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2017 AMC 10B Problem 11

2017 AMC 10B Problem 11AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 3 min · position P11-15

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?

A)
$10\%$
B)
$12\%$
C)
$20\%$
D)
$25\%$
E)
$33\frac{1}{3}\%$

Answer

Key insight

With 100 students, 12 like dancing but say they dislike it and 36 dislike it and say so; the requested fraction is 12 out of 48.

Solution

Suppose there are $100$ students.

$60$ like dancing. Of these, $80\%$ , or $48$ , say they like it, and the other $12$ say they dislike it.
$40$ dislike dancing. Of these, $90\%$ , or $36$ , say they dislike it, and the other $4$ say they like it.

Students who say they dislike dancing: $12 + 36 = 48$ . Among them, the ones who actually like it number $12$ . The fraction is
$\frac{12}{48} = \frac14 = 25\%.$

The answer is $\boxed{\textbf{(D)}\ 25\%}$ .

Why this works

This is a conditional-probability question dressed as percentages: the denominator is not all students but only the group that says "dislike." A concrete population of $100$ turns nested percents into whole numbers, and a two-by-two table (like/dislike versus say like/say dislike) makes the right denominator visible.

The trap

Dividing 12 by 100 (or by the 60 who like dancing) instead of by the 48 students who say they dislike it.

Common mistakes

Dividing 12 by 100 (or by the 60 who like dancing) instead of by the 48 students who say they dislike it.
Mixing up the two "rest" groups and using $4$ (dislike but say like) in the numerator, giving $\frac{4}{48}$ or a similar wrong fraction.

Techniques

Test small/specific values or special cases to find or verify the answer