Two discounts, one surprise

Shoppers were puzzled this week by a shop sign: twenty per cent off, and then ten per cent off that. Many believed the shop was giving thirty per cent off.

Sign: $20\%$ off, then $10\%$ off again. Many thought: $30\%$ off.

Price $P=100$; discounts $20\%$, then $10\%$.

Take a coat that costs one hundred tokens. The first discount removes twenty tokens, and the price drops to eighty.

A coat costs $100$ tokens. After the first discount it costs $100\times 0.8=80$.

$P_1=0.8P=80$.

The second discount is ten per cent of eighty, which is eight, so the coat finally costs seventy-two tokens.

After the second: $80\times 0.9=72$ tokens.

$P_2=0.9P_1=72$.

The total discount is twenty-eight tokens, that is twenty-eight per cent, not thirty. The shopkeeper explains: the second discount is taken from a smaller price.

The total discount is $100-72=28$ tokens, so $28\%\ne 30\%$.

Total discount $\frac{P-P_2}{P}=0.28$, and $0.28\ne 0.3$.

One more surprise: if the two discounts are given in the opposite order, you still pay seventy-two tokens, because multiplication does not care about order.

The order does not matter: $0.8\times 0.9=0.9\times 0.8=0.72$.

Swap them: $0.8\cdot 0.9=0.9\cdot 0.8$.