Multiplying and dividing fractions
Multiplying fractions
To multiply two fractions together, multiply the numeratorThe top part of a fraction. For â…ť , the numerator is 5. together and multiply the denominatorThe bottom part of a fraction. For â…ť, the denominator is 8, which represents 'eighths'. together.
Example 1
Work out \(\frac{3}{5} \times \frac{2}{3}\).
\(\frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15}\)
\(\frac{6}{15}\) can be simplified to \(\frac{2}{5}\) (by dividing the numerator and denominator by 3).
If the fractions to be multiplied contain mixed numbers, first convert them to improper fractionsIn improper fractions (or top-heavy fractions) the numerator is larger than the denominator, eg 7/5 , 9/4 . and then multiply the numerators together and multiply the denominators together.
Example 2
Work out \(2 \frac{1}{3} \times 1 \frac{1}{2}\).
\(2 \frac{1}{3} = \frac{7}{3}\) (\(\frac{2 \times 3 + 1}{3}\)) and \(1 \frac{1}{2} = \frac{3}{2}\) (\(\frac{1 \times 2 + 1}{2}\))
\(2 \frac{1}{3} \times 1 \frac{1}{2}\) is the same as \(\frac{7}{3} \times \frac{3}{2}\).
\(\frac{7}{3} \times \frac{3}{2} = \frac{7 \times 3}{3 \times 2} = \frac{21}{6}\) which can be simplifyA fraction is simplified when there are no more common factors shared by the numerator and denominator. For example, the fraction 8/10 will simplify to 4/5 by removing a common factor of 2. to \(\frac{7}{2}\) (by dividing the numerator and denominator by 3) which should be converted to a mixed number as the question contains mixed numbers.
\(\frac{7}{2} = 3 \frac{1}{2}\) (divide the numerator by the denominator).
This fraction cannot be simplified any further, so this is the final answer.
Dividing fractions
To divide two fractions, multiply the first fraction by the reciprocalThe reciprocal of a number is 1 divided by that number. The reciprocal of a fraction is that fraction turned upside down, eg the reciprocal of 3/4 is 4/3. of the second fraction. This means simply that the divide sign is swapped for a multiply sign, and the second fraction is flipped upside down.
Example
Work out \(\frac{3}{5} \div \frac{2}{3}\).
This is the same as \(\frac{3}{5} \times \frac{3}{2}\)
Keep the first fraction the same, change the divide sign to a multiply and write the second fraction as a reciprocal (flip it upside down).
The sum is now:
\(\frac{3}{5} \times \frac{3}{2} = \frac{3 \times 3}{5 \times 2} = \frac{9}{10}\)