Two boys on bicycles start cycling from the same place at the same time. one cyclist travels 15 miles per hour, the other travels 12 miles per hour, if they travel in the same direction, how many hours will it take before they are 9 miles apart?

The best way to do these is to draw a pic which I can't do here, and then do a table, which I also can't do...so I'll do my best to explain this to you.  D = rt is the equation that baffles absolutely every student everywhere.  Distance = rate times time.  We know that boy1 travels at a rate of 15 and that boy2 travels at a rate of 12.  We are looking for t, time.  We know from the problem that the distance they are apart is 9 miles and we want to know how long it took them to be 9 miles apart.  t is the same for both.  They both travel the same amount of time, but since one is going faster than the other he will of course go farther.  Therefore, the distance is the thing that takes some deciphering.  If boy1 is going faster than boy2 he will go farther in time t than boy2 will.  We are told in fact that the distance he goes is 9 miles farther.  So if boy1 travels d, then boy2 travels d - 9, which is 9 miles less.  So we have t the same for both.  By the transitive property, if t is the same for both boys, we can set the distance and rate for each equal to one another.  But since d=rt, then t=d/r.  So our equation looks like this: .  But wait, you say...I thought we were looking for the time it took them to get 9 miles between them, not the distance they traveled...hold on.  We'll get there.  Cross-multiply to find that d = 45.  Now that we have d, we can go back to our original equations for both boys and solve for t.  Remember that t is the same so we only need to sub d into one of them.  d=15t, and 45 = 15t.  Solving for t we get that the time it took them to get 9 miles between them was 3 hours.  Phew! I hate those distance problems!