What is X% of Y? Multiply Y by X and divide by 100. 20% of 250 is 50.
X is what percent of Y? Divide X by Y and multiply by 100. 50 out of 250 is 20%.
X is Y% of what? Divide X by Y and multiply by 100. If 50 is 20% of something, that something is 250.
| Percent | Fraction | Shortcut |
|---|---|---|
| 5% | 1/20 | divide by 20 |
| 10% | 1/10 | move the decimal one place |
| 12.5% | 1/8 | halve three times |
| 20% | 1/5 | divide by 5 |
| 25% | 1/4 | halve twice |
| 33.3% | 1/3 | divide by 3 |
| 50% | 1/2 | halve |
| 75% | 3/4 | halve, then add half again |
X% of Y always equals Y% of X. That makes awkward calculations easy: 4% of 75 is hard, but 75% of 4 is obviously 3. Likewise 18% of 50 is simply 50% of 18, which is 9.
A rate moving from 2% to 3% has risen by one percentage point and by 50 percent. Both statements describe the same change. Percentage points express the absolute gap; percent expresses the relative change against the starting value.
A 50% rise followed by a 50% fall leaves you 25% down, because the second change is measured against a larger base. To reverse an X% increase you need a decrease of X ÷ (100 + X) × 100 percent. To recover from an X% decrease you need an increase of X ÷ (100 − X) × 100 percent — which is why a 50% loss requires a 100% gain to break even.
Successive changes never add. Two 30% discounts do not make 60% off; they make 51%, because the second applies to the already-reduced price. The general form is to multiply the remaining fractions: 0.7 × 0.7 = 0.49, so 51% comes off.
The same applies to growth. Three consecutive years of 10% growth is not 30% but 33.1%, since 1.1 × 1.1 × 1.1 = 1.331. Over long periods this gap becomes the entire story — it is the reason compounding matters and the reason simple annual averages overstate real performance.
Two mistakes recur. The first is averaging rates from groups of different sizes. A 90% pass rate in a class of 10 and a 50% rate in a class of 100 do not average to 70% — the combined rate is 9 plus 50 out of 110, or about 54%. Weight by group size or the answer is meaningless.
The second is averaging growth rates arithmetically. A year of +20% followed by −10% averages to +5% on paper but delivers 8% over two years, an annualised 3.92%. Compounded series need the geometric mean: (final ÷ initial) raised to the power of one over the number of periods, minus one.
Break the target into pieces you can compute easily. For 15% of 80: ten percent is 8, half of that is 4, so 15% is 12. For 35% of 60: 10% is 6, so 30% is 18, and 5% is 3, giving 21.
Reversing the operands often helps too, since X% of Y equals Y% of X. Sixteen percent of 25 looks awkward; 25% of 16 is plainly 4. Tips are the everyday application — 20% is a tenth doubled, and 15% is a tenth plus half a tenth.