Last updated: 2026-05-09

How stacked discounts work

A stacked discount is a sequence of percent-off discounts applied one after another to the same price, where each discount is taken off whatever amount remains after the previous one. Because the discounts compound rather than add, the combined reduction is always multiplicative: "30% off" followed by "an extra 20% off" is 44% off, not the 50% the two percentages sum to.

Effective discount

The effective discount is the single percent-off rate that would produce the same final price as a stack of discounts. For example, 30% off followed by 20% off has an effective discount of 44% — identical to a one-time 44% off.

Mechanism

A single percent-off discount $d$ on a price $P$ leaves a price of $P \cdot (1 - d)$. When discounts stack, each successive discount applies to whatever price survived the previous one, so two stacked discounts $d_1$ and $d_2$ leave:

In general, for $n$ stacked discounts:

The effective discount — the single percent-off rate that produces the same final price — is therefore:

This is always strictly less than $\sum d_i$ whenever there are at least two non-zero discounts, because $(1-a)(1-b) = 1 - a - b + ab > 1 - a - b$ for any $a, b > 0$. The cross term $ab$ is the gap between the naive sum and the effective rate.

Worked example

A $100 item carries 30% off and an extra 20% off:

The total saved is $44, not $50, and the effective discount is 44%, not 50%. A straight 50% off would leave $50 — $6 less than the stacked result.

Common stacking patterns

Estimating the effective discount

For two stacked discounts $d_1$ and $d_2$, the gap between the naive sum $d_1 + d_2$ and the effective discount is exactly $d_1 \cdot d_2$. The exact shortcut formula for two discounts is:

For three or more discounts, higher-order cross terms appear and the product $d_1 \cdot d_2$ becomes an approximation (though a close one). Some examples:

• 30% + 20% → subtract $0.30 \times 0.20 = 0.06$ → 44% off (exact).
• 40% + 30% → subtract $0.12$ → 58% off (exact).
• 25% + 25% → subtract $0.0625$ → 43.75% off (exact).

For a stack of $n$ equal discounts of rate $d$, the effective rate is $1 - (1 - d)^n$, which is always strictly below $n \cdot d$. Five 10% discounts stack to $1 - 0.9^5 = 41\%$, not 50%.

Order of percent-only stacks

Multiplication commutes, so the order of percent-off discounts does not change the final price: $(1 - 0.30)(1 - 0.20) = (1 - 0.20)(1 - 0.30) = 0.56$. Whether a store discount or a coupon is applied first, the result is the same — for percentage discounts.

Order does matter when percent-off and dollar-off coupons are mixed. On a 100-dollar item, "10 off then 20% off" yields 90 × 0.80 = 72; "20% off then 10 off" yields 80 − 10 = 70. The dollar-off coupon is worth more when applied to the higher pre-percentage price. This calculator handles the percent-only case; dollar-off coupons must be worked out separately.

Sales tax

In most US states, sales tax is computed on the post-discount price, which is what this calculator does when the tax rate is non-zero. A few jurisdictions (and most VAT regimes outside the US) tax the pre-discount sticker price; in those cases tax is computed separately on the original amount and added to the discounted total.

Why retailers stack discounts

Stacking persists for three structural reasons rather than as a trick:

1. Headline arithmetic. "30% + 20%" reads as 50% even though the effective rate is 44%, so the advertised numbers appear larger than the realized discount.
2. Anchoring. A "30% off" sale tag plus "an extra 20% at the register" presents two separate price reductions, each anchored against the original price.
3. Margin control. A flat 50% off is a single fixed concession. Stacking the same headlines yields a smaller real discount and lets each layer be adjusted independently.

The underlying operation is ordinary compound multiplication; the calculator shows the actual final price alongside the headline rate.