Two cyclists, alan and brian, are racing around oval track of length 450m on the same direction simultaneously from the same point. alan races around the track in 45 seconds before brian does and overtakes him every 9 minutes. what are their rates, in meters per minute?

Step-by-step explanation:

Let n represent the number of laps that Alan completes in 9 minutes. Then n-1 is the number of laps Brian completes, and the difference in their lap times in minutes per lap is ...

  9/(n -1) - 9/n = 3/4 . . . . . minutes

Multiplying by (4/3)n(n -1), we get

  12n -12(n -1) = n(n -1)

  12 = n(n -1)

Solutions to this are n=4 and n=-3. We are only interested in the positive solution, n = 4. Then Alan's speed in m/min is ...

  (4·450 m)/(9 min) = 200 m/min

Brian completes 3 laps in that 9-minute time, so his rate is ...

  (3·450 m)/(9 min) = 150 m/min

so, converting firstly the mixed fractions to improper fractions and then "adding".