Equivalent fractions

Same amount, different name

Fold a strip of paper in half and shade one half. Now unfold it and fold it again so the strip has four equal parts. The shaded paper has not changed, but it now covers two of the four parts. One half and two quarters are the same amount of paper. They are equivalent fractions.

Equivalent means equal in value. The fractions 1/2, 2/4, 3/6, 4/8 and 5/10 all name the same point on a number line and the same amount of a whole. They look different because the whole has been cut into a different number of equal parts, and the numerator has grown to match.

The multiply rule

Here is a quick way to make an equivalent fraction. Choose any whole number greater than zero, then multiply both the numerator and the denominator by it. The value does not change. Start with 1/2 and multiply the top and the bottom by 2: 1 x 2 = 2 on top, and 2 x 2 = 4 on the bottom. You get 2/4.

Why does it work? Multiplying the denominator by 2 cuts every part of the whole in half, so there are twice as many parts. Multiplying the numerator by 2 takes twice as many of those smaller parts. Twice as many parts, each half the size, covers exactly the same amount. The share is unchanged.

The rule also runs backwards. If the numerator and the denominator can both be divided by the same number, do it. Divide 6/8 by 2 on top and bottom and you get 3/4. When no smaller numbers divide both, the fraction is in its simplest form. The simplest form of 4/8 is 1/2.

How to check your work

A useful check is to draw the two fractions as bars or put them on a number line. Equivalent fractions shade the same amount and land on the same point. If one bar covers more, the fractions are not equivalent. You can also multiply across: in 1/2 and 2/4, the cross products are 1 x 4 = 4 and 2 x 2 = 4. Equal cross products mean equivalent fractions. In 1/2 and 2/3 the cross products are 1 x 3 = 3 and 2 x 2 = 4, which are not equal, so the fractions are not equivalent.