A man flies a small airplane from Fargo to Bismarck, North Dakota --- a distance of 180 miles. Because he is flying into a head wind, the trip takes him 2 hours. On the way back, the wind is still blowing at the same speed, so the return trip takes only 1 hour 12 minutes. What is his speed in still air, and how fast is the wind blowing?

We are tasked with finding the airplane's speed in still air (denoted as \( x \)) and the wind speed (denoted as \( y \)). 

### Step 1: Defining variables and equations
- Distance from Fargo to Bismarck: \( d = 180 \) miles.
- Time to fly from Fargo to Bismarck (against the wind): \( t_1 = 2 \) hours.
- Time to fly from Bismarck to Fargo (with the wind): \( t_2 = 1 \, \text{hour} \, 12 \, \text{minutes} = 1.2 \, \text{hours} \).

#### Equations based on speed and distance
- When flying against the wind, the effective speed of the airplane is \( x - y \), and the time is 2 hours, so the first equation is:
  \[
  \frac{180}{x - y} = 2
  \]
  Simplifying this equation:
  \[
  x - y = \frac{180}{2} = 90 \quad \text{(Equation 1)}
  \]

- When flying with the wind, the effective speed of the airplane is \( x + y \), and the time is 1.2 hours, so the second equation is:
  \[
  \frac{180}{x + y} = 1.2
  \]
  Simplifying this equation:
  \[
  x + y = \frac{180}{1.2} = 150 \quad \text{(Equation 2)}
  \]

### Step 2: Solving the system of equations
We now have the system of two equations:
\[
x - y = 90 \quad \text{(1)}
\]
\[
x + y = 150 \quad \text{(2)}
\]

Add these two equations:
\[
(x - y) + (x + y) = 90 + 150
\]
\[
2x = 240
\]
\[
x = 120
\]

Now substitute \( x = 120 \) into either equation. Using Equation (1):
\[
120 - y = 90
\]
\[
y = 120 - 90 = 30
\]

### Final Answer:
- The airplane's speed in still air is \( 120 \) miles per hour.
- The wind speed is \( 30 \) miles per hour.

Would you like more details, or have any questions?

### Related Questions:
1. How would the problem change if the wind speed doubled?
2. What if the return trip took 1 hour instead of 1.2 hours?
3. How much time would the plane take if there was no wind at all?
4. If the wind speed increased to 40 mph, what would the new times be for each leg of the trip?
5. What would the plane's total travel time be if the wind speed was zero?

### Tip:
When dealing with wind-related problems, remember that wind affects the speed differently depending on whether it's a headwind or a tailwind, so adjust the relative speed accordingly.

Solve an algebraic problem where a man flies an airplane from Fargo to Bismarck with a headwind, determining his speed in still air and the wind speed. Learn how to apply the distance-speed-time relationship and solve systems of equations with step-by-step instructions. Suitable for Grades 9-10.

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