Profit and Loss: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction to Profit and Loss

Profit and Loss is an everyday mathematical concept that plays a vital role in commerce and business. For competitive exams like UPSC and MPSC (CSAT), Profit and Loss is a high-yield topic. Questions from this area often test your logical thinking, your grasp of percentages, and your ability to process commercial scenarios such as discounts, markup prices, and faulty weights.

At its core, this topic deals with the buying and selling of goods. A firm understanding of terminologies like Cost Price (CP), Selling Price (SP), Marked Price (MP), and Discount is essential to solve these problems effortlessly. A strong foundation in Percentages (as covered in our previous module) will act as a superpower when tackling Profit and Loss questions.

Core Concepts, Terminologies, and Formulas

1. Important Terminologies

• Cost Price (CP): The price at which an article is purchased.
• Selling Price (SP): The price at which an article is sold.
• Marked Price (MP) / Label Price: The price printed on the article or the tag.
• Discount: The reduction offered by a merchant on the Marked Price.
• Profit / Gain: Occurs when the Selling Price is greater than the Cost Price (SP > CP).
• Loss: Occurs when the Cost Price is greater than the Selling Price (CP > SP).

2. Basic Formulas

• Profit = SP – CP
• Loss = CP – SP
• Profit Percentage = (Profit / CP) × 100%
• Loss Percentage = (Loss / CP) × 100%

Crucial Note: Profit and Loss percentages are always calculated on the Cost Price (CP) unless the question explicitly asks you to calculate it on the Selling Price.

3. Formulas with Discount and Marked Price

• Discount = MP – SP
• Discount Percentage = (Discount / MP) × 100%

Crucial Note: Discount is always given on the Marked Price (MP).

4. Quick Formula connecting MP, CP, Profit%, and Discount%

MP / CP = (100 + Profit%) / (100 – Discount%)

This single formula is a time-saver for questions where three variables are given and the fourth is asked.

Solved Examples with Step-by-Step Explanations

Example 1: Basic Profit Calculation

Question: A man buys a bicycle for ₹1,200 and sells it for ₹1,500. Find his profit percentage.

Solution:

Step 1: Identify given values. CP = ₹1,200; SP = ₹1,500.

Step 2: Since SP > CP, there is a profit. Profit = SP – CP = 1500 – 1200 = ₹300.

Step 3: Calculate Profit Percentage = (Profit / CP) × 100.

Step 4: Profit % = (300 / 1200) × 100 = (1/4) × 100 = 25%.

Answer: The man makes a profit of 25%.

Example 2: Finding SP using Profit Percentage

Question: A shopkeeper bought a chair for ₹800. At what price should he sell it to gain 15%?

Solution:

Step 1: CP = ₹800, Desired Gain% = 15%.

Step 2: SP = CP × [(100 + Gain%) / 100]

Step 3: SP = 800 × (115 / 100) = 8 × 115 = ₹920.

Alternative approach: 15% of 800 = (15/100) × 800 = 120. So, SP = CP + Gain = 800 + 120 = ₹920.

Answer: He should sell the chair for ₹920.

Example 3: Concept of Marked Price and Discount

Question: The marked price of an article is ₹2,000. A shopkeeper offers a discount of 10% and still makes a profit of 20%. What is the cost price of the article?

Solution:

Step 1: Use the master formula: MP / CP = (100 + Profit%) / (100 – Discount%).

Step 2: Substitute the values: 2000 / CP = (100 + 20) / (100 – 10).

Step 3: 2000 / CP = 120 / 90 = 4 / 3.

Step 4: CP = (2000 × 3) / 4 = 1500.

Answer: The cost price of the article is ₹1,500.

Example 4: Successive Discounts

Question: Find the single equivalent discount for two successive discounts of 20% and 10%.

Solution:

Step 1: Use the successive change formula: x + y + (xy/100). Since these are discounts, x = -20 and y = -10.

Step 2: Net change = -20 – 10 + ((-20 × -10) / 100).

Step 3: Net change = -30 + (200 / 100) = -30 + 2 = -28%.

Answer: The single equivalent discount is 28%.

Pro-tips to Avoid Common Mistakes

• Never Calculate Profit/Loss on SP: Unless specified otherwise, profit and loss are strictly calculated on the Cost Price.
• Discount is on MP: Do not calculate the discount on the Cost Price or Selling Price. It is always calculated on the Marked Price (Label Price).
• False Weight Tricks: In problems where a merchant uses a faulty weight (e.g., uses 900g instead of 1kg), the cost price is the value of the actual goods given (900g), and the selling price is the value charged (1kg). Profit % = (Error / True Weight – Error) × 100.
• Identical Profit/Loss Percentage: If two items are sold at the same Selling Price, one at an x% profit and the other at an x% loss, the overall transaction always results in a loss. The loss percentage is (x² / 100)%.

Practice Questions

1. A shopkeeper sells an article for ₹540, incurring a loss of 10%. Find the cost price of the article.
2. By selling 33 meters of cloth, a merchant gains the selling price of 11 meters. Find his gain percentage.
3. A dealer marks his goods 25% above the cost price and allows a discount of 10%. Find his profit percentage.
4. Two bicycles were sold for ₹3990 each. The first was sold at a 5% gain and the second at a 5% loss. What is the overall gain or loss percentage?
5. A dishonest dealer professes to sell his goods at cost price, but uses a false weight of 950 grams for a kilogram. Find his gain percentage.