Profit, Loss and Discount

Overview

Profit, Loss and Discount form the backbone of commercial mathematics and appear consistently in HP TET Mathematics sections. These concepts test your ability to apply percentage calculations to real-world trading scenarios—buying, selling, calculating gains or losses, and understanding how discounts affect final prices.

For HP TET, you must be comfortable with quick mental calculations involving cost price, selling price, marked price, and successive discounts. Questions typically involve straightforward formula application but may include twists like dishonest dealers, successive discounts, or finding the cost price when profit/loss percentages are given. Mastering this topic also strengthens your foundation for teaching commercial mathematics to elementary students using relatable market examples.

The key to success is understanding the relationships between different prices and always being clear about which price forms the base for percentage calculations—a common source of errors.

Key Concepts

• **Cost Price (CP)**: The price at which an article is purchased or manufactured. This is always the base for calculating profit or loss percentage.

• **Selling Price (SP)**: The price at which an article is sold to a customer. Comparing SP with CP determines profit or loss.

• **Marked Price (MP)**: The price displayed on an article (also called list price or catalogue price). Discount is always calculated on MP, not on CP.

• **Profit occurs when SP > CP**; Loss occurs when SP < CP. The difference gives the absolute profit or loss amount.

• **Discount is a reduction from Marked Price** to arrive at Selling Price. A seller may still make profit even after giving discount if MP is sufficiently above CP.

• **Successive Discounts**: When multiple discounts are applied one after another, each discount is calculated on the reduced price from the previous discount, not on the original MP.

• **Break-even Point**: When SP = CP, there is neither profit nor loss.

Formulas / Key Facts

**Basic Profit and Loss:**

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

**Finding SP from CP:**

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

**Finding CP from SP:**

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

**Discount:**

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

**Successive Discounts:**

• For discounts of a% and b%, equivalent single discount = a + b − (ab/100)
• Or calculate stepwise: First discount gives new price, second discount applies on that.

**Dishonest Dealer:**

• If a dealer uses false weights, Profit % = (Error / True Weight − Error) × 100

**Key Fact:** Profit/Loss percentage is ALWAYS calculated on Cost Price. Discount percentage is ALWAYS calculated on Marked Price.

Worked Examples

**Example 1: Basic Profit Calculation** A shopkeeper buys a chair for ₹800 and sells it for ₹920. Find the profit percentage.

*Solution:*

• CP = ₹800, SP = ₹920
• Profit = 920 − 800 = ₹120
• Profit % = (120 / 800) × 100 = 15%

**Example 2: Finding CP when Loss% is Given** An article was sold for ₹1,700 at a loss of 15%. Find the cost price.

*Solution:*

• Loss of 15% means SP = 85% of CP
• CP = SP × 100 / (100 − 15)
• CP = 1700 × 100 / 85 = ₹2,000

**Example 3: Marked Price and Discount** A shirt has a marked price of ₹600. The shopkeeper gives a 20% discount and still makes 20% profit. Find the cost price.

*Solution:*

• SP = MP × (100 − 20) / 100 = 600 × 80 / 100 = ₹480
• Profit is 20%, so SP = 120% of CP
• CP = 480 × 100 / 120 = ₹400

**Example 4: Successive Discounts** Find the single equivalent discount for successive discounts of 20% and 10%.

*Solution:*

• Equivalent discount = 20 + 10 − (20 × 10 / 100)
• = 30 − 2 = 28%

*Verification:* On ₹100, after 20% discount: ₹80. After 10% on ₹80: ₹72. Total discount = ₹28 = 28%.

Common Mistakes

• **Calculating profit% on SP instead of CP** → Always remember: Profit and Loss percentages are calculated on Cost Price, the investment amount.

• **Calculating discount% on CP instead of MP** → Discount is a reduction from the displayed price (MP). SP after discount may still be above or below CP.

• **Adding successive discounts directly** → Two discounts of 10% and 20% do NOT equal 30%. Apply the formula or calculate stepwise. The combined discount is always less than the sum.

• **Confusing "profit on selling" with "profit on cost"** → Some tricky questions say "profit is 25% of SP." This is different from "25% profit." Convert to CP-based percentage before solving.

• **Ignoring the base in ratio problems** → When told "SP of 10 articles = CP of 12 articles," set up the ratio carefully. Here, SP/CP = 12/10, meaning 20% profit.

Quick Reference

• Profit% = (SP − CP) / CP × 100 — always on CP
• Discount% = (MP − SP) / MP × 100 — always on MP
• Successive discounts a% and b% = (a + b − ab/100)%
• No profit no loss means SP = CP
• If SP = CP × 1.25, profit is 25%
• If SP = CP × 0.85, loss is 15%