2017 AMC 10B Problems/Problem 11
Contents
Problem
At Typico High School, 
of the students like dancing, and the rest dislike it. Of those who like dancing, 
say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 
say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
Solution
of the people that claim that they like dancing actually dislike it, and 
of the people that claim that they dislike dancing actually dislike it. Therefore, the answer is 
.
Solution 2
Assume WLOG that there are 100 people. Then 60 of them like dancing, 40 dislike dancing. Of the ones that like dancing, 48 say they like dancing and 12 say they dislike it. Of the ones who dislike dancing, 36 say they dislike dancing and 4 say they like it. We want the ratio of students like it but say they dislike it to the total amount of students that say they dislike it. This is 
. We choose 
Solution 3 (Solution 2 but organized w table)
WLOG, assume that there are a total of 
students at Typico High School. We make a chart:
![\[\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & & & \\\hline \text{Says they don't like dancing} & & & \\\hline \text{Total} & & & 100 \\\hline \end{tabular}\]](http://latex.artofproblemsolving.com/0/0/e/00e0840c407a2b7befb90f0284eed20090036c84.png)
We know that 
of the students like dancing (since of is ), so we fill that in:
![\[\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & & & \\\hline \text{Says they don't like dancing} & & & \\\hline \text{Total} & 60 & & 100 \\\hline \end{tabular}\]](http://latex.artofproblemsolving.com/4/1/4/414807861c35918830138b892ce704b54c5a0f08.png)
of those kids say that they like dancing, so that's 
kids who like dancing and say that they like dancing. The other 
kids like dancing and say that they do not.
![\[\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & & \\\hline \text{Says they don't like dancing} & 12 & & \\\hline \text{Total} & 60 & & 100 \\\hline \end{tabular}\]](http://latex.artofproblemsolving.com/d/c/f/dcf590b7fec89806c347b0d94b4f22f127105caf.png)
students do not like dancing. of those students say that they do not like it, which is 
of them.
![\[\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & & \\\hline \text{Says they don't like dancing} & 12 & 36 & \\\hline \text{Total} & 60 & 40 & 100 \\\hline \end{tabular}\]](http://latex.artofproblemsolving.com/5/f/9/5f90e56b1010445c1c9f65d65e21bf2c4a3764e5.png)
At this point, one can see that there are 
total students who say that they do not like dancing. of those actually like it, so that is 
For the sake of completeness, let us fill out the rest of the table.
![\[\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & 4 & 52 \\\hline \text{Says they don't like dancing} & 12 & 36 & 48 \\\hline \text{Total} & 60 & 40 & 100 \\\hline \end{tabular}\]](http://latex.artofproblemsolving.com/9/7/4/974f9dd034103c939ed723223424f741ae1f62d4.png)
~Technodoggo
Video Solution 1
~savannahsolver
Video Solution 2
https://www.youtube.com/watch?v=5YbrPLGz5uM
~Math4All999
See Also
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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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