The ratio of speeds of two trains is $5:6$. The second train covers $450\text{km}$ in $5$ hours. What is the speed of the first train?

First, find the speed of the second train:
$$\text{Speed of second train}=\frac{\text{Distance}}{\text{Time}}=\frac{450}{5}=90\text{km/h}$$

The ratio of the speeds of the two trains is given as $5:6$.
Let the speed of the first train be $5x$ and the second train be $6x$.

$$6x=90⟹x=\frac{90}{6}=15$$

Therefore, the speed of the first train:

$$\text{Speed of first train}=5x=5\times 15=75\text{km/h}$$