CAT 2023 Slot 1 · Quant · Question 14

Let the two travel for T hours before meeting, and let their speeds be u for Arvind and v for Surbhi.

At the meeting point, the stretch Arvind still has left is exactly the stretch Surbhi has already covered, namely vT.

He covers that stretch in 6 hours, so 6u=vT.

Likewise the stretch Surbhi has left is the one Arvind has covered, uT, and she takes 24 hours, so 24v=uT.

Dividing the two relations, \frac{6u}{24v}=\frac{vT}{uT}=\frac{v}{u}.

So \frac{{u}^{2}}{{v}^{2}}=\frac{24}{6}=4, giving \frac{u}{v}=2.

With Arvind at 54 km/h, Surbhi travels at \frac{54}{2}=27 km/h.

Substituting into 6u=vT: 6\times 54=27T, so T=\frac{324}{27}=12 hours.

The total distance is covered by both together in those 12 hours: \left( 54+27 \right)\times 12.

That is 81\times 12=972 km.

Shortcut. Two standard results settle this instantly: the speed ratio is \sqrt{\frac{24}{6}}=2, and the time to the meeting is \sqrt{6\times 24}=12 hours. Then the distance is just \left( 54+27 \right)\times 12=972.