The 6% that quietly vanishes at checkout.
You use a 20% coupon on an item that's already 30% off. Is that 50% off in total?
Discounts don't stack by addition. They stack by multiplication.
Experiment
Hands-on experiment
Predict first — a $1,000 item is 30% off, and you have a 20% extra coupon. The final price?
Put it on the register yourself
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
See '30% off + 20% coupon' and everyone reflexively thinks 50%. Do the math and it's 44%. Where did the 6% go?
Because the second discount of 20% applies not to the list price but to the already-discounted price. As the base shrinks, so does the real size of the second discount.
This structure explains why a store advertising '50% + 50%' doesn't hand you free goods, and why the more discounts stack, the wider the gap between the advertised number and what you feel at checkout.
Misconception
Common misconceptions
30% off + 20% off = 50% off.
100 → 70 → 56. That's 44% off in total. The second discount's base is the already-reduced 70, so the 20% is really only worth 14.
50% + 50% off means free.
Half of a half is a quarter. That's 75% off, with 25% left to pay. No matter how many discounts stack, you never reach 100% — multiplication approaches 0 but never gets there.
Formula
Writing it as math
The 'vanished 6%' from the experiment is, in formula terms, the product of the two rates. The essence of stacked discounts: what multiplies is the fraction that remains.
The fraction remaining after discounts
After a% off then b% off, what remains is the product of (1−a) and (1−b). The key: it's the remaining fractions, not the discount rates, that multiply.
The total discount rate
It falls short of the naive sum a+b by exactly ab. For 30%+20%, 0.3×0.2 = 6% goes missing, leaving 44% — the very 6% that vanished in the experiment.
Order doesn't matter
Multiplication commutes. Whichever coupon you apply first, the final price is the same.
In Real Life
Where you meet it in real life
Stacked coupons in online stores
'Member discount + card discount + season coupon' never adds up to the sum of the advertised numbers. Multiply the remaining fractions and you get the real checkout price.
The outlet's 'extra discount'
'An extra 30% off the already-40%-off price' is 58% off, not 70%. The later the discount, the smaller its base — and the smaller its real effect.
A stock falling day after day
Falling 10% a day for 5 days is a 41% drop, not 50%. Declines follow the same multiplicative structure as discounts.
Stacked exchange fees
Pay a 2% exchange fee twice and you've paid 3.96%, not 4%. Small rates look additive, but the gap widens as they grow.
Math Behind
The math behind this
Related lab
Logarithms
If discounts stack by multiplication, a tool already exists to turn those multiplications into additions — the logarithm. See why logs are so convenient in a world of successive rates.
Go to the Logarithms lab →Connection
Labs connect
Previous lab
Percent
Want to nail down percent — the basic language of discounts — first? Head to the previous lab.
← Percent labRecommended next
VAT
If discounts are 'multiplication that subtracts', VAT is 'multiplication that adds'. So why does undoing it require division rather than subtraction?
Go to the VAT lab →