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Effect of Replacement (Average)

Introduction

In some average problems, one or more values are replaced with new values. This directly affects the total sum, and hence the average. These questions test whether you can quickly calculate the replaced value or its effect on the average.

The logic is simple: New total - Old total = New value - Replaced value. Multiply the change in average by the number of terms to find the net change in total.

Pattern: Effect of Replacement (Average)

Pattern

Old Total = Average × Number of terms

New Total = New Average × Number of terms

Net Change = New Total - Old Total

Replaced Value = New Value - Net Change

Step-by-Step Example

Question

The average of 8 numbers is 20. One number is replaced by 30, and the new average becomes 21. Find the replaced number.

Options:

  • A) 20
  • B) 22
  • C) 24
  • D) 26

Solution

  1. Step 1: Find the old total

    The average of 8 numbers is 20. Total = Average × Number of terms = 20 × 8 = 160.

  2. Step 2: Find the new total after replacement

    The new average is 21. Total = 21 × 8 = 168.

  3. Step 3: Compute the net change in total

    Difference = New Total - Old Total = 168 - 160 = 8.

  4. Step 4: Relate net change to the replaced number

    Net change comes from replacing the old number with 30. So, New Value - Old Value = 8 → 30 - Old Value = 8.

  5. Step 5: Solve for the old (replaced) number

    Old Value = 30 - 8 = 22.

  6. Final Answer:

    22 → Option B

  7. Quick Check:

    If we remove 22 and add 30 → New total = 160 - 22 + 30 = 168. New average = 168 ÷ 8 = 21 ✅ Matches the question.

Quick Variations

If average decreases: The replaced number is greater than the new number.

If more than one replacement: Net change = (Sum of new values - Sum of replaced values).

Trick to Always Use

  • Step 1: Multiply average × count to find old total.
  • Step 2: Multiply new average × count to find new total.
  • Step 3: Difference in totals = (new value - replaced value).
  • Step 4: Solve for the replaced value.

Summary

Summary

  • Old Total = Average × Number of items.
  • New Total = New Average × Number of items.
  • Difference in totals = New Total - Old Total = New Value - Replaced Value.
  • Replaced Value = New Value - Net Change.

Example to remember:
Old total = 160 (8 × 20). New total = 168 (8 × 21). Net change = 8. Replaced number = 30 - 8 = 22.

Practice

(1/5)
1. The average of 5 numbers is 18. If one number 12 is replaced by 22, what will be the new average?
easy
A. 20
B. 19
C. 18
D. 17

Solution

  1. Step 1: Calculate old total

    Old total = 5 × 18 = 90.

  2. Step 2: Apply replacement

    New total = 90 - 12 + 22 = 100.

  3. Step 3: Compute new average

    New average = 100 ÷ 5 = 20.

  4. Final Answer:

    20 → Option A

  5. Quick Check:

    Increase in total = 10 → 10 ÷ 5 = +2 → 18 + 2 = 20 ✅
Hint: Change in average = (change in total) ÷ number of terms.
Common Mistakes: Failing to subtract the removed value before adding the new one.
2. The average of 7 numbers is 15. If one number 10 is replaced by 24, what will be the new average?
easy
A. 16
B. 17
C. 18
D. 15

Solution

  1. Step 1: Calculate old total

    Old total = 7 × 15 = 105.

  2. Step 2: Apply replacement

    New total = 105 - 10 + 24 = 119.

  3. Step 3: Compute new average

    New average = 119 ÷ 7 = 17.

  4. Final Answer:

    17 → Option B

  5. Quick Check:

    Increase = 14 → 14 ÷ 7 = +2 → 15 + 2 = 17 ✅
Hint: Net change ÷ count = change in average.
Common Mistakes: Dividing by wrong count or forgetting to remove the replaced value.
3. The average of 6 numbers is 20. If one number 30 is replaced by 18, what is the new average?
easy
A. 18
B. 19
C. 20
D. 21

Solution

  1. Step 1: Calculate old total

    Old total = 6 × 20 = 120.

  2. Step 2: Apply replacement

    New total = 120 - 30 + 18 = 108.

  3. Step 3: Compute new average

    New average = 108 ÷ 6 = 18.

  4. Final Answer:

    18 → Option A

  5. Quick Check:

    Total drop = 12 → 12 ÷ 6 = 2 → 20 - 2 = 18 ✅
Hint: If new value < old value, average drops by (difference ÷ n).
Common Mistakes: Adding the difference instead of subtracting.
4. The average of 10 numbers is 25. One number 35 is replaced by 15. Find the new average.
medium
A. 24
B. 23
C. 25
D. 22

Solution

  1. Step 1: Calculate old total

    Old total = 10 × 25 = 250.

  2. Step 2: Apply replacement

    New total = 250 - 35 + 15 = 230.

  3. Step 3: Compute new average

    New average = 230 ÷ 10 = 23.

  4. Final Answer:

    23 → Option B

  5. Quick Check:

    Total decrease = 20 → 20 ÷ 10 = 2 → 25 - 2 = 23 ✅
Hint: Compute total change first, then divide by the number of items.
Common Mistakes: Not handling subtraction and addition correctly when replacing.
5. The average weight of 8 persons is 60 kg. One person of weight 70 kg is replaced by another of weight 58 kg. Find the new average weight.
medium
A. 59
B. 59.5
C. 58.5
D. 60

Solution

  1. Step 1: Calculate old total

    Old total = 8 × 60 = 480.

  2. Step 2: Apply replacement

    New total = 480 - 70 + 58 = 468.

  3. Step 3: Compute new average

    New average = 468 ÷ 8 = 58.5.

  4. Final Answer:

    58.5 → Option C

  5. Quick Check:

    Drop = 12 → 12 ÷ 8 = 1.5 → 60 - 1.5 = 58.5 ✅
Hint: Difference ÷ count = change in average.
Common Mistakes: Using wrong divisor or rounding too early.

Mock Test

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