Successive discounts

Learning outcome

Calculate successive discounts, compare equivalent offers and reverse the calculation without confusing the changing percentage bases.

Why successive discounts multiply

With successive discounts, each reduction applies to the price immediately before it. The first uses marked price (MP); the second uses the already reduced price, not MP again. Unless otherwise stated, this is what “successive” means.

A discount of a% retains the fraction 1 - a/100. Two successive discounts therefore give:

SP = MP × (1 - a/100) × (1 - b/100).

The equivalent single discount is the total reduction expressed as a percentage of the original MP. For two discount rates a% and b%, its percentage is a + b - (a × b)/100. The subtracted term corrects the overcount that occurs when the rates are merely added: the second discount acts on a smaller price.

For three or more stages, multiply all retained fractions, then subtract the resulting fraction from 1. Multiply by 100 to express the overall discount as a percentage.

All prices here are positive; discount rates are nonnegative and below 100%. Calculations are exact, with no added fees or intermediate rounding. Under these assumptions, exchanging the order of percentage discounts leaves the final price unchanged because multiplication is commutative, although the separate discount amounts can change.

To recover MP, divide SP by the product of the retained fractions. To find an unknown later discount, compare the final price with the price immediately before that discount.

Worked examples

Example 1 — Two reductions. MP is ₹2,800, with discounts of 15% and then 10%. First price = 2800 × 0.85 = ₹2,380. Final SP = 2380 × 0.90 = ₹2,142. Total discount = ₹658, so equivalent discount = (658 ÷ 2800) × 100 = 23.5%, not 25%.

Example 2 — Three reductions. MP is ₹3,600, with successive discounts of 20%, 10% and 5%. Prices become 3600 × 0.80 = ₹2,880, then 2880 × 0.90 = ₹2,592, then 2592 × 0.95 = ₹2,462.40. Retained fraction = 0.80 × 0.90 × 0.95 = 0.684. Equivalent discount = (1 - 0.684) × 100 = 31.6%.

Example 3 — Reverse the discounts. SP is ₹2,772 after discounts of 12% and 25%. Retained fraction = 0.88 × 0.75 = 0.66. Therefore MP = 2772 ÷ 0.66 = ₹4,200. Multiplying ₹4,200 by both retained fractions reproduces ₹2,772.

Common mistakes

Do not add successive discount rates. Do not apply the second rate to MP. When comparing offers, use the same MP; a larger equivalent discount means a lower SP only for that common base.

Practice questions

1. MP is ₹1,750. Find SP after successive discounts of 8% and 5%.
2. For the same MP, compare discounts of 25% then 16% with one discount of 35%. Which gives the lower SP?
3. SP is ₹2,376 after discounts of 10% and 12%. Find MP.
4. MP is ₹2,400. After a 25% discount and a second discount, SP is ₹1,476. Find the second discount rate.

Worked answers

1. First price = 1750 × 0.92 = ₹1,610. Final SP = 1610 × 0.95 = ₹1,529.50.
2. Retained fraction = 0.75 × 0.84 = 0.63, giving 37% discount. The successive offer is cheaper: its discount is 2 percentage points larger, saving an additional 2% of MP.
3. MP = 2376 ÷ (0.90 × 0.88) = 2376 ÷ 0.792 = ₹3,000.
4. After the first discount, price = 2400 × 0.75 = ₹1,800. The next reduction is 1800 - 1476 = ₹324, so its rate = (324 ÷ 1800) × 100 = 18%.