**CAT Arithmetic Mastery: Percentages, Profit & Loss, Discount**

CAT Arithmetic Mastery: Percentages, Profit & Loss, Discount

(A Premium Study Guide for 99+ Percentile Aspirants)

What This Is

Percentages, Profit & Loss (P&L), and Discounts form the core of CAT Arithmetic, appearing in 10–15% of QA questions (3–5 questions per slot). These topics test speed, accuracy, and logical reasoning—not just formula application. Mastering them gives you easy marks (since they’re formula-light) and time savings (since they’re often solvable in <2 minutes).

Real-CAT Example:
A shopkeeper marks up an article by 40% and then offers a 10% discount. If the selling price is ₹504, what is the cost price? (Answer: ₹400, but the trap is assuming markup is on selling price—it’s always on cost price.)

Key Concepts & Techniques

1. Percentage Change Formula
2. Formula: (New Value - Original Value) / Original Value × 100
3. When to use: Any question involving increase/decrease (e.g., salary hikes, population growth).

4. Pro tip: For successive changes, use multiplicative factors (e.g., +20% → ×1.2, -10% → ×0.9).

5. Cost Price (CP), Selling Price (SP), Profit/Loss

6. Profit % = (SP - CP)/CP × 100
7. Loss % = (CP - SP)/CP × 100

8. When to use: All P&L questions. Always identify CP first—it’s the base for calculations.

9. Marked Price (MP) and Discount

10. MP = CP × (1 + Markup%)
11. SP = MP × (1 - Discount%)
12. When to use: Questions with marked price + discount (e.g., "A shopkeeper offers 20% off on a shirt marked at ₹1200").

13. Trap: Discount is always on MP, not CP.

14. Successive Percentage Changes

15. Formula: Net Change = a + b + (a×b)/100 (for two changes).
16. When to use: Questions like "Price increased by 10%, then decreased by 20%. Net change?"

17. Shortcut: Use 100 → 110 → 88 (100 × 1.1 × 0.8 = 88 → 12% decrease).

18. Profit/Loss on Sales (Multiple Items)

19. Total CP = CP₁ + CP₂ + ...
20. Total SP = SP₁ + SP₂ + ...
21. When to use: Questions like "A sells to B at 20% profit, B sells to C at 25% profit. Total profit?"

22. Key: Profit is always on CP, not cumulative SP.

23. False Weights & Dishonest Dealers

24. If a dealer uses 900g for 1kg: He gains (1000 - 900)/900 × 100 = 11.11%.
25. When to use: Questions like "A shopkeeper cheats by 10% while buying and selling. Net profit?"

26. Formula: Net Profit % = (Cheat% + Cheat%) + (Cheat% × Cheat%)/100.

27. Option Elimination (For MCQs)

28. When to use: If stuck, plug in answer choices to reverse-engineer.
29. Example: If CP is asked, test options to see which fits the given SP/profit%.

Step-by-Step Strategy

Step 1: Identify the Base (CP/MP)

• For P&L: Always start with CP (the original cost).
• For Discount: Start with MP (the marked price before discount).

Step 2: Translate % into Multipliers

• +20% → ×1.2
• -15% → ×0.85
• Use this for successive changes (e.g., 10% increase + 20% decrease → ×1.1 × 0.8 = 0.88 → 12% decrease).

Step 3: Write the Equation

• P&L: SP = CP × (1 ± Profit%/100)
• Discount: SP = MP × (1 - Discount%/100)
• Markup: MP = CP × (1 + Markup%/100)

Step 4: Solve for the Unknown

• If CP is unknown, express everything in terms of CP.
• If SP is given, work backwards.

Step 5: Check for Traps

• Is the discount on MP or CP?
• Is the profit on CP or SP?
• Are there successive changes?

Step 6: Verify with Options (MCQs)

• Plug in answer choices to eliminate wrong options quickly.

Fully Worked CAT-Style Example

Question:
A shopkeeper marks up an article by 50% and then offers a 20% discount. If he still makes a 20% profit, what is the ratio of the cost price to the marked price?

Solution (Using Strategy):

1. Identify Base:
2. Let CP = x.

3. MP = x × 1.5 (50% markup).

4. Apply Discount:

5. SP = MP × (1 - 0.2) = 1.5x × 0.8 = 1.2x.

6. Given Profit:

7. Profit = 20% of CP → SP = 1.2x.

8. But from Step 2, SP = 1.2x → Consistent!

9. Find Ratio CP:MP:

10. CP = x, MP = 1.5x → CP:MP = 1:1.5 = 2:3.

Answer: 2:3

Common Mistakes

1. Mistake: Assuming discount is on CP.
2. Why it happens: Confusing markup (on CP) with discount (on MP).

3. Correct approach: Discount is always on MP, not CP.

4. Mistake: Adding percentages directly.

5. Why it happens: Thinking 10% + 20% = 30% (wrong for successive changes).

6. Correct approach: Use multiplicative factors (1.1 × 1.2 = 1.32 → 32% increase).

7. Mistake: Ignoring false weights.

8. Why it happens: Not accounting for cheating in buying/selling.

9. Correct approach: Use (Cheat% + Cheat%) + (Cheat% × Cheat%)/100.

10. Mistake: Misidentifying the base (CP vs MP).

11. Why it happens: Not reading the question carefully.
12. Correct approach: Always underline whether the % is on CP or MP.

CAT Traps & Time Management

Traps:

1. "Profit on SP" vs "Profit on CP"
2. Trap: If profit is given as % of SP, convert it to % of CP first.

3. Example: "Profit is 25% of SP" → Let SP = 100, Profit = 25 → CP = 75 → Profit % on CP = (25/75) × 100 = 33.33%.

4. Successive Discounts

5. Trap: Two discounts of 10% and 20% ≠ 30%.

6. Correct: 100 → 90 → 72 → 28% total discount.

7. Markup + Discount = Profit?

8. Trap: Assuming markup - discount = profit (wrong if markup is not on CP).
9. Correct: Always start with CP.

Time Management:

• Easy questions: <1 minute (direct formula application).
• Moderate questions: 1–2 minutes (successive changes, false weights).
• Hard questions: 2–3 minutes (multiple steps, traps).
• If stuck: Skip and return—don’t waste time on one question.

Quick Practice

1. Question:
   A man sells two articles for ₹4,000 each. On one he gains 25%, and on the other he loses 25%. What is his overall profit/loss %?
   Answer: 6.25% loss
   Explanation: Let CP of first = x → SP = 1.25x = 4000 → x = 3200. CP of second = y → SP = 0.75y = 4000 → y = 5333.33. Total CP = 3200 + 5333.33 = 8533.33. Total SP = 8000 → Loss = 533.33 → 6.25% loss.

2. Question:
   A shopkeeper offers a 10% discount on an item marked at ₹800. If he still makes a 20% profit, what is the cost price?
   Answer: ₹600
   Explanation: SP = 800 × 0.9 = 720. Profit = 20% → CP = 720 / 1.2 = 600.

Last-Minute Cram Sheet

1. Profit % = (SP - CP)/CP × 100 (Always on CP).
2. Discount is on MP, not CP.
3. Successive % changes: Multiply factors (e.g., +10% → ×1.1, -20% → ×0.8).
4. False weights: Net profit = (Cheat% + Cheat%) + (Cheat% × Cheat%)/100.
5. If profit is % of SP: Convert to % of CP first.
6. Two discounts of a% and b%: Net discount = a + b - (a×b)/100.
7. Markup is on CP, discount is on MP.
8. If SP is same for two items with profit/loss %: Overall loss = (x²)/100 (where x = % profit/loss).
9. Always start with CP—it’s the base for all calculations.
10. ⚠️ Trap: "Profit is 20% of SP" ≠ "Profit is 20% of CP". Convert first!

Final Tip: Practice 50+ questions from past CAT papers (2017–2023) to internalize traps and shortcuts. Speed comes from pattern recognition, not memorization. Good luck! ?