Time, Speed and Distance Question 4 - CAT 2023
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:
Step-by-Step Explanation
Let the distance between port A and port B be d, the speed of the first boat be b, the speed of the slower boat be s, and the river current speed be r.
Analyze Boat 1:
Downstream:
d/(b + r) = 2 ⟹ b + r = d/2
Upstream:
d/(b − r) = 3 ⟹ b − r = d/3
Subtracting the two equations allows us to solve for r:
2r = d/2 − d/3 = d/6
r = d/12 ⟹ d = 12r
Analyze Boat 2 (Slower boat round trip):
The total round trip time is 6 hours:
d/(s + r) + d/(s − r) = 6
Substitute d = 12r into the expression:
12r/(s + r) + 12r/(s − r) = 6
Solving gives:
s² − 4rs − r² = 0
Using the quadratic formula to solve for s in terms of r:
s = (4r + √(16r² + 4r²))/2
= (4r + √20r²)/2
= (2 + √5)r
Calculate the downstream time for the slower boat from A to B:
Time = d/(s + r)
= 12r/[(2 + √5)r + r]
= 12/(3 + √5)
Rationalize the denominator by multiplying the numerator and denominator by (3 − √5):
Time = 12(3 − √5)/(9 − 5)
= 12(3 − √5)/4
= 3(3 − √5) hours
Correct Option: D
💡 FundaMakers Tip:
When handling multi-boat river problems, express all individual speeds as ratios relative to the river current r. Finding d = 12r early on simplifies the variables down to a single system, allowing the final fractional expression to cancel out smoothly.