Aptitude · Profit, Loss and Discount · Successive discount

10% then 5% is not 15% off

Q: Why is 10% then 5% not a 15% discount?

Because the 5% is taken off the price the 10% already reduced. On a marked price of ₹200 the first discount leaves ₹180, and 5% of ₹180 is ₹9 rather than ₹10. The total discount is ₹29, which is 14.5% of ₹200.

Q: Give the formula for the single equivalent discount.

a + b − ab/100 for two successive discounts of a% and b%. For 10% and 5% that is 15 − 0.5 = 14.5%. The subtracted term is the second discount that was not applied to the first discount's amount.

Q: What does the ab/100 term actually represent?

The discount you did not get on the part already discounted. For ₹200 with 10% then 5%, it is 5% of the ₹20 already removed, which is ₹1 — exactly the difference between ₹170 and ₹171.

Q: Does the order of the discounts matter?

No. The final price is the marked price times both multipliers, and multiplication commutes, so 20% then 10% gives the same price as 10% then 20%. Questions that emphasise the order are checking whether you know it makes no difference.

Q: Find the single discount equivalent to 20% and 10%.

28%. Either 20 + 10 − 200/100 = 28, or note that the multipliers 0.8 and 0.9 give 0.72, so 72% of the price survives and 28% was discounted.

Q: How do you handle three successive discounts?

Chain them in pairs. For 10%, 20% and 25%, combine the first two into 28%, then combine 28% with 25%: 28 + 25 − 7 = 46%. Alternatively multiply 0.9 × 0.8 × 0.75 = 0.54, so 46% was discounted. Never feed three numbers into the two-discount formula.

Q: Which is better for the customer: successive 20% and 10%, or a flat 30%?

The flat 30%. The successive pair is equivalent to only 28%, so the single 30% discount is deeper. You can answer this without any prices at all, which is what makes it a good exam question.

Q: What do two successive 50% discounts come to?

75%, not 100%. Half of a half is a quarter, so a quarter of the price survives. This is the clearest demonstration that discounts cannot add — if they did, the article would be free.

Q: An article marked ₹15,000 sold for ₹8,400 after two discounts, the first being 30%. Find the second.

20%. After 30% the price is ₹10,500, and 8400/10500 = 0.8, so 80% survived the second discount and it was therefore 20%. The key move is dividing rather than working in rupees off the marked price.

Q: Can a successive discount ever equal the sum of its parts?

Only if one of the discounts is zero, since the correction term ab/100 is otherwise strictly positive. So for any two genuine discounts the equivalent is strictly less than the sum — which makes “is my answer below the sum?” a complete check rather than a heuristic.

Two discounts one after the other do not add, because the second is taken off a price the first has already reduced. The gap is small on small percentages and large on big ones, and examiners put the added figure in the options every time.

Chain two or three discounts and watch them fail to add →

Successive discounts of a% and b% equal a single discount of a + b − ab/100. The subtracted term is why they never simply add.

01 The idea

The second discount is charged on a smaller price

A book is marked at ₹200 and the seller offers 10% off, then a further 5% off. It is tempting to call that 15% and pay ₹170. The actual price is ₹171, and understanding the one-rupee difference is the whole lesson.

Work it in order. Ten per cent of ₹200 is ₹20, so the price becomes ₹180. The second discount is five per cent of ₹180, not of ₹200 — that is ₹9, giving ₹171. The total discount is ₹29, which on ₹200 is 14.5%, not 15%.

So the second discount is worth less than its face value, because the base it acts on has already shrunk. The shortfall is exactly the discount-on-the-discount: 5% of the ₹20 already taken off, which is ₹1. That is where the missing rupee went.

In percentage terms that correction is ab/100, giving the single equivalent discount a + b − ab/100. It is always less than a + b, so a successive-discount answer is always smaller than the sum — which is a useful check before you even calculate.

Each discount after the first is taken on a price the earlier ones already reduced. So the equivalent single discount is always less than the sum of the parts.

Successive discountsTwo or more discounts applied one after another, each on the price left by the previous one. Also written as “a discount of a% and then a further b%”.

Single equivalent discountThe one discount that would produce the same final price. Equal to a + b − ab/100 for two discounts.

Multiplier formWriting each discount as (100 − d)/100 and multiplying. The safest route, because it extends to any number of discounts without a new formula.

02 Worked example

₹200, 10% off then 5% off, three ways

The source question, worked by all three standard methods so you can see they agree. The marked price of a book is ₹200 and the seller gives successive discounts of 10% and 5%. Find the selling price and the total discount.

Method 1 — step by stepTake 10% off ₹200, then 5% off whatever is left.200 − 20 = 180, then 180 − 9 = ₹171

Method 2 — multipliersA 10% discount leaves 90%, and a 5% discount leaves 95%. Multiply the two.200 × 90/100 × 95/100 = ₹171

Method 3 — the single equivalentCombine the two percentages first, then apply once.10 + 5 − (10×5)/100 = 15 − 0.5 = 14.5%

Apply the equivalent discount14.5% off ₹200, or equivalently 85.5% of it.200 × 85.5/100 = ₹171 ⇒ discount = ₹29

Find the missing rupeeAdding the discounts would have given ₹170. The extra rupee is the second discount not applied to the first discount's amount.5% of the ₹20 already removed = ₹1 ⇒ 171, not 170

All three methods give ₹171, and each is worth having. Step-by-step is safest when you are unsure; the multiplier form is fastest and extends to any number of discounts; the single-equivalent formula is what you need when the question asks for the overall discount percentage rather than a price. Choose by what the final line asks for.

03 The method

The formula, chaining, and the order question

One formula for two discounts, and a method for three or more that avoids inventing a new one.

For two discounts: equivalent = a + b − ab/100. In multiplier form, which generalises: SP = MP × (100−a)/100 × (100−b)/100 × …

For three or more, chain them in pairs. Combine the first two into a single equivalent, then combine that result with the third. For 10%, 20% and 25%: 10 and 20 give 28%, then 28 and 25 give 28 + 25 − 7 = 46%. Never try to apply the two-discount formula to three numbers at once. The order never matters — multiplication commutes, so 10% then 20% equals 20% then 10%.

Discounts	Sum (wrong)	Equivalent (right)
10% and 5%	`15%`	`14.5%`
20% and 10%	`30%`	`28%`
10% and 5% and…	—	chain in pairs
12% and 25%	`37%`	`34%`
20% and 30%	`50%`	`44%`
10%, 20% and 25%	`55%`	`46%`
50% and 50%	`100%`	`75%`

05 Cheat sheet

Successive discounts on one page

One formula, one general method, and the properties worth knowing before you calculate.

Case	Route	On 10% and 5%
Two discounts, equivalent%	`a + b − ab/100`	15 − 0.5 = 14.5%
Any number, final price	`MP × ∏(100−d)/100`	200 × 0.9 × 0.95
Three or more, equivalent%	`chain in pairs`	—
Order of the discounts	`irrelevant`	10 then 5 = 5 then 10
Always compared with the sum	`equivalent < a + b`	14.5 < 15
Two 50% discounts	`75%, not 100%`	—
Adding the discounts	`always wrong`	would give ₹170

The answer is always less than the sumThe correction term ab/100 is always positive, so the equivalent discount is always smaller than a + b. If your answer exceeds the sum, you have added rather than combined.

Order does not matterThe multipliers commute, so 20% then 10% gives exactly the same price as 10% then 20%. Questions that stress the order are testing whether you know it is irrelevant.

Chain three in pairsCombine two, then combine the result with the third. Applying the two-discount formula to three numbers at once has no meaning.

06 Where & why

Where this shows up

One of the most reliably set discount questions, precisely because the intuitive answer is wrong and sits in the options.

Bank PO · SSC CGL

Two discounts, price or overall% wanted

Both forms appear. If the question wants a percentage use the formula; if it wants a price use the multipliers.

TCS NQT · Accenture

“Which offer is better?”

Comparing successive discounts against a single one. 20% and 10% is 28%, so a flat 30% offer beats it — a question you can answer without any price.

Three-discount versions

Chained in pairs

10%, 20% and 25% coming to 46% is a standard mains-level item. The trap answer of 55% is the sum.

Retail advertising

“Extra 10% off sale prices”

Exactly this arithmetic. The advertised total is never the sum, which is why the wording is always “extra off sale price” rather than a single bigger number.

Before computing anything, note that your answer must come out below the sum of the discounts. That one check catches the commonest error in the chapter and costs nothing.

07 Interview questions

What gets asked

Ten, and the last two are the ones that separate a formula from an understanding.

Why is 10% then 5% not a 15% discount?

Because the 5% is taken off the price the 10% already reduced. On a marked price of ₹200 the first discount leaves ₹180, and 5% of ₹180 is ₹9 rather than ₹10. The total discount is ₹29, which is 14.5% of ₹200.

Give the formula for the single equivalent discount.

a + b − ab/100 for two successive discounts of a% and b%. For 10% and 5% that is 15 − 0.5 = 14.5%. The subtracted term is the second discount that was not applied to the first discount's amount.

What does the ab/100 term actually represent?

The discount you did not get on the part already discounted. For ₹200 with 10% then 5%, it is 5% of the ₹20 already removed, which is ₹1 — exactly the difference between ₹170 and ₹171.

Does the order of the discounts matter?

No. The final price is the marked price times both multipliers, and multiplication commutes, so 20% then 10% gives the same price as 10% then 20%. Questions that emphasise the order are checking whether you know it makes no difference.

Find the single discount equivalent to 20% and 10%.

28%. Either 20 + 10 − 200/100 = 28, or note that the multipliers 0.8 and 0.9 give 0.72, so 72% of the price survives and 28% was discounted.

How do you handle three successive discounts?

Chain them in pairs. For 10%, 20% and 25%, combine the first two into 28%, then combine 28% with 25%: 28 + 25 − 7 = 46%. Alternatively multiply 0.9 × 0.8 × 0.75 = 0.54, so 46% was discounted. Never feed three numbers into the two-discount formula.

Which is better for the customer: successive 20% and 10%, or a flat 30%?

The flat 30%. The successive pair is equivalent to only 28%, so the single 30% discount is deeper. You can answer this without any prices at all, which is what makes it a good exam question.

What do two successive 50% discounts come to?

75%, not 100%. Half of a half is a quarter, so a quarter of the price survives. This is the clearest demonstration that discounts cannot add — if they did, the article would be free.

An article marked ₹15,000 sold for ₹8,400 after two discounts, the first being 30%. Find the second.

Hard

(a) A shop currently offers successive discounts of 20% and 20%. It wants to replace them with a single discount that leaves the customer paying exactly the same. Find it. (b) The shop instead wants to advertise “40% off” while keeping its revenue per item unchanged from the successive scheme. By what percentage must it first raise the marked price? (c) Show that no pair of successive discounts, each under 100%, can ever be equivalent to a single 100% discount, and say what that means about the product of the two multipliers.

Follow-up

Part (c) asks for the structural fact underneath the whole lesson: the multipliers are strictly positive, so their product is strictly positive and the price can never reach zero. That is the same reason repeated replacement never empties a vessel, which is worth noticing across chapters.

Show the hint

For (b), work out what fraction of the marked price the customer pays under each scheme and set the two revenues equal.