The basic formula is .





Let's call the distance travelled by the motorcycle , the

motorcycle's rate , and the time spent on the motorcycle .

Hence, for the motorcycle part of the trip we have .

Since we are given the motorcycle's speed, we can write .





Likewise for the car part of the trip we have , and

substituting the given speed we have .





We also know that the total distance is 540 miles, so we can say:







And we know that the total time is 11 hours, so we can say:







From here we need to find a way to express a relationship that is an equation

in one variable.





 =>  and

  => 





Next we can substitute these two expressions for  and 

into the equation that we developed to describe the car part of the trip, thus:













Simplifying and collecting like terms gives us:













Recalling the equation that we developed for the motorcycle part of the trip:

, we now have two different expressions for  in terms of  that we can equate, since .











Solving:











Now we know that the motorcycle part of the trip took 6 of the 11 hours total

time.  Substituting this value into the original equation for the motorcycle

portion we get:







Giving us 240 miles for the distance traveled by motorcycle.





Check the answer



The car part of the trip must have taken  hours, so the car

distance must have been  miles.





 which was the given total distance.  Answer checks. 

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