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241. TO REDUCE FRACTIONS TO THEIR LEAST COMMON DENOMINATOR.

ANALYSIS.

1. Reduce,, §, and to their least common denomi

nator.

The common denominator found by the foregoing rule is a common multiple of the denominators of the given fractions, but not always the least common multiple, aud consequently not always the least common denominator. The least common multiple of the denominators, 3, 4, 8 and 12 is 24(238), which may be divided into thirds Fourths, eighths and twelfths; for the new numerators we must therefore take such parts of 24 as are denoted by the given fractions; and this is done by dividing 24 by each of the denominators (2—8, 24—6, 24—3, and 24—2), and multiplying the quotients by the respective numerators, (8×18, 6×3-18, 3×5-15, and 2X11-2), and the new numerators 48, 18, 15 and 22) written over 24, the common denominator, give, 14, and for the new fractions,

18

having the least possible common denominator. Hence,

242. To reduce fractions of different denominators to equïvalent fractions having the least common denominators.

RULE.--Reduce the several fractions to their least terms (235). Find the least common multiple of all the denominators for a common denominator. Divide the common denominator by the denominators of the several fractions, and multiply the quotients by the respective numerators, and the products will be the new numerators required.

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5. Express and

dollar in the least possible

similar parts of a dollar.

Ans. $§ and .

6. Reducer,

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243. REDUCTION OF FRACTIONS. (52)

ANALYSIS.

1. What part of a shilling 24 of a pound?

Pounds are reduced to shillings by multiplying them by 20 (138), and X20 (220), and 88

1. What part of a pound is of a shilling?

To change shillings to pounds, divide them by 20 (138). 20)(= (221), and

(235). of a pound, then, is of a shilling is, then, of a shilling.

DESCENDING,

244. To change fractions of higher into those of a lower denomination.

RULE.-Reduce the numer ator to the lower denomination by Art. 139, and write it over the given denominator.

24 (235).
of a pound.

ASCENDING.

245. To change fractions of a lower into those of a higher denomination.

RULE.-Multiply the denominator by the number which is required to make one of the next higher denomination, and so on (140); and write the last product under the given numerator.

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246. To reduce fractions to integers of a lower denomination, and the reverse.

ANALYSIS.

1. Reduce & of a pound to shillings and pence.

£3×20=60s. and 60s.—7s.; but s.X12 d., and 48d.Gd. Then £7s. 6d. Hence,

247. To reduce fractions to

1. Reduce 7s. 6d. to the fraction of a pound.

7s. 6d. 90d. £120s, 240d.; then 7s. 6d.££. Hence,

248. To reduce integers to

integers of a lower denomina-fractions of a higher denomina

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249.

ADDITION OF FRACTIONS.

ANALYSIS.

1. What is the sum of 3 of a dollar and of a dollar? As both the fractions are 9ths of the same unit, the magnitude of the parts is the same in both-the number of parts, 3 and 4, may therefore be added as whole numbers, and their sum, 7, written over 9, thus, Z, expresses the sum of two given fractions.

2. What is the sum of of a yard and 3 of a yard?

As the parts denoted by the given fractions are not similar, we cannot add them by adding their numerators, 3 and 2, because the answer would be neither nor ; but if we reduce them to a common denominator, becomes, and, (240). Now each fraction denotes parts of the same unit, which are of the same magnitude, namely, 24ths; their numerators, 8 and 9, may therefore be added; and their sum, 17, being written pver 24 we have of a yard for the sum of and of a yard.

250. To add fractional quantities.

RULE.-Prepare them, when necessary, by changing compound fractions to single ones (224), mixed numbers to improper fractions (218), fractions of different integers to those of the same (247,248), and the whole to a common denominator (240); and then the sum of the numeraters written over the common denominator, will be the sum of the fractions required.

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251,

252.

VULGAR FRACTIONS.

251. SUBTRACTION OF FRACTIONS.

ANALYSIS.

1. What is the difference between of a dollar and of a dollar?

103

To evidently expresses 2 tenths more than 3 tenths; then is the differenee.

2. What is the difference between yard?

of a yard and † of a

Here we cannot subtract from, for the same reason that we could not add them (49). We therefore reduce them to a common denominator, (24924), and then the difference of the numerators (9—8—1), written over 24, the common denominator, gives for the difference of the fractions.

RULE. Prepare the fractions as for addition (250), and then the difference of the numerators written over the common denominator will be the difference of the fractions required.

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252. RULE OF THREE IN VULGAR FRACTIONS.

RULE.-Prepare the fractions by reduction, if necessary, and state the question by the general rule (198); invert the first term, and then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the new numerator, written over the new denominator, will be the answer required.

QUESTIONS FOR PRACTICE.

1. If oz. cost £7, what | is gyd. wide, will line 131 will loz. cost? yards of cloth that is 24 yds. wide?

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