2012 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2012 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 12B solutions, or check the answer key.

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Concepts:rate system of equations

Difficulty rating: 1440

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?

$36$

$40$

$42$

$48$

$52$

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.

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