A plane traveled 315 miles each way to Cairo and back.  The trip there was with the wind.  It
took 3 hours.  The trip back was into the wind.  The trip back took 5 hours.  Find the speed of the
plane in still air and the speed of the wind.


To solve this problem, we need to use the concept of relative speed. Let:

- \( p \) represent the speed of the plane in still air (in miles per hour),
- \( w \) represent the speed of the wind (in miles per hour).

### Given:
- The distance to Cairo and back is 315 miles each way.
- The trip with the wind took 3 hours, and the trip against the wind took 5 hours.

The speed of the plane with the wind is \( p + w \), and the speed of the plane against the wind is \( p - w \). Using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), we can set up two equations based on the time for each leg of the trip.

1. **With the wind:**
   \[
   p + w = \frac{315}{3} = 105 \, \text{miles per hour}
   \]

2. **Against the wind:**
   \[
   p - w = \frac{315}{5} = 63 \, \text{miles per hour}
   \]

### Now, we have the system of equations:
\[
\begin{aligned}
p + w &= 105 \, \text{(1)} \\
p - w &= 63 \, \text{(2)}
\end{aligned}
\]

### Solving the system:
Add the two equations together:
\[
(p + w) + (p - w) = 105 + 63
\]
\[
2p = 168
\]
\[
p = 84
\]
Now, substitute \( p = 84 \) into equation (1):
\[
84 + w = 105
\]
\[
w = 105 - 84 = 21
\]

### Solution:
- The speed of the plane in still air is **84 miles per hour**.
- The speed of the wind is **21 miles per hour**.

Would you like any additional details or clarifications on the solution?

Here are 5 related questions to expand on this topic:
1. What would happen if the wind speed increased by 10 mph?
2. How would the travel time change if the plane's speed in still air was 100 mph?
3. Can you calculate the total time for the entire round trip with the wind and against the wind?
4. What would the wind speed be if the trip with the wind took 4 hours instead of 3?
5. How would the plane’s speed affect fuel consumption if the wind speed remains constant?

**Tip:** When solving problems involving relative speeds, always break the problem into different segments (e.g., with wind vs. against wind) to create simple equations.

A plane traveled 315 miles each way to Cairo and back. The trip there was with the wind. It took 3 hours. The trip back was into the wind. The trip back took 5 hours. Find the speed of the plane in still air and the speed of the wind.

Learn how to solve a system of equations to find the speed of a plane in still air and the speed of the wind. A plane travels 315 miles each way, taking 3 hours with the wind and 5 hours against it. Suitable for students in Grades 9-11.

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