Fractions

Adding and subtracting fractions

To add or subtract fractions, the pieces have to be the same size. When the denominators match, add or subtract the numerators. When they don’t, rewrite both fractions with a common denominator first.

Why the denominator has to match

The denominator says how big each piece is; the numerator says how many pieces there are. $\frac{2}{7} + \frac{3}{7}$ is two sevenths plus three sevenths: five sevenths, $\frac{5}{7}$ . The size of the pieces does not change, so the denominator stays $7$ .

Halves and thirds are different sizes, so $\frac{1}{2} + \frac{1}{3}$ cannot be added directly. Cut both into sixths: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .

Worked examples

Same denominator

Problem: $\frac{2}{7} + \frac{3}{7}$

The denominators match: add the numerators, keep the denominator.
$\frac{2 + 3}{7} = \frac{5}{7}$

Result: $\frac{5}{7}$

Different denominators

Problem: $\frac{1}{4} + \frac{1}{6}$

Find a common denominator: the least common multiple of $4$ and $6$ is $12$ .
$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$
Add the numerators.
$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Result: $\frac{5}{12}$

Subtraction

Problem: $\frac{5}{6} - \frac{1}{4}$

Common denominator $12$ .
$\frac{10}{12} - \frac{3}{12}$
Subtract the numerators.
$\frac{7}{12}$

Result: $\frac{7}{12}$

Mixed numbers

Problem: $1\frac{2}{3} + \frac{3}{4}$

Write the mixed number as a fraction.
$1\frac{2}{3} = \frac{5}{3}$
Common denominator $12$ .
$\frac{20}{12} + \frac{9}{12} = \frac{29}{12}$
Back to a mixed number.
$\frac{29}{12} = 2\frac{5}{12}$

Result: $2\frac{5}{12}$

Common mistakes

Mistake 1 · Adding the denominators too: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$ .
Why it happens
The denominator is the size of the pieces, not a count to add.
What to do instead
Make the pieces the same size first: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .
Mistake 2 · Changing only the denominator: $\frac{1}{4} = \frac{1}{12}$ .
Why it happens
Multiplying the denominator by $3$ makes each piece three times smaller.
What to do instead
Multiply the numerator by the same number: $\frac{1}{4} = \frac{3}{12}$ .
Mistake 3 · Subtracting mixed numbers part by part: $3\frac{1}{4} - 1\frac{3}{4} = 2\frac{2}{4}$ .
Why it happens
$\frac{1}{4}$ is smaller than $\frac{3}{4}$ , so the fractions got subtracted the wrong way round ( $\frac{3}{4} - \frac{1}{4}$ ), and the answer came out too big.
What to do instead
Turn both into fractions: $\frac{13}{4} - \frac{7}{4} = \frac{6}{4} = 1\frac{1}{2}$ .

Practice

1. Calculate $\frac{3}{8} + \frac{1}{8}$ .
Hint
Same denominator.
Show answer
$\frac{4}{8} = \frac{1}{2}$
2. Calculate $\frac{2}{3} + \frac{1}{5}$ .
Hint
The common denominator is $15$ .
Show answer
$\frac{10}{15} + \frac{3}{15} = \frac{13}{15}$
3. Calculate $\frac{7}{10} - \frac{1}{4}$ .
Hint
The least common denominator is $20$ .
Show answer
$\frac{14}{20} - \frac{5}{20} = \frac{9}{20}$
4. Calculate $2\frac{1}{2} - \frac{3}{4}$ .
Hint
Write $2\frac{1}{2}$ as $\frac{5}{2}$ .
Show answer
$\frac{10}{4} - \frac{3}{4} = \frac{7}{4} = 1\frac{3}{4}$

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