Question

A boat takes 10 hours to travel 30 km upstream and 44 km downstream, but it takes 13 hours to travel 40 km upstream and 55 km downstream. Find the speed of the boat in still water and the speed of the stream.

Answer 1

Let the speed of the boat in still water be x km/hr and the speed of the stream by y km/hr.

Therefore, the speed of the boat downstream = (x + y) km/hr and the speed of the boat upstream

= (x - y) km/hr

Now, time = `"distance"/"speed"`

Therefore, time taken by the boat to cover 30 km upstream = `30/(x -  y)` hours and the time taken by the boat to cover 24km downstream = `44/(x + y)` hours.

But the total time is taken by the boat to cover 30 km upstream and 44 km downstream is 10 hours.

∴ `30/(x-y) + 44/(x + y) = 10` ...(i)

similarly by second condition,

`40/(x-y) + 55/(x - y) = 13` ....(ii)

substituting `1/(x - y) = a` and `1/(x - y) = b` in equation (i) and (ii)

∴ 30a + 44b = 10 ....(iii)

40a + 55b = 13 ....(iv)

Equation (iii) x (iv) and eqation (iv) x (iii), we get

120a + 176b = 40 ....(v)

120a + 165b = 39 ...(vi)

equation (v) – equation (vi) , we get

11b = 1

`b = 1/11`

substituting `b = 1/11` in equation (v), we get

`120a + 176(1/11) = 40`

120a = 40 - 16

120a = 24

`a = 24/120`

`a = 1/5`

Now,`1/(x - y) = 1/5` and `1/(x+y) = 1/11`

x - y = 5 and x + y = 11

x + y = 11 .....(vii)

x - y = 5 ......(viii)

Adding equation (vii) and equation (viii) , we get

2x = 16

x = 8

Subsidity x = 8 in equation (vii) we get y = 3

∴ speed of the boat in still water is 8 km/hr and speed of the stream is 3 km/hr