Ratio and Proportion

A ratio compares two or more amounts. If a recipe uses 4 cups of flour for every 6 cups of water, the ratio of flour to water is $4 : 6$. The colon is read "to", and the order matters — $4 : 6$ is not the same comparison as $6 : 4$.

To simplify a ratio, divide every part by a common factor — just as with fractions. $4 : 6$ both divide by 2, giving $2 : 3$.

The Stage 8 step up is handling ratios with units. Before you simplify, make sure every part is in the same unit.

So $2\text{ m} : 50\text{ cm}$ must first become $200\text{ cm} : 50\text{ cm}$, which simplifies to $4 : 1$. Writing $2 : 50$ straight away would be wrong — the units were not the same. A ratio is in simplest form when the only number dividing every part is 1.

A very common task is splitting an amount into a given ratio. The method has three clear steps.

Suppose $96 is shared between three friends in the ratio $3 : 4 : 5$.

1. Add the parts to find the total number of shares: $3 + 4 + 5 = 12$.
2. Find one share by dividing the amount by that total: $96 \div 12 = 8$.
3. Multiply each person's parts by the value of one share.

So the shares are 3 \times 8 = \24$,$4 \times 8 = $32$and$5 \times 8 = $40$.

A brilliant check: the shares should add back to the original amount. $24 + 32 + 40 = 96$, so it works. Ratio also links neatly to fractions — with 12 shares in total, the three friends receive $\frac{3}{12}$, $\frac{4}{12}$ and $\frac{5}{12}$ of the money.

Not every ratio question hands you the total. Sometimes you are told one person's share, or the difference between two shares, and must work back.

Suppose money is shared in the ratio $2 : 5$ and the larger share is $35. The 5 parts are worth $35, so one part is 35 \div 5 = \7$. Now the smaller share is$2 \times 7 = $14$, and the total is$7 \times (2 + 5) = $49$.

The same idea handles a difference. If two shares are in the ratio $3 : 7$ and one friend gets $20 more than the other, the difference of $7 - 3 = 4$ parts is worth $20, so one part is 20 \div 4 = \5$.

The golden rule for any ratio problem: find the value of one part first. Once you know one part, every share is a quick multiplication.

Two quantities are in direct proportion when they change together at the same rate. Double one and the other doubles; halve one and the other halves.

If 5 identical books weigh 2 kg, then 10 books weigh 4 kg and 15 books weigh 6 kg — the weight keeps in step with the number of books.

There are two clear signs of direct proportion. First, when one quantity is zero, the other is zero too — 0 books weigh 0 kg. Second, the ratio between the two quantities stays constant — for the books, weight $\div$ number is always $0.4$ kg per book.

That constant ratio is the engine of every proportion calculation. Spotting that two quantities are directly proportional tells you straight away that the unitary method, coming next, will solve the problem.

The most reliable way to solve a direct proportion problem is the unitary method. The word "unitary" comes from "unit", meaning one. The idea is simple: find the value of one item first, then scale up.

Say 6 pens cost $9 and you want the price of 10 pens.

1. Find one unit: one pen costs 9 \div 6 = \1.50$.
2. Scale up: 10 pens cost 10 \times 1.50 = \15$.

The same "find one, then scale" approach handles recipes, speeds, currency exchange and best-buy comparisons. To compare a 500 g pack at $4 with a 750 g pack at $5.40, find the price per gram of each — the smaller price per gram is the better value.

Whenever a question gives you one quantity and asks about a different quantity in proportion, reach for the unitary method.

Ratio and proportion are among the most practical skills in maths:

• Scale drawings and maps use ratio to shrink real distances down to a page, such as a scale of $1 : 50\,000$.
• Similar shapes in geometry have sides in equal ratios — proportion is what makes one a clean enlargement of another.
• Science is full of proportion, from mixing solutions to comparing speeds and densities.
• Everyday decisions lean on the unitary method — finding the best-value pack at the shop is a proportion question in disguise.

If a later topic about scaling or comparing feels tricky, it is often a ratio or proportion step underneath. The "find one part, then scale" habit is worth practising until it feels natural.