AMC 10 · 2019 · #3

Grade 6 arithmetic

Archetype Overlap / Complement Counting └ Sub-archetype Two-Attribute Table: Fill or Optimize a Cell

percentagesystems-of-equationsratio-proportionlinear-equations-two-var identify-subproblemscasework ↑ Prerequisites: percentagelinear-equations-two-var

📏 Medium solution 💡 3 insights

Problem

A high school has 500 students total. Among seniors, 40% play a musical instrument. Among non-seniors, 30% do NOT play. Across the whole school, 46.8% of students do NOT play. How many non-seniors play an instrument?

Pick an answer.

(A)

(B)

154

(C)

186

(D)

220

(E)

266

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): split into (a) find total non-players, (b) split them into 'senior non-players' and 'non-senior non-players' using the per-group percentages, (c) find the non-senior total, (d) take 70% to get non-senior players. Tool #8 (Units): every number is 'students' or 'percent of students' — translate carefully. Tool #6 (Guess and Check): plug in each answer choice as 'non-seniors who play', back out the non-senior total (÷ 0.7), then the senior total (500 - that), then check the total non-player count matches 234 — saves algebra entirely.

1STEP 1

The whole school is 500 and 46.8% don't play, so non-players total 0.468 · 500 = 234.

non-players total = 0.468 · 500 = 234

2STEP 2

Let N be non-seniors, seniors = 500 - N; the two non-playing groups sum to the total: 0.6(500 - N) + 0.3 N = 234.

0.6(500 - N) + 0.3 N = 234

3STEP 3

Test (B): non-seniors = 154 ÷ 0.7 = 220; non-players = 0.6·280 + 0.3·220 = 234 — match!

Try (B): N = 154 / 0.7 = 220; non-players = 0.6 · 280 + 0.3 · 220 = 168 + 66 = 234 ✓

4STEP 4

Every other choice fails — (A)(C)(D) give a non-whole N, and (E) 266 gives non-players 186 ≠ 234 — so the answer is 154.

Only (B) gives whole N AND matches 234 non-players → (B)

Answer

154

Final tally: 280 seniors and 220 non-seniors sum to 500 — good. Senior players: 0.4 · 280 = 112; senior non-players: 168. Non-senior players: 0.7 · 220 = 154; non-senior non-players: 66. Total non-players: 168 + 66 = 234 = 0.468 · 500 — matches. Answer 154 is consistent on every check.

💡Key takeaway

This AMC 10 problem only needs Grade 6 percent reasoning you already know — find 234 total non-players, then test the choices: 154 non-senior players means 220 non-seniors and 280 seniors, giving 168 + 66 = 234 non-players. Match — (B)!

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