Adding and subtracting fractions

You can add two halves of a chapati to get one whole. You can subtract one quarter from three quarters and get two quarters. Adding and subtracting fractions is simple , as long as the pieces are the same size.

Concept

Same denominator

If two fractions have the same denominator, just add or subtract the numerators and keep the denominator:

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

This makes sense visually: $3$ quarters plus $2$ quarters is $5$ quarters. The pieces are the same size; we just count more (or fewer) of them.

Different denominators

Different-sized pieces cannot be added directly. You first have to make them the same size by converting both fractions to a common denominator:

1. Find a common multiple of the two denominators (the LCM is best, but any common multiple will do).
2. Convert each fraction to an equivalent with that denominator.
3. Now add (or subtract) the numerators.

Example. $\frac{1}{4} + \frac{1}{6}$.

• LCM of $4$ and $6$ is $12$.
• $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{6} = \frac{2}{12}$.
• Add: $\frac{3 + 2}{12} = \frac{5}{12}$.

Simplify the final answer

After adding or subtracting, always check whether the answer can be simplified.

For example, $\frac{1}{6} + \frac{1}{2} = \frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}$.

When the answer exceeds $1$

A fraction whose numerator is bigger than (or equal to) its denominator is called improper. For example, $\frac{7}{4}$ or $\frac{11}{5}$.

You can convert an improper fraction to a mixed number by dividing: $\frac{7}{4} = 1 \frac{3}{4}$, because $7 = 4 \times 1 + 3$.

Two friendly worked sums

• $\frac{2}{3} + \frac{1}{6}$: LCM $= 6$. $\frac{2}{3} = \frac{4}{6}$. Sum: $\frac{4 + 1}{6} = \frac{5}{6}$.
• $\frac{5}{8} - \frac{1}{4}$: LCM $= 8$. $\frac{1}{4} = \frac{2}{8}$. Diff: $\frac{5 - 2}{8} = \frac{3}{8}$.

Worked examples

Example 1. Add $\frac{3}{7} + \frac{2}{7}$.

• Same denominator: $\frac{3 + 2}{7} = \frac{5}{7}$.

Example 2. Subtract $\frac{7}{9} - \frac{4}{9}$.

• $\frac{7 - 4}{9} = \frac{3}{9} = \frac{1}{3}$ (simplified).

Example 3. Add $\frac{1}{2} + \frac{1}{3}$.

• LCM $= 6$. $\frac{1}{2} = \frac{3}{6}$, $\frac{1}{3} = \frac{2}{6}$.
• Sum: $\frac{5}{6}$.

Example 4. Subtract $\frac{2}{5} - \frac{1}{4}$.

• LCM $= 20$. $\frac{2}{5} = \frac{8}{20}$, $\frac{1}{4} = \frac{5}{20}$.
• Diff: $\frac{3}{20}$.

Example 5. Add $\frac{5}{6} + \frac{3}{8}$.

• LCM $= 24$. $\frac{5}{6} = \frac{20}{24}$, $\frac{3}{8} = \frac{9}{24}$.
• Sum: $\frac{29}{24}$. As mixed number: $1 \frac{5}{24}$.

Example 6. Word problem. Raju has $\frac{3}{4}$ kg of rice and uses $\frac{1}{3}$ kg for lunch. How much rice is left?

• $\frac{3}{4} - \frac{1}{3}$. LCM $= 12$. $\frac{3}{4} = \frac{9}{12}$, $\frac{1}{3} = \frac{4}{12}$.
• Left: $\frac{9 - 4}{12} = \frac{5}{12}$ kg.

Try it yourself

1. Compute $\frac{4}{9} + \frac{3}{9}$.
2. Compute $\frac{11}{12} - \frac{5}{12}$.
3. Compute $\frac{1}{2} + \frac{2}{5}$.
4. Compute $\frac{3}{4} - \frac{1}{6}$.
5. Compute $\frac{1}{2} + \frac{1}{3} + \frac{1}{4}$.
6. A jug holds $1 \frac{1}{2}$ litres. You pour out $\frac{3}{4}$ litre. How much is left?
7. A recipe needs $\frac{2}{3}$ cup of milk and $\frac{1}{4}$ cup more. Total milk?
8. Investigate: find two different fractions whose sum is exactly $1$.

Activity

Chapati share. Three friends arrive, and you have $\frac{1}{2}$ of a chapati to give the first, $\frac{1}{4}$ to the second, and $\frac{1}{8}$ to the third. How many chapatis do you give in total? Did you give more than one whole chapati or less? Try the same with $\frac{1}{2} + \frac{1}{3} + \frac{1}{6}$.