UGC NET Paper 1 Unit 5 Topic 3: Mathematical Aptitude - Averages & Mixtures

📊 Averages & Mixtures: UGC NET Mastery Guide

Crack 3-5 questions/year with these data interpretation and problem-solving techniques. Includes alligation method, weighted averages, and real NET exam patterns.

🔑 Core Formula: Average = Sum of Observations / Number of Observations
Key Variations:

• Total Sum = Average × Number of items
• New Average when adding/removing items
• Weighted Average = (w₁x₁ + w₂x₂) / (w₁ + w₂)

1. Averages: Advanced Concepts

🔹 NET Exam's Favorite Average Problems

Type	Solution Approach	Example
Missing Data	Use total sum concept	Avg of 4 tests=80. First 3=85,76,72. Find 4th test? Total=320 → 4th=320-(85+76+72)=87
Inclusion/Exclusion	Adjust total sum and count	Class avg=40 for 30 students. If 2 leave, new avg=42. Find avg of departed students? Original total=1200 → New total=28×42=1176 → Departed sum=24 → Avg=12
Weighted Average	Account for different group sizes	Group A (10 people, avg=50) + Group B (20 people, avg=70) Combined avg=(10×50 + 20×70)/30=63.33

📚 NET 2021 Question Analysis:

"A researcher's average publication is 2.5 papers/year for first 6 years and 3.2 papers/year for next 4 years. What's the overall average?"

Solution:

1. First period total = 6 × 2.5 = 15 papers
2. Second period total = 4 × 3.2 = 12.8 papers
3. Overall average = (15 + 12.8) / 10 = 2.78 papers/year

2. Mixtures: The Alligation Method

📌 Understanding Cheaper vs. Dearer in Alligation

Cheaper Component (C):

• The ingredient with lower value (price/percentage/strength)
• Example: In mixing ₹30/kg and ₹60/kg rice, ₹30 is cheaper
• In acid solutions: 20% concentration is cheaper than 40%

Dearer Component (D):

• The ingredient with higher value
• Example: In same rice problem, ₹60 is dearer
• In medications: 80% pure drug is dearer than 50% diluted version

💡 Key Rule: Alligation always compares how much each component deviates from the desired mixture (M):
Ratio = (D - M) : (M - C)

⚖️ Visual Alligation Technique

⚖️ Interactive Alligation Diagram

Cheaper
(C=₹30)

Mixture
(M=₹50)

Dearer
(D=₹60)

Ratio = (60-50) : (50-30) = 10:20 = 1:2

Formula:

• Cheaper Quantity / Dearer Quantity = (D - M) / (M - C)
• Where: C = Cheaper item value, D = Dearer item value, M = Mixture value

Example: Mix ₹40/kg and ₹70/kg rice to get ₹55/kg mixture
Ratio = (70-55):(55-40) = 15:15 = 1:1

🔹 Mixture Problem Types

Scenario	Approach	NET Exam Frequency
Two-item mixtures	Direct alligation	High (2-3/year)
Three+ items	Step-wise alligation	Medium (1/year)
Replacement	Fraction replacement concept	Low (Every 2 years)

Advanced Mixture Formula:

When x liters are replaced from V liters of solution with C₁ concentration to C₂ concentration:

Final Concentration = C₁(1 - x/V) + C₂(x/V)

Example: 10L of 20% alcohol. If 2L replaced with pure alcohol:
New concentration = 20%(8/10) + 100%(2/10) = 16% + 20% = 36%

3. Combined Averages & Mixtures

🧪 NET 2022 Case Study:

"A lab mixes three acid solutions: 20% (5L), 30% (10L), and 40% (15L). Calculate: (a) Overall concentration (b) How much 50% solution to add for 35% mix?"

Solution:

1. Weighted average:
   (20×5 + 30×10 + 40×15) / (5+10+15) = (100+300+600)/30 = 33.33%
2. Alligation method:
   Existing (33.33%) + New (50%) → Target (35%)
   Ratio = (50-35):(35-33.33) = 15:1.67 ≈ 9:1
   For 30L existing, add (30×1)/9 ≈ 3.33L

🔍 Data Interpretation Shortcut

When NET gives tabular data:

Batch	Quantity	Purity%
A	200g	65%
B	300g	80%

Fast calculation:

• Total pure substance = (200×0.65) + (300×0.8) = 130 + 240 = 370g
• Total mixture = 500g
• Overall % = 370/500 × 100 = 74%

4. NET Exam Practice Arena

Level 1: Direct Formulas (1 min each)

1. Class avg=42 for 25 students. If teacher's score (58) included, new avg? (Ans: (1050+58)/26=42.62)
2. In what ratio mix ₹30/kg and ₹50/kg rice for ₹45/kg? (Ans: (50-45):(45-30)=5:15=1:3)

Level 2: Application (2-3 mins)

3. Average age of 10 researchers is 32. When 2 leave, avg increases by 3. If one departed is 25, find the other's age. (Ans: Original total=320 → New total=8×35=280 → Departed sum=40 → Other=40-25=15)

Level 3: Analysis (4 mins)

4. Solution A (40% salt, 100ml) mixed with B (60% salt, 150ml). If 50ml of mix replaced with pure water, final concentration? (Ans: Initial mix=(40+90)/250=52% → After replacement=52%(200/250)+0%(50/250)=41.6%)

🧠 Memory Anchors

Alligation Mnemonic: "Dear Minus Cheap" (D - M / M - C)
Average Trap: "Adding higher value → Avg ↑" (and vice versa)
Mixture Rule: "The middle value is always closer to larger quantity"

🚀 Exam Strategy

Situation	Action	Time Save
Complex averages	Assume base value (e.g., 100)	30-40 seconds
Decimal heavy calcs	Multiply all terms by 10/100	Prevents errors
Verification needed	Check with extreme values	1 min sanity check

💡 Pro Tip: For replacement problems, remember x/V remains constant - if 20L out of 100L replaced (1/5), then next replacement of 20L from new 100L still replaces original (1/5)² quantity.