Fifteen successive percentage discounts of 2%, 4%, 6%, 8%, … are applied to an…

Concept

A percentage discount reduces the current price, not the original price. Therefore, successive discounts combine multiplicatively.

After discounts d1, d2, …, dn, the remaining-price factor is (1 − d1/100)(1 − d2/100)…(1 − dn/100). The effective discount exceeds 50% exactly when the remaining price is below 50% of the original price.

Application

1. Take the original price as 100. The successive discount rates are 2%, 4%, 6%, 8%, 10%, 12%, 14%, 16%, and so on.

2. After the first seven discounts, the remaining price is 100 × 0.98 × 0.96 × 0.94 × 0.92 × 0.90 × 0.88 × 0.86 = 55.4162. Thus, the effective discount is 100 − 55.4162 = 44.5838%, which is below 50%.

3. The eighth discount is 16%. Applying it gives a remaining price of 55.4162 × 0.84 = 46.5496. Hence, the effective discount is 100 − 46.5496 = 53.4504%, which exceeds 50%.

Cross-check

Seven discounts leave more than 50% of the original price, while eight discounts leave less than 50%. Therefore, the threshold is crossed for the first time at the eighth discount.

The effective discount first exceeds 50% after the 8th successive discount.