Math Core

You already know that $\dfrac{2}{8} + \dfrac{3}{8} = \dfrac{5}{8}$ , because eighths plus eighths make eighths. But what is $\dfrac{1}{2} + \dfrac{1}{3}$ ? Halves and thirds are different sizes, so you can't just count them together. The trick is to cut them into pieces that are the same size.

Why you need like pieces

Imagine $\dfrac{1}{2}$ of a granola bar and $\dfrac{1}{3}$ of an identical bar. A half is a big piece and a third is a smaller one. Saying "1 piece plus 1 piece is 2 pieces" tells you nothing, because the pieces don't match.

Now cut both bars into sixths. Each half splits into 3 sixths, and each third splits into 2 sixths:

\frac{1}{2} = \frac{3}{6} \qquad\qquad \frac{1}{3} = \frac{2}{6}

Now every piece is a sixth, so you can count: 3 sixths plus 2 sixths is 5 sixths.

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

A jump of 3/6 (one half), then a jump of 2/6 (one third), lands on 5/6.

Finding a common denominator

A common denominator is a number that both denominators divide into evenly. A good way to find one is to list multiples of each denominator and look for the first match.

For fourths and sixths:

Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 6: 6, 12, 18, 24

The smallest match is 12, so twelfths work. (24 would also work, but the numbers get bigger.)

To rename a fraction, multiply the numerator and denominator by the same number. That makes the pieces smaller without changing the amount.

\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \qquad\qquad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}

Adding fractions with unlike denominators

Find a common denominator (a multiple of both denominators).
Rename each fraction as an equivalent fraction with that denominator.
Add the numerators. Keep the denominator.
Simplify, or write an improper fraction as a mixed number, if you like.

Tip

Multiplying the two denominators always gives a common denominator. For $\dfrac{1}{4} + \dfrac{1}{6}$ that is 24. It works, but the smallest common multiple (12) keeps the numbers easier.

Worked example: One denominator is a multiple of the other

Find $\dfrac{1}{4} + \dfrac{3}{8}$ .

8 is already a multiple of 4, so use eighths. Rename only the fourths:

\frac{1}{4} = \frac{2}{8}

Then add:

\frac{2}{8} + \frac{3}{8} = \frac{5}{8}

Worked example: Rename both fractions

Find $\dfrac{2}{3} + \dfrac{1}{5}$ .

Multiples of 3: 3, 6, 9, 12, 15. Multiples of 5: 5, 10, 15. Use fifteenths.

\frac{2}{3} = \frac{10}{15} \qquad\qquad \frac{1}{5} = \frac{3}{15}

\frac{10}{15} + \frac{3}{15} = \frac{13}{15}

Worked example: The sum is more than 1

Find $\dfrac{3}{4} + \dfrac{5}{6}$ .

The common denominator is 12.

\frac{3}{4} = \frac{9}{12} \qquad\qquad \frac{5}{6} = \frac{10}{12}

\frac{9}{12} + \frac{10}{12} = \frac{19}{12}

12 twelfths make 1 whole, and 7 twelfths are left, so $\dfrac{19}{12} = 1\dfrac{7}{12}$ .

Does it make sense? Both fractions are a bit less than 1, so the sum should be a bit less than 2. $1\dfrac{7}{12}$ fits.

Common mistake

Never add the denominators. $\dfrac{1}{2} + \dfrac{1}{3}$ is not $\dfrac{2}{5}$ . That answer is smaller than $\dfrac{1}{2}$ , and adding more can't make a smaller amount! The denominator names the size of the pieces, and that size stays the same when you add.

Practice

Practice 1

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

Type a number.

Practice 2

Fill in the missing numerator: $\dfrac{3}{5} = \dfrac{?}{15}$ .

Type a number.

Practice 3

What is the smallest common denominator for $\dfrac{1}{6}$ and $\dfrac{3}{8}$ ?

Practice 4

Find $\dfrac{1}{5} + \dfrac{1}{2}$ .

Type a number.

Practice 5

Find $\dfrac{2}{3} + \dfrac{1}{4}$ .

Type a number.

Practice 6

Jada says $\dfrac{1}{3} + \dfrac{1}{6} = \dfrac{2}{9}$ . What is her mistake?

Practice 7

Find $\dfrac{3}{4} + \dfrac{3}{8}$ . Write your answer as a fraction or a mixed number.

Type a number.

Practice 8

Find $\dfrac{5}{6} + \dfrac{2}{5}$ . Write your answer as a fraction or a mixed number.

Type a number.