Profit and Loss — Study Notes

Overview

Profit and Loss is a core arithmetic topic in SSC MTS Paper 1, typically contributing 2–4 questions in the Numerical and Mathematical Ability section. This topic tests your ability to calculate buying price, selling price, discounts, and profit or loss percentages in commercial transactions. Questions range from straightforward single-step calculations to multi-step problems involving marked price, successive discounts, and cost price recovery.

Mastery of this topic requires fluency with percentage calculations and the ability to translate word problems into mathematical relationships. Most errors arise from confusing the base (cost price vs. selling price) when calculating percentages or mishandling discount sequences. The good news: once you internalize the core formulas and their logical connections, these questions become highly predictable and scorable in under a minute each.

Expect questions on finding profit/loss percent, calculating cost price or selling price when one is given, determining marked price and discount relationships, and solving problems involving false weights or faulty measuring devices. Practice is key—work enough problems to recognize patterns instantly during the exam.

Key Concepts

• **Cost Price (CP)**: The price at which an article is purchased or manufactured. All profit and loss calculations use CP as the reference base unless stated otherwise.

• **Selling Price (SP)**: The price at which an article is sold to the customer. Profit occurs when SP > CP; loss occurs when SP < CP.

• **Profit and Loss Amounts**: Profit = SP − CP (when SP > CP). Loss = CP − SP (when CP > SP). These are absolute amounts in rupees.

• **Profit % and Loss %**: Always calculated on Cost Price unless explicitly stated otherwise. Profit % = (Profit / CP) × 100. Loss % = (Loss / CP) × 100.

• **Marked Price (MP)**: The label price or list price printed on an article before any discount is applied. Retailers mark up goods above CP to allow room for discounts while still making profit.

• **Discount**: The reduction offered on the marked price. Discount = MP − SP. Discount % = (Discount / MP) × 100. Note that discount percentage is always on Marked Price, not Cost Price.

• **Successive Discounts**: When multiple discounts are applied one after another, they compound. Two discounts of d₁% and d₂% are not simply (d₁ + d₂)%. The effective single discount = d₁ + d₂ − (d₁ × d₂)/100.

• **Break-even and No Profit No Loss**: When SP = CP, there is neither profit nor loss. This state is called break-even or no profit no loss situation.

Formulas / Key Facts

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

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

3. **SP = CP × (100 + Profit %) / 100** (when profit) SP = CP × (100 − Loss %) / 100 (when loss)

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

5. **Marked Price and Discount:** SP = MP − Discount = MP × (100 − Discount %) / 100

6. **If MP is x% above CP:** MP = CP × (100 + x) / 100

7. **Successive Discounts of d₁% and d₂%:** Single Equivalent Discount = d₁ + d₂ − (d₁ × d₂) / 100 Final SP = MP × [(100 − d₁)/100] × [(100 − d₂)/100]

8. **If an article is sold at a profit of x% and then at a loss of x%, there is always a net loss of (x²/100)%.**

9. **False Weight/Measure Problem:** If a seller claims to sell at cost price but uses a faulty weight giving less quantity, the profit % = [(Error / True Weight − Error) × 100]

10. **Break-even: When SP = CP, Profit % = 0% and Loss % = 0%.**

Worked Examples

**Example 1: Basic Profit %** 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 % = (10 / 40) × 100 = 25% **Answer: 25%**

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**Example 2: Finding CP from SP and Loss %** An article is sold for ₹450 at a loss of 10%. What was the cost price?

*Solution:* SP = 450, Loss % = 10 CP = (SP × 100) / (100 − Loss %) CP = (450 × 100) / (100 − 10) = 45000 / 90 = 500 **Answer: ₹500**

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**Example 3: Marked Price, Discount and Profit** A trader marks his goods 30% above cost price and allows a discount of 10%. Find his profit percentage.

*Solution:* Let CP = 100 MP = CP + 30% of CP = 100 + 30 = 130 Discount = 10% of MP = 10% of 130 = 13 SP = MP − Discount = 130 − 13 = 117 Profit = SP − CP = 117 − 100 = 17 Profit % = (17 / 100) × 100 = 17% **Answer: 17%**

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**Example 4: Successive Discounts** A customer gets successive discounts of 20% and 10% on a marked price of ₹1000. What is the final selling price?

*Solution:* MP = 1000 First discount 20%: Price after first discount = 1000 × (80/100) = 800 Second discount 10%: Final SP = 800 × (90/100) = 720

*Alternate method using formula:* Single Equivalent Discount = 20 + 10 − (20 × 10)/100 = 30 − 2 = 28% SP = 1000 × (72/100) = 720 **Answer: ₹720**

Common Mistakes

1. **Calculating Profit % on SP instead of CP**: Students often compute (Profit / SP) × 100 instead of (Profit / CP) × 100. Always remember: unless stated otherwise, profit and loss percentages are on Cost Price. *Fix:* Memorize that the denominator in Profit % and Loss % is always CP.

2. **Adding successive discounts directly**: Applying 20% + 10% = 30% discount is wrong. Discounts compound, not add linearly. *Fix:* Use the formula: d₁ + d₂ − (d₁ × d₂)/100 or apply each discount sequentially.

3. **Confusing Discount % base**: Discount % is always calculated on Marked Price, not Cost Price. Mixing these up scrambles the entire calculation. *Fix:* Write down MP, apply discount to get SP, then relate SP and CP separately.

4. **Misapplying the CP formula direction**: When given SP and Profit %, students sometimes multiply instead of divide or use the wrong sign in the denominator (100 + Profit % vs. 100 − Loss %). *Fix:* Sketch a number line: CP → (add profit) → SP or CP → (subtract loss) → SP. Reverse the operation to find CP.

5. **Ignoring the base in percentage change problems**: When the problem says "x% more than CP" or "y% less than MP," students miss identifying which quantity is the base (100%). *Fix:* Underline the base in the problem statement before calculating percentages.

Quick Reference

• Profit % and Loss % are always on **Cost Price** (CP) unless otherwise mentioned.
• SP = CP + Profit = CP − Loss. Rearrange to find any unknown.
• Discount is on **Marked Price** (MP), not CP.
• Successive discounts d₁% and d₂%: Single discount = d₁ + d₂ − (d₁ × d₂)/100.
• If an item is sold at x% profit then x% loss, net effect is a loss of (x²/100)%.
• Break-even: SP = CP → no profit, no loss.