The one question every percent deserves.
If something rises 30% and then falls 30%, is it back where it started?
A percent isn't a size — it's a ratio relative to a base. Change the base and the same % becomes a different amount.
Experiment
Hands-on experiment
Predict first — a $1,000 stock rises 30%, then falls 30%. Is it back at $1,000?
Take the round trip yourself
Start
$1,000
Now
$1,000
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
"The stock fell 30%, then rose 30%" — so it broke even? Intuition says yes, but you're actually down 9%.
The illusion has one cause: the first 30% and the second 30% have different bases. A percent isn't an absolute size — it's a ratio against whatever the current base value is.
Discounts, interest rates, returns, inflation, approval ratings — most of the world's numbers speak in %, and losing track of the base leads to exactly backwards judgments. That's why every % deserves the question: 'percent of what?'
Misconception
Common misconceptions
Down 30% then up 30% puts you back where you started.
100 → 70 → 91. The second 30% is computed on 70, not 100, so you end up 9 short. Rising by the same % you fell never gets you back to even.
If interest rates go from 2% to 3%, they 'rose 1%'.
They rose 1 percentage point (%p) — as a ratio, that's a 50% rise. Your interest burden became 1.5x. Confuse % and %p and you completely misread the size of a change.
Formula
Writing it as math
Writing the experiment's 'why the round trip doesn't return home' as formulas makes it clear: percent change is multiplication, not addition.
A p% increase = multiplying by (1 + p/100)
'Up 30%' doesn't mean adding 30 — it means multiplying by 1.3. The world of percent is a world of multiplication.
The round-trip experiment, unmasked
Up by r then down by r leaves you with 1−r² of the original. With r=0.3 that's 0.91 — a 9% loss. The bigger r is, the bigger the r² bite.
% vs %p
Subtract the difference directly and you get %p (points); divide as a ratio and you get %. The same change can be told in two languages, and which one gets used completely changes the impression.
In Real Life
Where you meet it in real life
The stock-loss recovery trap
Recovering from a 50% loss takes a 100% gain, not 50%. The illusion that 'rising by what it fell' is enough delays stop-loss decisions.
Interest rates in the news
'The base rate rose 0.5%p' and 'interest burden up 25%' can be two tellings of the same event. Distinguishing %p from % is how you read the news accurately.
The base value in sale ads
'Up to 70% off' is measured against the list price. Mark the list price up first, and the real discount shrinks. Always ask: percent of what?
Inflation rates
When inflation 'falls' from 5% to 2%, prices are still rising. Distinguish a change in the rate of change from a change in the value itself.
Math Behind
The math behind this
Related lab
Functions
'Up p%' is a function that multiplies x by (1+p/100). See percent as a function and it becomes natural why applying it repeatedly means multiplication.
Go to the Functions lab →Connection
Labs connect
Recommended next
Discounts
Now that you know percent is multiplication, it's time to see what happens when two discounts stack. Why isn't 30% + 20% equal to 50%?
Go to the Discounts lab →