Arjun and Vaibhav are partners sharing profits in the ratio of $3:2$. They admitted Rahul as a new partner for $\frac{1}{5}$ share in the future profits of the firm. The new partner acquired his share from the old partners in the old ratio. Calculate the new profit sharing ratio of Arjun, Vaibhav and Rahul.

Let the old ratio between Arjun and Vaibhav be $3:2$.Rahul's share = $\frac{1}{5}$ Rahul acquires his share from Arjun and Vaibhav in their old ratio ($3:2$):Share sacrificed by Arjun = $\frac{1}{5}\times \frac{3}{5}=\frac{3}{25}$ Share sacrificed by Vaibhav = $\frac{1}{5}\times \frac{2}{5}=\frac{2}{25}$ Now compute the new shares:Arjun's New Share = $\frac{3}{5}-\frac{3}{25}=\frac{15-3}{25}=\frac{12}{25}$ Vaibhav's New Share = $\frac{2}{5}-\frac{2}{25}=\frac{10-2}{25}=\frac{8}{25}$ Rahul's Share = $\frac{1}{5}=\frac{5}{25}$ Comparing the shares, the new profit sharing ratio is $12:8:5$.