Averages • Topic 4 of 4

Replacement Problems

Replacement problems are the CAT favourite where one member of a group leaves and another joins, and the average shifts. The count stays the same, so the entire change in total comes from the swap: total change = new value − old value = (change in average) × count. If the average weight of 8 people increases by 2.5 kg when a new person replaces one weighing 65 kg, the total rose by 8 × 2.5 = 20 kg, so the newcomer weighs 65 + 20 = 85 kg. The same logic runs in reverse to find the leaver. A close cousin is the "added or removed without replacement" case, where the count changes too — there you must use total = average × count for both the before and after states and subtract. The single most common trap is multiplying the change in average by the wrong count (use the unchanged group size for a pure swap).

✅ Solved examples

1. The average weight of 8 people increases by 2.5 kg when a new person replaces one weighing 65 kg. Find the new person’s weight.
Total rise = 8 × 2.5 = 20 kg. New weight = 65 + 20 = 85 kg.
2. A cricketer’s average for 10 innings is 40. In the 11th innings he scores 80. Find his new average.
Old total = 400. New total = 480 over 11 innings ⇒ 480/11 ≈ 43.64.
3. The average age of a team of 11 falls by 6 months when a 30-year-old is replaced by a younger player. Find the new player’s age.
Total falls by 11 × 0.5 = 5.5 years. New age = 30 − 5.5 = 24.5 years.
4. A batsman has an average of 32 after 15 innings. How many runs must he score in the 16th innings to raise his average to 34?
Old total = 480. Required new total = 34 × 16 = 544. Runs needed = 544 − 480 = 64.

✏️ Practice — try these, take hints as needed

1. The average of 6 numbers increases by 4 when one number is replaced by 50. Find the replaced number.
Total rise = 6 × 4 = 24.
New − old = 24.
Old = 50 − 24.
26
2. The average weight of 5 boys is 40 kg. A boy of 30 kg leaves and one of 50 kg joins. New average?
Count stays 5.
Total changes by 50 − 30 = 20.
New average = 40 + 20/5.
44 kg
3. A player averages 36 over 20 innings. What must he score next to make it 38?
New total = 38 × 21.
Old total = 36 × 20.
Subtract.
78
4. The average age of 40 students rises by 3 months when a teacher of age 45 joins. Find the original average age (count becomes 41).
This adds a member, not a swap.
40A + 45 = 41(A + 0.25).
Solve for A.
34.75 years
5. The average salary of 12 employees drops by ₹500 when a manager earning ₹40,000 is replaced. Find the replacement’s salary.
Total drop = 12 × 500 = 6000.
New = old − 6000.
40000 − 6000.
₹34,000

📝 Topic test — 8 questions

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