SPEC R9 • RATIO, PROPORTION & RATES OF CHANGE
Percentage Change: Multipliers Do the Work
Increases, decreases, compound changes, and comparisons — all handled in one multiplication step. A high-frequency topic worth 2–4 marks on every paper at both tiers.
Why Multipliers Matter
The phrase "percentage change" hides a simple idea: a percentage is just a multiplier. Once you switch to multipliers, every percentage problem becomes a one-step calculation.
What You'll Master
Percentage as parts per hundred — and as a fraction or decimal
Increase and decrease multipliers
Finding the percentage change between two values (increase or decrease)
Expressing one quantity as a percentage of another
Comparing two quantities using percentages
Compound (successive) percentage changes
Simple interest in financial contexts
Percentages greater than 100%Higher
Exam Signal
AQA examiner reports show compound percentage questions are "very well answered" when students use multipliers — and lose marks when they add percentages instead (e.g., writing 3% + 8% = 11% earns just SC1 out of 3).
Percentage: Parts per Hundred
Definition — Percentage
A percentage is a fraction with denominator 100. The word literally means "per hundred". So 45% means 45 out of 100, which equals the fraction 45/100 and the decimal 0.45.
Three Forms of the Same Number
PercentageDecimalFraction25%0.251/440%0.402/537.5%0.3753/8Convert: % → decimal divide by 100. Decimal → % multiply by 100.
Increase Multiplier: × (1 + p/100)
Key Rule — Increase
To increase a quantity by p%, multiply by (1 + p/100). This is the increase multiplier — one multiplication replaces "find the percentage, then add it on".
Worked Examples
Increase 120 by 15% → 120 × 1.15 = 138Increase 80 by 25% → 80 × 1.25 = 100Increase 240 by 8% → 240 × 1.08 = 259.20
Memory Hook
"1" keeps the original, "p/100" adds the extra. So 15% increase = 1 (the whole) + 0.15 (the extra) = 1.15. One multiplication replaces two steps.
Paper 1 Non-Calculator Trick
On Paper 1, break awkward percentages into building blocks:
10% of a number = divide by 10
1% of a number = divide by 100
5% of a number = half of 10%
Example: 15% of 240 = 10% + 5% = 24 + 12 = 36. No calculator needed.
Decrease Multiplier: × (1 − p/100)
Key Rule — Decrease
To decrease a quantity by p%, multiply by (1 − p/100). The whole (1) stays, but the percentage is subtracted.
Worked Examples
Decrease 120 by 15% → 120 × 0.85 = 102Decrease 640 by 15% → 640 × 0.85 = 544Decrease 200 by 8% → 200 × 0.92 = 184
The Classic Trap
For a decrease of 8%, the multiplier is 0.92 (which is 1 − 0.08), not 0.08. Students often multiply by 0.08 and get a tiny number — check yours makes sense.
Percentage Change Formula
Formula
% change = (change ÷ original) × 100The key word is original — never the new value. Works for both increases and decreases.
EX
Increase — coat £60 → £75
Change = 75 − 60 = 15
% change = (15 ÷ 60) × 100 = 25% increase
Check: 25% of £60 = £15, so £60 + £15 = £75 ✓
EX
Decrease — shoes £80 → £60
Change = 80 − 60 = 20
% change = (20 ÷ 80) × 100 = 25% decrease
Check: 25% of £80 = £20, so £80 − £20 = £60 ✓. Same formula, with the word "decrease" on the answer.
Most Common Error
Dividing by the new value. For £60 → £75: weak students write 15 ÷ 75 = 20%. The denominator must always be the original (£60), giving 25%.
Express One Quantity as a % of Another
Key Rule
% = (part ÷ whole) × 100Make sure both quantities are in the same unit before dividing.
Worked Examples
QuestionWorkingAnswer12 as a % of 80(12/80) × 10015%4 as a % of 25(4/25) × 10016%50 cm as a % of 2 m(50/200) × 10025%
Comparing Quantities
Percentage gives you a fair way to compare different-sized groups. "4 out of 25" and "16 out of 100" are both 16% — same proportion, expressed in a comparable form.
Compare Two Quantities Using Percentages (R9)
Spec R9 explicitly requires you to compare two quantities using percentages. The method: convert each to a percentage, then compare.
1
Convert each to a percentage using (part ÷ whole) × 100.
2
Compare the percentages and state a conclusion in words.
EX
Compare Pass Rates
At School A, 28 out of 35 students passed. At School B, 42 out of 50 students passed. Which school had the better pass rate?
School A: (28 ÷ 35) × 100 = 80%
School B: (42 ÷ 50) × 100 = 84%
School B had the better pass rate, by 4 percentage points.
Why Percentages for Comparison?
Raw numbers (28 vs 42) mislead because the cohorts differ in size. Percentages scale both to "out of 100" — a fair comparison.
Compound Change: Multiply the Multipliers
Key Rule — Successive Changes
When two (or more) percentage changes are applied in sequence, multiply the multipliers together. Do not add the percentages.
Worked Examples
20% up, 10% down: 1.20 × 0.90 = 1.08 (8% net increase)20% up, 20% down: 1.20 × 0.80 = 0.96 (4% net decrease)3% up, 8% up (Higher): 1.03 × 1.08 = 1.1124 (11.24% net, NOT 11%)
Top Examiner Error
Students add the percentages: "3% + 8% = 11%, multiply by 1.11". The AQA Jun24 Paper 2 Q10 mark scheme awards SC1 (only one mark out of three) for that approach. The correct multiplier is 1.03 × 1.08.
Worked Example — House Prices
EX
4 Marks — Non-calculator style
A house is bought for £200 000. Its value increases by 10% in the first year, then decreases by 10% in the second year. What is its value at the end of the two years?
M
Apply the two multipliers in sequence: × 1.10 for Year 1, × 0.90 for Year 2. A 10% rise followed by a 10% fall is not "no change".
C
Year 1: 200 000 × 1.10 = £220 000.
C
Year 2: 220 000 × 0.90 = £198 000.
E
Value at end of two years = £198 000 — a net 1% decrease from the original. The 10% decrease in Year 2 applied to the larger post-increase figure of £220 000, so the fall in pounds is bigger than the rise was.
Simple Interest (R9)
Spec R9 explicitly requires simple interest in financial mathematics. In simple interest, the same amount of interest is paid each year — based on the original amount only.
Simple Interest Method
Interest per year = principal × (rate/100). Multiply by the number of years for total interest.
EX
Simple Interest
£500 is invested at 4% simple interest per year for 3 years. Find the total interest earned and the final balance.
Interest per year = 500 × 0.04 = £20
Total interest over 3 years = 3 × 20 = £60
Final balance = 500 + 60 = £560
Simple vs Compound
Simple interest uses the original principal every year. Compound interest (Lesson 3.08) uses the new balance each year — earning interest on interest. For short periods the difference is small; for long periods it's huge.
Percentages Greater Than 100% Higher
Spec R9 Higher content requires you to work with percentages greater than 100%. This includes multipliers above 2.00 and percentage changes above 100%.
Interpretation
150% of a quantity = 1.5 × the quantity. So 150% of 80 = 1.5 × 80 = 120.
Increase by 250% means multiplier = 1 + 2.50 = 3.50. So 80 becomes 80 × 3.50 = 280 (3.5× the original).
EX
% change over 100%
A company's revenue grew from £40,000 to £120,000 in a year. Find the percentage increase.
Change = 120,000 − 40,000 = £80,000
% change = (80,000 ÷ 40,000) × 100 = 200%
The revenue tripled — a 200% increase means the new value is 3× the original.
Examiner Alert — Dividing by the Wrong Value
Most Common % Change Error
A coat goes from £60 to £75. Weak student writes:% change = (15 ÷ 75) × 100 = 20% ✗The denominator must be the original (£60), not the new price:
Correct Method
% change = (15 ÷ 60) × 100 = 25% ✓Verify the answer: 25% of £60 = £15, so £60 + £15 = £75 ✓. The calculation is consistent with the original price, which is why original is the right denominator.
Habit to Build
Say it out loud: "change divided by original". Write the word "original" on the denominator every time until it becomes automatic.
Examiner Alert — Adding Percentages in Compound Change
AQA Jun24 P2 Q10
A house is worth £90,000. It rises by 3% in year 1 and 8% in year 2. Find its value after 2 years. The examiner report noted this was "very well answered" — but students who wrote £90,000 × 1.11 (adding 3% + 8%) earned only SC1 (one mark out of three).
Correct vs Incorrect
Weak (1 mark)Strong (3 marks)90,000 × 1.11 (adds percentages)90,000 × 1.03 × 1.08 (multiplies multipliers)The correct multiplier is 1.03 × 1.08 = 1.1124, not 1.11.
Why It Matters
The 8% in Year 2 applies to the increased value after Year 1, not the original. Only multiplying the multipliers captures that.
Practice Question
A flat is bought for £150,000. Its value rises by 4% in the first year, then by 6% in the second year.
(a) Work out the value of the flat at the end of the two years. [3 marks]
(b) Explain why the overall percentage change is not 10%. [1 mark]
M
(a) Apply multipliers 1.04 (Year 1) and 1.06 (Year 2) to £150,000. (b) Compare the actual combined multiplier to 1.10.
C
(a) Year 1: 150,000 × 1.04 = £156,000. Year 2: 156,000 × 1.06 = £165,360.
C
(b) Combined multiplier = 1.04 × 1.06 = 1.1024 → net change is 10.24%, not 10%.
E
(a) £165,360. (b) The 6% Year 2 increase applied to £156,000 (the larger post-Year-1 value), not to £150,000, so the total increase is slightly bigger than 10%.
Percentage Pitfalls: Quick Recap
TrapFix
Multiplier for decrease is just p/100 (e.g. 0.08)Multiplier for decrease is 1 − p/100 (e.g. 0.92)
Dividing change by the new valueDivide change by the original value
Adding percentages for compound changeMultiply the multipliers : 1.03 × 1.08, not 1.11
"20% up then 20% down = no change"1.20 × 0.80 = 0.96 = 4% net decrease
Confusing "150% of" with "increase by 150%"150% of = × 1.50. Increase by 150% = × 2.50.
Compound interest method used for simple interestSimple: same amount of interest each year (on the original)
Lesson Summary
✓ Multipliers Do the Work
Increase by p% → × (1 + p/100). Decrease by p% → × (1 − p/100). One multiplication, one step.
✓ Percentage Change Formula
% change = (change ÷ original) × 100. Always divide by the original value, never the new.
✓ Compound Change
Successive changes → multiply the multipliers. Never add the percentages. 1.03 × 1.08 = 1.1124, not 1.11.
✓ Simple Interest
Same interest each year, based on the original principal. Compound interest is separate (Lesson 3.08).
✓ Over 100% Higher
"150% of" = × 1.50. "Increase by 150%" = × 2.50. Percentage changes above 100% are valid — the new value is more than double the original.
Next: Why Multipliers Matter