GREATEST COMMON DIVISOR OF FRACTIONS. 196. The Greatest Common Divisor of two or more fractions is the greatest number which will exactly divide each of them, giving a whole number for a quotient. NOTE. The definition of an exact divisor, (128), is general, and applies to fractions as well as to integers. 197. In the division of one fraction by another the quotient will be a whole number, if, when the divisor is inverted, the two lower terms may both be canceled. This will be the case when the numerator of the divisor is exactly contained in the numerator of the dividend, and the denominator of the divisor exactly contains, or is a multiple of, the denominator of the dividend. Hence, I. A fraction is an exact divisor of a given fraction when its numerator is a divisor of the given numerator, and its denominator is a multiple of the given denominator. And, II. A fraction is a common divisor of two or more given fractions when its numerator is a common divisor of the given numerators, and its denominator is a common multiple of the given ..denominators. Therefore, III. The greatest common divisor of two or more given fractions is a fraction whose numerator is the greatest common divisor of the given numerators, and whose denominator is the least common multiple of the given denominators. 5 and 15? 1. What is the greatest common divisor of, 2, ANALYSIS. The greatest common divisor of 5, 5, and 15, the given numerators, is 5. The least common multiple of 6, 12, and 16, the given denominators, is 48. Therefore the greatest common divisor of the given fractions is, Ans. (III). 198. From these principles and illustrations, we derive the following RULE. Find the greatest common divisor of the given numerators for a new numerator, and the least common multiple of the given denominators for a new denominator. This fraction will be the greatest common divisor sought. NOTE.-Whole and mixed numbers must first be reduced to improper fractions, and all fractions to their lowest terms. EXAMPLES FOR PRACTICE. 14, and 28? Ans. 135 7 1. What is the greatest common divisor of 7, 14, 2. What is the greatest common divisor of 31, 15, and 34? 3. What is the greatest common divisor of 4, 2, 2, and ? 4. What is the greatest common divisor of 109 and 1224? 5. What is the length of the longest measure that can be exactly contained in each of the two distances, 182 feet and 571 feet? Ans. 23 feet. 6. A merchant has three kinds of wine, of the first 134 gallons, of the second 1281 gallons, of the third 115 gallons; he wishes to ship the same in full casks of equal size; what is the least number he can use without mixing the different kinds of wine? How many kegs will be required? Ans. 59. LEAST COMMON MULTIPLE OF FRACTIONS. 199. The Least Common Multiple of two or more fractions is the least number which can be exactly divided by each of them, giving a whole number for a quotient. 200. Since in performing operations in division of fractions the divisor is inverted, it is evident that one fraction will exactly contain another when the numerator of the dividend exactly contains the numerator of the divisor, and the denominator of the dividend is exactly contained in the denominator of the divisor Hence, I. A fraction is a multiple of a given fraction when its numerator is a multiple of the given numerator, and its denominator is a divisor of the given denominator. And II. A fraction is a common multiple of two or more given fractions when its numerator is a common multiple of the given numerators, and its denominator is a common divisor of the given denominators. Therefore, III. The least common multiple of two or more given fractions is a fraction whose numerator is the least common multiple of the given numerators, and whose denominator is the greatest common divisor of the given denominators. NOTE. The least whole number that will exactly contain two or more given fractions in their lowest terms, is the least common multiple of their numerators, (193, Note 2). 1. What is the least common multiple of 1, 2, and 15? ANALYSIS. The least common multiple of 3, 5, and 15, the given numerators, is 15; the greatest common divisor of 4, 12, and 16, the given denominators, is 4. Hence, the least common multiple of the given fractions is 15 = 33, Ans. (III). 201. From these principles and illustrations we derive the following RULE. Find the least common multiple of the given numerators for a new numerator, and the greatest common divisor of the given denominators for a new denominator. This fraction will be the least common multiple sought. NOTE.-Mixed numbers and integers should be reduced to improper fractions, and all fractions to their lowest terms, before applying the rule. EXAMPLES FOR PRACTICE. 1. What is the least common multiple of 3, 6, 14, and . 7 35 2. What is the least common multiple of 4, 38, 3 Ans. 11. e and 48? 100 6 4, 4, 5, 4, 7, 8, 3. What is the least common multiple of 233, 133, and 63? 4. What is the least common multiple of 1, 3, and? Ans. 2520. feet in circum- 5. The driving wheels of a locomotive are 15,5 ference, and the trucks 93 feet in circumference. must the train move, in order to bring the wheel and truck in the same relative positions as at starting? Ans. 4593 feet. PROMISCUOUS EXAMPLES. 1. Change of to an equivalent fraction having 135 for its denominator. 135 Ans. 75 2. Reduce, 1, §, and 11 to equivalent fractions, whose denominators shall be 48. 6 3. Find the least common denominator of 1, 9, 2, 7%, 3 of 1, 6. What number multiplied by of ğ × 32, will produce ? 8. What number diminished by the difference between 2 and 7 of itself, leaves a remainder of 144? Ans. 2831. 9. A person spending, , and of his money, had $119 left; how much had he at first? 29 10. What will of 107 cords of wood cost, at of $42 per cord? Ans. $31. 11. There are two numbers whose difference is 25,7%, number is of the other; what are the numbers? and one Ans. 633 and 892. -12. Divide $2000 between two persons, so that one shall have as much as the other. Ans. $1125 and $875. 13. If a man travel 4 miles in of an hour, how far would he travel in 14 hours at the same rate? Ans. 10 miles. 14. At $a yard, how many yards of silk can be bought for $10%? 15. How many bushels of oats worth $3 a bushel, will pay for of a barrel of flour at $7 a barrel? 2 3 16. If g of a bushel of barley be worth of a bushel of corn, and corn be worth $3 per bushel, how many bushels of barley will $13 Buy? Ans. 18. \17. If 48 is of some number, what is of the same number?(.' 18. If cloth 14 yards in breadth require 20 yards in length to make a certain number of garments, how many yards in length will cloth of a yard wide require to make the same? 19. A gentleman owning of an iron foundery, sold of his share for $25703; how much was the whole foundery worth? Ane. $51411. 20. Suppose the cargo of a vessel to be worth $10,000, and 4 of of of the vessel be worth of 3 of 15 of the is the whole value of the ship and cargo? cargo; what Ans. $22000. 21. A gentleman divided his estate among his three sons as follows: to the first he gave of it; to the second of the remainder. The difference between the portions of first and second was $500. What was the whole estate, and how much was the third son's share? Whole estate, $12000. Third son's share, $2500. Ans. 22. If 71⁄2 tons of hay cost $60, how many tons can be bought for $78, at the same rate? 23. If a person agree to do a job of work in 30 days, what part of it ought he to do in 16 days? Ans. 11 24. A father divided a piece of land among his three sons; to the first he gave 124 acres, to the second of the whole, and to the third as much as to the other two; how many acres did the third have? Ans. 49 acres. 25. If of 6 bushels of wheat cost $42, how much will # of 1 bushel cost? 26. A man engaging in trade lost of his money invested, after which he gained $740, when he had $3500; how much did he lose? Ans. $1840. 27. A cistern being full of water sprung a leak, and before it could be stopped, § of the water ran out, but as much ran in at the same time; what part of the cistern was emptied? Ans. |