Profit and Loss — Study Notes

Overview

Profit and Loss is a high-weightage topic in SSC GD Elementary Mathematics. Nearly every year, you will face 2–4 direct questions on cost price (CP), selling price (SP), marked price (MP), discount, and profit/loss percentage. These problems test your ability to quickly manipulate percentages and understand commercial arithmetic.

The core concept is simple: a shopkeeper buys goods at CP and sells at SP. The difference determines profit or loss. Real-world complications like marked price, discount, and successive transactions add layers, but the fundamentals remain unchanged. Mastering the standard formulas and recognizing problem patterns will fetch you easy marks.

Focus on speed and accuracy. Most questions require 2–3 step calculations. Practice mental percentage conversions (like 20% = 1/5, 25% = 1/4) to save precious seconds during the exam.

Key Concepts

• **Cost Price (CP)** is the amount paid to acquire an article. It includes purchase price plus any additional expenses like transport or repair (called overhead expenses).

• **Selling Price (SP)** is the amount received when the article is sold. Profit or loss is always calculated on the CP, not the SP.

• **Profit** occurs when SP > CP. Profit = SP − CP. Profit percentage = (Profit / CP) × 100.

• **Loss** occurs when SP < CP. Loss = CP − SP. Loss percentage = (Loss / CP) × 100.

• **Marked Price (MP)** is the price tag displayed on the article, often higher than CP. The seller announces discounts on this marked price.

• **Discount** is a reduction given on the marked price. Discount = MP − SP. Discount percentage is always calculated on MP: Discount% = (Discount / MP) × 100.

• In real transactions, first apply discount to MP to get SP, then compare SP with CP to find profit or loss. The two steps are independent.

• **Successive transactions**: If an article passes through multiple hands (A → B → B → C), calculate each person's gain/loss separately using their individual CP and SP.

Formulas / Key Facts

1. **Profit = SP − CP**; **Loss = CP − SP**

2. **Profit% = (Profit / CP) × 100 = [(SP − CP) / CP] × 100**

3. **Loss% = (Loss / CP) × 100 = [(CP − SP) / CP] × 100**

4. **SP = CP × (100 + Profit%) / 100** when profit is given.

5. **SP = CP × (100 − Loss%) / 100** when loss is given.

6. **CP = (SP × 100) / (100 + Profit%)** when SP and profit% are given.

7. **CP = (SP × 100) / (100 − Loss%)** when SP and loss% are given.

8. **Discount = MP − SP**

9. **Discount% = (Discount / MP) × 100 = [(MP − SP) / MP] × 100**

10. **SP = MP × (100 − Discount%) / 100**

11. If an article is sold at x% profit and then at y% loss, net effect on CP: First find intermediate SP, treat it as new CP for second transaction.

12. **Break-even**: SP = CP means no profit, no loss.

Worked Examples

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

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

**Example 2:** An article is marked at ₹800. A discount of 15% is given. If the shopkeeper still makes 20% profit, find the cost price.

*Solution:* MP = ₹800, Discount% = 15% SP = MP × (100 − 15) / 100 = 800 × 85/100 = ₹680 Now SP = ₹680 and Profit% = 20% Using CP = (SP × 100) / (100 + Profit%) CP = (680 × 100) / 120 = 68000 / 120 = ₹566.67 (or ₹566.66 repeating, often rounded to ₹566.67 or keep as 1700/3)

In exam, if options are whole numbers, recheck calculation. Here exact CP = 1700/3 ≈ ₹566.67.

**Example 3:** A trader sells an item at 10% loss. If he had sold it for ₹50 more, he would have gained 5%. Find CP.

*Solution:* Let CP = x At 10% loss: SP₁ = x × (100 − 10)/100 = 0.9x At 5% profit: SP₂ = x × (100 + 5)/100 = 1.05x Given: SP₂ − SP₁ = 50 1.05x − 0.9x = 50 0.15x = 50 x = 50 / 0.15 = 50 × (100/15) = 5000/15 = ₹333.33 (or 1000/3)

Exact CP = ₹333.33 (commonly written as ₹333⅓ or accepted as ₹333.33).

Common Mistakes

1. **Calculating profit/loss percentage on SP instead of CP** → Always remember: Profit% and Loss% are fractions of CP, not SP. Formula uses CP in denominator.

2. **Calculating discount percentage on CP or SP instead of MP** → Discount% is always (Discount / MP) × 100. MP is the reference, not CP or SP.

3. **Confusing marked price with cost price** → MP is the label price; CP is what the shopkeeper paid. They are usually different. Don't assume MP = CP.

4. **Mixing up successive profit/loss in chain transactions** → When A sells to B at 10% profit and B sells to C at 20% profit, calculate separately. B's CP = A's SP. Don't directly multiply percentages unless using the combined percentage formula correctly.

5. **Forgetting to convert percentage to decimal or fraction** → Writing 20% profit as 20 instead of 0.20 or 20/100 leads to wrong SP or CP. Always convert: 20% = 20/100 = 0.2.

Quick Reference

• **Profit = SP − CP; Loss = CP − SP** (always on CP base)
• **Profit% = [(SP − CP)/CP] × 100; Loss% = [(CP − SP)/CP] × 100**
• **SP when profit: SP = CP(100 + P%)/100; SP when loss: SP = CP(100 − L%)/100**
• **Discount = MP − SP; Discount% on MP, not CP**
• **If SP and profit% given, CP = (SP × 100)/(100 + profit%)**
• **Chain transactions: treat each sale independently, previous SP becomes next CP**