Let’s assume,
The speed of the boat in still water as x km/hr
And,
The speed of the stream as y km/hr
We know that,
Speed of the boat in upstream = (x – y) km/hr
Speed of the boat in downstream = (x + y) km/hr
So,
Time taken to cover 30 km upstream = 30/ (x − y) hr [∵ time = distance/ speed]
Time taken to cover 44 km downstream = 44/ (x + y) hr [∵ time = distance/ speed]
It’s given that the total time of journey is 10 hours. So, this can expressed as
30/ (x – y) + 44/ (x + y) = 10 …….. (i)
Similarly,
Time taken to cover 40 km upstream = 40/ (x – y) hr [∵ time = distance/ speed]
Time taken to cover 55 km downstream = 55/ (x + y) hr [∵ time = distance/ speed]
And for this case the total time of the journey is given as 13 hours.
Hence, we can write
40/ (x – y) + 55/ (x + y) = 13 ……. (ii)
Hence, by solving (i) and (ii) we get the required solution
Taking, 1/ (x – y) = u and 1/ (x + y) = v in equations (i) and (ii) we have
30u + 44v = 10
40u + 55v = 10
Which may be re- written as,
30u + 44v – 10 = 0 ……. (iii)
40u + 55v – 13 = 0……… (iv)
Solving these equations by cross multiplication we get,
Now, 1/ (x – y) = 2/10
⇒ 1 x 10 = 2(x – y)
⇒ 10 = 2x – 2y
⇒ x – y = 5 ……. (v)
And,
1/ (x + y) = 1/11
⇒ x + y = 11 ……. (vi)
Again, solving (v) and (vi)
Adding (v) and (vi), we get
2x = 16
⇒ x = 8
Using x in (v), we find y
8 – y = 5
⇒ y = 3
Therefore, the speed of the boat in still water is 8 km/hr and the speed of the stream is 3 km/hr.
