Equivalent Fractions

Cut every piece again

Take a bar cut into 2 equal pieces and color 1 of them. Half of the bar is colored: $\frac{1}{2}$1/2.

Now cut every piece in 2. The bar has 4 pieces instead of 2, and the colored half is now 2 of those pieces. Nothing was colored in and nothing was rubbed out, so the same amount of the bar is colored. It has a new name: $\frac{2}{4}$2/4.

The bar opens at $\frac{1}{2}$1/2. Cut every piece in 2, and the colored part is 2 of the 4 pieces: $\frac{1}{2}=\frac{2}{4}$1/2 = 2/4.

One amount, many names

$\frac{1}{2}$1/2 and $\frac{2}{4}$2/4 are equivalent fractions: two names for the same amount. Cut every piece of the half in 3 instead, and the bar has 6 pieces with 3 of them colored, so $\frac{1}{2}=\frac{3}{6}$1/2 = 3/6. Cut every piece in 4, and $\frac{1}{2}=\frac{4}{8}$1/2 = 4/8.

Each cut multiplies both numbers by the same number. Cutting every piece in 3 makes 3 times as many pieces, so the denominator is multiplied by 3. It also makes 3 times as many colored pieces, so the numerator is multiplied by 3 too: 1 × 3 = 3 and 2 × 3 = 6, and $\frac{1}{2}=\frac{3}{6}$1/2 = 3/6.

Every row is the same whole. The shading of $\frac{1}{2}$1/2, $\frac{2}{4}$2/4, $\frac{3}{6}$3/6 and $\frac{4}{8}$4/8 stops at the same line.

Any fraction

The same cut works on every fraction. Start with $\frac{2}{3}$2/3 and cut every third into 4 pieces. There are now 3 × 4 = 12 pieces, and the 2 colored thirds have become 2 × 4 = 8 colored pieces, so $\frac{2}{3}=\frac{8}{12}$2/3 = 8/12.

To make an equivalent fraction, multiply the numerator and the denominator by the same number.

The bar opens at $\frac{2}{3}$2/3. Cut every third into 4 pieces: $\frac{2}{3}=\frac{8}{12}$2/3 = 8/12, and the colored part has not moved.

Both numbers, and only by multiplying

The usual mistake is to change only the denominator. To write $\frac{1}{2}$1/2 in sixths, the denominator 2 is multiplied by 3, so the numerator is multiplied by 3 as well, and $\frac{1}{2}=\frac{3}{6}$1/2 = 3/6. Writing $\frac{1}{6}$1/6 keeps 1 colored piece while the pieces get smaller, so less of the bar is colored.

Adding the same number to both does not work either. Add 1 to the numerator and the denominator of $\frac{1}{2}$1/2 and you get $\frac{2}{3}$2/3, which is more than a half.

$\frac{1}{6}$1/6 falls short of the half line and $\frac{2}{3}$2/3 passes it, so neither one is equivalent to $\frac{1}{2}$1/2.