Percentages

A comprehensive lesson on percentages — converting between fractions, decimals and percentages, finding a percentage of a number, and calculating percent change.

What Is a Percentage?

A percentage is a fraction with denominator 100.

$25\% = \frac{25}{100} = 0.25$

A percentage bar: how much is shaded out of 100 equal parts.

Converting: Percent ↔ Decimal ↔ Fraction

Percentage	Decimal	Fraction
50%	0.5	1/2
25%	0.25	1/4
10%	0.1	1/10
1%	0.01	1/100
33.3%	0.333	1/3

Finding a Percentage of a Number

$P\% \text{ of } N = \frac{P}{100} \times N$

Mental maths shortcut:

10% = divide by 10
5% = half of 10%
20% = double of 10%
1% = divide by 100

Expressing One Number as a Percentage of Another

$\frac{\text{part}}{\text{whole}} \times 100$

27 out of 60: $\frac{27}{60} \times 100 = 45\%$

Percent Increase and Decrease

$\%\text{ change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$

New value after a change: $\text{New} = \text{Old} \times \left(1 + \frac{P}{100}\right) \quad \text{(increase)}$

Warning Always divide by the ORIGINAL value when calculating percent change.

A phone cost €240. After a 15% discount, what is the sale price?

Discount: $0.15 \times 240 = 36\text{ EUR}$

Sale price: $240 - 36 =$ €204

Or directly: $240 \times 0.85 = 204\text{ EUR}$

Real world: VAT on a purchase

A laptop costs €600 before VAT. With a 20% VAT rate, the tax is $600 \times 0.20 = 120\text{ EUR}$ , making the total €720. Consumers and businesses both need percentages to check whether a quoted price includes or excludes VAT.

Real world: exam results and grade boundaries

A student scores 68 out of 80 on a test. As a percentage: $\frac{68}{80} \times 100 = 85\%$ . Grade boundaries are set as percentages, so converting raw marks to percentages is essential for comparing results across different papers.

Real world: shop discounts

A jacket originally priced at €120 is reduced by 35%. Discount $= 120 \times 0.35 = 42\text{ EUR}$ . Sale price $= 120 - 42 = 78\text{ EUR}$ , or directly $120 \times 0.65 = 78\text{ EUR}$ . Using the multiplier method is faster and less prone to error.

Tip Use the multiplier shortcut: a 15% discount means multiply by 0.85; a 7% increase means multiply by 1.07. One step instead of two.

Practice Problems

What is 12% of 350?

Show answer

0.12 × 350 = 42.

A score went from 60 to 75. What is the percent increase?

Show answer

(75−60)/60 × 100 = 25%.

After a 20% increase a value is 84. What was the original?

Show answer

84 / 1.20 = 70.