Two trains running in opposite directions cross a man standing on the platform in $27\text{seconds}$ and $17\text{seconds}$ respectively, and they cross each other in $23\text{seconds}$. The ratio of their speeds is:

Let the speed of the first train be $x$ and the speed of the second train be $y$.
1. Length of the first train = $\text{Speed}\times \text{Time}=27x$
2. Length of the second train = $\text{Speed}\times \text{Time}=17y$

When moving in opposite directions, their relative speed is $(x+y)$. The time taken to cross each other completely is $23\text{seconds}$:

$$\text{Time}=\frac{\text{Total Length}}{\text{Relative Speed}}$$

$$23=\frac{27x+17y}{x+y}$$

$$23(x+y)=27x+17y$$

$$23x+23y=27x+17y$$

Rearrange terms to isolate the variables:

$$23y-17y=27x-23x$$

$$6y=4x$$

$$\frac{x}{y}=\frac{6}{4}=\frac{3}{2}$$

The ratio of their speeds is $3:2$.