Understanding Profit and Loss: Concepts, Formulas & Shortcuts

Understanding Profit and Loss: Concepts, Formulas, and Shortcuts

Profit and Loss is an essential topic in quantitative aptitude, relevant for competitive exams and real-life applications. It deals with transactions involving buying and selling goods, where understanding these concepts can help you calculate profits, losses, and discounts effectively.

Key Concepts of Profit and Loss

Before diving into examples and formulas, let’s clarify some fundamental terms:

1. Cost Price (CP)

The price at which goods are purchased.

2. Selling Price (SP)

The price at which goods are sold.

3. Profit

Occurs when the Selling Price (SP) exceeds the Cost Price (CP).

Profit=SP−CP\text{Profit} = \text{SP} – \text{CP}

4. Loss

Occurs when the Selling Price (SP) is less than the Cost Price (CP).

Loss=CP−SP\text{Loss} = \text{CP} – \text{SP}

5. Profit Percentage

The percentage of profit relative to the Cost Price (CP).

Profit%=Profit×100CP\text{Profit\%} = \frac{\text{Profit} \times 100}{\text{CP}}

6. Loss Percentage

The percentage of loss relative to the Cost Price (CP).

Loss%=Loss×100CP\text{Loss\%} = \frac{\text{Loss} \times 100}{\text{CP}}

7. Marked Price (MP)

The price displayed on the product. Discounts are applied on the Marked Price (MP).

8. Discount

The reduction given on the Marked Price (MP).

Discount=MP−SP\text{Discount} = \text{MP} – \text{SP}

Important Formulas of Profit and Loss

Here are the formulas you need to solve profit and loss problems:

$Gain=SP−CP\text{Gain} = \text{SP} – \text{CP}$
$Loss=CP−SP\text{Loss} = \text{CP} – \text{SP}$
$SP with Gain%=(100+Gain%100)×CP\text{SP with Gain\%} = \left( \frac{100 + \text{Gain\%}}{100} \right) \times \text{CP}$
$SP with Loss%=(100−Loss%100)×CP\text{SP with Loss\%} = \left( \frac{100 – \text{Loss\%}}{100} \right) \times \text{CP}$
$CP with Gain%=100100+Gain%×SP\text{CP with Gain\%} = \frac{100}{100 + \text{Gain\%}} \times \text{SP}$
$CP with Loss%=100100−Loss%×SP\text{CP with Loss\%} = \frac{100}{100 – \text{Loss\%}} \times \text{SP}$

Advanced Concepts

1. Successive Discounts

If two discounts of

a%a\%

and

b%b\%

are applied successively:

Total Discount%=a+b−a×b100\text{Total Discount\%} = a + b – \frac{a \times b}{100}

2. False Weight

If a trader uses false weights to sell goods at cost price, the effective gain is: $Gain%=ErrorTrue Value – Error×100\text{Gain\%} = \frac{\text{Error}}{\text{True Value – Error}} \times 100$

Solved Examples

Example 1: Successive Discounts

Question: The marked price of a shirt is Rs. 1,000. A shopkeeper offers a 30% discount, followed by a 20% discount. What is the final selling price?Solution:

Total Discount%=30+20−30×20100=44%\text{Total Discount\%} = 30 + 20 – \frac{30 \times 20}{100} = 44\%

SP=MP−Discount=1000−(44%×1000)=Rs.560\text{SP} = \text{MP} – \text{Discount} = 1000 – (44\% \times 1000) = Rs. 560

Example 2: Profit Tripling

Question: If the selling price is doubled, the profit triples. Find the profit percentage.Solution: Let the Cost Price (CP) =

xx

, and Selling Price (SP) =

yy

3(y−x)=2y−x→y=2×3(y – x) = 2y – x \rightarrow y = 2x

Profit%=y−xx×100=100%\text{Profit\%} = \frac{y – x}{x} \times 100 = 100\%

Example 3: Real-Life Scenario

Question: A trader buys apples at Rs. 100/kg and sells them at Rs. 120/kg. What is the profit percentage?Solution:

Profit=120−100=Rs.20\text{Profit} = 120 – 100 = Rs. 20

Profit%=20100×100=20%\text{Profit\%} = \frac{20}{100} \times 100 = 20\%

Profit and Loss Shortcuts

Two Transactions with Same Profit and Loss Percentage: Loss percentage = $x2100\frac{x^2}{100}$ , where $x%x \%$ is the gain or loss.
Trader Selling at False Weight: $Gain%=x100−x×100\text{Gain\%} = \frac{x}{100 – x} \times 100$ , where $x%x \%$ is the weight error.

Conclusion

Mastering Profit and Loss requires understanding core concepts, formulas, and shortcuts. Regular practice will help you solve problems efficiently in competitive exams and real-life scenarios.