Percentage Mistakes: Discounts, VAT and Margins

Percent of what? Changes do not reverse

A price that rises 50% and then falls 50% does not return to where it started. From 100 it rises to 150, and 50% of 150 is 75, so it ends at 75, a 25% loss. The second change is measured from a different base. The same holds for investments: a fall of 20% needs a rise of 25% to recover (80 × 1.25 = 100), and a fall of 50% needs a rise of 100%.

The safe way to combine changes is to multiply factors rather than add percentages. A rise of 10% is × 1.10 and a fall of 10% is × 0.90; together they give 0.99, a 1% loss overall, not zero.

Stacked discounts multiply

"20% off, plus an extra 10% at the till" is not 30% off. The second discount applies to the already reduced price: 100 becomes 80, and 10% off 80 is 72, a total discount of 28%. As factors, 0.80 × 0.90 = 0.72. The order does not matter, so 10% then 20% also gives 72.

Coupons worth a fixed amount behave differently from percentages. A 10 coupon used before a 20% discount saves 10 × 0.8 = 8 of the final price, while the same coupon used after the discount saves the full 10. If you can choose, apply percentage discounts first and fixed amounts last.

Taking tax out of a price

Where prices include VAT or sales tax, the tax is a percentage of the price before tax, not of the price you see. With 10% tax, a price of 11,000 is 10,000 plus 1,000 of tax; working out 10% of 11,000 gives 1,100, which is wrong. The correct steps are to divide by 1 plus the rate (11,000 ÷ 1.10 = 10,000) and subtract.

The same goes for any rate. At 20% VAT, a price of 120 contains 20 of tax (120 ÷ 1.20 = 100), not 24. A quick check for the tax share of a tax-inclusive price is rate ÷ (1 + rate): 1/11 of the price at 10%, 1/6 at 20%.

Margin is not markup

Markup is profit as a percentage of cost; margin is profit as a percentage of the selling price. Buy at 80 and sell at 100, and the profit of 20 is a 25% markup (20 ÷ 80) but a 20% margin (20 ÷ 100). Margin can never reach 100%, while markup has no upper limit.

Mixing them up loses money. To reach a 30% margin on something that costs 70, the price is 70 ÷ (1 − 0.30) = 100. Adding 30% to the cost gives 91, a margin of only 23%. When a supplier, a platform or a partner quotes a percentage, ask which base it uses.

Percent and percentage points

When a rate itself changes, there are two ways to describe it. An interest rate moving from 4% to 5% is up one percentage point, but up 25% in relative terms, because 1 is a quarter of 4. Both are correct; they answer different questions, and headlines often choose whichever sounds larger.

The same care applies to risks. A risk that rises from 1 in 1,000 to 2 in 1,000 has doubled, a 100% increase, while the absolute change is 0.1 percentage points. Reporting only the relative change makes small risks sound alarming, and reporting only the absolute change can hide a real effect, so look for both.

• Ask what each percentage is a percentage of
• Combine successive changes by multiplying factors, not adding percentages
• Remove included tax by dividing by 1 plus the rate
• Say whether a profit figure is margin or markup
• Use percentage points for changes in a rate, and say which you mean
• Check a result by working backwards to the starting number