Percentages

Updated 7 October 2026

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.

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

What is the percentage flip?

The percentage flip says that x% of y always equals y% of x, so swap the numbers and work out the easier one. For 16% of 25, swap to 25% of 16, which is a quarter of 16, so 16% of 25 = 4. It works because both mean the same multiplication: 16% of 25 is 16 × 25 ÷ 100, and 25% of 16 is 25 × 16 ÷ 100, and both give 400 ÷ 100 = 4. The flip helps only when one side becomes a friendly percentage such as 25% or 50%. That is the case that changes the steps: in 13% of 17, the flip gives 17% of 13, which is no easier, so work it out directly as 13 × 17 ÷ 100. To check a flip, estimate the original side: 16% is a little more than 15%, and 15% of 25 is between 3 and 4, so an answer of 4 is the right size.

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

How do you work out percentage change?

Percentage change is the change divided by the original amount, multiplied by 100. When a price rises from 80 to 100, the change is 20 and the original is 80, so the percentage change is 20 ÷ 80 × 100 = 25%. The original is the reference because the change is measured against where you started. That is why the same gap gives a different answer in reverse: a fall from 100 to 80 is 20 ÷ 100 × 100 = 20%, because this time the change is measured against 100. For a fall, subtract the new amount from the original to find the change, divide by the original as before, and call the result a decrease. The common mistake is dividing by the new amount, which makes the rise look smaller than it is. To check, apply the percentage to the original: 25% of 80 = 20, and 80 + 20 = 100.

$40\to 50$40 → 50: the change is 10, the original is 40, so 10 / $40=0.25\to a25$40 = 0.25 → a 25% increase.

$50\to 40$50 → 40: the change is 10, the original is 50, so 10 / $50=0.2\to a20$50 = 0.2 → a 20% decrease.

Percentages do not stack

100 rises 10% $\to 110$→ 110. It then falls 10% $\to 99$→ 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.

How do you calculate an increase or discount in one step?

To increase or decrease an amount by a percentage in one step, multiply it by the percentage you end up with, written as hundredths. A 20% rise leaves 100% + 20% = 120% of the original, so increasing 40 by 20% gives $40\times \frac{120}{100}=48$40 × 120/100 = 48. A 25% discount leaves 100% − 25% = 75%, so a price of 80 with 25% off becomes $80\times \frac{75}{100}=60$80 × 75/100 = 60. It works because the original amount is 100%, and adding or taking away the change before you multiply means you never have to work out the change on its own. For two changes in a row, multiply the two multipliers, and never add the percentages. A 20% rise followed by a 20% fall gives $\frac{120}{100}\times \frac{80}{100}=\frac{96}{100}$120/100 × 80/100 = 96/100, which is a 4% loss overall, not zero. To check a one-step answer, find the change separately once: 20% of 40 is 8, and 40 + 8 = 48.

How do you find the original amount after a percentage change?

To find the original amount after a percentage change, divide the new amount by the multiplier the change used. After a 20% rise, a ticket costs 84. The new price is 120% of the original, so the original is $84÷\frac{120}{100}=70$84 ÷ 120/100 = 70. It works because the rise multiplied the original by $\frac{120}{100}$120/100, and dividing by the same number undoes it. The usual mistake is taking the percentage of the new amount: the 20% was a share of the original, not of 84, so taking 20% of 84 away does not give 70. A fall works the same way with a smaller multiplier: after 20% off, divide by 0.8, because the sale price is 80% of the original. A price that already includes tax is a rise in disguise, so its multiplier is 100% plus the tax rate. To check, apply the change to your answer: 20% of 70 is 14, and 70 + 14 = 84.

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.

How do you work out a tip or a discount?

To work out a tip or a discount, start from 10%, which is the amount divided by 10, and build the percentage you need from it. For a 15% tip on a bill of 60, 10% is 6, half of that is 3, and 6 + 3 = 9, so the tip is 9. A discount uses the same pieces and then subtracts: 20% off 60 is 2 × 6 = 12 off, so you pay 60 − 12 = 48. It works because 10% is easy to find, and most everyday percentages are made of tens and fives. Two discounts in a row do not add. After 20% off, a further 10% off is taken from the lower price, so you pay 80% of 90% = 72% of the original, and the total saving is 28%, not 30%. To check a discount, add the saving back to the price paid: 48 + 12 = 60.

• 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.

Which fractions are worth knowing as percentages?

The fractions worth knowing as percentages are $\frac{1}{2}=50$1/2 = 50%, $\frac{1}{4}=25$1/4 = 25%, $\frac{3}{4}=75$3/4 = 75%, $\frac{1}{5}=20$1/5 = 20%, $\frac{1}{10}=10$1/10 = 10%, $\frac{1}{8}=12.5$1/8 = 12.5%, $\frac{1}{3}=33.3$1/3 = 33.3% and $\frac{2}{3}=66.7$2/3 = 66.7%. Each percentage is the fraction written as hundredths: $\frac{1}{4}=\frac{25}{100}$1/4 = 25/100, and 25 out of 100 is 25%. Knowing them turns a percentage question into a fraction question, and fractions of friendly numbers are quick: 75% of 40 is $\frac{3}{4}$3/4 of 40, which is 40 ÷ 4 × 3 = 30. Other percentages build from these, so $\frac{3}{5}$3/5 is three lots of 20%, which makes 60%. A third has a percentage that never ends, so 66.7% for two thirds is rounded: $\frac{2}{3}$2/3 of 60 is exactly 40, while the rounded percentage of 60 comes out slightly more. For thirds, use the fraction. To check any of them, multiply the percentage by the bottom number of its fraction: 25% × 4 = 100%, one whole.

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%.