Math Core

You have $\dfrac{3}{4}$ of a cup of milk and a recipe uses $\dfrac{1}{3}$ of a cup. How much is left? To take away pieces of different sizes, you use the same trick as adding: rename both fractions so the pieces match.

Same idea, opposite direction

Subtracting fractions works just like adding them. First make the denominators the same, then subtract the numerators.

For $\dfrac{3}{4} - \dfrac{1}{3}$ , the multiples of 4 and 3 first meet at 12, so use twelfths:

\frac{3}{4} = \frac{9}{12} \qquad\qquad \frac{1}{3} = \frac{4}{12}

\frac{9}{12} - \frac{4}{12} = \frac{5}{12}

There is $\dfrac{5}{12}$ of a cup left.

Start at 9/12 and jump back 4/12. You land on 5/12.

Subtracting fractions with unlike denominators

Find a common denominator.
Rename each fraction with that denominator.
Subtract the numerators. Keep the denominator.
Simplify if you like.

Simplifying your answer

Sometimes the answer can be written with smaller numbers. If the numerator and denominator share a factor, divide both by it. For example, $\dfrac{6}{10}$ becomes $\dfrac{3}{5}$ when you divide both by 2. Both fractions name the same amount, so either one is correct.

Worked example: One denominator is a multiple of the other

Find $\dfrac{5}{6} - \dfrac{1}{2}$ .

6 is a multiple of 2, so rename the half in sixths: $\dfrac{1}{2} = \dfrac{3}{6}$ .

\frac{5}{6} - \frac{3}{6} = \frac{2}{6}

2 and 6 share a factor of 2, so $\dfrac{2}{6} = \dfrac{1}{3}$ .

Worked example: Rename both fractions

Find $\dfrac{4}{5} - \dfrac{2}{3}$ .

Use fifteenths.

\frac{4}{5} = \frac{12}{15} \qquad\qquad \frac{2}{3} = \frac{10}{15}

\frac{12}{15} - \frac{10}{15} = \frac{2}{15}

Worked example: Check with a benchmark

Find $\dfrac{7}{8} - \dfrac{1}{6}$ , then check that the answer is reasonable.

Multiples of 8: 8, 16, 24. Multiples of 6: 6, 12, 18, 24. Use twenty-fourths.

\frac{7}{8} = \frac{21}{24} \qquad\qquad \frac{1}{6} = \frac{4}{24}

\frac{21}{24} - \frac{4}{24} = \frac{17}{24}

Check: $\dfrac{7}{8}$ is close to 1 and $\dfrac{1}{6}$ is small, so the answer should be a little less than 1 but more than $\dfrac{1}{2}$ . Since $\dfrac{12}{24} = \dfrac{1}{2}$ , the answer $\dfrac{17}{24}$ is between $\dfrac{1}{2}$ and 1. It makes sense.

Common mistake

Don't subtract the tops and the bottoms separately. $\dfrac{3}{4} - \dfrac{1}{3}$ is not $\dfrac{2}{1}$ , which equals 2! Taking some away from $\dfrac{3}{4}$ has to leave less than $\dfrac{3}{4}$ . Rename first, then subtract only the numerators.

Tip

Check a subtraction by adding. If $\dfrac{3}{4} - \dfrac{1}{3} = \dfrac{5}{12}$ , then $\dfrac{5}{12} + \dfrac{1}{3}$ should equal $\dfrac{3}{4}$ . It does: $\dfrac{5}{12} + \dfrac{4}{12} = \dfrac{9}{12} = \dfrac{3}{4}$ .

Practice

Practice 1

Find $\dfrac{1}{2} - \dfrac{1}{4}$ .

Type a number.

Practice 2

Find $\dfrac{4}{5} - \dfrac{1}{2}$ .

Type a number.

Practice 3

Find $\dfrac{5}{6} - \dfrac{1}{3}$ .

Type a number.

Practice 4

Find $\dfrac{1}{3} - \dfrac{1}{4}$ .

Type a number.

Practice 5

Which is the best estimate for $\dfrac{9}{10} - \dfrac{1}{8}$ ?

Practice 6

Find $\dfrac{5}{8} - \dfrac{1}{3}$ .

Type a number.

Practice 7

Find the missing number: $\dfrac{1}{2} + ? = \dfrac{2}{3}$ .

Type a number.

Practice 8

Find $\dfrac{9}{10} - \dfrac{8}{15}$ .

Type a number.