AMC 10 · 2019 · #3

Grade 6 arithmetic

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

📏 Medium solution 💡 3 insights

Problem

In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$

Pick an answer.

(A)

(B)

154

(C)

186

(D)

220

(E)

266

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Total students: $500$; Seniors: $40\%$ play, so $60\%$ do not; Non-seniors: $30\%$ do not play, so $70\%$ play; Whole school: $46.8\%$ do not play, i.e., $0.468 \cdot 500 = 234$ non-players; Answer choices: (A) $66$, (B) $154$, (C) $186$, (D) $220$, (E) $266$

Unknowns: The number of non-seniors who play an instrument

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units, #6 Guess and Check, #3 Eliminate Possibilities

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 ($\div 0.7$), then the senior total ($500 - $ that), then check the total non-player count matches $234$ — saves algebra entirely.

Execute — Answer: B

#7 Identify Subproblems 6.RP.A.3 Step 1

Find the total number of students who do NOT play.
Whole school is $500$, and $46.8\%$ don't play.
Compute $0.468 \cdot 500$ by noting $0.468 \cdot 500 = 0.468 \cdot 1000 / 2 = 468 / 2 = 234$.

$$\text{non-players total} = 0.468 \cdot 500 = 234$$

💡 Percent of total $=$ fraction of $500$ — compute it once and lock it in.

#7 Identify Subproblems 6.EE.B.7 Step 2

Let $N$ be the number of non-seniors.
Then seniors $= 500 - N$.
Senior non-players $= 60\%$ of $(500 - N) = 0.6(500 - N)$.
Non-senior non-players $= 30\%$ of $N = 0.3 N$.
Their sum must equal $234$.

$$0.6(500 - N) + 0.3 N = 234$$

💡 Add up the two non-playing groups and match the total.

#6 Guess and Check 6.RP.A.3 Step 3

Test the answer choices by guess-and-check.
Each choice gives 'non-seniors who play' $= 0.7 N$, so $N = \text{choice} / 0.7$.
Try (B) $154$: $N = 154 / 0.7 = 220$.
Then seniors $= 500 - 220 = 280$, senior non-players $= 0.6 \cdot 280 = 168$, non-senior non-players $= 0.3 \cdot 220 = 66$.
Sum: $168 + 66 = 234$ — match!

Try (B): $N = 154 / 0.7 = 220$; non-players $= 0.6 \cdot 280 + 0.3 \cdot 220 = 168 + 66 = 234\ \checkmark$

💡 Plug a candidate in, work both sides, watch for $234$.

#3 Eliminate Possibilities 6.RP.A.3 Step 4

Confirm by quickly killing the other options.
(A) $66$ gives $N = 94.3$, not whole — out.
(C) $186$ gives $N \approx 265.7$ — out.
(D) $220$ gives $N \approx 314.3$ — out.
(E) $266$ gives $N = 380$, then non-players $= 0.6 \cdot 120 + 0.3 \cdot 380 = 72 + 114 = 186 \ne 234$.
Only (B) fits, so the answer is $\boxed{154}$.

Only (B) gives whole $N$ AND matches $234$ non-players $\;\Rightarrow\; \textbf{(B)}$

💡 One clean match — the rest fail the whole-number or total-non-player test.

[1] #7 6.RP.A.3 Find the total number of students who do NOT play. Whole school is $500$, and $4

[2] #7 6.EE.B.7 Let $N$ be the number of non-seniors. Then seniors $= 500 - N$. Senior non-playe

[3] #6 6.RP.A.3 Test the answer choices by guess-and-check. Each choice gives 'non-seniors who p

[4] #3 6.RP.A.3 Confirm by quickly killing the other options. (A) $66$ gives $N = 94.3$, not who

Review

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

Alternative: Tool #13 (Algebra): expand $0.6(500 - N) + 0.3 N = 234$ to $300 - 0.6 N + 0.3 N = 234$, so $-0.3 N = -66$, giving $N = 220$ non-seniors and $0.7 \cdot 220 = 154$ playing. Same answer in three lines.

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Computing percentages of student counts: $46.8\%$ of $500 = 234$, $70\%$ of $220 = 154$, $60\%$ of $280 = 168$, $30\%$ of $220 = 66$.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Setting up $0.6(500 - N) + 0.3 N = 234$ to match total non-players across the two groups.)

⭐ 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)!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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