Equivalent fractions

Cut a pizza in half — that's $\frac{1}{2}$ . Now cut each half in half again — you've got 4 quarters, and 2 of them still cover half the pizza. So:

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

We say these two fractions are equivalent. They look different on paper, but they mean exactly the same amount.

What makes fractions equivalent

Two fractions are equivalent when both the numerator and the denominator get multiplied (or divided) by the same number.

Start with $\frac{1}{2}$ . Multiply top and bottom by 2:

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

Multiply by 3:

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

Multiply by 4:

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

All of these are the same one-half. You can make as many as you want.

💡 The rule has one part: what you do to the top, you must also do to the bottom.

Going the other way — simplifying

You can also divide top and bottom by the same number to make a fraction look simpler.

\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}

This is called simplifying (or "reducing") the fraction. The value doesn't change — only the look.

A fraction is in its simplest form when there's no whole number bigger than 1 that divides both top and bottom.

$\frac{3}{4}$ is simplest. You can't find a number that divides both 3 and 4 (apart from 1).
$\frac{4}{8}$ is not simplest. Both 4 and 8 are divisible by 4: $\frac{4}{8} = \frac{1}{2}$ .

Equivalent fractions on a fraction wall

A fraction wall has rows for $\frac{1}{1}$ , $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{4}$ , … Each row is split into that many equal pieces.

Read along the wall, and you can see at a glance:

$\frac{1}{2}$ lines up exactly with $\frac{2}{4}$ , with $\frac{3}{6}$ , with $\frac{4}{8}$ and with $\frac{5}{10}$ .
$\frac{1}{3}$ lines up with $\frac{2}{6}$ and with $\frac{3}{9}$ .
$\frac{2}{3}$ lines up with $\frac{4}{6}$ .
$\frac{3}{4}$ lines up with $\frac{6}{8}$ and $\frac{9}{12}$ .

If you find a fraction wall on paper, keep it handy — it makes equivalent fractions much easier to see.

A worked example

Is $\frac{6}{9}$ equivalent to $\frac{2}{3}$ ?

Find a number that divides 6 → 2 and 9 → 3. That number is 3. Since dividing both top and bottom by 3 turns $\frac{6}{9}$ into $\frac{2}{3}$ , the two fractions are equivalent.

Fill in the missing number: $\frac{3}{5} = \frac{?}{20}$

The bottom went from 5 to 20 — that's multiplying by 4. So multiply the top by 4 too: $3 \times 4 = 12$ . Answer: $\frac{12}{20}$ .

A common trap

Some children try to add the same number to top and bottom:

\frac{1}{2} = ? \frac{1 + 2}{2 + 2} = \frac{3}{4}

But $\frac{3}{4}$ is not the same as $\frac{1}{2}$ — three out of four slices is much more than one out of two. The rule for equivalence is multiply or divide, never add or subtract.

What's next

Comparing fractions — which fraction is bigger
Worked examples — adding, subtracting, mixed numbers
Back to the introduction

Try it out

🟰 Equivalent fractions
🎨 Shade the fraction