How to calculate percentages in your head, step by step

A percentage is a fraction of 100

Percent means per hundred. 15% of something is 15 hundredths of it, so every percentage question is a multiplication by a small fraction. What makes percentages hard in the head is rarely the concept. It is that multiplying by 0.15 directly means holding several partial results at once, and DeStefano and LeFevre’s review of working memory in mental arithmetic describes exactly that holding as the expensive part. The methods below all do the same thing: they replace one awkward multiplication with two or three easy ones whose results you can keep.

Start with 10%

Ten percent is the anchor for everything else, because finding it only means moving the decimal point one place to the left. 10% of 340 is 34. 10% of 72 is 7.2. 10% of 6,500 is 650. Make this step automatic before anything else, and say the result out loud if that helps you hold it. Most percentage mistakes made in the head are errors of place value, a result ten times too big or too small, rather than errors of arithmetic.

Then 5% and 1%

Five percent is half of ten percent, and one percent is ten percent divided by ten again. Starting from 340: 10% is 34, 5% is 17 and 1% is 3.4. With these three blocks you can assemble almost any whole percentage by adding or subtracting. 15% of 340 is 34 + 17 = 51. 20% is 34 × 2 = 68. 9% is 34 − 3.4 = 30.6. Nothing here requires more than halving and moving a decimal point.

Build awkward percentages from blocks

Take 17% of 250. Break 17 into 10 + 5 + 2. Ten percent of 250 is 25, five percent is 12.5 and two percent is 5, so the total is 25 + 12.5 + 5 = 42.5. The same breakdown works for any percentage; choose the split with the fewest steps. 19% is usually faster as 20% minus 1% than as 10% + 5% + 4%. Estimate first: 17% is a little under a fifth, and a fifth of 250 is 50, so an answer near 42 is plausible while 4.25 or 425 is not.

Swap the numbers when one is friendlier

x% of y is always equal to y% of x, because both are x × y ÷ 100. This turns some unpleasant problems into trivial ones. 8% of 25 is the same as 25% of 8, which is 2. 16% of 50 is 50% of 16, which is 8. 12% of 75 is 75% of 12, which is 9. Whenever one of the two numbers is a friendly percentage such as 50, 25, 20 or 10, try the swap before doing anything else.

Know a few fractions by heart

Some percentages are faster as fractions. 50% is a half, 25% a quarter, 20% a fifth, 12.5% an eighth, and 33.3% is roughly a third. So 25% of 88 is 88 ÷ 4 = 22, and 12.5% of 64 is 64 ÷ 8 = 8. You do not need a long table. The handful that correspond to halving, quartering and dividing by five cover most everyday cases, and they double as checks for the block method.

Discounts: work out what you pay

A 30% discount on 64 means you pay 70% of 64. You can calculate the discount and subtract it, since 30% of 64 is 19.2 and 64 − 19.2 = 44.8, or go straight to what you pay: 70% of 64 is 7 × 6.4 = 44.8. The second route is one step shorter and avoids a subtraction with decimals, which is where many errors hide. The same idea works for any reduction: a 15% discount means paying 85%.

Two discounts do not add up

A 20% discount followed by a further 10% is not a 30% discount. You pay 80% of the price and then 90% of that, so 0.8 × 0.9 = 0.72 of the original, which is a 28% reduction. On a price of 50, you pay 36 rather than 35. The gap is small at these sizes and grows with larger percentages, and it is the most common mistake people make with stacked offers. Multiply the factors; never add the percentages.

Tips without a calculator

For a 15% tip on a bill of 46, take 10%, which is 4.6, add half of it, 2.3, and you have 6.9. For 20%, double the 10%: 9.2. For 18%, take 20% and subtract 1% twice: 1% is 0.46, so 9.2 − 0.92 = 8.28, which you might round to a convenient amount. If exactness does not matter, round the bill first: 15% of 50 is 7.5, a quick upper bound for the tip on 46.

Adding tax or a surcharge

To add 20% to 85, you can calculate 20% separately and add it, 85 + 17 = 102, or multiply by 1.2 in one go: 85 × 1.2 = 102. Both are fine. What matters is choosing one routine and using it consistently. The multiplier view pays off when several changes stack, because you can multiply the factors together exactly as with discounts, and it prepares you for the reverse problem below.

Reverse percentages: finding the original

If a price including 20% tax is 96, the price before tax is not 96 minus 20% of 96. That would give 76.8, and adding 20% back to 76.8 gives 92.16, not 96. The price including tax is 120% of the original, so divide by 1.2: 96 ÷ 1.2 = 80. Check it: 20% of 80 is 16, and 80 + 16 = 96. Reverse percentages are where the mental shortcut most often gives a confident wrong answer, so always check by going forward again.

Percentage change

The change from 80 to 92 is an increase of 12, and 12 out of 80 is 15%. From 120 down to 90 is a decrease of 30, and 30 out of 120 is 25%. The base is always the starting value. That is why a rise of 25% followed by a fall of 25% does not bring you back to where you started: 100 becomes 125, 25% of 125 is 31.25, and 125 − 31.25 leaves 93.75.

Percent versus percentage points

If a rate goes from 4% to 5%, it has risen by one percentage point, and by 25% in relative terms, because 1 is a quarter of 4. News reports, loan offers and survey results mix these two up constantly, sometimes by accident and sometimes because the bigger-sounding figure is more convenient. When you hear about a change in a rate, ask which of the two is meant before judging whether it is large.

Practise the blocks, not the tricks

The fastest way to get comfortable is to drill the building blocks until they need no thought: 10%, 5% and 1% of assorted numbers, and a few swaps. Short, frequent sessions spread over days tend to be retained better than one long session, which is what Cepeda and colleagues found across many studies of distributed practice. Mix percentages with other operations rather than doing twenty in a row, so that you also practise recognising which method a problem needs.

Where Numvolt fits

Numvolt’s sessions train the four operations that every percentage method above breaks down into: addition, subtraction, multiplication and division. Zen mode has no timer and shows the method behind every correction, which suits the stage where the blocks are still being learned, and timed sprints of 15, 30 or 60 seconds show whether the steps have become quick. For percentages themselves, the receipts, bills and price labels of an ordinary week remain the best practice material.

Further reading

• DeStefano & LeFevre (2004), The role of working memory in mental arithmetic, European Journal of Cognitive Psychology
• Cepeda et al. (2006), Distributed practice in verbal recall tasks: a review and quantitative synthesis, Psychological Bulletin