Multiplication and division

Multiply fractions

Fractions are multiplied by multiplying numerators with numerators and denominators with denominators. In general:

$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Hint

You multiply a fraction with a rational number by multiplying the numerator with the number and maintaining the denominator.

$\frac{a}{b}\cdot c=\frac{a\cdot c}{b}$

Hint

Sometimes, it is faster to shorten even before calculating. Simply, shorten crosswise. Example:$\frac{7}{10}\cdot\frac{15}{14}=\frac{1\cdot\rlap{\backslash}\color{green}{7}}{2\cdot\rlap{\backslash}\color{blue}{5}}\cdot\frac{3\cdot\rlap{\backslash}\color{blue}{5}}{2\cdot\rlap{\backslash}\color{green}{7}}$ $=\frac34$

If the other option is easier, you can continue to calculate as in the example below.

Examples

Calculate and shorten if necessary

$\frac{7}{10}\cdot\frac{15}{14}=\frac{7\cdot15}{10\cdot14}=\frac{\rlap{\backslash}7\cdot3\cdot\rlap{\backslash}5}{2\cdot\rlap{\backslash}5\cdot2\cdot\rlap{\backslash}7}=\frac34$

$\frac{1}{16}\cdot4=\frac{4}{16}=\frac{\rlap{\backslash}4}{4\cdot\rlap{\backslash}4}=\frac{1}{4}$

$\frac{a+b}{x}\cdot\frac{a-b}{x}=\frac{(a+b)\cdot(a-b)}{x\cdot x}$ $=\frac{a^2-b^2}{x^2}$ (Binomial formulas)

Divide fractions

You divide a fraction by multiplying the reciprocal of the fraction. In general:

$\frac{a}{b}:\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{a\cdot d}{b\cdot c}$

Hint

You divide a fraction with a rational number (except 0) by keeping the numerator and multiplying the denominator with the number.

$\frac{a}{b}:c=\frac{a}{b\cdot c}$

Examples

Calculate and shorten if necessary

$\frac{5}{16}:\frac{1}{4}=\frac{5}{16}\cdot\frac41=\frac{20}{16}$ $=\frac{5\cdot\rlap{\backslash}4}{4\cdot\rlap{\backslash}4}=\frac54$

$\frac{2}{5}:3=\frac{2}{5\cdot3}=\frac{2}{15}$

$\frac{3a^2}{2b^2}:\frac{2a}{3b}=\frac{3a^2}{2b^2}\cdot\frac{3b}{2a}$ $=\frac{9a^2b}{4ab^2}=\frac{9a}{4b}$