Percentage change is not percentage points

Two quantities, one symbol

The percent sign is doing two jobs, and they produce different numbers.

A central bank raises a rate from 4% to 5%.

• The percentage point change is 1. That is subtraction: 5 minus 4.
• The percentage change is 25%. That is division: the rise of 1 relative to the starting 4.

Both are correct. They describe the same event and they are not interchangeable, and when the base is small the gap between them becomes enormous — a move from 0.5% to 1% is half a percentage point and a doubling.

The convention that resolves it is to say percentage point for the subtraction and per cent for the ratio, and never to leave it to context.

Percentages multiply, they do not add

This is the part that catches people with numbers they care about.

A price falls 50%, then rises 50%. It is not back where it started — it is at 75% of the original. Each percentage was applied to a different base: the fall took half of 100, the rise added half of 50.

100  ->  -50%  ->  50  ->  +50%  ->  75

To recover from a 50% fall you need a 100% rise. From a 20% fall, a 25% rise. From a 90% fall, a 900% rise. The recovery required grows far faster than the loss that caused it, which is why the asymmetry matters most exactly where it is least intuitive.

Reversing an increase is a division

Related, and just as common:

wrong:   original = final - 20% of final
right:   original = final / 1.20

A price that rose 20% to 120 started at 100. Subtracting 20% from 120 gives 96, which is four short. The error is the percentage squared — 4% here, 25% when reversing a 50% rise — so it is small enough to miss on a small change and severe on a large one.

This is exactly the arithmetic behind removing tax from a total, which is why the same mistake shows up on receipts and invoices.

What “lowest terms” is actually for

Reducing a fraction does not change its value. 20/120, 10/60 and 1/6 are one number written three ways.

What reduction changes is what a person can do with it. A sixth of 120 is obviously 20. Twenty one-hundred-and-twentieths of 120 is the same answer and nobody can see it.

That is the whole purpose: comparability and mental arithmetic. It is also why tax authorities publish a reduced fraction beside a rate rather than the division — and why fractions that refuse to reduce are worth knowing about, since they are the cases where no shortcut exists and the calculator is the honest answer.

The same profit, two percentages

The most expensive instance of “what is this a percentage of” is in pricing, where one number has two names depending on what it is divided by.

Sell something costing 60 for 100. The profit is 40, and that single fact is a 40% profit margin and a 66.67% markup — margin divides by the price, markup divides by the cost. Both are correct, neither is a rounding of the other, and the gap widens as the numbers rise: a 50% markup is a 33.3% margin, a 100% markup is a 50% margin.

The error, and it is a common one, is setting a target in one and applying it in the other. Aiming for a 40% margin by adding 40% to the cost gives a price of 84, a profit of 24 and a margin of 28.57% — sixteen less on every unit, from an arithmetic mistake that produces a completely plausible price.

It is the same “divide, do not subtract” shape as reversing an increase, for the same reason: a margin is measured against the price, which is the number you do not have yet.

1. margin = profit / price

   The definition. The profit is measured against the price, which is the number you do not have yet.

2. price - cost = profit

   And the profit is whatever the price leaves after the cost.

3. price x (1 - margin) = cost

   Substitute and rearrange. The cost is the share of the price the margin does not take.

4. price = cost / (1 - margin)

   Divide. A 40% margin on a cost of 60 is 60 / 0.6, which is 100 — not the 84 that adding 40% gives.

Selling price

100

Markup needed

66.67%

Profit

40

Target margin

40%

Every value

Target margin	Selling price	Markup needed	Profit
5%	63.16	5.26%	3.16
10%	66.67	11.11%	6.67
15%	70.59	17.65%	10.59
20%	75	25.00%	15
25%	80	33.33%	20
30%	85.71	42.86%	25.71
35%	92.31	53.85%	32.31
40%	100	66.67%	40
45%	109.09	81.82%	49.09
50%	120	100.00%	60
55%	133.33	122.22%	73.33
60%	150	150.00%	90
65%	171.43	185.71%	111.43
70%	200	233.33%	140
75%	240	300.00%	180
80%	300	400.00%	240

Margin also has a ceiling of 100% it can never reach, because the profit cannot exceed the price. Markup has no ceiling at all. Two percentages, one profit, and almost nothing in common at the top of the range.

One implementation trap

Reducing a fraction means dividing both parts by their greatest common divisor, and the greatest common divisor is defined on integers.

So a fraction derived from a decimal has to be scaled to whole numbers first. 13.5/113.5 has to become 135/1135 before it can be reduced to 27/227. Reduce before scaling and the answer is silently wrong — while every whole-number case keeps working perfectly, which is what makes the bug so easy to ship.