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? 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 2. 5. A mixed number is an integer with a fraction; as 7, 5%. 6. A compound mixed fraction is one, whose numerator or denominator, or both, is a mixed number; as 71⁄2 or 42 7. The 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 Therefore 2 will measure or divide 98 and 114. Q. E. D. that of their sum. 2. What is the greatest common measure of 56 and 168? 3. What is the greatest common measure of 4. What is the greatest common measure of Ans. 56. 96 and 128? Ans. 32. 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. OPERATION. 4)=4)} Ans. NOTE. That is equal to 48 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 18. Q. E. D. 16 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. 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? 4. How many halves in 11? In 21? In 31? 5. How many quarters in 14? In 5? In 6? In 41? In 12? In 24? In 2? In 13? In 24? In 2? 6. How many sevenths in 1? OPERATION. 1793 5 88 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 much 85 fifths; to which, if we add 3 fifths, the amount will be 88 fifths. Hence we deduce the following 5 Ans. RULE. Multiply the whole number by the denominator of the fraclion, and to the product add the numerator, and place their sum over the denominator of the fraction. 8. Reduce 16 to an improper fraction. 9. Reduce 14 to an improper fraction. 10. Reduce 126 to an improper fraction. 11. Reduce 14911 to an improper fraction. 12. Reduce 161 to an improper fraction. 13. Reduce 1715 to an improper fraction. 14. Reduce 98 to an improper fraction. 15. Reduce 116 to an improper fraction. 16. Reduce 718 to an improper fraction. 17. Reduce 100109 to an improper fraction. 18. Reduce 478 to an improper fraction. 19. Reduce 871, to an improper fraction. 20. Reduce 167109 to an improper fraction. 21. Reduce 613101 to an improper fraction. 22. Reduce 159100 to an improper fraction. 23. Reduce 999,99 to an improper fraction. 307 1000 CASE IV. To reduce improper fractions to integers or mixed numbers. 1. How many dollars in 2 half dollars? In 4? In 5? 2. How many dollars in 5 quarters? In 6? In 7? In 8? 3. How many dollars in 16 eighths? In 24? In 30? 4. Reduce to a mixed fraction. OPERATION. 19)117(6 Ans. 114 3 To apply this question, we may suppose a certain number of dollars to have been cut into 19 equal parts each, and we wish to know how many dollars 117 of these parts contain. To effect this, we must divide 117 by 19. Hence the following RULE. Divide the numerator by the denominator, and if there be a remainder, place it over the denominator at the right hand of the integer. 5. Reduce 167 to a mixed number. 6. Reduce 1631 to a mixed number. 116 Ans. 11. Ans. 1476. 7. Reduce 131 to a mixed number. 8. Reduce to a mixed number. Ans. 71. Ans. 3134. |