Math Core

If you eat 2 slices of a pizza and your friend eats 3 slices, together you ate 5 slices. Adding fractions works the same way when the pieces are the same size: you just count the pieces.

Adding pieces of the same size

Suppose a pan of brownies is cut into 8 equal pieces. You take $\dfrac{3}{8}$ of the pan and your sister takes $\dfrac{2}{8}$ . How much of the pan did you take together?

The pieces are all eighths. 3 eighths plus 2 eighths is 5 eighths:

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

On a number line, start at $\dfrac{3}{8}$ and jump forward 2 eighths:

Start at 3/8. A jump of +2 means 2 eighths. You land on 5/8.

Think of "eighths" as a unit, like "apples." 3 apples plus 2 apples is 5 apples. 3 eighths plus 2 eighths is 5 eighths.

Same denominator: add or subtract the numerators

When two fractions have the same denominator, add or subtract the numerators. Keep the denominator the same.

\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d} \qquad\qquad \frac{a}{d} - \frac{b}{d} = \frac{a - b}{d}

Subtracting pieces of the same size

Subtracting means taking pieces away. If you have $\dfrac{7}{10}$ of a liter of juice and drink $\dfrac{4}{10}$ of a liter, you have

\frac{7}{10} - \frac{4}{10} = \frac{3}{10}

of a liter left. On a number line, jump back:

Start at 7/10. A jump of −4 means back 4 tenths. You land on 3/10.

Common mistake

Never add the denominators. $\dfrac{3}{8} + \dfrac{2}{8}$ is not $\dfrac{5}{16}$ . Sixteenths are smaller pieces than eighths, so $\dfrac{5}{16}$ is less than what you started with. That can't be right when you added more! The pieces stay eighths, so the answer is $\dfrac{5}{8}$ .

Worked example: Adding

Find $\dfrac{2}{6} + \dfrac{3}{6}$ .

Both fractions are sixths. Add the numerators: $2 + 3 = 5$ .

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

Worked example: Subtracting from one whole

A rope is 1 whole meter long. Ana cuts off $\dfrac{3}{5}$ of a meter. How much rope is left?

Write 1 whole as fifths: $1 = \dfrac{5}{5}$ . Then subtract:

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

There is $\dfrac{2}{5}$ of a meter left.

Worked example: A sum greater than 1

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

Add the numerators: $3 + 2 = 5$ , so the sum is $\dfrac{5}{4}$ .

Since $\dfrac{4}{4} = 1$ , you can also say $\dfrac{5}{4} = 1 + \dfrac{1}{4}$ . That's a little more than one whole.

Worked example: A word problem

Jake walked $\dfrac{5}{12}$ of a mile to school and $\dfrac{4}{12}$ of a mile to the park. How much farther did he walk to school than to the park?

"How much farther" means subtract:

\frac{5}{12} - \frac{4}{12} = \frac{1}{12}.

He walked $\dfrac{1}{12}$ of a mile farther to school.

Tip

Check your answer by estimating. If you add two fractions that are each less than $\dfrac{1}{2}$ , the sum must be less than 1. If you subtract, your answer must be smaller than the fraction you started with.

Practice

Practice 1

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

Type your answer like 2/3.

Type a number.

Practice 2

Find $\dfrac{7}{8} - \dfrac{5}{8}$ .

Type your answer like 2/3.

Type a number.

Practice 3

Ben says $\dfrac{2}{6} + \dfrac{1}{6} = \dfrac{3}{12}$ . What mistake did he make?

Practice 4

Lily read $\dfrac{3}{10}$ of a book on Monday and $\dfrac{4}{10}$ of it on Tuesday. What fraction of the book has she read?

Type your answer like 2/3.

Type a number.

Practice 5

A pitcher is full of lemonade. The class drinks $\dfrac{2}{3}$ of it. What fraction of the lemonade is left?

Type your answer like 2/3.

Type a number.

Practice 6

Find $\dfrac{5}{4} + \dfrac{1}{4}$ .

Type your answer like 2/3.

Type a number.

Practice 7

Find $\dfrac{11}{12} - \dfrac{3}{12} - \dfrac{4}{12}$ .

Type your answer like 2/3.

Type a number.

Practice 8

Fill in the missing numerator: $\dfrac{2}{8} + \dfrac{?}{8} = \dfrac{5}{8}$ .

Type a number.