Percentage, Profit and Loss — Study Notes

Overview

Percentage, profit and loss is one of the highest-scoring and most frequently tested topics in the SOF IMO Everyday Mathematics section. These problems appear across Classes 8–10 and form the backbone of commercial arithmetic questions. Students must master converting fractions and decimals to percentages, computing profit/loss from cost price (CP) and selling price (SP), understanding marked price (MP) and discount, and solving multi-step problems involving successive percentages or transaction chains.

This topic directly tests your ability to apply percentages to real-life scenarios: shopkeeper transactions, discount sales, commission-based earnings and instalment payments. A strong grasp of this topic not only secures marks in Everyday Mathematics but also builds the foundation for Compound Interest and Ratio-Proportion problems. The key is speed and accuracy—most IMO questions on this topic can be solved in under 90 seconds if you know the right formula and method.

Expect 3–5 questions from this topic in the IMO paper, often combined with word-problem reasoning or multi-step calculations. Practice converting word problems into mathematical expressions and always double-check whether the question asks for profit, loss, percentage or actual amount.

Key Concepts

• **Percentage** means "per hundred" and is denoted by %. To convert a fraction to percentage, multiply by 100. To convert percentage to fraction, divide by 100.

• **Cost Price (CP)** is the price at which an article is purchased. **Selling Price (SP)** is the price at which it is sold. Profit or loss is always calculated on CP unless stated otherwise.

• **Profit** occurs when SP > CP. Profit = SP − CP. **Loss** occurs when SP < CP. Loss = CP − SP.

• **Marked Price (MP)** is the printed or list price on an article. **Discount** is the reduction offered on MP. Selling Price after discount: SP = MP − Discount.

• **Successive percentages** are not additive. If two discounts or changes are applied one after another, you must apply them sequentially, not by adding percentages. For example, 10% discount followed by 20% discount is **not** 30% discount.

• **Overhead expenses** or additional costs increase the effective CP. Always add these to the purchase price before calculating profit or loss.

• In problems involving profit/loss percentage, always identify the base: profit % and loss % are calculated on CP, while discount % is calculated on MP.

• When an article is sold at a profit of x%, the SP is (100 + x)% of CP. When sold at a loss of x%, SP is (100 − x)% of CP.

Formulas / Key Facts

**Basic Percentage Conversions:**

• Fraction to percentage: (a/b) × 100 %
• Percentage to fraction: (x/100)
• Decimal to percentage: multiply by 100 (e.g., 0.25 = 25%)

**Profit and Loss:**

• Profit = SP − CP; Loss = CP − SP
• Profit % = (Profit / CP) × 100
• Loss % = (Loss / CP) × 100
• SP = CP × (100 + Profit %) / 100
• SP = CP × (100 − Loss %) / 100
• CP = (SP × 100) / (100 + Profit %) or (SP × 100) / (100 − Loss %)

**Discount:**

• Discount = MP − SP
• Discount % = (Discount / MP) × 100
• SP = MP × (100 − Discount %) / 100

**Successive Percentages:**

• Net percentage change after a% increase and b% decrease: a − b − (ab/100)
• If two changes a% and b% are applied successively: Final Value = Initial Value × (1 ± a/100) × (1 ± b/100)

**Important Facts:**

• If CP of x articles = SP of y articles, then Profit % = [(x − y)/y] × 100
• If an article is sold at x% profit and then at y% loss, net effect depends on successive percentage formula
• A shopkeeper uses faulty weights: effective profit depends on error percentage in weight measurement

Worked Examples

**Example 1: Basic Profit Percentage** A shopkeeper buys a pen for ₹40 and sells it for ₹50. Find the profit percentage.

*Solution:* CP = ₹40, SP = ₹50 Profit = SP − CP = 50 − 40 = ₹10 Profit % = (Profit / CP) × 100 = (10 / 40) × 100 = 25%

**Example 2: Discount and Marked Price** A shirt is marked at ₹800. A shopkeeper offers a 15% discount and still makes a profit of 20%. Find the CP of the shirt.

*Solution:* MP = ₹800, Discount = 15% SP = MP × (100 − 15)/100 = 800 × 85/100 = ₹680 Profit = 20%, so SP = CP × (100 + 20)/100 680 = CP × 120/100 CP = (680 × 100)/120 = ₹566.67

**Example 3: Successive Discounts** A customer gets two successive discounts of 10% and 20% on a jacket marked ₹1000. What is the final selling price?

*Solution:* First discount = 10% on ₹1000 = ₹100 Price after first discount = 1000 − 100 = ₹900 Second discount = 20% on ₹900 = ₹180 Final SP = 900 − 180 = ₹720

*Alternate shortcut:* Net percentage = 100 × (90/100) × (80/100) = 72% of MP SP = 1000 × 72/100 = ₹720

**Example 4: Loss and Recovery** A man sells an article at a loss of 10%. To make a profit of 10%, he should increase the selling price by how much percent?

*Solution:* Let CP = 100 At 10% loss, SP₁ = 90 At 10% profit, SP₂ = 110 Required increase = 110 − 90 = 20 Percentage increase on SP₁ = (20/90) × 100 = 22.22% (approx)

Common Mistakes

**Mistake 1:** Adding successive percentages directly. **Wrong:** Two discounts of 10% and 20% = 30% total discount. **Correct:** Apply sequentially. Net discount = 100 − (90% of 80%) = 100 − 72 = 28%. Not 30%.

**Mistake 2:** Calculating profit/loss on SP instead of CP. **Wrong:** If SP = 120 and profit = 20, then profit % = (20/120) × 100. **Correct:** Profit % is always (Profit/CP) × 100. Find CP first: CP = 120 − 20 = 100, so profit % = 20%.

**Mistake 3:** Confusing marked price with selling price. **Wrong:** Discount is given on SP. **Correct:** Discount is always on MP. SP = MP − Discount.

**Mistake 4:** Ignoring overhead expenses in CP. **Wrong:** CP = purchase price only. **Correct:** If transport or repair costs are mentioned, add them to purchase price to get effective CP.

**Mistake 5:** Using wrong base for percentage increase/decrease. **Wrong:** Increasing 100 by 50% then decreasing by 50% returns to 100. **Correct:** 100 → +50% → 150 → −50% → 75. Final value is 75, not 100. Different bases!

Quick Reference

• Profit = SP − CP; Loss = CP − SP
• Profit % = (Profit/CP) × 100; Loss % = (Loss/CP) × 100
• SP = CP × (100 ± Profit or Loss %)/100
• Discount % is on MP, not SP: SP = MP × (100 − Discount %)/100
• Successive discounts: multiply the remaining percentages sequentially
• When two articles are sold at same SP, one at x% profit and other at x% loss, there is always a net loss = (x²/100)% of CP