Exercise solution 6, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) To find how many containers of a given size can be filled, we use division. If you know the total amount and the amount in each bottle, then:

\[ \text{number of bottles}=\frac{\text{total amount of water}}{\text{capacity of one bottle}} \]

For example, if someone has \(9\) litres of juice and each bottle holds \(\frac{3}{5}\) litre, the calculation is \(9 \div \frac{3}{5}\).

B) A mixed number is a whole number and a fraction written together, such as \(4\frac{2}{3}\). Before dividing, it is usually easier to convert a mixed number into an improper fraction.

To do this, multiply the whole number by the denominator, then add the numerator:

\[ a\frac{b}{c}=\frac{ac+b}{c} \]

Example:

\[ 5\frac{1}{4}=\frac{5\times 4+1}{4}=\frac{21}{4} \]

C) Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

So:

\[ \frac{m}{n}\div \frac{p}{q}=\frac{m}{n}\times \frac{q}{p} \]

Example:

\[ \frac{7}{3}\div \frac{2}{9}=\frac{7}{3}\times \frac{9}{2} \]

D) When multiplying fractions, multiply the numerators together and the denominators together:

\[ \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd} \]

Then simplify if possible. Example:

\[\begin{aligned} \frac{4}{7}\times \frac{21}{8} &= \frac{4\times 21}{7\times 8} \\ &= \frac{84}{56} \\ &= \frac{3}{2} \end{aligned}\]

E) A final answer about the number of bottles should make sense in the context. If the calculation gives a whole number, that means the water fills an exact number of bottles. If it does not give a whole number, you must think carefully about whether the question is asking for completely full bottles only, or whether a partly filled bottle is allowed.

Always check that your answer matches the units: here, litres are being divided by litres, so the result is a number of bottles.