CAT 2025 Lesson : Profit & Loss - CP, SP & Profit

This is an important chapter in Arithmetic which tests one's logical thinking. Data Interpretation cases involving application of basic profit & loss concepts (especially the fixed cost/variable cost structure) are also common. While the basic concepts are quite simple, different scenarios can be integrated to make the questions challenging. The examples in this chapter, which cover most question types, will prepare you for quick understanding of the question/logic, selection of the right approach and application of the same.

1. Cost Price, Selling Price & Profit

The Selling Price (SP) is the price at which a good or commodity is sold.

The Cost Price (CP) is the price at which the good or commodity is purchased.

Typically, one purchases at a lower price and sells at a higher price to make a profit. The amount by which the Selling Price exceeds the Cost Price is called Profit.

Profit = SP - CP
⇒ SP = CP + Profit
⇒ CP = SP - Profit

∴ If any two of SP, CP or Profit are known, the third can be computed.

When CP is more than SP, profit is negative. This is called a Loss.
(Note: Loss is written as a positive value. So, profit of $- 5000$ is a loss of $5,000$)

Example 1

If the cost price of an item was Rs. $450$ and the shopkeeper made a profit of $10 \%$ when he sold it, what price did he sell it at?

Solution

Profit is a percentage of the Cost Price (or purchase price).

Profit $= \dfrac{10}{100} \times 450 = \text{Rs.} 45$

Selling Price $=$ Cost Price + Profit $= 450 + 45 =$ Rs. $495$

Answer: $495$

2. Profit & Loss as Percentages

Sellers often determine the Selling Price by adding a desired Profit to the Cost Price. This desired Profit is usually a percentage of the Cost Price (and not the Selling Price).

Therefore, unless explicitly stated otherwise, Profit/Loss $\%$ is a percentage of the Cost Price.

[Only if the question explicitly specifies that the profit/loss is to be computed as a percentage of the Selling Price or any other benchmark, will you have to treat it otherwise. Questions of this kind are uncommon.]

$\text{Profit} \% =$ $\dfrac{\text{Profit}}{\text{CP}} \times 100 \% =$ $\dfrac{(\text{SP} - \text{CP})}{\text{CP}} \times 100 \% =$ $\left(\dfrac{\text{SP}}{\text{CP}} - 1\right) \times 100 \%$

Once again, a negative profit $\%$ is written as a loss $\%$. For instance, a profit of $-5 \%$ is a loss of $5 \%$.

Example 2

A merchant purchased $100$ kg of rice at Rs. $80$/kg. $10 \%$ of the stock wast lost due to rodents. He sold $40$ kg at Rs. $90$/kg and the rest at Rs. $70$/kg. What was his overall profit or loss $\%$?

Solution

Total Cost Price $= 100 \times 80 =$ Rs. $8,000$

$10$ kg of rice was lost to rodents and $40$ kg was sold at Rs.$90$/kg. ∴ The remaining $50$ kg was sold at Rs. $70$/kg.

Total Selling Price $= (40 \times 90) + (50 \times 70) =$ Rs. $7100$

Profit $\% = \dfrac{7100 - 8000}{8000} \times 100 \% = -11.25 \%$

∴ Loss $\%$ is $11.25 \%$.

Answer: $11.25 \%$

3. Calculating SP or CP with Profit/Loss percentage

As mentioned earlier, Profit % is a percentage of the Cost Price. Also we know that SP = CP + Profit.

$\text{SP} = \text{CP} + \text{CP} \times \dfrac{p}{100}$

⇒ $\text{SP} = \text{CP} \times \left( 1 + \dfrac{p}{100} \right)$

Example 3

Ram bought a book for Rs. $250$ and sold it at a loss of $25 \%$. Ram sold another book for Rs. $250$, and made a profit of $25 \%$ on this sale. What is the overall profit or loss from the sale of the two books?

(1) No Profit/Loss            (2) Loss of Rs. $12.50$           
(3) Profit of Rs. $10$            (4) Profit of Rs. $25$           

Solution

Book 1
CP $= 250$ and Loss $= 25 \%$

Loss $= \dfrac{25}{100} \times 250 =$ Rs. $\bm{62.50}$

Book 2
SP $= 250$ and Profit $= 25 \%$

$\text{SP} = \text{CP} \times \left( 1 + \dfrac{p}{100} \right)$

⇒ $250 = \text{CP} \times \left(1 + \dfrac{25}{100} \right) ⇒ 250 = \text{CP} \times 1.25$

⇒ $\text{CP} =$ Rs. $200$

Profit $= 250 - 200 =$ Rs. $\bm{50}$

Overall Loss $= 62.50 - 50 =$ Rs. $12.50$

Answer: (2) Loss of Rs. $12.50$

Example 4

A merchant sells a product at Rs. $464$ and, thereby, achieves a $16 \%$ profit. At what price did the merchant purchase the product?

Solution

SP $= 464$ , Profit $= 16 \%$

$\text{SP} = \text{CP} \times \left(1 + \dfrac{p}{100} \right)$

$⇒ 464 = \text{CP} \times \left(1 + \dfrac{16}{100} \right)$

$⇒ \text{CP} = 464 \times \dfrac{100}{116} = 400$

Answer: Rs. $400$