Josh and mike live 13km apart. josh started to ride his bicycle towards mikes house. a little later mike started to ride his bicycle towards josh's house. when they met, josh had ridden for twice the length of time as mike and at four-fifths of mike's rate. how many kilometers had mike ridden when they met?

Remark
The general formula for this is distance = rate * time or more briefly d = r * t

Step One 
Fill out as much as possible for each bike rider.

Mike
d = d
r =  r
t = t

Josh
d1 = 13 - d
r1 =  4/5 * r
t1 = 2 * t

Equation
r1 * t1 + r * t = 13

substitute and solve
4/5 r * 2t + r*t = 13
(8/5) r * t + r*t = 13    Take out the common factor of rt
rt(8/5 + 1 ) = 13         change 1 to 5/5 and add.       
rt(8/5 + 5/5) = 13        Divide by 13/5    
rt(13/5) = 13
rt = 13/(13/5) =  Invert the bottom fraction and multiply.
rt =  Note that the 13s cancel.

Answer
Mike traveled 5 km.
rt = 5 <<<< Answer

Notice that you only need rt together. You don't need to know exactly what either one of them is.

Every so often someone comes up with a unique question using d=r*t. You are the lucky person today. Thanks for posting.