Aptitude · Profit, Loss and Discount · Marked price

A 50% markup and a 20% discount is not a 30% profit

Once a marked price enters, there are three prices in play and two percentages taken on different bases. Markup is on cost; discount is on the marked price. Because the bases differ, the percentages never simply cancel — and that gap is the shopkeeper's margin.

Set a markup and a discount and watch them fail to cancel →

Markup is a percentage of the cost price. Discount is a percentage of the marked price. Different bases, so they never cancel.

01 The idea

Three prices, in the order they happen

A shopkeeper pays ₹400 for an article. He puts a label on it reading ₹600 — a 50% markup on what he paid. Then he advertises “20% off” and sells it for ₹480. Three prices, and each is derived from the one before it.

The cost price is ₹400, the marked price is ₹600, and the selling price is ₹480. His profit is ₹80 on an outlay of ₹400, which is 20%. Notice that this is neither the 50% he marked up nor the 30% you get by subtracting the two percentages.

The reason is the bases. The 50% was taken on ₹400; the 20% was taken on the larger ₹600. A percentage of a bigger number is worth more, so the discount eats into the markup by more than its face value suggests — and yet the seller still ends up ahead, because the markup applied first and compounded into the base the discount worked on.

There is a general consequence worth internalising: an equal markup and discount always leaves a loss. Mark up 20% and discount 20% and you end at 96% of cost, not 100%. Students assume they cancel; they never do, in either direction.

Marked price = CP × (100 + markup)/100. Selling price = MP × (100 − discount)/100. Two multipliers, applied in that order.

Marked price (MP)The label or list price, sometimes called the tag price. It exists to be discounted from, and is set as a markup on the cost price.

MarkupThe percentage by which the marked price exceeds the cost price. Also called “above cost” in exam phrasing.

DiscountThe percentage taken off the marked price to reach the selling price. Never off the cost price — that is the golden rule of this half of the chapter.

02 Worked example

₹400 cost, marked up 50%, discounted 20%

The same article as the previous lesson, now with a label on it. An article costing ₹400 is marked 50% above cost and then sold at a discount of 20%. Find the selling price and the profit percentage.

Mark up on the cost priceFifty per cent above ₹400 makes the label price 150% of cost.MP = 400 × 150/100 = ₹600

Discount off the marked priceTwenty per cent off ₹600, not off ₹400. The base has changed, which is the crux of the whole lesson.SP = 600 × 80/100 = ₹480

Compare the selling price with the costHe paid ₹400 and received ₹480.profit = 480 − 400 = ₹80

Turn it into a percentage on costEighty rupees on an outlay of four hundred.80 / 400 × 100 = 20% profit

Do it again without the cost priceBoth steps are multipliers, so they combine — and the cost price cancels out entirely.(150 × 80)/100 = 120 ⇒ SP is 120% of CP ⇒ 20% profit

The last step is the one to keep: a markup of m% then a discount of d% gives a profit of (100+m)(100−d)/100 − 100 per cent, with no cost price required. Here 150 × 80 / 100 = 120, so 20% profit. That is why exam questions of this type can give you no rupee figure at all and still have a unique answer.

03 The method

The golden rule, and why equal percentages lose

One rule about bases, and one consequence of it that is worth being able to state instantly.

MP = CP(100 + m)/100, then SP = MP(100 − d)/100. Combined: SP = CP × (100+m)(100−d)/10000, so the profit percentage is (100+m)(100−d)/100 − 100.

Equal markup and discount always lose. Mark up x% and discount x% and you land at (100+x)(100−x)/100 = 100 − x²/100 per cent of cost — a loss of x²/100 per cent. At 20% each that is a 4% loss, not break-even. The discount is taken on a larger base than the markup was, and the difference never quite recovers.

Markup	Discount	Net effect on cost
50%	20%	`+20% profit`
25%	20%	`0% break even`
20%	20%	`−4% loss`
10%	10%	`−1% loss`
40%	30%	`−2% loss`
100%	50%	`0% break even`
Adding the two	`never valid`	different bases

05 Cheat sheet

Marked price on one page

Two multipliers and the combinations worth recognising instantly.

Case	Formula	On CP 400, +50%, −20%
Marked price	`CP(100+m)/100`	400 × 1.5 = 600
Selling price	`MP(100−d)/100`	600 × 0.8 = 480
Profit% direct	`(100+m)(100−d)/100 − 100`	150×80/100 = 120 → 20%
Break even	`(100+m)(100−d) = 10000`	+25% with −20%
Equal m and d	`loss of m²/100 %`	20/20 → 4% loss
Discount off CP	`never`	discount is on MP
Adding m and d	`never`	different bases

Discount is always on the marked priceThis is the golden rule. If a question mentions a discount at all, find the marked price first — the discount has nowhere else to be applied.

The cost price cancels outProfit per cent from a markup and a discount needs no rupee figure. That is why these questions often give none, which students read as missing information.

Equal percentages loseMark up x% and discount x% and you are down x²/100 per cent. Never assume a markup and an equal discount return you to cost.

06 Where & why

Where this shows up

This is the commercial reality behind every “50% off” sign, which is why exams and interviews both like it.

Bank PO · SSC CGL

Markup and discount, profit wanted

The standard form and often given with no rupee figures at all, since the cost price cancels. Students who need a number to start with get stuck.

TCS Digital · Infosys

Reverse: find the markup needed

“What markup allows a 20% discount and still yields 20% profit?” Solve (100+m)(80) = 12000 for m.

Real retail

Why list prices are inflated

The marked price exists so a discount can be advertised while the seller stays profitable. The arithmetic here is exactly the pricing decision a shop makes.

Interviews

“If I mark up 20% and discount 20%, where am I?”

Down 4%. It is a quick test of whether you think about bases or just about percentages, and the wrong answer is the intuitive one.

The habit to carry: whenever two percentages appear in one question, ask what each is a percentage of before doing anything with them. In this chapter they are almost never percentages of the same thing.

07 Interview questions

What gets asked

Ten, and the equal-percentage question is the one most likely to come up in an interview.

What is the marked price?

The label or list price, set as a markup on the cost price. It exists so that a discount can be advertised from it. The chain is cost price, then marked price, then selling price, and each is derived from the one before.

Is a discount taken on the cost price or the marked price?

Always the marked price. That is the golden rule of the discount half of this chapter. If a question gives a discount, you must find the marked price before you can apply it.

An article costing ₹400 is marked 50% above cost and sold at 20% off. Find the profit percentage.

20%. The marked price is 400 × 1.5 = ₹600, and the selling price is 600 × 0.8 = ₹480. That is a profit of ₹80 on ₹400, so 20%. Note it is neither 50% nor 50 − 20 = 30%.

Why can't you just subtract the discount from the markup?

Because they are percentages of different amounts. The markup is a percentage of the cost price and the discount is a percentage of the larger marked price, so the same numerical percentage is worth more as a discount. Subtracting treats them as if they shared a base.

Mark up 20% and then discount 20%. Where do you end up?

At a 4% loss. The multipliers are 1.2 and 0.8, and 1.2 × 0.8 = 0.96, so you land at 96% of cost. In general an equal markup and discount of x% leaves a loss of x²/100 per cent, and it is always a loss, never break-even.

Give the one-line formula for profit from a markup and a discount.

Profit% = (100 + m)(100 − d)/100 − 100. For a 50% markup and a 20% discount that is 150 × 80/100 − 100 = 120 − 100 = 20%. No cost price is needed, which is why these questions often give none.

What markup allows a 20% discount and still leaves a 20% profit?

Fifty per cent. You need (100 + m) × 80 = 120 × 100 = 12000, so 100 + m = 150 and m = 50%. This reverse form is common, and it is one equation once you have the combined multiplier.

What condition makes a markup and discount exactly break even?

(100 + m)(100 − d) = 10000. So a 25% markup with a 20% discount breaks even, as does a 100% markup with a 50% discount. Recognising a break-even pair on sight saves the whole calculation.

A shop offers 20% off and still makes 25% profit. What is the marked price as a percentage of cost?

Hard

(a) Find the condition on m and d for a markup of m% followed by a discount of d% to exactly break even, and give three whole-number pairs that satisfy it. (b) A shop marks up 50% and wants to run the deepest discount it can while still making at least 10% profit. Find the largest whole-number discount it can offer. (c) Explain why, for any fixed markup, there is a discount above which the shop necessarily makes a loss, and give that discount for a 50% markup.

Follow-up

Part (c) asks for the structure rather than a number: the combined multiplier is strictly decreasing in d, so there is exactly one crossing point, and everything beyond it loses money. This is the reasoning behind every real “up to X% off” cap.

Show the hint

Work with the product (100+m)(100−d) and compare it against 10000 for break-even and against 11000 for a 10% profit.