The arithmetic here is not hard and almost nobody gets it wrong through arithmetic. The errors come from answering a different question to the one asked, and from one property of percentages that is genuinely counter-intuitive.
The four questions
What is X% of Y? Multiply and divide by a hundred. 15% of 240 is 36.
X is what percent of Y? Divide and multiply by a hundred. 36 out of 240 is 15%.
What is the percentage change from A to B? The difference, divided by the starting value. From 40 to 50 is (50 − 40) ÷ 40 = 25%.
What was it before the change? Divide by one plus the change. This is the one people get wrong, and it has its own section below.
The middle two look similar and are opposites. Reading the question twice costs less than any of the rest of this page.
Always divide by where you started
Percentage change is measured against the original figure, never the new one.
From 40 to 50 is a 25% increase, because the change of 10 is a quarter of 40. It is not a 20% increase, which is what you get by dividing by 50 — and 20% is the answer to a different and rarer question: how much of the new figure the change represents.
The asymmetry surprises people. Going from 40 to 50 is a 25% rise; coming back from 50 to 40 is a 20% fall. Same two numbers, same gap, two different percentages, both correct.
Why changes never cancel
Start with 100. Add 50% and you have 150. Take 50% off and you have 75.
You are not back at 100 and you never will be, because the second percentage was taken from a larger base. The rise added 50; the fall removed 75.
The order makes no difference — 100 down 50% is 50, up 50% is 75. Either way you finish below where you began, and this holds for any pair of equal-sized opposite changes.
To actually reverse a 50% fall you need a 100% rise. To reverse a 20% fall you need 25%. The larger the fall, the more disproportionate the recovery, which is the entire arithmetic of why a portfolio down 50% needs to double.
Working backwards from a discounted price
An item costs 80 after 20% off. What was it before?
The instinct is to add 20% to 80, giving 96. That is wrong, and it is wrong for the reason above: the 20% was taken from the original price, and 20% of the original is more than 20% of the discounted one.
The correct move is to divide. 80 is 80% of the original, so the original is 80 ÷ 0.8 = 100. Check it: 20% off 100 is 80.
The same applies to removing tax from a total, which is the version most people meet at work. A total of 120 including 20% tax contains 100, not 96 — divide by 1.2, do not subtract 20%.
Percent against percentage points
An interest rate rises from 4% to 6%.
That is a rise of 2 percentage points. It is also a rise of 50 percent, because 2 is half of 4. Both statements are accurate and they sound wildly different, which is why the ambiguity is used deliberately in headlines and advertising.
When the quantity being measured is itself a percentage — an interest rate, an unemployment figure, a conversion rate, a market share — say which you mean. It is the one place where being pedantic is the same as being clear.
Averaging percentages
You usually cannot. Two shops with 50% and 10% conversion rates do not average to 30% if one had ten visitors and the other had ten thousand.
Percentages carry no information about the size they came from, so averaging them treats a sample of ten as equal to a sample of ten thousand. Go back to the underlying counts, add those, and divide once.
The short version
Divide by where you started. Reverse a discount by dividing, not by adding back. Equal opposite changes never cancel. And when the thing changing is itself a percentage, say whether you mean percent or percentage points.
A calculator that shows all four questions at once is worth more than one that makes you pick, because picking wrongly is the actual failure mode.