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CASE III. § 142. When the fractions are of different denom. inations.

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 f of a £. to of a shilling.

of a £.=f of Y=y of a shilling: Then, 40+1=4+1}=**s=\'s=14s 2d. Or, the f of a shilling might have been reduced to the fraction of a £. thus,

of j;=rty of a £.= of a £. Then, f +=+*+*=*] of a £.: which being reduced by $ 136, gives 14s 2d.

Ans. 14s 2d. 2. Add f of a yard to of an inch.

Ans. 741yds. or 1474inches. 3. Add } of a week, 1 of a day, and į of an hour together.

Ans. 2da. 144hr. 4. Add of a cwt., 8416. and 37602. together.

Ans. 2qr. 1726. 1370z. 5. Add 17 miles, fo furlongs, and 30 rods together.

Ans. lm. 3fur. 18rd. NOTE. $ 143. The value of each of the fractions may be found separately, and their several values then added.

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

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of a year = of 302 days=219 days į of a week=1 of 7 days

2 days, 8 hours of a day =

3 hours.

Ans. 221da. 11hr. 7. Add f of a yard, of a foot, and of a mile together.

Ans. 1540yd. 2ft. Iin. 8. Add of a cwt., y lb. 13oz. and į of a cwt. 61b. together.

Ans. Icwt. lqr. 2716. 13oz.

QUESTIONS.

$138. What is addition of integer numbers ? What is ad. dition 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 ad. dition be performed ?

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

§ 143. In what other way may they bo 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 pla

sho 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 it's and 3%} ?

Ans.

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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 onės, and all the fractions to a common denominator : then subtract them as in Case I. Ex. 1. What is the difference between 1 and } ?

Here, 441=-==1 answer. 2. What is the difference between 121 of f and 2?

Ans. th:

3. What is the difference between 21 of a £., and its of a £. ?

Ans. £2 6s. CASE III. $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 4 of a £., and of a shilling?

of a shilling = of brit of a £. Then, t=28=1# of a £=9s 8d. 2. What is the difference between f of a day or of a second ?

Ans. 11hr. 59m. 59

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

Ans. 10ft. 114in. 4. From 17 of a lb. troy weight, take of an ounce.

Añs. 126. 8oz. 16pwt. 16gr. 5. What is the difference between is of a hogshead, and of a quart?

Ans. 16gal. 2gt. Ipt. 39% gi.

QUESTIONS. $ 144. What does Subtraction of Vulgar Fractions teach?

g 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.

Ø 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 f, by 4, it is necessary to increase the fraction as many times as there are units in 4; which, as shown in Ø 114 and $ 117, may be done either by multiply. ing 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=11=2. Or to Xb='=f=2 Ans.

2. Multiply it by 12.

Ans. 315.

3. Multiply 47 by 7.

Ans. 64.

Ø 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, 1 times; that is, to take 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 4 by 4. Here

is to be taken times. But #=5X+; hence is to be multiplied by 5, and then of the product taken. But * x 5=Y, and Y-7573=} 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 denominator. Ex. 1. Multiply 1 of if by 84.

of ?=; and 8f=Y:
Hence, xv=i.
2. Multiply 57 by :

Ans. H=

Ans. Na

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