Simplifying and Equivalent Fractions

Imagine cutting a pizza in half. You have 1 out of 2 slices — that’s one half. Now imagine a new pizza, cut into 6 equal slices, and you take 3. You’ve still got half the pizza. One half and three sixths are the same amount.

We can take the original fraction, one half, and triple the number of pieces and the total — and we get three sixths. Still half the pizza. So we can multiply the top and bottom of a fraction by the same number, and the value stays the same.

And similarly, we can divide both by the same number to get a simpler version of the same fraction.

Take six fifteenths. What number goes into both 6 and 15? Both 6 and 15 divide by 3. Six divided by 3 is 2, fifteen divided by 3 is 5. So six fifteenths simplifies to two fifths.

Let’s simplify eight twelfths. Eight and twelve are both even, so we can divide by 2 — that gives us four sixths. They’re still both even, so let’s divide by 2 again — and we get two thirds. Nothing divides into both 2 and 3, so we’re fully simplified.

You might have spotted that the highest common factor of 8 and 12 is 4. So dividing both by 4 gives us two thirds in one step.

Sometimes we need to find a missing number in an equivalent fraction. Three quarters equals what over twenty? Look at the bottom — 4 times 5 is 20. So we multiply the top by 5 as well. 3 times 5 is 15. The answer is fifteen twentieths.

Are four sixths and ten fifteenths equivalent? Let’s simplify both. Four sixths — divide by 2 — two thirds. Ten fifteenths — divide by 5 — also two thirds. They simplify to the same fraction, so yes — they’re equivalent.

What fraction of this shape is shaded? Write your answer in its simplest form. 6 out of 9 squares are shaded, so that’s six ninths. Now let’s simplify — both 6 and 9 divide by 3. That gives us two thirds.