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Ratio Proportion and Rates of ChangeHigherRatio Proportion and Rates of ChangePercentagesCompound Interest
AQA GCSE · Question 12 · Ratio Proportion and Rates of Change
The population of a country is now 67,200,000.
The population is predicted to:
- increase by 1% per year for 6 years
- and then decrease by 1.2% per year for 2 years.
Work out the predicted population of the country 8 years from now. Give your answer to 3 significant figures.
The population of a country is now 67,200,000.
The population is predicted to:
- increase by 1% per year for 6 years
- and then decrease by 1.2% per year for 2 years.
Work out the predicted population of the country 8 years from now. Give your answer to 3 significant figures.
How to approach this question
1. Determine the multiplier for a 1% increase.
2. Apply this multiplier for 6 years to the initial population using powers.
3. Determine the multiplier for a 1.2% decrease.
4. Take the result from step 2 and apply the decrease multiplier for 2 years.
5. It is best to do this as a single calculation to avoid rounding errors.
6. Round your final answer to 3 significant figures.
Full Answer
1. **Calculate the population after the 6-year increase.**
An increase of 1% is a multiplier of 1.01.
Population after 6 years = 67,200,000 × (1.01)⁶
= 67,200,000 × 1.06152015...
= 71,334,154.3...
2. **Calculate the final population after the 2-year decrease.**
A decrease of 1.2% is a multiplier of (1 - 0.012) = 0.988.
Final population = (Population after 6 years) × (0.988)²
= 71,334,154.3... × (0.988)²
= 71,334,154.3... × 0.976144
= 69,631,833.1...
Alternatively, in one calculation:
Final Population = 67,200,000 × (1.01)⁶ × (0.988)² = 69,631,833.1...
3. **Round the answer to 3 significant figures.**
The first three significant figures are 6, 9, 6. The next digit is 3, so we round down.
Answer = 69,600,000.
This is a compound percentage change problem. We can solve it by using multipliers.
Step 1: Find the multiplier for the 1% increase.
An increase of 1% means we have 100% + 1% = 101% of the original amount. As a decimal, this is 1.01.
Step 2: Find the multiplier for the 1.2% decrease.
A decrease of 1.2% means we have 100% - 1.2% = 98.8% of the original amount. As a decimal, this is 0.988.
Step 3: Set up the calculation for the 8-year period.
The population starts at 67,200,000.
It increases by 1% for 6 years, so we multiply by (1.01)⁶.
Then it decreases by 1.2% for 2 years, so we multiply by (0.988)².
The full calculation is:
Final Population = 67,200,000 × (1.01)⁶ × (0.988)²
Step 4: Calculate the result.
Final Population = 69,631,833.108...
Step 5: Round to 3 significant figures.
The first three significant figures are 6, 9, and 6. The fourth digit is 3, which is less than 5, so we round down.
Final Population ≈ 69,600,000.
Common mistakes
✗ Using simple interest instead of compound interest (i.e., calculating 6% increase and 2.4% decrease and applying them once).
✗ Incorrect multipliers, e.g., using 0.01 instead of 1.01, or 1.2 instead of 0.988.
✗ Rounding the answer prematurely after the first 6 years.
✗ Incorrectly rounding to 3 significant figures, e.g., writing 69.6 or 69,632,000.
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