This problem involves understanding the concepts of speed in still water and the effect of the stream's speed when moving upstream and downstream. The key is that the speed of the stream either helps (downstream) or hinders (upstream) the rower's speed in still water.

Understanding Boat and Stream Speed

When a rower is moving in a river:

• Downstream: The speed of the stream adds to the speed of the rower in still water. This makes the effective speed faster, so it takes less time to cover a certain distance.
• Upstream: The speed of the stream opposes the speed of the rower in still water. This makes the effective speed slower, so it takes more time to cover the same distance.

Calculating Upstream and Downstream Speeds

We are given the distance and time taken for both upstream and downstream journeys. We can use the basic formula: Speed = Distance / Time.

Upstream Journey:

• Distance = 40 km
• Time = 8 hours
• Upstream Speed = Distance / Time = $\frac{40 \text{ km}}{8 \text{ hours}} = 5 \text{ km/h}$

Downstream Journey:

• Distance = 40 km
• Time = 5 hours
• Downstream Speed = Distance / Time = $\frac{40 \text{ km}}{5 \text{ hours}} = 8 \text{ km/h}$

Setting Up Equations

Let:

• r be the rower's speed in still water (in km/h).
• s be the speed of the stream (in km/h).

Based on our understanding of upstream and downstream speeds, we can write two equations:

1. Upstream Speed = Rower's speed in still water - Speed of stream

$\qquad r - s = 5 \quad \text{(Equation 1)}$

2. Downstream Speed = Rower's speed in still water + Speed of stream

$\qquad r + s = 8 \quad \text{(Equation 2)}$

Solving for Rower's Speed in Still Water

We now have a system of two linear equations with two variables (r and s). We want to find the value of r (rower's speed in still water). We can solve this system by adding the two equations:

Add Equation 1 and Equation 2:

$\qquad (r - s) + (r + s) = 5 + 8$

$\qquad r - s + r + s = 13$

The -s and +s terms cancel out:

$\qquad 2r = 13$

Now, divide by 2 to find r:

$\qquad r = \frac{13}{2}$

$\qquad r = 6.5$

So, the rower's speed in still water is 6.5 km/h.

We can also find the speed of the stream if needed by substituting the value of r back into either equation. Using Equation 2:

$\qquad 6.5 + s = 8$

$\qquad s = 8 - 6.5$

$\qquad s = 1.5$

The speed of the stream is 1.5 km/h.

Final Answer Calculation

We calculated the rower's speed in still water to be 6.5 km/h.

This matches one of the given options.

Revision Table: Boat and Stream Problems

Here's a quick summary of key formulas for solving boat and stream problems:

Additional Information: Types of Speed Problems

Problems involving speed, distance, and time are common in quantitative aptitude. Understanding how relative speeds work in different scenarios is crucial.

• Basic Speed Problems: Simple application of Distance = Speed × Time.
• Relative Speed: When two bodies are moving. Speed adds if moving in opposite directions, subtracts if moving in the same direction.
• Boat and Stream Problems: A specific type of relative speed problem where the moving medium (stream) affects the speed of the object (boat/rower).
• Trains Problems: Often involve relative speed and considering the length of the trains or platforms.

Mastering these concepts helps in solving a wide range of speed-related questions.