Percentages as Operators Multipliers, Fractions & Percentage Problems

SPEC N12 · NUMBER — PERCENTAGES

Percentages as Operators

Multipliers · Fractions · Percentage Problems

Converting, finding, expressing, increasing, decreasing

What is a Percentage?

PERCENTAGE

A fraction with denominator 100.

45% = 45/100 = 0.45

MULTIPLIER

A single number used to apply a percentage change.

1.15 increases by 15% | 0.85 decreases by 15%.

Percentages appear on all three papers — including Paper 1 (non-calculator). Every method in this lesson must work without a calculator.

Spec Ref N12

⭐ High priority — percentage questions appear in every series. Multiplier method is expected at Higher. Non-calculator breakdown is tested on Paper 1.

Converting: %, Decimal, Fraction

ConversionMethodExample

% → DecimalDivide by 10045% = 0.45

Decimal → %Multiply by 1000.73 = 73%

% → FractionPlace over 100, simplify40% = 40/100 = 2/5

Fraction → %Multiply by 1003/8 = 37.5%

Percentage of a Quantity

A% of B = (A ÷ 100) × B

Multiplier method (fastest — works on any paper):

Example — Calculator

Find 15% of £240.

Multiplier = 0.15 → 0.15 × 240 = £36

Example — Definition

"A% of B" = (A/100) × B

15% of 240 = (15/100) × 240 = 0.15 × 240 = 36

⚠️ EXAMINER ALERT: Students write 15% of 240 = 240 ÷ 15 = 16. Wrong — division is reversed.

Always multiply by the decimal: 0.15 × 240.

Paper 1: Non-Calculator Method

When no calculator is allowed, build up the percentage from easy parts.

Method — Build from 10%

Find 15% of £240 (no calculator)

10% of 240 = 24 (÷10)

5% of 240 = 12 (half of 10%)

1% of 240 = 2.4 (÷100)

15% = 10% + 5% = 24 + 12 = £36

Method — Build from 1%

Find 17% of £500

1% of 500 = 5

17% = 17 × 5 = £85

Paper 1 Tip

Find 10% by dividing by 10. Find 5% by halving the 10% answer. Find 1% by dividing by 100. Build any percentage from these building blocks.

Percentage Increase

To increase a value by p%, multiply by a multiplier greater than 1:

Multiplier = 1 + p/100

Increase by 25%

Multiplier = 1 + 0.25 = 1.25

80 × 1.25 = 100

Increase by 3%

Multiplier = 1 + 0.03 = 1.03

£90,000 × 1.03 = £92,700

Percentage Decrease

To decrease a value by p%, multiply by a multiplier less than 1:

Multiplier = 1 − p/100

Decrease by 15%

Multiplier = 1 − 0.15 = 0.85

120 × 0.85 = 102

Decrease by 12%

Multiplier = 1 − 0.12 = 0.88

NOT 0.12 — that would remove 88%!

⚠️ EXAMINER ALERT — Wrong decrease multiplier. Students write: decrease by 12% → × 0.12. This is wrong — 0.12 only keeps 12% of the value (removes 88%).

What to write instead: Multiplier for decrease = 1 − p/100 = 1 − 0.12 = 0.88.

One Quantity as a % of Another

Percentage = (part ÷ whole) × 100%

Example

12 out of 80 as a percentage:

(12 ÷ 80) × 100 = 0.15 × 100 = 15%

Percentage profit or loss — always divide by the original (cost price):

Percentage change = (change ÷ original) × 100%

Example

Cost price £40, selling price £52.

Change = 52 − 40 = 12

% profit = (12 ÷ 40) × 100 = 30%

⚠️ EXAMINER ALERT — dividing by the wrong value. Students divide the change by the new value: (12 ÷ 52) × 100 = 23%. Always divide by the original.

MCE Worked Example

Q: A shop increases the price of a coat from £60 to £75. Find the percentage increase. 3 marks

M Percentage increase = (change ÷ original) × 100

Identify: change = 75 − 60 = 15; original = 60

C Change = 75 − 60 = 15

(15 ÷ 60) × 100

= 0.25 × 100 = 25

EPercentage increase = 25%

Check: 25% of 60 = 15. £60 + £15 = £75 ✓

Mark Scheme Note

M1: correct formula (change ÷ original) stated or implied. M1: correct substitution (15 ÷ 60). A1: 25%. The sanity check earns no marks but prevents silly errors.

Practice Question

A jacket costs £85. In a sale it is reduced by 20%.

Work out the sale price of the jacket. 2 marks

M:

Decrease by 20% → multiplier = 1 − 0.20 = 0.80

C:

85 × 0.80 = 68

E:

Sale price = £68

Check: 20% of 85 = 17. 85 − 17 = 68 ✓

Compound Percentage Changes

When a value undergoes two or more separate percentage changes, multiply the multipliers — do not add or subtract the percentages.

Final = Original × multiplier₁ × multiplier₂ × …

Example — Jun24 P2 Q10

A house worth £90,000 increases by 3% in year 1 and 8% in year 2.

Find its value after 2 years.

Year 1 multiplier: 1.03 | Year 2 multiplier: 1.08

90,000 × 1.03 × 1.08 = 90,000 × 1.1124 = £100,116

AQA Examiner 2024

"Very well answered — most students used multipliers 1.03 and 1.08 correctly." Students who added the percentages (3 + 8 = 11%) received SC1 only.

Common Error: Adding Percentages

⚠️ EXAMINER ALERT — adding percentages instead of multiplying multipliers.

A value increases by 20%, then decreases by 20%. Students think it returns to the original.

What students writeWhat to write instead

+20% then −20% = 0% change. Original returned.1.20 × 0.80 = 0.96 → 4% net decrease

3% + 8% = 11% total increase → × 1.111.03 × 1.08 = 1.1124 → 11.24% total increase

Why it goes wrong

Adding 20% and subtracting 20% would only cancel if both percentages referred to the same base amount. They don't. The second change is applied to the already-changed value, so the percentages are not symmetric.

Practice Question

A car is worth £12,000.

In year 1 it loses 15% of its value.

In year 2 it loses 10% of its value at the start of year 2.

Work out the value of the car after 2 years. 3 marks

M:

Year 1 decrease of 15% → multiplier = 0.85

Year 2 decrease of 10% → multiplier = 0.90

Multiply multipliers: 12,000 × 0.85 × 0.90

C:

12,000 × 0.85 = 10,200

10,200 × 0.90 = 9,180

E:

Value after 2 years = £9,180

(Not £12,000 × 0.75 = £9,000 — that would assume both % apply to the same base.)

The Multiplier Number Line

All percentage changes map to a single multiplier:

Increase by p%

Multiplier = 1 + p/100

Always > 1

Decrease by p%

Multiplier = 1 − p/100

Always between 0 and 1

Common Errors — Don't Lose These Marks

Students often writeCorrect versionWhy it matters

15% of 240 = 240 ÷ 15240 × 0.15 = 36Division is reversed — always multiply

% change = (change ÷ new) × 100% change = (change ÷ original ) × 100Standard error — loses 1–2 marks per question

Decrease by 12%: × 0.12× 0.88 (= 1 − 0.12)0.12 keeps only 12% of the value

+20% then −20% = no change1.20 × 0.80 = 0.96 (−4%)Compound changes: always multiply multipliers

Exam Tip

After any percentage calculation, do a quick sanity check. If a price increases by 25% and your answer is lower than the original, something is wrong. Always ask: is my answer bigger or smaller than expected?

Practice Question

In a class of 32 students, 24 passed their mock exam.

What percentage of students passed? 2 marks

M:

Percentage = (part ÷ whole) × 100

Part = 24, whole = 32

C:

(24 ÷ 32) × 100 = 0.75 × 100 = 75

E:

75% of students passed.

Paper 1 Non-Calculator Practice

Without a calculator, work out 35% of £480. 2 marks

M:

Build from 10%: 10% of 480 = 48 | 5% of 480 = 24 | 1% of 480 = 4.8

C:

35% = 30% + 5%

30% of 480 = 3 × 48 = 144

5% of 480 = 24

35% = 144 + 24 = 168

E:

35% of £480 = £168

Alternative: 35% = 0.35 × 480 = 168 ✓

Summary

Key Points — Lesson 1.07

Percentage = fraction with denominator 100. 45% = 45/100 = 0.45.

"A% of B" = (A/100) × B. Multiplier method: 0.15 × 240 = 36.

Increase by p%: multiply by (1 + p/100). Decrease by p%: multiply by (1 − p/100).

% change = (change ÷ original ) × 100%. Always divide by the original.

Compound changes: multiply multipliers. Never add percentages.

Formulae to remember

% of quantity: (A/100) × B

Increase multiplier: 1 + p/100

Decrease multiplier: 1 − p/100

% change: (change ÷ original) × 100

Top 2 Examiner Errors

1. Dividing % change by the new value, not the original.

2. Using × 0.12 for a 12% decrease instead of × 0.88.

Exam Tip

For any compound percentage question: write down each multiplier, then multiply them all by the starting value. One product, one answer — much faster than working step-by-step and less likely to introduce rounding errors.