Question

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream 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 = $\frac{12}{\left(x-y\right)}$ hrs
Time taken to cover 40 km downstream = $\frac{40}{\left(x+y\right)}$ hrs
Total time taken = 8 hrs
∴ $\frac{12}{\left(x-y\right)}$ + $\frac{40}{\left(x+y\right)}$ = 8 ....(i)

Again, we have:
Time taken to cover 16 km upstream = $\frac{16}{\left(x-y\right)}$ hrs
Time taken to cover 32 km downstream = $\frac{32}{\left(x+y\right)}$ hrs
Total time taken = 8 hrs

∴ $\frac{16}{\left(x-y\right)}$ + $\frac{32}{\left(x+y\right)}$ = 8 ....(ii)
Putting $\frac{1}{\left(x-y\right)}=u$ and $\frac{1}{\left(x+y\right)}=v$ 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
⇒ $u=\frac{2}{8}=\frac{1}{4}$
On substituting $u=\frac{1}{4}$ in (iii), we get:
40v = 5
⇒ $v=\frac{5}{40}=\frac{1}{8}$
Now, we have:
$u=\frac{1}{4}$
⇒ $\frac{1}{x-y}=\frac{1}{4}\Rightarrow x-y=4$ ....(v)
$v=\frac{1}{8}$
⇒ $\frac{1}{x+y}=\frac{1}{8}\Rightarrow x+y=8$ ....(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 = 6 km/h
And, speed of the stream = 2 km/h