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REDUCTION OF FRACTIONS TO A GIVEN DENOMINATOR.

29. Suppose we wish to change to an equivalent fraction having 6 for its denominator.

It is obvious, that if we first multiply by 6, and then divide the product by 6, its value will not be altered. By this means we find that =

X6_24
6

6'

Hence, to reduce a fraction to an equivalent fraction hav. ng 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 de. nominator.

In this example, we first multiply by 8, which gives 24 ;

24

8

therefore, placing 8 under we get for the fraction required.

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

nominator.

Ans.

33

12'

3. Reduce to an equivalent fraction having 7 for its

Ans. 97%.

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to fractions having 12 for

10,

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

Ans.

to fractions having 100 for 11, 91, 71, and 517

15

100°

REDUCTION OF DENOMINATE FRACTIONS.

30. A denominate fraction is a fraction of a number of a particular denomination. Thus of a foot, of a yard, of 18 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 one pound, it follows that of a pound is the same as 20 times of a shilling, and this

1

is also the same as 12 times 20 times of a penny; which, in turn, is 4 times 12 times 20 times of a pound sterling is equivalent to

of a farthing.

0

of a farthing. Hence,

of 20 of 12 of

Again, let us reduce 3 of a farthing to an equivalent fraction of a pound sterling. In this case we must use the reciprocals of,,, we thus find that of a farthing is equivalent to ofofof of a pound sterling.

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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, 161=33 feet, 40 rods, and 8 furlongs; .. our compound fraction is of of ‚ of 1, 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 denominator we get

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1

X X 8123340 2

16. Therefore, 3 of an inch is equivalent to o

of a mile.

2. Reduce TT 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 60 of 0, which, multiplied by the given fraction, becomes 11320 of 24 of of; this becomes, after canceling like factors, 45 of a second.

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

Ans.

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

Ans.

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

6. Reduce of a mile to the fraction of a foot.

Ans. 3360

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Ans. 1760 feet.

7. Reduce of of of a yard to the fraction of a mile. Ans. Jo

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.

10. Reduce of 11. Reduce of 12. Reduce avoirdupois.

of 4 yards to the fraction of an inch. of a farthing to the fraction of a shilling. of an ounce to the fraction of a pound

32. To find what fractional part one quantity is of an another of the same kind, but of different denominations.

Suppose we wish to know what part of 1 yard 2 feet 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=1.

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 4s. 1d. is 2s. 6d. ?

In this example the quantities when reduced become £3. 4s. 1d.=769d.; and 2s. 6d.=30d.; therefore is the fractional part which 2s. 6d. is of £3. 4s. 1d.

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

Ans.

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

Ans. sis

Ans.

4. What part of $700 is $5.30 ?
5. What fractional part of 2 hogsheads is 3 pints?

6. What part of $3 is 24 cents?

5

Ans•;36•

Ans. To

7. What part of 10 shillings 8 pence is 3 shillings 1 penny?

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8. What part of 100 acres is 63 acres 2 roods and 7 rods

of land?

Ans. 1887.

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