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-steam . Determine the speed of the stream and that of the boat in still water .
Answers
Given
In 10 hours the boat covers a distance of 30 km upstream and 44 km downstream. ------(1)
And in 13 hours the boat covers a distance of 40 km upstream and 55 km downstream. ------(2)
Now let the speed of boat in still water = x km/h
and let the speed of stream = y km/h
Thus downstream speed = x+y and
upstream speed = x- y
We know
Speed = Distance/ Time
Time = Distance/ Speed.
For condition 1 we have
(Upstream distance/ speed upstream) +(downstream distance/ speed downstream) = 10 hour
(30/x-y)+(44/x+y) = 10 -----(A)
For condition 2
(40/x-y) + (55/x+y) = 13 -----(B)
considering (1/x-y) = a and (1/x+y) = b -------(3)
Thus A and B can be written as,
30 a + 44 b = 10 -----(C)
40 a + 55 b = 13 -----(D)
Subtracting C and D we get
C × 4 ----> 120 a +176 b = 40
D × 3 ----> 120 a + 165 b =39
b = 1/11 -----(E)
Putting E in C we get,
30 a + 44(1/11) = 10
a = 6/30 = 1/5
Reducing the above found values in 3 we get,
(1/x - y) = 1/5 (1/x+y) =1/11
x -y = 5 x + y = 11
x = 5+ y 5 + y + y =11
x = 5+ 3 y = 6/2 = 3 km/h
x = 8 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.