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5|, which, written under the proposed numerator, gives — as the fraction required.
2. Eeduce | to a fraction of equal value having 12 for a denominator.
Operation. Since the proposed denominator, 12,
l 2 >s V* of tne given denominator, 9, we
— X 8 find ^ of the given numerator, 8, for
9 1 Of numerator of the proposed fraction;
I 2 "= Y2~ -V ot" 8 — 10l, which, written over
0 ^ ^ the proposed denominator, gives -^ias the fraction required.
Rule. — Take of both terms of the given fraction such a fractional part as the proposed numerator, or denominator, is of the given numerator, or denominator, and the result will be the required fraction.
3. Change to a fraction whose numerator shall be 34.
4. Change 3f to a fraction whose numerator shall be 9.
5. Reduce 4 to a fraction whose numerator shall be 5.
6. Reduce \% to a fraction having 12 for its denominator.
7. Change f to fifteenths. , Ans. T9V
8. Reduce J to halves.
9. Reduce to thirty-fifths. Ans. -^A
10. J. Holton owns -Jf of a wood-lot, and his brother of the same lot; what fraction whose denominator shall be 12 will express the part each owns? Ans.
A COMMON NUMERATOR.
245i° A Common numerator of two or more fractions is a common multiple of their numerators.
246i° To reduce fractions to a common numerator. Ex. 1. Change f, f, f, and T9,j to other fractions of the same value, having a common numerator. Ans. £|, §§, |g.
36, least common multiple of the numerators, = common nuAj6- of 4 = 48, new denominator. f = jl y of 5 = 45, new denominator. t = 4?
of 7 = 42, new denominator. ® = if ^ of 10 = 40, new denominator, -fa = J|J
We find the least common multiple of all the numerators, which is 36, for the common numerator; and to obtain the several new denominators we take such a part of the given denominators, respectively, as the common numerator, 36, is of each given numerator. Thus, both terms of each fraction being proportionably increased, its value is not changed.
Rule. —-Find the least common multiple of the given numerators for a common numerator. Take, for the new denominator of each fraction, respectively, such a part of its given denominator as the common numerator is of its given numerator.
Note. — Compound fractions, or whole and mixed numbers, must be reduced to simple fractions, and all to their lowest terms, before finding the common numerator.
2. Eeduce §, £, and f to other fractions of equal value having a common numerator. Ans. §, \, }J, fjf, §£.
3. Change 2£, and If to fractions having a common numerator.
4. A can travel round a certain island, which is 50 miles in circumference, in 4T4S days, B in 6f days, and C in 6f days. If they all set out from the same point, and travel round the island the same way, in how many days will they all meet at the point from which they started, and how many times will each have gone round the island?
Ans. They will meet in 320 days; A will have gone round the island 75 times; B, 50 times; and C, 48 times.
GREATEST COMMON DIVISOR OF FRACTIONS.
247i The greatest common divisor of two or more fractions is the greatest number that will divide each of them, and give a whole number for the quotient.
248i To find the greatest common divisor of two or more fractions.
Ex. 1. What is the greatest common divisor of 2. and 5^. Ans.
,h, 2f, 5^ - ^ v» ¥ = H. W, W
Greatest common divisor of the numerators = 4 > °'^8ji^TMj Least common denominator of the fractions = 45 j required.
Having reduced the fractions to equivalent fractions having the least common denominator, we find the greatest common divisor of the numerators 12, 100, and 240 to be 4. Now, since the 12, 100, and 240 represent forly-fifllis, their greatest common divisor is not 4, a whole number, but 4 forty-fifihs; therefore we write the 4 over the least common denominator, 45, and have ^ as the answer.
Rule.—Reduce the fractions, if necessary, to their least common denominator. The greatest common divisor of the numerators, written over the least common denominator, will give the greatest common divisor required.
2. What is the greatest common divisor of f, and H? Ans. ifc.
3. What is the greatest common divisor of 12f, 9£, and 8£?
4. What is the greatest common divisor of f, and £?
5. What is the greatest common divisor of 3J, and 2-^?
6. A farmer has 33J bushels of corn, 67£ bushels of rye, 70J bushels of wheat. He wishes to put this grain, without mixing, into the largest bags, each of which shall contain the same quantity. Required the number of bags and the quantity each will contain.
Ans. The capacity of each bag, 3| bushels; and the number of bags, 51.
7. I have three fields; the first contains 7S-^T acres, the second 88-^ acres, the third 139|a acres. Required the largest-sized house-lots of the same extent into which the three fields can be divided, and also the number of lots.
Ans. Size of each lot, 7T4r acres; number of lots, 41.
LEAST COMMON MULTIPLE OF FRACTIONS.
249. The least common multiple of two or more fractions is the least number that can be divided by each of them, and give a whole number for the quotient.
250. To find the least common multiple of two or more fractions.
Ex. 1. What is the least common multiple of £, T6e, and 2^? Ans- 8i
3 6 9 1 3 3 33
I, TB' ^TB — j' Bi TS"
Least common multiple of the numerators = 33 (Least com1 . — = 84 < '"on multi
Greatest common divisor of denominators = 4 4 J ple required.
Having reduced the fractions to their simplest form, we find the least common multiple of the numerators, 3, 3, and 33, to be 33. Now, since the 3, 3, and 33 are, from the nature of a fraction, dividends, of which their respective denominators, 4, 8, and 16, are the divisors (Art. 216), the least common multiple of the fractions is not 33, a whole number, but so many fractional parts of the greatest common divisor of the denominators. This common divisor we find to be 4, which, written as the denominator of the 33, gives ^ = as the least number that can be exactly divided by the given fractions.
Rule. — Re/luce 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. Or,
Reduce the fractions, if necessary, to their least common denominator. Then find the least common multiple of the numerators, and write it over the least common denominator.
Note. — The least whole number that will contain two or more fractions an exact whole number of times, is the least common multiple of their numerators.
2. What is the least common multiple of \, f, and f?
Ans. 3£ = 24.
3. Find the least number that 3ig, 1\, and 5£ will divide without a remainder. Ans. 15f.
4. What is the least common multiple of %, f, and T9ff?
5. What is the smallest sum of money with which I could purchase a number of sheep at $ 2£ each, a number of calves at $ 4^ each, and a number of yearlings at $ 9$ each? and how many of each could I purchase with this money?
Ans. $ 112£; 50 sheep; 25 calves; 12 yearlings.
6. There is a certain island 80 miles in circumference. A, B, and C agree to travel round it. A can walk 3£ miles in an hour, B 4§ miles, and C 5\ miles. They start from the same point and travel round the same way, and continue their travelling 8 hours a day, until they shall all meet at the point from which they started. In how many days will they all meet, and how far will each have travelled?
Ans. In 17| days; A 480m., B 640m., and C 720m. 7. How. many times the least common multiple of 3|, 4jj, and 5^, is the least whole number that 3£, 4jj, and 54; will exactly divide.
251. A Denominate Fraction is one in which the unit of the fraction is a denomination of a compound number; as, § of a pound, f of a mile, and J of a gallon.
REDUCTION OF DENOMINATE FRACTIONS.
252. Reduction of denominate fractions is the process of changing fractions from the unit of one denomination to that of another, without altering their value.
253. To reduce a denominate fraction from a higher denomination to a lower.
Ex. 1. Reduce of a pound to a fraction of a penny.
Ans. $ d.
X 20 = 20 20 X 12 = 240 , _ 3 , . 640 6408,5 640 640 ~ 8
f) 1 X #0 X I# = 3 , . Since 20s. makeapound.
<J?0 8 there wil1 be 20 times a9
ri^ Q many shillings as pounds,
JWe or^s.; and since 12d.
make a shilling, there will be 12 times as many pence as shillings,
$m = t<i
Rule. — Multiply the r/iven fraction by the name numbers that would be employed in the reduction of whole numbers to the lower denomination required.
2. Reduce yjVff of a pound to the fraction of a farthing.
3. Reduce ^^xs of a pound troy to the fraction of a grain.
4. Reduce J-^-j of a pound, apothecaries' weight, to the fraction of a scruple.
5. Reduce ^fijn of a cwt. to the fraction of an ounce.