# 2017 AMC 10B Problems/Problem 11

## Problem

At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ 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? $\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 12\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 25\%\qquad\textbf{(E)}\ 33\frac{1}{3}\%$

## Solution $60\% \cdot 20\% = 12\%$ of the people that claim that they like dancing actually dislike it, and $40\% \cdot 90\% = 36\%$ of the people that claim that they dislike dancing actually dislike it. Therefore, the answer is $\frac{12\%}{12\%+36\%} = \boxed{\textbf{(D) } 25\%}$.

## 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 $\frac{12}{12+36}=\frac{12}{48}=\frac{1}{4}$. We choose $\boxed{\textbf{(D) } 25\%}$

## Solution 3 (Solution 2 but organized w table)

WLOG, assume that there are a total of $100$ students at Typico High School. We make a chart: $$\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & & & \\\hline \text{Says they don't like dancing} & & & \\\hline \text{Total} & & & 100 \\\hline \end{tabular}$$

We know that $60$ of the students like dancing (since $60\%$ of $100$ is $60$), so we fill that in: $$\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & & & \\\hline \text{Says they don't like dancing} & & & \\\hline \text{Total} & 60 & & 100 \\\hline \end{tabular}$$ $80\%$ of those $60$ kids say that they like dancing, so that's $48$ kids who like dancing and say that they like dancing. The other $12$ kids like dancing and say that they do not. $$\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & & \\\hline \text{Says they don't like dancing} & 12 & & \\\hline \text{Total} & 60 & & 100 \\\hline \end{tabular}$$ $40$ students do not like dancing. $90\%$ of those $40$ students say that they do not like it, which is $36$ of them. $$\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & & \\\hline \text{Says they don't like dancing} & 12 & 36 & \\\hline \text{Total} & 60 & 40 & 100 \\\hline \end{tabular}$$

At this point, one can see that there are $12+36=48$ total students who say that they do not like dancing. $12$ of those actually like it, so that is $\dfrac{12}{48}=\dfrac14=\boxed{\textbf{(D) }25\%}.$

For the sake of completeness, let us fill out the rest of the table. $$\begin{tabular}[t]{|c|c|c|c|}\hline & \text{Likes dancing} & \text{Doesn't like dancing} & \text{Total} \\\hline \text{Says they like dancing} & 48 & 4 & 52 \\\hline \text{Says they don't like dancing} & 12 & 36 & 48 \\\hline \text{Total} & 60 & 40 & 100 \\\hline \end{tabular}$$

~Technodoggo

## Video Solution

~savannahsolver

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