A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined them after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed amount A, B and C?

The share of A in the business is $3\left(12\right)x=36x$

The share of B in the business for one year is $5\left(12\right)x=60x$

The share of $C$ in the business with A and B is $6\left(5x\right)=30x$

The proportion should profit at the end of one year be distributed amount $A, B$ and $C$ is $36x:60x:30x=\boxed{6:10:5}$