Ratio and Proportion: Exam Practice

This final lesson in the Ratio, Proportion and Rates of Change topic brings together all the skills covered in the previous nine lessons. It provides a range of mixed exam-style questions, highlights common mistakes, and offers strategies for tackling multi-step problems on the AQA GCSE Mathematics papers.

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Key Formulae and Methods to Remember

Before attempting exam questions, make sure you are confident with these core methods:

Topic	Key Formula / Method
Simplifying ratios	Divide all parts by the HCF
Dividing in a ratio	Total parts, find one part, multiply
Direct proportion	y = kx (find k first)
Inverse proportion	y = k/x (find k first)
Percentage change	(Change / Original) x 100
Reverse percentages	Divide by the multiplier
Compound interest	Amount = P x (1 + r/100) to the power of n
Depreciation	Value = P x (1 - r/100) to the power of n
Speed	Speed = Distance / Time
Density	Density = Mass / Volume
Pressure	Pressure = Force / Area

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Worked Exam Questions

Question 1: Ratio (3 marks)

A box contains red, blue and green counters. The ratio of red to blue is 3 : 5. The ratio of blue to green is 2 : 1. There are 50 counters in total. How many are green?

Step 1: Combine the ratios. Blue appears in both — find LCM of 5 and 2 = 10.

• Red : Blue = 3 : 5 becomes 6 : 10
• Blue : Green = 2 : 1 becomes 10 : 5

Step 2: Combined ratio: Red : Blue : Green = 6 : 10 : 5

Step 3: Total parts = 6 + 10 + 5 = 21

Step 4: One part = 50 / 21 — this does not give a whole number, so let us re-read the question.

Actually, let us recalculate. With 50 counters and a combined ratio of 6 : 10 : 5:

• Total parts = 21
• This would require 50/21 per part, which is not a whole number.

Let us reconsider: if the ratios are exactly as stated, then 50 counters with ratio 6 : 10 : 5 means there cannot be exactly 50. Let us adjust the question for a clean answer.

Revised: There are 63 counters in total.

• One part = 63 / 21 = 3
• Green = 5 x 3 = 15 green counters

Exam Tip: When combining ratios, always make the shared quantity equal in both ratios using the LCM. Then read all three values directly across.

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Question 2: Dividing in a Ratio (3 marks)

Sarah and Tom share some money in the ratio 4 : 7. Tom receives 18 pounds more than Sarah. How much money was shared in total?

• Difference in parts = 7 - 4 = 3 parts
• 3 parts = 18 pounds
• 1 part = 6 pounds
• Total parts = 4 + 7 = 11
• Total money = 11 x 6 = 66 pounds

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Question 3: Percentage Change (3 marks)

A shop increases the price of a jacket from 45 pounds to 54 pounds. Calculate the percentage increase.

• Change = 54 - 45 = 9 pounds
• Percentage increase = (9 / 45) x 100 = 20%

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Question 4: Reverse Percentage (3 marks)

After a 12% increase, a gym membership costs 89.60 pounds per month. What was the original monthly cost?

• 89.60 represents 112% of the original
• Original cost = 89.60 / 1.12 = 80 pounds

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Question 5: Compound Interest (4 marks)

7,500 pounds is invested at 2.5% compound interest per year. Calculate the value of the investment after 5 years. Give your answer to the nearest penny.

• Multiplier = 1.025
• Amount = 7,500 x 1.025 to the power of 5
• Amount = 7,500 x 1.131408...
• Amount = 8,485.56 pounds

Interest earned = 8,485.56 - 7,500 = 985.56 pounds

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Question 6: Compound Measures (3 marks)

A metal block has a mass of 3.24 kg and dimensions 15 cm x 12 cm x 9 cm. Calculate the density of the metal in g/cm cubed.

Step 1: Convert mass to grams: 3.24 kg = 3,240 g Step 2: Volume = 15 x 12 x 9 = 1,620 cm cubed Step 3: Density = 3,240 / 1,620 = 2 g/cm cubed

graph TD
    A[Given: mass in kg and dimensions in cm] --> B[Convert mass: kg to g]
    B --> C[Calculate volume: l x w x h]
    C --> D[Density = mass / volume]
    D --> E[State units: g/cm cubed]

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Question 7: Proportion Equation [H] (4 marks)

y is inversely proportional to the square of x. When x = 4, y = 5. Find the value of y when x = 10.

1. y = k / (x squared)
2. 5 = k / 16, so k = 80
3. y = 80 / (x squared)
4. When x = 10: y = 80 / 100 = 0.8

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Question 8: Growth and Decay [H] (4 marks)

The value of a vintage guitar increases by 6% per year. It is currently worth 2,400 pounds. After how many years will its value first exceed 3,200 pounds?

Multiplier = 1.06

Year	Value (pounds)
1	2,544.00
2	2,696.64
3	2,858.44
4	3,029.95
5	3,211.74

The value first exceeds 3,200 after 5 years.

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Question 9: Speed, Distance and Time (4 marks)

A train travels from London to Edinburgh, a distance of 630 km. The train departs at 09:15 and arrives at 13:45. Calculate the average speed in km/h.

Step 1: Journey time = 13:45 - 09:15 = 4 hours 30 minutes = 4.5 hours Step 2: Average speed = 630 / 4.5 = 140 km/h

Exam Tip: Convert journey times to decimals of hours before dividing. Use the 24-hour clock to avoid errors with am/pm.

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Question 10: Multi-Step Problem (5 marks)

A painter mixes red and white paint in the ratio 2 : 5 to make pink paint. She has 3 litres of red paint and 10 litres of white paint. What is the maximum amount of pink paint she can make?

Step 1: Work out how much paint each colour allows.

• If she uses all 3 litres of red: 2 parts = 3 litres, so 1 part = 1.5 litres. White needed = 5 x 1.5 = 7.5 litres. Total = 3 + 7.5 = 10.5 litres.
• If she uses all 10 litres of white: 5 parts = 10 litres, so 1 part = 2 litres. Red needed = 2 x 2 = 4 litres. But she only has 3 litres of red — not enough.

Step 2: Red paint is the limiting factor. Using all 3 litres of red:

• Total pink paint = 10.5 litres

She will have 10 - 7.5 = 2.5 litres of white paint left over.

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