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GREATEST COMMON DIVISOR OF FRACTIONS.
167. To find the greatest common divisor of two or more fractions.
Ex. 1. What is the greatest common divisor of $, , and 1 ?
$, , 14, 19; 26. Greatest common divisor of the numerators 2 greatest com
mon divisor reLeast common denominator of the fractions
quired. 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 is 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.
and 13? 2. What is the greatest common divisor of , ,
and 18? 3. What is the greatest common divisor of 13, 4, zet,
4. What is the greatest common divisor of 18, 24, 4, and 54 ?
5. There is a three-sided lot, of which one side is 166şft., another side 156fft., and the third side 208fft. 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 fft., and all the panels to be of equal length ?
LEAST, COMMON MULTIPLE OF FRACTIONS. 168. To find the least common multiple of fractions. Ex. 1. What is the least common multiple of 13, 14, and 57?
Ans. = 107.
167. The rule for finding the greatest common divisor of fractions ? Why, in the operation, was the divisor 2 writton over the denominator 45 ?
, 11, 51=1, 3, 44 Least common multiple of the numerators 21 least
mon multiple Greatest common divisor of the denominator 2
required. 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 = 104 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 if ?
Ans. 44. 3. What is the least number that can be exactly divided by I's, 21, 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 $ 16 a bushel; a number of bushels of corn, at $ $ a bushel; a number of bushels of rye, at $ 11 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. $ 22}; 72 bushels of oats; 36. bushels of corn;
15 bushels of
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 Ğ 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. 51 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 mul. tiple of the fractions ?
1. What are the contents of a field 7625 rods in length, and 18 rods in breadth ?
Ans. 8A. 3R. 304p. 2. What are the contents of 10 boxes which are 7 feet long, 1 feet wide, and 10 feet in height?
Ans. 169}} cubic feet. 3. From Yr of an acre of land there were sold 20 poles and 200 square feet. What quantity remained ?
Ans. 22075ft. 4. What cost 1} of an acre at $ 1.75 per square rod ?
Ans. $ 236.92 4. 5. What cost is of a ton at $ 154 per cwt. ?
Ans. $ 49.7313 6. What is the continued product of the following numbers: 149, 114, 54, and 10+?
Ans. 9184. 7. From 7 of a cwt. of sugar there was sold 4 of it; what is the value of the remainder at $ 0.12% per pound?
Ans. $ 3.18%. 8. What cost 194 barrels of flour at $ 7% per barrel ?
Ans. $ 1434 9. Bought a piece of land that was 47 -f rods in length, and 2976 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. 136633 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 84 cents per pound ?
Ans. $ 2.0415 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 122 cents per pound, and what does Green make on his bargain ?
Value of what remains, $ 13.194. Ans.
12. What cost 144 of an acre at $ 144 per acre ? Ans. $ 2. 13. Multiply of 1 of 1 by 1 of 1 of 1
14. What are the contents of a board 114 inches long, and 41 inches wide ?
Ans. 491. square inches.
15. Mary Brown had $ 17.871: half of this sum was given to the missionary society, and 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 123, and the remainder multiplied by 104, that the product shall be 50 ?
Ans. 810. 17. What number must be multiplied by 7), that the product may be 20 ?
Ans. 243 18. What are the contents of a box 8 feet long, 3:11 feet wide, and 211 feet high ?
Ans. 68 143 $ foet. 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
field ? Ans. I A. OR. 29p.
REDUCTION OF FRACTIONS OF DENOMINATE NUMBERS.
169. To reduce from a higher to a lower denomination.
Ex 1. Reduce zToo of a pound to the fraction of a farthing.
OPERATION BY CANCELLATION.
Since 20s. make a pound, 1 X 20 X 12 X 4
there must be 20 times as many
ffar. shillings as pounds ; we there2160
fore multiply zibo by 20, and 9
obtain zidos. ; and since 12d. make a shilling, there will be 12 times as many pence as shillings ; hence we multiply zib, by 12, and obtain . Again, since 4far. make a penny, there will be 4 times as many farthings as pence; we therefore multiply to by 4, and obtain far.
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 prin. ciple from reduction of whole denominate numbers ?
EXAMPLES FOR PRACTICE.
2. Reduce tabo of a pound to the fraction of a farthing.
Ans. 26 3. What part of a penny is 45 of a shilling ? 4. What part of a grain is yety of a pound Troy?
Ans. 4. 5. What part of an ounce is 172 g of a cwt. ? 6. Reduce 1320 of a furlong to the fraction of a foot.
Ans. 1 foot is 58080
of an acre 7. What part of a square
? Ans. *
Ans 2 8. What part of a second is getoo of a day?
Ans. G. 9. What part of a peck is 'ts of a bushel ?
Ans. 10. What part of a pound is zoo 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.
OPERATION BY CANCELLATION.
1 will be į as many pence as far
£. things; therefore we divide the 9 X4 X 12 X 20 2160
by 4, and obtain d. And since 12d. make a shilling, there will be it as many shillings as pence; hence we divided by 12, and obtain 25. Again, since 20s. make a pound, there will be to as many pounds as shillings; therefore we divide fa by 20, and obtain 3840 € 2160 £ for the answer. 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 4 of a grain Troy to the fraction of a pound.
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 ?