Multiplication and Division of Fractions

Find the product of ${\displaystyle \frac{a}{b}}\times {\displaystyle \frac{c}{d}}$.

Let ${\displaystyle \frac{a}{b}}=x$, and ${\displaystyle \frac{c}{d}}=y$.

Then $a=bx$, and $c=dy$.

The product of these two equations is

ac = bdxy. lint Divide by $bd$, acbd = xy. [1] lint But ab × cd = xy. [1] lint Therefore ab × cd = acbd.

find the product of two fractions, therefore,

Find the product of the numerators for the required numerator, and the product of the denominators for the required denominator.

In like manner, $\frac{a}{b}\times \frac{c}{d}\times \frac{e}{f}=\frac{ac}{bd}\times \frac{e}{f}=\frac{ace}{bdf}.$

136. Reciprocals.

If the product of two numbers is equal to $1$, each of the numbers is called the reciprocal of the other.

The reciprocal of ${\displaystyle \frac{a}{b}}$ is ${\displaystyle \frac{b}{a}}$, for ${\displaystyle \frac{b}{a}}\times {\displaystyle \frac{a}{b}}={\displaystyle \frac{ba}{ab}}=1$.

The reciprocal of a fraction, therefore, is the fraction inverted.

Since ${\displaystyle \frac{a}{b}}÷{\displaystyle \frac{a}{b}}=1$, and ${\displaystyle \frac{b}{a}}\times {\displaystyle \frac{a}{b}}=1$, it follows that

To divide by a fraction is the same as to multiply by its reciprocal.

Divide by a Fraction, therefore,

Invert the divisor and multiply.

Every mixed expression should first be reduced to a fraction, and every integral expression should be written as a fraction having $1$ for the denominator. Both terms of each fraction should be expressed in their prime factors, and if a factor is common to a numerator and denominator, it should be cancelled, as the cancelling of a common factor before the multiplication is evidently equivalent to cancelling it after the multiplication.

1. Find the product of ${\displaystyle \frac{3a^{2}b}{2x^{2}y}}\times {\displaystyle \frac{6x{y}^2}{7ab}}\times {\displaystyle \frac{7abc}{9a^{2}b{y}^2}}$. $\frac{3a^{2}b}{2x^{2}y}\times \frac{6x{y}^2}{7ab}\times \frac{7abc}{9a^{2}b{y}^2}=\frac{3\times 6\times 7a^3{b}^{2}cx{y}^2}{2\times 7\times 9a^3{b}^2{x}^2{y}^3}=\frac{c}{xy}.$

2. Find the product of ${\displaystyle \frac{ab-b^2}{a+b}}\times {\displaystyle \frac{ab+b^2}{a^2-b^2}}$. $\frac{ab-b^2}{a+b}\times \frac{ab+b^2}{a^2-b^2}=\frac{b(a-b)}{(a+b)}\times \frac{b(a+b)}{(a-b)(a+b)}=\frac{b^2}{a+b}.$