Math Core

Cut a sandwich in half and eat one piece. Now cut an identical sandwich into fourths and eat two pieces. You ate the same amount both times! Different fractions can name the same amount.

Same size, different names

Look at these two bars. They are the same size. The top one is cut into halves, and the bottom one is cut into fourths.

Top: 1 half is shaded. Bottom: 2 fourths are shaded. The shaded parts are the same size.

The shaded parts cover exactly the same amount. So one half and two fourths are equal:

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

Definition

Equivalent fractions

Equivalent fractions are fractions that name the same amount of the same whole. They are the same size, even though they have different numbers.

Equivalent fractions on a number line

Line up number lines of the same length. Fractions that land on the same spot are equivalent.

All three dots are at the same point, so $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{4}{8}$ .

Same point, same amount

Two fractions are equivalent when they are at the same point on a number line, or cover the same amount of the same whole.

Here are some equivalent fractions you will use a lot:

Fraction	Equivalent fractions
$\dfrac{1}{2}$	$\dfrac{2}{4}$ , $\dfrac{3}{6}$ , $\dfrac{4}{8}$
$\dfrac{1}{3}$	$\dfrac{2}{6}$
$\dfrac{2}{3}$	$\dfrac{4}{6}$
$\dfrac{1}{4}$	$\dfrac{2}{8}$
$\dfrac{3}{4}$	$\dfrac{6}{8}$

Tip

Notice a pattern: when the pieces are cut twice as small, you need twice as many of them. $\dfrac{1}{3}$ is 1 third, and each third splits into 2 sixths, so $\dfrac{1}{3} = \dfrac{2}{6}$ .

Worked example: Thirds and sixths

Is $\dfrac{2}{3}$ equal to $\dfrac{4}{6}$ ?

Put a thirds line above a sixths line:

Both dots are at the same point. Yes, $\dfrac{2}{3} = \dfrac{4}{6}$ .

Worked example: Find the missing number

Fill in the blank: $\dfrac{3}{4} = \dfrac{?}{8}$ .

Each fourth splits into 2 eighths. So 3 fourths is $3 \times 2 = 6$ eighths. The answer is $\dfrac{6}{8}$ .

Worked example: Pizza check

Omar ate $\dfrac{2}{8}$ of a pizza. Zoe ate $\dfrac{1}{4}$ of the same size pizza. Who ate more?

Cut each fourth of Zoe's pizza in half. Now her pizza has 8 equal slices, and her 1 fourth is 2 of those slices. So $\dfrac{1}{4} = \dfrac{2}{8}$ , and they ate the same amount.

Common mistake

Equivalent fractions only work for the same size whole. Half of a small pizza is not the same amount as half of a giant pizza.

Practice

Practice 1

Which fraction is equivalent to $\dfrac{1}{2}$ ?

Practice 2

Fill in the missing numerator: $\dfrac{1}{3} = \dfrac{?}{6}$ .

Type a number.

Practice 3

Fill in the missing numerator: $\dfrac{1}{2} = \dfrac{?}{8}$ .

Type a number.

Practice 4

These two number lines are the same length. Which fractions are equivalent?

Practice 5

Fill in the missing numerator: $\dfrac{2}{3} = \dfrac{?}{6}$ .

Type a number.

Practice 6

Lily says $\dfrac{3}{4}$ and $\dfrac{3}{8}$ are equivalent because they both have a 3 on top. Is she right?

Practice 7

Max ran $\dfrac{2}{4}$ of a mile. Rosa ran the same distance, but her map measures in eighths of a mile. How many eighths of a mile did Rosa run?

Type a number.