2.7 Effect on Average When a Member Joins or Leaves

Case A — A member joins the group: If a new member with value x joins a group of n members (average A), and the new average becomes A’: x = A’ × (n+1) - A × n Alternatively, useful shortcut: if the average increases by d upon joining, x = A + (n+1) × d because the new member must be d more than the old average, plus make up for pulling all n old members up by d each.

Case B — A member leaves the group: If a member with value x leaves a group of n members (average A), and the new average of the remaining (n-1) members becomes A’: x = A × n - A’ × (n-1)

Solved Example 19: The average age of 30 students in a class is 12 years. If the age of the teacher is included, the average becomes 12.5 years. Find the teacher’s age.

Using x = A’(n+1) − An: x = 12.5×31 − 12×30 = 387.5 − 360 = 27.5 years.

Solved Example 20: The average weight of 24 students in a class is 35 kg. When one student leaves, the average weight of the remaining 23 students becomes 34.8 kg. Find the weight of the student who left.

x = An − A’(n−1) = 35×24 − 34.8×23 = 840 − 800.4 = 39.6 kg.

Solved Example 21: The average marks of 40 students in a class is 50. If 5 students with an average of 60 marks join the class, find the new average.

New total = 40×50 + 5×60 = 2000 + 300 = 2300. New count = 45. New average = 2300/45 = 51.11 (approx).

Solved Example 21(a): The average salary of 15 employees in an office is ₹28,000. If two employees, each drawing ₹20,000, resign, find the new average salary of the remaining 13 employees.

Old total = 15×28000 = 4,20,000. Total salary of the two who left = 2×20,000 = 40,000. New total = 4,20,000 − 40,000 = 3,80,000. New count = 13. New average = 3,80,000/13 = ₹29,230.77 (approx).

Solved Example 21(b): The average age of a group of 25 friends is 24 years. If 5 more friends, whose average age is 30 years, join the group, find the new average age of all 30 friends.

New total = 25×24 + 5×30 = 600+150 = 750. New count = 30. New average = 750/30 = 25 years.

Solved Example 21(c): The average age of a group of 20 workers is 35 years. A worker aged 50 years leaves the group, and simultaneously two new workers of equal age join the group, taking the total strength back up to 21. If the new average age of the group is 36 years, find the age of each new worker.

This tests the edge case where a member leaves and multiple new members join at the same time, changing the group size in a single combined step. Old sum = 20×35 = 700. After the worker aged 50 leaves: sum = 700−50 = 650 (19 workers remain). Let each new worker’s age be x; two of them join: new sum = 650+2x, new count = 21. New average: (650+2x)/21 = 36 → 650+2x = 756 → 2x = 106 → x = 53 years.

Solved Example 21(d): The average salary of 25 employees in a company is ₹30,000. Three employees, drawing ₹25,000, ₹28,000, and ₹32,000, resign, and simultaneously five new employees join with an average salary of ₹35,000. Find the new average salary of the company (27 employees).

Old total = 25×30,000 = 7,50,000. Sum of the three who resigned = 25,000+28,000+32,000 = 85,000. Remaining total = 7,50,000−85,000 = 6,65,000 (22 employees). Sum of the five new joinees = 5×35,000 = 1,75,000. New total = 6,65,000+1,75,000 = 8,40,000 (27 employees). New average = 8,40,000/27 = ₹31,111.11 (approx).