At the dinner table my mother handed me her phone: the fund she holds rose 10% last month and fell 10% this month, and her conclusion was "that means it has not moved, it will come back". My aunt bought a coat where the tag price was cut by 20% and then a 30% coupon was stacked on at checkout, and she thought that made it half price. My uncle saw "interest rates up 2%" in the news and asked me whether you simply add 2% to the old number. All three claims are wrong in the same place: a percentage is not an isolated number; a denominator sits underneath it, and that denominator shifts with the previous change. Below I take the three sums apart, with the rule of judgment at the end.
Up 10% then down 10%: the denominator changes hands
On the way up, the denominator is the amount before the rise. On the way back down, the denominator becomes the post-rise amount, which is larger. So the same 10% adds 10% of the smaller base and subtracts 10% of the larger base, and the absolute amount removed exceeds the amount added. On that basis, up 10% then down 10% leaves 99.0000% of the original, a net change of -1.0000%. The steeper the rise, the wider the gap: up 20% and back the same percentage leaves 96.0000% (net -4.0000%), up 30% leaves 91.0000% (net -9.0000%), up 50% leaves 75.0000% (net -25.0000%). The error in "it went up and came back down, so I am even" is treating the two rounds' denominators as the same thing.
The drop needed to break even is not the same as the rise itself. The formula I use to check this is drop = rise / (1 + rise), where the 1 + rise in brackets is the denominator that has moved. For 10%, 0.10 / 1.10 = 9.0909%: after a 10% rise you need a 9.0909% fall to get back to the original value, not a 10% fall. The five cases below have four columns with different denominators, so identify the denominator before reading a number.
| Rise (denominator: pre-rise amount) | Fall needed to break even (denominator: post-rise amount) | Left after an equal-percentage fall (vs original) | Net change (vs original) |
|---|---|---|---|
| 10% | 9.0909% | 99.0000% | -1.0000% |
| 20% | 16.6667% | 96.0000% | -4.0000% |
| 30% | 23.0769% | 91.0000% | -9.0000% |
| 50% | 33.3333% | 75.0000% | -25.0000% |
| 100% | 50.0000% | 0.0000% | -100.0000% |
The last row is jarring: up 100% then down 100% leaves 0.0000%, a net -100.0000%, while breaking even from that rise only takes a 50.0000% fall, because the post-rise amount is already twice the original.
The same round trip, a different denominator at each end
Writing the amounts on paper as 4 -> 2 and 2 -> 4 makes it clearest: from 4 to 2 is (2 - 4) / 4 = -50.00%, and from 2 to 4 is (4 - 2) / 2 = 100.00%. The numerator is the same absolute difference either way; one denominator is 4 and the other is 2, so the same round trip is recorded as -50.00% at one end and 100.00% at the other. That is also where "a fall by half needs a double to recover" comes from arithmetically, the mirror image of a 100% rise needing a 50.0000% fall in the table.
Discounts stack round by round: 20% + 30% is not 50%
My aunt's purchase was two rounds: the first takes 20% off, and what remains is the denominator for the second, so the 30% coupon takes 30% of the reduced price. That works out to paying 56.00% and a total discount of 44.00%, not the 50% she claimed. Adding the two percentages directly quietly applies the second one to the original price as well, which conjures up a chunk of discount out of nothing.
There are not many numbers of this kind, but each round needs its denominator fixed first, which is exactly where I used to slip whenever I did it by hand. Now I state the mode for the round before computing: the share step is "a is what percent of b", the comparison step is "how many percent did it change from a to b", and there are two more, adding or subtracting a percentage, and finding the whole from a part. Each of the four computes one direction at a time, and the denominator is decided by the step I choose. Percentage Calculator writes out the formula for the selected mode as well, and the comparison mode also gives the percentage needed to get back to the original value, which are the two numbers I check. Everything runs locally in the browser.
Percentage points, cumulative growth, completion rate: three measures that get mixed up
- Percentage points: an interest rate going from 3% to 5% has an absolute difference of 2 percentage points, while measured against the original 3% as denominator it is 66.6667%. The two statements are not the same thing.
- Cumulative: a base of 10,000 rising 15% twice gives 13225.00, a cumulative gain of 32.2500%, not the 30% you get by adding the two 15%s.
- Completion rate: 80 of a planned 50 is 160.00%, and 40 of 50 is 80.00%. Here the denominator is pinned to the planned quantity, which is why it can exceed 100%.
The cumulative case deserves a note. Two consecutive rises are still reliable because there are only two rounds; over many rounds, "what would that be per period" is not something a single one-direction conversion covers, because it is a period-by-period compounding measure. For that I use Compound Interest Calculator, which handles amounts that roll up period by period, a different level from the card above. As for the interest rate passage, if you want to push the change through to repayments, what you need is monthly payment and total interest, which belongs to Mortgage Calculator and is not the same as converting percentage points.
Q: Up 10% then down 10%: why is that not break-even?
Because the two rounds have different denominators: adding 10% uses the pre-rise amount, subtracting 10% uses the larger post-rise amount, so more is removed than added, leaving 99.0000% and a net -1.0000%. Getting back to the original value really takes a 9.0909% fall.
Q: Can two discounts simply be added together?
No. 20% and 30% in sequence means paying 56.00%, a total discount of 44.00%; the second denominator is what is left after the first discount, not the original price.
Q: When should I say percentage points instead of percent?
When the quantity being compared is itself a percentage. Writing 3% to 5% as "up 2%" reads as 2% times 3%, when the intended meaning is an absolute difference of 2 percentage points; expressed relatively it is 66.6667%.
Q: Can this arithmetic tell me whether to buy?
No, it only handles measurement. Every number above is a ratio between two observations of the same quantity; none of it is a forecast or an allocation recommendation. My only criterion is this: fix the denominator first, then talk about the size of the move.
Back to the three claims at the table: "it will come back" does not hold for the fund, since after a 10% rise breaking even needs a 9.0909% fall, and an equal-percentage fall leaves only 99.0000% of the original; the coat was not half price but 56.00% paid and 44.00% off; and "rates up 2%" on an original 3% is either 2 percentage points or 66.6667%, not a blur of the two. From now on, whenever I see a percentage, I ask one question first: what quantity is this divided by, before the change or after it, planned or actual. Get the denominator wrong and every step after it is wrong too; that is how I judge whether a claim is credible.