Work out percentages quickly — what is X% of Y, and more.
Almost every percentage problem is one of three forms, and mixing them up is the usual source of error.
What is X% of Y? Multiply Y by X and divide by 100. 15% of 200 is 200 × 15 ÷ 100 = 30.
X is what percent of Y? Divide X by Y and multiply by 100. 30 out of 200 is 15%.
X is Y% of what? Divide X by Y and multiply by 100. If 30 is 15% of something, that something is 200.
The trick is identifying which number is the whole. In "20% of students", the students are the whole. In "the price rose by 20%", the original price is the whole — not the new one.
This distinction causes more confusion than any other. If an interest rate moves from 2% to 3%, that is a rise of one percentage point and also a rise of 50 percent. Both statements are correct and they describe the same event with wildly different-looking numbers.
News reports use percentage points precisely to avoid this ambiguity. When you see "the rate rose 0.25 points", it means the absolute difference, not a quarter of one percent of the previous value.
A value that rises 50% and then falls 50% does not return to where it started. Starting at 100, it goes to 150, then to 75 — a net loss of 25%. The reason is that the second change is measured against a different base.
To undo 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.
This is why a share price that halves needs to double to break even, and why successive discounts never simply add up. Thirty percent off followed by a further twenty percent off is a 44% discount, not 50%. Use the discount calculator when stacking offers.