2017 AMC 10B Problem 11

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}\%

Separate actual preference from reported preference

The denominator is everyone who says they dislike dancing

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 .

Problem 11 in other years