Question 200936
Let w = speed of the wind (in mph)



First, let's set up the equation involving the trip against the wind:



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



{{{8=(18-w)t}}} Plug in {{{d=8}}} and {{{r=18-w}}} (the wind slows him down, so you must subtract "w" from 18)



{{{8/(18-w)=t}}} Divide both sides by {{{18-w}}}.



{{{t=8/(18-w)}}} Rearrange the equation



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



{{{12=(18+w)t}}} Plug in {{{d=12}}} and {{{r=18+w}}} (the wind speeds him up, so you must add "w" to 18)



{{{12/(18+w)=t}}} Divide both sides by {{{18+w}}}.



{{{t=12/(18+w)}}} Rearrange the equation


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Now recall that he "can bicycle 8 mi against the wind in the same time it takes to bicycle 12 miles with the wind". So this means that the times "t" are the same variable. This means that we can plug one equation into another:



{{{t=12/(18+w)}}} Start with the given equation.



{{{8/(18-w)=12/(18+w)}}} Plug in {{{t=8/(18-w)}}}



{{{8(18+w)=12(18-w)}}} Cross multiply



{{{144+8w=216-12w}}} Distribute.



{{{8w=216-12w-144}}} Subtract {{{144}}} from both sides.



{{{8w+12w=216-144}}} Add {{{12w}}} to both sides.



{{{20w=216-144}}} Combine like terms on the left side.



{{{20w=72}}} Combine like terms on the right side.



{{{w=(72)/(20)}}} Divide both sides by {{{20}}} to isolate {{{w}}}.



{{{w=18/5}}} Reduce.



{{{w=3.6}}} Divide (either use a calculator or long division)



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Answer:


So the solution is {{{w=3.6}}} which means that the speed of the wind is 3.6 mph.



Note: if you are expecting a whole number for an answer, I would verify that you have the correct problem written down.