A plane has a cruising speed of 200 miles per hour when there is no wind. At this speed, the plane flew 500 miles with the wind in the same amount of time it flew 300 miles against the wind. Find the speed of the wind. Speed of the wind: miles/hour

To find the speed of the wind, let’s denote it as \( w \) miles per hour. When the plane is flying with the wind, its effective speed is \( 200 + w \) miles per hour, and against the wind, it is \( 200 - w \) miles per hour. The time taken to fly with the wind for 500 miles is: \[ \text{Time with wind} = \frac{500}{200 + w} \] And the time taken to fly against the wind for 300 miles is: \[ \text{Time against wind} = \frac{300}{200 - w} \] Since both times are equal, we can set up the following equation: \[ \frac{500}{200 + w} = \frac{300}{200 - w} \] Cross-multiplying gives: \[ 500(200 - w) = 300(200 + w) \] Expanding both sides results in: \[ 100000 - 500w = 60000 + 300w \] Combining like terms yields: \[ 100000 - 60000 = 500w + 300w \] \[ 40000 = 800w \] Thus, we can solve for \( w \): \[ w = \frac{40000}{800} = 50 \] So, the wind speed is \( 50 \) miles per hour.