Study Notes: Profit and Loss

Overview

Profit and Loss is a cornerstone topic in Railway Group D mathematics, consistently appearing in 3–5 questions per exam. The concept revolves around buying and selling goods, calculating gains or losses, and understanding pricing strategies involving discounts and marked prices. Mastery of this topic is essential because it forms the foundation for more complex commercial mathematics and directly impacts your ability to solve real-world transaction problems under time pressure.

The exam tests your ability to quickly compute profit or loss percentages, work backward from selling price to cost price, handle successive discounts, and navigate problems involving marked price and discount combinations. Most questions require 2–3 steps of calculation, so fluency with the basic formulas and percentage conversions is critical. Students who can mentally calculate common profit/loss percentages (like 10%, 20%, 25%) gain significant time advantages.

You must be comfortable switching between absolute values (profit in rupees) and percentages (profit %), understanding when a seller makes a profit versus a loss, and applying discount formulas in the correct sequence. This topic integrates closely with Percentage, so ensure you're comfortable with percentage-to-fraction conversions before diving deep here.

Key Concepts

• **Cost Price (CP)** is the price at which an article is purchased or the total expense incurred to acquire/produce it. All profit or loss calculations reference this base value.

• **Selling Price (SP)** is the price at which an article is sold to the buyer. The relationship between SP and CP determines whether there's profit or loss.

• **Profit** occurs when SP > CP; the profit amount is (SP − CP), and the profit percentage is always calculated on the Cost Price: Profit % = [(SP − CP)/CP] × 100.

• **Loss** occurs when SP < CP; the loss amount is (CP − SP), and loss percentage is Loss % = [(CP − SP)/CP] × 100, again based on Cost Price.

• **Marked Price (MP)** is the label price or list price printed on an article before any discount is applied; it's typically higher than the seller's intended selling price.

• **Discount** is the reduction offered on the Marked Price. Discount amount = MP − SP, and Discount % = [(MP − SP)/MP] × 100, calculated on Marked Price.

• **Successive Discounts** are multiple discounts applied one after another. Two discounts of x% and y% do NOT simply add up; the net effect is less: Net discount = [x + y − (xy/100)]%.

• In problems involving both MP and CP, the seller's profit is calculated from CP to final SP, while discount is calculated from MP to SP. Never confuse these reference points.

Formulas / Key Facts

**Basic Relationships:**

• SP = CP + Profit (when there's profit)
• SP = CP − Loss (when there's loss)
• If SP > CP, there is profit; if SP < CP, there is loss; if SP = CP, no profit no loss

**Percentage Formulas:**

• Profit % = [(SP − CP)/CP] × 100
• Loss % = [(CP − SP)/CP] × 100
• SP = CP × (100 + Profit %)/100 (when profit)
• SP = CP × (100 − Loss %)/100 (when loss)
• CP = [SP × 100]/(100 + Profit %) (when profit)
• CP = [SP × 100]/(100 − Loss %) (when loss)

**Discount Formulas:**

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

**Successive Discount:**

• For two successive discounts of d₁% and d₂%: Single equivalent discount % = d₁ + d₂ − (d₁ × d₂)/100
• After first discount d₁% on MP, new price = MP × (100 − d₁)/100; then apply d₂% on this new price

**Special Cases:**

• If an article is sold at x% profit and then at y% loss, net effect on CP: Final multiplier = (100 + x)/100 × (100 − y)/100
• When two articles are sold at the same SP, one at x% profit and other at x% loss, there is always an overall loss = 2(x/10)² %

Worked Examples

**Example 1: Basic Profit Calculation** A shopkeeper buys a pen for ₹80 and sells it for ₹100. Find the profit percentage.

*Solution:* CP = ₹80, SP = ₹100 Profit = SP − CP = 100 − 80 = ₹20 Profit % = (Profit/CP) × 100 = (20/80) × 100 = 25%

**Example 2: Finding CP from Loss %** An article is sold for ₹850 at a loss of 15%. Find the cost price.

*Solution:* SP = ₹850, Loss % = 15% Using CP = [SP × 100]/(100 − Loss %) CP = (850 × 100)/(100 − 15) = 85000/85 = ₹1000

**Example 3: Marked Price and Discount** An article with marked price ₹500 is sold at 20% discount. If the cost price is ₹350, find the profit percentage.

*Solution:* MP = ₹500, Discount = 20% SP = MP × (100 − Discount %)/100 = 500 × 80/100 = ₹400 CP = ₹350 Profit = SP − CP = 400 − 350 = ₹50 Profit % = (50/350) × 100 = 100/7 ≈ 14.29%

**Example 4: Successive Discounts** A shirt with marked price ₹2000 is offered at two successive discounts of 10% and 20%. Find the final selling price.

*Solution:* Method 1 (Step-by-step): After first discount of 10%: Price = 2000 × 90/100 = ₹1800 After second discount of 20% on ₹1800: Price = 1800 × 80/100 = ₹1440

Method 2 (Formula): Single equivalent discount = 10 + 20 − (10 × 20)/100 = 30 − 2 = 28% SP = 2000 × 72/100 = ₹1440

Common Mistakes

**Mistake 1: Calculating profit/loss % on SP instead of CP** Wrong thinking: "I bought at ₹100, sold at ₹120, so profit % = 20/120 × 100" Correct fix: Always use CP as the base. Profit % = (20/100) × 100 = 20%. The denominator must be Cost Price.

**Mistake 2: Adding successive discounts directly** Wrong thinking: "20% discount + 10% discount = 30% total discount" Correct fix: Use the formula or apply discounts sequentially. Two discounts of 20% and 10% give 28% net discount, not 30%, because the second discount applies to the reduced price.

**Mistake 3: Confusing MP and SP when calculating discount** Wrong thinking: Calculating discount percentage using CP as the base Correct fix: Discount % is always (MP − SP)/MP × 100. The base is Marked Price, not Cost Price. Profit is based on CP; discount is based on MP.

**Mistake 4: Wrong formula selection when working backwards** Wrong thinking: Using CP = SP × (100 + Profit %)/100 when profit is given Correct fix: When profit is given and you need CP, use CP = [SP × 100]/(100 + Profit %). The denominator shifts when solving for CP.

**Mistake 5: Ignoring the sign in loss calculations** Wrong thinking: Treating loss % the same as profit % in formulas Correct fix: In loss scenarios, use (100 − Loss %) not (100 + Loss %). When SP < CP, subtract the loss percentage from 100 in your calculations.

Quick Reference

• **Profit = SP − CP; Loss = CP − SP** — Always reference Cost Price for gain/loss
• **Profit % and Loss % always calculated on CP** — Never use SP or MP as base for profit/loss %
• **Discount % calculated on MP** — Base is Marked Price, not CP or SP
• **Successive discounts formula: d₁ + d₂ − (d₁ × d₂)/100** — Never add discounts directly
• **When two items sold at same SP with equal % profit and loss, net result is always loss** — Loss = 2(x/10)² %
• **Quick multipliers:** 10% profit means SP = 1.1 × CP; 25% loss means SP = 0.75 × CP