A motorboat takes 4 hours to travel 192 kilometers going upstream. The return trip takes 3 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?

Let's solve this problem by using the formula:

\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

### Let:
- \( b \) = speed of the boat in still water (in km/h)
- \( c \) = speed of the current (in km/h)

#### Upstream:
- When the boat is going upstream, the current slows it down, so its effective speed is \( b - c \).
- Time taken upstream: 4 hours
- Distance traveled: 192 kilometers
- Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we get:
  \[
  192 = (b - c) \times 4
  \]
  Simplifying:
  \[
  b - c = \frac{192}{4} = 48 \quad \text{(Equation 1)}
  \]

#### Downstream:
- When the boat is going downstream, the current helps it, so its effective speed is \( b + c \).
- Time taken downstream: 3 hours
- Distance traveled: 192 kilometers
- Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we get:
  \[
  192 = (b + c) \times 3
  \]
  Simplifying:
  \[
  b + c = \frac{192}{3} = 64 \quad \text{(Equation 2)}
  \]

### Now, solve the system of equations:
1. \( b - c = 48 \)
2. \( b + c = 64 \)

#### Adding the two equations:
\[
(b - c) + (b + c) = 48 + 64
\]
\[
2b = 112
\]
\[
b = \frac{112}{2} = 56
\]

#### Substitute \( b = 56 \) into Equation 1:
\[
56 - c = 48
\]
\[
c = 56 - 48 = 8
\]

### Final Answer:
- The speed of the boat in still water, \( b \), is **56 km/h**.
- The speed of the current, \( c \), is **8 km/h**.

Would you like further details or have any questions? Here are 5 related questions you can explore:

1. How would the problem change if the upstream and downstream times were the same?
2. What if the boat traveled a different distance downstream but took the same time?
3. How do we calculate average speed if the times for upstream and downstream are different?
4. How can we calculate the total time taken for a round trip considering current?
5. What is the significance of knowing the speed of the current in real-world navigation?

**Tip:** Always double-check units and verify if speeds or times are given in consistent measures before solving!

Learn how to determine the speed of a boat in still water and the speed of the current using algebraic methods. This example problem involves solving a system of equations based on upstream and downstream travel times. Suitable for students in Grades 8-10.

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