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# A man travels by a motor boat down a river to his office and back. Wit

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Math Expert
Joined: 02 Sep 2009
Posts: 64068
A man travels by a motor boat down a river to his office and back. Wit  [#permalink]

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02 Apr 2020, 08:50
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Difficulty:

55% (hard)

Question Stats:

46% (01:27) correct 54% (02:03) wrong based on 13 sessions

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A man travels by a motor boat down a river to his office and back. With the speed of the river unchanged, if he doubles the speed of his motor boat, then his total travel time gets reduced by 75%. The ratio of the original speed of the motor boat to the speed of the river is:

A. √6 : √2
B. √7 : 2
C. 2√5 : 3
D. 3 : 2
E. √5 : 3

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Re: A man travels by a motor boat down a river to his office and back. Wit  [#permalink]

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02 Apr 2020, 21:22
Bunuel wrote:
A man travels by a motor boat down a river to his office and back. With the speed of the river unchanged, if he doubles the speed of his motor boat, then his total travel time gets reduced by 75%. The ratio of the original speed of the motor boat to the speed of the river is:

A. √6 : √2
B. √7 : 2
C. 2√5 : 3
D. 3 : 2
E. √5 : 3

Let the speed of river = $$u$$, speed of the boat = $$v$$ & distance between office and home = $$d$$

Speed during downstream = $$v + u$$ & that of upstream = $$v - u$$
--> time taken, $$t = \frac{d}{v + u} + \frac{d}{v - u}$$
--> $$t = d[\frac{1}{v + u} + \frac{1}{v - u}] = d[\frac{v - u + v + u}{(v - u)(v + u)}]$$
--> $$t = \frac{2vd}{v^2 - u^2}$$ ....... (1)

If he doubles the speed of his motor boat, then his total travel time gets reduced by 75%
--> Time taken = $$\frac{t}{4} = \frac{d}{2v + u} + \frac{d}{2v - u}$$
--> $$t = 4d[\frac{1}{2v + u} + \frac{1}{2v - u}] = 4d[\frac{2v - u + 2v + u}{(2v - u)(2v + u)}]$$
--> $$t = \frac{16vd}{4v^2 - u^2}$$ ....... (2)

From (1) & (2),
$$\frac{2vd}{v^2 - u^2} = \frac{16vd}{4v^2 - u^2}$$
--> $$\frac{1}{v^2 - u^2} = \frac{8}{4v^2 - u^2}$$
--> $$4v^2 - u^2 = 8v^2 - 8u^2$$
--> $$7u^2 = 4v^2$$
--> $$\frac{v^2}{u^2} = \frac{7}{4}$$
--> $$\frac{v}{u} = \frac{\sqrt{7}}{2}$$

Option B
Re: A man travels by a motor boat down a river to his office and back. Wit   [#permalink] 02 Apr 2020, 21:22