TheCATExam
Sign in

CAT 2023 Slot 3 — QA Question 9

Boats and StreamsEasy

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?

Answer & solution

  • A

    3(3 + √5)

  • B

    3(√5 - 1)

  • 3(3 - √5)

  • D

    12(√5 - 2)

Solution

Easy

Fix a convenient distance between A and B, then read each leg as distance=speed×time\text{distance}=\text{speed}\times\text{time}. Use the faster boat's two legs to pin down the river speed rr, then use the slower boat's round-trip time to solve for its speed ss, and finally compute its downstream time A→\toB.

Let river, faster-boat and slower-boat still-water speeds be r,f,sr,f,s (km/hr). Take the distance AB=12AB=12 km (a number that divides cleanly).

1

Faster boat: A→\toB is downstream (2 h), B→\toA is upstream (3 h).

f+r=122=6(downstream)f−r=123=4(upstream)\begin{aligned} &f+r=\frac{12}{2}=6 \quad\text{(downstream)}\\ &f-r=\frac{12}{3}=4 \quad\text{(upstream)} \end{aligned}
2

Subtract to find the river speed:

(f+r)−(f−r)=6−4⇒ 2r=2 ⇒ r=1\begin{aligned} &(f+r)-(f-r)=6-4\\ &\Rightarrow\ 2r=2\ \Rightarrow\ r=1 \end{aligned}
3

Slower boat: round trip B→\toA→\toB takes 6 h. Upstream speed =s−1=s-1, downstream speed =s+1=s+1:

12s−1+12s+1=6⇒ 12 (2s)s2−1=6⇒ s2−1=4s ⇒ s2−4s−1=0\begin{aligned} &\frac{12}{s-1}+\frac{12}{s+1}=6\\ &\Rightarrow\ \frac{12\,(2s)}{s^2-1}=6\\ &\Rightarrow\ s^2-1=4s\ \Rightarrow\ s^2-4s-1=0 \end{aligned}
4

Solve the quadratic and keep the positive root:

s=4±16+42=2±5⇒ s=2+5(reject negative)\begin{aligned} &s=\frac{4\pm\sqrt{16+4}}{2}=2\pm\sqrt5\\ &\Rightarrow\ s=2+\sqrt5 \quad\text{(reject negative)} \end{aligned}
5

Time for the slower boat, A→\toB (downstream): divide distance by s+1s+1 and rationalise:

t=12(2+5)+1=123+5=12 (3−5)(3+5)(3−5)=12 (3−5)9−5=3 (3−5)\begin{aligned} &t=\frac{12}{(2+\sqrt5)+1}=\frac{12}{3+\sqrt5}\\ &=\frac{12\,(3-\sqrt5)}{(3+\sqrt5)(3-\sqrt5)}=\frac{12\,(3-\sqrt5)}{9-5}\\ &=3\,(3-\sqrt5) \end{aligned}
t=3 (3−5) hourst=3\,(3-\sqrt5)\ \text{hours}
CAT 2023 Slot 3 QA Q9: A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Anot — Solution | TheCATExam