When Amar walks from $P$ to $Q$ and returns from $Q$ to $P$ on a cycle, it takes him $29$ minutes. By walking both ways, it takes him $49$ minutes. If he rides a cycle both ways, then what will be the total time (in minutes)?

Let the time taken to walk one way be $W$ and the time taken to cycle one way be $C$.From the first condition (walking one way + cycling one way):
$$W+C=29\text{— (Equation 1)}$$
From the second condition (walking both ways):
$$2W=49⟹W=24.5\text{ minutes}$$
Substitute the value of $W$ into Equation 1:
$$24.5+C=29$$
$$C=29-24.5=4.5\text{ minutes}$$
We need to find the total time taken if he cycles both ways ($2C$):
$$2C=2\times 4.5=9\text{ minutes}$$