2017 AMC 10B Problem 11

P11-15~3 minPrint

Ratios, Percents & Averages Conditional Probability & States Test small/specific values or special cases to find or verify the answer

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?

Show solution without answering

Key insight

Picture 100 students. 12 like dancing but say they dislike it, and 36 dislike it and say so. Of the 48 who say dislike, 12 actually like it.

Solution

Percents of percents are easier with real numbers, so imagine the school has exactly $100$ students.

1. $60\%$ like dancing: $60$ students.
- $80\%$ of them say they like it: $48$ students.
- The other $12$ say they dislike it.
2. The other $40$ students dislike dancing.
- $90\%$ of them say they dislike it: $36$ students.
- The other $4$ say they like it.
3. Put it in a table:

	say "like"	say "dislike"
really like	$48$	$12$
really dislike	$4$	$36$

4. The question is about students who say they dislike dancing. That is the "say dislike" column: $12 + 36 = 48$ students.
5. Of these $48$ , the ones who really like dancing number $12$ . The fraction is
$\frac{12}{48} = \frac14 = 25\%.$

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

Why this works

The key is the phrase "of students who say they dislike dancing": the bottom of the fraction is only that group, not the whole school. Choosing $100$ students turns every percent into a whole number, and a two-by-two table shows exactly which group to divide by.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Counting & Probability, Chapter 8: Basic Probability Techniques (§8.5)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Intermediate Counting & Probability, Chapter 11: Conditional Probability