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In daily life, when we buy and sell goods, the difference between the price at which we buy and the price at which we sell results in either a profit or a loss. Understanding how to calculate these amounts is essential both in real-world commerce and in competitive exams.
For example, if a shopkeeper purchases sugar at 40 INR per kilogram and sells it at 45 INR per kilogram, he makes a profit. On the other hand, if the selling price is less than the cost price, there is a loss.
These ideas are the foundation of the topic of Profit and Loss, and mastering them helps you solve many related numerical problems with confidence.
Before diving into calculations, let's clarify some fundamental terms:
These relationships can be visualized as a simple bar chart:
Figure: Cost Price is 70 units high (e.g., 700 INR), Selling Price is 110 units (e.g., 1100 INR) illustrating profit as the extra area (orange).
Note: If the selling price bar is shorter than the cost price bar, the difference represents loss.
To compare transactions easily or understand the magnitude of profit or loss relative to the original investment, we use percentages.
Profit Percentage measures profit relative to cost price:
Similarly, Loss Percentage measures loss relative to cost price:
Why cost price and not selling price? Because the amount invested (CP) is the baseline amount, and profit or loss is calculated on that investment.
| Aspect | Formula | Example Values | Result |
|---|---|---|---|
| Profit | \( \text{Profit} = SP - CP \) | CP = 500 INR, SP = 600 INR | Profit = 600 - 500 = 100 INR |
| Profit% | \( \text{Profit\%} = \frac{100}{500} \times 100 \) | Profit = 100 INR, CP = 500 INR | Profit% = 20% |
| Loss | \( \text{Loss} = CP - SP \) | CP = 800 INR, SP = 720 INR | Loss = 80 INR |
| Loss% | \( \text{Loss\%} = \frac{80}{800} \times 100 \) | Loss = 80 INR, CP = 800 INR | Loss% = 10% |
In real commerce, goods are often priced with a marked price (also called list price or maximum retail price - MRP). Sellers may offer discounts to buyers to attract sales, reducing the selling price and affecting profit and loss.
Discount is a reduction from the marked price. If the marked price is \(P\) and the discount is \(d\%\), the selling price after discount is:
Markup is an increase over the cost price indicating how much above the cost price the item is listed. If the markup percentage is \(m\%\), marked price is:
For example, if a shopkeeper buys an item at 1500 INR (CP) and marks it up by 20%, then:
\( P = 1500 \times (1 + 0.20) = 1800 \) INR.
Later, he may offer a discount on this marked price, changing the actual selling price.
Successive Discounts are given one after another, not added. For two discounts \(d_1\%\) and \(d_2\%\), the net price is:
graph TD P[Original Price] -->|Apply first discount d1%| P1[Price after d1%] P1 -->|Apply second discount d2%| P2[Final Price after d1% and d2%] P2 --> Output[Net Selling Price] style P fill:#2196F3,color:#fff style P1 fill:#4CAF50,color:#fff style P2 fill:#FF9800,color:#fff
The formula for two successive discounts is:
This is important because simply adding discounts overestimates the total discount.
Used when selling price is greater than cost price.
Used when selling price is less than cost price.
Calculates profit as a percentage of cost price.
Calculates loss as a percentage of cost price.
Finds selling price when profit percentage is known.
Finds selling price when loss percentage is known.
Calculates final price after two successive discounts.
Step 1: Identify Cost Price (CP) and Selling Price (SP).
CP = 500 INR; SP = 600 INR
Step 2: Calculate Profit.
Profit = SP - CP = 600 - 500 = 100 INR
Step 3: Calculate Profit Percentage.
\[ \text{Profit\%} = \left( \frac{100}{500} \right) \times 100 = 20 \% \]
Answer: Profit is 100 INR and profit percentage is 20%.
Step 1: Identify CP and SP.
CP = 800 INR; SP = 720 INR
Step 2: Calculate Loss.
Loss = CP - SP = 800 - 720 = 80 INR
Step 3: Calculate Loss Percentage.
\[ \text{Loss\%} = \left( \frac{80}{800} \right) \times 100 = 10 \% \]
Answer: Loss is 80 INR and loss percentage is 10%.
Step 1: Identify the original price and discounts.
Price = 1000 INR; Discounts = 10% and 5%
Step 2: Calculate price after first discount.
\[ P_1 = 1000 \times \left(1 - \frac{10}{100}\right) = 1000 \times 0.90 = 900 \text{ INR} \]
Step 3: Calculate price after second discount.
\[ P_2 = 900 \times \left(1 - \frac{5}{100}\right) = 900 \times 0.95 = 855 \text{ INR} \]
Answer: The final selling price after successive discounts is 855 INR.
Step 1: Calculate the marked price (MP) with markup.
\[ MP = CP \times \left(1 + \frac{20}{100}\right) = 1500 \times 1.20 = 1800 \text{ INR} \]
Step 2: Calculate selling price after 10% discount.
\[ SP = MP \times \left(1 - \frac{10}{100}\right) = 1800 \times 0.90 = 1620 \text{ INR} \]
Step 3: Calculate the loss.
Loss = CP - SP = 1500 - 1620 = -120 \text{ INR}
Since loss is negative, re-check signs: Here, SP is 1620 INR, CP is 1500 INR, so SP > CP means profit, not loss. The problem states selling at loss because markup is 20% but discount reduces SP below CP?
Check calculation again:
MP = 1500 + 20% of 1500 = 1500 + 300 = 1800 INR
Discount of 10% on MP = 1800 - 180 = 1620 INR selling price
CP = 1500 INR
SP (1620 INR) is greater than CP (1500 INR), so this should be profit, not loss, contradicting problem statement.
Assume problem wording means "still sells at loss" due to some additional factor. Let's double-check.
Suppose shopkeeper markup by 20% but discount is 30%, to produce loss:
Calculate SP for 30% discount:
SP = 1800 x (1 - 0.30) = 1260 INR which is less than CP
Loss = 1500 - 1260 = 240 INR
Loss % = \( \frac{240}{1500} \times 100 = 16\% \)
Let's solve assuming discount is 10%, as given:
Loss = CP - SP = 1500 - 1620 = -120 (profit of 120 INR)
Answer: There is a profit of 120 INR, profit percentage is:
\[ \text{Profit \%} = \left(\frac{120}{1500}\right) \times 100 = 8\% \]
Note to students: Always double-check information and calculations in problem statements and data.
Step 1: Calculate total cost price (CP).
CP = 10 kg x 40 INR/kg = 400 INR
Step 2: Calculate total selling price (SP).
SP = 10 kg x 45 INR/kg = 450 INR
Step 3: Calculate profit.
Profit = SP - CP = 450 - 400 = 50 INR
Answer: The trader makes a total profit of 50 INR on the 10 kg of rice.
When to use: While calculating profit or loss percentages to avoid mistakes.
When to use: To save time when calculating SP from CP and profit/loss percentage.
When to use: When solving problems with multiple discounts to avoid overestimating the discount.
When to use: In real-world and exam word problems involving weights and prices.
When to use: When under time pressure to quickly verify answers.
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