The average weight of some persons in a group is 76 kg. If 15 persons with average weight 72 kg join the group or 5 persons with average weight 84 kg leave the group, the average weight of the persons in the group in both cases is the same. How many persons were there in the group, initially?

Let the initial number of persons be n.
Initial total weight = 76n.

Case 1: 15 persons with average weight 72 kg join

New total weight=76n+15×72=76n+1080
New average
1
​
=
n+15
76n+1080
​

Case 2: 5 persons with average weight 84 kg leave

New total weight=76n−5×84=76n−420
New average
2
​
=
n−5
76n−420
​

Given that both averages are equal:

n+15
76n+1080
​
=
n−5
76n−420
​

Using deviation concept:
In case 1, net deficit = 15×(76−72)=60. So average decreases by
n+15
60
​
.
In case 2, net deficit created by leaving = 5×(84−76)=40. So average decreases by
n−5
40
​
.

Since the new average is the same in both cases, the decrease in average must be equal:

n+15
60
​
=
n−5
40
​

n+15
3
​
=
n−5
2
​

3(n−5)=2(n+15)
3n−15=2n+30⟹n=45