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CASE III.

142. When the fractions are of different denominations.

RULE.

Reduce the fractions to the same denomination. Then reduce all the fractions to a common denominator, and then add them as in Case 1.

Ex. 1. Add of a £. to of a shilling.

Then, Or, the

of a £3 of 24 of a shilling:

+40 +75=2&s=&s=14s 2d.
of a shilling might have been reduced

to the fraction of a £. thus,

of of a £.4 of a £.

Then, +=+=4} of a £.: which being reduced by 136, gives 14s 2d.

2. Add of a yard to of an inch.

Ans. 14s 2d.

Ans. yds. or 14 inches. 3. Add of a week, † of a day, and together.

of an hour

Ans. 2da. 14hr.

4. Add of a cwt., 82b. and 3oz. together.

Ans. 2gr. 1776. 15oz.

5. Add 1 miles, furlongs, and 30 rods together. Ans. 1m. 3fur. 18rd.

NOTE. 143. The value of each of the fractions may be found separately, and their several values then added.

Ex. 6. Add of a year, of a week, and † of a day together.

[blocks in formation]

of a mile

Ans. 1540yd. 2ft. 9in.

7. Add of a yard, of a foot, and

together.

8. Add 3 of a cwt., 4lb. 13oz. and of a cwt. 61b.

together.

Ans. lcwt. 1qr. 27lb. 13oz.

QUESTIONS.

§ 138. What is addition of integer numbers? What is addition of fractions? Can fractions be added while they have different units? If they have the same unit can they be ad. ded if the parts are unlike? What then is necessary before fractions can be added?

§ 139. How do you add when the fractions have a common "denominator?

§ 140. How do you add when the fractions have different denominators?

§ 141. When there are mixed numbers, how may the addition be performed?

§ 142. How do you add when the fractions are of different denominations?

§ 143. In what other way may they be added?

SUBTRACTION OF VULGAR FRACTIONS.

§ 144. Subtraction of Vulgar Fractions teaches how to take a less fraction from a greater.

CASE I.

145. When the fractions are of the same denomination, and have a common denominator.

RULE.

Subtract the less numerator from the greater and place the difference over the common denominator.

Ex. 1. What is the difference between ? and ? 3-2=1: hence is the answer.

2. What is the difference between 1 and ? Ans. 385.

CASE II.

146. When the fractions are of the same denomination, but have different denominators.

RULE.

Reduce mixed numbers to improper fractions, compound fractions to simple ones, and all the fractions to a common denominator: then subtract them as in Case I.

Ex. 1. What is the difference between and ? Here, ---- answer.

2. What is the difference between 121 of 1 and 2? Ans.

3. What is the difference between 2 of a £., and of a £.?

CASE III.

Ans. £2 6s.

147. When the fractions are of different denominations.

RULE

Reduce the fractions to the same denomination : then reduce them to a common denominator, after which subtract as in Case I.

Ex. 1. What is the difference between of a £., and of a shilling?

Then,

of a shilling

of of a £. 3848 of a £=9s 8d.

60

2. What is the difference between of a day

of a second?

Ans. 11hr. 59m. 590

3. What is the difference between f of a rod and of an inch?

Ans. 10ft. 114in.

4. From 1 of a lb. troy weight, take of an ounce. Ans. 1lb. 8oz. 16pwt. 16gr.

5. What is the difference between of a hogshead, and of a quart?

Ans. 16gal. 2qt. 1pt. 3gi.

QUESTIONS.

6 144. What does Subtraction of Vulgar Fractions teach? 145. How do you subtract when the fractions are of the same denomination, and have a common denominator? § 146. How do you subtract when the fractions have dif. ferent denominators?

§ 147. What do you do when the fractions are of different denominations?

MULTIPLICATION OF VULGAR FRACTIONS.

§ 148. When we multiply by a whole number we repeat the multiplicand as many times as there are units in the multiplier, § 28.

Therefore, if it be required to multiply a fraction, as, by 4, it is necessary to increase the fraction as many times as there are units in 4; which, as shown in § 114 and § 117, may be done either by multiplying the numerator, or dividing the denominator by 4.

CASE I.

§ 149. To multiply a fraction by a whole number.

RULE.

Multiply the numerator, or divide the denominator by the whole number.

Ex. 1. Multiply by 6.

×6==2. "Or fr×6=ors=f=2 Ans.

2. Multiply by 12.

3. Multiply 17 by 7.

Ans. 3.

Ans. 6.

150. When we multiply by a fraction it is required to repeat the multiplicand as many times as there are units in the fraction.

For example, to multiply 8 by is to repeat 8, times; that is, to take 2 of 8, which is 2.

Hence, when the multiplier is less than 1 we do not take the whole of the multiplicand, but only such a part of it as the fraction is of unity. For example, if the multiplier be one half of unity, the product will be half the multiplicand: if the multiplier be of unity, the product will be one third of the multiplicand, &c.

Ex. 1. Let it be required to multiply by . Here is to be taken times. But =5×4; hence is to be multiplied by 5, and then of the product taken. But 5, and ÷7=15=4} § 116. Hence, for multiplying one fraction by another we have the following

RULE.

§ 151. Reduce all the mixed numbers to improper fractions, and all compound fractions to simple ones: then multiply the numerators together for a numerator, and the denominators together for a denomi

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Ans. 1.

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