How to Read Percentage Change Without Getting Fooled

Percentages are the most common way numbers are used to mislead — usually without anyone intending to. Four ideas cover nearly all of it.

1. Percentage change is not symmetrical

A 50% fall followed by a 50% rise does not return you to the start:

• 100 → falls 50% → 50
• 50 → rises 50% → 75

Each percentage applies to a different base. To recover from a fall of *p* you need a rise of p ÷ (1 − p):

This is why a portfolio down 50% needs to double, and why "recovered half its losses" usually means much less than half.

2. Percent and percentage point are different

An interest rate rising from 4% to 6% has increased by:

• 2 percentage points
• 50 percent

Both correct, wildly different in feel. Whenever a report says a rate "rose 50%", check whether it means points — the difference is frequently the entire story.

3. Reversing a percentage means dividing

A jacket costs £96 after 20% off. The original was not £96 × 1.20 = £115.20.

The £96 is 80% of the original:

original = 96 ÷ 0.80 = £120

Check: 20% of 120 is 24, and 120 − 24 = 96. ✓

The same applies to tax: to strip 20% VAT from £120, divide by 1.20 to get £100. Multiplying by 0.80 gives £96, which is wrong. This is the most common percentage error in ordinary financial life.

The general rules are worth memorising as divisions:

And a useful sanity check: reversing a discount always gives a number larger than the one you started with, while reversing an increase always gives a smaller one. If your answer moves the wrong way, you multiplied where you should have divided.

4. Successive percentages multiply

Three 10% rises are not 30%:

1.10 × 1.10 × 1.10 = 1.331 → 33.1%

And a 20% rise followed by a 20% fall is not zero:

1.20 × 0.80 = 0.96 → a 4% loss

Order does not matter; the loss appears either way. Over many periods this compounding effect becomes larger than the individual changes.

5. Stacked discounts do not add

"30% off, then an extra 20% off at the till" is not 50% off. The second discount applies to the already-reduced price:

0.70 × 0.80 = 0.56 → 44% off

A £200 coat becomes £140, then £112. The 50% version would be £100. Six percent of the original price is where that argument ends, and the order of the two discounts makes no difference to the total.

The same multiplication handles tax on top of a discount. £200 less 30%, plus 20% VAT: 200 × 0.70 × 1.20 = £168. Compare with the discount calculator and the percent-off calculator.

6. Markup and margin are not the same number

Both describe the gap between cost and price, measured against different bases:

• Margin = (price − cost) ÷ price
• Markup = (price − cost) ÷ cost

Buy at £60, sell at £100. Margin is 40 ÷ 100 = 40%. Markup is 40 ÷ 60 = 66.7%. Same transaction, two correct answers nineteen points apart.

This is where real money is lost. Applying a "50% markup" to an £80 item gives £120, a margin of 40 ÷ 120 = 33.3% — not 50%. A business that prices for 50% markup while budgeting for 50% margin is short by a sixth of its revenue on every sale. The margin calculator converts between them.

Averaging percentages

The average of percentage changes is usually not the overall change. Returns of +50% then −50% average to 0% but leave you down 25%. The correct measure is the geometric mean: multiply the growth factors and take the nth root.

Here the factors are 1.50 and 0.50, product 0.75, and √0.75 = 0.866 — an average of −13.4% per period, which compounds to the actual −25%. The arithmetic mean of volatile percentages always overstates the result, and the gap widens with volatility. Three years of +10% compounds to 1.10³ = 1.331, and the geometric mean correctly returns 1.331^(1/3) − 1 = 10%.

Averaging percentages across unequal groups needs weighting too. A 90% pass rate in a class of 10 and a 50% rate in a class of 90 is not a 70% overall rate; it is (9 + 45) ÷ 100 = 54%. Use the weighted average calculator when the groups differ in size.

Quick mental methods

• 10% — shift the decimal one place
• 5% — half of 10%
• 15% — 10% plus half again
• Swap trick — a% of b = b% of a. 8% of 25 is awkward; 25% of 8 is

obviously 2.

• Reverse a round discount — to undo 20% off, add a quarter: £96 + £24 = £120.

To undo 25% off, add a third. To undo 50% off, double.

A short checklist

Before accepting any percentage claim, ask three questions:

1. Percent of what? The base determines the answer entirely.
2. Percent or percentage point? Especially for rates, shares and

probabilities.

1. Is it being added when it should be multiplied? Discounts, growth rates

and tax all compound.

Check anything consequential with the percentage calculator, or the percentage increase calculator when you have two figures and need the change between them.