Percentages, interest and recovering the original
Identify the reference whole, use percentage multipliers and distinguish simple from repeated growth.
Core and Extended: percentages, change, simple and compound interest. Reverse percentages are Extended; related growth models follow in E1.17.
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
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%.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources