Percentage Discount and Successive Discounts — Worked Examples
"20% off, plus an extra 10% off" is not 30% off. It is 28% off. That gap is the whole content of this article, and it is worth more than it looks: successive percentage change is a standard question in school maths and in every aptitude test, and it is the same algebra that turns up in compound interest, population growth and dilution problems in chemistry.
The formulas
Single equivalent discount = d₁ + d₂ − (d₁ × d₂ ÷ 100)
What the terms mean
- Marked price (MP) — the listed or "before" price. Discounts are always a percentage of this.
- Sale price (SP) — what you pay after the discount, before tax.
- Single equivalent discount — the one discount that would produce the same sale price as the sequence. This is what you quote when comparing offers.
Discounts multiply, they do not add. Each discount applies to whatever price is left after the previous one, and that price is already smaller — which is exactly why the total is less than the sum.
Worked example 1 — a single discount
A jacket is marked ₹2,400 with 25% off.
Discount = 2400 × 25 ÷ 100 = ₹600
Sale price = 2400 − 600 = ₹1,800
Faster route: paying 75% of the price, so 2400 × 0.75 = ₹1,800 in one step. Learn to work with the surviving fraction (0.75) rather than the discount (0.25) — it makes the successive case trivial.
Worked example 2 — the 20% + 10% question
A ₹5,000 item carries 20% off, then a further 10% off at the counter.
Step by step:
After the first discount: 5000 × 0.80 = ₹4,000
After the second: 4000 × 0.90 = ₹3,600
In one line: 5000 × 0.80 × 0.90 = 5000 × 0.72 = ₹3,600
Single equivalent discount:
Formula route: 20 + 10 − (20 × 10 ÷ 100) = 30 − 2 = 28%
Check: 1 − 0.72 = 0.28 = 28% ✓
A flat 30% would have given 5000 × 0.70 = ₹3,500 — ₹100 cheaper. The second discount was only ever applied to ₹4,000, not to ₹5,000, and 10% of the missing ₹1,000 is that ₹100.
Worked example 3 — does the order matter?
Same item, 10% first and then 20%:
5000 × 0.90 = ₹4,500, then 4500 × 0.80 = ₹3,600
Identical, because 0.90 × 0.80 = 0.80 × 0.90. Multiplication is commutative, so the order of discounts never changes the final price. If a shop insists otherwise, it is applying the second discount to something other than the discounted price.
Worked example 4 — three discounts
₹8,000 with 10%, then 10%, then 5%.
Surviving fraction = 0.90 × 0.90 × 0.95 = 0.7695
Sale price = 8000 × 0.7695 = ₹6,156
Single equivalent discount = (1 − 0.7695) × 100 = 23.05%
Naive addition would have said 25%. The three-discount case is where the two-discount formula stops being worth memorising — just multiply the surviving fractions.
Worked example 5 — working backwards
A book marked ₹1,740 was bought for ₹1,479. What was the discount percentage?
Discount amount = 1740 − 1479 = ₹261
Discount % = 261 ÷ 1740 × 100 = 15%
Check: 1740 × 0.85 = ₹1,479 ✓
The base is always the marked price. Dividing by the sale price instead gives 261 ÷ 1479 × 100 = 17.65%, which is the wrong answer to a different question.
The trap in reverse: a rise and an equal fall do not cancel
A price is raised 20%, then reduced 20%. Back to where it started?
1.20 × 0.80 = 0.96 — the price ends 4% below the original.
The increase applied to the smaller original; the decrease applied to the larger inflated price. The general result: raise by x% and then cut by x% and you always end at (1 − x²/10 000) of the original, whatever the order.
Discount and GST together
The discount is applied to the marked price, and tax is then charged on the discounted value shown on the invoice.
₹2,000 marked, 10% discount, then 18% GST:
2000 × 0.90 = ₹1,800; GST = 1800 × 0.18 = ₹324; total = ₹2,124.
Worked example 6 — finding the marked price from the sale price
The other reverse question: you know what was paid and the discount rate, and you want the original price. Divide, do not add the percentage back on.
An item was bought for ₹1,479 after a 15% discount. What was it marked at?
MP = 1479 ÷ 0.85 = ₹1,740
The wrong route: adding 15% to the sale price gives 1479 × 1.15 = ₹1,700.85, which is ₹39.15 short. The 15% was taken off the larger marked price, so adding 15% back to the smaller sale price cannot return you to it — the same asymmetry as removing tax from an inclusive price.
Discount, cost price and profit — three different bases
The chapter that contains discounts also contains profit and loss, and questions deliberately combine them. The rule that keeps you straight: discount is a percentage of the marked price; profit and loss are percentages of the cost price.
A shopkeeper buys an item for ₹1,200, marks it at ₹1,800 and sells it at a 20% discount. What is the profit percentage?
Sale price = 1800 × 0.80 = ₹1,440
Profit = 1440 − 1200 = ₹240
Profit % = 240 ÷ 1200 × 100 = 20%
The discount was 20% and the profit was also 20% — a coincidence of these numbers, not a rule, and exactly the trap such questions are built around. The two percentages are measured against different bases (₹1,800 and ₹1,200), so they are not comparable quantities even when they happen to print the same.
Quick reference — single equivalent discounts
| Advertised as | Surviving fraction | Actual single discount |
|---|---|---|
| 10% + 10% | 0.90 × 0.90 = 0.81 | 19% |
| 20% + 10% | 0.80 × 0.90 = 0.72 | 28% |
| 25% + 25% | 0.75 × 0.75 = 0.5625 | 43.75% |
| 50% + 50% | 0.50 × 0.50 = 0.25 | 75% (never 100%) |
| 10% + 10% + 5% | 0.90 × 0.90 × 0.95 = 0.7695 | 23.05% |
The last row of the pattern is the useful sanity check: repeated percentage discounts can never reach 100%, however many you apply, because you are always multiplying by something greater than zero.
Common mistakes
- Adding successive discounts. 20 + 10 = 30 is wrong; the answer is 28.
- Using the sale price as the base when finding a discount percentage. Discount is always a percentage of the marked price.
- Believing a rise and an equal fall cancel. They leave you 4% down for 20%, 1% down for 10%.
- Confusing discount with profit or loss percentage. Discount is reckoned on the marked price; profit and loss on the cost price. Three different bases in one chapter, and mixing them is the classic exam trap.
- Rounding at each stage. Multiply the fractions first, round once at the end.
- Forgetting tax. The discounted price is not the final bill.
Where this maths appears
| Context | Use |
|---|---|
| School maths (Class 7–10) | Profit, loss and discount; successive percentage change |
| Aptitude and banking exams | Single equivalent discount; marked price from sale price |
| Commerce | Trade discount, cash discount, invoice preparation |
| Science, indirectly | Serial dilution and repeated decay use exactly the same repeated-multiplication logic |
That last row is worth pausing on. Diluting a solution ten-fold three times leaves 1/1000 of the original concentration, for precisely the same reason that three discounts multiply. Percentages are one idea wearing several uniforms.
Check a shop offer in seconds. The Finance and Everyday section of the calculator suite includes a discount tool, alongside GST and EMI calculators, so you can test the single-equivalent-discount table above against real prices.
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