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Profit-loss

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Introduction to Profit and Loss

In everyday life, buying and selling goods or services are common activities. Understanding whether you have earned money or lost money from a transaction is important-not only in running a business but also for individual financial decisions and competitive exams. The concepts of profit and loss help measure this earning or losing.

Simply put, if you buy something at a certain price and sell it for more, you make a profit. Conversely, if you sell it for less, you incur a loss. These concepts form the basis of many arithmetic problems, especially in entrance exams for undergraduate programs in India where quick and accurate calculations are crucial.

We will explore these ideas from the ground up, define all terms clearly, learn useful formulas, and practice with real-world examples, all using the Indian Rupee (Rs.) and metric system.

Basic Definitions

Let's define four key terms that form the foundation of profit and loss calculations.

  • Cost Price (CP): The price at which an article or item is bought. For example, if you buy a bicycle for Rs.8,000, the cost price is Rs.8,000.
  • Selling Price (SP): The price at which the article or item is sold. If you sell the bicycle for Rs.9,000, the selling price is Rs.9,000.
  • Profit: When the selling price is more than the cost price, the difference is called profit. It means you earned money.
    Mathematically, profit = SP - CP, if SP > CP.
  • Loss: When the selling price is less than the cost price, the difference is called loss. It means you lost money.
    Mathematically, loss = CP - SP, if SP < CP.

To visualize these relationships, consider the following balance scale illustration:

CP SP If SP > CP -> Profit If SP < CP -> Loss

This scale shows the cost price on one side and selling price on the other. When SP is heavier (greater), it tips in favor of profit. If CP is heavier, it tips toward loss.

Profit and Loss Formulas

Once you understand the basic terms, the next step is to quantify profit or loss precisely and to express them as percentages. Percentage helps us compare earnings or losses relative to how much was spent.

Quantity Formula Description
Profit Amount \( \text{Profit} = \text{SP} - \text{CP} \) Positive difference when SP > CP
Loss Amount \( \text{Loss} = \text{CP} - \text{SP} \) Positive difference when SP < CP
Profit Percentage \( \text{Profit %} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \) Profit expressed as a percentage of cost price
Loss Percentage \( \text{Loss %} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 \) Loss expressed as a percentage of cost price
Selling Price with Profit \( \text{SP} = \text{CP} \times \left( 1 + \frac{\text{Profit %}}{100} \right) \) Find SP if CP and profit % are known
Selling Price with Loss \( \text{SP} = \text{CP} \times \left( 1 - \frac{\text{Loss %}}{100} \right) \) Find SP if CP and loss % are known
Cost Price from SP and Profit % \( \text{CP} = \frac{\text{SP}}{1 + \frac{\text{Profit %}}{100}} \) Find CP when SP and profit % are given
Cost Price from SP and Loss % \( \text{CP} = \frac{\text{SP}}{1 - \frac{\text{Loss %}}{100}} \) Find CP when SP and loss % are given

Why use percentage profit/loss?

Using percentages standardizes profit or loss relative to the investment made (cost price). This makes comparison easier between transactions of different sizes, whether you bought a bicycle or a smartphone.

Worked Examples

Example 1: Basic Profit Calculation Easy
Calculate the profit and profit percentage when the cost price is Rs.800 and the selling price is Rs.1,000.

Step 1: Calculate Profit = SP - CP = Rs.1,000 - Rs.800 = Rs.200.

Step 2: Calculate Profit Percentage = (Profit / CP) x 100 = (200 / 800) x 100 = 25%.

Answer: Profit = Rs.200 and Profit Percentage = 25%.

Example 2: Basic Loss Calculation Easy
Find the loss and loss percentage if an item bought for Rs.1,200 is sold for Rs.1,000.

Step 1: Calculate Loss = CP - SP = Rs.1,200 - Rs.1,000 = Rs.200.

Step 2: Calculate Loss Percentage = (Loss / CP) x 100 = (200 / 1,200) x 100 ≈ 16.67%.

Answer: Loss = Rs.200 and Loss Percentage ≈ 16.67%.

Example 3: Selling Price from Profit Percentage Medium
If the cost price of an article is Rs.500 and the trader makes a profit of 20%, find the selling price.

Step 1: Use formula SP = CP x (1 + Profit % / 100).

Step 2: Substitute values: SP = 500 x (1 + 20/100) = 500 x 1.20 = Rs.600.

Answer: Selling Price = Rs.600.

Example 4: Cost Price from Loss Percentage Medium
A shop sells an article for Rs.2,000 at a loss of 10%. Find the cost price.

Step 1: Use formula CP = SP / (1 - Loss % / 100).

Step 2: Substitute values: CP = 2,000 / (1 - 10/100) = 2,000 / 0.9 ≈ Rs.2,222.22.

Answer: Cost Price ≈ Rs.2,222.22.

Example 5: Profit and Loss with Discount Hard
A shopkeeper marks an article at Rs.1,500 but offers a 10% discount. If the cost price is Rs.1,200, calculate the profit or loss.

Step 1: Calculate the discount amount = 10% of Rs.1,500 = \( 0.10 \times 1,500 = Rs.150 \).

Step 2: Calculate the selling price after discount = Marked Price - Discount = Rs.1,500 - Rs.150 = Rs.1,350.

Step 3: Find Profit or Loss = SP - CP = Rs.1,350 - Rs.1,200 = Rs.150.

Step 4: Since SP > CP, this is a profit of Rs.150.

Step 5: Calculate profit percentage = (150 / 1,200) x 100 = 12.5%.

Answer: The shopkeeper makes a profit of Rs.150, which is 12.5% of the cost price.

Example 6: Mixed Profit and Loss Problem Hard
A trader gains 12% on one article and loses 12% on another, both articles costing the same. Find the overall gain or loss percentage.

Step 1: Assume cost price of each article = Rs.100 for simplicity.

Step 2: Calculate SP of first article = 100 + 12% of 100 = 100 + 12 = Rs.112.

Step 3: Calculate SP of second article = 100 - 12% of 100 = 100 - 12 = Rs.88.

Step 4: Total cost price = Rs.200; total selling price = 112 + 88 = Rs.200.

Step 5: Since total SP = total CP, there is neither gain nor loss overall.

Answer: The trader breaks even, with 0% overall gain or loss.

Formula Bank

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

Tips & Tricks

Tip: Always use 100 as the base in percentage calculations.

When to use: To convert profit or loss amounts to percentages quickly by comparing to cost price.

Tip: Remember that profit or loss = 0 means selling price equals cost price exactly.

When to use: To quickly check if a transaction breaks even.

Tip: For successive percentage changes (like discount then profit), multiply the adjustment factors.

When to use: When dealing with multiple percentage increases or decreases, e.g., price after discount then profit.

Tip: Use the formula SP = CP x (1 ± profit/loss % / 100) to find selling price without calculating profit or loss amount separately.

When to use: To save time during exams when profit or loss percentages are given.

Tip: To find cost price given selling price and profit %, divide SP by (1 + profit %/100).

When to use: Useful when cost price is unknown but SP and profit % are known.

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: By definition, profit and loss percentages are relative to cost price; using selling price gives incorrect results.
❌ Assuming profit without checking whether selling price is actually greater than cost price.
✓ First determine if SP > CP (profit) or SP < CP (loss) before calculating.
Why: Confusing profit and loss causes sign errors and incorrect calculations.
❌ Using percentage values directly in formulas without converting to decimals.
✓ Always divide the percentage value by 100 before multiplying or dividing.
Why: Using whole percentages causes values to be inflated or reduced incorrectly.
❌ Ignoring the discount on marked price before calculating profit or loss.
✓ Subtract discount from marked price to get the actual selling price before calculations.
Why: Discount affects effective selling price; neglecting it results in wrong profit or loss values.
❌ Mixing different currency units or measurement units in calculations.
✓ Ensure all values are in the same currency (Rs.) and consistent units.
Why: Different units lead to logically inconsistent or incorrect numeric answers.
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