Percentage change: five traps that make correct arithmetic look wrong

The formula, and the assumption hiding inside it

Percentage change is the difference divided by the original, times one hundred. That is the whole of it. Almost every mistake below comes not from the arithmetic but from which number you put underneath the line.

• Percentage change = (new value − original value) ÷ original value × 100

Trap one: a zero baseline has no percentage change

Going from 0 to 50 is not a 100% increase, not a 5000% increase, and not 0%. It has no percentage expression at all, because the fraction has a zero denominator and division by zero is undefined.

This matters in real reporting. A product line that sold nothing last month and fifty units this month has no growth rate — only an absolute increase. Tools that silently print 0% or 100% here are lying, and the lie is dangerous in the specific direction it usually goes: a 0% is read as 'no change' when the truth is 'new'.

When a report shows 0% growth on a category, check the baseline. If it was zero, the figure is not a measurement, it is a placeholder for 'cannot be calculated'.

Trap two: successive percentages do not add

This is the one that catches almost everyone. A price rises 20%, then falls 20%. Intuition says you are back where you started. You are not.

The loss is 4%, and the reason is that the second percentage was applied to a bigger base than the first. The 20% fall came off $120, not $100, so it removed $24 while the rise had added only $20.

Multiply the multipliers, never add the percentages. 1.20 × 0.80 = 0.96, and 0.96 is a 4% loss. This holds for any chain of changes, in any order.

A consequence worth remembering: a 50% loss requires a 100% gain to recover. Down 50%, up 50%, and you are still down 25%.

Trap three: percent and percentage points

If a conversion rate moves from 4% to 6%, it has risen by 2 percentage points. It has also risen by 50%. Both statements are true and they describe different things, and political and commercial arguments are routinely won by choosing the flattering one.

• Percentage points: the arithmetic difference between two percentages — 6% minus 4% is 2 points.
• Percent: the relative change — 2 ÷ 4 is a 50% increase.

Neither is dishonest on its own. The dishonesty is in the omission. If you write 'up 50%' without saying of what, a reader will assume points. Say which you mean, every time, and the ambiguity disappears.

Trap four: reversing a percentage means dividing, not adding

An item is on sale at $80 after a 20% discount. What was the original price? The instinctive answer is to add 20% to $80, which gives $96. That is wrong.

The discount was taken off the original price, so the original is the base. To reverse it you divide: $80 ÷ 0.80 = $100. Adding 20% instead gives $96, because 20% of $80 is smaller than 20% of the original.

The same reversal applies to any percentage change: to undo an increase, divide by (1 + rate); to undo a decrease, divide by (1 − rate). Never multiply by the complement and call it a reversal.

This is why 'we will take 15% off the invoice and you take 15% off your fee' does not split the difference. Two reductions off two different bases are not equal halves of one discount.

Trap five: a percentage of what

Every percentage has a base, and a percentage quoted without its base is not information. A 10% discount off list, off the discounted price, or off the total including tax are three different amounts, and all three are described as '10% off' in ordinary speech.

The markup and margin confusion is the same error in a different costume: the same profit is 50% of cost and 33.3% of price, and calling both '50%' guarantees that two people leave a meeting with different numbers in their heads.

Trap six: averaging percentages that sit on different bases

Two consecutive months of growth, 10% then 30%. Average them for 20% and the figure is wrong for the same reason the up-then-down case is wrong: the two percentages were measured against different starting values.

A base of 100 growing 10% reaches 110. Growing 30% from there reaches 143. That is a 43% increase across two months, and the true monthly equivalent is about 19.6% rather than the 20% the average suggests.

• Average the multipliers, not the percentages, when the periods are sequential.
• Average the percentages only when each is measured against the same base.
• For a rate of return, use the geometric mean — the nth root of the total multiplier — rather than the arithmetic average.

This is why investment performance is quoted as an annualised compound figure rather than an average of annual returns. Averaging a series of returns reliably overstates them, and the gap widens the more the returns vary.

Check any of these in a few seconds

The Percentage & Discount Calculator handles increase, decrease, the reverse case and successive changes, and it shows the baseline it used for each one — which is the detail that decides whether the answer is right.