2009 AMC 10B Problem 23

Question

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every

90

seconds, and Robert runs clockwise and completes a lap every

80

seconds. Both start from the start line at the same time. At some random time between

10

minutes and

11

minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

\dfrac{1}{16}

\dfrac18

\dfrac{3}{16}

\dfrac14

\dfrac{5}{16}

Answer: C

Concepts:geometric probability distance rate and time

Difficulty rating: 1920

Small hint:

The picture covers

\dfrac18

of a lap on each side of the start line

Big hint:

Find each runner’s window of time inside that region during the

10

th minute, then overlap them

Solution:

The picture spans

\dfrac18

lap on each side of the start. After

600

seconds Rachel is

30

seconds short of the line; running

\dfrac14

lap in

22.5

seconds, she is in view between

30-11.25=18.75

and

30+11.25=41.25

seconds of the

10

th minute.

After

600

seconds Robert is

40

seconds from the line; running

\dfrac14

lap in

20

seconds, he is in view between

30

and

50

seconds.

Both appear between

30

and

41.25

seconds, a window of length

11.25

out of

60,

so the probability is

\dfrac{11.25}{60}=\dfrac{3}{16}.

Thus, the correct answer is C.