How to Multiply, Divide, Add, and Subtract Fractions

A fraction represents a numerator divided by a nonzero denominator. The four operations follow two different patterns: multiplication and division work directly with the fraction parts, while addition and subtraction require a common denominator. The examples below show how to apply each rule and simplify your answer.

Multiplying fractions

To multiply two fractions, multiply the numerators together and the denominators together:

$$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}, \qquad b,d\ne 0.$$

Then reduce the result if the numerator and denominator have a common factor. You can also cancel common factors before multiplying, which often keeps the numbers smaller.

Example: multiply and simplify

Calculate $\frac{3}{4}\cdot\frac{10}{9}$.

First cancel common factors across the numerators and denominators: $3$ and $9$ share a factor of $3$, and $10$ and $4$ share a factor of $2$.

$$\frac{3}{4}\cdot\frac{10}{9} =\frac{1}{2}\cdot\frac{5}{3} =\frac{5}{6}.$$

This is the same as multiplying first to get $30/36$ and then simplifying to $5/6$.

Dividing fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of $c/d$ is $d/c$, so

$$\frac{a}{b}\div\frac{c}{d} =\frac{a}{b}\cdot\frac{d}{c}, \qquad b,c,d\ne 0.$$

The divisor must not be zero. A fraction $c/d$ equals zero when $c=0$, and division by zero is undefined.

Example: divide and simplify

Calculate $\frac{5}{6}\div\frac{10}{9}$.

Keep the first fraction, change division to multiplication, and flip the second fraction:

$$\frac{5}{6}\div\frac{10}{9} =\frac{5}{6}\cdot\frac{9}{10} =\frac{45}{60} =\frac{3}{4}.$$

You can cancel before multiplying: $5$ and $10$ reduce to $1$ and $2$, while $9$ and $6$ reduce to $3$ and $2$. The product is then $3/4$.

Adding and subtracting fractions

You can combine numerators only when the fractions have the same denominator. With a common denominator, add or subtract the numerators and keep the denominator:

$$\frac{a}{m}+\frac{b}{m}=\frac{a+b}{m}, \qquad \frac{a}{m}-\frac{b}{m}=\frac{a-b}{m}, \qquad m\ne 0.$$

If the denominators differ, rewrite the fractions as equivalent fractions with a common denominator. The least common multiple of the denominators is often the smallest convenient choice, but any common multiple works.

Example: add fractions with different denominators

Calculate $\frac{2}{3}+\frac{1}{4}$. A common denominator is $12$:

$$\frac{2}{3}+\frac{1}{4} =\frac{8}{12}+\frac{3}{12} =\frac{11}{12}.$$

The numerators change when you make equivalent fractions: multiplying the denominator by $4$ means multiplying the numerator by $4$ as well.

Example: subtract fractions with different denominators

Calculate $\frac{5}{6}-\frac{1}{4}$. The least common denominator is $12$:

$$\frac{5}{6}-\frac{1}{4} =\frac{10}{12}-\frac{3}{12} =\frac{7}{12}.$$

If the larger fraction is subtracted from the smaller one, the answer is negative. For example, $\frac{1}{4}-\frac{5}{6}=-\frac{7}{12}$.

Mixed numbers and negative fractions

A mixed number such as $2\frac{1}{3}$ is usually easiest to calculate with after converting it to an improper fraction:

$$2\frac{1}{3}=\frac{2\cdot 3+1}{3}=\frac{7}{3}.$$

Then use the ordinary fraction rules. For instance,

$$2\frac{1}{3}\cdot\frac{3}{7} =\frac{7}{3}\cdot\frac{3}{7}=1.$$

A negative sign applies to the whole fraction: $-\frac{2}{5}=\frac{-2}{5}$. Keep track of signs just as you would with integers: a negative times a positive is negative, and a negative times a negative is positive. For addition and subtraction, preserve the signs when combining the numerators.

Common mistakes

• Adding denominators: In general, $\frac{1}{2}+\frac{1}{3}$ is not $\frac{2}{5}$. Use a common denominator: $\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$.
• Flipping the wrong fraction in division: Keep the first fraction unchanged and take the reciprocal of the second one only.
• Changing only the denominator: To make an equivalent fraction, multiply or divide both numerator and denominator by the same nonzero number.
• Cancelling across addition: You may cancel common factors in a product, but not across terms being added or subtracted. For example, the $2$ in $\frac{2+4}{2}$ cannot be cancelled with just one term; simplify the whole numerator first.
• Forgetting to simplify: A result such as $30/36$ is correct, but its simplified form is $5/6$.

Before finishing, check that your denominator is nonzero and that your answer has the expected sign. For an addition or subtraction problem, a common denominator is essential; for multiplication or division, it is not.