Stacked discounts: 50% off plus 20% off is not 70% off

A rail is marked 50% off. At the till there is a sign: take an extra 20% off the sale price. Almost everyone reads that as 70% off, and almost everyone is wrong by a wide margin.

It is 60% off. The gap between what people expect and what they pay is the reason this post exists.

Discounts multiply, they do not add

A percentage off is easier to handle as the fraction you still pay. Taking 50% off means you pay 1 − 0.50 = 0.50 of the price. Taking 20% off means you pay 1 − 0.20 = 0.80.

Apply one after the other and those fractions multiply:

0.50 × 0.80 = 0.40

You pay 40% of the original, so the discount is 1 − 0.40 = 0.60, or 60% off. The second discount is 20% of the already reduced price, not 20% of the ticket, and that is the whole story.

On a real tag

Take a jacket at $80.00.

1. First reduction: 80.00 × 0.50 = $40.00.
2. Extra 20% off that: 40.00 × 0.80 = $32.00.

You pay $32.00 and save 80.00 − 32.00 = $48.00, which is 48.00 ÷ 80.00 = 60% of the ticket.

If the signs really had meant 70% off, you would pay 80.00 × 0.30 = $24.00. The difference between the expectation and the till is 32.00 − 24.00 = $8.00 on an $80 item — a tenth of the original price, on one garment.

The order does not matter

Run the same two reductions the other way round:

1. 80.00 × 0.80 = $64.00
2. 64.00 × 0.50 = $32.00

The same $32.00. That is not a coincidence, it is multiplication: 0.50 × 0.80 and 0.80 × 0.50 are the same number. So when a shop tells you which discount it applies before the other, for percentage-off reductions that ordering changes nothing at all.

Stacking three of them

The rule extends by just multiplying more factors. A hypothetical 30% seasonal reduction, a 20% clearance markdown and a 10% code:

0.70 × 0.80 × 0.90 = 0.504

You pay 50.4% of the ticket, so the combined discount is 49.6% — not the 60% you get by adding. On a $250.00 price: 250.00 × 0.504 = $126.00, saving 250.00 − 126.00 = $124.00. Check it the other way: 250.00 × 0.496 = $124.00. Same.

A useful sanity check falls straight out of this: stacked percentage discounts can never reach 100% off. Two 50% reductions give 0.50 × 0.50 = 0.25, so you pay a quarter and save 75%, not everything. Each cut takes a share of a smaller number than the last one.

The exception: a fixed amount in the stack

Everything above holds because both offers are percentages. Mix in a flat amount — $20.00 off, a voucher, a gift card — and the order changes the answer.

A $120.00 basket, with a $20 voucher and a 25% reduction:

• Voucher then percentage: (120.00 − 20.00) × 0.75 = 100.00 × 0.75 = $75.00.
• Percentage then voucher: (120.00 × 0.75) − 20.00 = 90.00 − 20.00 = $70.00.

A $5.00 difference on the same two offers. The gap is exactly the discount applied to the voucher, 20.00 × 0.25 = $5.00: in the first order the percentage cut eats a quarter of your voucher. Taking the percentage off before the fixed amount is the better outcome for you, and which one happens is set by the checkout, not by you. It is worth reading the receipt.

Doing it at the rail

The free Discount Calculator takes one price and one percentage and returns what you pay and what you save. For a stack, run it twice: put the price and the first percentage in, then feed the result back in as the price with the second percentage. The number that comes out the far end is the one on the till display.

Two things it deliberately does not do. It does not add tax — whatever tax applies where you are shopping goes on afterwards, at your own local rate. And it has no opinion about whether the reduced price is good value, because a 60% reduction from a price nobody was paying is still just a number on a tag.