How to Calculate Percentages Without Reaching for a Calculator

Percentages are the arithmetic you actually use. Not algebra, not long division — the ability to look at “30% off” and know in two seconds whether it’s worth walking to the other shop.

Most people can do the easy ones and freeze on the rest. That’s not a maths problem; it’s a problem of never having been shown that there are only four questions, and that each has a shape you can recognise.

The four questions

Every percentage problem you’ll meet in real life is one of these:

1. What is 15% of 240? — You know the percentage and the whole, and you want the part.
2. 36 is what percent of 240? — You know the part and the whole, and you want the percentage.
3. It went from 50 to 65 — by how much? — You know two values and you want the change between them.
4. I paid $68 after 15% off. What was the original? — You know the result and want to work backwards.

Recognising which one you’re in is 80% of the work. The arithmetic afterwards is easy.

Question one: a percentage of a number

Divide the percentage by 100, multiply by the number. 15 ÷ 100 = 0.15, and 0.15 × 240 = 36.

In your head, build it out of 10% and 5%:

• 10% — move the decimal one place left. 10% of 240 is 24.
• 5% — half of 10%. So 12.
• 15% — add them. 36.

This decomposition handles most of what you need. 20% is 10% doubled. 25% is a quarter — divide by four. 30% is 10% times three.

And one genuinely useful trick: x% of y always equals y% of x. If someone asks for 4% of 75, flip it — 75% of 4 is obviously 3. This works every time and feels like cheating.

Question two: what percentage is this?

Divide the part by the whole, multiply by 100. 36 ÷ 240 = 0.15, so 15%.

This is the one for marks out of a total, a line item’s share of a budget, or how much of a target you’ve hit. The order matters: it’s always part ÷ whole, and getting it backwards gives you a number over 100% that should immediately look wrong.

Question three: percentage change

((new − old) ÷ old) × 100.

From 50 to 65: (65 − 50) ÷ 50 = 0.3, so a 30% increase.

Note the denominator. Change is always measured against where you started, which is why the same absolute move gives different percentages depending on direction. This is not a quirk — it’s the whole reason the next section exists.

Question four: working backwards

You know a result and the percentage it represents. Divide, don’t multiply.

You paid $68 after a 15% discount. You paid 85% of the original. So 68 ÷ 0.85 = $80.

The tempting mistake is adding 15% back to $68, which gives $78.20. That’s wrong, because the 15% was taken off the larger number. Percentages are not symmetric, and this is where it bites.

The three mistakes that cost money

A discount and an equal increase don’t cancel

Take 20% off 100 and you have 80. Add 20% to 80 and you get 96, not 100. The two percentages are calculated against different bases — 100 and 80 — so they aren’t the same size.

This matters when a shop raises a price by 20% and then advertises 20% off. You are not back where you started; you are paying more than the original.

Stacked discounts don’t add up

Two successive 10% discounts are not 20% off. They’re 19%: 0.9 × 0.9 = 0.81, so you pay 81% of the original.

The gap widens as the discounts get bigger. Three 20% discounts sound like 60% off but are actually 48.8%. To combine discounts, multiply what remains — never add what’s removed.

Percentage points are not percent

If an interest rate goes from 2% to 3%, that is a rise of one percentage point and a 50% increase.

Both are true. Both describe the same event. And whoever is writing the headline will choose whichever sounds more dramatic. When you see a percentage describing a change in another percentage, always ask which one is meant — the difference can be a factor of fifty.

A worked example: is the bulk deal actually cheaper?

A 500g jar costs $4.50. The 750g jar costs $6.30. Which is better value?

Price per gram is the honest comparison: 4.50 ÷ 500 = $0.009 and 6.30 ÷ 750 = $0.0084. The larger jar is cheaper per gram — by (0.009 − 0.0084) ÷ 0.009 = 6.7%.

Worth buying if you’ll use it. A 6.7% saving on something that goes off before you finish it is a 100% loss.

The 15-minute practice that sticks

Do the maths before you look at the label. Estimate 20% off the price while you’re still walking towards the shelf. Work out the tip before the card machine offers you three buttons. Guess the percentage change in a news headline before reading the second paragraph.

You’ll be wrong for about a week. Then you won’t be, and you’ll never need to think about it again.