Work out percentages of a number, ratios and percentage change.
Almost every percentage problem is one of three, and knowing which you are facing is most of the work.
The word "of" almost always signals multiplication, and "is" signals equals. Reading "what is 15% of 240" as "what = 0.15 x 240" turns the sentence directly into arithmetic.
Change is always measured against the starting value, which is the detail people get wrong.
A price rising from 80 to 100 is a 25% increase, because the 20 gain is measured against 80. Falling from 100 to 80 is a 20% decrease, because the same 20 is now measured against 100. The same absolute change gives different percentages depending on direction, which is why a 50% loss requires a 100% gain to recover.
| Loss | Gain needed to break even |
|---|---|
| 10% | 11.1% |
| 25% | 33.3% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |
This distinction is routinely mangled in news reporting and it changes the meaning entirely. If an interest rate moves from 4% to 6%, that is a rise of 2 percentage points but a 50% increase in the rate itself. Both statements are true and they describe the same event.
The rule is that "percentage points" measures the arithmetic difference between two percentages, while "percent" measures the relative change. A poll moving from 40% to 44% support gained 4 points, or 10%. Whichever a headline chooses usually reveals what it wants you to feel.
A 10% increase followed by a 10% decrease does not return you to the start. Starting at 100: up 10% is 110, down 10% of that is 99. You end 1% below where you began, and the order does not matter.
The same logic governs successive discounts — 20% off then 10% off is 28% off, not 30% — and successive growth rates. Multiply the factors; never add the percentages.
A percentage without its base is close to meaningless. "Risk increased by 200%" sounds alarming; if the risk went from 1 in a million to 3 in a million, the absolute change is negligible. This is the difference between relative and absolute risk, and it is the most common way statistics are used to overstate a finding.
Two more traps worth knowing. Percentages of different-sized groups cannot be averaged — a 90% rate among 10 people and a 50% rate among 100 combine to 53.6%, not 70%. And percentages above 100% are perfectly valid for growth but not for proportions of a whole: something can grow by 300%, but it cannot be 300% of the total it is part of.
Percentage points measure the arithmetic gap between two percentages; percent measures relative change. A rate moving from 4% to 6% rose by 2 percentage points, which is a 50% increase in the rate.
Because the second percentage applies to a different base. From 100, up 10% gives 110; down 10% of 110 is 11, leaving 99. Percentage changes multiply rather than add.
Divide the part by the whole and multiply by 100. So 63 out of 180 is 63/180 = 0.35, which is 35%.
Because the gain is measured against the reduced amount. Losing half of 100 leaves 50, and getting back to 100 means doubling 50 — a 100% increase. The deeper the loss, the more disproportionate the recovery.
Yes. Percentages are commutative because both are the same multiplication. Both equal 4, and swapping them is often the quickest route to a mental answer.