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?

Answer:Alan: 200 m/minBrian: 150 m/minStep-by-step explanation:    Given : 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. To find : What are their rates, in meters per minute?Solution : Let n represent the number of laps that Alan completes in 9 minutes. Then n-1 is the number of laps Brian completes.45 seconds = 3/4 minutes.The difference in their lap times in minutes per lap is[tex]\frac{9}{(n -1)}-\frac{9}{n} =\frac{3}{4}[/tex]Solving the equation we get,[tex]\frac{9n-9n+9}{n(n -1)} =\frac{3}{4}[/tex][tex]9\times 4=3\times n(n -1)[/tex][tex]36=3\times n(n -1)[/tex][tex]n^2-n-12=0[/tex][tex]n^2-4n+3n-12=0[/tex][tex]n(n-4)+3(n-4)=0[/tex][tex](n-4)(n+3)=0[/tex][tex]n=4,-3[/tex]Neglecting n=-3So, n=4Then Alan's speed in m/min is[tex]S=\frac{D}{T}[/tex][tex]S_a=\frac{4\times 450}{9}[/tex][tex]S_a=200 m/min[/tex]Brian completes 3 laps in that 9-minute time, so his rate is[tex]S_b=\frac{3\times 450}{9}[/tex][tex]S_b=150 m/min[/tex]Therefore, Alan: 200 m/minBrian: 150 m/min