Antonio flies from houston to philadelphia, a distance of about 1340 miles. his plane travels with the wind and takes 2 hours and 20 minutes. at the same time, paul is on a plane from philadelphia to houston. since his plane is heading against the wind, paul's flight takes 2 hours and 50 minutes. what was the speed of the wind in miles per hour?

I'll pick variables and set up a system. In this case, I'll use "p" for "the plane's speedometer reading (apparent speed) " and "w" for "the windspeed". When the plane is going "with" the wind (when it has a "tailwind"), the two speeds will add together; when the plane is going "against" the wind (when it has a "headwind"), the windspeed will be subtracted from the plane's speedometer reading (that is, from the engines' actual output).

In each case, the "distance" equation will be " (the combined speed) times (the time at that speed) equals (the total distance travelled) ":

with the wind: (p + w) (2.33) = 1340

against the wind: (p - w) (2.833) = 1340

Rather than multiply through, I notice that, if I divide off the 2.33 and the 2.833, I'll have a system that's already set for solving by addition

p + w = 575.1

p - w = 472.99

Then, by adding down, 2p = 1048 so p = 524, and w must then be 51.1

wind speed is 51.1 mph