IV. To reduce fractions having different denominators to equiv alent fractions having a common denominator.

It is evident that both terms of the first fraction may be mul

tiplied by df giving

adf
bd

and that this operation does not

bef bdf

change the value of the fraction (Art. 67).

In like manner both terms of the second fraction may be

multiplied by bf, giving ; also, both terms of the fraction

adf bef
bdf bdf

bde

and

bdf

If now we examine the three fractions

we see that they have a common denominator, bdf, and that each numerator has been obtained by multiplying the numerator of the corresponding fraction by the product of all the denominators except its own. Since we may reason in a similar manner upon any fractions whatever, we have the following

RULE.

Multiply each numerator into the p: auct of all the denomina tors except its own, for new numerators, and all the denominators together for a common denominator.

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.

VI. To subtract one fraction from another.

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 numerator of the minuend, and place the difference over the common denominator.

VII. To multiply one fractional quantity by another.

a

C

Let represent any fraction, and any other fraction; and

b

d

let it be required to find their product.

a

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

ас

b

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

but this product is d times too

a

b

great, since we multiplied.

by a quantity d times too great. Hence, to obtain the true

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.