**CAT Arithmetic Mastery: Averages, Mixtures & Alligation**

CAT Arithmetic Mastery: Averages, Mixtures & Alligation

(A Premium Study Guide for 99+ Percentile Aspirants)

What This Is

Averages, Mixtures, and Alligation (AMA) form ~15-20% of CAT Quant questions—often disguised as word problems in Data Interpretation (DI) or Logical Reasoning (LR) sets. Mastering this topic means: ✅ Speed: Solve in <2 minutes with structured steps.
✅ Accuracy: Avoid traps like weighted averages vs. simple averages.
✅ Versatility: Apply to profit/loss, speed-distance, and even geometry problems.

Real CAT-Style Example: "A shopkeeper mixes two varieties of rice costing ₹30/kg and ₹50/kg in the ratio 2:3. If he sells the mixture at ₹48/kg, what is his profit percentage?" (This tests alligation + profit/loss in one question—exactly what CAT does.)

Key Concepts & Techniques

1. Simple Average (Mean)
2. What: Sum of values ÷ Number of values.
3. When: When all items contribute equally (e.g., marks of 5 students).

4. Pro Tip: For consecutive numbers, average = (First + Last) ÷ 2.

5. Weighted Average

6. What: Average where items have different weights (e.g., marks in 3 subjects with different credit hours).
7. Formula:
   [
   \text{Weighted Avg} = \frac{\sum (\text{Value} \times \text{Weight})}{\sum \text{Weights}}
   ]

8. When: When quantities are not equally important (e.g., mixture problems, profit/loss with different quantities).

9. Alligation (Rule of Mixtures)

10. What: A shortcut to find the ratio of two components in a mixture when their individual averages and the final average are known.
11. When: Only for two-component mixtures (e.g., two types of rice, two solutions).

12. How:

    
    
    • Write the individual averages (A and B) and the final average (M).
    • The ratio of quantities = (|M - B| : |M - A|).
    • Memory Trick: "Cheaper : Dearer = (Dearer - Mean) : (Mean - Cheaper)".

13. Replacement Problems

14. What: A portion of a mixture is removed and replaced with another component (e.g., replacing milk with water).
15. When: Questions like "A 20L milk solution has 10% water. How much water should be added to make it 25% water?"

16. Formula:
    [
    \text{Final Quantity} = \text{Initial Quantity} \times \left(1 - \frac{\text{Replaced Quantity}}{\text{Initial Quantity}}\right)^n
    ]
    (where (n) = number of replacements)

17. Alligation for Profit/Loss

18. What: Treat cost prices (CP) as "averages" and selling price (SP) as the "final average."
19. When: Questions like "A shopkeeper sells two items at the same SP, gaining 10% on one and losing 10% on the other. What is his overall profit/loss?"

20. Key Insight: If SP is the same, overall profit/loss depends on the ratio of quantities sold.

21. Average Speed (Harmonic Mean)

22. What: For two equal distances at speeds (S_1) and (S_2), average speed = (\frac{2S_1S_2}{S_1 + S_2}).

23. When: Only when distances are equal (e.g., up and down a hill).

24. Trick: "Alligation with Zero"

25. What: If one component is free (e.g., water in a milk-water mixture), its "average" = 0.
26. When: Questions like "How much water should be added to 10L of 20% milk to make it 10% milk?"

Step-by-Step Strategy (Follow This Every Time)

Step 1: Identify the Type

• Mixture? → Alligation.
• Replacement? → Replacement formula.
• Profit/Loss? → Treat CP as averages.
• Simple average? → Sum ÷ Count.

Step 2: Assign Variables

• Let quantities be (x) and (y) (or use ratios).
• For alligation, write A (cheaper), B (dearer), and M (mean).

Step 3: Apply the Right Formula

• Alligation: (|M - B| : |M - A|) → Ratio.
• Weighted Avg: (\frac{\sum (\text{Value} \times \text{Weight})}{\sum \text{Weights}}).
• Replacement: Final = Initial × ((1 - \frac{\text{Replaced}}{\text{Initial}})^n).

Step 4: Solve for the Unknown

• Use the ratio to find quantities.
• Cross-check with answer choices (if MCQ).

Step 5: Verify Units & Traps

• Ensure units match (e.g., kg vs. liters).
• Check if the question asks for ratio, quantity, or percentage.

Fully Worked CAT-Style Example

Question: "A milkman mixes milk (₹40/L) and water (₹0/L) in the ratio 3:1. He sells the mixture at ₹36/L. What is his profit percentage?"

Solution (Using the Strategy):

1. Identify Type: Mixture + Profit/Loss → Alligation + Profit formula.
2. Assign Variables:
3. Milk (A) = ₹40/L, Water (B) = ₹0/L.
4. Mixture ratio = 3:1 → Total parts = 4.
5. Cost Price (CP) of mixture = (\frac{3 \times 40 + 1 \times 0}{4} = ₹30/L).
6. Apply Alligation:
7. CP = ₹30/L, SP = ₹36/L.
8. Profit = SP - CP = ₹6/L.
9. Profit Percentage:
   [
   \text{Profit %} = \left(\frac{6}{30}\right) \times 100 = 20\%.
   ]
10. Verify:
11. Alligation ratio: (|30 - 0| : |30 - 40| = 30:10 = 3:1) (matches given ratio).

Answer: 20%.

Common Mistakes

1. Mistake: Using simple average instead of weighted average.
2. Why: Assuming all quantities are equal (e.g., averaging marks without considering credit hours).

3. Correct Approach: Always check if weights are given.

4. Mistake: Misapplying alligation for >2 components.

5. Why: Alligation only works for two components.

6. Correct Approach: For 3+ components, use weighted average.

7. Mistake: Ignoring "replacement" in mixture problems.

8. Why: Forgetting that a portion is removed before adding the new component.

9. Correct Approach: Use the replacement formula.

10. Mistake: Confusing profit/loss alligation.

11. Why: Treating SP as CP or vice versa.

12. Correct Approach: CP = "average," SP = "final average."

13. Mistake: Not checking units.

14. Why: Mixing kg and liters in the same problem.
15. Correct Approach: Convert all units to the same type.

CAT Traps & Time Management

Traps to Avoid

1. "Hidden Zero" Trap:
2. Example: "A 10% milk solution is mixed with water to make it 5% milk." → Water = 0% milk.

3. How to Spot: Look for "water," "free," or "0%."

4. "Replacement vs. Addition" Trap:

5. Example: "A 20L solution has 10% alcohol. If 5L is removed and replaced with water, what is the new %?" → Replacement, not addition.

6. How to Spot: Keywords: "removed," "replaced," "drained."

7. "Profit/Loss Alligation" Trap:

8. Example: "A shopkeeper sells two items at the same SP, gaining 10% on one and losing 10% on the other." → Overall loss (not 0%).
9. How to Spot: Same SP but different CP.

Time Management

• Simple Average: 30–45 sec.
• Alligation: 1–1.5 min.
• Replacement Problems: 1.5–2 min.
• Profit/Loss + Alligation: 2 min.

Rule: If stuck >2 min, mark and move on.

Quick Practice

1. Question:
   "The average weight of 10 students is 50 kg. If the teacher (60 kg) is included, what is the new average?"
   Answer: 50.91 kg (Sum = 10×50 + 60 = 560; New avg = 560/11).

2. Question:
   "Two varieties of tea costing ₹200/kg and ₹300/kg are mixed in the ratio 3:2. What is the cost of the mixture per kg?"
   Answer: ₹240/kg (Alligation: (|250 - 300| : |250 - 200| = 50:50 = 1:1 → Wait, no! Given ratio is 3:2 → Weighted avg = (\frac{3×200 + 2×300}{5} = ₹240)).

Last-Minute Cram Sheet (10 One-Liners)

1. Alligation Ratio: Cheaper : Dearer = (Dearer - Mean) : (Mean - Cheaper).
2. Replacement Formula: Final = Initial × ((1 - \frac{\text{Replaced}}{\text{Initial}})^n).
3. Profit/Loss Alligation: CP = "average," SP = "mean."
4. Average Speed: For equal distances, harmonic mean = (\frac{2S_1S_2}{S_1 + S_2}).
5. Water = 0%: Treat as "free" component in alligation.
6. Weighted Avg: (\frac{\sum (\text{Value} \times \text{Weight})}{\sum \text{Weights}}).
7. Same SP Trap: If SP is equal, overall profit/loss depends on ratio of quantities.
8. Consecutive Numbers: Avg = (First + Last) ÷ 2.
9. Alligation for 3+ Components: Use weighted average, not alligation.
10. Units Check: Always match kg, liters, etc.

⚠️ Trap Distinction: - Mixture Addition → New quantity = Old + Added.
- Mixture Replacement → New quantity = Old × ((1 - \frac{\text{Replaced}}{\text{Old}})).

Final Tip: Practice 10 alligation + 5 replacement + 5 profit/loss problems daily for 2 weeks. Speed comes from pattern recognition, not formulas. ?