Percentage models, reverse changes and large increases
Identify the comparison base and distinguish percentage change, percentage points and repeated multipliers.
What you will tackle
- Interpret percentage growth/decay factors, including changes of at least 100%
- Calculate forward, reverse and successive percentage changes
- Apply percentage models to tax, interest, tips and discounts
- Distinguish percentage points from relative percentage change
A percentage needs a reference amount
Increasing A by p% gives A(1 + p/100). A 150% increase gives 2.5A, not 1.5A: the increase itself is 1.5A. A rate rising from 20% to 30% changes by 10 percentage points and by 50% relative to its original rate.
Reverse or chain whole multipliers
To recover a price before 8% tax, divide the final price by 1.08. A 20% reduction followed by a 20% increase multiplies by 0.8·1.2 = 0.96. For compound growth, repeat a multiplier; for simple interest, interest is based on the original principal each period.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
A population of 80 increases by 150%. Find the new population. A rate rises from 20% to 30%; give both kinds of change.
Show a starting hint
Separate the original amount from the increase.
Compare with the explained answer
- The growth multiplier is 1 + 1.5 = 2.5.
- 80 × 2.5 = 200.
- The rate increases by 10 percentage points; 10/20 = 50% relative change.
Answer / response criteria: 200; 10 percentage points; 50% relative increase.
Independent practice
Independent 1
A jacket costs $86.40 after a 20% discount and 8% tax on the discounted price. Find the original price.
Compare with the explained answer
- The combined multiplier is 0.8 × 1.08 = 0.864.
- If P is the original price, 0.864P = 86.4.
- P = 100; checking gives 100 × 0.8 × 1.08 = 86.4.
Answer / response criteria: $100.
Application practice
Application 1
Compare $500 invested at 10% annual simple interest and 10% annual compound growth for two years. State which quantity is repeatedly used as the reference.
Compare with the explained answer
- Simple interest adds 500 × 0.1 = 50 each year, giving 600.
- Compounding gives 500 × 1.1² = 605.
- Simple interest keeps the original principal as its base; compounding uses the growing balance.
Answer / response criteria: $600 simple; $605 compound; compounding is $5 higher.
What can you explain now?
Compare $500 invested at 10% annual simple interest and 10% annual compound growth for two years. State which quantity is repeatedly used as the reference.
Compare with the explanation
$600 simple; $605 compound; compounding is $5 higher.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide