jane took 30 min to drive her boat upstream to​ water-ski at her favorite spot. Coming back later in the​ day, at the same boat​ speed, took her 15 min. If the current in that part of the river is 4 km per​ hr, what was her boat speed in still​ water?

Let's solve the problem step by step.

### Definitions and Variables:
- Let \( v_b \) be the boat's speed in still water (in km/h).
- The speed of the current is \( v_c = 4 \) km/h.
- The time taken to go upstream (against the current) is \( t_u = 30 \) minutes = \( 0.5 \) hours.
- The time taken to go downstream (with the current) is \( t_d = 15 \) minutes = \( 0.25 \) hours.

### Formulas:
1. **Upstream speed**: The effective speed of the boat going upstream is \( v_b - v_c \).
2. **Downstream speed**: The effective speed of the boat going downstream is \( v_b + v_c \).

### Distance Calculation:
Since the distance traveled upstream and downstream is the same, we can set up the following equation:

\[
\text{Distance upstream} = \text{Distance downstream}
\]

\[
(v_b - 4) \times 0.5 = (v_b + 4) \times 0.25
\]

### Solve for \( v_b \):
Now, distribute and solve the equation:

\[
0.5v_b - 2 = 0.25v_b + 1
\]

To eliminate the fractions, multiply the entire equation by 4:

\[
2v_b - 8 = v_b + 4
\]

Subtract \( v_b \) from both sides:

\[
v_b - 8 = 4
\]

Add 8 to both sides:

\[
v_b = 12 \text{ km/h}
\]

### Final Answer:
Jane's boat speed in still water is **12 km/h**.

Would you like further details on this solution or have any questions?

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**Related Questions:**
1. What is the total distance Jane traveled during her trip?
2. If the current speed increased to 5 km/h, how would that affect Jane's travel times?
3. What would Jane's travel time be upstream if the current was 6 km/h instead of 4 km/h?
4. How much time would it take Jane to travel 10 km downstream with the current speed of 4 km/h?
5. If Jane doubled her boat speed in still water, how would her upstream and downstream times change?

**Tip:** Understanding how to manipulate and solve equations with different variables is key in word problems, especially those involving rates and time.

Learn how to calculate the boat speed in still water for Jane's river journey using speed, distance, and time principles. This problem involves understanding relative motion with a river current and solving for the boat's speed in still water. Suitable for students in Grades 7-9.

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