Solved Examples on Profit & Loss - Quantitative Techniques for CLAT PDF

Introduction

The concept of profit and loss is used to measure the gain or loss in a business transaction. Whenever we purchase and sell an article we either earn a profit or incur a loss depending on the relation between the purchase price and selling price.

Basic terms

Cost price (CP): The price at which an article is purchased.

Sale price (SP): The price at which an article is sold.

Overhead charges: Additional charges incurred apart from the actual cost price, for example freight, rent, salaries, etc.

Profit (when SP > CP): The positive difference when the selling price is more than the cost price.

Loss (when SP < CP): The positive difference when the selling price is less than the cost price.

Important formulae

• Profit = Sale price - Cost price
  P = SP - CP
• Loss = Cost price - Sale price
  L = CP - SP
• Profit percent = (Profit / Cost price) × 100
  (P / CP) × 100
• Loss percent = (Loss / Cost price) × 100
  (L / CP) × 100

Notes

• Profit or loss is always calculated on the cost price.
• If an article is sold at a gain of x%, then SP = (100 + x)% of CP.
• If an article is sold at a loss of x%, then SP = (100 - x)% of CP.

Worked Examples

Example: 1) A person buys an article for Rs. 50 and sells it for Rs. 75. What will be his gain percent?

Solution:

CP = Rs. 50

SP = Rs. 75

Profit = SP - CP = 75 - 50 = Rs. 25

Gain percent = (Profit / CP) × 100 = (25 / 50) × 100 = 50%

Example 2: A person buys an umbrella for Rs. 450 and sells it for Rs. 350/-. Find his loss %.

Solution:

CP = Rs. 450

SP = Rs. 350

Loss = CP - SP = 450 - 350 = Rs. 100

Loss percent = (Loss / CP) × 100 = (100 / 450) × 100 = 200/9 %

Example 3: Find the sale price if CP is Rs. 160 and gain is 20%.

Solutions:

SP = (100 + 20)% of CP = 120% of CP

SP = (120 / 100) × 160 = Rs. 192

Example 4: Find sale price when CP is Rs. 160 and loss is 20%.

Solution:

SP = (100 - 20)% of CP = 80% of CP

SP = (160 × 80) / 100 = Rs. 128

Example 5: Find CP when SP Rs. 64 and loss is 20%

Solution:

Loss = 20% ⇒ SP = 80% of CP

CP = SP × 100 / 80

Example 6: Find CP when SP = 192 and profit is 20%

Solutions:

Profit = 20% ⇒ SP = 120% of CP

CP = SP × 100 / 120

Example 7: A chair is bought for Rs. 150 and sold at a gain of 8%. Find the selling price.

Solutions:

SP = (100 + 8)% of 150 = 108% of 150

Example 8: A bicycle is bought for Rs. 1500 and sold at a loss of 6%. Find the selling price.

Solutions:

CP = Rs. 1500

Loss = 6% ⇒ SP = 94% of CP

= Rs. 1410

Example 9: A table is sold for Rs. 1440 and there is a loss of 10%. At what price should it be sold to gain 10%.

Solutions:

Given SP = Rs. 1440, loss = 10%

CP = SP × 100 / (100 - 10) = 1440 × 100 / 90

CP = Rs. 1600

To gain 10%: SP = (100 + 10)% of CP = 110% of 1600

Example 10: Ravi lost 20% by selling a watch at Rs. 1536. What will be his gain percent if he sells if for Rs. 2040.

Solution:

Given SP1 = Rs. 1536, loss = 20%

CP = SP1 × 100 / 80

CP = Rs. 1920

New SP = Rs. 2040

Gain = SP - CP = 2040 - 1920 = Rs. 120

Gain percent = (120 / 1920) × 100

Example 11: A shopkeeper sells an article at a profit of 20%. If he had bought it at 20% less and sold for Rs. 10 less, he would have gained 25%. Find the cost price of the article.

Solution:

Let actual cost price be Rs. 100x

Actual selling price = 120% of 100x = Rs. 120x

If bought at 20% less, new cost price = 80% of 100x = Rs. 80x

If sold for Rs. 10 less and gain is 25%, new selling price = 125% of 80x = Rs. 100x

According to the question: 120x - 100x = 10

20x = 10

x = 1/2

Actual CP = 100 × (1/2) = Rs. 50

∴ The Cost Price of the article is Rs. 50.

Example 12: A person sells apples at 10 for a rupee and gains 20%. How many apples did he buy for a rupee.

Solution:

Quantity sold = 10 apples for Re. 1

Profit = 20% on the transaction

Profit on 10 apples = 20% of 10 apples = 2 apples (in terms of quantity saved)

So he originally bought 10 + 2 = 12 apples for Re. 1

Example 13: A shopkeeper mixes 160 kg of rice at Rs. 27 per kg with 240 kg of rice at Rs. 32 per kg and sells the mixture to gain 20%. What is the sale rate of the mixture.

Solutions:

Cost of 160 kg @ Rs. 27/kg = 160 × 27 = Rs. 4320

Cost of 240 kg @ Rs. 32/kg = 240 × 32 = Rs. 7680

Total cost = 4320 + 7680 = Rs. 12000

Total weight = 160 + 240 = 400 kg

Gain = 20% ⇒ SP total = 120% of 12000

Sale price per kg = (SP total) / 400

= Rs. 36 per kg

Example 14: A man buys certain no. of oranges at the rate of 3 per rupee and the same number at 4 per rupee. He mixes then together and sells at 7 for two rupees. Find his gain or loss%.

Solution:

LCM of 3 and 4 is 12

Assume he purchases 12 oranges at 3 for Re. 1 ⇒ Money spent = Rs. 4

And 12 oranges at 4 for Re. 1 ⇒ Money spent = Rs. 3

Total oranges purchased = 24

Total CP = Rs. 4 + Rs. 3 = Rs. 7

He sells 24 oranges at the rate of 7 for Rs. 2 ⇒ SP = (24 / 7) × 2

Loss = CP - SP = 7 - (SP)

Percentage loss = (Loss / CP) × 100

Technique to solve problems in a fast way

If a person sells two articles, one at a loss of x% and another at a gain of x%, then there is always a loss to the seller and the loss is equal to

Example 15: A man sells two horses for Rs. 4000 each. On one of the horses he loses 20% while on the other he gains 20%. Find his gain / loss percent in the whole transaction.

Solutions:

Loss percent for one and gain percent for another being equal in magnitude ⇒ overall loss is given by the formula above

Example 16: The profit on selling an article for Rs. 1196 is equal to the loss on selling the same article for Rs. 1056. The cost price of the article is:

1) 1143    2) 1134    3) 1156    4) 1126    5) None of these 

Solutions:

When profit at one selling price equals loss at another selling price, the cost price is the arithmetic mean of the two sale prices

CP = (1196 + 1056) / 2

= Rs. 1126

Example 17: On selling an article for Rs. 264 a man loses 4%. He should sell the article for how much so that his gain is 12%.

Solution:

Given SP = Rs. 264, loss = 4%

CP = SP × 100 / (100 - 4)

Now desired gain = 12% ⇒ Required SP = CP × (100 + 12) / 100

Example 18: If the selling price of 10 pens is the same as the cost price of 8 pens. Find the gain or loss percent.

Solution:

Assume SP of 10 pens = CP of 8 pens = Rs. 100

CP of 1 pen = 100 / 8 = Rs. 12.50

SP of 1 pen = 100 / 10 = Rs. 10

Loss per pen = CP - SP = 12.50 - 10 = Rs. 2.50

Loss percent = (Loss / CP) × 100 = (2.50 / 12.50) × 100 = 20%

Example 19: On selling 100 mangoes, a person gains the SP of 20 mangoes. Find his gain percent.

Solution:

Let SP of 100 mangoes = Rs. 100

He saves SP of 20 mangoes = Rs. 20 ⇒ Profit = Rs. 20

CP = SP of 100 - Profit = 100 - 20 = Rs. 80

Gain percent = (20 / 80) × 100 = 25%

Example 20: On selling a shaving machine for Rs. 1530, the loss is 10%. What will be the gain percent if the machine is sold for Rs. 1819.

Solutions:

First SP1 = Rs. 1530, loss = 10%

CP = Rs. 1700

Second SP2 = Rs. 1819

Gain = SP2 - CP = 1819 - 1700 = Rs. 119

Example 21: A trader allows a discount of 15% of the marked price on his article. How much above the cost price should he mark the price to gain 19%.

Solution:

Let the marked price be Rs. 100

Discount = 15% ⇒ SP = 100 - 15 = Rs. 85

To achieve 19% profit: CP must be such that SP = 119% of CP

So CP = SP / 1.19 = 85 / 1.19

If CP is Rs. 100, the marked price should be

= 140 ⇒ therefore the mark-up is 40% above CP

He should mark the price 40% more than the CP.

Example 22: I purchased an item for Rs. 8200 and sold it at a gain of 25%. From the sale price of the item I purchased another item and sold it at a loss of 20%. What will be the overall Gain or loss.

Solutions:

First CP = Rs. 8200

Gain = 25% ⇒ SP1 = 125% of 8200

SP1 = Rs. 10250

He buys another item for Rs. 10250 and sells at a loss of 20% ⇒ SP2 = 80% of 10250

SP2 = Rs. 8200

Overall there is no gain or loss.

Example 23: The cost price of an item is two third of its selling price. What is the gain / loss percent?

Solution:

Let CP = Rs. 100

Given CP = (2/3) × SP ⇒ SP = (3/2) × CP

SP = Rs. 150

Gain = SP - CP = 150 - 100 = Rs. 50

Gain percent = (50 / 100) × 100 = 50%

Example 24: Apples are bought at 6 for Rs. 5 and sold at 8 for Rs. 11. Find the gain or loss per cent.

Solution:

Take LCM of 6 and 8 = 24 apples

CP of 24 apples = (24 / 6) × 5 = 4 × 5 = Rs. 20

SP of 24 apples = (24 / 8) × 11 = 3 × 11 = Rs. 33

Gain = SP - CP = 33 - 20 = Rs. 13

Gain percent = (13 / 20) × 100

Example 25: Find a single discount equivalent to three successive discounts of 10%, 20% and 40%.

Solution:

Start with Rs. 100

After 10% discount: amount = 100 - 10 = Rs. 90

After 20% discount on Rs. 90: amount = 90 - 20% of 90 = 90 - 18 = Rs. 72

After 40% discount on Rs. 72: amount = 72 - 40% of 72 = 72 - 28.8 = Rs. 43.2

Total discount = 100 - 43.2 = 56.8%

Concluding remarks

These standard methods and formulae cover most routine profit and loss problems encountered in competitive and school-level quantitative reasoning. For multi-item or mixture problems, convert quantities to a common base (LCM) and compare total cost and total selling price. For successive discounts and successive gains/losses, work multiplicatively on a base amount (usually Rs. 100) to find the net effect.