Two successive discounts of 10% and 15% are equivalent to a single discount of

Concept — When two discounts are applied one after the other, they do not add. The second discount is charged on the price that is left after the first one, so the two price factors multiply: a discount of a% followed by b% leaves (1 - a/100) x (1 - b/100) of the marked price. The single discount that has exactly the same effect is therefore a + b - (a x b)/100 percent.

Application — for discounts of 10% and 15%:

1. Fraction of the price left after the 10% discount: 1 - 0.10 = 0.90.

2. Fraction left after the further 15% discount: 0.90 x (1 - 0.15) = 0.90 x 0.85 = 0.765.

3. Fraction of the price removed in total: 1 - 0.765 = 0.235.

4. Written as a percentage, the single equivalent discount is 23.5%.

Cross-check — track a marked price of Rs 100 through the two stages:

A reduction of Rs 23.50 on a marked price of Rs 100 is 23.5%, which matches the multiplication above. Applying the two discounts in the reverse order gives 0.85 x 0.90 = 0.765 as well, so the single equivalent discount does not depend on which discount is given first.

Contrast — why simply adding the two rates is wrong:

• Adding the rates gives 10 + 15 = 25%, which would leave 0.75 of the marked price; that charges the 15% against the full marked price instead of against the reduced Rs 90.

• The correction is the product term (10 x 15)/100 = 1.5 percentage points, so 25 - 1.5 = 23.5%.

• The same shortcut works for any two successive discounts a% and b%: a + b - (a x b)/100.