CAT 2023 Slot 3 QA Q13 Time, Speed & Distance

Question 13 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is

A
\(3(3-\sqrt{5})\)
B
\(12(\sqrt{5}-2)\)
C
\(3(3+\sqrt{5})\)
D
\(3(\sqrt{5}-1)\)

Answer: (A) 3(3-√(5)) Time, Speed & Distance MCQ

Solution

Take AB = 12 units. Boat 1: downstream speed 6, upstream 4, so boat speed 5 and river 1 (A to B is downstream).
Boat 2 with speed c: \(\frac{12}{c-1}+\frac{12}{c+1}=6 \Rightarrow 24c = 6(c^2-1) \Rightarrow c^2-4c-1=0 \Rightarrow c=2+\sqrt5\approx4.24\).
Boat 2 is slower. Its downstream time A to B \(= \frac{12}{3+\sqrt5} = \frac{12(3-\sqrt5)}{4} = 3(3-\sqrt5)\).