Stacked discounts: why 20% off then 10% off isn't 30% off

Take 100. Cut it by 20% and you’re at 80. Cut that by a further 10% and you land on 72, not 70. The two discounts don’t add up to 30% off — they combine to an effective 28% off — because the second percentage is taken from the smaller, already-reduced number, not from the original 100.

The example in full

Start at 100.

• First discount, 20% off: 100 minus 20 is 80.
• Second discount, 10% off: the 10% is now calculated on 80, not on 100. Ten per cent of 80 is 8, so 80 minus 8 is 72.

The second discount was worth 8, not 10, purely because its base had shrunk from 100 to 80 before it was applied. Combine the two steps and you’ve paid 72% of the original price — an effective single discount of 28%, not the 30% that adding 20 and 10 would suggest.

The general rule

Successive percentage discounts multiply; they don’t add. Each one applies to whatever the previous step left behind, so the order of operations matters more than the arithmetic of addition ever would.

Try it with a different pair of numbers to see the pattern hold. Start at 60. Take 20% off first: that leaves 48. Then take 25% off that 48: 25% of 48 is 12, so 48 minus 12 is 36. Two discounts that look like “20% then 25%” — a combined 45 percentage points on paper — actually take you from 60 down to 36, an effective discount of 40%, not 45%.

The shortcut is to multiply the “keep” fractions rather than add the “off” percentages. Keeping 80% and then keeping 90% means keeping 80% of 90%, which is 72% — the number arrives at the same place without needing to track any specific pair of figures.

Why shops stack

A “further 10% off already-reduced sale prices” reads to most shoppers as if it adds straight onto the first discount, landing near 30% off, when it’s actually costing the retailer noticeably less than that framing implies.

Checking any stack

The reliable way to check any run of discounts is to chain the multiplications step by step, exactly as above: apply the first percentage to the starting price, then apply the second to whatever that first step produced, and so on for as many discounts as are stacked. A calculator that takes a starting price and a list of percentages will do this chaining for you and show the effective single discount at the end, which is the number worth comparing against any “combined” figure printed on a sign.

This same logic — that percentages compound rather than add when applied in sequence — is the same mechanism behind percentage change calculations generally, and it’s why reverse percentages (working back from a final price to find the original) need the discounts unwound in the opposite order, not simply added back.