How do you work out a percentage of a number in your head?

Start from 10%, which is the number with the decimal point moved one place left, and build the rest from it. 10% of 240 is 24, so 20% is 48, 5% is 12, and 35% is 24 + 48 + 12 read as 10 + 20 + 5. Almost every percentage you meet is a small sum of 10%, 5% and 1%.

What is the percentage flip?

x% of y always equals y% of x, and one side is usually far easier. 4% of 25 looks awkward, but 25% of 4 is 1. 18% of 50 becomes 50% of 18, which is 9. Both sides are the same multiplication - x times y divided by 100 - so the flip is free, and it costs nothing to check whether the other order is nicer.

How do you calculate percentage change?

Divide the change by the ORIGINAL amount, then multiply by 100. From 40 to 50 is a change of 10 on an original of 40, so 10 divided by 40 is 0.25, a 25% increase. The most common error is dividing by the new number instead - going from 40 to 50 is a 25% rise, but coming back from 50 to 40 is a 20% fall, and the two are genuinely different.

How do you find the original price before a discount?

Divide, do not add the percentage back. If 20% was taken off and the price is now 60, then 60 is 80% of the original, so the original is 60 divided by 0.8, which is 75. Adding 20% to 60 gives 72 and is wrong, because the 20% was a share of the larger original price, not of what is left.

Why can you not just add two percentages together?

Because the second one is a share of a different amount. A 10% rise followed by a 10% fall does not return you to where you started: 100 becomes 110, and 10% of 110 is 11, so you land on 99. Each percentage is measured against whatever the total was at the time it applied.

Percentages

Percent means per hundred. 35% is 35 out of every 100 — a fraction with the bottom already agreed on, which is exactly why percentages caught on. Two fractions with different denominators are hard to compare; two percentages never are.

That is also the key to doing them in your head. You are never really calculating a percentage; you are building it out of a few you already know.

Build everything from 10%

10% is the number with the decimal point moved one place to the left. Every other percentage is a short sum of pieces built from it.

Want	Do this	Example (of 240)
10%	move the point one place	24
1%	move the point two places	2.4
20%	double 10%	48
5%	half of 10%	12
15%	10% + 5%	36
25%	quarter — halve, halve again	60
35%	10% + 20% + 5%	84

17% of 300 → 10% is 30, 5% is 15, 1% is 3, so 30 + 15 + 3 × 2 = 51.

Percentages are additive as long as they are shares of the same amount. 10% and 5% of the same number really do combine into 15% of it, so a percentage can be assembled out of easy parts rather than computed in one go. This is where nearly all mental percentage work comes from.

The percentage flip

x% of y always equals y% of x — and one side is usually far easier.

4% of 25 is awkward; 25% of 4 is 1. 18% of 50 → 50% of 18 = 9. 12% of 50 → 50% of 12 = 6.

Both sides are the same multiplication, x × y / 100, read in a different order. Multiplication does not care about order, so the flip is free — and it costs nothing to glance at the other side before you start.

Percentage change

Divide the change by the original, then multiply by 100.

40 → 50: the change is 10, the original is 40, so 10 / 40 = 0.25 → a 25% increase.

50 → 40: the change is 10, the original is 50, so 10 / 50 = 0.2 → a 20% decrease.

Those two are not a contradiction, and noticing why is most of what understanding percentages means. A percentage is always a share of something, and the two directions have different somethings. Ten is a quarter of 40 and a fifth of 50. Whenever a percentage looks wrong, the first question is: a percentage of what?

Percentages do not stack

100 rises 10% → 110. It then falls 10% → 99, not 100.

The fall was 10% of 110, which is bigger than 10% of 100. The same asymmetry is why a 50% loss needs a 100% gain to recover.

Increases and discounts in one step

Do not work out the part and then add or subtract it — go straight to the final amount with a multiplier.

Change	Multiply by	Example
+10%	1.1	80 → 88
−10%	0.9	80 → 72
+20%	1.2	80 → 96
−25%	0.75	80 → 60
−30%	0.7	80 → 56

A 30% discount leaves 70% behind, and 70% of something is 0.7 × it. Going to the multiplier directly halves the work and removes the step most likely to go wrong. It also makes several changes in a row easy: two 10% rises is × 1.1 × 1.1 = × 1.21, which is the 21% — not 20% — that stacking predicts.

Reverse percentages

The awkward one, and the one worth practicing: you know the price after the change and want the price before.

A coat costs 60 after 20% off. 60 is 80% of the original, so the original is 60 ÷ 0.8 = 75.

A bill is 108 including 8% tax. 108 is 108% of the pre-tax amount, so it is 108 ÷ 1.08 = 100.

Adding the percentage back is the classic error — 60 + 20% of 60 gives 72, which is not 75. The 20% was a share of the original price, not of the reduced one, so the only way back is to undo the multiplication you now know was applied. If forward is × 0.8, backward is ÷ 0.8.

The everyday ones

A 15% tip on 43. 10% is 4.30, half of that is 2.15, so about 6.45.
A third off 27. A third is 9, so 18. Fraction-shaped discounts are faster as fractions.
20% off, then 10% off. × 0.8 × 0.9 = × 0.72, so 28% off — not 30%.
"Up to 70% off". Multiply by 0.3 to see the best case in one move.

The fractions worth knowing by sight

Recognizing these removes the arithmetic entirely.

Percentage	Fraction	In practice
50%	1/2	halve it
25%	1/4	halve twice
75%	3/4	halve twice, take three
33.3%	1/3	divide by 3
20%	1/5	divide by 5
12.5%	1/8	halve three times

More on where these come from in the guide to fractions, and the rest of the mental shortcuts are in mental math tricks.

Everything on this page is a percentage applied once. Applied repeatedly it becomes interest, depreciation and inflation — compound growth, effective rates, loan repayments and annuities are worked through in decimals, percentages and interest.

The mistakes worth naming

Dividing by the new amount in a percentage change. The original is the reference, always.
Adding the discount back to reverse it. Divide by the multiplier instead.
Adding successive percentages. Multiply the multipliers; 20% then 10% is 28%, not 30%.
Confusing a percentage with a percentage point. A rate going from 4% to 5% is up one percentage point — and up 25%.

Practice these in the game

Math Challenge has a Percentages topic that drills the core move — a percentage of a number — starting at 10%, 25% and 50% of friendly numbers and widening to the less-memorized ones as you get them right. That is the piece everything on this page is built out of, and it is the piece worth making automatic.

The percentage flip is also one of the 36 illustrated Magic Tricks lessons, which teach the reasoning first and then drop you into practice on it. That one is part of the Pro set rather than the free starter tricks — the Percentages topic above needs no purchase.

Your turn

Three to try — tap what you get.

25% of 80

15% of 200

A $40 shirt drops by 10%. New price?

Practice percentages free →