A man travels three-fifth of a distance $AB$ at a speed of $3x$ and the remaining distance at a speed of $2y$. If he goes from $B$ to $A$ and returns at a uniform speed of $5z$ in exactly the same total time, then which one of the following relations is correct?

Let the total distance $AB$ be $D$.
* Distance for the first part = $\frac{3}{5}D$, covered at speed $3x$.

$${\text{Time}}_1=\frac{\frac{3}{5}D}{3x}=\frac{D}{5x}$$

* Distance for the remaining part = $D-\frac{3}{5}D=\frac{2}{5}D$, covered at speed $2y$.

$${\text{Time}}_2=\frac{\frac{2}{5}D}{2y}=\frac{D}{5y}$$

Total time taken for the forward journey from $A$ to $B$:

$${\text{Total Time}}_{\text{forward}}=\frac{D}{5x}+\frac{D}{5y}$$

For the round trip from $B$ to $A$ and back to $B$, the total distance covered is $2D$ at a uniform speed of $5z$:

$${\text{Total Time}}_{\text{round}}=\frac{2D}{5z}$$

Given that the times are equal:

$$\frac{D}{5x}+\frac{D}{5y}=\frac{2D}{5z}$$

Dividing the entire equation by $\frac{D}{5}$:

$$\frac{1}{x}+\frac{1}{y}=\frac{2}{z}$$