$A$ bike covers a certain distance at a speed of $64 \, km/h$ in $8 \, hours$. If the bike were to cover the same distance in approximately $6 \, hours$,at what approximate speed should the bike travel? (in $km/h$)

Two men $A$ and $B$ walk from $P$ to $Q$ at a distance of $21 \text{ km}$ at speeds of $3 \text{ km/h}$ and $4 \text{ km/h}$ respectively. $B$ reaches $Q$ and returns immediately and meets $A$ at $R$. The distance from $P$ to $R$ is (in $\text{km}$):

Two trains are moving in the same direction at $50 \text{ km/hr}$ and $30 \text{ km/hr}$. The faster train crosses a man sitting in the slower train in $18 \text{ seconds}$. Find the length of the faster train in meters.

Seeta and Geeta cycled towards each other,one from point $A$ and the other from point $B$,respectively. Seeta left point $A$ $6 \, hours$ later than Geeta left point $B$,and it turned out on their meeting that Seeta had travelled $12 \, km$ less than Geeta. After their meeting,they kept cycling with the same speed,and Seeta arrived at $B$ $8 \, hours$ later and Geeta arrived at $A$ $9 \, hours$ later. Find the speed of the faster cyclist in $km/hr$.

$A$ bus takes $15\, hours$ to travel from Delhi to Chandigarh. The bus travels for one-third of the total time at a speed of $80\, km/h$. It travels for $40\%$ of the remaining time at a speed of $70\, km/h$. It travels for $\frac{2}{3}$ of the remaining time at a speed of $85\, km/h$,and the rest of the time at a speed of $100\, km/h$. What is the total distance covered by the bus?

The average speed of a train is $1 \frac{3}{7}$ times the average speed of a car. The car covers a distance of $588 \text{ km}$ in $6 \text{ hours}$. How much distance will the train cover in $13 \text{ hours}$? (in $\text{km}$)