Percentages Explained: Parts, Changes, and Common Traps

One relationship, three questions

Every basic percentage problem uses the same relationship: part = percent ÷ 100 × whole. You can be asked for any one of the three quantities. “What is 15% of 80?” asks for the part. “12 is what percent of 80?” asks for the percent. “12 is 15% of what?” asks for the whole.

Dividing a percent by 100 turns it into a multiplier: 15% becomes 0.15, 150% becomes 1.5. Once you think of percentages as multipliers, most of the harder problems below become straightforward.

Percent change is measured from the starting value

To find a percent change, divide the difference by the original value: (new − old) ÷ old × 100. Going from 50 to 65 is a 30% increase because the 15-unit rise is 30% of the starting 50.

The starting point matters, which is why changes are not symmetric. A price that falls 20% and then rises 20% doesn't return to where it started: 100 → 80 → 96. The rise is 20% of the smaller number. To undo a 20% fall you need a 25% rise.

A change from zero has no meaningful percentage, because there is nothing to divide by. Reporting it as “infinite” growth tells you nothing; quote the absolute change instead.

Stacked discounts multiply — they don't add

“20% off, plus an extra 10% at the register” is not 30% off. Each discount applies to the already-reduced price, so you pay 0.8 × 0.9 = 0.72 of the original — a combined 28% discount.

The same multiplication applies to anything compounded in sequence: successive pay raises, investment returns, or price increases. Two 10% increases are a 21% increase in total, not 20%.

Percent versus percentage points

When a rate itself is a percentage, there are two ways to describe its change. If a loan rate goes from 4% to 5%, it rose by one percentage point — but by 25% in relative terms, because 1 is a quarter of 4.

Both statements are true, and headlines sometimes pick whichever sounds more dramatic. When you read about a change in a rate, check which one is meant.

Quick mental shortcuts

• 10% of a number: move the decimal point one place left (10% of 84.50 is 8.45).
• 5% is half of 10%; 15% is 10% plus 5%; 20% is 10% doubled.
• X% of Y equals Y% of X — so 8% of 50 is the same as 50% of 8, which is 4.
• To add a percent, multiply by (1 + percent ÷ 100); to remove one, divide by it. Taking 8.25% tax out of a total means dividing by 1.0825, not subtracting 8.25%.