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

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