Fraction arithmetic

Adding and subtracting fractions

Fractions with the same denominator can be added (or subtracted) by adding (or subtracting) the numerators.

For instance, \frac{2}{9} + \frac{3}{9} = \frac{5}{9} or \frac{6}{11} - \frac{4}{11} = \frac{2}{11}.

If two fractions do not have the same denominator, then find a common denominator by making equivalent fractions.

Example

Work out \frac{4}{7} + \frac{1}{3}.

Work out the common denominator by looking for the lowest common multiple of 7 and 3. This is 21 ( 7 \times 3 = 21).

Write the equivalent fractions using 21 as the new common denominator.

\frac{4}{7} = \frac{12}{21}

\frac{1}{3} = \frac{7}{21}

So: \frac{4}{7} + \frac{1}{3} = \frac{12}{21} + \frac{7}{21} = \frac{19}{21}

This is the final answer as the fraction cannot be simplified.

Question

Work out 2 \frac{2}{5} - \frac{3}{4}.

This sum contains a mixed number ( 2 \frac{2}{5}), which must be converted to an improper fraction.

  • Change the whole number part to a fraction using the denominator of the fraction part. So 5 = \frac{10}{5}
  • Add on the fraction part: \frac{10}{5} + \frac{2}{5} = \frac{12}{5}

So 2\frac{2}{5} = \frac{12}{5}

This gives: 2 \frac{2}{5} - \frac{3}{4} = \frac{12}{5} - \frac{3}{4}

Now look for a common denominator by looking for the lowest common multiple of 5 and 4 which is 20.

Write equivalent fractions using 20 as the common denominator.

\frac{12}{5} = \frac{48}{20}

\frac{3}{4} = \frac{15}{20}

So: \frac{12}{5} - \frac{3}{4} = \frac{48}{20} - \frac{15}{20} = \frac{33}{20}

The question was asked in mixed number format, so the answer should be given as a mixed number. \frac{33}{20} is an improper fraction so it can be written as a mixed number.

Divide the numerator by the denominator:

\frac{33}{20} = 1 \frac{13}{20}

This fraction cannot be simplified further, so this is the final answer.