Difference between revisions of "2012 AMC 10B Problems/Problem 13"

Latest revision as of 20:42, 21 August 2023

Problem

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?

$\textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52$

Solution 1

Let $s$ be the speed of the escalator and $c$ be the speed of Clea. Using $d = v t$ , the first statement can be translated to the equation $d = 60c$ . The second statement can be translated to $d = 24(c+s)$ . Since the same distance is being covered in each scenario, we can set the two equations equal and solve for $s$ . We find that $s = \dfrac{3c}{2}$ . The problem asks for the time it takes her to ride down the escalator when she just stands on it. Since $t = \dfrac{d}{s}$ and $d = 60c$ , we have $t = \dfrac{60c}{\dfrac{3c}{2}} = 40$ seconds. Answer choice $\boxed{B}$ is correct.

Solution 2

Let $s$ be the speed of the escalator and $c$ be the speed of Clea. Then without loss of generality, assume that the length of the escalator be 1. Then $c=\dfrac{1}{60}$ and $c+s=\dfrac{1}{24}$ , so $s=\dfrac{1}{24}-\dfrac{1}{60}=\dfrac{1}{40}$ . Thus the time it takes for Clea to ride down the operating escalator when she just stands on it is $\dfrac{1}{\dfrac{1}{40}}=\boxed{\textbf{(B)}\ 40}$ .

Solution 3

WLOG, let the length of the escalator be $120$ ft. So, Clea's rate is $120/60=2$ feet per second (fps $*$ ), and the escalator and Clea's rate is $120/24=5$ fps. So, the escalator's rate is $5-2=3$ fps. Therefore, $120/3=\boxed{\textbf{(B)}\ 40}$ .

$*$ not to be confused with frames per second.

~MrThinker

Video Solution

https://youtu.be/JGENoSEAqGk

~savannahsolver

@@ Line 1: / Line 1: @@
-== Solution ==
+== Problem ==
+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?
-Let s be the speed of the escalator and c be the speed of Clea. Using d = v t, the first statement can be translated to the equation d = 60c. The second statement can be translated to d = 24(c+s). Since the same distance is being covered in each scenario, we can set the two equations equal and solve for s to find s = 3c/2. The problem asks for the time it takes her to ride down the escalator when she just stands on it. Since t = d/s and d = 60c, we have t = 60c/(3c/2) = 40 seconds. Answer choice B is correct.
+<math> \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 </math>
+== Solution 1==
+Let <math>s</math> be the speed of the escalator and <math>c</math> be the speed of Clea. Using <math>d = v t</math>, the first statement can be translated to the equation <math>d = 60c</math>. The second statement can be translated to <math>d = 24(c+s)</math>. Since the same distance is being covered in each scenario, we can set the two equations equal and solve for <math>s</math>. We find that <math>s = \dfrac{3c}{2}</math>. The problem asks for the time it takes her to ride down the escalator when she just stands on it. Since <math>t = \dfrac{d}{s}</math> and <math>d = 60c</math>, we have <math>t = \dfrac{60c}{\dfrac{3c}{2}} = 40</math> seconds. Answer choice <math>\boxed{B}</math> is correct.
+== Solution 2==
+Let <math>s</math> be the speed of the escalator and <math>c</math> be the speed of Clea. Then [[without loss of generality]], assume that the length of the escalator be 1. Then <math>c=\dfrac{1}{60}</math> and <math>c+s=\dfrac{1}{24}</math>, so <math>s=\dfrac{1}{24}-\dfrac{1}{60}=\dfrac{1}{40}</math>. Thus the time it takes for Clea to ride down the operating escalator when she just stands on it is <math>\dfrac{1}{\dfrac{1}{40}}=\boxed{\textbf{(B)}\ 40}</math>.
+==Solution 3==
+WLOG, let the length of the escalator be <math>120</math> ft. So, Clea's rate is <math>120/60=2</math> feet per second (fps<math>*</math>), and the escalator and Clea's rate is <math>120/24=5</math> fps. So, the escalator's rate is <math>5-2=3</math> fps. Therefore, <math>120/3=\boxed{\textbf{(B)}\ 40}</math>.
+<math>*</math> not to be confused with frames per second.
+~MrThinker
+==Video Solution==
+https://youtu.be/JGENoSEAqGk
+~savannahsolver
+==See Also.==
+{{AMC10 box|year=2012|ab=B|num-b=12|num-a=14}}
+{{MAA Notice}}

2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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