A good start by Jia and Jeremy from Raffles Institution, Singapore

To balance a bar balance, equal weights are required on each side of the pivot.

Taking a as the distance of the 3 weight from the pivot, and b as the distance of the 2 weight from the pivot. And using each column from the pivot as a distance of one, the weight on the left of the pivot has a moment of 40.

So 3a+2b must equal (that is, balance) 40

Matching pairs of a & b could be:

a = 5 and b = 12.5
or
a = 4 and b = 14

And some good algebra reasoning from Joan in Edinburgh

If we want a less than b it would be good to find where a = b, that means that the two weights are together at the same place.

If a = b then a and b have to be 8
(from 5a = 40 or 5b = 40 whichever you prefer)

But we want a less than b, so a can't be more than 8 .
The 3 weight can go as near to the pivot as we like and b will just have to get bigger to keep the balance.
If the 3 weight does go to zero b will have to be 20.

20 is the furthest out from the pivot that the 2 weight goes if it keeps balanced with the 4 weight on the other side.