2009 AMC 12B Problem 18

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}

\dfrac{1}{8}

\dfrac{3}{16}

\dfrac{1}{4}

\dfrac{5}{16}

Answer: C

Concepts:geometric probability distance rate and time

Difficulty rating: 1930

Small hint:

The picture spans

\tfrac{1}{8}

lap on each side of the start; find when each runner is within that arc during the

10

th minute

Big hint:

Rachel is in view for

18.75\text{-}41.25

s and Robert for

30\text{-}50

s; take the overlap over

60

s

Solution:

The picture covers the arc within

\dfrac{1}{8}

lap of the start on each side. At

600

s Rachel has run

6\tfrac{2}{3}

laps,

30

s short of the line; a quarter lap takes her

22.5

s, so she is in view between

30 - 11.25 = 18.75

s and

30 + 11.25 = 41.25

s of the

10

th minute.

At

600

s Robert is

40

s from the line; a quarter lap takes

20

s, so he is in view between

30

and

50

s. Both appear between

30

and

41.25

s, a window of

11.25

s out of

60,

giving probability

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

Thus, the correct answer is C.