Flying with the wind, a small plane flew 384 mi in 2 h. Flying against the wind, the plane could fly only 366 mi in the same amount of time. Find the rate of the plane in calm air and the rate of the wind.
rate of plane    	
 mph
rate of wind	
 mph


To solve this problem, we need to set up two equations based on the information provided. Let's define:

- \( p \) as the rate of the plane in calm air (in mph),
- \( w \) as the rate of the wind (in mph).

### Step 1: Write equations for both scenarios
1. **Flying with the wind:**
   When the plane is flying with the wind, the wind helps it, so the effective speed of the plane is \( p + w \). The plane covers 384 miles in 2 hours. Using the formula:
   \[
   \text{distance} = \text{rate} \times \text{time}
   \]
   we can set up the equation:
   \[
   384 = (p + w) \times 2
   \]
   Simplifying:
   \[
   384 = 2p + 2w \quad \text{(1)}
   \]
   Dividing by 2:
   \[
   192 = p + w \quad \text{(2)}
   \]

2. **Flying against the wind:**
   When flying against the wind, the wind slows down the plane, so the effective speed is \( p - w \). The plane covers 366 miles in 2 hours. Using the same formula for distance, we set up the equation:
   \[
   366 = (p - w) \times 2
   \]
   Simplifying:
   \[
   366 = 2p - 2w \quad \text{(3)}
   \]
   Dividing by 2:
   \[
   183 = p - w \quad \text{(4)}
   \]

### Step 2: Solve the system of equations
Now, we have two equations:
1. \( p + w = 192 \)
2. \( p - w = 183 \)

We can solve this system of equations by adding them:
\[
(p + w) + (p - w) = 192 + 183
\]
\[
2p = 375
\]
\[
p = \frac{375}{2} = 187.5
\]

Now substitute \( p = 187.5 \) into one of the original equations, say \( p + w = 192 \):
\[
187.5 + w = 192
\]
\[
w = 192 - 187.5 = 4.5
\]

### Final Answer:
- The rate of the plane in calm air \( p = 187.5 \) mph.
- The rate of the wind \( w = 4.5 \) mph.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the equation change if the wind speed was doubled?
2. What if the plane took 3 hours for both flights instead of 2?
3. How does the wind speed affect the overall time for the journey?
4. Can you solve for the time it would take the plane to cover 500 miles with and against the wind?
5. How would you check if the answers are reasonable in real-world scenarios?

**Tip:** Always double-check the units you're working with, especially in problems involving speed, distance, and time.

Flying with the wind, a small plane flew 384 mi in 2 h. Flying against the wind, the plane could fly only 366 mi in the same amount of time. Find the rate of the plane in calm air and the rate of the wind.

Learn how to find the speed of a plane in calm air and the speed of the wind using systems of linear equations. This problem demonstrates the application of distance, rate, and time formulas and is suitable for Grades 9-11.

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