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Weighted Average (Combined Groups)

Introduction

Aptitude questions में अक्सर ऐसे problems आते हैं जहाँ दो या ज़्यादा groups को मिलाकर पूरा समूह का average निकालना होता है।

यह सिर्फ सामान्य arithmetic average नहीं होता - हर group का size (items या people की संख्या) एक weight की तरह काम करता है, इसलिए हमें weighted average use करना पड़ता है।

Pattern: Weighted Average (Combined Groups)

Pattern

Combined average निकालने का तरीका यह है कि हर group के average को उसके number से multiply करें, सब जोड़ें और total संख्या से divide करें।

Formula:
Combined Average = (n₁a₁ + n₂a₂ + n₃a₃ + …) ÷ (n₁ + n₂ + n₃ + …)
यहाँ n = group के items की संख्या, a = उस group का average है।

Step-by-Step Example

Question

एक class में 30 लड़कों की average age 15 years है, और 20 लड़कियों की average age 13 years है। पूरे class का average age क्या होगा?

Options:

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

Solution

  1. Step 1: Formula लिखें

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

  2. Step 2: मान पहचानें

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

  3. Step 3: weighted sum compute करें

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

  4. Step 4: total size निकालें

    30 + 20 = 50.

  5. Step 5: Combined average निकालें

    Combined Average = 710 ÷ 50 = 14.2.

  6. Final Answer:

    14.2 years → Option C

  7. Quick Check:

    क्योंकि boys का average (15) ज्यादा है और girls का (13) कम, पूरे class का average इनके बीच आता है - 14.2 सही है ✅

Quick Variations

  • 1. दो groups को मिलाना (जैसे classes, sections, batches)।
  • 2. तीन या ज़्यादा groups को मिलाना - उसी formula का विस्तार करें।
  • 3. कभी-कभी total average दिया होता है और missing group size पूछा जाता है - वही formula rearrange करके solve करें।

Trick to Always Use

  • Step 1: numerator को “average × group size” के रूप में लिखें।
  • Step 2: सभी group totals जोड़ें।
  • Step 3: कुल size से divide करें।
  • Step 4: Check करें कि result smallest और largest group averages के बीच है।

Summary

Summary

Weighted Average (Combined Groups) pattern में आपको:

  • हर group के average को उसके size से multiply करना है।
  • सभी weighted sums जोड़ने हैं।
  • कुल items की संख्या से divide करना है।
  • नतीजा minimum और maximum group averages के बीच होना चाहिए।

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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