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REDUCTION OF FRACTIONS OF CONCRETE NUMBERS.

112. Our attention has hitherto been confined to fractions considered generally, without regard to the particular species of their units, and it remains to apply what has been said to such concrete quantities as constitute the principal subjects of practical computation.

113. To reduce a given quantity to the fraction of another given quantity. In order to render possible the division of one concrete quantity by another concrete quantity, it is necessary that both should be in the same denomination. Hence the following rule:

RULE XL.

Bring the proposed quantities into the same (not necessarily the lowest) denomination, and take the result of the quantity that is to be reduced for the numerator, and the other result for the denominator of the required fraction.

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Here 168. 5d. 197 pence, and £1240 pence. Therefore, the 197 pence in 168. 5d. are to be divided by the 240 pence; or, 1 is the required fraction of £1. The reason for this is as follows:-Since I contains 240 pence, and 16s. 5d. contains 197 pence, if the pound be divided into 240 equal parts and 197 of them be taken, these 197 parts will be represented by 16s. 5d.; but the fraction represents that the pound has been divided into 240 equal parts and 197 of them taken;

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therefore, 20 = 4 × =, the required fraction of £1.

Ex. 3. Reduce half-a-crown to the fraction of half-a-guinea.

Reducing them to pence we have the required fraction; but reducing to sixpences we have the same fraction in lower terms = ÁÅ.

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NOTE.- expresses what is called the ratio of 2s. 6d. to 1os. 6d.

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therefore, 41=4×?==71 half-a-crown.

Ex. 6. What fraction of £3 Reducing both the quantities and £2 158. 3d. contains 221.

10s. 6d. is 2 158. 3d.?

to threepenny pieces, we find that £3 10s. 6d. contains 282, Hence, if £3 10s. 6d. be divided into 282 parts, and 221 of these parts are taken, the result will be £2 158. 3d. Hence, £2 158. 3d. is the value of of £3 108. 6d. The answer then will be 23.

Ex. 7. Express 3 weeks 4 days 6 hours as the fraction of a year of 365 days.

3 weeks 4 days 6 hours 25 days days,

and the year consists of 365 days: =1461 days;

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I. 6s. 8d.; 12s. 6d.; 18s. 4d.; 178. 11d.; 198. 10d., and £1 138. 113d. Iff. of £1.

2.

38. 4d. of 16s. 8d.; 7}d. of 28.; £4 os. 6d. of £7 148.; d. of 6s. 8d.; £1 38. 4d. of

£9 6s. 8d.; 6s. 84d. of 1d.; 68. 74d. of 78. 9d.; 38. 4d. of half-a-guinea; 16s. of £200.

3.

2 cwt. I qrs. 16 lbs. of a ton; 14h 15m of 3 days; 1 ton of 3 cwt. 3 qrs. 21 lbs.; 20 ft. 7 in. of 1 mile; 17 lbs. of 1 qr. 143 lbs. ; 23 tons 11 cwt. I qr. 1 lb. 1 oz. of 25 tons 6 cwt. 3 qrs. 10 lbs.; 439 mls. 7 fur. 211 yds. of 445 mls. 21 yds.; 2 wks. 5 dys. 7 hrs. 27 mins. of a day.

4.

126 yds. 2 ft. 6 in. of a mile; 6 cubic ft. Ico cubic in. of 1 cubic yd.; 6 ft. 3 in. of 13 ft. 8 in.; 14 yd. of 1 in.; 1 ft. in. of sq. yd.

Reduce to lowest terms the following fractions

217 mls. 5 fur. 18 pls. 2 yds. 2 ft. 1 in.
506 mls. 2 fur. 23 pls. 1 yd. 2 ft. 7 in.

1011 tons 18 cwt. 3 qrs. 10 lbs. 13 oz.

£19 16s. 7d.

5.

£20 168.8.

75 tons 7 cwt. o qrs. 27 lbs. 1 oz.

6.

208 sq. miles 181 acres o ch. 93 yds. 4 ft.
767 acres 9 ch. 279 yds. 4 ft.

;

258 dys. 9 hrs. 19 mins. 57 secs.
446 dys. 7 hrs. 23 mins. 33 secs.
in. is of a league.

7. What fraction of 5 mls. 5 fur. 7 poles 11

114. The value of a fraction of a concrete number is easily determined in terms of the same or lower denominations. For since of a pound may be obtained either by dividing one pound into 3 equal parts, and taking 2 of these parts, or by taking the concrete unit twice, viz., 2 pounds, and dividing it into 3 equal parts; therefore we have

of £1 of 2 of 408. = 138. 4d., and of a mile of 3 miles, &c.

=

Hence

=

To find the value of the fraction of a concrete quantity in terms of the same or lower denominator.

RULE XLI.

Multiply the given concrete quantity of the numerator of the fraction, and divide the product by the denominator, reducing the remainder (if any) always to the lower denominations, as in compound division.

N

EXAMPLES.

Find the value of of a pound; that is, how many shillings and pence are in of a pound.

Ex. I.

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Compound fractions must be reduced to simple ones before the application of this rule; but in the case of mixed numbers it is best to multiply separately and add the result to that obtained by the rule for the fractional parts.

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Since, we have (§ 106) to multiply by . We may do this as in Ex. 1, or (which is more convenient) by first dividing by 2, which gives of the quantity, and then dividing this half by 2, which gives of it, and adding the two results together, we shall have of it. tons cwt. qrs. lbs.

tons cwt.
I 13

qrs.
O

lbs. 15

13 O 15

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1. £; £*; £1%; & of £27 148. od.; § of £31 108. od.; 19 of £5 18. 11d.; #4 of 22 of 8 148. 2d. ÷ 8.

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4. of 1 ton 15 cwts.; of 3 tons; of 4 tons 16 cwt.; 1s of 3 tons 15 cwt.

5. I cwt. 2 qrs. 13 lbs. X 34; 9 of 1 ton 14 cwt. 3 qrs.; 1889888888 of 8288181 square miles.

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20 cwts. 3 qr8. 7 lbs. × 5; £38 28. 6d. ÷ 318; 2 × 31 of 97 of £721 178. 6d.

6.

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of £270; 2 × 31 of 94 of £721 178. 6d; # of of of a ton.

133

39

8. 5 times ÷ 38 × 1 × 84 times 613 of £972; (94 of 119 ÷ 3) × 41 times 140 tons 12 cwt. I qr.

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115. A fraction of a concrete number may be reduced to the fraction of another concrete number of a higher or lower denomination, by means of the principle employed in the reduction of integers from one denomination to another (see § 54, Rules XI and XII, pages 39 and 41).

EXAMPLE.

Ex. Let the fraction be 4, where the unit is one pound; then if it be required to find the fraction when the unit is one farthing; then since any integer number of pounds is reduced to farthings by multiplying the number of pounds by 20, and the product by 12, and this last by 4, so any fraction of a pound is reduced to the fraction of a farthing by multiplying the numerator by 20 X 12 X 4 or 960, that is, in order to retain the same absolute value, we must have 20 X 12 X 4 times as great a fraction as the original one: hence

6=1×3° x 12 x 1= 1920 farthings,

and the value of the unit in the latter fraction being th part of that in the former, the same absolute value is retained by taking 960 times as many parts in the latter as in the former.

Again, reversing the operation we have

1920 far. = 1920 × 1 × 1 × = 1929 = £4,

the divisors 4, 12, 20 being inverted, according to the rule laid down in Division of Fractions (Rule XXXIX, page 83).

The preceding examples enable us to lay down the following rules for reducing a fraction of any given quantity to a fraction of another given quantity.

RULE XLII.

1o. If the given fraction has to be reduced from a higher to a lower denomination.

Multiply the given fraction by those numbers which in whole numbers would reduce the denomination in which the fraction stands to the denomination required.

20. But if it is required to reduce a given fraction of a lower to a higher denomination.

Divide the given fraction by those numbers which in whole numbers would reduce the denomination in which the fraction stands to the denomination required.

Or we may reduce a number, or fraction, of one name to a fraction of another denominator by the following

RULE XLIII.

Express by (Rule XL) the first quantity as a fraction of the second, and the fraction required will then be found by reducing the resulting compound fraction to a simple one.

Ex. I.

EXAMPLES.

Reduce of a pound to the fraction of a penny.

Here we have a given fraction of a higher denomination, viz., the denomination of pounds, and it has to be expressed as an equivalent fraction of a lower denomination, viz., pence. Hence we multiply the given fraction by 20 and 12.

Thus, x 20 × 12 of a penny = 480, or 18o of a penny.

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The divisor is inverted, and the process is the same as in Multiplication of Fractions.

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Ex. 3. Reduce of an hour to the fraction of a day.

hour=4=1×4=1 of a day.

The divisor is inverted, and we proceed as in Multiplication of Fractions.

Or (by Rule XL), 1h of 1d.' . 7 = 7 of 14 = 1 of 1a.

=

Ex. 4. Reduce 1 to the fraction of rom.

== X

17 15

== 6 of 10m.

5

1h 6 times 10m, therefore multiply the fraction by f, the product is an equivalent fraction of 10.

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