# A boat goes 30 km upstream and 44 km downstream in 10 hours .In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.

1
by tavsubhavi

2016-04-08T18:43:32+05:30
Let the speed of boat in still water = x km/h
Let the speed of stream = y km/h

Obviously, x>y.  (Otherwise the question cannot be solved)

Upstream                                         Downstream

velocity, v = x - y km/h                       v = x + y km/h
distance, d = 30 km                           d = 44 km

velocity = distance/time                     v = d/t
∴ v = d/t                                            ∴x + y =  44/t
∴ x - y = 30/t                                     ∴t = 44 / (x+y) hours
∴ time t = 30 / (x-y) hours
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Now, the boat takes 10 hours to travel 30 km upstream and 44 km downstream.

∴ 30 / (x-y) + 44 / (x+y)  = 10 ------------------- (1)

Similarly the boat takes 13 hours to travel 40 km upstream and 55 km downstream
∴ 40 / (x-y) + 55 / (x+y) = 13 ---------------- (2)

Let 1/(x-y) = a   and  1/(x+y) = b

So, for equation (1)

30a + 44b = 10
∴2 (15a + 22b) = 10
∴15a + 22b = 5 ---------------(3)

For equation (2)
40a + 55b = 13 ---------------(4)

Solving equations (3) and (4) by Elimination method.

15a + 22b = 5     Equation (3) * 8
40a + 55b = 13   Equation (4) * -3

∴120a + 176b = 40
-120a - 165b = -39      Adding both equations
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∴ 11b = 1
b = 1/11

Putting b = 1/11 in equation (3)
∴ 15a + 22 (1/11) = 5
∴ 15a + 2 = 5
∴15a = 5 - 2
∴15a = 3
∴a = 3/15
a = 1/5

Now,
a = 1/5                                      and     b = 1/11
∴1/(x-y) = 1/5                                     ∴1/(x+y) = 1/11
∴x - y = 5 ---------(5)                           ∴x + y = 11 ------------(6)

Solving equations (5) and (6) by Elimination Method,

x - y = 5
x + y = 11                 Adding (5) and (6)
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∴2x = 16
x = 8

Putting x = 8 in equation (6)
∴ 8 + y = 11
∴y = 11 - 8
y = 3

Thus,
Speed of boat in still water = x = 8 km/h
Speed of stream = y = 3 km/h

But it might have helped you
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