GREATEST COMMON DIVISOR OF FRACTIONS. 167. To find the greatest common divisor of two or more fractions. Ex. 1. What the greatest common divisor of , &, and 1? OPERATION. = †, 3, 13 — 28, 18, 28. Greatest common divisor of the numerators Least common denominator of the fractions Having reduced the fractions to equivalent fractions with the least common denominator (Art. 141), we find the greatest common divisor of the numerators 20, 30, and 36, to be 2. (Art. 124.) Now, since 20, 30, and 36 are forty-fifths, their greatest common divisor is not 2, a whole number, but so many forty-fifths. Therefore we write the 2 over the common denominator 45, and have as the answer. RULE. - Reduce the fractions, if necessary, to the least common denominator. Then find the greatest common divisor of the numerators, which, written over the least common denominator, will give the greatest common divisor required. EXAMPLES FOR PRACTICE. 2. What is the greatest common divisor of, g, and 11? Ans. 3. What is the greatest common divisor of 13, 4, 1, and 1§? Ans. 23. 4. What is the greatest common divisor of 18, 21, 4, and 51? Ans. 8. 5. There is a three-sided lot, of which one side is 166ft., another side 1564ft., and the third side 2084ft. What must be the length of the longest rails that can be used in fencing it, allowing the end of each rail to lap by the other ft., and all the panels to be of equal length? Ans. 1011ft. LEAST COMMON MULTIPLE OF FRACTIONS. 168. To find the least common multiple of fractions. Ex. 1. What is the least common multiple of, 1, and 54? Ans. 10. = 167. The rule for finding the greatest common divisor of fractions? Why, in the operation, was the divisor 2 written over the denominator 45 ? OPERATION. 1, 11, 51 = 4, 32, 21. Least common multiple of the numerators 21 com least = 2 Having reduced the fractions to their lowest terms, we find the least common multiple of the numerators, 1, 3, and 21, to be 21. (Art. 128.) Now, since the 1, 3, and 21 are, from the nature of a fraction, dividends of which their respective denominators, 6, 2, and 4, are the divisors (Art. 132), the least common multiple of the fractions is not 21, a whole number, but so many fractional parts of the greatest common divisor of the denominators. This common divisor we find to be 2, which, written as the denominator of the 21, gives 21 = 101 as the least number that can be exactly divided by the given fractions. ― RULE. - Reduce the fractions, if necessary, to their lowest terms. Then find the least common multiple of the numerators, which, written over the greatest common divisor of the denominators, will give the least common multiple required. NOTE. Another method is to reduce the fractions, if necessary, to their least common denominator, and then finding the least common multiple of the numerators, and writing that over the least common denominator. EXAMPLES FOR PRACTICE. 2. What is the least common multiple of 18, 4, and 15? Ans. 42. 3. What is the least number that can be exactly divided by 1, 2, 5, 61, and ? Ans. 95. 4. What is the smallest sum of money for which I could purchase a number of bushels of oats, at $ a bushel; a number of bushels of corn, at $ a bushel; a number of bushels of rye, at $1 a bushel; or a number of bushels of wheat, at $24 a bushel; and how many bushels of each could I purchase for that sum? Ans. $221; 72 bushels of oats; 36 bushels of corn; 15 bushels of rye; 10 bushels of wheat. 5. There is an island 10 miles in circuit, around which A can travel in of a day, and B in 7 of a day. Supposing them each to start together from the same point to travel around it in the same direction, how long must they travel before coming together again at the place of departure, and how many miles will each have traveled? Ans. 5 days; A 70 miles; B 60 miles. 168. The rule for finding the least common multiple of fractions? Why is not the least common multiple of the numerators the least common multiple of the fractions? MISCELLANEOUS EXERCISES. 1. What are the contents of a field 76 rods in length, and 18 rods in breadth ? Ans. 8A. 3R. 30 p. 2. What are the contents of 10 boxes which are 7 feet long, 12 feet wide, and 14 feet in height? 3. From of 200 square feet. Ans. 169 cubic feet. an acre of land there were sold 20 poles and What quantity remained? 1 Ans. 22075ft. 4. What cost of an acre at $ 1.75 per square rod? Ans. $236.923. 5. What cost of a ton at $15 per cwt.? Ans. $49.7313. 6. What is the continued product of the following numbers: 14, 11, 5, and 104? Ans. 9184. of it; what is 7. From 7 of a cwt. of sugar there was sold the value of the remainder at $0.123 per pound? Ans. $3.182. 8. What cost 19 barrels of flour at $ 73 per barrel ? Ans. $143. 9. Bought a piece of land that was 475 rods in length, and 29 in breadth; and from this land there were sold to Abijah Atwood 5 square rods, and to Hazen Webster a piece that was 5 rods square; how much remains unsold? Ans. 136683 square rods. 10. From a quarter of beef weighing 1753lb. I gave John Snow of it; of the remainder I sold to John Cloon. What is the value of the remainder at 82 cents per pound? Ans. $2.0413. Ans. 11. Alexander Green bought of John Fortune a box of sugar containing 475lb. for $30. He sold of it at 8 cents per pound, and of the remainder at 10 cents per pound. What is the value of what still remains at 12 cents per pound, and what does Green make on his bargain? Value of what remains, $13.193. Green's bargain, $ 16.973. of an acre at $14 per acre? { 12. What cost Ans. $2. 13. Multiply of of 14 by 1 of 17 of 18. Ans. To 14. What are the contents of a board 112 inches long, and 41 Ans. 491 square inches. inches wide? 15. Mary Brown had $17.871: half of this sum was given to the missionary society, and g of the rema.nder she gave to the Bible society; what sum has she left? Ans. $3.571. 16. What number shall be taken from 122, and the remainder multiplied by 10%, that the product shall be 50? Ans. 8. 17. What number must be multiplied by 7%, that the product may be 20? Ans. 24. 18. What are the contents of a box 8 feet long, 311 feet wide, and 2 feet high? Ans. 68 feet. 19. On of my field I plant corn; on of the remainder I sow wheat; potatoes are planted on of what still remains; and I have left two small pieces, one of which is 3 rods square, and the other contains 3 square rods. How large is my field? Ans. 1A. OR. 29p. REDUCTION OF FRACTIONS OF DENOMINATE NUMBERS. 169. To reduce from a higher to a lower denomination. Ex 1. Reduce ZTʊ of a pound to the fraction of a farthing. Ans. far. OPERATION. 20 X 12 240 2160 2160 = ; OPERATION BY CANCELLATION. 1 × 20 × 12 × 4 Since 20s. make a pound, there must be 20 times as many far. shillings as pounds; we there fore multiply by 20, and obtains.; and since 12d. 20 make a shilling, there will be 12 times as many pence as shillings; hence we multiply by 12, and obtain d. 216 Again, since 4far. make a penny, there will be 4 times as many farthings as pence; we therefore multiply 24 by 4, and obtain 60 40 2160 far. = #far., Ans. RULE. Multiply the given fraction by the same numbers that would be employed in reduction of whole numbers to the lower denomination required. 169. The rule for reducing a fraction of a higher denomination to the fraction of a lower? Explain the operations? Does this process differ in principle from reduction of whole denominate numbers? EXAMPLES FOR PRACTICE. 2. Reducer of a pound to the fraction of a farthing. Ans. Ans. 3. What part of a penny is of a shilling? 4. What part of a grain is go of a pound Troy? 4 = ; 36 5. What part of an ounce is 172 of a cwt.? Ans. 7. What part of a square foot is 5080 of an acre? 8. What part of a second is gʊʊ of a day? 9. What part of a peck is of a bushel? 10. What part of a pound is goʊ of a cwt.? 170. To reduce from a lower to a higher denomination, Ex. 1. Reduce of a farthing to the fraction of a pound. Ans. 160 - 4 4 ; 36 X 12 432 OPERATION. = 4 OPERATION BY CANCELLATION. £. Since 4far. make a penny, there 4 1 will be as many pence as farthings; therefore we divide the 9 X 4 X 12 X 20 2160 by 4, and obtain d. And since 12d. make a shilling, there will be as many shillings as pence; hence we divide by 12, and obtains. Again, since 20s. make a pound, there will be as many pounds as shillings; therefore we divide by 20, and obtain 40 £ = 2160£ for the answer. 4 Ans.. Ans.. £ = Ans. . Ans . Ans.. Ans. 1 2160 £. RULE. Divide the given fraction by the same numbers that would be employed in reduction of whole numbers to the higher denomination required. EXAMPLES FOR PRACTICE. 2. Reduce of a grain Troy to the fraction of a pound. Ans. Toogo 170. Do you multiply or divide to reduce a fraction of a lower denomina tion to the fraction of a higher? What is the rule ? |