How to Solve Average Weight Replacement Problems
Solve average weight replacement aptitude problems with the New = Old + (Δaverage × n) formula, a worked example and practice questions.
Expected Interview Answer
When one member of a group is replaced by another and the average weight changes, the weight of the new member equals the old member’s weight plus (change in average × number of people in the group), because the entire shift in total weight is caused by that single swap.
If the average rises by d when a person weighing W is replaced, the total weight must have risen by d × n for a group of n people, and since only one person changed, the new person’s weight is W + d×n. If the average falls, the same formula applies with a negative d, effectively subtracting d×n from W. This is a direct corollary of Sum = Average × Count: only the total moves, and it moves by exactly the average’s change multiplied by the group size, since count is unaffected by a like-for-like replacement. The formula collapses replacement problems into one arithmetic step instead of forming two separate total-weight equations.
- One formula — New = Old + (Δaverage × n) — replaces two-step total tracking
- Works identically whether the average rises or falls
- Directly derived from Sum = Average × Count, so it generalizes beyond weight
AI Mentor Explanation
A team’s average strike rate over 11 batters rises by 2 runs-per-100-balls when a batter with a known rate is swapped for a new one. The entire rise, spread across all 11 batters, must come from the new batter alone, so the incoming batter’s rate equals the outgoing rate plus 2×11 — that direct “swap shifts the total by delta times headcount” logic is exactly the average-weight replacement formula applied to strike rate instead of weight.
Worked example
Group size and delta
- n = 8
- Δ average = +2.5kg
Outgoing weight
- 65 kg
New weight
- 65 + 2.5×8 = 85 kg
Step-by-Step Explanation
Step 1
Identify n and Δaverage
Note the group size and how much the average changed (sign matters).
Step 2
Identify the outgoing weight
The known weight of the person being replaced.
Step 3
Apply the formula
New weight = Old weight + (Δaverage × n).
Step 4
Sanity-check the sign
A rising average means the new value is heavier; a falling average means lighter.
What Interviewer Expects
- Correct derivation of New = Old + (Δaverage × n) from Sum = Average × Count
- Correct sign handling for a rising versus falling average
- Recognizing that only the total changes, not the count, in a like-for-like swap
- Ability to reverse the formula to solve for the change in average given both weights
Common Mistakes
- Forgetting to multiply the change in average by the group size
- Using the wrong sign when the average decreases
- Confusing group size before and after (it never changes in a replacement)
- Trying to average the two weights directly instead of using the total-shift formula
Best Answer (HR Friendly)
“I use one formula: the new person’s weight equals the old person’s weight plus the change in average times the number of people. That is because the entire shift in the group’s total weight has to come from that single swap, spread evenly across everyone when you compute the average — so multiplying the average change back out by the headcount tells you exactly how much heavier or lighter the new person is.”
Follow-up Questions
- How would you solve for the change in average if both weights are known?
- What if two people are replaced at once instead of one?
- How does this formula relate to the general Sum = Average × Count identity?
- What happens to this formula if a person joins rather than replaces someone?
MCQ Practice
1. The average weight of 10 people increases by 3kg when a person weighing 60kg is replaced by a new person. The new person’s weight is?
New weight = 60 + 3×10 = 90 kg.
2. The average weight of 6 people decreases by 1.5kg when a person weighing 80kg is replaced. The new person’s weight is?
New weight = 80 − 1.5×6 = 80 − 9 = 71 kg.
3. In a replacement problem, if the group size were unknown, which value could NOT be determined from Δaverage and the outgoing weight alone?
New weight = Old + Δaverage × n requires knowing n; without it, the new weight cannot be found.
Flash Cards
Replacement formula for average weight? — New weight = Old weight + (Δaverage × group size n).
Why does this formula work? — Only the total weight shifts, by exactly Δaverage × n, since one swap causes the entire change.
Sign convention for a falling average? — Use a negative Δaverage; the new value is lighter than the old.
Does group size change in a replacement? — No — replacement swaps one member for another, count stays fixed.