CAT 2022 Slot 2 · Quant · Question 22

South and west are at right angles, so the two ships' paths form the legs of a right triangle.

Let the slower ship travel at v km/h, so the faster one travels at v+6.

After two hours the slower has covered 2v km and the faster 2\left( v+6 \right) km.

The distance between them is the hypotenuse, which is 60 km.

By Pythagoras, {\left( 2v \right)}^{2}+{\left( 2v+12 \right)}^{2}={60}^{2}.

Expanding, 4{v}^{2}+4{v}^{2}+48v+144=3600.

Collecting terms, 8{v}^{2}+48v-3456=0.

Dividing by 8, {v}^{2}+6v-432=0.

The discriminant is 36+1728=1764, whose square root is 42.

So v=\frac{-6+42}{2}=18, taking the positive root.

The slower ship travels at 18 km/h — the second option.

Checking: the distances are 36 km and 48 km, and {36}^{2}+{48}^{2}=1296+2304=3600={60}^{2}.

Shortcut. A hypotenuse of 60 with legs in a whole-number ratio points at the 3-4-5 triple scaled by 12, giving legs 36 and 48. Those are two hours of travel, so the speeds are 18 and 24 — differing by 6, exactly as required.