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Proportion is a relationship where two ratios are equal, written as a:b = c:d or a/b = c/d. Free GCSE AQA Maths notes with step-by-step methods and 12 FAQs.
Understanding Proportion as Equality of Ratios is part of Ratio and Proportion Fundamentals in GCSE AQA Higher Maths.
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GCSE AQA Higher
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Proportion is a relationship where two ratios are equal, written as a:b = c:d or a/b = c/d. Free GCSE AQA Maths notes with step-by-step methods and 12 FAQs.
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Use this lesson to stay inside Ratio and Proportion Fundamentals while connecting the detail back to the wider Ratio, proportion and rates of change topic.
Read the core explanations from Understanding Proportion as Equality of Ratios before testing yourself from memory.
You've been working with ratios for a while now. But here's where it gets powerful — when two ratios are equal, we call this a proportion.
Think of it like this: if you can make orange squash with 1 part concentrate to 4 parts water, then to make a bigger batch, you need 2 parts concentrate to 8 parts water. The ratios 1:4 and 2:8 are equal — that's proportion in action.
Proportion appears everywhere in real life. Recipe scaling, map distances, currency exchange rates — they all rely on this fundamental idea that equal ratios create predictable relationships.
The key insight is recognising when you have a proportion. If you know three values in the relationship a:b = c:d, you can always find the fourth.
When we write a proportion as fractions, the relationship becomes clearer. If $\frac{a}{b} = \frac{c}{d}$, then we have a powerful tool for solving problems.
The most reliable method for solving proportions is cross multiplication. This technique transforms the proportion into a simple equation you can solve.
Cross multiplication works because we're essentially multiplying both sides of the equation by both denominators. This eliminates the fractions and gives us a straightforward equation to solve.
Let's tackle a classic proportion problem. The key is setting up the ratios correctly and then using cross multiplication to find the unknown value.
Worked example: Finding a Missing Value in a Proportion
If 3 pencils cost £1.20, how much do 8 pencils cost?
Answer: £3.20
Notice how we kept the units organised — pencils over money on both sides. This prevents confusion and helps you spot errors.
Worked example: Solving a Ratio Proportion
The ratio of boys to girls in a class is 3:5. If there are 15 girls, how many boys are there?
Answer: 9 boys
You can always check your answer by substituting back. Does 9:15 simplify to 3:5? Yes, because $\frac{9}{15} = \frac{3}{5}$ when we divide both by 3.
Cross multiplication is reliable, but sometimes there are quicker approaches. When you spot the pattern, you can use scaling factors or unitary methods.
Worked example: Using Scale Factors
If 4 apples cost £2.40, find the cost of 10 apples using a scale factor.
Answer: £6.00
The unitary method is another powerful approach. Find the cost of one unit first, then multiply by the number you need.
Worked example: Using the Unitary Method
If 6 chocolate bars cost £4.50, find the cost of 9 chocolate bars.
Answer: £6.75
Choose your method based on the numbers involved. If the scale factor is obvious, use it. If finding one unit is straightforward, try the unitary method. Otherwise, cross multiplication never fails.
Proportion problems appear constantly in everyday situations. Maps, recipes, currency exchange, and speed calculations all use proportional reasoning.
Map scales are classic proportion problems. If 1 cm represents 5 km on a map, then 3.5 cm represents how many kilometres? You set up the proportion and solve.
Worked example: Map Scale Problem
On a map, 2 cm represents 15 km. What distance does 7 cm represent?
Answer: 52.5 km
Currency exchange uses proportion too. If £1 = $1.25, then £80 equals how many dollars? The exchange rate gives you one ratio, and you need to find the equivalent amount.
Recipe scaling is another common application. If a recipe for 4 people uses 300g flour, how much flour do you need for 6 people? Set up the proportion: people to flour.
The graph below shows how proportional relationships create straight lines through the origin. This visual check helps you verify your answers.
In this graph, y = 1.5x represents a proportion where y is always 1.5 times x. If you know one value, you can read off the other. This is direct proportion in action.
Proportion questions appear regularly in GCSE exams, often disguised as real-world problems. The key is recognising the proportional relationship and setting up your ratios correctly.
Examiners love to test whether you can identify which quantities are proportional. Sometimes they'll give you extra information that's not needed — focus on the two quantities that change together.
Always check your answer makes sense. If 3 items cost £6, then 6 items should cost £12, not £3. A quick sense-check catches most errors.
Remember that proportion is fundamentally about equal ratios. Whether you use cross multiplication, scale factors, or the unitary method, you're working with the same mathematical relationship. Master this concept and you'll handle any proportion problem with confidence.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Proportion is a relationship where two ratios are equal, written as a:b = c:d or a/b = c/d. For example, 1 part concentrate to 4 parts water is in proportion with 2 parts concentrate to 8 parts water, because 1:4 and 2:8 are equal.
Set up the proportion with the same units in corresponding positions, then cross multiply. If 3 pencils cost £1.20, then 3/1.20 = 8/x gives 3x = 9.60, so 8 pencils cost £3.20.
A ratio compares quantities, while a proportion is when two ratios are equal, written as a:b = c:d or a/b = c/d. For example, 1:4 and 2:8 are equal ratios, so they are in proportion.
Cross multiplication works because it multiplies both sides of the equation a/b = c/d by both denominators, which clears the fractions and gives ad = bc. This leaves a straightforward equation to solve.
Cross multiplication solves a proportion by multiplying diagonally across the equals sign: if a/b = c/d, then ad = bc. For boys to girls of 3:5 with 15 girls, 3/5 = x/15 gives 45 = 5x, so there are 9 boys.
Divide the new number of items by the original number to find the scale factor, then multiply the cost by it. If 4 apples cost £2.40, the scale factor to 10 apples is 10 ÷ 4 = 2.5, so 10 apples cost £2.40 × 2.5 = £6.00.
Keep map distance over real distance on both sides, then cross multiply. If 2 cm represents 15 km, then 2/15 = 7/x gives 2x = 105, so 7 cm represents 52.5 km.
8 pencils cost £3.20. Setting up the proportion 3/1.20 = 8/x and cross multiplying gives 3x = 8 × 1.20 = 9.60, so x = 9.60 ÷ 3 = 3.20.
The method for a proportion problem depends on the numbers: use a scale factor if it is obvious, the unitary method if finding one unit is straightforward, and cross multiplication otherwise, because it never fails. All three use the same idea of equal ratios.
A proportional relationship gives a straight line through the origin. For y = 1.5x, y is always 1.5 times x, so knowing one value lets the other be read off the line.
A proportion answer is sense-checked by asking whether it fits the pattern. If 3 items cost £6, then 6 items should cost £12, not £3, and a quick check like this catches most errors.
Proportion questions sometimes give extra information that is not needed, so the focus should be on the two quantities that change together. Setting up the ratios with the same units in the same positions avoids confusion.