Every percent question, answered live — X% of Y, what percent one number is of another, percent change, and adding or subtracting a percentage.
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"200 minus 20%" answers a sale price instantly; "plus" adds VAT or sales tax — no mental gymnastics needed.
Percent change between two figures — revenue, weight, followers — with the formula shown so you can cite it.
"42 is what percent of 60" turns marks into percentages in a keystroke — handy for grading and study goals.
The distinction that causes the most misreporting is between a percentage and a percentage point. If an interest rate moves from 2 percent to 3 percent, that is a rise of one percentage point — and also a rise of 50 percent, because 3 is half again as much as 2. Both statements are true and they describe the same change.
This is routinely exploited. "Unemployment rose 25 percent" sounds alarming; if the rate went from 4 percent to 5 percent, the percentage-point change is one. Neither figure is false, but they carry very different impressions, and which one is quoted usually reveals what the writer wants you to conclude.
The convention worth adopting is to state percentage points for changes in something already measured as a percentage, and to reserve percent change for absolute quantities. Where you must use percent change on a rate, say so explicitly.
A 50 percent increase followed by a 50 percent decrease does not return you to where you started. Starting at 100, a 50 percent rise gives 150, and a 50 percent fall from 150 gives 75. The reason is that each percentage applies to a different base — the second is computed on the larger figure.
The asymmetry grows with size. A 50 percent loss requires a 100 percent gain to recover; an 80 percent loss requires a 400 percent gain. This is why investment losses are far harder to recover from than equivalent-sounding gains suggest, and why drawdown is reported separately from return.
The same trap appears in discounts. A 20 percent discount followed by a further 10 percent is not 30 percent off — it is 28 percent, because the second discount applies to the already-reduced price. Sequential discounts always total less than their sum.
Recovering an original value from a result is where sign errors cluster. If a price after a 20 percent discount is 80, the original is not 80 plus 20 percent — that gives 96. It is 80 divided by 0.8, which gives 100. The discount was taken from the original, so you must divide by the multiplier rather than adding the percentage back.
The same applies to tax. A total of 120 including 20 percent VAT contains 100 of net value and 20 of tax, found by dividing by 1.2 — not by taking 20 percent of 120, which gives 24 and is wrong. This is a genuinely common invoicing error.
The general rule: to remove a percentage that was applied, divide by (1 + rate) for an increase or (1 − rate) for a decrease. Averaging percentages has a related pitfall — the mean of several percentages is only meaningful if each applies to the same base size, which is why a weighted average is usually what you actually want.
Percent change divides by the starting value — going from 1 to 10 is a 900% increase, and from 0 to anything is undefined. For tiny baselines, absolute numbers are more meaningful.
If 80 is the price after a 20% discount, the original is 80 ÷ (1 − 0.20) = 100 — not 80 + 20%. Reversing means dividing, not adding back.
Yes — all fields accept decimals and negative numbers, and the working line always shows what was computed.
If a rate moves from 2% to 3%, that is one percentage point and also a 50 percent increase. Use percentage points for changes in something already expressed as a percentage.
Each percentage applies to a different base. To reverse a 20 percent discount, divide by 0.8 rather than multiplying by 1.2 — the discount was taken from the larger original figure.