A motorboat covers a given distance in 6 hours moving downstream on a river. It covers the same distance in 10 hours moving upstream. The time it takes to cover the same distance in still water is
A
9 hours
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B
7.5 hours
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C
6.5 hours
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D
8 hours
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Solution
The correct option is B 7.5 hours Let Vbw and Vw be the velocities of motorboat and water in the river respectively. Let the distance covered by the motorboat in both the case be d.
Case 1: While moving downstream d6=Vbw+Vw .... (i)
Case 2: While moving upstream d10=Vbw−Vw ..... (ii)
Dividing (i) by (ii) 5/3=(Vbw+Vw)/(Vbw−Vw) ⇒5Vbw−5Vw=3Vbw+3Vw ⇒2Vbw=8Vw
Time taken in still water =d/Vbw
Putting value of Vw in (i) d6=Vbw+Vbw4d6=(5Vbw)4 ⇒dVbw=52×3=7.5 hours
Alternative :
Let t1 be the time taken by the motorboat to travel some distance in still water.
Let t2 be the time taken by the motorboat to cover same distance when flowing with the river.
Let t3 and t4 be the time taken by the motorboat to cover same distance while moving downstream and upstream respectively. ⟹t3=(t1t2)(t1+t2)=6
and t4=(t1t2)(t2−t1)=10 ⇒(t1+t2)(t2−t1)=106=53⇒t1t2=14⇒t1=t24
Hence, 6=(4t21)(5t1)⇒t1=7.5 hours