Let’s solve the given problems step by step:

Problem Details:

• Speed upstream: 8 km/h
• Speed downstream: 16 km/h

We aim to calculate:

1. The speed of the boat in still water.
2. The speed of the stream.
3. The mathematical concept used in the problem.
4. The average speed of the stream and the boat in still water.

1. Speed of the Boat in Still Water:

The speed of the boat in still water $V_b$ is the average of the upstream and downstream speeds: $V_b = \frac{\text{Speed downstream} + \text{Speed upstream}}{2}$ $V_b = \frac{16 + 8}{2} = \frac{24}{2} = 12 \, \text{km/h}$

Thus, the speed of the boat in still water is 12 km/h (option c).

2. Speed of the Stream:

The speed of the stream $V_s$ is half the difference between the downstream and upstream speeds: $V_s = \frac{\text{Speed downstream} - \text{Speed upstream}}{2}$ $V_s = \frac{16 - 8}{2} = \frac{8}{2} = 4 \, \text{km/h}$

Thus, the speed of the stream is 4 km/h (option b).

3. Mathematical Concept:

This problem involves the relative speed concept, where:

• Speed downstream = $\text{Speed of boat in still water} + \text{Speed of stream}$,
• Speed upstream = $\text{Speed of boat in still water} - \text{Speed of stream}$.

Thus, the concept used is Relative Speed.

4. Average Speed of Stream and Boat in Still Water:

The average speed of the stream and boat in still water is: $\text{Average Speed} = \frac{V_b + V_s}{2}$ $\text{Average Speed} = \frac{12 + 4}{2} = \frac{16}{2} = 8 \, \text{km/h}$

Thus, the average speed is 8 km/h.

Final Answers:

1. Speed of the boat in still water: 12 km/h (option c).
2. Speed of the stream: **4 km/h