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Tricky Shifts in Average

Introduction

In aptitude tests, you will often find questions where the average changes when a new value is added or removed from the group. This is called the shift in average.

Such problems look confusing, but they can be solved quickly once you understand the pattern.

Pattern: Tricky Shifts in Average

Pattern

The key idea is: Average × Number of Terms = Total Sum.

When one item is added/removed, the difference in the new average is directly linked to how much the new item differs from the old average.

If average increases: New number is greater than old average.
If average decreases: New number is smaller than old average.

Step-by-Step Example

Question

The average age of 10 students is 20 years. When one more student joins, the average increases by 2 years. Find the age of the new student.

Options:

  • A) 40
  • B) 38
  • C) 42
  • D) 44

Solution

  1. Step 1: Compute the old total

    Old total = 10 × 20 = 200.

  2. Step 2: Determine the new average

    New average = 20 + 2 = 22.

  3. Step 3: Compute the new total

    New total = 11 × 22 = 242.

  4. Step 4: Find the contribution of the new student

    Age of new student = New total - Old total = 242 - 200 = 42.

  5. Final Answer:

    42 years → Option C

  6. Quick Check:

    (200 + 42) ÷ 11 = 242 ÷ 11 = 22 → matches the new average ✅

Quick Variations

Case 1: One number added → Check if new number is higher/lower than old average.

Case 2: One number removed → New average shifts depending on the removed number.

Case 3: Average of group increases/decreases by a fixed value → Use total difference method.

Trick to Always Use

  • Step 1: Total = Average × Count.
  • Step 2: Compare totals before and after change.
  • Step 3: Difference = Contribution of new/removed item.
  • Step 4: Solve directly without recalculating average each time.

Summary

Summary

  • Compute totals using Total = Average × Number of terms before and after the change.
  • Difference between totals equals the value of the added/removed item.
  • If average increases, the added item is greater than the original average; if it decreases, it's smaller.
  • Use quick checks by recomputing (old total ± item) ÷ new count to confirm the new average.

Example to remember:
Old total 200 (10 × 20). New total 242 (11 × 22). Difference 42 → age of new student = 42.

Practice

(1/5)
1. The average age of 6 students is 12 years. A new student joins, and the average increases by 1 year. Find the age of the new student.
easy
A. 18
B. 19
C. 20
D. 21

Solution

  1. Step 1: Compute the old total age

    Total age of 6 students = 6 × 12 = 72.

  2. Step 2: Compute the new total with increased average

    New average = 13 → New total = 7 × 13 = 91.

  3. Step 3: Find the age of the new student

    Age of new student = 91 - 72 = 19.

  4. Final Answer:

    19 years → Option B

  5. Quick Check:

    (72 + 19)/7 = 91/7 = 13 ✅
Hint: Multiply average by number of students to get total sum quickly.
Common Mistakes: Using difference of averages directly without adjusting totals.
2. The average of 10 numbers is 25. If one number is removed, the average becomes 24. Find the removed number.
easy
A. 34
B. 35
C. 36
D. 37

Solution

  1. Step 1: Compute the original total

    Total of 10 numbers = 10 × 25 = 250.

  2. Step 2: Compute the new total after removal

    Total of remaining 9 numbers = 9 × 24 = 216.

  3. Step 3: Subtract to find the removed number

    Removed number = 250 - 216 = 34.

  4. Final Answer:

    34 → Option A

  5. Quick Check:

    (250 - 34)/9 = 216/9 = 24 ✅
Hint: Subtract new total from original total to find the removed number.
Common Mistakes: Dividing by wrong count (forgetting one number removed).
3. The average marks of 5 students is 60. A new student joins and the average increases to 62. Find the marks of the new student.
easy
A. 70
B. 72
C. 74
D. 76

Solution

  1. Step 1: Compute original total marks

    Total of 5 students = 5 × 60 = 300.

  2. Step 2: Compute new total marks with increased average

    Total of 6 students = 6 × 62 = 372.

  3. Step 3: Subtract to find new student's marks

    Marks of new student = 372 - 300 = 72.

  4. Final Answer:

    72 → Option B

  5. Quick Check:

    (300 + 72)/6 = 372/6 = 62 ✅
Hint: Difference in average × new total students = extra marks added.
Common Mistakes: Multiplying difference by old count instead of new count.
4. The average of 8 students is 15. One student leaves, and the average rises to 16. Find the age of the student who left.
medium
A. 6
B. 7
C. 8
D. 9

Solution

  1. Step 1: Compute original total age

    Total of 8 students = 8 × 15 = 120.

  2. Step 2: Compute new total age

    Total of remaining 7 students = 7 × 16 = 112.

  3. Step 3: Find age of student who left

    Age of student who left = 120 - 112 = 8.

  4. Final Answer:

    8 years → Option C

  5. Quick Check:

    (120 - 8)/7 = 112/7 = 16 ✅
Hint: Leaving a low value increases the average.
Common Mistakes: Using 8 instead of 7 when recalculating new total.
5. The average of 20 students is 25. If one new student joins, the average decreases by 1. Find the marks of the new student.
medium
A. 2
B. 4
C. 6
D. 8

Solution

  1. Step 1: Compute the original total marks

    Total marks of 20 students = 20 × 25 = 500.

  2. Step 2: Compute new total marks with decreased average

    New average = 24 → Total for 21 students = 21 × 24 = 504.

  3. Step 3: Find marks of new student

    Marks of new student = 504 - 500 = 4.

  4. Final Answer:

    4 marks → Option B

  5. Quick Check:

    (500 + 4)/21 = 504/21 = 24 ✅
Hint: When average decreases, new member is below the old average.
Common Mistakes: Using wrong total count (20 instead of 21).

Mock Test

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