A, B and C invested their capitals in the ratio 3 : 4 : 6. However their share of profit are equal. The duration of their investments must be in the ratio ?

Answer: Option A

Let their investments be Rs. 3x for p months
Rs. 4x for q months and Rs. 6x for r months respectively
Then,
$$\begin{array}{rl}& 3xp:4xq:6xr=1:1:1\\ & \Rightarrow 3p:4q:6r=1:1:1\\ & \text{So,}3p=4q\\ & \iff q=\frac{3p}{4}\\ & \text{And, }4q=6r\\ & \iff r=\frac{2q}{3}=\left(\frac{2}{3}\times \frac{3}{4}p\right)=\frac{p}{2}\\ & \therefore p:q:r\\ & =p:\frac{3p}{4}:\frac{p}{2}\\ & =4:3:2\end{array}$$