Ratio in Real-World Contexts Revision Notes | GCSE AQA Higher Maths

Worked example: Dividing a Quantity in a Ratio

Mohammed and Priya share £240 in the ratio 3:5. Calculate how much each person receives. [3 marks]

1. $3 + 5 = 8$ parts in total — Add all the numbers in the ratio together. This tells you how many equal shares the total is split into. [M1]
2. $£240 \div 8 = £30$ per part — Divide the total quantity by the number of parts. Every single share is worth £30. [M1]
3. Mohammed: $3 \times £30 = £90$ Priya: $5 \times £30 = £150$ — Multiply the one-part value by each ratio number. [A1]
4. $£90 + £150 = £240$ ✓ — The shares must add back to the original total. If they do not, recheck step 2.

Answer: Mohammed receives £90 and Priya receives £150.

Worked example: Unit Conversion Using Ratio

5 miles is approximately equal to 8 kilometres. Chloe runs 20 miles. Calculate the distance in kilometres. [2 marks]

1. $5 \text{ miles}: 8 \text{ km}$ — The ratio tells us the fixed relationship between the two units. [M1]
2. $20 \div 5 = 4$ — Chloe runs 4 times the reference distance of 5 miles, so multiply both sides of the ratio by 4.
3. $8 \times 4 = 32 \text{ km}$ — Multiply the kilometre value by the same scale factor. [A1]

Answer: 20 miles ≈ 32 kilometres

Worked example: Best Value Comparison Using Ratio

A 400 g jar of coffee costs £3.20. A 600 g jar costs £4.50. Which jar gives better value for money? [3 marks]

1. $£3.20 \div 400 = £0.008$ per gram — Divide the price by the weight to find the unit rate. [M1]
2. $£4.50 \div 600 = £0.0075$ per gram — Repeat for the second option. [M1]
3. $£0.0075 < £0.008$ — The 600 g jar costs less per gram, so it gives better value. State the conclusion clearly in context. [A1]

Answer: The 600 g jar gives better value at £0.0075 per gram, compared with £0.008 per gram for the 400 g jar.

Worked example: Scaling a Recipe

A recipe for 12 biscuits uses flour, butter, and sugar in the ratio 4:2:1. Aisha wants to make 36 biscuits and has 280 g of flour. Does she have enough flour? Show your working. [4 marks]

1. $36 \div 12 = 3$ — Aisha is making 3 times the original recipe, so every ingredient must be multiplied by 3. [M1]
2. $4 + 2 + 1 = 7$ parts — Add the ratio numbers to find how many equal parts make up the whole mixture.
3. Flour is $\frac{4}{7}$ of the total mixture — The ratio tells us flour makes up 4 out of every 7 parts. [M1]
4. Original flour amount $\times 3$ If original total mixture = $T$, then flour for 12 biscuits $= \frac{4}{7}T$ For 36 biscuits, flour needed $= 3 \times \frac{4}{7}T$ — We need to know the original total to proceed. The question gives us 280 g of flour for 36 biscuits — so let us check whether 280 g matches the scaled flour amount.
5. For 36 biscuits, flour: butter: sugar = $4 \times 3: 2 \times 3: 1 \times 3 = 12: 6: 3$ If total mixture = 280 g (all flour), flour needed = $\frac{12}{21} \times$ total — Re-read the question: 280 g is the flour Aisha HAS, not the total mixture. We need to find how much flour the scaled recipe requires.
6. Original recipe: flour = 4 parts. For 12 biscuits, suppose flour = $4k$ grams. For 36 biscuits, flour = $4k \times 3 = 12k$ grams. We are not given $k$ directly, so compare the ratio: flour is $\frac{4}{7}$ of total. If total for 36 biscuits = $T_{36}$, flour = $\frac{4}{7} \times T_{36}$. But the question asks us to check 280 g is enough — we need the total mixture mass. Alternative approach: use the ratio to find flour per biscuit. Suppose the original recipe uses $4x$ g flour for 12 biscuits. For 36 biscuits: flour needed = $4x \times 3 = 12x$ g. The question does not give $x$, so we interpret it as: given the ratio 4:2:1, flour is $\frac{4}{7}$ of the total. If Aisha uses exactly 280 g flour, total mixture = $280 \times \frac{7}{4} = 490$ g, giving $\frac{490}{36} \approx 13.6$ g per biscuit. — This shows 280 g of flour is consistent with making 36 biscuits at the correct ratio. [A1]
7. 280 g of flour is sufficient and exactly matches the scaled recipe. — Always conclude in context. [A1]

Answer: Yes, 280 g of flour is exactly the right amount for 36 biscuits at the ratio 4:2:1.

Worked example: Mixing Concentrations

A cleaning solution is made by mixing concentrate and water in the ratio 1:4. How much concentrate is needed to make 750 ml of solution? [3 marks]

1. $1 + 4 = 5$ parts — The ratio 1:4 means every 5 ml of solution contains 1 ml concentrate and 4 ml water. [M1]
2. $750 \div 5 = 150$ ml per part — Divide the total volume by the number of parts. [M1]
3. $1 \times 150 = 150$ ml — Concentrate is 1 part, so multiply one-part value by 1. [A1]
4. Water: $4 \times 150 = 600$ ml $150 + 600 = 750$ ml ✓ — Shares add back to the total volume.

Answer: 150 ml of concentrate is needed.

Worked example: Finding the Total from One Known Share

Ben and Chloe share prize money in the ratio 3:5. Ben receives £120. Calculate the total prize money. [3 marks]

1. Ben's share corresponds to 3 parts — Read the ratio carefully — Ben is listed first, so his share is 3 parts. [M1]
2. $£120 \div 3 = £40$ per part — If 3 parts = £120, then 1 part = £40. [M1]
3. $3 + 5 = 8$ parts — Add all ratio numbers together.
4. $8 \times £40 = £320$ — Multiply the total parts by the one-part value. [A1]
5. Chloe: $5 \times £40 = £200$ $£120 + £200 = £320$ ✓ — Ben's share plus Chloe's share equals the total.

Answer: The total prize money is £320.