AMC 10 · 2017 · #11

Grade 7 rate-ratio
Archetype Overlap / Complement Counting └ Sub-archetype Two-Attribute Table: Fill or Optimize a Cell
percentageconditional-probabilityratio-proportion easier-related-problemsystematic-enumeration ↑ Prerequisites: percentage
📏 Medium solution 💡 2 insights
Problem
At a school, 60% of students like dancing and 40% dislike it. Of the likers, 80% say they like it and 20% say they dislike it. Of the dislikers, 90% say they dislike it and 10% say they like it. Among all the students who SAY they dislike dancing, what fraction actually like it?

Pick an answer.

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

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

Tool #2 (Make a Systematic List): every student falls into one of four boxes — like/say-like, like/say-dislike, dislike/say-dislike, dislike/say-like — so I list all four and count each. Tool #9 (Solve an Easier Related Problem): instead of juggling percentages, I pretend there are exactly 100 students so each percent becomes a plain count. Tool #16 (Change Focus): the question only cares about students who SAY they dislike dancing, so I narrow my attention to just that group at the end.

1STEP 1

Use 100 students

Set the whole school to exactly 100 students, so 60 like dancing and 40 dislike it.

likers=60, dislikers=40
2STEP 2

Split the likers

Of the 60 likers, 48 say 'like' and 12 say 'dislike' even though they truly like it.

0.80× 60=48 (say like), 0.20× 60=12 (say dislike)
3STEP 3

Split the dislikers

Of the 40 dislikers, 36 say 'dislike' and 4 say 'like'.

0.90× 40=36 (say dislike), 0.10× 40=4 (say like)
4STEP 4

Gather the 'say dislike' group

Everyone who says 'dislike' is the 12 real likers plus the 36 real dislikers, giving 48 in total.

12+36=48 say they dislike dancing
5STEP 5

Take the fraction

Of those 48, exactly 12 actually like dancing, so the fraction is 12/48=1/4=25%, choice (D).

12/48=1/4=25% → (D)
Answer
25%
Check that the four boxes cover all 100 students: 48+12+36+4=100, good. The 'say dislike' group is 12+36=48, and 12 of them really like dancing, giving 12/48=25%. That is more than 1/10 but well below half, which fits the intuition that most people who say they dislike it truly do — so 25% is reasonable.
💡Key takeaway

Pretend there are 100 students, sort them into four boxes by what they feel and what they say, then take 'actually like it' over 'everyone who says dislike' — that is 12/48=25%, choice (D).

  • Use 100 students
  • Split the likers
  • Split the dislikers
  • Gather the 'say dislike' group
  • Take the fraction

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