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2n ESO · Fractions Notes 16 Sep 2026

6. Adding and subtracting fractions

The five steps for adding and subtracting fractions with different denominators.

The steps

Procedure

1. Simplify the fractions.

2. Work out the least common multiple of the denominators.

3. Scale every fraction up so that $D = \operatorname{lcm}$.

4. Add or subtract the numerators.

5. Simplify the result.

If the fractions already share a denominator, steps 1, 2 and 3 are not needed: go straight to step 4.

In step 4 the denominator stays put: only the numerators are operated on.

Example

$$\dfrac{3}{8} - \dfrac{5}{12} + \dfrac{5}{6} - \dfrac{28}{24}$$

1. Simplify what we can: $\dfrac{28}{24} = \dfrac{7}{6}$.

$$\dfrac{3}{8} - \dfrac{5}{12} + \dfrac{5}{6} - \dfrac{7}{6}$$

2. Least common multiple of the denominators: $\operatorname{lcm}(8, 12, 6) = 24$.

3. Scale every fraction up to denominator $24$.

$$\dfrac{9}{24} - \dfrac{10}{24} + \dfrac{20}{24} - \dfrac{28}{24}$$

4. Operate on the numerators: $9 - 10 + 20 - 28 = -9$.

$$\dfrac{-9}{24}$$

5. Simplify by dividing by $3$.

$$\dfrac{-9}{24} = -\dfrac{3}{8}$$

Here simplifying first did not change the least common multiple, but it usually keeps the numbers we work with much smaller.