Two fractions are equivalent when they show the same amount — they just slice the whole into a different number of pieces. Half a bar is half a bar whether you call it one piece out of two or two pieces out of four:

Every one of these covers exactly the same length. If you already know what a fraction is — a count of equal parts of a whole — then equivalent fractions are simply different ways to name one and the same amount.

The rule

To make an equivalent fraction, multiply the top and the bottom by the same number. You cut every piece into the same number of smaller pieces, so you end up with more slices but exactly the same shaded length:

It works in reverse too: divide the top and bottom by the same number and you glue the small pieces back into bigger ones. The key is that the top and bottom always change together by the same factor — that is what keeps the amount the same.

See it built

Two bars of the same length. The top bar shows \frac{m}{n}; the bottom bar splits every part in two to show \frac{2m}{2n}. Watch the shaded length stay exactly the same. Step through it — each reload shows a fresh example.

See it explained

Sal Khan shows how the same amount of pizza can be cut into different numbers of slices.