Percentages, interest and recovering the original
Identify the reference whole, use percentage multipliers and distinguish simple from repeated growth.
Section C1.13 / E1.13 · Percentage calculations
Core and Extended: percentages, change, simple and compound interest. Reverse percentages are Extended; related growth models follow in E1.17. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Convert percentages to decimals and fractions.
- Multiply decimal quantities and use powers.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What decimal multiplier represents a 7% increase?
What you will learn
- Calculate percentages, percentage changes and profit or loss.
- Use simple and compound interest models.
- Recover an original amount in Extended percentage problems.
Name the whole before calculating
To find 18% of 250, use 0.18 × 250 = 45. To express 45 as a percentage of 250, use (45/250) × 100% = 18%. These reverse calculations work because the same reference whole is used. A value can exceed 100%: 300 is 120% of 250.
Percentage change compares the difference with the original amount. If a price rises from $80 to $92, the increase is $12 and the percentage increase is 12/80 × 100% = 15%. For profit percentage based on cost, divide profit by cost price. Using selling price as the denominator answers a different question, a profit margin.
A multiplier keeps the whole and its change together
An increase of 15% keeps 100% and adds 15%, so multiply by 1.15. A reduction of 15% keeps 85%, so multiply by 0.85. A $240 item discounted by 15% costs $204. For a deposit, calculate the stated fraction of the total and subtract it to find the outstanding amount.
Equal percentage increases and decreases do not cancel, because the second percentage applies to a changed whole. Multiplying by 1.2 then 0.8 gives 0.96: a net 4% decrease. A move from 20% to 25% is five percentage points, but the relative increase is 5/20 × 100% = 25%.
Simple and compound interest use different bases
In a simple-interest model, each period adds the same percentage of the original principal. For $600 at 4% per year over three years, yearly interest is $24 and total interest is $72. The final amount is $672. Match the number of periods to the stated rate before applying I = Prn/100.
In a compound model, each period multiplies the current amount by the same growth factor. For $600 at 4% for three years, the final amount is 600(1.04)³ = $674.9184, reported as $674.92. Interest is final amount minus the original principal. These are mathematical models with stated fixed rates and no extra fees or deposits.
Extended: reverse the multiplier
If a reduced price of $204 is 85% of an unknown original, write 0.85 × original = 204, then divide by 0.85 to get $240. Adding 15% to $204 uses the wrong whole and gives $234.60. Draw the original as 100% and label the amount you actually know before choosing the operation.
Repeated changes multiply their factors. A 10% reduction followed by a 5% reduction retains 0.9 × 0.95 = 0.855 of the original, a 14.5% overall reduction. To recover an original after both changes, divide the final amount by 0.855. Distinguish a compound amount from the interest or change alone.
Pause and explain
An item costs $60 and sells for $75. What is profit as a percentage of cost?
Worked example
A $250 item is reduced by 12%. Find the sale price, then recover the original from that sale price.
Show the worked solution
- The retained share is 100% − 12% = 88%, so the multiplier is 0.88.
- Sale price = 250 × 0.88 = $220.
- Extended reverse check: 220/0.88 = $250, the original.
Answer $220 sale price; $250 original
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the price after a 15% reduction from $240, then find the remaining balance after paying a 20% deposit on that reduced price.
Give me a hint
Use 0.85 for the sale and 0.8 for the outstanding share.
Compare my reasoning
- Reduced price = 240 × 0.85 = $204.
- Deposit = 204 × 0.2 = $40.80.
- Balance = 204 − 40.80 = $163.20.
Sale $204; deposit $40.80; balance $163.20.
Look for these in your work
- Applied the deposit to the reduced price.
- Checked deposit and balance add to $204.
Compare final amounts for $800 at 5% per year over three years under simple and annual compound interest, with no other changes.
Give me a hint
Simple interest uses the original $800 each year; compound interest uses the growing amount.
Compare my reasoning
- Simple yearly interest = 800 × 0.05 = $40; final amount = 800 + 3 × 40 = $920.
- Compound amount = 800 × 1.05³ = $926.10.
- The compound amount is $6.10 larger because later interest includes earlier interest.
Simple $920; compound $926.10.
Look for these in your work
- Distinguished interest from final amount.
- Used the correct base for each model.
Extended: a price is $153 after a 15% discount. Find its original price.
Give me a hint
The known $153 represents 85%, not 100%.
Compare my reasoning
- Write 0.85 × original = 153.
- Original = 153/0.85 = $180.
- Check: $180 less 15% is $153.
$180
Look for these in your work
- Divided by the retained multiplier.
- Checked by applying the original discount.
A shop increases a price by 20%, then reduces the new price by 20%. A customer says this restores the original. Test the claim using a multiplier and a $150 example.
Give me a hint
The percentage bases change between the two steps.
Compare my reasoning
- Combined multiplier = 1.2 × 0.8 = 0.96.
- $150 becomes $180, then $144.
- The final price is 96% of the original, a 4% decrease; the claim is false.
False: a 4% net decrease; $150 becomes $144.
Look for these in your work
- Multiplied the sequential change factors.
- Explained why equal rates do not mean equal money changes.
Common mistakes
- Dividing a percentage change by the final amount.
- Adding a discount percentage to reverse a reduction.
- Reporting the compound final amount as interest alone.
What can you explain now?
A value increases from 48 to 60. Find the percentage increase.
Compare with the explanation
25%
The change is 12, measured against the original 48: 12/48 × 100% = 25%.
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Make it stick
Calculate a 12% increase followed by a 12% decrease using multipliers. Explain the difference between simple interest, compound amount and total interest.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.