Ratio and Proportion
The Skill
A ratio compares quantities of the same kind. It tells us how much of one thing there is compared to another.
Writing Ratios
Ratios can be written as:
- $3 : 5$ (using a colon)
- $\frac{3}{5}$ (as a fraction)
- "3 to 5"
Simplifying Ratios
Simplify ratios like fractions — divide all parts by their highest common factor: $$12 : 18 = 2 : 3$$ (dividing both by 6)
Sharing in a Ratio
To share an amount in a given ratio:
- Add the ratio parts to find the total number of parts
- Divide the amount by the total parts to find one part
- Multiply to find each share
Example: Share £40 in the ratio 3 : 5
- Total parts: $3 + 5 = 8$
- One part: $40 \div 8 = £5$
- First share: $3 \times 5 = £15$
- Second share: $5 \times 5 = £25$
Given One Quantity, Find the Other
If you know one quantity in a ratio, find the "multiplier":
Example: The ratio of boys to girls is 2 : 3. There are 12 boys. How many girls?
- $2 \rightarrow 12$ means multiplier is $12 \div 2 = 6$
- Girls: $3 \times 6 = 18$
The Traps
Common misconceptions and how to avoid them.
Dividing by one ratio part instead of the total "The Part-Total Mixup"
The Mistake in Action
Share £60 in the ratio 2 : 3
Wrong:
- $60 \div 2 = 30$
- $60 \div 3 = 20$
- Shares are £30 and £20
Why It Happens
Students divide the total by each individual ratio number rather than finding the total number of parts first.
The Fix
Always find the TOTAL parts first.
The ratio 2 : 3 means there are $2 + 3 = 5$ parts in total.
Then:
- One part = $60 \div 5 = £12$
- First share = $2 \times 12 = £24$
- Second share = $3 \times 12 = £36$
Check: $24 + 36 = 60$ ✓
Spot the Mistake
Share £60 in the ratio 2 : 3
$60 \div 2 = 30$
$60 \div 3 = 20$
Shares are £30 and £20
Click on the line that contains the error.
Getting the ratio the wrong way round "The Flip"
The Mistake in Action
In a class, there are 3 girls for every 2 boys. Write the ratio of boys to girls.
Wrong: 3 : 2
Why It Happens
Students write the numbers in the order they appear in the question without checking what ratio is actually being asked for.
The Fix
Circle what you're asked for in the question.
"3 girls for every 2 boys" means:
- Girls : Boys = 3 : 2
But the question asks for Boys : Girls, so flip it:
- Boys : Girls = 2 : 3
Always label your ratio to check it matches what's asked.
Spot the Mistake
In a class, there are 3 girls for every 2 boys.
Write the ratio of boys to girls.
Answer: 3 : 2
Click on the line that contains the error.
Giving shares to the wrong people "The Swap Error"
The Mistake in Action
Ali and Ben share sweets in the ratio 3 : 7. There are 40 sweets.
Wrong:
- Ali gets $7 \times 4 = 28$ sweets
- Ben gets $3 \times 4 = 12$ sweets
Why It Happens
Students correctly calculate the values but assign them to the wrong people.
The Fix
Keep names with their numbers throughout.
Ali : Ben = 3 : 7
- Total parts = $3 + 7 = 10$
- One part = $40 \div 10 = 4$ sweets
- Ali (first in ratio) gets $3 \times 4 = 12$ sweets
- Ben (second in ratio) gets $7 \times 4 = 28$ sweets
Spot the Mistake
Ali and Ben share sweets in the ratio 3 : 7. There are 40 sweets.
Total parts = 10, One part = 4 sweets
Ali gets $7 \times 4 = 28$ sweets
Ben gets $3 \times 4 = 12$ sweets
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Tom and Sarah share £120 in the ratio 3 : 5. How much does each person receive?
Solution
Step 1: Find the total number of parts. $$3 + 5 = 8 \text{ parts}$$
Step 2: Find the value of one part. $$120 \div 8 = £15 \text{ per part}$$
Step 3: Calculate each share.
- Tom (3 parts): $3 \times 15 = £45$
- Sarah (5 parts): $5 \times 15 = £75$
Step 4: Check your answer. $$45 + 75 = 120 \checkmark$$
Answer: Tom gets £45, Sarah gets £75
Question
Red and blue paint are mixed in the ratio 2 : 5. If 6 litres of red paint are used, how much blue paint is needed?
Solution
Step 1: Identify what we know.
- Ratio is Red : Blue = 2 : 5
- Red paint = 6 litres
Step 2: Find the multiplier. The ratio says 2 parts red, but we have 6 litres of red. $$\text{Multiplier} = 6 \div 2 = 3$$
Step 3: Apply the multiplier to find blue. $$\text{Blue} = 5 \times 3 = 15 \text{ litres}$$
Answer: 15 litres of blue paint
Level 2: Scaffolded
Fill in the key steps.
Question
Three friends share a lottery win of £600 in the ratio 1 : 2 : 3. How much does each person receive?
Level 3: Solo
Try it yourself!
Question
A recipe for 4 people needs 300g of flour. How much flour is needed for 6 people?
Show Solution
Method 1: Unitary method For 1 person: $300 \div 4 = 75g$ For 6 people: $75 \times 6 = 450g$
Method 2: Scale factor Scale factor: $6 \div 4 = 1.5$ Flour needed: $300 \times 1.5 = 450g$
Answer: 450g
Question
Money is shared between Amy and Beth in the ratio 4 : 7. Beth receives £35. How much does Amy receive?
Show Solution
Beth's share represents 7 parts = £35
Value of one part: $35 \div 7 = £5$
Amy's share (4 parts): $4 \times 5 = £20$
Answer: Amy receives £20
Examiner's View
Mark allocation: Ratio questions are typically 2-4 marks.
Common errors examiners see:
- Dividing by one ratio part instead of the total
- Confusing which person gets which share
- Not giving all required answers
- Forgetting to check answers sum to the original total
What gains marks:
- Clearly showing the total parts calculation
- Writing "1 part = ..." explicitly
- Checking: do your answers add up to the original amount?
AQA Notes
AQA often uses worded context problems (recipes, mixing paint, sharing money). Read carefully to identify what's being compared.