Question

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. 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/h and the speed of the stream be y km/h.
Then we have
Speed upstream = (x – y) km/hr
Speed downstream = (x + y) km/hr
Time taken to cover 12 km upstream = `12/((x−y)` hrs
Time taken to cover 40 km downstream = `40/((x+y))` hrs
Total time taken = 8 hrs
`∴ 12/((x−y)) + 40/((x+y)) = 8 `              ……….(i)
Again, we have:

Time taken to cover 16 km upstream = `16/((x−y))` hrs
Time taken to cover 32 km downstream = `32/((x+y)` hrs
Total time taken = 8 hrs
`∴ 16/((x−y) )+ 32/((x+y)) = 8`     ……….(ii)
Putting `1/((x−y)) = u and 1/((x+y))` = v in (i) and (ii), we get:
12u + 40v = 8
3u + 10v = 2                              ………(a)
And, 16u + 32v = 8
⇒2u + 4v = 1                           ……….(b)
On multiplying (a) by 4 and (b) by 10, we get:
12u + 40v = 8                     …….(iii)
And, 20u + 40v = 10           …….(iv)
On subtracting (iii) from (iv), we get:
8u = 2
⇒ u = `2/8 = 1/4`
On substituting u = `1/4` in (iii), we get:
40v = 5
⇒ v =` 5/40 = 1/8`
Now, we have:
u = `1/4`
⇒ `1/((x−y)) = 1/4 ⇒ x – y = 4`                        ……..(v)
v = `1/8`
⇒` 1/((x+y)) = 1/8 ⇒ x + y = 8`                   ……..(vi)
On adding (v) and (vi), we get:
2x = 12
⇒ x = 6
On substituting x = 6 in (v), we get:
6 – y = 4
y = (6 – 4) = 2
∴ Speed of the boat in still water = 6km/h
And, speed of the stream = 2 km/h