Percentage Change vs Percentage Points

Percentage change measures how much a value moved relative to where it started:

percentage change = (new − old) ÷ old × 100

The denominator is always the old (starting) value. A price that moves from 80 to 100 rose by (100 − 80) ÷ 80 = 25%. A price that moves from 100 to 80 fell by (80 − 100) ÷ 100 = −20%. The same $20 gap produces two different percentages because the starting point is different. This asymmetry is the root of many mistakes.

A percentage point is a unit of difference between two percentages. If an interest rate goes from 4% to 5%, it rose by 1 percentage point. Relative to the starting rate, it rose by (5 − 4) ÷ 4 = 25%. Both statements are true and answer different questions.

Careless wording causes confusion. “Unemployment rose 2%” is ambiguous: it could mean from 5% to 7% (2 points, a 40% relative rise) or from 5.00% to 5.10% (2% relative, 0.1 points). When you read or write about rates, say “points” for differences between percentages and “percent” for relative change.

To increase a number by p%, multiply by (1 + p ÷ 100). To decrease, multiply by (1 − p ÷ 100). These multipliers are the key to everything else.

• Increase 80 by 25%: 80 × 1.25 = 100.
• Decrease 100 by 20%: 100 × 0.80 = 80.
• Add 20% tax to 50: 50 × 1.20 = 60.

To undo a percentage, divide by the multiplier, do not subtract the percentage back.

• An item is on sale for $80 after 20% off. The original price is 80 ÷ 0.80 = $100, not 80 × 1.20 = $96.
• A total of $60 includes 20% tax. The pre-tax amount is 60 ÷ 1.20 = $50, and the tax is $10 — not 20% of 60, which would be $12.
• To get from 80 back to 100 you need a 25% rise, because 20 is 25% of 80. A 20% fall needs a 25% rise to recover; a 50% fall needs a 100% rise.

Successive changes multiply their multipliers; they do not add.

• +50% then −50%: 1.5 × 0.5 = 0.75, a net change of −25%, not 0.
• +10% then −10%: 1.1 × 0.9 = 0.99, a net −1%.
• A 20% discount followed by a further 10% off the sale price: 0.8 × 0.9 = 0.72, a total of 28% off, not 30%.

The same logic applies to investment returns. A fund that gains 50% in one year and loses 50% the next is down 25% overall. That is also why average annual return and compound annual growth rate (CAGR) are different numbers.

When a value grows from a start to an end over several years, CAGR is the constant yearly rate that would produce the same result: (end ÷ start)^(1 ÷ years) − 1. A value that rises from 100 to 150 over five years has a CAGR of 1.5^0.2 − 1 ≈ 8.45% a year. Dividing the total 50% rise by five to get 10% would overstate it, because compounding is ignored.

To find p% of x, compute x × p ÷ 100. To find what percent a is of b, compute a ÷ b × 100. To find the number when you know that p% of it is a, compute a ÷ (p ÷ 100). These three operations cover most everyday questions: how much is a 15% tip on 40 ($6), what percent of 250 is 40 (16%), and what number has 15% equal to 6 (40).

• Using the new value as the denominator for percentage change.
• Subtracting a percentage to undo it instead of dividing by the multiplier.
• Adding chained percentages together.
• Saying “percent” when you mean “percentage points”.
• Comparing percentages of different bases (for example, 10% of a small number against 5% of a large one) without noting the bases.

The Percentage Change calculator takes an old and new value and returns the relative change. The Multi-Step Percentage Change tool applies several changes in sequence and shows the net effect. The Discount Calculator handles sale prices and stacked discounts. Results are arithmetic, not advice, but they let you check a headline quickly.