NumberFoundation + Higher★ Core topic
Percentages.
Change, increase, decrease, reverse.
A percentage is a fraction of 100. Use multipliers to switch between percentages, decimals and fractions. For % change: subtract, divide, multiply by 100. For reverse %: divide by the multiplier.
I · Key definitions
Percentage
A fraction out of 100.
25% = 25/100 = 1/4
Multiplier
Decimal version of the total ratio.
+20% → × 1.20 · −15% → × 0.85
% change
How much something has grown or shrunk relative to original.
(new − old)/old × 100
Compound
Interest applied to new total each time, not just original.
3% per year over 5 years → × 1.03⁵
Depreciation
A decrease each period (cars, equipment).
15% per year → × 0.85 each year
Reverse %
Finding the original amount before a % change.
£42 after 30% off — find original
II · The multiplier cheat sheet
INCREASE — multiply by:
+5%× 1.05
+10%× 1.10
+15%× 1.15
+20%× 1.20
+25%× 1.25
DECREASE — multiply by:
−5%× 0.95
−10%× 0.90
−15%× 0.85
−20%× 0.80
−25%× 0.75
Rule: start at 1.00 (= 100%), then add or subtract the percentage (as a decimal). £80 increased by 15% = 80 × 1.15 = £92.
III · The three formulas you must know
% change
% change = (new − old) / old × 100
£50 → £65 : (65−50)/50 × 100 = 30% increase
Compound interest / growth
Final = Original × (multiplier)ⁿ
where n = number of periods
£500 at 3% for 4 yr: 500 × 1.03⁴ = £562.75
Reverse % (finding original)
Original = Final ÷ multiplier
£42 after 30% off → 42 ÷ 0.70 = £60
Simple interest
I = PRT/100
P=principal, R=rate, T=time (years)
£500, 5%, 3 yr → I = 500×5×3/100 = £75
IV · Worked examples
Increase by a percentage
2 marks
Increase £80 by 15%.
i.Multiplier for +15% = 1 + 0.15 = 1.15
ii.80 × 1.15 = 92
£80 + 15% = £92
Compound interest
3 marks
Alma invests £2000 at 4% compound interest per year. How much after 5 years?
i.Multiplier = 1.04
ii.Apply 5 times: 2000 × 1.04⁵
iii.= 2000 × 1.2166529… = 2433.305…
£2433.31 (to the nearest penny)
Reverse percentage
3 marks
A laptop is £480 after a 20% discount. Find the original price.
i.20% off → multiplier = 1 − 0.20 = 0.80
ii.£480 = original × 0.80
iii.Original = 480 ÷ 0.80 = £600
Original price = £600
V · Common mistakes & examiner tips
Common mistakes
Using 0.15 instead of 1.15 for a 15% increase. Multiplier is total, not just the change.
Dividing by the new price in % change. Always divide by the original.
Multiplying by the % back to reverse. No — divide by the multiplier.
Compound vs simple mixed up. Compound = multiplier raised to n. Simple = PRT/100.
20% down then 20% up ≠ original. 0.80 × 1.20 = 0.96 = 4% smaller.
Examiner tips
Always write the multiplier first — gets the method mark even if arithmetic slips.
Work out 10% first by moving decimal one place (mental arithmetic).
Check reverse % answers by doing the change forward.
For compound, use the ^ button on your calculator.
Always round money to 2 d.p. unless stated otherwise.