Multiplying Fractions

Multiplying fractions is the friendliest fraction operation there is: multiply straight across. Top times top, bottom times bottom. $\dfrac{2}{5} \cdot \dfrac{3}{7} = \dfrac{2 \cdot 3}{5 \cdot 7} = \dfrac{6}{35}$ . No common denominator, no rewriting — just two small multiplications.

In symbols, $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d}$ . The only finishing touch is to simplify the answer if it isn't already in simplest form.

Multiply straight across

Multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. That's the entire rule.

Students who just learned to add fractions often reach for a common denominator here — you do not need one. Common denominators are an addition-and-subtraction tool. Multiplication works straight across every time.

Whole numbers are fractions too

To multiply a whole number by a fraction, write the whole number over $1$ first. $5 \cdot \dfrac{2}{3}$ becomes $\dfrac{5}{1} \cdot \dfrac{2}{3} = \dfrac{10}{3}$ . Once everything is a fraction, the straight-across rule takes over.

This also matches the meaning: $5 \cdot \dfrac{2}{3}$ is five copies of two-thirds, which is ten thirds.

Simplify — before or after

You can multiply first and simplify the answer at the end: $\dfrac{3}{4} \cdot \dfrac{2}{9} = \dfrac{6}{36} = \dfrac{1}{6}$ .

Or you can cancel a common factor between any numerator and any denominator before multiplying — the $3$ cancels into the $9$ , and the $2$ cancels into the $4$ , leaving $\dfrac{1}{2} \cdot \dfrac{1}{3} = \dfrac{1}{6}$ . Canceling first keeps the numbers small, which means fewer arithmetic slips. Both routes land on the same answer.

Worked examples

Example 1: straight across

Multiply $\dfrac{2}{3} \cdot \dfrac{4}{5}$ .

Multiply the numerators

2 \cdot 4 = 8

Multiply the denominators

3 \cdot 5 = 15

8

and

15

share no common factor — already simplest form

\dfrac{8}{15}

Answer: $\dfrac{8}{15}$

Example 2: simplify the product

Multiply $\dfrac{3}{4} \cdot \dfrac{2}{9}$ .

Multiply straight across

\dfrac{3 \cdot 2}{4 \cdot 9} = \dfrac{6}{36}

Divide top and bottom by the GCF,

6

\dfrac{6}{36} = \dfrac{1}{6}

Answer: $\dfrac{1}{6}$

Example 3: a whole number times a fraction

Multiply $5 \cdot \dfrac{2}{3}$ .

Write the whole number over

1

\dfrac{5}{1} \cdot \dfrac{2}{3}

Multiply straight across

\dfrac{5 \cdot 2}{1 \cdot 3} = \dfrac{10}{3}

Answer: $\dfrac{10}{3}$

Try one yourself

Common questions

Do I need a common denominator to multiply fractions?

No. Common denominators are only for adding and subtracting. To multiply, go straight across: numerator times numerator, denominator times denominator.

Can I simplify before I multiply?

Yes — cancel any common factor between a numerator and a denominator first. In $\dfrac{3}{4} \cdot \dfrac{2}{9}$ , the $3$ and $9$ share a factor of $3$ , and the $2$ and $4$ share a factor of $2$ . Canceling early keeps the numbers small; the answer comes out the same.

Why is my answer smaller than the fractions I started with?

Multiplying by a fraction less than $1$ takes a part of something. $\dfrac{1}{2} \cdot \dfrac{1}{3}$ means half of a third, which is $\dfrac{1}{6}$ — smaller than either piece. That's normal, not a mistake.

What do I do with a mixed number like

2\dfrac{1}{2}

Convert it to an improper fraction first: $2\dfrac{1}{2} = \dfrac{5}{2}$ . Then multiply straight across as usual.

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Multiplying Fractions

Multiply straight across

Whole numbers are fractions too

Simplify — before or after

Worked examples

Example 1: straight across

Example 2: simplify the product

Example 3: a whole number times a fraction

Try one yourself

Common questions

Keep learning

Simplifying Fractions

Adding & Subtracting Fractions

Dividing Fractions