How to Calculate Percentages: The Three Questions, the Shortcuts, and the Traps

Percentages cause more everyday arithmetic mistakes than anything else I see, and not because the maths is hard. It’s because three different questions all use the word “percent” and each needs a different calculation. “What is 20% of 150?” “150 is what percent of 600?” “What’s the percent change from 120 to 150?” Get the wrong one and a discount, a tip, a tax return or a KPI report comes out wrong. This guide separates the three, shows the shortcuts, and covers the two traps (percentage points and reverse percentages) that catch people who otherwise know what they are doing.

The three questions

Question	Formula	Example
What is P% of X?	X × P ÷ 100	20% of 150 = 150 × 0.20 = 30
X is what percent of Y?	X ÷ Y × 100	150 ÷ 600 × 100 = 25%
Percent change from A to B	(B − A) ÷ A × 100	(150 − 120) ÷ 120 × 100 = 25% increase

Notice that the third one divides by the starting value, not the ending one. A price rising from 120 to 150 is a 25% increase; the same price falling from 150 to 120 is a 20% decrease. Percentages are not symmetric, and that asymmetry is the source of a lot of misleading headlines. The Online Percentage Calculator has a mode for each question so you never have to remember which way round the division goes.

Mental shortcuts that actually work

• 10% is “move the decimal”. 10% of 340 is 34. Everything else builds from there: 5% is half of that (17), 20% is double (68), 15% is 34 + 17 = 51.
• Swap them. X% of Y equals Y% of X. 8% of 50 feels awkward; 50% of 8 is obviously 4.
• 1% then multiply. 1% of 2,400 is 24, so 7% is 168.
• Tips and tax: 15% tip on $48: 10% is 4.80, half again is 2.40, total 7.20. A 17.5% VAT: 10% + 5% + 2.5%.

Trap 1: percentages vs. percentage points

If a mortgage rate goes from 4% to 5%, it rose by 1 percentage point but by 25 percent (1 ÷ 4). Both statements are true; a headline saying “rates up 25%” is technically correct and deliberately alarming. Whenever the thing being measured is itself a percentage (interest rates, conversion rates, unemployment, market share), say “points” for the difference and “percent” for the relative change, and ask which one you are being told.

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Trap 2: reverse percentages

A jacket costs $84 after a 30% discount. What was the original price? The instinct is to add 30% back: 84 × 1.30 = 109.20. Wrong. The $84 is 70% of the original, so the original is 84 ÷ 0.70 = $120. The same logic applies to anything quoted “after” a percentage: a salary after a 5% raise, a price including 20% VAT (divide by 1.20, do not subtract 20%), a figure net of fees. The rule: to undo a percentage, divide by (1 ± rate); never add or subtract the rate itself.

This is the exact mistake behind under-invoicing for payment fees, which is why the PayPal fee calculator has a reverse mode.

Trap 3: stacking percentages

A 20% discount followed by a further 10% off is not 30% off. The second discount applies to the already-reduced price: 100 → 80 → 72, so 28% off in total. Likewise a 10% loss followed by a 10% gain leaves you at 99, not 100. To combine percentage changes, multiply the factors: 0.80 × 0.90 = 0.72. This matters for anything compounding: investment returns, inflation over several years, sequential markdowns.

Percent error, percent difference and other cousins

• Percent error (lab work, estimates): |measured − true| ÷ true × 100. Always relative to the accepted value.
• Percent difference (comparing two measurements, neither “true”): |A − B| ÷ ((A + B) ÷ 2) × 100. Relative to their average.
• Percentile is not a percentage of a value; it is a rank. Being in the 90th percentile means scoring higher than 90% of the group, whatever the score was.
• Basis points: 1 bp = 0.01 percentage point. Finance uses them to avoid the points-vs-percent confusion; “up 25 bps” is unambiguous.

Frequently asked questions

How do I calculate a percentage of a number quickly?

Find 10% by moving the decimal one place left, then scale. For 35% of 80: 10% is 8, so 30% is 24 and 5% is 4, total 28.

How do I find the original price before a discount?

Divide the sale price by (1 − discount). $84 after 30% off: 84 ÷ 0.70 = $120.

Is a 50% increase followed by a 50% decrease back to the start?

No. 100 → 150 → 75. Percentage changes are not symmetric.

What is the difference between percent and percentage points?

Points measure the arithmetic gap between two percentages (4% to 5% is 1 point). Percent measures relative change (that same move is a 25% increase).