Subtracting Fractions

Same shape as adding, with two extra rules:

• Keep track of order. 2/3 − 1/4 is not the same as 1/4 − 2/3.
• The answer can be negative. We accept negatives — write a minus sign in front of the numerator: -1/2, not 1/-2.

The recipe

1. Find the common denominator (use LCM if denominators differ).
2. Scale each fraction to that denominator.
3. Subtract the numerators in order. Keep the denominator.
4. Simplify by dividing top and bottom by their GCF.
5. If the result is negative, the sign goes on the numerator.

Worked examples

Example A — bigger minus smaller, same denom

5/8 - 2/8
= (5 - 2)/8
= 3/8

Example B — needs scaling, positive result

3/4 - 1/6
LCM(4, 6) = 12
3/4 = 9/12
1/6 = 2/12
9/12 - 2/12 = 7/12

Example C — smaller minus bigger, negative result

1/3 - 1/2
LCM(3, 2) = 6
1/3 = 2/6
1/2 = 3/6
2/6 - 3/6 = -1/6

Example D — full simplify after subtraction

7/9 - 1/3
LCM(9, 3) = 9
1/3 = 3/9
7/9 - 3/9 = 4/9    (4 and 9 share no factor; lowest terms)

Example E — result of zero

4/5 - 4/5 = 0/5 = 0

We accept 0/1 or 0 for zero answers.

Sign conventions

The minus sign always lives on the numerator, never on the denominator, never on both.

Try these (answers below)

1. 5/6 - 1/3
2. 1/2 - 5/6
3. 3/4 - 3/8
4. 2/9 - 7/9

1. 1/2
2. -1/3
3. 3/8
4. -5/9