Percentages without mistakes

Percentages look like the easiest maths there is, yet they cause more misleading headlines, wrong discounts, and broken spreadsheets than almost anything else. The trouble is rarely the arithmetic. It is choosing the wrong base: what the percentage is a percentage of. This guide covers the three basic questions, then the traps that catch even careful readers.

The three basic questions

Nearly every percentage problem is one of three questions, and the percentage calculator has a mode for each:

• What is x% of y? 20% of 150 = 0.20 × 150 = 30.
• What percentage is x of y? 45 of 60 = 45 ÷ 60 × 100 = 75%.
• What is the percentage change from a to b? From 80 to 100 = (100 − 80) ÷ 80 × 100 = +25%.

The third question is where most mistakes happen, because the base is the starting value. Going from 100 back down to 80 is a 20% decrease, not 25%. The same absolute change of 20 is a different percentage depending on where you start.

Trap 1: increases and decreases do not cancel

A price that rises 50% and then falls 50% does not return to where it started: 100 → 150 → 75. A share that falls 50% needs to rise 100% to recover. The general rule is that a fall of p% needs a rise of p ÷ (1 − p) to undo it:

• Down 10% needs up 11.1%.
• Down 25% needs up 33.3%.
• Down 50% needs up 100%.

Successive percentage changes multiply rather than add. Two 10% increases make 1.1 × 1.1 = 1.21, a 21% increase, not 20%.

Trap 2: reversing a percentage

If a price is 120 after a 20% increase, the original was 120 ÷ 1.2 = 100, not 120 − 20% = 96. The same applies to taxes: a price of 121 that includes 21% VAT has a net price of 121 ÷ 1.21 = 100 and tax of 21. Taking 21% of 121 (25.41) overstates the tax. The sales tax calculator has an inclusive mode for exactly this.

Trap 3: percentage points vs. percent

When an interest rate goes from 4% to 5%, it has risen by one percentage point, which is a 25 percent increase in the rate. Both statements are true, but they sound very different. Reports that want a change to look large tend to use the relative figure, and reports that want it to look small use points. When you read “up 50%” about something that was already a percentage, check which one is meant.

Trap 4: margin is not markup

In pricing, a 25% markup on cost is only a 20% margin on the selling price, because the two use different bases: cost 80, price 100, profit 20. The profit margin calculator shows both, and the pricing guide explains why the difference matters for your prices.

Trap 5: small bases produce huge percentages

“Sales up 300%” sounds impressive until you learn they went from 2 to 8. Percentage changes from very small numbers are volatile and often misleading, and change from zero is undefined, since any increase is infinitely large in relative terms. Report the absolute numbers alongside the percentage whenever the base is small.

Trap 6: averaging percentages

If one shop converts 10% of 1,000 visitors and another converts 50% of 10 visitors, the overall rate is not the average of 10% and 50% (30%). It is (100 + 5) ÷ 1,010 = 10.4%. Average the underlying counts, not the percentages, unless every group is the same size. The average calculator is the right tool for the counts. Combining the rates needs the totals.

Mental shortcuts

• 10% of anything: move the decimal point one place left.
• 5% is half of 10%, and 15% is 10% plus 5%.
• x% of y equals y% of x: 8% of 50 is the same as 50% of 8, which is 4.
• To add a percentage, multiply by (1 + p): add 21% by multiplying by 1.21.
• To remove an included percentage, divide by (1 + p): remove 21% VAT by dividing by 1.21.

For ratios rather than percentages, such as recipes, mixtures, and screen proportions, the ratio calculator keeps relationships exact, and its share percentages show each part’s portion of the whole.