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.
Answers
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
Answer:
Speed of stream = 3 km / hr.
Speed of boat in still water = 8 km / hr.
Step-by-step explanation:
Let the speed of the boat in still water be a km / hr and stream be b km / hr
For upstream = a - b
For downstream = a + b
We know :
Speed = Distance / Time
Case 1 .
10 = 30 / a - b + 44 / a + b
Let 1 / a - b = x and 1 / a + b = y
30 x + 44 y = 10 ... ( i )
Case 2 .
13 = 40 / a - b + 55 / a + b
40 x + 55 y = 13 ... ( i )
Multiply by 4 in ( i ) and by 3 in ( ii )
120 x + 176 y = 40
120 x = 40 - 176 y ... ( iii )
120 x + 165 y = 39
120 = 39 - 165 y ... ( iv )
From ( iii ) and ( iv )
40 - 176 y = 39 - 165 y
11 y = 1
y = 1 / 11
120 x = 40 - 176 y
120 x = 40 - 176 / 11
x = 1 / 5
Now :
1 / a - b = 1 / 5
a - b = 5
a = 5 + b ... ( v )
1 / a + b = 1 / 11
a + b = 11
a = 11 - b ... ( vi )
From ( v ) and ( vi )
11 - b = 5 + b
2 b = 6
b = 3
a = 5 + b
a = 5 + 3
a = 8
Hence we get answer.