Two friends start at the same time from cities P and Q towards cities Q and P respectively. After meeting at some point between P and Q, they reach their destinations in 54 and 24 minutes respectively. In what time did the one who moved from Q to P complete his journey?

Let the two friends be “M” who will be travelling from P to Q and “N” who will be travelling from Q to P.

Let us consider the speed of M be “x” and the speed of N be “y”. Also, let them meet after “t” minutes.

Case 1: Before M & N meets each other

Distance traveled by $$M = x \times(t)$$

Distance traveled by $$N= y\times(t)$$

Case 2: After M & N crosses each other

Distance traveled by $$M = x \times(54)$$

Distance traveled by $$N= y\times(24)$$

Now,  

(Distance traveled by M & N before meeting each other) = (Distance traveled by M & N after crossing each other)

∴ $$x \times t = x\times 54$$ ….. (i)

And, $$y \times t = y \times24$$ …. (ii)

Multiplying equation (i) & (ii), we get

$$xy \times t^2 = xy \times 54 \times 24$$

or, t² = 1296

or, t = 36

Since we are required to find the time taken by N to travel the entire journey from city Q to P, therefore, N would take 36 + 24 = 60 minutes to complete his entire journey.