What is a percentage?
A percentage is a number out of 100. The sign % means "divide by 100". So 7% = 7/100 = 0.07.
Changing between fractions, decimals and percentages
- Percent → decimal: divide by 100 (35% = 0.35).
- Decimal → percent: multiply by 100 (0.6 = 60%).
- Fraction → percent: divide top by bottom, then × 100 (3/8 = 0.375 = 37.5%).
- Percent → fraction: write over 100 and simplify (45% = 45/100 = 9/20).
Proportion as a percentage
A proportion is a part compared with the whole. If 18 of 24 students walk to school, the proportion is 18/24 = 0.75 = 75%.
Finding a percentage of an amount
"x% of A" = A × x/100. For 15% of 360: 360 × 0.15 = 54.
A quick mental way: 10% is one tenth, 5% is half of 10%, 1% is one hundredth. 15% of 360 = 36 + 18 = 54.
One number as a percentage of another
Write it as a fraction of the whole, then × 100. 12 out of 40 = 12/40 × 100 = 30%. Use the same unit for both numbers first (45 min out of 2 h = 45/120 = 37.5%).
Percentage change and multipliers
Percentage change = (new − original) ÷ original × 100. A plus answer is an increase; a minus answer is a decrease. Always divide by the original value.
Multipliers
Increase by r% → multiply by (1 + r/100). Decrease by r% → multiply by (1 − r/100). Examples: +8% → × 1.08; −30% → × 0.7. A multiplier does the change in one step, and it tells you the rate: × 0.85 means a 15% decrease.
Percentage points
If a rate goes from 20% to 25%, it rose by 5 percentage points, but by 25% of its old value (5 ÷ 20). News reports often mix these up.
Successive and reverse percentages
Successive changes: multiply the multipliers. +10% then −10%: 1.1 × 0.9 = 0.99, so overall −1%. +20% then +20% is × 1.44 = +44%, not +40%.
Repeated change (compound growth): r% per year for n years → × (1 + r/100)ⁿ. ₹10,000 at 5% a year for 3 years: 10,000 × 1.05³ = ₹11,576.25.
Reverse percentage: you know the value after the change. Divide by the multiplier. A price after 18% tax is ₹590, so before tax it was 590 ÷ 1.18 = ₹500. Do not take 18% off 590.
Key formulas and definitions
- x% = x ÷ 100
- x% of A = A × x/100
- A as % of B = (A ÷ B) × 100
- Percentage change = (new − original) ÷ original × 100
- Increase r%: × (1 + r/100); decrease r%: × (1 − r/100)
- Successive changes: multiply the multipliers
- Repeated change: final = original × (1 ± r/100)ⁿ
- Reverse: original = new ÷ multiplier
Worked examples
1. Write 3/5 as a percentage and as a decimal.
3 ÷ 5 = 0.6. 0.6 × 100 = 60%.
2. Find 35% of 240 m.
240 × 0.35 = 84 m. (Check: 10% = 24, 30% = 72, 5% = 12, total 84.)
3. A class has 32 students and 12 wear glasses. What percentage wear glasses?
12 ÷ 32 = 0.375, × 100 = 37.5%.
4. A bus fare rises from ₹40 to ₹46. Find the percentage increase.
Change = 6. 6 ÷ 40 × 100 = 15% increase.
5. A laptop costs €800. It is reduced by 15%, then by a further 10%. Find the final price and the overall percentage change.
Multipliers 0.85 and 0.9. 800 × 0.85 × 0.9 = €612. Overall multiplier 0.765, so a 23.5% decrease (not 25%).
6. After a 12% rise, a town has 28,000 people. How many were there before?
Multiplier 1.12. 28,000 ÷ 1.12 = 25,000 people.
Common mistakes
- Dividing by the new value in percentage change. Always divide by the original.
- Adding successive percentages: +20% then +20% is +44%, not +40%.
- Reverse percentage by subtracting: to undo +18%, divide by 1.18; do not take 18% off.
- Mixing percent and percentage points: 20% → 25% is +5 points but a 25% rise.