V. To add fractions together.

Quantities cannot be added together unless they have the same unit. Hence, the fractions must first be reduced to equivalent ones having the same fractional unit; then the sum of the numerators will designate the number of times this unit is to be taken. We have, therefore, for the addition of frac tions the following

RULE.

Reduce the fractions, if necessary, to a common denominator: then add the numerators together and place their sum over the common denominator.

EXAMPLES.

bx dxf = bdf

the new numerators.

the common denominator.

adf, cbf ebd adfcbfebd
+ =
bdfbdfbdf

the sum.

bdf

Ans. a+b+

C

2abx 3cx2 bc

Reduce the fractional quantities to equivalent ones, having the same fractional unit; the difference of their numerators will express how many times this unit is taken in one fraction more than in the other. Hence the following

RULE.

I. Reduce the fractions to a common denominator.

II. Subtract the numerator of the subtrahend from the numer ́ator of the minuend, and place the difference over the common denominator.

VII. To multiply one fractional quantity by another.

α

с

bc

Let represent any fraction, and any other fraction; and

b

d

let it be required to find their product.

If, in the first place, we multiply by e, the product will

ас

b

be obtained by multiplying the numerator by c, (Art. 65); b'

but this product is d times too great, since we multiplied a by a quantity d times too great. Hence, to obtain the true

b

product we must divide by d, which is effected (Art. 66) by multiplying the denominator by d. We have then,

I. Cancel all factors common to the numerator and denomi

nator.

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

5. It is required to find the product of b+

and

α

VIII. To divide one fraction by another.

a

с

Let represent the first, and the second fraction; then

b

he division may be indicated thus.

d

If now we multiply both numerator and denominator of this

d

complex fraction by which will not change the value of the

ad

fraction (Art. 67), the new numerator will be and the new

bc'

This last result we see might have been obtained by inverting the terms of the divisor and multiplying the dividend by the resulting fraction. Hence, for the division of fractions, we have the following

RULE.

Invert the terms of the divisor and multiply the dividend by the resulting fraction.