📐 Concept diagram

00-04 — Percentages

Phase: 0 — Arithmetic & Number Foundations Subject: 00-04 Prerequisites: 00-03 — Decimals Next subject: See the curriculum overview for Phase 1 subjects

Learning Objectives

By the end of this subject, you will be able to:

1. Express any percentage as a fraction (with denominator 100) and as a decimal, and convert freely between all three representations
2. Calculate a percentage of a quantity, and express one quantity as a percentage of another
3. Calculate percentage increases and decreases, including successive percentage changes
4. Solve reverse percentage problems (finding the original amount before a percentage change)
5. Apply the compound interest formula to calculate growth over multiple periods, and distinguish simple from compound interest

Core Content

1. What Is a Percentage?

The word percent comes from Latin per centum, meaning "per hundred" or "out of 100."

A percentage is a fraction with denominator 100.

$x% = x/100
$

Concrete analogy: If you score 85% on a test, that means you got 85 marks out of every 100 possible — equivalent to 85/100 of the total.

The symbol "%" is shorthand for "/100." Whenever you see % in a calculation, mentally replace it with "÷ 100."

2. Converting Between Percentages, Fractions, and Decimals

These three representations are different ways of writing the same number. The key conversions:

$Percentage  ←→  Fraction  ←→  Decimal
    ↓               ↓              ↓
  x% = x/100    x/100 = x÷100   x/100 = x÷100
$

Percentage → Decimal

Rule: Divide by 100 (move the decimal point 2 places left).

$85% = 85 ÷ 100 = 0.85
7%  = 7 ÷ 100  = 0.07
150% = 150 ÷ 100 = 1.5
0.5% = 0.5 ÷ 100 = 0.005
$

Why moving the decimal two places works: Dividing by 100 is the same as dividing by 10 twice. Each division by 10 shifts the decimal point one place left.

Decimal → Percentage

Rule: Multiply by 100 (move the decimal point 2 places right).

$0.85 = 0.85 × 100 = 85%
0.07 = 0.07 × 100 = 7%
1.5  = 1.5 × 100  = 150%
0.005 = 0.005 × 100 = 0.5%
$

Percentage → Fraction

Rule: Write over 100 and simplify.

$85% = 85/100 = (÷5) = 17/20
25% = 25/100 = (÷25) = 1/4
12.5% = 12.5/100 = 125/1000 = (÷125) = 1/8
33 1/3% = (100/3)/100 = 100/300 = 1/3
$

Fraction → Percentage

Rule: Convert the fraction to a decimal (divide numerator by denominator), then multiply by 100.

$3/4 = 3 ÷ 4 = 0.75 = 75%
2/5 = 2 ÷ 5 = 0.4 = 40%
1/3 = 1 ÷ 3 = 0.333... = 33.333...% = 33 1/3%
$

Common percentages worth memorising:

3. Finding a Percentage of a Quantity

Formula:

$x% of a quantity = (x/100) × quantity
$

Equivalently: convert the percentage to a decimal and multiply.

Example 1: Find 15% of £240

$Method 1: Fraction method
  15% of 240 = (15/100) × 240 = (3/20) × 240 = 3 × 12 = 36

Method 2: Decimal method
  15% = 0.15
  0.15 × 240 = 36
$

Answer: £36

Example 2: Find 17.5% of £360

$17.5% = 0.175
0.175 × 360 = 63
$

Mental shortcuts (break down the percentage):

$Find 32% of 450:

10% of 450 = 45
30% of 450 = 45 × 3 = 135
1% of 450 = 4.5
2% of 450 = 4.5 × 2 = 9

32% = 30% + 2% = 135 + 9 = 144
$

4. Expressing One Quantity as a Percentage of Another

Formula:

a as a percentage of b = (a ÷ b) × 100%

Example: In a class of 32 students, 12 are left-handed. What percentage is left-handed?

$(12 ÷ 32) × 100% = 0.375 × 100% = 37.5%
$

Check: 37.5% of 32 = 0.375 × 32 = 12 ✓

5. Percentage Increase

When a quantity goes up by a percentage, the new value is:

New value = Original + (percentage × Original)
          = Original × (1 + percentage as decimal)

Derivation:

$Original = P
Increase = r% of P = (r/100) × P
New = P + (r/100)P = P × (1 + r/100)
$

The factor (1 + r/100) is called the multiplier.

Formula:

New value = Original × (1 + r/100) where r is the percentage increase.

Example: A coat originally costs £80. In a sale, its price increases by 15%. What is the new price?

$Increase = 15% of £80 = 0.15 × 80 = £12
New price = £80 + £12 = £92

Using the multiplier:
New price = £80 × (1 + 0.15) = £80 × 1.15 = £92
$

Answer: £92

6. Percentage Decrease

When a quantity goes down by a percentage:

$New value = Original × (1 − r/100)
$

Derivation: Same as increase, but subtract:

$New = P − (r/100)P = P × (1 − r/100)
$

The factor (1 − r/100) is the multiplier for a decrease.

Example: A laptop originally costs £650. It's discounted by 20%. What is the sale price?

$Decrease = 20% of £650 = 0.20 × 650 = £130
Sale price = £650 − £130 = £520

Using the multiplier:
Sale price = £650 × (1 − 0.20) = £650 × 0.80 = £520
$

Answer: £520

7. Successive Percentage Changes

When multiple percentage changes happen one after another, you apply each multiplier in sequence. The overall effect is NOT simply the sum of the percentages.

Key insight: A 10% increase followed by a 10% decrease does NOT bring you back to the original. The decrease applies to the NEW (larger) amount.

Formula for successive changes:

$Final = Original × m₁ × m₂ × m₃ × ...
$

where each m is the multiplier for one change (1 + r/100 for increase, 1 − r/100 for decrease).

Example 1: A share price of £200 increases by 25% then falls by 20%. What is the final price?

$After increase: £200 × 1.25 = £250
After decrease: £250 × 0.80 = £200

Overall multiplier: 1.25 × 0.80 = 1.00
Final: £200 × 1.00 = £200
$

Answer: £200 — the price returns to the original because (1.25)(0.80) = 1.00 exactly.

Example 2: A salary of £30,000 increases by 5% for three consecutive years. What is the final salary?

Multiplier for one increase: 1.05
Overall: 1.05 × 1.05 × 1.05 = 1.05³

1.05² = 1.1025
1.05³ = 1.1025 × 1.05 = 1.157625

Final salary = £30,000 × 1.157625 = £34,728.75

Common pitfall: Some people add: 5% + 5% + 5% = 15%, so salary = £30,000 × 1.15 = £34,500. But the actual increase is 15.7625% — bigger because each 5% applies to the PREVIOUSLY increased amount (compounding). This is the same principle as compound interest.

⚠️ THIS IS CRITICAL — Understanding that successive percentage multipliers multiply (not add) is essential for compound interest, inflation, population growth, radioactive decay, and any growth/decay model in later math and science subjects.

8. Reverse Percentages

A reverse percentage problem gives you the value AFTER a percentage change and asks: what was the original?

Formula:

$Original = Final ÷ multiplier
$

where the multiplier is (1 + r/100) for an increase or (1 − r/100) for a decrease.

Derivation:

$Final = Original × multiplier
→ Original = Final ÷ multiplier
$

Example 1 (increase): After a 15% pay rise, Anya earns £34,500. What was her original salary?

$This is an increase, so multiplier = 1 + 15/100 = 1.15
Original = £34,500 ÷ 1.15 = £30,000
$

Check: £30,000 × 1.15 = £34,500 ✓

Example 2 (decrease): A TV is reduced by 30% in a sale and now costs £420. What was the original price?

$This is a decrease, so multiplier = 1 − 30/100 = 0.70
Original = £420 ÷ 0.70 = £600
$

Check: £600 × 0.70 = £420 ✓

Common pitfall: Students often try to "add back" the percentage to the final value. For Example 1: "34,500 − 15% = 34,500 − 5,175 = 29,325." WRONG! The 15% in the problem was 15% of the ORIGINAL, not 15% of the final. 15% of £30,000 = £4,500, which added to £30,000 gives £34,500. But 15% of £34,500 = £5,175, which is a different number. You cannot apply the percentage to the wrong base.

How to distinguish forward from reverse problems: - Forward: "What is 20% of...", "Increase ... by 15%", "Decrease ... by 10%" → multiply - Reverse: "After a 15% increase the price is...", "...is 80% of the original" → divide

9. Simple Interest vs Compound Interest

Interest is money earned (or paid) on an investment (or loan). The original amount is called the principal (P).

Simple Interest

With simple interest, you earn interest ONLY on the original principal. The interest does NOT earn interest itself.

Simple interest after n years = P × r × n

where:  P = principal (original amount)
        r = annual interest rate (as a decimal)
        n = number of years

Total amount after n years:

$A = P + P × r × n = P(1 + rn)
$

Example: £1,000 invested at 5% simple interest for 3 years.

$Interest per year = £1,000 × 0.05 = £50
Total interest = £50 × 3 = £150
Total amount = £1,000 + £150 = £1,150
$

Compound Interest

With compound interest, each year's interest is ADDED to the principal. The next year's interest is calculated on this new, larger amount. This is "interest on interest."

Amount after n years = P × (1 + r)ⁿ

where:  P = principal
        r = annual interest rate (as a decimal)
        n = number of compounding periods (years)

Derivation:

$Year 0: A₀ = P
Year 1: A₁ = P + P×r = P(1 + r)
Year 2: A₂ = P(1 + r) + P(1 + r)×r = P(1 + r)(1 + r) = P(1 + r)²
Year 3: A₃ = P(1 + r)³
...
Year n: Aₙ = P(1 + r)ⁿ
$

This is the same successive percentage increase pattern from Section 7. Each year is a r% increase applied to the previous year's total.

Example: £1,000 invested at 5% compound interest for 3 years.

$Year 1: £1,000 × 1.05 = £1,050
Year 2: £1,050 × 1.05 = £1,102.50
Year 3: £1,102.50 × 1.05 = £1,157.625

Using the formula: £1,000 × (1.05)³ = £1,000 × 1.157625 = £1,157.63 (to nearest penny)
$

Comparison: Simple interest gave £1,150; compound gave £1,157.63. The difference (£7.63) is the "interest on interest." Over longer periods, this difference becomes much larger.

Compound interest formula for ANY compounding frequency:

$A = P × (1 + r/k)^(k×n)
$

where k is the number of times interest compounds per year (k=12 for monthly, k=4 for quarterly, k=365 for daily).

For Phase 0, we focus on annual compounding (k=1), which simplifies to A = P(1 + r)ⁿ.

⚠️ THIS IS CRITICAL — The compound interest formula introduces exponentiation, which is the gateway to exponential functions. This formula P(1+r)ⁿ is a specific case of the more general exponential growth model: y = a × bˣ, which you'll encounter in Phase 3 (Algebra) and beyond in calculus, finance, biology, and physics.

Key Terms

• Finding a percentage of a quantity
• Percentage increase and decrease
• Reverse percentages
• Simple interest

Worked Examples

Example 1: Multiple Representations

Problem: Express 62.5% as (a) a simplified fraction and (b) a decimal.

Solution:

$(a) Fraction:
  62.5% = 62.5/100 = 625/1000
  Simplify: GCF(625, 1000) = 125
  625 ÷ 125 = 5
  1000 ÷ 125 = 8
  Fraction: 5/8

(b) Decimal:
  62.5% = 62.5 ÷ 100 = 0.625
$

Answer: (a) 5/8, (b) 0.625

Example 2: Finding a Percentage of a Quantity — Multi-Step

Problem: A restaurant bill is £84. You want to leave a 12.5% tip. How much is the tip, and what is the total paid?

Solution:

$12.5% of £84:
  Method 1 — Decimal: 0.125 × 84 = £10.50
  Method 2 — Fraction: 12.5% = 1/8, so £84 × 1/8 = £84 ÷ 8 = £10.50

Total paid = £84.00 + £10.50 = £94.50
$

Answer: Tip = £10.50, Total = £94.50

Example 3: Reverse Percentage

Problem: After a 40% discount, a jacket costs £78. What was the original price?

Solution:

$This is a decrease of 40%, so the multiplier is 1 − 0.40 = 0.60.
The £78 represents 60% of the original price.

Original = Final ÷ multiplier
         = £78 ÷ 0.60
         = £130

Check: £130 × 0.60 = £78 ✓
       The discount was £130 − £78 = £52
       £52/£130 = 0.40 = 40% ✓
$

Answer: £130

Example 4: Successive Percentage Changes

Problem: A population of 50,000 increases by 8% in the first year, then decreases by 5% in the second year. What is the population after two years, and what is the overall percentage change?

Solution:

$Year 1: 50,000 × 1.08 = 54,000
Year 2: 54,000 × 0.95 = 51,300

Overall multiplier: 1.08 × 0.95 = 1.026

Overall change: (1.026 − 1) × 100% = +2.6%

Note: 8% − 5% = 3%, but the actual change is 2.6%.
The 5% decrease applies to 54,000 (a bigger number than 50,000), 
so it's a bigger absolute decrease than 5% of the original would be.
$

Answer: 51,300 people; overall increase of 2.6%

Example 5: Compound Interest

Problem: £5,000 is invested in a savings account paying 3.5% compound interest per year. How much will the account be worth after 4 years? Give your answer to the nearest penny.

Solution:

$P = 5000
r = 3.5% = 0.035
n = 4

A = P × (1 + r)ⁿ
A = 5000 × (1.035)⁴

Calculate (1.035)⁴:
(1.035)² = 1.035 × 1.035 = 1.071225
(1.035)⁴ = 1.071225 × 1.071225 = 1.147523... (let's be precise)

1.035⁴:
1.035 × 1.035 = 1.071225
1.071225 × 1.035 = 1.108717875
1.108717875 × 1.035 = 1.147523000625

A = 5000 × 1.147523000625 = 5737.615003125

Rounded to nearest penny: £5,737.62

Total interest = £5,737.62 − £5,000.00 = £737.62
$

Answer: £5,737.62

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Convert 17/20 to a percentage.

Problem 2: Find 32% of £850.

Problem 3: In a school of 720 students, 162 are in Year 11. What percentage of the school is in Year 11?

Problem 4: A bicycle originally costing £320 is reduced by 15% in a sale. What is the sale price?

Problem 5: After receiving a 12% pay cut, Maria now earns £24,640. What was her original salary?

Problem 6: A company's profits increase by 20% to £360,000, then decrease by 15%. What is the final profit?

Problem 7: £2,000 is invested at 4% compound interest per year. How much will the investment be worth after 5 years? (Round to nearest penny.)

Summary

1. Percent means "per hundred": x% = x/100 = x ÷ 100 as a decimal. Converting between percentages, fractions, and decimals is a matter of multiplying or dividing by 100 and simplifying
2. Finding a percentage of a quantity uses the multiplier x/100 or the decimal equivalent; expressing one quantity as a percentage of another uses (part ÷ whole) × 100%
3. Percentage increase and decrease use multipliers: (1 + r/100) for increases, (1 − r/100) for decreases. Apply each successive change by multiplying the multipliers — never simply add or subtract the percentages
4. Reverse percentages find the original by dividing the final amount by the multiplier. Do NOT apply the percentage to the final value — the percentage is always of the original
5. Simple interest earns interest only on the principal: A = P(1 + rn); compound interest earns interest on interest: A = P(1 + r)ⁿ. Compound interest grows faster and is the foundation of exponential growth models

Pitfalls

• Adding and subtracting successive percentages instead of multiplying multipliers. A 10% increase followed by a 10% decrease does NOT return to the original. The overall multiplier is 1.10 × 0.90 = 0.99 (1% lower). The changes multiply, they never simply add or subtract.
• Applying the percentage to the wrong base in reverse problems. After a 25% increase the price is £500 — the original is £500 ÷ 1.25 = £400, NOT £500 × 0.75 = £375. The 25% in the problem was 25% of the original, not 25% of the final amount.
• Confusing percentage points with percent. If interest rates rise from 4% to 5%, that's a 1 percentage point increase but a 25% increase (1/4 = 0.25). The two measures are numerically very different.
• Using simple interest when compound interest is called for (or vice versa). Compound interest uses exponentiation (A = P(1+r)ⁿ); simple interest uses multiplication (A = P(1+rn)). For multi-year problems the difference is substantial.
• Writing percentages over the wrong denominator. 0.5% is 0.5/100 = 5/1000 = 1/200, not 5/100 = 1/20. The "%" symbol means "÷ 100", so the numerator itself can be fractional.

Quiz

Answer each question, then read the explanation for your choice.

Q1: What is 0.08 as a percentage?

A) 0.08% B) 0.8% C) 8% D) 80%

Q2: Find 15% of £340.

A) £5.10 B) £51.00 C) £510.00 D) £34.00

Q3: A shirt priced at £45 is reduced by 30%. What is the sale price?

A) £13.50 B) £31.50 C) £15.00 D) £42.00

Q4: After a 20% increase, a painting is valued at £9,600. What was its original value?

A) £7,680 B) £8,000 C) £11,520 D) £8,200

Q5: A salary increases by 10% then by 5% in successive years. What is the single overall percentage increase?

A) 15% B) 15.5% C) 14.5% D) 50%

Q6: £4,000 is invested at 3% compound interest per year for 2 years. How much interest is earned?

A) £120.00 B) £240.00 C) £243.60 D) £247.20

Q7: What is 2/5 expressed as a percentage?

A) 25% B) 40% C) 20% D) 2.5%

Q8: A car depreciates (loses value) by 15% each year. If it was bought for £18,000, what is its value after 2 years?

A) £12,600 B) £13,005 C) £13,500 D) £15,300

Q9: 45 out of 180 students walk to school. What percentage is this?

A) 20% B) 25% C) 45% D) 40%

Q10: Which of the following statements is true about 0.9̄ (0.999...)?

A) 0.9̄ is slightly less than 1 B) 0.9̄ as a percentage is 99.9% C) 0.9̄ = 100% D) 0.9̄ as a percentage is 99.999...%

Next Steps

You have now completed Phase 0 — Arithmetic & Number Foundations. You have a solid grounding in whole numbers, fractions, decimals, and percentages.

To continue the curriculum, refer to the curriculum overview for Phase 1 subjects, which build on these foundations and typically cover: - Negative numbers and the number line - Ratios and proportions - Basic measurement and unit conversions - Introduction to algebraic thinking

The specific Phase 1 sequence depends on the curriculum pathway you are following. Check the main index for the recommended next subject.

Q5: A salary increases by 10% then decreases by 10%. The final salary, compared to the original, is:

A) The same B) 1% higher C) 1% lower D) 10% lower

Answer: C) 1% lower

Overall multiplier = 1.10 × 0.90 = 0.99. Final is 99% of original, so 1% lower. The 10% decrease applies to a larger amount.