Equivalent Fractions: Same Amount, Different Numbers

Two names for one amount

A fraction has a top number and a bottom number.

The bottom number says how many equal parts the whole is cut into. The top number says how many of those parts you have.

Now look at these two bars. They are the same length.

The shaded parts stop at the same place. So $\tfrac{1}{2}$ and $\tfrac{2}{4}$ are the same amount, written two ways.

Fractions like this are called equivalent fractions. The word equivalent means equal in value.

Why cutting again does not change the amount

Start with half a bar. Now cut every part in half again.

You still have the same amount of shading. Nothing was added. Nothing was taken away. You only made more cuts.

But the numbers change:

• The parts are now twice as small, so there are twice as many. The bottom number doubles.
• Your shaded piece got cut in two, so you now hold two pieces. The top number doubles.

Both numbers doubled. That is the whole rule:

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

Cut again and it keeps working:

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

Every bar is shaded to the halfway mark. Only the number of cuts changed.

The rule

To build an equivalent fraction, multiply the top and the bottom by the same number.

$$\frac{a}{b} = \frac{a \times n}{b \times n}$$

You can also divide the top and the bottom by the same number. That works for the same reason, because it puts the cuts back together again.

Here is why multiplying by the same number is safe. Any number over itself, like $\tfrac{2}{2}$ or $\tfrac{5}{5}$, equals $1$. Multiplying by $1$ never changes an amount.

$$\frac{3}{5} \times \frac{2}{2} = \frac{6}{10}$$

You multiplied by $1$ in a clever costume. The amount stayed put. The numbers changed.

Worked examples

Why cross multiplying works

Take $\tfrac{3}{7}$ and $\tfrac{12}{28}$. Give both the same bottom number by multiplying each one by the other’s bottom number:

$$\frac{3}{7} = \frac{3 \times 28}{7 \times 28} \qquad \frac{12}{28} = \frac{12 \times 7}{28 \times 7}$$

Both bottom numbers are now $7 \times 28$. When two fractions share a bottom number, they are equal only if the top numbers match. So the whole test comes down to comparing $3 \times 28$ with $12 \times 7$.

That is the cross product test. It is not a trick. It is what you get after making the bottom numbers match.

Where you will use this

Equivalent fractions are the tool behind almost everything else with fractions.

Task	What you do
Simplify a fraction	find the equivalent one with the smallest numbers
Compare two fractions	rewrite both with the same bottom number
Add or subtract fractions	rewrite both with the same bottom number, then add the tops

So the work you do here pays off in every lesson that follows.

Practice problems