Question

A streamer goes downstream and covers the distance between two ports in $5$ hours while it covers the same distance upstream in $6$ hours. If the speed of the stream is $1$ km/h, find the speed of the streamer in the still water and the distance between two ports.

Answer 1

Hint: If the speed of the boat is $x$ km/h and the speed of the stream is $y$ km/h, then the speed of the boat when it is moving in the direction of the stream or we can say when it is moving downstream, is $(x + y)$ km/h and if the boat is moving against the direction of the stream or we can say when it is moving upstream, then the speed of the boat is $(x - y)$ km/h.

Complete step by step answer:
 It is given that the speed of the stream is $1$ km/h. And for a certain distance between two ports a streamer goes downstream and covers the distance between two ports in $5$ hours while it covers the same distance upstream in $6$ hours.
Consider the speed of the streamer to be $x$ km/h and the distance between the two ports to be $D$ km.
Since, we are given that the speed of the boat is $1$ km/h, then if the streamer goes against the stream, that is, if it goes upstream,
Then the speed of the streamer when moving upstream is $(x - 1)$ km/h
And if the streamer moves in the direction of the stream, that is, if he goes downstream, then the speed of the streamer is $(x + 1)$ km/h.
For a certain distance $D$ between the ports, time taken by the streamer to move upstream is $6$ hr and time taken by the streamer to move downstream is $5$ hr.
Using the relation between speed, distance and time, we get
For the streamer moving upstream to cover $D$ km distance with speed $(x - 1)$ km/h in $6$ hours,
$
  (x - 1) = \dfrac{D}{6} \\
  D = 6(x - 1) - - - - - - - - - - - - - - - - (1) \\
 $
For the streamer moving downstream to cover $D$ km distance with speed $(x + 1)$ km/h in $5$ hours,
$
  (x + 1) = \dfrac{D}{5} \\
  D = 5(x + 1) - - - - - - - - - - - - - - - - (2) \\
 $
From equation (1) and (2), we have
 $
  6(x - 1) = 5(x + 1) \\
  6x - 6 = 5x + 5 \\
 $
Solve the above equation for $x$
$
  6x - 5x = 5 + 6 \\
  x = 11 \\
 $
So, the speed of the streamer is $11$ km/h.
Now we need to determine the distance between the two ports, using the value of $x$.
From equation $(1)$
$D = 6(x - 1)$
Put $x = 11$
$
  D = 6(11 - 1) \\
  D = 6 \times 10 \\
  D = 60 \\
 $
So, the distance between the two ports comes out to be $60$ km.
Hence, we found the speed of the steamer is $11$ km/h and the distance between the two ports is $60$km.

Note:
To define speed, we know that speed is the rate of change of the distance. So, when we need to determine the distance when we are provided with time and speed, then distance can be evaluated by multiplying the speed with the time.