2012 AMC 12B Problem 9

Question

It takes Clea

60

seconds to walk down an escalator when it is not operating, and only

24

seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?

36

40

42

48

52

Answer: B

Concepts:rate system of equations

Difficulty rating: 1440

Small hint:

Let the escalator length be fixed and write Clea’s walking rate and the escalator’s rate

Big hint:

From

24(x+r)=60x

find

r

in terms of

x,

then use

rt=60x

Solution:

Let

x

be Clea’s walking rate and

r

the escalator’s rate, with the length equal to

60x.

Walking on the moving escalator gives

24(x+r)=60x,

so

r=\tfrac32x.

Standing takes time

t

with

rt=60x,

so

\tfrac32xt=60x

and

t=40

seconds.

Thus, the correct answer is B.