Question 197828
Let w = speed of the wind (in miles per hour)



Note: if a plane's speed in still air is 180 mph, then the speed with the wind "w" is 180+w (since going with the wind speeds up the plane) and the speed against the wind is 180-w (since going against the wind slows the plane down)



{{{d=rt}}} Start with the distance-rate-time formula



{{{7=(180+w)t}}} Plug in {{{d=7}}} and {{{r=180+w}}} (see above)



{{{7/(180+w)=t}}} Divide both sides by 180+w.



{{{t=7/(180+w)}}} Rearrange the equation
 


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{{{d=rt}}} Go back to the distance-rate-time formula



{{{5=(180-w)t}}} Plug in {{{d=5}}} and {{{r=180-w}}} (see note above)



{{{5=(180-w)(7/(180+w))}}} Since the times are equal, this means that we can plug in {{{t=7/(180+w)}}}



{{{5(180+w)=(180-w)7}}} Multiply both sides by 180+w.



{{{5(180+w)=7(180-w)}}} Rearrange the terms.



{{{900+5w=1260-7w}}} Distribute.



{{{5w=1260-7w-900}}} Subtract {{{900}}} from both sides.



{{{5w+7w=1260-900}}} Add {{{7w}}} to both sides.



{{{12w=1260-900}}} Combine like terms on the left side.



{{{12w=360}}} Combine like terms on the right side.



{{{w=(360)/(12)}}} Divide both sides by {{{12}}} to isolate {{{w}}}.



{{{w=30}}} Reduce.



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Answer:


So the solution is {{{w=30}}} which means that the speed of the wind is 30 mph.