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Weighted Average (Combined Groups)

Introduction

In aptitude questions, you may see problems where two or more groups are combined, and you are asked to find the average of the combined group.

This is not just a simple arithmetic average - instead, we must use a weighted average, where each group’s size (number of items or people) acts as the weight.

Pattern: Weighted Average (Combined Groups)

Pattern

The combined average is calculated by multiplying each group’s average by the number of terms, adding them up, and dividing by the total number of terms.

Formula:
Combined Average = (n₁a₁ + n₂a₂ + n₃a₃ + …) ÷ (n₁ + n₂ + n₃ + …)
where n = number of items in the group, a = average of the group.

Step-by-Step Example

Question

The average age of 30 boys in a class is 15 years, and the average age of 20 girls is 13 years. Find the average age of the whole class.

Options:

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

Solution

  1. Step 1: Write the formula

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

  2. Step 2: Identify the values

    n₁ = 30, a₁ = 15 n₂ = 20, a₂ = 13

  3. Step 3: Compute the total weighted sum

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

  4. Step 4: Compute the total size

    30 + 20 = 50.

  5. Step 5: Calculate the combined average

    Combined Average = 710 ÷ 50 = 14.2.

  6. Final Answer:

    14.2 years → Option C

  7. Quick Check:

    Since boys’ average (15) is higher and girls’ average (13) is lower, the combined average must lie between them → 14.2 is correct ✅

Quick Variations

  • 1. Combining two groups (e.g., classes, sections, batches).
  • 2. Combining three or more groups - extend the same formula.
  • 3. Questions may ask for a missing group size if total average is given.

Trick to Always Use

  • Step 1: Write numerator as “average × group size”.
  • Step 2: Add all group totals.
  • Step 3: Divide by total size.
  • Step 4: Check if result lies between smallest and largest group averages.

Summary

Summary

The Weighted Average (Combined Groups) pattern requires you to:

  • Multiply each group’s average by its size.
  • Add all weighted sums.
  • Divide by the total number of items.
  • Ensure final value lies between the minimum and maximum group averages.

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.

Mock Test

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