FRACTIONAL SUBTRACTION.

Rule. Prepare the fractions as for Addition, and find the difference of the numerators.

Observe. If, when the given fractions are mixed with whole numbers, the fraction belonging to the less whole number is greater than the other fraction, an integer of the greater whole number must be taken from it, and reduced into the same sort of parts, which are then to be added to the less fraction.

Also, as in Addition, the values of the given fractions may be used in the subtraction.

The numerator of the difference between two aliquot parts is equal to the difference between the denominators, and the denominator is their product.

FRACTIONAL MULTIPLICATION.

Rule. Multiply the numerators together for the numerator of the product, and the denominators together for the denominator of the product.

Observe 1. To use this rule, mixed numbers and whole numbers should be reduced or expressed as improper fractions, but compound fractions need not be reduced.

Observe 2. The same directions apply to the division or cancelment of the terms of the fractions, as have been given for the reduction of compound fractions.

Observe 3. When the multiplier is a whole number, the calculation is performed by multiplying the numerator by that whole number, but when the multiplier is exactly contained in the denominator, the calculation may be performed by dividing the denominator by the whole number.

FRACTIONAL DIVISION.

Rule. Invert the divisor and use it as a multiplier.*

Observe. The reduction of whole and mixed numbers (as well as the division or cancelment of the numerators and denominators, after the divisor has been inverted) is to be practised as in Fractional Multiplication.

Also, When the divisor is a whole number, it may be used either as a divisor to the numerator, or as a multiplier to the denominator.

When both terms of the fractional divisor will act as exact divisors upon the corresponding terms of the dividend, they may be used as such.

When only one term is an exact divisor, it may be used as a divisor, and the quotient is then to be multiplied by the other term of the divisor.

When both terms have the same denominator, the quotient will be formed of the division of the numerator of the dividend by the numerator of the divisor.

Ex. 1. Divide 1 by .

7 ÷ 8 = 4 × 4 = 14 = 211. Ex. 2. Divide 12 by 7 and by 31. 47, and

Ex. 3. Divide 32 by 4, by 5, and by . 334 = ¥ ÷ ÷ = V × 1

155 X 7

4

For the multiplication and division of integral quantities by fractional numbers,

see pages 52, 53, and 54.

FRACTIONAL PROPORTION.

Rule. Reduce, if requisite, the three terms into simple fractions, and the first and third terms into fractions of the same integer.

Then, for Direct Proportion, multiply the second term by the number of the third, and divide the product by the number of the first, or invert the number of the first term, and multiply the three numbers together.

For Inverse Proportion, invert the number of the third term, and multiply the numbers of the three terms together.

EXAMPLE 1.

If 2 of a yard cost £ what will 12 of an ell of 5 quarters of a yard cost?

2 yd. yd. 1 ell of yd. yd.

If yd.

5

=

£

ig yd.?

× πT = 2 $ £ s. 11 4 d

When all the terms are not fractionally expressed, it is not particularly necessary to use this form, as in the following Example. EXAMPLE 2.

If 2 yds. of Cloth cost £1 7 6 what will be the cost of 11g yds.?

yds. If 2

£ S. d.

yds. 11?

2) 15 19 81*

11 ) 63 18 9

£5 16 3

Or, we may reduce the first and third terms into the same sort

of parts, and then rejecting the denominators, use the numerators

as whole numbers: thus,

The divisor and dividend are here multiplied by 4, to get rid of the fraction in the divisor.

Or, as 93-22 nds are 4 and 5-22 nds, we may work thus,

Lastly, to use the simple fractional form, we have,

8

£ 1 × 93 X 1 = ¦¦ £ = £ 5 16 3.

EXERCISES IN DIRECT PROPORTION.

Ex. 1. What is the cost of 5 oz. of Gold, when 13 oz. cost

£ 61?

Ex. 2. What is the worth of a Ship, when of

Answer £22 15 32.

Answer £1630 14 5.

Answer £ 4 3 10.

£3821?
Ex. 3. If lb. cost of a shilling, what will
91

cost?

EXERCISES IN INVERSE PROPORTION.

cost

of an cwt.

Ex. 4. If 12 inches broad require 12 inches long to make a foot square, what length will 5 inches broad require?

Ex. 5. If Calico yard wide require 13 yards long, what length will Calicoyard require?

Answer 11 yards.

Ex. 6. If by working 13 hours per day a work requires 174 days, how many days will be required, if only 10 hours per day Answer 23 days.

be employed.