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Percentage Calculator

Four calculations with the working shown, and the one asymmetry behind almost every percentage mistake.

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The mistake worth knowing

Percentages are not symmetrical, and almost every percentage error traces back to that one fact. Each step is calculated on whatever the number became, not on where it started.

Why minus 20 percent then plus 20 percent does not return to the start Three bars: 100, then 80 after a 20 percent cut, then 96 after a 20 percent rise. The rise is calculated on the smaller number. −20%+20%100start80after −20%96after +20% The +20% is taken on 80, not on 100 — so you land 4 short.
This is why "it was reduced 20%, so add 20% to get the original price" gives the wrong answer. To go backwards you divide: 80 ÷ 0.8 = 100.
The same arithmetic runs stacked discounts and compounding returns. 20% off then a further 10% off is 28% off, not 30% — multiply the survivals, 0.8 × 0.9 = 0.72. And a fund down 50% has to double just to break even.

Percent versus percentage points

When the number being measured is itself a percentage, there are two honest ways to describe a change to it, and they give very different-sounding answers.

A mortgage rate moving from four per cent to six per cent, described two ways The same change is a rise of two percentage points and a rise of fifty per cent. Both statements are accurate. A mortgage rate, before and after 0%2%4%6%8% 4%6% 2 percentage points — the gap on this axis 50% higher — the gap as a share of where it started Six is half as much again as four, so the relative rise is fifty per cent. Both headlines are true. Ask which one a number is before reacting to it.
This is the most common reporting error in coverage of interest rates, unemployment and polling — and it is usually the larger-sounding of the two that gets printed.

Questions

What is the reverse-percentage trap?

Taking a percentage off and then putting the same percentage back on does not return you to where you started, because the second calculation runs on the new, smaller number. A 20% cut then a 20% rise leaves you at 96% of the original. It matters most when working backwards from a sale price: to find the pre-discount price of an item reduced 20%, divide by 0.8 rather than multiplying by 1.2.

Do stacked discounts add up?

No. 20% off then a further 10% off is not 30% off — it is 0.8 × 0.9 = 0.72, so 28% off. The second discount applies to the already-reduced price. The same compounding applies to successive price rises and to investment returns.

What's the difference between percent and percentage points?

If a rate rises from 4% to 6%, that is a two percentage-point rise but a 50% increase. Both are correct and they describe different things — mixing them up is one of the most common errors in reporting on interest rates, unemployment and polling.

References

  1. NIST — Guide for the Use of the International System of Units
  2. Office for National Statistics — style guide on percentages and percentage points

Reviewed September 2026.