Why do a 20% increase and 20% decrease not cancel out?

Because each percentage is applied to a different base. Starting at £100: +20% gives £120, then −20% of £120 gives £96 — not £100. The percentages use different denominators. To return to exactly £100 after a 20% increase, you need approximately a 16.67% decrease (£120 × 0.833 = £100).

What is the formula to find the original price after a percentage change?

Divide the new price by (1 + percentage/100). If a house is worth £330,000 after a 10% increase: original = £330,000 ÷ 1.10 = £300,000. For a decrease: if a car is worth £18,000 after a 25% depreciation: original = £18,000 ÷ 0.75 = £24,000.

How do I calculate the percentage change for multiple years of inflation?

Multiply the multipliers together. For three years of 3% inflation on a £100 item: £100 × 1.03 × 1.03 × 1.03 = £100 × 1.0927 = £109.27. Don't add the percentages (3 + 3 + 3 = 9% → £109 is slightly wrong). The compounding effect creates a difference that grows with time.

What is the correct base for calculating a pay rise percentage?

Always use the original (before-raise) salary as the base. If your salary goes from £30,000 to £32,500, the increase is: (32,500 − 30,000) ÷ 30,000 × 100 = 8.3%. Using the new salary would give (2,500 ÷ 32,500) × 100 = 7.7% — the same absolute raise looks smaller when using the higher base.

How to Calculate Percentage Increase and Decrease

Percentage increase and decrease appear constantly in everyday life — house price changes, pay rises, inflation rates, discount calculations, and statistics in news headlines. Despite this, many people get the formula slightly wrong, particularly when applying multiple percentage changes in sequence. This guide gives you the exact formulas, common examples, and the key pitfalls to avoid.

Percentage Increase Formula

% Increase = [(New Value − Old Value) ÷ Old Value] × 100

Example: House Price Rise

A house was valued at £280,000 last year. It's now worth £315,000. What is the percentage increase?

% Increase = [(315,000 − 280,000) ÷ 280,000] × 100
= [35,000 ÷ 280,000] × 100
= 0.125 × 100 = 12.5%

Percentage Decrease Formula

% Decrease = [(Old Value − New Value) ÷ Old Value] × 100

Example: Sale Price Reduction

A laptop originally costs £1,200. In a sale it's reduced to £900. What's the percentage decrease?

% Decrease = [(1,200 − 900) ÷ 1,200] × 100
= [300 ÷ 1,200] × 100
= 0.25 × 100 = 25%

One Formula for Both

You can use a single formula that works for both increases and decreases:

% Change = [(New − Old) ÷ Old] × 100

A positive result is an increase; a negative result is a decrease. This is the most versatile version and is what our percentage calculator uses.

Finding the New Value After a % Change

New value = Old value × (1 + % change ÷ 100)

For an increase of 8%: multiply by 1.08
For a decrease of 15%: multiply by 0.85
For an increase of 100%: multiply by 2 (doubles the value)

Example: A salary of £28,000 rises by 4.5%. New salary = £28,000 × 1.045 = £29,260

Common Errors to Avoid

Error 1: Confusing the Base Value

Always divide by the original (old) value, not the new one. Students and even professionals sometimes use the wrong base.

Example mistake: "Prices rose by £50, from £200 to £250. Increase = 50 ÷ 250 = 20%." This is wrong. Correct: 50 ÷ 200 = 25%.

Error 2: Percentage Increases Don't "Cancel Out"

A 20% increase followed by a 20% decrease does NOT return you to the original value:

Start: £100
After 20% increase: £100 × 1.20 = £120
After 20% decrease: £120 × 0.80 = £96

You end up at £96, not £100. This is why market indices that "recover" their percentage losses don't necessarily recover their absolute value.

Error 3: Successive Percentage Changes

To apply multiple percentage changes, multiply the multipliers together:

Inflation of 3% for 3 consecutive years on a £50 item:
£50 × 1.03 × 1.03 × 1.03 = £50 × (1.03)³ = £50 × 1.0927 = £54.64

Not: £50 × (1 + 0.03 × 3) = £50 × 1.09 = £54.50. The compound version is correct; the simple addition underestimates the effect.

Real-World Applications

Inflation

UK CPI inflation for April 2025 is approximately 3.5%. If your salary hasn't risen by at least 3.5%, your real purchasing power has fallen — even if your nominal salary increased.

Investment Returns

A stock portfolio rises 15% in year one and falls 8% in year two. Net change:
1.15 × 0.92 = 1.058 → +5.8% overall

Population Statistics

A town's population grows from 45,000 to 52,650 over 5 years. % change = (52,650 − 45,000) ÷ 45,000 × 100 = +17%

Percentage Points vs Percentage Change

This distinction matters enormously in financial and political reporting. If interest rates rise from 4% to 5%, that is:

1 percentage point increase (the absolute difference)
25% increase in the interest rate (the relative change: 1 ÷ 4 × 100)

Both statements are correct; they measure different things. Be precise when reading news headlines — "X% increase in interest rates" could mean either, depending on context.

Summary

% Change = [(New − Old) ÷ Old] × 100. Always use the original value as the base. When applying sequential percentage changes, multiply the multipliers — don't add the percentages. Distinguish between percentage points (absolute change) and percentage change (relative change). Our calculator handles all of this and produces clear breakdowns of the working.