The core relationship behind every percentage
A percentage is just a fraction out of 100: p% means p ÷ 100. Every percentage question is really the equation value = (p ÷ 100) × base, with one of the three quantities — value, percent, or base — missing and the other two given.
Once you can see which quantity is missing, the formula to use follows automatically, because it's the same equation solved for a different variable.
Finding p% of a number
This is the most common case: 'what is 20% of 150?' The formula is result = base × p ÷ 100. Here, 150 × 20 ÷ 100 = 30.
It's used for discounts, tips, tax additions, and interest charges — anywhere a rate is applied to an amount to get a portion of it.
Finding what percent one value is of another
This answers 'x is what percent of y?' — for example, '30 is what percent of 150?' The formula is percent = (x ÷ y) × 100. Here, (30 ÷ 150) × 100 = 20%.
This is the reverse of the first case: instead of knowing the percent and finding the amount, you know two amounts and want the percent that relates them.
Finding the base, and percentage change
If you know a result and the percent that produced it, solve for the base instead: base = value × 100 ÷ p. For example, if 30 is 20% of some number, the number is 30 × 100 ÷ 20 = 150.
Percentage change is a related but distinct question: how much did a value move, relative to where it started? The formula is percentage change = (new − old) ÷ |old| × 100. A move from 100 to 125 is a 25% increase; a move from 100 to 75 is a 25% decrease. The key difference from the first formula is the denominator — change is always measured against the original (old) value, not the new one.
Frequently asked questions
Why do I divide by the old value in percentage change, not the new one?
Percentage change measures how far a value moved relative to where it started. Dividing by the starting value answers 'what fraction of the original did this move by'; dividing by the ending value would answer a different question.
Can percentage change be more than 100%?
Yes. If a value more than doubles, the increase is over 100% — for example, going from 50 to 120 is a 140% increase, since (120 − 50) ÷ 50 × 100 = 140.
Is a 20% increase followed by a 20% decrease back to the original value?
No. A 20% increase on 100 gives 120. A 20% decrease is then applied to 120, not 100, giving 96 — not back to 100. Percentage changes are not simply reversible because the base changes between steps.
What's the difference between percentage and percentage points?
A percentage change is relative (a rate moving from 10% to 15% is a 50% percentage increase), while a percentage-point change is the plain difference (10% to 15% is a 5-percentage-point change). The two describe the same move very differently, so it matters which one is being reported.