Two trains are moving in the same direction at $50 \, km/h$ and $30 \, km/h$. The faster train crosses a man in the slower train in $18 \, seconds$. Find the length of the faster train (in $m$).

$A$ thief is stopped by a policeman from a distance of $150\, m$. When the policeman starts the chase,the thief also starts running. Assuming the speed of the thief as $7\, km/h$ and that of the policeman as $9\, km/h$,how far would the thief have run before he is overtaken by the policeman? (in $m$)

Two stations $A$ and $B$ are $100 \, km$ apart on a straight line. One train starts from $A$ at $7 \, am$ and travels towards $B$ at $20 \, km/h$ speed. Another train starts from $B$ at $8 \, am$ and travels towards $A$ at $25 \, km/h$ speed. At what time will they meet?

$A$ man can reach a certain place in $30 \, \text{hours}$. If he reduces his speed by $\frac{1}{15}^{th}$, he goes $10 \, \text{km}$ less in that same time. Find his speed in $\text{km/hr}$.

$A$ person travels from $P$ to $Q$ at a speed of $40 \, km/h$ and returns from $Q$ to $P$ by increasing his speed by $50\%$. What is his average speed for the entire journey? (in $km/h$)

$A$ policeman starts to chase a thief. When the thief goes $10 \, \text{steps}$, the policeman moves $8 \, \text{steps}$. $5 \, \text{steps}$ of the policeman are equal to $7 \, \text{steps}$ of the thief. The ratio of the speeds of the policeman and the thief is