Multiplication & Division with Fractions (SQA National 5 Maths): Revision Note

Evaluate $3\frac{3}{5}\times \frac{2}{9}$.

Give your answer in simplest form.

Answer:

Change $3\frac{3}{5}$ into an improper fraction

• Multiply the denominator by the whole part and add to the numerator

• Write this value over the denominator

$3\frac{3}{5}=\frac{3\times 5+3}{5}=\frac{18}{5}$

So

$3\frac{3}{5}\times \frac{2}{9}=\frac{18}{5}\times \frac{2}{9}$

A factor of 9 can be cancelled from the 18 and the 9 before multiplying

$\begin{array}{rcl}\frac{18}{5}\times \frac{2}{9}& =& \frac{2\times 9}{5}\times \frac{2}{1\times 9}\\ & =& \frac{2\times \cancel{9}}{5}\times \frac{2}{1\times \cancel{9}}\\ & =& \frac{2}{5}\times \frac{2}{1}\end{array}$

Multiply the numerators together and the denominators together

 $$\begin{gathered} =\frac{2\times 2}{5\times 1} \\ =\frac{4}{5} \end{gathered}$$

There is no further cancelling that can be done, so the answer is in simplest form

$\frac{4}{5}$

If you didn't cancel the factors of 9 before multiplying in the Worked Example, the answer would proceed

$\frac{18}{5}\times \frac{2}{9}=\frac{36}{45}$

You would then need to simplify that to $\frac{4}{5}$ to get full marks.

You can remember this process as “Keep Change Flip".

• Keep the fraction before the division sign the same

• Change the divide sign to a times

• Flip the fraction that is after the division sign

Evaluate $2\frac{5}{6}÷\frac{7}{9}$.

Give your answer in simplest form.

Answer:

Change $2\frac{5}{6}$ into an improper fraction

• Multiply the denominator by the whole part and add to the numerator

• Write this value over the denominator

$2\frac{5}{6}=\frac{2\times 6+5}{6}=\frac{17}{6}$

So

$2\frac{5}{6}÷\frac{7}{9}=\frac{17}{6}÷\frac{7}{9}$

Turn the division into a multiplication using 'Keep Change Flip'

• 'Flip' the second fraction and turn ÷ into ×

$\frac{17}{6}÷\frac{7}{9}=\frac{17}{6}\times \frac{9}{7}$

A factor of 3 can be cancelled from the 9 and the 6 before multiplying

$\begin{array}{rcl}\frac{17}{6}\times \frac{9}{7}& =& \frac{17}{2\times 3}\times \frac{3\times 3}{7}\\ & =& \frac{17}{2\times \cancel{3}}\times \frac{3\times \cancel{3}}{7}\\ & =& \frac{17}{2}\times \frac{3}{7}\end{array}$

Multiply the fractions

$$\begin{gathered} =\frac{17\times 3}{2\times 7} \\ =\frac{51}{14} \end{gathered}$$

There is no further cancelling that can be done, so the answer is in simplest form

Otherwise, the question does not specify a particular form for the answer, so you can give it as either that improper fraction, or as a mixed number

$\frac{51}{14}\left(\mathrm{or}3\frac{9}{14}\right)$

If you didn't cancel the factors of 3 before multiplying in the Worked Example, the answer would proceed

$\frac{17}{6}\times \frac{9}{7}=\frac{153}{42}$

You would then need to simplify that to $\frac{51}{14}$ to get full marks.