How to Calculate Percentage Increase and Decrease

Percentages break people's intuition in one specific way: up and back down by the same percentage never returns you to where you started. Here is the formula, and the four places it is most often applied wrong.

MathBy Jul 7, 20265 min read
How to Calculate Percentage Increase and Decrease — ListCalc

The two formulas

% change = (New − Old) ÷ Old × 100
% of a number = Part ÷ Whole × 100

A positive result is an increase, a negative result a decrease. The critical detail: the old value is always the denominator.

FromToWorkingResult
405010 ÷ 40+25%
5040−10 ÷ 50−20%
20026060 ÷ 200+30%
1,200900−300 ÷ 1,200−25%

Why up and down never cancel

Rows one and two above use the same two numbers and give different answers. That is not an error — the base moved.

Recovering from a loss
−10% loss+11.1%
−25% loss+33.3%
−50% loss+100%
Gain required just to get back to even

This asymmetry is why investment returns are so unforgiving of large drawdowns.

Every percentage type in one tool.Percentage calculator →

Reverse percentages: finding the original

A jacket costs $80 after 20% off. What was the list price?

The wrong move is adding 20% to $80, giving $96. The 20% was taken from the original, so $80 represents 80% of it:

$80 ÷ 0.80 = $100

The same logic works for tax-inclusive prices: a $120 total including 20% VAT contains $100 of goods and $20 of tax, because $120 ÷ 1.20 = $100.

Percent vs percentage points

These are different units and mixing them is the most common statistical sleight of hand you will meet.

ChangeIn percentage pointsIn percent
Rate 5% → 7%+2 points+40%
Share 20% → 25%+5 points+25%
Fee 1% → 1.5%+0.5 points+50%
Both descriptions of each row are true. Whenever a headline reports a percentage change in something already measured as a percentage, check which unit is being used before reacting to it.

Stacking percentages

Discounts do not add — they multiply. A 20% discount followed by a further 10% off is not 30% off:

$100 × 0.80 = $80, then $80 × 0.90 = $72 — a 28% total discount. The same applies to successive price rises, and to margin and markup, where multiplying rather than adding is the difference between a profitable price and a loss-making one.

Run your own numbers

Percentage Profit margin ROI

FAQ

What is the formula for percentage increase?
(New − Old) ÷ Old × 100. Going from 40 to 50: (50 − 40) ÷ 40 × 100 = 25% increase. The old value is always the denominator — that is the step people skip.
Why is a 25% increase reversed by only a 20% decrease?
Because the base changes. The 25% rise was measured against 40, but the fall back is measured against the larger 50. Ten is 25% of 40 but only 20% of 50. This is why a stock that drops 50% needs a 100% gain to break even.
What is the difference between percent and percentage points?
If a rate rises from 5% to 7%, that is a 2 percentage-point increase but a 40% relative increase. News reports and financial products routinely mix these up, usually in whichever direction sounds more dramatic.
How do I find the original price before a discount?
Divide by (1 − discount). An item costing $80 after 20% off was originally $80 ÷ 0.80 = $100. Do not add 20% back to $80 — that gives $96 and is wrong, because the discount was taken from the larger original figure.
How do I calculate what percentage one number is of another?
Part ÷ Whole × 100. If 37 of 200 orders were returned, that is 37 ÷ 200 × 100 = 18.5%.

Sources

Primary references used for the figures and rules on this page.

  1. Consumer Price Index — Bureau of Labor Statistics
  2. Handbook of Methods — Bureau of Labor Statistics