Subtracting Fractions

Counting down

Work out $\frac{5}{6}-\frac{1}{6}$5/6 − 1/6. Both fractions are in sixths, so the pieces already match. Start with 5 colored sixths and take 1 of them away: 5 − 1 = 4 sixths are left, so $\frac{5}{6}-\frac{1}{6}=\frac{4}{6}$5/6 − 1/6 = 4/6.

Subtract the numerators and keep the denominator. The piece taken away is a sixth, and the pieces left are sixths too, so the denominator stays 6. When a question asks for simplest form, finish the job: $\frac{4}{6}=\frac{2}{3}$4/6 = 2/3.

5 sixths are colored, and 1 sixth is to be taken away.

Start at $\frac{5}{6}$5/6 and step back one sixth: $\frac{5}{6}-\frac{1}{6}=\frac{4}{6}$5/6 − 1/6 = 4/6.

Unlike denominators

$\frac{3}{4}-\frac{1}{2}$3/4 − 1/2 has quarters and a half, pieces of different sizes. Rewrite the half in quarters first: cut the half into 2 pieces, and $\frac{1}{2}=\frac{2}{4}$1/2 = 2/4. Now both fractions are in quarters, so count down: 3 − 2 = 1, and $\frac{3}{4}-\frac{1}{2}=\frac{3}{4}-\frac{2}{4}=\frac{1}{4}$3/4 − 1/2 = 3/4 − 2/4 = 1/4.

When neither denominator is in the other one's times table, rewrite both. For $\frac{5}{6}-\frac{1}{4}$5/6 − 1/4, the lowest common multiple of 6 and 4 is 12, so $\frac{5}{6}=\frac{10}{12}$5/6 = 10/12 and $\frac{1}{4}=\frac{3}{12}$1/4 = 3/12. Then 10 − 3 = 7, and $\frac{5}{6}-\frac{1}{4}=\frac{7}{12}$5/6 − 1/4 = 7/12.

Written in quarters, the half is 2 of the 4 pieces. 3 quarters take away 2 quarters leaves 1 quarter.

Cut every sixth into 2 and every quarter into 3, and both bars are in twelfths. 10 twelfths take away 3 twelfths leaves 7 twelfths.

$\frac{5}{6}$5/6 is 10 twelfths and $\frac{1}{4}$1/4 is 3 twelfths. Take the twelfths away one at a time: after 3 of them, 7 twelfths are left.

Two slips to check for

One slip is to subtract the tops and the bottoms: $\frac{3}{4}-\frac{1}{2}=\frac{2}{2}$3/4 − 1/2 = 2/2. But $\frac{2}{2}$2/2 is a whole bar, more than the $\frac{3}{4}$3/4 you started with, and taking something away cannot leave more. Only the numerators are subtracted, and only once both fractions are in quarters.

The other slip is to take 1 straight off the top, $\frac{3}{4}-\frac{1}{2}=\frac{2}{4}$3/4 − 1/2 = 2/4, without rewriting the half. The 1 in $\frac{1}{2}$1/2 counts one half, not one quarter, and one half is 2 quarters.

Adding back checks a subtraction: $\frac{1}{4}+\frac{2}{4}=\frac{3}{4}$1/4 + 2/4 = 3/4, so $\frac{3}{4}-\frac{2}{4}=\frac{1}{4}$3/4 − 2/4 = 1/4 is right.

Taking away from a whole

To take a fraction away from one whole, write the whole with the same denominator. One whole is 6 sixths, so $1-\frac{5}{6}=\frac{6}{6}-\frac{5}{6}=\frac{1}{6}$1 − 5/6 = 6/6 − 5/6 = 1/6.