Steamer Speed Calculation

This problem involves calculating the speed of a steamer in still water given its travel times downstream and upstream, and the speed of the stream.

Understanding the Concepts

• Let the speed of the steamer in still water be $s$ km/h.
• Let the speed of the stream be $v$ km/h.
• The speed of the steamer downstream (with the current) is $(s + v)$ km/h.
• The speed of the steamer upstream (against the current) is $(s - v)$ km/h.
• The distance covered is the same in both directions.
• Distance = Speed × Time.

Setting Up the Equations

We are given:

• Time downstream, $t_{down} = 6$ h
• Time upstream, $t_{up} = 8$ h
• Speed of the stream, $v = 3$ km/h

Let $d$ be the distance between the two ports.

Downstream distance: $d = (s + v) \times t_{down}$

Upstream distance: $d = (s - v) \times t_{up}$

Since the distance $d$ is the same:

$ (s + v) \times t_{down} = (s - v) \times t_{up} $

Solving for Steamer Speed ($s$)

1. Substitute the known values into the equation:

   $ (s + 3) \times 6 = (s - 3) \times 8 $

2. Expand both sides of the equation:

   $ 6s + 18 = 8s - 24 $

3. Rearrange the terms to solve for $s$:

   $ 18 + 24 = 8s - 6s $

   $ 42 = 2s $

4. Calculate the value of $s$:

   $ s = \frac{42}{2} $

   $ s = 21 $

The speed of the steamer in still water is 21 km/h.