If of a ton cost 12 dollars, what did a ton moo 84 is of what number? of a ton cost? cost? 47. Bought of a barrel of pork for 14 dollars, what was the cost of of a barrel? If of a barrel cost 2 dollars, what was the price of a barrel? 14 is } of what number? 48. Bought of a barrel of fish for 18 dollars, what did of a barrel cost? If of a barrel cost 9 dollars, what was the value of a barrel? 18 is 3 of what number? 49. If of an acre value of of an acre? is the value of an acre? of land cost 42 dollars, what is the If of an acre cost 6 dollars, what 42 is of what number? 12 50. If of a bushel of apples cost 35 cents, what will of a bushel cost? If of a bushel cost 5 cents, what is the value of a bushel? 35 is of what number? 51. If of a bushel of corn cost 63 cents, what is of a bushel worth? If of a bushel cost 9 cents, what is the value of a bushel? 63 is of what number? 52. If of a pound of saleratus cost 8 cents, what is of a pound worth? If of a pound cost 2 cents, what does a pound cost? 53. If of a load of hay is sold for 14 dollars, what is of a load worth? If of a load cost 2 dollars, what will 3 loads cost? 14 is of what number? 54. If 10 cents will buy of a pound of figs, how many cents will buy a pound? 55. If 12 dollars will buy of a cwt. of dollars will it take to buy a cwt.? 57. 9 is 58. 12 is 59. 4 is 60. 3 is 61. 15 is 62. 18 is 63. 20 is sugar, how many 56. 8 is of what number? 7? 6? 11? 12? 15? 19? 20? of what number? 6? 7? 8? 15? 17? 18? 23? of what number? 13? 16? 17? 19? 25? 27? of what number? 11? 7? 16? 17? 19? 21? of what number? 4? 5? 6? 8? 9? 11? 18? of what number? 16? 18? 20? 25? 26? 30? of what number? 3? 4? 5? 6? 7? 8? 11? of what number? 10? 8? 18? 21? 27? 28? of what number? 27? 28? 29? 33? 37? 38? of what number? 11? 16? 17? 18? 27? 36? of what number? 28? 29? 30? 37? 38? 61? of what number? 36? 42? 58? 59? 61? 72? 68. 30 is of what number? 35? 40? 45? 55? 67? 85? 69. If of a bushel of salt cost 42 cents, how many pounds of raisins at 6 cents a pound, would it take to purchase a bushel? 64. 21 is 65. 25 is 66. 24 is 67. 27 is 70. If of a load of hay cost 18 dollars, how many barrels of cider at 3 dollars a barrel would it take to purchase a load? 18 is of how many times 3? 71. Bought of a yard of cloth for 60 cents; how many pounds of sugar at 9 cents would pay for one yard? 60 is of how many times 9? 72. A man sold a pair of oxen for 48 dollars, which was of what they cost him; he had paid for them in calves at 6 dollars a head. How many calves did it take to pay for the oxen? 73. 28 is 74. 12 is 75. 32 is 76. 60 is 77. 72 is 78. 64 is 79. 50 is 80. 36 is 00 of how many times 7? 8? 9? 10? 11? 12? 13? VULGAR FRACTIONS. FRACTIONS are parts of an integer. VULGAR FRACTIONS are expressed by two numbers, called the Numerator and Denominator; the former above, and the latter below a line. Thus ; { Numerator 7 Denominator 11 The Denominator shows into how many parts the integer, or whole number, is divided. The Numerator shows how many of these parts are taken. 1. A proper fraction is one, whose numerator is less than the denominator; as. 2. An improper fraction is one, whose numerator exceeds or is equal to the denominator; as or . 3. A simple fraction has a numerator and denominator only; as, . 4. A compound fraction is a fraction of a fraction, connected by the word of; as of of of. 5. A mixed number is an integer with a fraction; as 7, 53. 6. A compound mixed fraction is one, whose numerator or denominator, or both, is a mixed number; as 7 or 42 7. The greatest common measure of two or more numbers is the largest number, that will divide them without a remainder. 8. The least common multiple of two or more numbers is the least number, that may be divided by them without a remainder. 9. A fraction is in its lowest terms, when no number but a unit will measure both its terms. 10. A prime number is that, which can be measured only by itself or a unit; as 7, 11, and 19. 11. A perfect number is equal to the sum of all its aliquot parts; as 6, 28, 496, &c. 12. A fraction is equal to the number of times the numerator will contain the denominator. 13. The value of a fraction depends on the proportion, which the numerator bears to the denominator. CASE I. To find the greatest common measure of two or more numbers, or to find the greatest number, that will divide two or more numbers. RULE. Divide the greater number by the less, and if there be a remainder, divide the last divisor by it, and so continue dividing the last divisor by the last remainder, until nothing remains, and the last divisor is the greatest common measure. If there be more than two numbers, find the greatest common measure of two of them, and then of that common measure and the other numbers. If it should happen that 1 is the common measure, the numbers are prime to each other, and are incommensurable. EXAMPLE. 1. What is the greatest common measure of 98 and 114? 98)114(1 98 16)98(6 Common measure 2)16(8 16 By this process, it is found that 2 is the greatest number, that will divide 98 and 114. NOTE. As 2 will divide 16, it will also divide 96, because it is a multiple of 16. It will also divide 98, because 98 is the sum of 96 and 2; and, as it will divide them separate, it will also united. Again 114 is equal to 9816 and as 2 will divide both these numbers, it will also that of their sum. Therefore 2 will measure or divide 98 and 114 Q. E. D. 2. What is the greatest common measure of 56 and 168? Ans. 56. 3. What is the greatest common measure of 96 and 128? Ans. 32. 4. What is the greatest common measure of 57 and 285? Ans. 57. 5. What is the greatest common measure of 169 and 175? Ans. 1. 6. What is the greatest common measure of 175 and 455? Ans. 35. 7. What is the greatest common measure of 169 and 866? Ans. 1. 8. What is the greatest common measure of 47 and 478? Ans. 1. 9. What is the greatest common measure of 84 and 1068? Ans. 12. 10. What is the greatest common measure of 75 and 165? Ans. 15. 11. What is the greatest common measure of 78, 234 and 468? Ans. 78. 12. What is the greatest common measure of 144, 485 and 25? Ans. 1. 13. What is the greatest common measure of 671, 2013 and 4026? Ans. 671. 14. What is the greatest common measure of 16, 20 and 24? Ans. 4. 15. What is the greatest common measure of 21, 27 and 81? Ans. 3. CASE II. To reduce fractions to their lowest terms 1. Reduce to its lowest terms. NOTE. That is equal to 4 may be demonstrated as follows:-16 is the same multiple of 1, that 48 is of 3, therefore 16 has the same ratio to 48, that 1 has to 3; and as the value of a fraction depends on the ratio, which the numerator has to the denominator, it is evident when their ratios are the same that their values are equal; therefore is equal to 16 Q. E. D. 48° Thus we see the propriety of the following RULE. Divide the numerator and denominator by any number, that will divide them both without a remainder; and so continue, until no number will divide them but unity. Or divide the numerator and denominator by their greatest common measure. 2. Reduce to its lowest terms. 3. Reduce to its lowest terms. to its lowest terms. 4. Reduce Ans.. 4. Ans. 3 Ans.. Ans. 4. Ans.. 422. Ans. 81 875⚫ Ans. 811 509. CASE III. To reduce mixed numbers to improper fractions. 1. How many halves in 2 apples? In 3? In 4? In 5? 2. How many quarters in 7 oranges? In 9? In 10? 3. How many fifths in 3 bushels? In 4? In 5? In 6? 4. How many halves in 1? In 21? In 3? In 4? 5. How many quarters in 14? In 12? In 24? In 2? 6. How many sevenths in 14? OPERATION. 173 5 88 5 The object in this question is to find how many fifths are contained in 173. This we obtain by multiplying 17 by 5, and adding 3 to the product. We may analyze this by saying, If 1 unit contain 5 fifths, 17 units will contain 17 times as Ans. much 85 fifths; to which, if we add 3 fifths, the amount will be 88 fifths. Hence we deduce the following = RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction. |