Aptitude · Profit, Loss and Discount

The percentage is always on what the seller paid

Profit and loss is a two-price chapter: what it cost and what it sold for. The arithmetic is trivial. The marks turn entirely on one convention — which of the two prices goes underneath when you turn the gap into a percentage.

Set a cost and a selling price and watch the percentage form →

Profit% and loss% are always computed on the cost price. Divide by the selling price and you have calculated something real, but not what was asked.

01 The idea

Two prices, one gap, and a fixed denominator

A shopkeeper buys an article for ₹400 and sells it for ₹500. He is ₹100 better off. That much is obvious, and it is also not an answer to any exam question, because ₹100 does not say whether the deal was good.

To judge it you compare the gain against the outlay. He risked ₹400 to make ₹100, so the gain is 100/400 = 25% of what he put in. That is the profit percentage, and the denominator is the cost price. Always.

It is worth seeing what the alternative gives, because it is a real quantity and it is the wrong answer. Dividing by the selling price gives 100/500 = 20%. That figure is the profit margin, which is what businesses actually quote — but in this chapter, “profit per cent” means on cost, and 20% will be sitting in the options to catch you.

The most useful way to hold all of this is as a single multiplier. A 25% profit means the selling price is 125% of the cost price. One number, and it works forwards and backwards: ₹400 × 1.25 = ₹500, and ₹500 ÷ 1.25 = ₹400. Almost every question in this chapter is one application of that idea.

Profit% = (SP − CP)/CP × 100. The cost price is underneath, because the cost price is what the seller risked.

Cost price (CP)What the seller paid to acquire or produce the article, including any extra expense such as transport or repair. It is the base of every percentage in this chapter.

Selling price (SP)What the buyer actually paid. Above the cost price it is a profit, below it a loss, and equal to it is breaking even.

Profit percentageThe gain as a percentage of the cost. Equivalent to saying SP = CP × (100 + p)/100, which is the form worth using.

02 Worked example

₹400 in, ₹500 out

This article carries the lesson and the next one. A shopkeeper buys an article for ₹400 and sells it for ₹500. Find the profit and the profit percentage.

Name the two pricesFour hundred is what he paid, so it is the cost price. Five hundred is what he received, so it is the selling price.CP = 400 SP = 500

Find the gap in rupeesThe selling price is higher, so this is a profit rather than a loss.profit = 500 − 400 = ₹100

Divide by the cost priceOne hundred as a share of the four hundred he risked. This is the step where the convention bites.100 / 400 × 100 = 25% profit

See what the wrong denominator givesDividing by the selling price instead gives a different number, and it is a plausible one.100 / 500 × 100 = 20% ← the margin, not the profit per cent

Rewrite it as one multiplierA 25% profit means the selling price is 125% of the cost. Check that it reproduces the numbers.400 × 125/100 = 500 ✓ and 500 × 100/125 = 400 ✓

The 20% is not a mistake in arithmetic — it is a mistake about what was asked. Both 25% and 20% describe this sale correctly, and only one of them is the profit percentage. When an option list contains two numbers this close together, that is usually what is being tested, and the check is to ask what you divided by.

03 The method

The four formulas, which are one formula

Everything below rearranges the same statement. Learn the multiplier form and the rest follow without memorising.

SP = CP × (100 + p)/100 for a profit of p%, and SP = CP × (100 − l)/100 for a loss of l%. Backwards: CP = SP × 100/(100 ± that percentage).

Divide, never subtract. To recover the cost from a selling price of ₹500 at 25% profit, divide by 1.25 to get ₹400. Subtracting 25% of ₹500 gives ₹375, which is wrong, because the 25% was a percentage of the cost and not of the selling price. This single error accounts for more lost marks in the chapter than anything else.

You know	You want	Do this
CP and SP	profit%	`(SP−CP)/CP × 100`
CP and profit%	SP	`CP × (100+p)/100`
SP and profit%	CP	`SP × 100/(100+p)`
SP and loss%	CP	`SP × 100/(100−l)`
CP and SP, SP smaller	loss%	`(CP−SP)/CP × 100`
Extra expense paid	true CP	`purchase + expense`
Subtracting p% from SP	`wrong`	gives 375, not 400

05 Cheat sheet

Profit and loss on one page

The first row is the definition and the rest are its rearrangements. The last two rows are the errors worth naming.

Case	Formula	On CP 400, SP 500
Profit%	`(SP−CP)/CP × 100`	100/400 = 25%
Loss%	`(CP−SP)/CP × 100`	—
SP from CP	`CP(100+p)/100`	400 × 1.25 = 500
CP from SP	`SP × 100/(100+p)`	500 / 1.25 = 400
Break even	`SP = CP`	p = 0
Margin (not asked for)	`(SP−CP)/SP`	100/500 = 20%
SP minus p% of SP	`wrong route to CP`	gives 375

Cost price is always the denominatorProfit per cent means on cost. The same sale is 25% profit and a 20% margin; only the first is what this chapter is asking for.

Extra expense joins the cost priceTransport, repair and making charges are part of what the seller paid, so they go into CP before any percentage is taken.

Reverse questions divideFrom a selling price back to a cost price, divide by the multiplier. Subtracting the percentage from the selling price uses the wrong base.

06 Where & why

Where this shows up

Profit and loss is one of the few chapters that appears in nearly every quantitative paper and also in most interviews about basic business sense.

TCS NQT · Accenture · Wipro

One or two direct questions

Usually CP and SP given with the percentage wanted, or the reverse. Options routinely include both the profit per cent and the margin.

Bank PO · SSC CGL

Reverse and multi-step versions

Selling price and percentage given, cost wanted — where dividing rather than subtracting is the whole question.

Later in this module

The base for markup, discount and dealer questions

Marked price, discount and false-weight problems all reduce to a cost price and a selling price with extra steps in between.

Interviews

“What's the difference between profit and margin?”

A real question with a clean answer: profit per cent is on cost, margin is on revenue. Being able to say which is which reads as commercial literacy.

If you take one habit from this lesson, make it the multiplier. Turning “25% profit” into “× 1.25” on sight removes most of the arithmetic and all of the confusion about which direction to go.

07 Interview questions

What gets asked

Ten, starting with the definition and ending with the distinction interviewers actually probe.

Define profit percentage.

The profit expressed as a percentage of the cost price: (SP − CP)/CP × 100. The cost price is the base because it is what the seller committed to the deal. On a ₹400 article sold for ₹500 that is 100/400 = 25%.

Why is it not calculated on the selling price?

Because the percentage is meant to measure return on what was risked, and what was risked is the cost. Dividing by the selling price gives 20% on the same sale — that is the profit margin, a genuine business figure, but not what this chapter means by profit per cent.

An article sells for ₹500 at a 25% profit. What did it cost?

₹400. The selling price is 125% of the cost, so divide: 500 × 100/125 = 400. Subtracting 25% of ₹500 gives ₹375, which is wrong because the 25% was a percentage of the cost, not of the selling price.

What is the fastest way to handle these questions?

Turn the percentage into a single multiplier. A 25% profit is × 1.25 forwards and ÷ 1.25 backwards. A 20% loss is × 0.8 and ÷ 0.8. One number, two directions, and no formula to misremember.

An article costing ₹800 is sold at a 15% loss. Find the selling price.

₹680. A 15% loss makes the selling price 85% of the cost, so 800 × 0.85 = 680. Losses work exactly like profits with the sign of the percentage flipped.

Where does an extra expense like transport go?

Into the cost price, before any percentage is taken. If a stone costs ₹4200 and ₹1600 is spent making it into a ring, the cost price is ₹5800, and profit is measured against that. Forgetting to add the expense inflates the profit percentage.

A stone bought for ₹4200 has ₹1600 spent on it and sells for ₹7200. Find the profit percentage.

About 24.14%. The true cost is 4200 + 1600 = ₹5800, and the profit is 7200 − 5800 = ₹1400. So 1400/5800 × 100 = 24.14%. Using ₹4200 as the base would give 71%, which is why the expense matters so much.

Can profit percentage exceed 100%?

Yes, easily. An article costing ₹500 sold at ₹1000 is a 100% profit, and anything above that goes higher. The margin, by contrast, can never reach 100%, because the profit is always a part of the selling price — which is a useful sanity check on which one you have computed.

Two articles are sold at the same price, one at 20% profit and one at 20% loss. What is the net result?

Hard

Two articles are sold at the same price, one at a profit of x% and the other at a loss of x%. (a) Show algebraically that the net result is always a loss, whatever the value of x. (b) Show that the loss is x²/100 per cent of the total cost. (c) Verify your formula against x = 20 using the numbers from the previous problem. (d) Explain in one sentence, without algebra, why it has to be a loss.

Follow-up

Part (d) is the one worth getting: the article sold at a loss must have had the larger cost price to reach the same selling price, so the x% loss is taken on a bigger base than the x% profit. Every exam question of this type is that sentence in disguise, and knowing it lets you answer without computing anything.

Show the hint

Let the common selling price be S. Write each cost price in terms of S and x, add them, and compare the total with 2S.