Writing and simplifying ratios, dividing a quantity in a given ratio, and solving direct proportion problems including the unitary method.

What this dot point is asking

AQA wants you to write and simplify ratios, divide a quantity in a given ratio, and solve direct (and simple inverse) proportion problems, often using the unitary method of finding the value of one unit first. Ratio and proportion thread through recipes, scale, currency conversion, best-buy comparisons and rates, so this is one of the most applied number topics in the specification.

Writing and simplifying ratios

A ratio such as $12 : 18$ is simplified by dividing both parts by their highest common factor, here $6$, giving $2 : 3$. Ratios must use the same units before simplifying: $50\,\text{cm} : 2\,\text{m}$ becomes $50 : 200 = 1 : 4$. A ratio can also be written in the form $1 : n$ by dividing both parts by the first: $4 : 10$ becomes $1 : 2.5$, which is handy for comparing.

Dividing a quantity in a ratio

To split an amount in a given ratio, count the total number of shares and find the value of one.

Some questions give one part and ask for the whole. If the ratio is $3 : 5$ and the smaller share is $\pounds 24$, then one share is $24 \div 3 = \pounds 8$, so the larger share is $5 \times 8 = \pounds 40$ and the total is $\pounds 64$.

The unitary method and direct proportion

The unitary method finds the value of a single unit, then scales to the quantity you want. If $6$ pens cost $\pounds 4.50$, one pen costs $4.50 \div 6 = \pounds 0.75$, so $10$ pens cost $10 \times 0.75 = \pounds 7.50$. This method also powers recipe scaling, currency conversion and best-buy comparisons (work out the cost per unit and compare).

Direct and inverse proportion in words

In direct proportion, doubling one quantity doubles the other (more workers, more pay). In inverse proportion, doubling one halves the other (more workers, less time), and the product stays constant. Recognising which type a worded problem is determines whether you multiply up or share out the constant total.

Best buys and recipe scaling

Two of the most common applied ratio questions are best-buy comparisons and recipe scaling. For a best buy, work out the cost per unit (or amount per pound) for each option and compare: a $750\,\text{g}$ pack at $\pounds 3$ costs $\dfrac{300}{750} = 0.4$ pence per gram, while a $1.2\,\text{kg}$ pack at $\pounds 4.20$ costs $\dfrac{420}{1200} = 0.35$ pence per gram, so the larger pack is better value. Always compare the same quantity and state which is cheaper with a reason.

For recipe scaling, set up a ratio between the given recipe and the amount you need. A recipe for $4$ people needing $300\,\text{g}$ of rice scales to $6$ people by finding one person's share, $300 \div 4 = 75\,\text{g}$, then multiplying by $6$ to get $450\,\text{g}$. The unitary method (the value of one) is the reliable engine for all of these.

Exam technique for ratio problems

Read carefully whether a question gives you a total to share, a single part, or a difference between parts. "The difference between the shares is $\pounds 12$" means one ratio-difference of parts equals $\pounds 12$, so a $5 : 2$ split with a difference of $3$ parts gives one part as $\pounds 4$. Laying out the shares clearly and checking they recombine to the stated total or difference catches most slips before they cost marks.