Profit, Loss and Discount for CUET General Aptitude

Q: How do you find the selling price or the cost price from a percentage?

Use a multiplier. A profit of p% means SP = CP × (1 + p/100), and a loss of l% means SP = CP × (1 - l/100). To get from the SP back to the CP, divide by the same multiplier.

Q: What happens when two items are sold at the same price, one at a profit and one at a loss?

If both sell at the same price, one at x% profit and the other at x% loss, the seller always makes an overall loss of x^2/100 percent. The item sold at a loss cost more, so the same percentage of it is a bigger amount.

Q: How do "buy x, get y free" offers work as a discount?

If all the items cost the same, the customer gets x + y items and pays for x, so the effective discount is y/(x + y) × 100 percent.

About 10 min to learn · 72 questions

Profit, loss and discount questions are percentages with a price tag attached. The arithmetic is easy; most marks are lost by dividing by the wrong base. This lesson covers cost, selling and marked price, profit and loss percentages, finding the cost from a percentage, overheads, equal selling prices, false weights, discounts, markup with discount, free-item offers and successive discounts, with worked examples and the traps beside them.

Syllabus area: General Mental Ability, Numerical Ability

On this page · 7 sections

01The core idea
02Profit and loss
03Discounts
04Exam shortcuts
05Common mistakes
06Quick revision
07Check your understanding

The core idea

Profit, loss and discount are percentages with a price tag. What decides the answer is the base: profit and loss are percentages of the cost price, and a discount is a percentage of the marked price. Divide by the wrong one and you land on a planted option.

The second habit is to think in multipliers. A 15% profit is $\times 1.15$ on the cost, a 12% discount is $\times 0.88$ on the marked price, and a markup followed by a discount is one multiplier after another. When no rupee amounts are given, start from 100.

Profit and loss

What do CP, SP, MP, profit and loss mean?

The cost price is what the seller pays, the selling price is what the buyer pays, and the marked price is the price on the tag before any discount. If the selling price is above the cost price, the seller makes a profit; below it, a loss.

Profit = SP − CP, when the SP is higher.
Loss = CP − SP, when the CP is higher.
Discount = MP − SP.

Key idea

Anything the seller spends to get the item ready to sell, such as transport, repairs, packing or stall rent, is an overhead, and it is added to the cost price.

How do you calculate profit percentage and loss percentage?

Divide the profit or loss by the cost price and multiply by 100. Profit and loss percentages are on the cost price unless the question clearly says "on the selling price".

Profit and loss percentage

\text{Profit \%} = \frac{\text{profit}}{\text{CP}} \times 100 \qquad \text{Loss \%} = \frac{\text{loss}}{\text{CP}} \times 100

Example 1 · Profit percentage

Ishaan buys a guitar for ₹6,400 and sells it for ₹7,200. Find his profit percentage.

Profit: $7{,}200 - 6{,}400 = 800$ .
Profit %: $\tfrac{800}{6{,}400} \times 100 = 12.5\%$ .

Answer

12.5%

Dividing by ₹7,200 instead gives about 11.1%, which is wrong.

How do you find the selling price or the cost price from a percentage?

Use a multiplier. A profit of $p\%$ means SP = CP $\times (1 + \tfrac{p}{100})$ , and a loss of $l\%$ means SP = CP $\times (1 - \tfrac{l}{100})$ . To get from the SP back to the CP, divide by the same multiplier.

Situation	Multiplier on CP	CP from SP
20% profit	$\times 1.20$	$\div 1.20$
15% profit	$\times 1.15$	$\div 1.15$
8% loss	$\times 0.92$	$\div 0.92$
25% loss	$\times 0.75$	$\div 0.75$

Example 2 · Cost from selling price

Mariam sells a rug for ₹2,070 at a profit of 15%. What did it cost?

SP = $1.15 \times$ CP.
CP = $2{,}070 \div 1.15 = 1{,}800$ .

Answer

₹1,800

Taking 15% off ₹2,070 gives ₹1,759.50, which is wrong: the 15% was taken on the cost, not the selling price.

How do overheads change the profit percentage?

Overheads raise the cost price, so they lower the profit percentage. Add every expense to the purchase price before you work out the profit.

Example 3 · Overheads

Ramesh buys a used motorbike for ₹27,000, spends ₹3,000 on tyres, and sells it for ₹33,600. Find his profit percentage.

CP: $27{,}000 + 3{,}000 = 30{,}000$ .
Profit: $33{,}600 - 30{,}000 = 3{,}600$ .
Profit %: $\tfrac{3{,}600}{30{,}000} \times 100 = 12\%$ .

Answer

12%

Leaving out the tyres gives about 24.4%, far too high.

What happens when two items are sold at the same price, one at a profit and one at a loss?

If both sell at the same price, one at $x\%$ profit and the other at $x\%$ loss, the seller always makes an overall loss of $\tfrac{x^2}{100}$ percent. The item sold at a loss cost more, so the same percentage of it is a bigger amount.

Example 4 · Equal selling prices

Two lamps are sold for ₹1,800 each, one at 10% profit and one at 10% loss. Find the overall result.

CP of the first: $1{,}800 \div 1.1 \approx 1{,}636.36$ . CP of the second: $1{,}800 \div 0.9 = 2{,}000$ .
Total CP is about ₹3,636.36 against a total SP of ₹3,600, a loss.
Loss %: $\tfrac{10^2}{100} = 1\%$ .

Answer

A loss of 1%

Watch out

The shortcut works only when the two percentages are equal. For, say, 25% profit and 20% loss, find each cost separately and compare the totals.

How do you solve "bought x for ₹a, sold y for ₹b" questions?

Find the cost of one item and the selling price of one item, then compare them. Never compare the numbers of items directly.

Example 5 · Buying and selling in lots

Nandini buys pencils at 8 for ₹20 and sells them at 6 for ₹18. Find her profit percentage.

CP of one: $20 \div 8 = 2.50$ . SP of one: $18 \div 6 = 3$ .
Profit %: $\tfrac{0.50}{2.50} \times 100 = 20\%$ .

Answer

20%

Exam shortcut

"The CP of $m$ items equals the SP of $n$ items": take each item's cost as ₹1, so $n$ items cost ₹ $n$ and sell for ₹ $m$ . The profit is $\tfrac{m - n}{n} \times 100$ percent. If the CP of 25 equals the SP of 20, the profit is $\tfrac{5}{20} = 25\%$ . Divide by $n$ , the number actually sold.

How do you find the gain of a dishonest dealer who uses a false weight?

Compare what the dealer actually gives with what he is paid for. The base is the quantity actually given: gain is the error divided by the true quantity given, times 100. A dealer who claims to sell at cost but gives 950 g for a kilogram gains $\tfrac{50}{950} \times 100$ , about 5.26%.

Example 6 · False weight with a loss

A trader sells at 5% below cost and gives 900 g for every kilogram. Find his gain or loss.

He collects 0.95 times the cost of 1,000 g, which is the cost of 950 g.
He gives only 900 g.
Gain: $\tfrac{950 - 900}{900} \times 100 \approx 5.56\%$ .

Answer

A gain of about 5.56%

He claims a loss but makes a gain.

Watch out

Combine a markup or a loss with a false weight by multiplying the two factors, not adding them. And divide by the weight actually given, not the weight claimed.

Discounts

What is marked price and how is discount calculated?

A discount is always a percentage of the marked price. After a discount of $d\%$ , the selling price is MP $\times (1 - \tfrac{d}{100})$ , and the marked price is SP $\div (1 - \tfrac{d}{100})$ .

Example 7 · A discount

A cricket bat marked at ₹2,800 is sold at a 15% discount. Find the selling price.

SP: $2{,}800 \times 0.85 = 2{,}380$ .
The discount is $2{,}800 - 2{,}380 = 420$ .

Answer

₹2,380

How do you find the profit when a trader marks up and then gives a discount?

Multiply the markup factor by the discount factor and compare the result with 1. Taking the cost as ₹100 makes it quick.

Example 8 · Markup, then discount

A shop marks goods 30% above cost and gives a 15% discount. Find the profit percentage.

Take CP = 100. Then MP = 130.
SP: $130 \times 0.85 = 110.50$ .
Profit: 10.5%.

Answer

10.5%

The tempting $30 - 15 = 15\%$ is wrong, because the 15% is taken on 130, not on 100.

Exam shortcut

To find the markup needed for a target profit, divide the target selling price by the discount factor. For a 10% profit after a 12% discount: MP = $110 \div 0.88 = 125$ , so mark 25% above cost.

How do "buy x, get y free" offers work as a discount?

If all the items cost the same, the customer gets $x + y$ items and pays for $x$ , so the effective discount is $\tfrac{y}{x + y} \times 100$ percent.

Offer	Received	Paid for	Effective discount
Buy 1 get 1	2	1	50%
Buy 2 get 1	3	2	$33\tfrac{1}{3}\%$
Buy 4 get 1	5	4	20%
Buy 5 get 3	8	5	37.5%

Watch out

Divide the free items by all the items received, not by the items paid for: buy 3 get 1 is 25% off, not $33\tfrac{1}{3}\%$ . If the free item has a different price, use values: the free item's value over the total value received.

What is a successive discount and how do you combine two discounts into one?

Successive discounts apply one after another, each on the price left by the one before. They are never simply added. Two discounts of $a\%$ and $b\%$ equal a single discount of $a + b - \tfrac{ab}{100}$ percent, and the order does not matter.

Two successive discounts

\text{Single discount} = a + b - \frac{ab}{100}

Example 9 · Combining discounts

Find the single discount equal to successive discounts of 30% and 10%.

$30 + 10 - \tfrac{30 \times 10}{100} = 37$ .
Check: $0.7 \times 0.9 = 0.63$ is paid, so 37% is taken off.

Answer

37%

Exam shortcut

For three discounts, multiply three factors: 10%, 20% and 25% leave $0.9 \times 0.8 \times 0.75 = 0.54$ paid, a single discount of 46%.

How do you compare two discount schemes?

Turn each scheme into the fraction of the marked price paid, and compare. The smaller fraction is better for the buyer, and the rupee gap is the difference between the fractions times the marked price.

A flat 40% leaves 0.60 paid, and 25% then 20% leaves $0.75 \times 0.8 = 0.60$ . The schemes are equal.
A flat 35% leaves 0.65 paid, so it is worse for the buyer than 25% then 20%.
To find a missing discount, divide the final fraction paid by the ones you know: 0.63 paid after a first discount of 10% means $0.63 \div 0.9 = 0.7$ , a second discount of 30%.

Exam shortcuts

Base 100. With no rupee values, take CP = 100, or MP = 100 for discount-only questions.
Profit equal to loss at two prices. The cost is exactly halfway between the two selling prices.
An extra ₹k changes the profit from $p\%$ to $q\%$ . Then $(q - p)\%$ of the cost is ₹k.
Profit on the SP. $r\%$ on the SP is $\tfrac{r}{100 - r} \times 100\%$ on the CP, so 20% on the SP is 25% on the CP.
Fractions. $\tfrac{1}{6} = 16\tfrac{2}{3}\%$ , $\tfrac{1}{8} = 12.5\%$ , $\tfrac{1}{9} = 11\tfrac{1}{9}\%$ , $\tfrac{1}{12} = 8\tfrac{1}{3}\%$ .

Common mistakes

Dividing the profit by the SP.
Taking a percentage off the SP to get the CP. Divide by the multiplier.
Forgetting overheads.
Adding successive discounts. 20% then 10% is 28%, not 30%.
Cancelling a markup against an equal discount. $1.2 \times 0.8 = 0.96$ , a 4% loss.
Thinking equal profit and loss percentages cancel. At the same SP they always give a loss.
The wrong base for a false weight.
"Paid for" as the base in a free-item offer.
Giving the MP when the SP is asked, or the other way round.

Quick revision

Profit and loss

Profit % and loss % on the CP
Profit $p\%$ : SP = CP $\times (1 + \tfrac{p}{100})$ . Back: divide
Overheads go into the CP
Same SP, $x\%$ up and $x\%$ down: loss $\tfrac{x^2}{100}\%$

Tricks of the trade

Lots: compare one item's CP and SP
CP of $m$ = SP of $n$ : profit $\tfrac{m - n}{n} \times 100\%$
False weight: error ÷ weight given

Discounts

Discount on the MP. SP = MP $\times (1 - \tfrac{d}{100})$
Markup then discount: multiply the factors
Buy $x$ get $y$ : $\tfrac{y}{x + y} \times 100\%$
Successive: $a + b - \tfrac{ab}{100}$

The multiplier method comes from percentages, and mixtures sold at a profit are worked in mixtures and alligation.

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