Successive Discount Formula: Turning Two or Three Discounts into One

A successive discount is a chain of reductions where each new percentage is applied to the price left after the previous one. A shop that advertises "20% off, plus an extra 10% at the counter" does not give 30% off the tag price. It first cuts the tag price by 20%, and then takes 10% off what remains.

Exam questions test whether you can turn such a chain into one equivalent single discount, work backwards to the marked price, or choose the better of two offers. All three tasks become quick once you think of each discount as a multiplier rather than an amount to subtract.

A discount of r% leaves (100 − r)% of the price, so the price is multiplied by (1 − r/100). A 20% discount multiplies by 0.80; a 15% discount multiplies by 0.85; a 5% discount multiplies by 0.95. A chain of discounts is simply a product of these multipliers.

For two discounts of a% and b%, the price kept is (1 − a/100)(1 − b/100). Expanding the brackets gives 1 − a/100 − b/100 + ab/10,000. The fraction removed is therefore a/100 + b/100 − ab/10,000, which, written as a percentage, is:

The term ab/100 is the part of the second discount that would have been charged on money already removed by the first. Subtracting it corrects the double-counting. Because the formula is symmetric in a and b, the order of the two discounts never changes the answer.

The same formula handles price rises if you write an increase as a negative discount, or more simply use the general rule for successive percentage changes: x + y + xy/100, where an increase is positive and a decrease is negative. A 30% rise followed by a 20% cut is 30 − 20 − 6 = a 4% net rise.

With three discounts the expanded formula becomes long, so use the multiplier method directly. Multiply the three retention factors, then subtract the product from 1.

A useful shortcut is to apply the two-discount formula twice. Combine 10% and 20% into 28%, then combine 28% and 30%: 28 + 30 − (28 × 30)/100 = 58 − 8.4 = 49.6%. Both routes agree, and the second one is easier to do in your head when the numbers are round.

When the selling price is given, divide by the product of the multipliers instead of multiplying.

To find a missing discount, divide the overall retention factor by the known one. If the overall discount is 40% and the first discount is 25%, the second multiplier is 0.60 ÷ 0.75 = 0.80, so the second discount is 20%.

Comparing a flat discount with a chain is a common question. Convert the chain to its equivalent single discount and compare the two percentages; the larger one is better for the buyer. To express the difference in rupees, multiply the gap in percentage points by the marked price.

Discount questions often combine with mark-up. If a trader marks goods 40% above cost and then allows a 25% discount, the selling price is 1.4 × 0.75 = 1.05 times the cost, a 5% profit. The same multiplier chain works for any sequence of mark-ups and discounts.

• ✘ Adding the discounts. Two 50% discounts leave 25% of the price, not zero.
• ✘ Applying the second discount to the marked price instead of the reduced price.
• ✘ Dividing by the wrong factor when working backwards. Use the product of all retention factors.
• ✘ Treating "buy 3, get 1 free" as 33⅓% off. You pay for 3 items and receive 4, so the discount is 1/4 = 25% of the total value.

For more percentage tools, see the Quant cheat code and the page on price rise and consumption cut, which uses the same multiplier idea in reverse.

Why can't I simply add two successive discounts?

The second discount is taken on a price that is already reduced, so it removes less money than the same percentage of the marked price. The overlap is ab/100, which is why the equivalent discount is a + b − ab/100 rather than a + b.

Does the order of successive discounts matter?

No. Each discount multiplies the price by a factor such as 0.8 or 0.9, and multiplication gives the same result in any order. A 20% then 10% discount and a 10% then 20% discount both leave 72% of the marked price.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

• Quant Cheat Code: 20 Must-Know Shortcuts & Formulas