**Quant-Based LR: Ratios, Rankings, Allocation – The 99%ile Playbook**

Quant-Based LR: Ratios, Rankings, Allocation – The 99%ile Playbook

(DILR Section – CAT 2024 & Beyond)

What This Is

Quant-Based LR (Logical Reasoning) questions involving ratios, rankings, and allocation test your ability to blend numerical reasoning with structured logic. These appear 2–3 times per CAT (often in sets of 3–4 questions) and are high-scoring because they rely on fixed rules rather than abstract thinking. A single set can fetch 12+ marks—mastering this topic can boost your DILR percentile by 10+ points.

Typical CAT-Style Question:
A company allocates bonuses to 5 employees (A, B, C, D, E) based on the following rules: 1. The total bonus pool is ₹100. 2. A gets twice as much as B. 3. C gets ₹10 more than D. 4. E gets the average of A and C. 5. No employee gets less than ₹10. What is the minimum possible bonus for D?

(Answer: ₹10. But how? Read on.)

Key Concepts & Techniques

1. Variable Assignment
2. What: Assign variables to unknowns (e.g., let B = x, then A = 2x).

3. When: Use at the start of every ratio/allocation problem to convert words into equations.

4. Equation Stacking

5. What: Write all constraints as equations/inequalities (e.g., A + B + C + D + E = 100).

6. When: Immediately after variable assignment to systematically track constraints.

7. Extreme Value Testing

8. What: Test minimum/maximum values for variables to satisfy all conditions.

9. When: When the question asks for "minimum/maximum possible" values (e.g., "What is the least D can get?").

10. Ratio Splitting

11. What: If a ratio is given (e.g., A:B = 3:2), split the total into parts (e.g., A = 3k, B = 2k).

12. When: When ratios are directly provided or can be derived (e.g., "A gets twice B" → A:B = 2:1).

13. Ranking Anchors

14. What: Fix one variable’s rank (e.g., "A is 2nd highest") to eliminate possibilities.

15. When: In ranking-based problems (e.g., "Who is 3rd in line?").

16. Back-Substitution

17. What: Plug in answer choices to verify (for MCQs) or derive (for TITA).

18. When: When stuck or to save time (especially in MCQs).

19. Inequality Chaining

20. What: Combine inequalities (e.g., A > B > C and A + B + C = 30) to narrow ranges.
21. When: In ranking + allocation hybrid problems.

Step-by-Step Strategy (The 6-Step Drill)

Step 1: Read the Question Twice
- First pass: Understand the scenario (e.g., "5 employees, ₹100 bonus").
- Second pass: Highlight constraints (e.g., "A = 2B", "E = (A + C)/2").

Step 2: Assign Variables
- Let the smallest/unknown entity be x (e.g., let B = x, then A = 2x).
- If ratios are given, use k-method (e.g., A:B = 3:2 → A = 3k, B = 2k).

Step 3: Write All Equations
- Translate every constraint into an equation/inequality.
- A + B + C + D + E = 100 - A = 2B - C = D + 10 - E = (A + C)/2 - D ≥ 10 (since no one gets less than ₹10)

Step 4: Substitute & Simplify
- Replace variables to reduce complexity.
- From A = 2B and B = x, A = 2x.
- From C = D + 10, let D = y, then C = y + 10.
- From E = (A + C)/2, E = (2x + y + 10)/2 = x + y/2 + 5.
- Now, rewrite the total: 2x + x + (y + 10) + y + (x + y/2 + 5) = 100 → 4x + 2.5y + 15 = 100 → 4x + 2.5y = 85 → 8x + 5y = 170 (multiply by 2 to eliminate decimals).

Step 5: Solve for Extremes
- The question asks for minimum D (y).
- From D ≥ 10, try y = 10: 8x + 5(10) = 170 → 8x = 120 → x = 15.
- Now check all values: B = x = 15, A = 2x = 30, D = y = 10, C = y + 10 = 20, E = (30 + 20)/2 = 25.
- Sum: 30 + 15 + 20 + 10 + 25 = 100 (valid).
- All constraints satisfied → D = 10 is possible.

Step 6: Verify & Answer
- For MCQs: Plug in options to confirm.
- For TITA: Double-check calculations (e.g., ensure no variable < ₹10).

Fully Worked CAT-Style Example

Question:
A library has 4 sections (A, B, C, D) with books in the ratio 3:4:5:6. The total books are 360. If 20 books are moved from C to B, what is the new ratio of A:B:C:D?

Step 1: Read & Highlight
- Initial ratio: A:B:C:D = 3:4:5:6.
- Total books = 360.
- 20 books moved from C to B.

Step 2: Assign Variables
- Let A = 3k, B = 4k, C = 5k, D = 6k.

Step 3: Write Equations
- 3k + 4k + 5k + 6k = 360 → 18k = 360 → k = 20.
- Initial books: A = 60, B = 80, C = 100, D = 120.

Step 4: Apply Change
- 20 books moved from C to B: C_new = 100 - 20 = 80 B_new = 80 + 20 = 100

Step 5: Find New Ratio
- New counts: A = 60, B = 100, C = 80, D = 120.
- Simplify ratio by dividing by 20: 60:100:80:120 = 3:5:4:6.

Answer: 3:5:4:6

Common Mistakes

1. Mistake: Ignoring "minimum/maximum" constraints.
2. Why it happens: Students solve for any valid solution, not the extreme one.

3. Correct approach: Always test boundaries (e.g., if D ≥ 10, try D = 10 first).

4. Mistake: Misassigning variables (e.g., letting A = x when B is the smaller one).

5. Why it happens: Random variable assignment without logic.

6. Correct approach: Let the smallest/unknown entity be x (e.g., if A = 2B, let B = x).

7. Mistake: Forgetting to simplify ratios.

8. Why it happens: Leaving answers as 60:100:80:120 instead of 3:5:4:6.

9. Correct approach: Always simplify ratios to lowest terms.

10. Mistake: Not checking all constraints.

11. Why it happens: Solving for one condition but violating another (e.g., D < 10).

12. Correct approach: Verify every constraint before finalizing the answer.

13. Mistake: Overcomplicating with too many variables.

14. Why it happens: Using x, y, z when k-method would suffice.
15. Correct approach: Use ratio splitting (A = 3k, B = 4k) for ratio problems.

CAT Traps & Time Management

1. Trap: Hidden Constraints
2. Example: "No employee gets less than ₹10" is easy to miss but critical.

3. Avoid: Underline all constraints in the first read.

4. Trap: Non-Integer Solutions

5. Example: A ratio problem might yield k = 1.5, but books must be integers.

6. Avoid: Check for integer solutions (e.g., k must be divisible by 2 if A = 3k and A must be integer).

7. Trap: "All of the Above" Distractors

8. Example: A question asks for the minimum value, but options include all possible values.
9. Avoid: Focus on the question’s exact ask (e.g., "minimum" vs. "maximum").

Time Guide:
- Easy set: 2–3 minutes.
- Medium set: 3–4 minutes.
- Hard set: 4–5 minutes (skip if stuck after 5 mins).

Quick Practice

Question 1:
Three friends (P, Q, R) share ₹120 in the ratio 2:3:5. If Q gives ₹10 to P, what is the new ratio? Answer: 3:2:5 Solution Path:
- Initial shares: P = 24, Q = 36, R = 60.
- After transfer: P = 34, Q = 26, R = 60.
- New ratio: 34:26:60 = 17:13:30 (Wait, this contradicts the answer! Correction:) - Correct Solution:
- Initial ratio 2:3:5 → Total parts = 10 → P = 24, Q = 36, R = 60.
- After Q → P: P = 34, Q = 26, R = 60.
- Simplify 34:26:60 → Divide by 2 → 17:13:30.
- Answer: 17:13:30 (The initial answer was wrong. Key Takeaway: Always simplify!)

Question 2:
In a race, A finishes before B, C finishes after D, and E finishes between A and C. Who could be 3rd? Answer: D or E Solution Path:
- Order constraints: A > B, D > C, A > E > C.
- Possible orders: 1. A > E > C > D > B (D is 4th) 2. A > E > D > C > B (D is 3rd) 3. D > A > E > C > B (E is 3rd) - 3rd position: D or E.

Last-Minute Cram Sheet (10 One-Liners)

1. Always assign variables to the smallest/unknown entity first.
2. Ratio splitting: A:B = 3:2 → A = 3k, B = 2k.
3. Minimum/maximum? Test boundary values (e.g., D = 10).
4. Check all constraints—missing one = wrong answer.
5. Simplify ratios to lowest terms (divide by GCD).
6. Non-integer solutions? Recheck variable assignment.
7. Ranking problems: Fix one rank to eliminate options.
8. Allocation problems: Write total equation first (A + B + C = 100).
9. MCQs: Plug in options to save time.
10. TITA: Double-check calculations—no room for error.

Final Tip: In the exam, skip if stuck after 5 minutes. These questions are high-reward but time-consuming—don’t let one set derail your entire DILR section. Practice 10–15 sets to build speed and accuracy. ?