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
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide