MENUMultiplication of Fractions
Multiplying fractions is the easiest of the four operations, because you don't need a common denominator. Multiply the tops, multiply the bottoms, and simplify. This page explains why that works, how to cancel before multiplying, and how to multiply whole numbers, mixed numbers, negative fractions and fractions with letters.
The rule
Multiply the numerators together and the denominators together. $$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$$
$$\frac{2}{3}\cdot\frac{3}{4}=\frac{2\cdot 3}{3\cdot 4}=\frac{6}{12}=\frac{1}{2}$$Why it works
Multiplying by a fraction means taking that fraction of a number: $\frac{2}{3}\cdot\frac{3}{4}$ is $\frac{2}{3}$ of $\frac{3}{4}$. Take a square and shade $\frac{3}{4}$ of it: 3 columns out of 4. Then take $\frac{2}{3}$ of the shaded part: 2 rows out of 3.
The square is now cut into $3\cdot 4=12$ small parts: the denominators multiplied. The dark part is $2\cdot 3=6$ of them: the numerators multiplied. So $\frac{2}{3}$ of $\frac{3}{4}$ is $\frac{6}{12}=\frac{1}{2}$.
Cancel first to keep the numbers small
Before you multiply, you may divide any numerator and any denominator by a common factor. It's the same as simplifying the answer, just done earlier. In $\frac{4}{9}\cdot\frac{15}{8}$:
- the numerator 4 and the denominator 8 share the factor 4: $4\div 4=1$ and $8\div 4=2$;
- the numerator 15 and the denominator 9 share the factor 3: $15\div 3=5$ and $9\div 3=3$.
Without cancelling you get $\frac{4\cdot 15}{9\cdot 8}=\frac{60}{72}$, which you then have to reduce to $\frac{5}{6}$ anyway.
Cancel a numerator only against a denominator, never two numerators or two denominators with each other.
Whole numbers
Write the whole number as a fraction over 1. Only the numerator gets multiplied:
$$3\cdot\frac{2}{5}=\frac{3}{1}\cdot\frac{2}{5}=\frac{6}{5}=1\frac{1}{5}$$A fraction of a number works the same way: $\frac{2}{3}$ of 12 is $\frac{2}{3}\cdot 12=\frac{24}{3}=8$. Cancelling first is quicker: $12\div 3=4$, then $2\cdot 4=8$.
If the whole number equals the denominator, the denominator cancels completely: $\frac{a}{b}\cdot b=a$, for example $\frac{5}{7}\cdot 7=5$.
Mixed numbers
Change mixed numbers to improper fractions first, then multiply:
$$\begin{aligned}2\frac{1}{2}\cdot 1\frac{1}{3}&=\frac{5}{2}\cdot\frac{4}{3}=\frac{20}{6}\\&=\frac{10}{3}=3\frac{1}{3}\end{aligned}$$Don't multiply the whole parts and the fractions separately. $2\cdot 1+\frac{1}{2}\cdot\frac{1}{3}=2\frac{1}{6}$ is wrong: $2\frac{1}{2}\cdot 1\frac{1}{3}$ is $2\frac{1}{2}$ plus a third of $2\frac{1}{2}$, which is more than 3.
Three or more fractions
Multiply all the numerators and all the denominators. Here the 2s and the 3s cancel:
$$\begin{aligned}\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}&=\frac{1\cdot 2\cdot 3}{2\cdot 3\cdot 4}\\&=\frac{6}{24}=\frac{1}{4}\end{aligned}$$Negative fractions
Multiply as usual and use the sign rules: different signs give a negative product, equal signs a positive one.
$$-\frac{2}{3}\cdot\frac{9}{10}=-\frac{18}{30}=-\frac{3}{5}$$ $$\left(-\frac{1}{2}\right)\cdot\left(-\frac{4}{7}\right)=\frac{4}{14}=\frac{2}{7}$$Powers of a fraction
A power is repeated multiplication, so raise the numerator and the denominator to the power:
$$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$$ $$\left(\frac{2}{3}\right)^2=\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}$$Algebraic fractions
The rule is the same. Factor every numerator and denominator first, cancel the common factors, then multiply what's left.
Example 1: $\frac{3b}{c}\cdot\frac{d}{2m}$
$$\frac{3b}{c}\cdot\frac{d}{2m}=\frac{3bd}{2cm}$$Example 2: $\frac{a}{r}\cdot\frac{h}{a}\cdot\frac{d}{y}$
$a$ is in a numerator and in a denominator, so it cancels:
$$\frac{\cancel{a}}{r}\cdot\frac{h}{\cancel{a}}\cdot\frac{d}{y}=\frac{dh}{ry}$$Example 3: $\frac{x^2-9}{x}\cdot\frac{2x}{x+3}$
Factor $x^2-9=(x-3)(x+3)$, then cancel $x$ and $x+3$:
$$\begin{aligned}&\frac{x^2-9}{x}\cdot\frac{2x}{x+3}\\&=\frac{(x-3)\cancel{(x+3)}}{\cancel{x}}\cdot\frac{2\cancel{x}}{\cancel{x+3}}\\&=2(x-3)\end{aligned}$$for $x\ne 0$ and $x\ne -3$, where the original fractions are not defined.
Example 4: $\frac{a+d}{y}\cdot\frac{my}{ah}$
Here $y$ cancels, but $a$ does not: it is a factor of the denominator $ah$, but only a term of the sum $a+d$.
$$\begin{aligned}\frac{a+d}{\cancel{y}}\cdot\frac{m\cancel{y}}{ah}&=\frac{(a+d)m}{ah}\\&=\frac{am+dm}{ah}\end{aligned}$$To divide fractions you turn the division into a multiplication, see division of fractions.
Common mistakes
- Looking for a common denominator. Multiplication doesn't need one; it only makes the numbers bigger.
- Multiplying mixed numbers part by part: $2\frac{1}{2}\cdot 1\frac{1}{3}\ne 2\frac{1}{6}$. Change them to improper fractions first.
- Multiplying the denominator by a whole number too: $3\cdot\frac{2}{5}=\frac{6}{5}$, not $\frac{6}{15}$.
- Cancelling a term of a sum: in $\frac{a+d}{y}\cdot\frac{my}{ah}$ the $a$ can't be cancelled.
Practice
Try each problem, then open it to check your answer.
1. $\frac{3}{5}\cdot\frac{2}{7}$
$\frac{3\cdot 2}{5\cdot 7}=\frac{6}{35}$
2. $\frac{5}{6}\cdot\frac{3}{10}$
Cancel 5 with 10 and 3 with 6: $\frac{1\cdot 1}{2\cdot 2}=\frac{1}{4}$
3. $4\cdot\frac{3}{8}$
$\frac{12}{8}=\frac{3}{2}=1\frac{1}{2}$
4. $1\frac{1}{2}\cdot 2\frac{2}{3}$
$\frac{3}{2}\cdot\frac{8}{3}=\frac{24}{6}=4$
5. $-\frac{3}{4}\cdot\frac{8}{9}$
$-\frac{24}{36}=-\frac{2}{3}$
6. $\left(\frac{3}{5}\right)^2$
$\frac{3^2}{5^2}=\frac{9}{25}$
7. $\frac{2a}{b}\cdot\frac{b^2}{6a}$
$\frac{2ab^2}{6ab}=\frac{b}{3}$
8. $\frac{x+1}{x}\cdot\frac{x^2}{x^2-1}$
$\frac{(x+1)\,x^2}{x\,(x-1)(x+1)}=\frac{x}{x-1}$
You can check any product of numeric fractions, step by step, with the fraction calculator below.
Fraction Calculators with Steps