Math walkthrough · the formula behind stacked discounts

The successive discount formula (and why it isn't a + b)

Successive discounts — 20% off, then 10% off that, then 5% off that — are a compound operation, and the arithmetic that feels right (adding the percentages) is wrong. The correct formula is a product, and its expansion shows exactly why the sum overstates every time.

The formula

Each discount keeps a fraction of the price, and the kept fractions multiply:

final price = P × (1 − a)(1 − b)(1 − c)…

The equivalent single discount — the one percentage that does the same job in one layer — is:

d_eq = 1 − (1 − a)(1 − b)(1 − c)…

Two layers expand as (1 − a)(1 − b) = 1 − a − b + ab. That last term — the interaction — is the whole story: the combined discount is the naive sum minus a·b, because the second discount applies to a price the first discount already shrank. Since a·b is positive, the sum always overstates.

Worked example: 20% + 10% + 5%

On a $100 base, so the factors are the percentages themselves:

1. After 20%: $100 × 0.8 = $80
2. After 10%: $80 × 0.9 = $72
3. After 5%: $72 × 0.95 = $68.40

Equivalent single discount: 1 − 0.684 = 31.6% — not the 35% the naive sum suggests. The 3.4-point gap is exactly the interaction terms the sum ignores. (The multiplication commutes, so the order of the layers never changes the final price — which is why a single equivalent discount exists at all. What order does change is eligibility: which coupons qualify to apply together is a rule question, covered in coupons on sale items.)

Verify it

For the everyday versions of this: one layer is mental math (the 20% off walkthrough), the reverse direction — recovering the original price — is the reverse-discount formula, and the calculator that runs any stack is the triple discount calculator.

Arithmetic only — register rounding, gift-card interplay, and which coupons a retailer actually lets you combine are separate questions answered by the coupon's terms and the retailer's policy.