Average: Replacement, Addition and Age Problems

Adding or removing a member

If a new value is added and the average changes, use sums:

• Average of 10 numbers = 50 (sum 500). An 11th number makes the average 52 → new sum 572 → new number = 72.
• Shortcut: new value = old average + (new count × change) = 50 + 11 × 2 = 72.

Replacement

When one person is replaced and the average of n people changes by d:

New person = Old person + n × d (use − if the average falls).

Example: average weight of 8 men rises by 2 kg when a 60 kg man is replaced → new man = 60 + 8 × 2 = 76 kg.

Wrongly recorded values

Correct sum = recorded sum − wrong value + correct value. Then divide again.

Age problems

• Average age of a family of 4 is 25 now → total 100. After 5 years, total = 100 + 4 × 5 = 120 (average 30).
• A baby joins: find total before and after, subtract.

Cricket-style questions

Batsman averages 40 in 10 innings. To make the average 42 after the 11th innings, he must score 40 + 11 × 2 = 62.

Analogy

Replacement is like swapping one player in a tug-of-war team. If the whole team suddenly pulls 2 kg harder per person, the new player must bring all that extra strength alone — the old player's strength plus 2 kg for every member of the team.