Equivalent fractions

Sometimes two fractions that look very different are actually the same amount. $\frac{1}{2}$ is the same size as $\frac{2}{4}$. They are equivalent fractions , different names for the same friend.

What is this?

Take a chapati. Cut it into 2 equal halves. Take 1 half , that is $\frac{1}{2}$.

Now take another chapati of the same size. Cut it into 4 equal pieces. Take 2 pieces , that is $\frac{2}{4}$.

If you put the two amounts side by side, they will cover exactly the same area. The chapati did not become more by cutting into smaller pieces; we just changed the way we counted.

We can make equivalent fractions easily by multiplying the top and bottom by the same number.

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

We can also go the other way , divide top and bottom by the same number. This gives the simplest name for the fraction.

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

When the top and bottom have no common factor except 1, the fraction is in its simplest form.

Worked examples

Example 1. Find two fractions equivalent to $\frac{2}{3}$.

• Multiply top and bottom by 2 → $\frac{4}{6}$.
• Multiply top and bottom by 3 → $\frac{6}{9}$.
• So $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$.

Example 2. Simplify $\frac{10}{15}$.

• Both 10 and 15 are divisible by 5.
• $\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$.
• Simplest form: $\frac{2}{3}$.

Example 3. Are $\frac{3}{4}$ and $\frac{9}{12}$ equivalent?

• Multiply 3 and 4 by 3 → $\frac{9}{12}$. Yes, they are equivalent.

Try it

1. Write two fractions equivalent to $\frac{1}{4}$.
2. Simplify $\frac{8}{12}$.
3. Is $\frac{5}{10}$ equal to $\frac{1}{2}$?
4. Fill the blank: $\frac{2}{5} = \frac{?}{10}$.
5. Find the simplest form of $\frac{6}{18}$.

Hands-on

Paper-strip match. Cut three equal paper strips. Fold the first into 2 equal parts, the second into 4 parts, the third into 8 parts. Shade $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{4}{8}$. Place them on top of each other , do the shaded parts match?

Fraction chart. Draw a chart showing $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}$ as bars of the same total length. Spot two pairs that look the same width when shaded equally , those are equivalent fractions.