Question

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?

$10\%$

$12\%$

$20\%$

$25\%$

$33\frac{1}{3}\%$

Solution:

Observe that of the $60\%$ of people that actually like dancing, only $80\%$ say they like dancing. This suggests that $48\%$ of the students say that they like dancing, and as such, $60\%-48\% = 12\%$ of the students who like dancing say they don't like it.

Then, we know that $90\%$ of the $40\%$ of people who don't like dancing say they don't like it, which is $36\%$ of the total student population.

This means the total amount of people who say they don't like dancing is $12\%+36\% = 48\%.$

We know then that the fraction of people who say they dislike dancing but actually like it is equal to: $\frac{12}{48} = \frac 14 = 25\% .$

Thus, the correct answer is D .

2017 AMC 10B Problem 11

Solution:

Problem 11 in Other Years