$P$ and $Q$ are $27 \, km$ apart. Two trains with speeds of $24 \, km/h$ and $18 \, km/h$ respectively start simultaneously from $P$ and $Q$ and travel in the same direction. They meet at a point $R$ beyond $Q$. The distance $QR$ is (in $km$):

$A$ car travels from $P$ to $Q$ at a constant speed. If its speed were increased by $10 \, km/h$,it would have taken $1 \, hour$ less to cover the distance. It would have taken a further $45 \, minutes$ less if the speed was increased by another $10 \, km/h$. The distance between the two cities is (in $km$):

On a tour,a man travels at the rate of $64 \text{ km/h}$ for the first $160 \text{ km}$,then travels the next $160 \text{ km}$ at the rate of $80 \text{ km/h}$. The average speed in $\text{km/h}$ for the first $320 \text{ km}$ of the tour is:

Walking at three-fourth of his usual speed, a man covers a certain distance in $2 \, \text{hours}$ more than the time he takes to cover the distance at his usual speed. The time taken by him to cover the distance with his usual speed is (in $\text{hours}$):

$A$ man starts running from point $P$ at $11:00$ $a.m.$ with a speed of $10 \, km/hr$. He runs for $2 \, hours$ and then takes a $1 \, hour$ rest. He continues this pattern until he is caught by another man who starts at $2:00 \, p.m.$ from point $P$ and runs non-stop at a speed of $15 \, km/hr$ toward the first man. At what time (in $p.m.$) will the first man be caught?

$A, B$ and $C$ start together from the same place to walk around a circular path of length $12 \, km$. $A$ walks at the rate of $4 \, km/h$,$B$ at $3 \, km/h$,and $C$ at $\frac{3}{2} \, km/h$. They will meet together at the starting place at the end of (in $h$):