2017 AMC 10B Problem 11

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?

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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 .