Then their ages at the start were 2n, 3n, and 4n for some constant n

After 5 years their ages were 2n+5, 3n+5, and 4n+5

And those ages must be 6k, 7k, and 8k, respectively, for some constant k.

2n+5 = 6k
3n+5 = 7k
4n+5 = 8k

Solving the system

 gives n=1.25, k=1

 gives n=1.25, k=1

 gives n=1.25, k=1

Their ages at the start were 2n, 3n, and 4n

That's 2(1.5) = 3, 3(1.5) = 4.5, 4(1.5) = 5

So their ages are 3, 4.5 and 5 at the start.

After 5 years their ages will be 8, 9.5, and 10.

So the youngest is ???

Edwin