Simplest Form

Put the pieces back together

This bar is cut into 8 equal pieces, and 6 of them are colored, so $\frac{6}{8}$6/8 of the bar is colored.

Join the pieces in pairs. The bar then has 4 pieces, each twice as big, and the 6 colored pieces make 3 of them. The colored part has not changed, so $\frac{6}{8}=\frac{3}{4}$6/8 = 3/4.

Joining the pieces in pairs halves both numbers: 6 ÷ 2 = 3 and 8 ÷ 2 = 4. Dividing the numerator and the denominator by the same number gives an equivalent fraction with bigger pieces.

Both bars have the same amount colored. The second has half as many pieces, and each one is twice as big.

Simplest form

No number bigger than 1 divides both 3 and 4, so $\frac{3}{4}$3/4 cannot be divided down any further. $\frac{3}{4}$3/4 is $\frac{6}{8}$6/8 in simplest form: the equivalent fraction with the smallest numbers, and so the biggest pieces.

Some fractions are in simplest form already. $\frac{7}{12}$7/12 cannot be simplified, because 7 and 12 have no common factor except 1.

In steps, or in one step

To simplify a fraction, divide the numerator and the denominator by a common factor, and keep going until their only common factor is 1. Take $\frac{8}{12}$8/12. Both numbers are even, so halve them: $\frac{8}{12}=\frac{4}{6}$8/12 = 4/6. Both are still even, so halve again: $\frac{4}{6}=\frac{2}{3}$4/6 = 2/3. 2 and 3 have no common factor but 1, so $\frac{2}{3}$2/3 is in simplest form.

The quickest way is to divide once by the highest common factor (HCF). The HCF of 8 and 12 is 4, and 8 ÷ 4 = 2 and 12 ÷ 4 = 3, so $\frac{8}{12}=\frac{2}{3}$8/12 = 2/3 in a single step.

The bar opens at $\frac{8}{12}$8/12. Take out the cuts by 2 and it shows $\frac{4}{6}$4/6; take them out by 4 and it shows $\frac{2}{3}$2/3. The colored part stays where it is.

Both numbers, all the way

Simplifying changes the name of a fraction, never its amount. The usual mistake is to divide only one of the two numbers. Divide only the numerator of $\frac{8}{12}$8/12 by 4 and you get $\frac{2}{12}$2/12, which is a much smaller part of the bar. Whatever divides the numerator must divide the denominator too.

The other mistake is to stop too early. $\frac{12}{18}$12/18 halves to $\frac{6}{9}$6/9, but 6 and 9 are both in the 3 times table, so divide again: $\frac{6}{9}=\frac{2}{3}$6/9 = 2/3. Dividing once by the HCF of 12 and 18, which is 6, gets there in one step.

$\frac{8}{12}$8/12 and $\frac{2}{3}$2/3 color the same length of the bar. Dividing only the numerator gives $\frac{2}{12}$2/12, which is much shorter.