Stacked Discounts Calculator

The formula this calculator uses

Final Price = Price x (1 - first) x (1 - second)

P

Original price before any discount — 200 in the standard case

a

First discount as a decimal — 0.20 for 20% off

b

Second discount as a decimal — 0.10 for 10% off

(1 - a)(1 - b)

Combined survival factor — 0.72 here, the mirror of a 0.28 discount

How to check the result by hand

1. 1

   Convert each discount into the share that survives

   Subtract each rate from 1. A 20% discount leaves 0.80 standing and a 10% discount leaves 0.90, so the two survival factors are 0.80 and 0.90. This one move removes the temptation to add the rates, because the factors are obviously meant to be multiplied.

2. 2

   Apply the first discount to the original price

   200 x 0.80 = 160.0, so the first cut of 20% removes 40. Keep this intermediate figure visible, because the second discount is measured against it rather than against the original 200.

3. 3

   Apply the second discount to the reduced price

   160.0 x 0.90 = 144.0, so the second cut of 10% removes only 16 rather than the 20 it would have taken from the original. The final price is 144.0 and the total saved is 200 - 144.0 = 56.0.

4. 4

   Multiply the two factors for the effective discount

   0.80 x 0.90 = 0.72 surviving, so 1 - 0.72 = 0.28 is the single rate equivalent to the chain. Checking it against the money confirms it: 200 x 0.28 = 56.0, the same saving reached step by step.

5. 5

   Compare with the naive sum and report the gap

   Adding the rates gives 0.20 + 0.10 = 0.30, which overstates the true 0.28 by 0.020000000000000018. Stating both the effective rate and that gap keeps the headline honest, and the gap is always the product of the two rates.