Two trains $A$ and $B$ start at the same time from Delhi and Patna respectively towards each other. They meet after $16 \, h$. If the distance between Delhi and Patna is $1872 \, km$ and train $B$ runs $27 \, km/h$ faster than train $A$,what is the speed of train $A$ in $km/h$?

$A$ train running at the speed of $84 \, km/h$ passes a man walking in the opposite direction at the speed of $6 \, km/h$ in $4 \, s$. What is the length of the train (in $meters$)?

$A$ car driver leaves Bangalore at $8:30 \, am$ and expects to reach a place $300 \, km$ from Bangalore at $12:30 \, pm$. At $10:30 \, am$, he finds that he has covered only $40 \%$ of the distance. By how much does he have to increase the speed of the car in order to keep up with his schedule? (in $km/h$)

$A$ man walks at the rate of $5 \, km/hr$ for $6 \, hours$ and at $4 \, km/hr$ for $12 \, hours$. The average speed of the man (in $km/hr$) is:

Heena and Renu are competing in a $100\, m$ race. Initially,Heena runs at twice the speed of Renu for the first $50\, m$. After the $50\, m$ mark,Heena runs at $\frac{1}{4}$ of her initial speed,while Renu continues to run at her original speed. If Renu catches up with Heena at a distance of $N$ meters from the finish line,then $N$ is equal to (in meters).

$A$ dog finds a cat at $25$ leaps away. The cat sees the dog coming towards it and starts running with the dog in hot pursuit. In every minute, the dog makes $5$ leaps and the cat makes $6$ leaps, and one leap of the dog is equal to $2$ leaps of the cat. Find the time in which the dog will catch the cat (in minutes).