CAT 2023 Slot 3 Solved Paper

Question : 57 of 66

Marks: +1, -0

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

Solution:

Let us assume the speed of the

1 {\ st}

boat is

{\b}

, the 2 nd boat is

{\ s}

, and the river's speed is

{\r}

.
Let '

d

' be the distance between A and B.

\;⇒ d=2(b+r)

and

d=3(b-r)

\;⇒b+r=d {/} 2

and

b-r=d {/} 3⇒r=d {/} 12

(subtracting both equations).
Now, it is given that

\;\;{d}/{s+r}+\;{d}/{s-r}=6

⇒\;\;{d}/{s+\;{d}/{12}}+\;{d}/{s-\;{d}/{12}}=6

⇒\;2 d s=6 (s^2-\;{d^2}/{144})

⇒\;144 s^2-48 d s-d^2=0

Solving the quadratic equation, we get:

\;s=d (\;{ (48+{√{48^2+4(144)}}) }/{2 × 144})

\;s=d (\;{1}/{6}+\;{{√{5}}}/{12})

Required value of

\;{d}/{s+r}

\;=\;{d}/{{d}/{6}+\;{{√{5}} d}/{12}+\;{d}/{12}}

\;=\;{12}/{3+{√{5}}}=\;{(12)(3-{√{5}})}/{4}

\;=3(3-{√{5}})

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