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Profit and loss

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Introduction to Profit and Loss

In everyday life, buying and selling goods is a common activity. Whether you are purchasing groceries, selling old books, or running a business, understanding how much money you gain or lose is essential. This section introduces the fundamental concepts of Profit and Loss, which are crucial for solving many competitive exam problems, especially in India where transactions are often measured in Indian Rupees (INR).

Profit and loss calculations help us determine whether a transaction is beneficial or not. For example, if you buy a bicycle for INR 5,000 and sell it for INR 6,000, you have made a profit. Conversely, if you sell it for INR 4,500, you have incurred a loss. These concepts are not only important for exams but also for practical financial decisions.

Basic Definitions and Relationships

Let's start by defining the key terms:

  • Cost Price (CP): The price at which an item is purchased.
  • Selling Price (SP): The price at which the item is sold.
  • Profit: The amount gained when the selling price is more than the cost price.
  • Loss: The amount lost when the selling price is less than the cost price.

Mathematically, these can be expressed as:

  • Profit occurs if \( SP > CP \)
  • Loss occurs if \( CP > SP \)
Cost Price (CP) Selling Price (SP) Profit Area Loss Area

Why is this important? Understanding these relationships helps you identify whether a transaction results in profit or loss, which is the first step in solving related problems.

Profit and Loss Formulas

Once you understand the basic terms, you can use formulas to calculate the exact amounts and percentages of profit or loss. These formulas are essential tools for solving problems efficiently.

Quantity Formula Description
Profit \( \text{Profit} = SP - CP \) Amount gained when selling price is higher than cost price
Loss \( \text{Loss} = CP - SP \) Amount lost when cost price is higher than selling price
Profit Percentage \( \text{Profit\%} = \left( \frac{\text{Profit}}{CP} \right) \times 100 \) Profit expressed as a percentage of cost price
Loss Percentage \( \text{Loss\%} = \left( \frac{\text{Loss}}{CP} \right) \times 100 \) Loss expressed as a percentage of cost price
Selling Price from Profit Percentage \( SP = CP \times \left(1 + \frac{\text{Profit\%}}{100}\right) \) Calculate SP when CP and profit percentage are known
Selling Price from Loss Percentage \( SP = CP \times \left(1 - \frac{\text{Loss\%}}{100}\right) \) Calculate SP when CP and loss percentage are known

Note: In all percentage calculations, the base is always the Cost Price (CP), not the selling price. This is a common source of confusion.

Profit

Profit = SP - CP

Used when selling price is greater than cost price

SP = Selling Price
CP = Cost Price

Loss

Loss = CP - SP

Used when cost price is greater than selling price

CP = Cost Price
SP = Selling Price

Profit Percentage

\[Profit\% = \left( \frac{Profit}{CP} \right) \times 100\]

Calculates profit as a percentage of cost price

Profit = SP - CP
CP = Cost Price

Loss Percentage

\[Loss\% = \left( \frac{Loss}{CP} \right) \times 100\]

Calculates loss as a percentage of cost price

Loss = CP - SP
CP = Cost Price

Selling Price from Profit Percentage

\[SP = CP \times \left(1 + \frac{Profit\%}{100}\right)\]

Find selling price when profit percentage is known

CP = Cost Price
Profit% = Profit Percentage

Selling Price from Loss Percentage

\[SP = CP \times \left(1 - \frac{Loss\%}{100}\right)\]

Find selling price when loss percentage is known

CP = Cost Price
Loss% = Loss Percentage

Worked Examples

Example 1: Calculating Profit and Profit Percentage Easy
Calculate the profit and profit percentage if an article is bought for INR 500 and sold for INR 600.

Step 1: Identify the Cost Price (CP) and Selling Price (SP).

CP = INR 500, SP = INR 600

Step 2: Calculate the profit using the formula:

\( \text{Profit} = SP - CP = 600 - 500 = 100 \) INR

Step 3: Calculate the profit percentage:

\( \text{Profit\%} = \left( \frac{100}{500} \right) \times 100 = 20\% \)

Answer: Profit is INR 100 and profit percentage is 20%.

Example 2: Calculating Loss and Loss Percentage Easy
Find the loss and loss percentage if an item is bought for INR 800 and sold for INR 700.

Step 1: Identify CP and SP.

CP = INR 800, SP = INR 700

Step 2: Calculate the loss:

\( \text{Loss} = CP - SP = 800 - 700 = 100 \) INR

Step 3: Calculate the loss percentage:

\( \text{Loss\%} = \left( \frac{100}{800} \right) \times 100 = 12.5\% \)

Answer: Loss is INR 100 and loss percentage is 12.5%.

Example 3: Finding Cost Price from Profit Percentage Medium
If an article is sold for INR 1320 at a profit of 10%, find the cost price.

Step 1: Let the cost price be \( CP \).

Step 2: Using the formula for selling price with profit percentage:

\( SP = CP \times \left(1 + \frac{\text{Profit\%}}{100}\right) \)

Given, \( SP = 1320 \), Profit% = 10%

So, \( 1320 = CP \times \left(1 + \frac{10}{100}\right) = CP \times 1.10 \)

Step 3: Solve for \( CP \):

\( CP = \frac{1320}{1.10} = 1200 \) INR

Answer: The cost price is INR 1200.

Example 4: Successive Profit and Loss Hard
An article is sold at a profit of 20% and then at a loss of 10% on the new price. Find the overall profit or loss percentage.

Step 1: Assume the cost price of the article is INR 100 (for easy calculation).

Step 2: After 20% profit, the new price becomes:

\( SP_1 = 100 \times (1 + \frac{20}{100}) = 100 \times 1.20 = 120 \) INR

Step 3: Then, a loss of 10% on this new price means the selling price becomes:

\( SP_2 = 120 \times (1 - \frac{10}{100}) = 120 \times 0.90 = 108 \) INR

Step 4: Compare final selling price with original cost price:

Since \( SP_2 = 108 > CP = 100 \), there is an overall profit.

Step 5: Calculate overall profit percentage:

\( \text{Profit\%} = \left( \frac{108 - 100}{100} \right) \times 100 = 8\% \)

Answer: Overall profit percentage is 8%.

Example 5: Discount and Marked Price Medium
An article is marked at INR 1500 and sold at a 10% discount. If the cost price is INR 1200, find the profit or loss percentage.

Step 1: Calculate the selling price after discount:

\( SP = \text{Marked Price} \times \left(1 - \frac{\text{Discount\%}}{100}\right) = 1500 \times (1 - 0.10) = 1500 \times 0.90 = 1350 \) INR

Step 2: Compare selling price with cost price:

CP = INR 1200, SP = INR 1350

Step 3: Calculate profit:

\( \text{Profit} = SP - CP = 1350 - 1200 = 150 \) INR

Step 4: Calculate profit percentage:

\( \text{Profit\%} = \left( \frac{150}{1200} \right) \times 100 = 12.5\% \)

Answer: Profit percentage is 12.5%.

Formula Bank

Profit
\[ \text{Profit} = SP - CP \]
where: \( SP \) = Selling Price, \( CP \) = Cost Price
Loss
\[ \text{Loss} = CP - SP \]
where: \( CP \) = Cost Price, \( SP \) = Selling Price
Profit Percentage
\[ \text{Profit\%} = \left( \frac{\text{Profit}}{CP} \right) \times 100 \]
where: \( \text{Profit} = SP - CP \), \( CP \) = Cost Price
Loss Percentage
\[ \text{Loss\%} = \left( \frac{\text{Loss}}{CP} \right) \times 100 \]
where: \( \text{Loss} = CP - SP \), \( CP \) = Cost Price
Selling Price from Profit Percentage
\[ SP = CP \times \left(1 + \frac{\text{Profit\%}}{100}\right) \]
where: \( CP \) = Cost Price, \( \text{Profit\%} \) = Profit Percentage
Selling Price from Loss Percentage
\[ SP = CP \times \left(1 - \frac{\text{Loss\%}}{100}\right) \]
where: \( CP \) = Cost Price, \( \text{Loss\%} \) = Loss Percentage

Tips & Tricks

Tip: Use the formula \( SP = CP \times (1 \pm \frac{\text{Profit\% or Loss\%}}{100}) \) to quickly find selling price without separately calculating profit or loss.

When to use: When profit or loss percentage and cost price are known.

Tip: For successive profit and loss problems, convert percentages to multipliers (e.g., 20% profit = 1.20, 10% loss = 0.90) and multiply sequentially instead of adding or subtracting percentages.

When to use: When multiple transactions with profit/loss percentages occur.

Tip: Always calculate profit and loss percentages based on the cost price, not the selling price.

When to use: To avoid confusion in percentage calculations.

Tip: Convert percentages to decimals for easier multiplication in complex problems.

When to use: When dealing with multiple percentage calculations or successive transactions.

Tip: Check if the problem involves discount or tax before applying profit and loss formulas, as these affect the effective selling price.

When to use: When marked price and selling price differ due to discounts or taxes.

Common Mistakes to Avoid

❌ Calculating profit or loss percentage based on selling price instead of cost price.
✓ Always calculate profit or loss percentage using cost price as the base.
Why: Profit and loss percentages are defined relative to cost price, not selling price.
❌ Confusing profit with loss and mixing up SP and CP values.
✓ Remember that profit occurs if \( SP > CP \) and loss if \( CP > SP \); label values carefully.
Why: Mislabeling leads to incorrect formula application and wrong answers.
❌ Ignoring the effect of discount when calculating profit or loss.
✓ Adjust selling price by subtracting discount before calculating profit or loss.
Why: Discount reduces the effective selling price, impacting profit or loss calculations.
❌ Applying successive profit and loss percentages by simply adding or subtracting percentages.
✓ Use multiplication of factors \( (1 \pm \frac{\text{percentage}}{100}) \) for successive calculations.
Why: Percentages compound multiplicatively, not additively.
❌ Using incorrect units or currency symbols in calculations.
✓ Always use INR and metric units consistently as per the problem statement.
Why: Ensures clarity and correctness in real-world context problems.
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