A motorboat travels 110 kilometers in 2 hours going upstream. It travels 154 kilometers going downstream in the same amount of time. What is the rate of the boat in still water and what is the rate of the current?

To solve this problem, we can set up a system of equations based on the given information. Let's denote:

- \( b \) as the rate of the boat in still water (in km/h).
- \( c \) as the rate of the current (in km/h).

### Step 1: Set Up the Equations
When the boat is traveling upstream, the effective speed of the boat is reduced by the speed of the current. Thus, the upstream speed is \( b - c \).

When the boat is traveling downstream, the effective speed of the boat is increased by the speed of the current. Thus, the downstream speed is \( b + c \).

We are given that:

- The boat travels 110 km upstream in 2 hours.
- The boat travels 154 km downstream in 2 hours.

Using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), we can write two equations:

1. **Upstream equation:**
   \[
   \frac{110}{2} = b - c
   \]
   Simplifies to:
   \[
   55 = b - c \quad \text{(Equation 1)}
   \]

2. **Downstream equation:**
   \[
   \frac{154}{2} = b + c
   \]
   Simplifies to:
   \[
   77 = b + c \quad \text{(Equation 2)}
   \]

### Step 2: Solve the System of Equations
We can solve these two equations simultaneously.

**Add Equation 1 and Equation 2:**

\[
55 + 77 = (b - c) + (b + c)
\]
\[
132 = 2b
\]
\[
b = 66 \, \text{km/h}
\]

**Subtract Equation 1 from Equation 2:**

\[
77 - 55 = (b + c) - (b - c)
\]
\[
22 = 2c
\]
\[
c = 11 \, \text{km/h}
\]

### Step 3: Conclusion
- The rate of the boat in still water \( b \) is **66 km/h**.
- The rate of the current \( c \) is **11 km/h**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the results change if the distances were different but the time remained the same?
2. How would you solve this problem if the boat traveled for 3 hours upstream instead of 2?
3. What if the current’s speed increased by 5 km/h? How would that affect the boat's travel times upstream and downstream?
4. How can you determine the time it would take the boat to travel a certain distance downstream with the same current?
5. How does the presence of current affect the energy consumption of the boat?

**Tip:** When solving problems involving rates and currents, always break down the problem into upstream and downstream components to form your equations.

Learn how to calculate the speed of a boat in still water and the rate of the current using a system of equations. This problem involves finding speeds based on distances traveled upstream and downstream in given times. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation