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Difference between revisions of "2006 AMC 10B Problems/Problem 13"

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== Problem ==
 
== Problem ==
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Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?  
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Joe and JoAnn each bought <math>12</math> ounces of coffee in a <math>16</math> ounce cup. Joe drank <math>2</math> ounces of his coffee and then added <math>2</math> ounces of cream. JoAnn added <math>2</math> ounces of cream, stirred the coffee well, and then drank <math>2</math> ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?  
  
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<math> \mathrm{(A) \ } \frac{6}{7}\qquad \mathrm{(B) \ } \frac{13}{14}\qquad \mathrm{(C) \ }1 \qquad \mathrm{(D) \ } \frac{14}{13}\qquad \mathrm{(E) \ } \frac{7}{6} </math>
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<math> \textbf{(A) } \frac{6}{7}\qquad \textbf{(B) } \frac{13}{14}\qquad \textbf{(C) }1 \qquad \textbf{(D) \ } \frac{14}{13}\qquad \textbf{(E) } \frac{7}{6} </math>
  
 
== Solution ==
 
== Solution ==
 
After drinking and adding cream, Joe's cup has <math>2</math> ounces of cream.  
 
After drinking and adding cream, Joe's cup has <math>2</math> ounces of cream.  
  
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After adding cream to her cup, JoAnn's cup had <math>14</math> ounces of liquid. By drinking <math>2</math> ounces out of the <math>14</math> ounces of liquid, she drank  
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After adding cream to her cup, JoAnn's cup had <math>14</math> ounces of liquid. By stirring and then drinking <math>2</math> ounces out of the <math>14</math> ounces of liquid, she drank  
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<math>\frac{2}{14}=\frac{1}{7}</math>th of the cream. So there is <math>2\cdot\frac{6}{7}=\frac{12}{7}</math> ounces of cream left.  
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<math>\frac{2}{14}=\frac{1}{7}</math>th of the cream. So there are <math>2\cdot\frac{6}{7}=\frac{12}{7}</math> ounces of cream left.  
  
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So the desired ratio is: <math> \frac{2}{\frac{12}{7}} = \frac{7}{6} \Rightarrow E </math>
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So the desired ratio is: <math>2 \div \frac{12}{7}= \boxed{\textbf{(E) }\frac{7}{6}}</math>.
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This is problem 6.2.5 in the Art of Problem Solving's <math>\textit{Introduction to Algebra}</math>
 
  
 
== See Also ==
 
== See Also ==

Latest revision as of 13:11, 26 January 2022

Problem

Joe and JoAnn each bought $12$ ounces of coffee in a $16$ ounce cup. Joe drank $2$ ounces of his coffee and then added $2$ ounces of cream. JoAnn added $2$ ounces of cream, stirred the coffee well, and then drank $2$ ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?

$\textbf{(A) } \frac{6}{7}\qquad \textbf{(B) } \frac{13}{14}\qquad \textbf{(C) }1 \qquad \textbf{(D) \ } \frac{14}{13}\qquad \textbf{(E) } \frac{7}{6}$

Solution

After drinking and adding cream, Joe's cup has $2$ ounces of cream.

After adding cream to her cup, JoAnn's cup had $14$ ounces of liquid. By stirring and then drinking $2$ ounces out of the $14$ ounces of liquid, she drank $\frac{2}{14}=\frac{1}{7}$th of the cream. So there are $2\cdot\frac{6}{7}=\frac{12}{7}$ ounces of cream left.

So the desired ratio is: $2 \div \frac{12}{7}= \boxed{\textbf{(E) }\frac{7}{6}}$.

See Also

2006 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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