ^, and 298 packages each requiring a stamp costing 21 ^. The firm sent 15 telegrams and 137 express packages. Fifty-one checks were given by the firm. How much revenue accrued to the government from the sale of stamps necessary for the business of that firm for that week ? 484. The taxes levied by the government upon imported goods are called Duties or Customs. All goods which come into the country must be brought in at certain places called Ports of Entry. At these places the government maintains custom houses, with officers who collect the duties. There are two kinds of duties, Specific and Ad Valorem. A duty of a certain per cent of the amount at which the goods were invoiced in the country from which they were im- ported is called an Ad Valorem Duty. The Latin phrase ad valorem means " according to value." Find the ad valorem duty of 100 yd. silk invoiced at $ .50 per yd., duty 55%. At the rates given, how much ad valorem duty would be paid by a firm of importers upon the following goods ? 485. 50 yd. of silk, invoiced at $ 1.25 per yd., duty 55%. 486. 500 pieces of ribbon, 10 yd. in a piece, at 75^ per yd., duty 40%. 487. 50 yd. of lace, at $ 2.25 per yd., duty 60%. TAXES 263 488. At 20%, what is the duty on 75 bales of wool, 400 Ib. each, invoiced at 25 ^ per pound ? 489. At 25%, what is the duty on 500 boxes of raisins, each containing 40 Ib., costing 6J cents per pound ? 490. A duty levied upon a certain quantity of goods, with- out reference to their value, is called a Specific Duty. If the specific duty is $ 2.25 per dozen pairs, how much is that duty on 600 pairs of gloves invoiced at 50 f a pair ? If they were invoiced at 75^ a pair, what would be the specific duty ? 491. Sometimes both specific and ad valorem duties are levied upon the same article. What is the duty on 30 pieces of carpet, 25 yd. each, in- voiced at $1.75 per yard, the specific duty being 25^ per yard, and the ad valorem duty 40% ? 492. What is the duty upon 800 Ib. of cigars, invoiced at $5 per pound, which pay a specific duty of $ 4.50 per pound and 25 % ad valorem ? 493. An importation of. silks from France was invoiced at 9324 fr. At 60% ad valorem, how much is the duty in American currency, $ 1 being considered equal to 5.18 fr. ? 494. What duty is paid by an American importer upon 600 doz. pairs of gloves invoiced at 60 fr. per dozen, if there is a specific duty of $2 per dozen pairs and an ad valorem duty of 40% ? 495. Persons are allowed to bring from abroad a limited amount of goods for their own use without paying duties upon them. An American lady brought home from Europe a silk dress pattern, upon which the duty was $ 31.75 ; ^ doz. pairs of kid gloves, upon which the duty was $ 2.25 per dozen pairs ; 20 yd. of lace worth f 2 per yard, upon which the duty was ad valorem; and 12 yd. Irish linen at 60^ per yard, ad 264 PERCENTAGE valorem duty 35 %. As these goods were for her own use, they were passed in duty free. How much less did the goods cost her than they would have cost had the duty been collected ? 496. If an importer buys 700 yd. of velvet at $1.50 per yard, pays an ad valorem duty of 60%, and sells it at $ 4 per yard, how much does he gain ? 497. How much is gained by an importer who buys 20 pieces of matting, 40 yd. in a piece, at $ .10 per yard, pays a duty of 25%, pays for transportation $60, and sells the matting at 35^ per yard? 498. A list of articles upon which duties must be paid, with the special duty upon each, is called a Tariff. Tariffs are changed from time to time by acts of Congress. An importer brought through the custom house $ 80,000 worth of cut glass when the duty was 35% ad valorem. He sold | of it at a profit of 25% upon invoice price plus the duty. The tariff upon glass was raised after his purchase to 60%. He sold the other half of his stock at a profit of 25% upon invoice price plus the new duty. How much more did he gain on the last half of his stock than on the first half ? 499. A New York firm imported goods invoiced at $64,000, upon which there was a duty of 12^% ad valorem. For how much must these goods be sold to give a profit of 20% ? 500. Soon after those goods were bought, the duty on that class of goods was changed to 25% ad valorem. Another firm imported $ 64,000 worth of the same kind of goods under the new tariff, and sold their goods at a profit of 20%. How much did they receive ? If the first firm sold their goods for the same amount as the second firm, how much more did they gain than the second firm gained ? 501. Mr. Gilman imported $ 100,000 worth of goods, the duty upon which was 30% ad valorem. If, after he sold J of them at a profit of 20%, this class of goods was put on the MISCELLANEOUS EXERCISES 265 free list, for how much could his competitors in business buy an amount of goods equal to what he had left on hand ? If he sold the rest of his goods for 20% more than that sum, would he gain or lose on the whole transaction, and how much ? 502. Make a problem in which an importer's business is injured or benefited by changes in the tariff. MISCELLANEOUS EXERCISES 1. Resolve into prime factors 6750. 7920. 2. Find the g. c. d. of 235 and 685. 3. Find the 1. c. m. of 8, 10, 12, 16, 18, 20. 4. Divide .012261 by 2.01. 5. 3V64 + 7V81=? 6. 2A/64 + 4^/125 = ? 7. If 1 qt. of nuts costs 11^, how many bushels can be bought for $ 13.20 ? 8. Find the interest of $ 1240 for 5 yr. 9 mo. 27 da. at 3%. 9. A wheel of a bicycle is 7 ft. in circumference. How many times does tjie wheel turn in going 10 rd. 1 yd. ? 10. John weighs 115 lb., and his cousin weighs 110 Ib. John's weight is what per cent of the sum of their weights? 11. Write the following decimally and as common fractions in their lowest terms: 13%. 18%. 22|%. 158%. 875%. 12. How much is 30% of 40 minus 16|% of 66 ? 13. Find 11% of 24 2 . 37^% of 16 2 . 62i% of 64 2 . 14. Find 12|% of 12 3 . 66f % of 9 3 . 87J% of 12 3 . 15. Square: f. .3. 1.2. 2f .06. ?i. 266 PERCENTAGE 16. Name four numbers between 100 and 200 that are perfect squares and give their square roots. 17. What number between 100 and 200 is a perfect cube ? 18. Every prime number greater than 10 must end with either 1, 3, 7, or 9. Give the reason. 19. The arc AB is 24 in. long. BC is 50% longer than AB. CD is 33% longer than BC. DA is 25% shorter than DC. How long is the circumference ? Diam- eter ? Radius ? Perimeter of the sector DO A? BOG'! DOC? AOB? c 20. A circumference which is 65 in. long is divided into two arcs, the smaller arc of which is 13 in. long. The smaller arc is what per cent of the greater ? 21. How many coins an inch in diam- eter could be placed in rows touching one another on a rectangle 4 in. by 3 in. ? Represent. 22. When the hour hand of a clock is at 3, what per cent of one revolution around the clock face has it made since 12 ? 23. At 4 P.M. the time past noon is what per cent of the time before midnight ? 24. The time past noon is what per cent of the time to mid- night at 2 P.M.? 8 P.M.? 1.30 P.M.? 25. Thomas wished to add the fractions ^, , and i. He first multiplied each term of each fraction by the product of all the denominators except its own. How were the three fractions then expressed? He then added these fractions and reduced their sum to its lowest terms. Was his process correct? Can you show a better way to find the sum of these fractions ? 26. Take Ex. 25, substituting the fractions, , -f, and Jy. MISCELLANEOUS EXERCISES 267 Fill blanks and solve : 27. A house worth dollars was insured for of its value, at per cent. What was the annual premium ? 28. Mr. A. held a life insurance policy for $ 2000, on which he paid an annual premium of $ 52. He was insured March 1, 1890, and died June 1, 1900. How much more did his heirs receive than he had paid out in premiums ? 29. A room 36 ft. long and 24 ft. wide is to be covered with carpet f yd. wide, at $ 1.10 per yard. How much will it cost if the strips run lengthwise of the room and each strip is turned in 4 in.? 30. Advance the following goods 15% in price: Caps at 30 t, coats at $ 8, shoes at f 1.25, gloves at 78 f, ties at 15 31. A fisherman caught herring enough to fill 500 barrels. He sold 35% of the catch, and kept the rest for a rise in price. How many barrels of herring did he keep ? 32. How many quarts of berries at 12-^ a quart would be required to pay for 9 yd. of cloth at 16^ a yard? 33. Two men traveled from the same point, one east, 45^ mi. ; the other west, 92f mi. How far apart were they ? 34. Two men started from the same point, and traveled in opposite directions. One man traveled at the rate of 7^ mi. per hour, the other at the rate of 6J mi. per hour. How far apart were they at the end of 1 hr.? Of 3 hr.? Represent. 35. John's uncle showed him a half eagle one morning, and promised to give him at night 25% of all of it that was not spent. At night his uncle reported that 100% of the money had been spent, but he gave him 75 $ instead. How much more or less would John have received if his uncle had spent only 50% of the value of the half eagle ? 40% ? 36. Nine is how much greater per cent of 144 than of 288 ? 268 PERCENTAGE 37. How much is gained on each tablet bought at the rate of $ 1 per dozen, and sold at 10 $ each ? 38. Mr. Hall earned $ 125 in one month, which was 62 of his earnings the next month. How much did he earn in both months ? 39. An apple tree bore 21 bu. of apples, which was 87 -J% of what the tree next to it bore. What was the difference in the yield of the two trees ? 40. An automobile started from New York, and ran 60 mi. the first day. On the next day its speed was 33 \] greater than on the first day, and on the third day it was 25% greater than on the second. How far was the automobile from New York at the end of the third day ? 41. What number plus 1% of itself equals 909 ? 2424 ? 42. Forty-five is 50% more than what number? 50% less than what number ? 43. A man had $ 654 in bank. He drew out 33% of it, and afterward drew out 25% of the remainder. How much had he left in bank ? 44. A man sold a wagon for $ 180, and gained 25%. What was the cost of the wagon ? 45. A man sold a wagon for $180 and lost 25%. What was the cost of the wagon ? 46. Near the close of summer the price of goods costing $1.10 per yard was cut to 95^ a yard. What per cent was lost? 47. Fifty yards of cloth were bought for $30. For what price per yard must they be sold to gain 25% ? 48. A house valued at $8000 was insured for f of its value at 1J%. What was the premium ? 49. On the day before Christmas Mary counted at a cer- tain corner 37 ladies who were carrying packages, and 13 who MISCELLANEOUS EXERCISES 269 had no packages. What per cent of the ladies that she counted had no packages? 50. Thirteen children were transferred from a class of 42. What per cent of them remained ? 51. In making peach marmalade, Mrs. Harland boiled 4 Ib. of peaches and 3 Ib. of sugar in a quart of water. Each pint of water weighed a pound. If 1 pt. of the water evaporated in cooking, what per cent of the marmalade was sugar ? Peaches ? 52. How wide is a rectangle 20 cm. long and equal to f of a square decimeter ? Represent. 53. In the rhomboid ABCD the line BC represents 10 ft. AB is 50% longer than BC. How long is the perimeter ? What per cent of FIG. 2. the perimeter is AD ? DC? 54. AB and DC are parallel. DC = 48 ft. AB = 25% of DC. BC = 250% of AB. AD = 100% of BC. Find the perimeter of the trapezoid. 55. A farm is in the shape of a trapezoid. The shorter parallel side is 16 rd. long. The longer parallel side is 121% longer. One of the non-parallel sides is 10 rd. and the other is 20% longer. Represent. Find the cost of fencing the farm at 75^ per rod. 56. What is a trapezoid? How does it differ from a rhomboid? 57. A four-sided plane figure which has no two sides parallel is called a Trapezium. Draw a trapezium. 58. How long is the perimeter of a trape- zium of which the side AB is 3J- in., the side BC 5 in., the side CD 6J in., and DA FIG. 4. 7|in? FIG. 3. 270 PERCENTAGE 59. A garden is fenced in the form of a trapezium. One side is 4 rd. 3 yd. 2 ft. 8 in. long, another side is 5 rd. 1 ft. 10 in. long, another side is 4 rd. 5 yd. 4 in. long. The other side is 6 rd. 2 yd. 6 in. long. How long is the fence ? 60. How long is the perimeter of a trapezium, the shortest side of which is 12 in. long, the next side 2 in. longer than the first, the next side 3 in. longer than the second, and the last 4 in. longer than the third ? B 61. In the trapezium ABCD, AD re- presents 8 ft. DC represents 50% more than AD. CB represents 33^% more than DC. BA represents 121% more than CB. How long is the perimeter ? 62. John had a kite frame in the shape of a trapezium having two short FlG 5 equal sides and two long equal sides. If a long side was 3 ft. long and a short side 33|% as long, what was the combined length of the sticks that made the frame, allowing -J- an inch for lapping the sticks at each angle ? 63. Plane figures bounded by four straight lines are called Quadrilaterals. You have learned six different kinds of quadrilaterals. Draw one of each kind and write its name upon it. 64. How long is the perimeter of a rhombus whose sides are each 1.7 in. ? 65. Find the perimeter of a rhomboid whose long sides are each 9.9 in. and whose short sides are each 5 in. less than a long side. 66. How long is the perimeter of a trapezoid if one of the parallel sides is 7.65 in., the other 8.45 in., and each of the non-parallel sides is 4.7 in. ? Kepresent. 67. Plane figures bounded by straight lines are Polygons. Name four kinds of polygons. MISCELLANEOUS EXERCISES 271 68. What name is given to a polygon of 3 sides ? 5 sides ? 6 sides? 8 sides? 10 sides? 69. Is a sector a polygon ? Explain. 70. Mr. K. bought 9 doz. pencils for $ 2.16. He sold them at $ .03 apiece. What per cent was gained ? 71. 500 sacks of coffee were bought for $300. At what price per sack must they be sold to gain 10% ? 72. 350 bottles of ink were bought for $ 21. For how much per bottle must they be sold to gain 66f % ? 73. 280 penknives cost $ 70. For how much apiece must they be sold to gain 25% ? 74. A dealer paid $ 24 for 300 slates. In selling them he gained 50 % . What was the selling price of each ? 75. Two gross of handkerchiefs were bought for $28.80. At what price apiece must they be sold to gain 30% ? 76. Mr. Fowler's salary was $ 1800 a year. Last year he paid $ 1175.50 for household expenses, $ 22.50 for life insur- ance, $ 15.75 for taxes, $ 178.25 for clothing, and $ 23.75 for incidentals. What per cent of his salary did he save ? 77. A grocer sold 630 heads of cabbage, which was 66|% of what he had. How many had he left ? 78. It cost $ 15 to build a certain fence, and $ 10 to paint it. The cost of painting was what per cent of the whole cost ? 79. A quart of water was added to 6 gallons -/^ /y- /$t/ OCT J8 19 J5 '- ' ; \9^ MftV 30 U?r?9 Rvi - --' J/I/V S 1t 40 ^ ^w / 7 ^U S 31 ^^^- /^/ ; LD 21-100m-7,'33 YB (7393 /, 1