Percentage change
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Last August and September, President Donald Trump asserted that his actions would reduce drug prices by as much as 1500%, and more recently claimed actual reductions by as much as 600%. On April 22, Elizabeth Warren questioned RFK Jr. about this. She registered a doubt about the mathematics of a reduction in price by greater than 100%, although she mainly focused on the fact that Costco's prices for some cited drugs are substantially less than those at Trump Rx.
The president pitched his Trump Rx website as the answer for Americans who are worried about healthcare costs. He claims that Trump Rx has reduced prices by as much as 600%, 600%, which I think means companies should be paying you to take their drugs.
A couple of days ago in the Oval Office, RFK Jr. left Costco out of it, and offered an odd defense of the president's percentage calculations.
I was reminded when the President was speaking
of a conversation that I had yesterday with one of the Democratic senators
who was questioning me during the hearing and
she was ridiculing President Trump for his math
and she was saying
"It's im- mathematically impossible
to have ((that)) a drug drop by six hundred percent cost"
which he had claimed.
And I said "well if the drug was a hundred dollars
a- and it raised the price to six hundred dollars
that would be a six hundred percent rise.
Well if it drops from six hundred to a hundred
that's a six hundred percent savings."
Last fall, a report on remedial math programs at the UC San Diego shocked some commenters with the proportions of students apparently unable to solve some very basic math problems. But the clips featured above underline the fact that this is not not a new issue.
As explained in the Wikipedia article on Relative Change, if V1 represents the old value and V2 the new one, the formula for Percentage Change is
So an increase in price from \$100 to \$600 is an a percentage change of
And a decrease in price from \$600 to \$100 is a percentage change of
As Senator Warren suggested, the only way to get a 600% decrease in price would be to make the V2 price negative, specifically -\$3000:
In other words, the company would need to pay you \$3000 to get the drug.
My experience teaching today's (generally smart and well prepared) college students is that many of them have conceptual problems with percentage change calculations, and in particular often need to be reminded of the just-cited asymmetry. So it's not surprising that Donald Trump and Robert F. Kennedy Jr. had similar problems; though it's worrying that none of their aides straightened them out.
The video clip for the Senate hearing:
And for the Oval Office press event:
Dave Emme said,
April 25, 2026 @ 1:32 pm
Um, close but no cigar!
(-4200 – 600) is NOT -3600
Which your calculation appears to assume.
(-4200 – 600) = -4800
Ask any 4-function calculator.
So ((-4200 – 600) / 600) x 100% = -800%
Or am I missing something?
Mark Liberman said,
April 25, 2026 @ 1:40 pm
@Dave Emme "Um, close but no cigar!"
Fixed now — I was never any good at mental arithmetic. So sympathy for RFK Jr. …
At least I had the equation right :-).
VVOV said,
April 25, 2026 @ 3:32 pm
I think the more succinct way to express what trump et al are *trying* to say is a “fold change”, that is:
A price change from $100 to $600 is a sixfold increase.
A price change from $600 to $100 is a sixfold decrease.
Jim Breen said,
April 25, 2026 @ 4:18 pm
The suffix "fold' simply means "multiple.. A reduction from 600 to 100 is NOT a sixfold decrease.
AntC said,
April 25, 2026 @ 4:47 pm
it's worrying that none of their aides straightened them out.
Yes this. We don't need mathematically (or linguistically or scientifically or militarily ..) literate politicians, we just need them to take advice from what is (or at least was) the greatest intelligence-gathering organisation in history. Instead, Trump would rather watch Fox so-called News and Twitter.
Kenny Easwaran said,
April 25, 2026 @ 4:51 pm
I've seen bits of this kerfuffle going around. Part of it seems to me to just be about the confusing ways to think about ratios.
A price change from $100 to $500 could be described as getting 400% more expensive, or 5 times more expensive, or going up by a factor of 5. I suppose you might even be able to say it's going up in price by 4 times.
A price change from $500 to $100 could naturally be described as getting 80% cheaper, or going down by a factor of 5. You might be able to say it's 5 times cheaper, though some would object. If someone were just describing this as 400% cheaper, I'd say this is just someone making the mistake of thinking that 400% more expensive followed by 400% cheaper should cancel, which I'd be willing to grant as a non-standard but natural usage.
But the actual prices I've seen quoted in any of these discussions don't match any of this.
(Much of the terminological confusion is parallel to other confusion about probabilities and odds – I think that "odds of 4 to 1" usually means probability 1/5 or 20%, but might occasionally mean probability 1/4 or 25%. It might also sometimes mean 4/5 or 80%, or even 3/4 or 75%. And I seem to remember that Language Log has had some posts in the past about the difficulty of interpreting "risk ratios" and "odds ratios" and other ways of calculating changes in odds.)
Julian said,
April 25, 2026 @ 5:20 pm
Editor here.
Percentages more than 100 are cognitively hard and I think should be avoided if at all possible.
Don't "say X is 500% more than y."
Say " X is six times bigger than y."
As for "a sixfold decrease", this is a bit like the common locution " X is ten times less than y" – when you mean "X is a tenth as big as y."
I know that's very common, but it always grates on me.
J.W. Brewer said,
April 25, 2026 @ 9:55 pm
Pace Julian, here's a maybe useful post from 2008 defending the empirical-existence-and-thus-grammatical-acceptability of phrases like "six times lower than." Which seems like it ought to be approximately synonymous with "600% less than" but is maybe somehow more tolerable than that as an established idiom, although there are of course peevers who say it's illogical or wrong …. Indeed, in the comment thread someone responds to a complaint that it's obviously just wrong when you try to write it out as an actual mathematical equation by saying more or less dude, it's an idiom so its semantics are non-compositional.
https://motivatedgrammar.wordpress.com/2008/12/21/having-defended-five-times-bigger-on-to-six-times-lower/
Note that this blogger finds the usage (as "ten times lower than" in the writing of David Hume, so not a recent development.
Coby said,
April 25, 2026 @ 10:01 pm
The only percentage calculation that's symmetric is the natural logarithm of the ratio, multiplied by 100.
VMartin said,
April 26, 2026 @ 4:00 am
Maybe it's a matter of time. The price rose 500% and then fell by 80%.
Robert Coren said,
April 26, 2026 @ 8:24 am
@Julian: I wouldn't say "X is six times bigger than y"; I would say "X is six times as big as Y". The very common "N times bigger" usage is kind of ambiguous.
Joshua K. said,
April 26, 2026 @ 10:10 am
I remember a radio commercial for a clinic which claimed to have reduced patients' post-surgery recovery time by 200%.
So let's say that previously, a patient who had surgery would have to remain in the hospital for two days.
a. If they reduced the recovery time by 50%, the patient could leave the hospital one day after surgery.
b. If they reduced the recovery time by 100%, the patient could leave the hospital immediately after surgery.
c. So if they reduced the recovery time by 200%, when could the patient leave after surgery? My guess would be two days before the surgery took place.
unekdoud said,
April 26, 2026 @ 1:13 pm
@Cody If you only mean equal increases and decreases cancel, the problematic version max(from,to)/min(from,to) (with the appropriate rescaling and sign) also does that.
Also, both of these are so much more difficult to calculate than the correct "relative change" version:
(a) I spent $25 last month and this month I spent (100-93)% as much.
(b) I spent $25 last month and this month I spent 100%/(100+93)% as much.
(c) I spent $25 last month and this month I spent exp(-93%) as much.
And all of this doesn't work if you have any negative numbers. Mixing addition/subtraction and greater/less becomes disastrous.
(d) Through buying candy I gained $10 of it in January, $1 in February, $0 in March, -$1 in April (I sold one), $1 in May. For each pair of months, what percentage did that gain increase/decrease by?
(If you use the phrase "10x less", the same issues apply.)
One more example, using both relative increase and decrease in the same sentence:
(e) We hired 60% more kitchen staff, but wait times only decreased by 40%.
Even without the loaded wording it might be misleading!
unekdoud said,
April 26, 2026 @ 1:17 pm
Sorry, without the dollar signs:
(a) I spent 25 last month, and this month I spent (100 −93) pct as much /
(b) I spent 25 last month and this month I spent 100/(100+93) as much /
(c) I spent 25 last month and this month I spent exp(-93%) as much.
(d) Through buying candy I gained 10 in January, 1 in February, 0 in March, −1 in April (I sold one), 1 in May.
Thomas Shaw said,
April 26, 2026 @ 1:22 pm
I tend to favor "increase/decrease by a factor of 6" for 100 to 600 or 600 to 100, respectively. I agree with Julian above that relative changes of more than 100% are generally grating, and I even avoid relative changes that are less than but approaching 100%, because the increase and decrease by say 80% are so asymmetrical. For even bigger changes I tend to reach for orders of magnitude — "increased by 3 orders of magnitude" to go from 100 to 100 thousand or "decreased by 3 orders of magnitude" to go from 100 to 0.1. Lots of biological data involving large ratios are presented as log fold change (sometimes log2 or log10, depending on the field).
MattF said,
April 26, 2026 @ 3:32 pm
What’s disturbing about the exchange with Senator Warren is that RFK Jr. couldn’t bring himself to say in public that Donald Trump had, obviously, made a mistake. This inability is far worse than any sort of misunderstanding about the allowable magnitude of a relative percentage change. The point of the exchange should be clear: Trump is infallible. Period.
Julian said,
April 26, 2026 @ 5:41 pm
@Mattf
I look forward to the Trump executive order declaring that pi is 3.
JPL said,
April 26, 2026 @ 5:59 pm
Dr Oz, at least, was laughing at RFK Jr's explanation of Trump's "different way of calculating". I haven't read the above comments, but it appears that Sen. Warren did not ask him, "Just out of curiosity, according to your boss's "different way of calculating", if the cost of the drug is 600 dollars, what would a one hundred percent decrease of that price be?" (The support of the Trump cult has a completely irrational basis.)
D.O. said,
April 27, 2026 @ 10:10 am
AFAIK difference in basis confusion is pretty common in inexpert finance evaluations. If stock prices rose yesterday by 1% and fallen today by 1%, they are less today then a day before yesterday, by a very small ammount, but still.
There is nothing sacred in calculation % change = (V(new)-V(old))/V(old)x100%. We could have defined it as "Trump-Kennedy change" = (V(new)-V(old))/min(V(old),V(new))x100% or to keep things even more symmetric, "commenter D.O. change" = 2x(V(new)-V(old))/(V(old)+V(new))x100%. But once given a definition its better to stick to it.
G Doorenbos said,
April 27, 2026 @ 3:19 pm
This reminds me of how currency inflation is sometimes reported with values of 1000% or more. When a currency's exchange rate plummets from, say, 1.10 to 0.10, apparently it is common to calculate this as (1.10 – 0.10)/0.10 = 1000%. (I.e., in the percentage change formula the divisor is V(new) instead of V(old).)
Michael Watts said,
April 27, 2026 @ 6:00 pm
This is false; I was annoyed when I took Intro to Microeconomics that the textbook defined an all-new way to report "percentages" which was specifically chosen to "solve" the "problem" that an increase of x% and a decrease of x% don't cancel out.
To have percentages where the numeric value of opposite increases and decreases cancel out, they are calculated as a (real) percentage of the midpoint between the original value and the new value. This means it is always the case that an increase of x% combined with a decrease of x% means no change, but it stops being the case that an increase of x% from K is the same size as a decrease of x% from K.
Michael Watts said,
April 28, 2026 @ 8:54 am
I'm not sure how else you think that would work. It's not an issue of using an atypical definition of percentages. When a currency's exchange rate plummets from "1 to the US$1.10" to "1 to the US$0.10", that means that it's inflated from "1 to the US$1.10" to "11 to the US$1.10".
You'll notice that the second set of quoted rates is the one with normal phrasing (though it'd be even more normal if we used a basis of US$1 rather than US$1.10), and also that 11 represents an increase of 1000% from 1. That's what 1000% inflation means.
Michael Watts said,
April 28, 2026 @ 8:58 am
Let me try that again:
Michael Watts said,
April 28, 2026 @ 9:06 am
One more shot:
Jonathan Smith said,
April 28, 2026 @ 9:55 pm
To go full pedant, e.g. "two times bigger" doesn't even make plain language sense. Let us say rather "two times [OR twice] as big [OR the size]." Cf. Chinese, where no one knows if e.g. "da liang bei" (lit. "big[er by] two times") means twice as big or three times as big or what (nerds say it's the latter.) Let us say rather "liang bei da", unambiguously (lit.) "two times [as] big."
Extending to percentages, here also let us make clear reference to original size, thus e.g. "130% as big [OR the size]" rather than "30%~130% bigger", "40% as big [OR the size]" rather than the criminal "60% smaller", etc.
Philip Taylor said,
April 29, 2026 @ 7:39 am
Why do you view "60% smaller" as criminal, Jonathan ? To my mind, "x% smaller" is valid for all $0 < x <= 100$.
Philip Taylor said,
April 29, 2026 @ 7:41 am
Sorry, that formula should have read "0 < <100".
Philip Taylor said,
April 29, 2026 @ 7:43 am
Oh b****r. One (clearly) can't copy and paste a mathematical "x" into a Mathjax formula. One last try — Sorry, that formula should have read "$0 < x <100"$.
Tom said,
April 29, 2026 @ 6:30 pm
I don't know what you-all are on about. I just hope Trump puts his math in an EO. Then I'll increase my purchases of drugs exponentially.
Jim said,
May 1, 2026 @ 3:39 pm
Given the widespread use of phrases like "This is three times as small as that" — which makes no sense to me, they mean 1/3 as large, but we contextually know what is intended — I found myself unable to get worked up over this math "error" after the first day of it.
We all knew a "600% decrease" meant "1/6 as much" (really an 84% or so decrease) and it didn't seem worth dwelling on something he would never be willing to correct. (Of course, with every second phrase from Trump being incorrect [or a lie], the math issue does need to be pointed out over and over, I guess.)
unekdoud said,
May 2, 2026 @ 7:58 am
One case where the correct (fixed base) relative change formula is better:
It is very common in role-playing games for critical hits to have both a percentage chance and a percentage bonus amount. For example, when you have a 10% chance to deal 400% more damage, it works out to a bonus of 40% to the average damage.
Imagine the same situation worded as a weakness debuff (treat the best hit as normal and the lower damage as a weaker hit) that causes a 90% chance to deal 80% less damage.
Here you can work out the end result almost immediately, whereas it would be much tougher using the alternative wordings.