Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward 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 miles had  Mike ridden when they met?
I need to find how many miles Mike had ridden when they met, so I will call that x. Then Josh must have ridden 13 - x miles when they met. I also know that Josh rode for twice the length of time as Mike, so if I let t be the time that Mike rode, then 2t is the time that Josh rode. I also know that Josh rode at four-fifths of Mike's rate, so if I let r be Mike's rate in miles per hour, then 4/5 r is Josh's rate. Now I can use the formula that distance equals rate times time, or d = rt, for both Josh and Mike. For Josh, I have 13 - x = (4/5 r)(2t), and for Mike, I have x = rt. I want to solve for x, so I will try to eliminate the other variables. I can use the second equation to solve for r in terms of x and t, and get r = x/t. Then I can substitute that into the first equation and get 13 - x = (4/5)(x/t)(2t). I can simplify this by canceling out the t's and multiplying by 5, and get 65 - 5x = 8x. Then I can add 5x to both sides and divide by 13, and get x = 65/13. This is the number of miles that Mike had ridden when they met. 