§ 109. 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.

EXAMPLES.

1. What is the difference between and ?

Here, ---3-3-1 answer.

Q. How do you subtract fractions which have the same unit but different denominators? What is the difference between one-half and one-third?

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

Ans.

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

4. From of 6, take 19 of

3

Ans. £

S.

5. From of of 7, take 3 of 5. 6. From 371, take 35 of 1.

Ans.

Ans. 363

CASE III.

§ 110. 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

EXAMPLES.

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

of a shilling

of

of a £. Then, -638-3-38 of a £=9s 8d.

30

60

Q. How do you subtract fractions which are of different denomi

nations?

2. What is the difference between of a day and of a second? Ans. 11hr. 59m. 59 sec. 3. What is the difference between 8 of a rod and of an inch? Ans. 10 ft. 11 in.

4. From 13 of a lb. troy weight, take of an ounce.

and of a quart?

Ans.

4

5. What is the difference between of a hogshead, Ans. 16gal. 2qt. 1pt. 375gi. of a shilling? Ans. s. d. Ans. 11pwt. 3gr.

6. From of a £ take 7. From oz. take pwt.

8. From 43cwt. take 4lb.

Ans. 4cwt. 1qr. 15lb. 1oz. 93dr.

MULTIPLICATION OF FRACTIONS.

$111. John gave much must he give for 2 apples? For 5? For 6? For 7? For 8 Charles gave of a cent for a peach? must he give for 2 peaches? For 3? For 6?

of a cent for an apple. How For 3 apples? For 4? For 9?

EXAMPLES.

1. Multiply the fraction by 4. When it is required to multiply a fraction by a whole number, it is required to increase the fraction as many times as there are units in the multiplier, which may be done. by multiplying the numerator

How much For 4 For 5?

(see § 80), or by dividing the denominator (see § 83).

CASE I.

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

RULE.

Multiply the numerator, or divide the denominator by the

whole number.

Q. How do you multiply a fraction by a whole number?

§ 113. NOTE. 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 3 is to repeat 8, 2 times; that is, to take 3 of 8, which is 6.

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. Hence, to multiply by a proper fraction does not imply increase, as in the multiplication of whole numbers.

Q. What is required when we multiply by a fraction? What is the product of 8 multiplied by one-half? By one-fourth? By oneeighth? By three-halves? By six-halves? What is the product of 9 multiplied by one-half? By one-third? By one-sixth? By one-ninth? When the multiplier is less than 1, how much of the multiplicand is taken? Does the multiplication by a proper fraction imply increase?

CASE II.

§ 114. To multiply one fraction by another.

1. Multiply by 5.

4

EXAMPLES.

In this example is to be taken times. That is, 3 is

first to be multiplied by 5 and the

product divided by 7, a result which is obtained by multiplying the numerators and denominators together.

Hence, we have the following

RULE.

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.

Q. What is the product of one-sixth by one-seventh ? Of threefourths by one-half? Of six-ninths by three-fifths? Give the general rule for the multiplication of fractions.

2. Multiply of by 8. We first reduce the compound fraction to the simple one, and then the mixed number to the

numerators and denominators together.

equivalent fraction 25; af

126

ter which, we multiply the

3. Multiply 51 by .

4. Multiply 12 by 2 of 9.

Ans. 810.

5. Multiply of 3 of 1 by 151.

Ans.

18'

6. Multiply by 3 of 9.

Ans.

7. Required the product of 6 by 3 of 5.

8. Required the product of 3 of 3 by 5 of 32. 9. Required the product of 32 by 411. 10. Required the product of 5, 3, 4 of 3 and 4.

Ans.

11. Required the product of 4, of and 18.

Ans. 9140

12. Required the product of 14, 5, 4 of 9 and 63.

Ans.

§ 115. NOTE. In multiplying by a mixed number, we may first multiply by the integer, then multiply by the fraction, and then add the two products together. This is the best method when the numerator of the fraction is 1.

10. Reduce of a tun to the fraction of a gill.

CASE III.

Ans. 40320 gill.

§ 100. To find the value of a fraction in integers of a less denomination.

RULE.

I. Reduce the numerator to the next lower denomination, and then divide the result by the denominator.

II. If there be a remainder, reduce it to the denomination still less, and divide again by the denominator. Proceed in the same way to the lowest denomination. The several quotients, being connected together, will form the equivalent denominate number.

Q. How much is one-half of a £? One-third of a shilling? Onehalf of a penny? How much is one-half of a lb. Avoirdupois? Onefourth of a ton? One-fourth of a cwt.? One-half of a quarter? Onefourth of a quarter? One-seventh of a quarter? One-fourteenth of a quarter? One-twenty-eighth of a quarter? How do you find the value of a fraction in terms of integers of a less denomination?

6. What is the value of 1⁄2 of a hogshead?

Ans. s d.

Ans. 52gal. 2gt.