2.6 Effect on Average When a Member is Replaced

This is a very high-frequency question type. The key insight: you don’t need to know either the old average or the new average of the whole group — only the change.

Rule: If a member with value O is replaced by a new member with value N, and this causes the average of the group (of size n) to increase by d: N - O = n × d N = O + n × d

If the average decreases by d instead: N = O - n × d

Solved Example 16: The average weight of 10 persons increases by 2 kg when a person weighing 50 kg is replaced by a new person. Find the weight of the new person.

N = O + n×d = 50 + (10×2) = 50 + 20 = 70 kg.

Solved Example 17: The average age of 8 members of a family decreases by 1 year when a member aged 40 years is replaced by a new member. Find the age of the new member.

N = O − n×d = 40 − (8×1) = 40 − 8 = 32 years.

Solved Example 18: In a cricket team of 11 players, the average score increases by 3 runs when a player who scored 25 runs is replaced by a new player. Find the new player’s score.

N = 25 + 11×3 = 25 + 33 = 58 runs.

Solved Example 18(a): The average of a group of 20 students in a test is 55 marks. If a student who scored 35 marks is replaced by a student who scored 75 marks, find the new average of the group.

Change in total = 75−35 = 40. Change in average = 40/20 = 2 (increase). New average = 55+2 = 57 marks.

Note this is the same replacement rule used in the reverse direction: here you are given both old and new values and must find the change in average, rather than being given the change in average and asked for the new value. Both directions use the identical relationship N−O = n×d.

Solved Example 18(b): The average of 6 numbers is 18. One of the numbers, 12, is replaced by another number, and the new average becomes 20. Find the number that replaced 12.

Change in average = 20−18 = 2 (increase). n = 6. New number = Old number + n×d = 12 + 6×2 = 12+12 = 24.

Solved Example 18(c): The average weight of 12 persons increases by 3 kg when two persons, weighing 45 kg and 55 kg, are simultaneously replaced by two new persons. Find the average weight of the two new persons.

This is the two-member edge case of the replacement rule: the total increase in the group’s sum is still n×d, but that increase must now be distributed correctly across the two replaced positions. Total increase in sum = n×d = 12×3 = 36. Sum of the two persons who were replaced = 45+55 = 100. Sum of the two new persons = 100 + 36 = 136. Average weight of the two new persons = 136/2 = 68 kg.

Solved Example 18(d): The average of 15 numbers is 25. One number is replaced by another number that is 20 more than twice the original number, and this raises the average by 4. Find the original number that was replaced.

Let the original number be O; the new number is N = 2O+20. By the replacement rule, N − O = n×d = 15×4 = 60. So (2O+20) − O = 60 → O + 20 = 60 → O = 40. (Check: New number = 2×40+20 = 100. N−O = 100−40 = 60 = 15×4 ✓.)