Three successive discounts of 10%, 20% and 30% on the marked price of an item…

Concept

When two or more percentage discounts are applied one after another, each discount acts on the amount left by the previous one, not on the original marked price. So discounts of a%, b% and c% applied successively leave a multiplier of (1 - a/100)(1 - b/100)(1 - c/100) of the marked price, and the single equivalent discount is 100% minus that surviving percentage. Because of this chaining, successive discount rates never simply add up.

Application

1. Let the marked price be Rs 1,000. Any starting value works, because the required answer is a percentage.

2. After the first discount of 10%: 1000 x (1 - 10/100) = 1000 x 0.9 = Rs 900.

3. After the second discount of 20%, applied to Rs 900: 900 x (1 - 20/100) = 900 x 0.8 = Rs 720.

4. After the third discount of 30%, applied to Rs 720: 720 x (1 - 30/100) = 720 x 0.7 = Rs 504.

5. Total reduction = 1000 - 504 = Rs 496, so the single equivalent discount = (496/1000) x 100 = 49.6%.

Cross-check

• Multiply the surviving fractions directly: 0.9 x 0.8 x 0.7 = 0.504, so 50.4% of the marked price still has to be paid and the discount is 100% - 50.4% = 49.6%.

• Combine the rates two at a time using a + b - ab/100: for 10% and 20%, 10 + 20 - (10 x 20)/100 = 28%; then combining 28% with 30%, 28 + 30 - (28 x 30)/100 = 49.6%.

• The plain total 10 + 20 + 30 = 60% overstates the reduction, because the second and third discounts act on already reduced amounts rather than on the full marked price.

Result

The three successive discounts of 10%, 20% and 30% are together equivalent to a single discount of 49.6%.