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Weighted Average (Combined Groups)

Introduction

aptitude கேள்விகளில், இரண்டு அல்லது அதற்கு மேற்பட்ட குழுக்கள் ஒன்றாக சேர்க்கப்பட்டு, combined group-ன் average கேட்கப்படும் கேள்விகள் அடிக்கடி வருகின்றன.

இது சாதாரண arithmetic average அல்ல - இங்கு weighted average பயன்படுத்த வேண்டும். இதில், ஒவ்வொரு குழுவின் அளவு (items அல்லது people எண்ணிக்கை) அந்த குழுவின் weight ஆக செயல்படும்.

Pattern: Weighted Average (Combined Groups)

Pattern

ஒவ்வொரு குழுவின் average-ஐ அதன் எண்ணிக்கையுடன் பெருக்கி, அவற்றை கூட்டி, மொத்த எண்ணிக்கையால் வகுத்தால் combined average கிடைக்கும்.

Formula:
Combined Average = (n₁a₁ + n₂a₂ + n₃a₃ + …) ÷ (n₁ + n₂ + n₃ + …)
இதில் n = குழுவின் எண்ணிக்கை, a = அந்த குழுவின் average.

Step-by-Step Example

Question

ஒரு வகுப்பில் 30 boys-ன் average age 15 years. 20 girls-ன் average age 13 years. முழு வகுப்பின் average age-ஐ கண்டறியுங்கள்.

Options:

  • A: 13.6 years
  • B: 14.0 years
  • C: 14.2 years
  • D: 15.0 years

Solution

  1. Step 1: Formula-ஐ எழுதுங்கள்

    Combined Average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

  2. Step 2: மதிப்புகளை அடையாளம் காணுங்கள்

    n₁ = 30, a₁ = 15
    n₂ = 20, a₂ = 13

  3. Step 3: Weighted sum-ஐ கணக்கிடுங்கள்

    (30 × 15) + (20 × 13) = 450 + 260 = 710.

  4. Step 4: மொத்த எண்ணிக்கையை காணுங்கள்

    30 + 20 = 50.

  5. Step 5: Combined average-ஐ கணக்கிடுங்கள்

    Combined Average = 710 ÷ 50 = 14.2.

  6. Final Answer:

    14.2 years → Option C

  7. Quick Check:

    boys average (15) அதிகம், girls average (13) குறைவு. எனவே combined average இவற்றுக்கிடையில் இருக்க வேண்டும் → 14.2 சரி ✅

Quick Variations

  • 1. இரண்டு குழுக்கள் சேர்த்தல் (classes, sections, batches).
  • 2. மூன்று அல்லது அதற்கு மேற்பட்ட குழுக்கள் - அதே formula-ஐ விரிவுபடுத்துங்கள்.
  • 3. சில கேள்விகளில் missing group size கேட்கப்படும், total average கொடுக்கப்பட்டிருந்தால்.

Trick to Always Use

  • Step 1: “average × group size” என numerator-ஐ எழுதுங்கள்.
  • Step 2: எல்லா group totals-ஐ கூட்டுங்கள்.
  • Step 3: மொத்த size-ஆல் வகுக்குங்கள்.
  • Step 4: பதில் smallest மற்றும் largest group averages-க்கு இடையில் உள்ளதா எனச் சரிபாருங்கள்.

Summary

Summary

Weighted Average (Combined Groups) கேள்விகளை தீர்க்க:

  • ஒவ்வொரு குழுவின் average-ஐ அதன் அளவுடன் பெருக்குங்கள்.
  • அனைத்து weighted sums-ஐ கூட்டுங்கள்.
  • மொத்த எண்ணிக்கையால் வகுக்குங்கள்.
  • Final value, குறைந்த மற்றும் அதிக group averages-க்கு இடையில் இருக்கிறதா என உறுதி செய்யுங்கள்.

Practice

(1/5)
1. Section A has 40 students with average marks 50. Section B has 60 students with average marks 70. What is the combined average marks of all students?
easy
A. 60
B. 62
C. 64
D. 65

Solution

  1. Step 1: Compute total for Section A

    Total for A = 40 × 50 = 2000.

  2. Step 2: Compute total for Section B

    Total for B = 60 × 70 = 4200.

  3. Step 3: Calculate grand total and total students

    Grand total = 2000 + 4200 = 6200; total students = 40 + 60 = 100.

  4. Step 4: Compute combined average

    Average = 6200 ÷ 100 = 62.

  5. Final Answer:

    62 → Option B

  6. Quick Check:

    62 lies between 50 and 70 and nearer to 70 since section B is larger → consistent ✅
Hint: Weighted average = (sum of (size×avg)) ÷ total size.
Common Mistakes: Taking simple mean (50+70)/2 = 60 without weights.
2. Group X has 40 people with average score 55 and Group Y has 60 people with average score 65. What is the average score of the combined 100 people?
easy
A. 60
B. 58.5
C. 62.5
D. 61

Solution

  1. Step 1: Compute total for Group X

    Total for X = 40 × 55 = 2200.

  2. Step 2: Compute total for Group Y

    Total for Y = 60 × 65 = 3900.

  3. Step 3: Add totals and population counts

    Grand total = 2200 + 3900 = 6100; total people = 100.

  4. Step 4: Compute combined average

    Average = 6100 ÷ 100 = 61.

  5. Final Answer:

    61 → Option D

  6. Quick Check:

    61 lies between 55 and 65 and is closer to 65 since group Y is larger → consistent ✅
Hint: The larger group's average pulls the combined average toward itself.
Common Mistakes: Using (55+65)/2 = 60 without considering group sizes.
3. The average age of 15 men is 40 years and that of 25 women is 30 years. Find the average age of the whole group.
medium
A. 33.75
B. 34
C. 32.5
D. 35

Solution

  1. Step 1: Compute total for men

    Total for men = 15 × 40 = 600.

  2. Step 2: Compute total for women

    Total for women = 25 × 30 = 750.

  3. Step 3: Add totals and counts

    Total people = 15 + 25 = 40; Grand total = 600 + 750 = 1350.

  4. Step 4: Compute combined average

    Average = 1350 ÷ 40 = 33.75.

  5. Final Answer:

    33.75 years → Option A

  6. Quick Check:

    Result is between 30 and 40, closer to 30 because women group is larger → consistent ✅
Hint: Compute total = size × average for each group, then divide grand total by total size.
Common Mistakes: Averaging the two averages without weighting by group sizes.
4. Twelve students scored an average of 72 marks and eighteen students scored an average of 66 marks. What is the combined average of these 30 students?
medium
A. 69
B. 67.8
C. 68.4
D. 70

Solution

  1. Step 1: Compute total for first group

    Total for first group = 12 × 72 = 864.

  2. Step 2: Compute total for second group

    Total for second group = 18 × 66 = 1188.

  3. Step 3: Add group totals and counts

    Grand total = 864 + 1188 = 2052; total students = 12 + 18 = 30.

  4. Step 4: Compute combined average

    Average = 2052 ÷ 30 = 68.4.

  5. Final Answer:

    68.4 → Option C

  6. Quick Check:

    68.4 is between 66 and 72 and weighted toward 66 (larger group) → consistent ✅
Hint: Multiply averages by respective counts, add totals, divide by combined count.
Common Mistakes: Rounding intermediate totals too early or forgetting to sum group totals.
5. A group of 25 employees has an average salary of 54,000 and another group of 15 employees has an average of 66,000. What is the combined average salary?
medium
A. 58,500
B. 60,000
C. 59,400
D. 57,800

Solution

  1. Step 1: Compute total for first group

    Total for first group = 25 × 54,000 = 1,350,000.

  2. Step 2: Compute total for second group

    Total for second group = 15 × 66,000 = 990,000.

  3. Step 3: Add totals and counts

    Grand total = 1,350,000 + 990,000 = 2,340,000; total employees = 40.

  4. Step 4: Compute combined average

    Average = 2,340,000 ÷ 40 = 58,500.

  5. Final Answer:

    58,500 → Option A

  6. Quick Check:

    58,500 is between 54,000 and 66,000 and weighted toward 54,000 (larger group) → consistent ✅
Hint: Compute group totals first (count×avg), then divide the sum by total count.
Common Mistakes: Averaging the averages directly (54,000+66,000)/2 without weights.

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