Multiply and Divide Fractions Calculator

Multiply the numerators together and the denominators together. Nothing else:

$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

No common denominator is needed. That rule belongs to addition and subtraction only, and dragging it into a multiplication just wastes time.

Why it works: $\frac{2}{3} \times \frac{4}{5}$ means "two thirds of four fifths". Cutting each fifth into three parts makes fifteenths, and you keep two of every three — so $8$ of the $15$.

A whole number is a fraction with denominator $1$: $6 = \frac{6}{1}$, so $6 \times \frac{2}{3} = \frac{12}{3} = 4$. Mixed numbers must be converted first: $2\frac{1}{4} = \frac{9}{4}$. Multiplying the whole parts and the fraction parts separately is wrong.

Notice that multiplying by a fraction less than $1$ makes things smaller, which is the opposite of what multiplication usually does with whole numbers. $\frac{2}{3}$ of $\frac{4}{5}$ has to be under $\frac{4}{5}$, and $\frac{8}{15} \approx 0.533$ is indeed under $0.8$ — a fast way to check any answer.

Any numerator can be cancelled against any denominator, across either fraction, before you multiply. It is exactly the simplifying you would do at the end, done early while the numbers are still small.

In $\frac{3}{8} \times \frac{4}{9}$:

• $3$ and $9$ share a factor of $3$: they become $1$ and $3$.
• $4$ and $8$ share a factor of $4$: they become $1$ and $2$.

$\frac{3}{8} \times \frac{4}{9} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$

The rule that makes this legal: you may only cancel top against bottom. Cancelling two numerators against each other, or two denominators, changes the value. Skipping the cancelling is never wrong — $\frac{12}{72}$ reduces to $\frac{1}{6}$ too — it just leaves bigger numbers to tidy up.

To divide by a fraction, multiply by its reciprocal — the same fraction upside down:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Dividing by $\frac{2}{5}$ asks "how many two-fifths fit in?", and since five halves of $\frac{2}{5}$ make a whole, dividing by $\frac{2}{5}$ is the same as multiplying by $\frac{5}{2}$.

The mistakes to watch for:

• Only the second fraction flips. $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}$, never $\frac{4}{3} \times \frac{5}{2}$.
• Flip before you cancel. Cancelling across a division sign is meaningless.
• Dividing can make the answer bigger. $\frac{3}{4} \div \frac{2}{5} = \frac{15}{8}$, which is more than you started with. That is correct whenever the divisor is under $1$.

To divide by a whole number, put it over $1$ first and then flip: $\frac{3}{4} \div 5 = \frac{3}{4} \times \frac{1}{5} = \frac{3}{20}$. In practice, dividing a fraction by a whole number just multiplies the denominator by it.