The problem states:

• The speed of the stream is $2 \, \text{mph}$.
• A boat travels $12 \, \text{miles}$ upstream and $16 \, \text{miles}$ downstream in the same amount of time.
• We are asked to find the speed of the boat in still water, denoted as $v_b \, \text{mph}$.

Let’s define the key relationships:

1. Upstream speed: The boat's speed against the current is $v_b - 2 \, \text{mph}$ (since the stream slows the boat down).
2. Downstream speed: The boat's speed with the current is $v_b + 2 \, \text{mph}$ (since the stream pushes the boat forward).
3. Time equation: Since the boat takes the same amount of time going upstream and downstream, we can equate the time it takes for both trips:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Thus, we can set up the following equation:

$\frac{12}{v_b - 2} = \frac{16}{v_b + 2}$

Solving this equation:

We'll cross-multiply to solve for $v_b$:

$12(v_b + 2) = 16(v_b - 2)$

Expanding both sides:

$12v_b + 24 = 16v_b - 32$

Now, solve for $v_b$:

$24 + 32 = 16v_b - 12v_b$ $56 = 4v_b$ $v_b = 14 \, \text{mph}$

Thus, the speed of the boat in still water is 14 mph.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. What would happen to the upstream speed if the stream’s speed increased to 4 mph?
2. How would the equation change if the boat traveled the same distance upstream and downstream?
3. What is the total time taken for the boat to complete the upstream and downstream trips?
4. How would the boat's speed in still water be affected if it traveled 10 miles upstream and 18 miles downstream?
5. What would the speed of the boat in still water be if the current was 3 mph instead of 2 mph?

Tip: Always set up the time equation carefully by ensuring you account for the effect of the current on both upstream and downstream speeds.