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Hence, to reduce a fraction to an equivalent one having a given denominator, we have this

RULE.

Multiply the fraction by the number which is to be the given denominator, (see Rule under Art. 25,) under which place the given denominator, and it will be the fraction required.

EXAMPLES.

1. Reduce to an equivalent fraction having 8 for its denominator.

In this example, we first multiply by 8, which gives

24; therefore, placing 8 under 24, we get

the fraction required.

1

2 3/
8

for

2. Reduce to an equivalent fraction having 12 for its denominator.

3TT Ans.

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5. Reduce, TT, ', and, to fractions having 100

for their common denominator.

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6. Reduce,,,, and, to fractions having 30 for their common denominator.

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30. A denominate fraction is a fraction of a number of a particular denomination. Thus, of a foot, of a yard, of a dollar, and § of a shilling are denominate fractions.

Reduction of denominate fractions is the changing of them from one denomination to another, without altering their values.

31. Suppose we wish to reduce of a pound sterling to an equivalent fraction of a farthing, we proceed as follows: Since there are 20 shillings in 1 pound, it follows that of a pound is the same as 20 times

0

of a shilling, and this is also the same as 12 times 20 times of a penny; which, in turn, is 4 times 12 times 20 times of a farthing. Hence, of a pound sterling is equivalent to 。 of 2o of 12 of † of a farthing.

Again, let us reduce of a farthing to an equivalent fraction of a pound sterling. In this case, we must use

;

the reciprocals of 20, 12, farthing is equivalent to sterling.

we thus find that of a

of

of, of of a pound

Hence, to reduce fractions of one denominate value to equivalent fractions of other denominate values, we have this

RULE.

I. When the given fraction is to be reduced to a higher denomination, multiply it by a compound fraction whose terms are the reciprocals of the successive denominate values, included between the denomination of the given fraction, and the one to which it is to be reduced.

II. When the given fraction is to be reduced to a lower denomination, then multiply it by a compound fraction whose terms have units for their denominators, and for numerators the successive denominate values included between the denomination of the given fraction and the one to which it is to be reduced.

EXAMPLES.

1. Reduce of an inch to the fraction of a mile. In this example, the different denominate values between an inch and a mile are 12 inches, 16232 feet, 40 rods, and 8 furlongs; .. our compound fraction is

of of of; which, multiplied by the given fraction, produces of of of of; canceling the 3 and 2 of the numerators, against a part of the 12 of the denominators, we get,

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Therefore, of an inch is equivalent to T of a

mile.

320

2. Reduce of a solar day to an equivalent fraction of a second.

In this example, the successive denominate values between a solar day and a second, are 24 hours, 60 minutes, and 60 seconds; therefore, our compound fraction is 24 of of; which, multiplied by the given fraction, becomes T of 24 of of; this becomes, after canceling like factors, 45 of a second.

1

3. Reduce 144 of a yard to the fraction of a mile.

Ans. 400.

4. Reduce of a gill to the fraction of a gallon.

Ans. 9.

5. Reduce 33 of a pound to the fraction of a ton.

Ans. 500.

6. Reduce of a mile to feet. Ans. 1760 feet. of of of a yard to the fraction of a Ans. 36960.

7. Reduce

mile.

8. Reduce of of 24 of a gallon to the fraction of a gill.

Ans. 4.

9. Reduce of of a hogshead of wine to the fraction of a gill. Ans. 1792-597 gills. 10. Reduce of of 42 yards to the fraction of an inch. Ans. 243-34 inches. 11. Reduce of of a farthing to the fraction of a shilling. Ans. 1816. 12. Reduce of an ounce to the fraction of a pound avoirdupois. Ans. 44.

32. To find what fractional part one quantity is of another of the same kind, but of different denominations. Suppose we wish to know what part of 1 yard 2 feet

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3 inches is; we reduce 1 yard to inches, which gives 1 yard 36 inches; we also reduce 2 feet 3 inches to inches, which gives 2 feet 3 inches=27 inches. Now it is obvious that 2 feet 3 inches is the same part of one yard that 27 is of 36, which is 7=4.

Hence, we deduce this

RULE.

Reduce the given quantities to the lowest denomination mentioned in either; then divide the number, which is to become the fractional part, by the other number.

EXAMPLES.

1. What part of £3 4 s. 1 d. is 2 s. 6 d.?

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30 d.; there

In this example, the quantities, when reduced, become £3 4 s. 1 d. 769 d.; and 2 s. 6 d. fore, is the fractional part which 2 s. 4 s. 1 d.

6 d. is of £3

2. What part of 3 miles 40 rods is 27 feet 9 inches?

Ans. 12.

3. What part of a day is 17 minutes 4 seconds?

4. What part of $700 is $5.30 ?
5. What fractional part of 2 hogsheads

6. What part of $3 is 2 cents? 7. What part of 10 shillings 8 pence 1 penny?

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8. What part of 100 acres is 63 acres, 2 roods, and

7 rods of land?

Ans. 1987.

160

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