2009 AMC 10A Problem 20

Question

Andrea and Lauren are

20

kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of

1

kilometer per minute. After

5

minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?

20

30

55

65

80

Answer: D

Concepts:distance rate and time rate

Difficulty rating: 1510

Small hint:

Let Lauren’s rate be

r;

then

r + 3r = 1

kilometer per minute

Big hint:

Find the distance left after

5

minutes, then the time for Lauren to cover it alone

Solution:

Let Lauren’s rate be

r

km/min. Then

r + 3r = 1,

so

r = \dfrac14.

In the first

5

minutes the gap shrinks by

5

km, leaving

15

km. Lauren covers this alone at

\dfrac14

km/min, taking

\dfrac{15}{\frac{1}{4}} = 60

minutes.

The total time is

5 + 60 = 65