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2019 AMC 10B Problem 3

2019 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 3 min · position P1-10

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?

A)
$66$
B)
$154$
C)
$186$
D)
$220$
E)
$266$

Answer

Key insight

Count non-players: 60% of seniors plus 30% of non-seniors equals 46.8% of 500, which pins down the split and then 70% of non-seniors play.

Solution

Work entirely with the students who do not play. Let $s$ be the number of seniors, so there are $500 - s$ non-seniors.

Seniors who do not play: $60\%$ of $s$ , or $0.6s$ .
Non-seniors who do not play: $30\%$ of $500 - s$ , or $0.3(500 - s)$ .
Total non-players: $46.8\%$ of $500$ , or $234$ .

So
$0.6s + 0.3(500 - s) = 234 \quad\Longrightarrow\quad 0.3s + 150 = 234 \quad\Longrightarrow\quad s = 280.$
There are $500 - 280 = 220$ non-seniors, and $70\%$ of them play: $0.7 \cdot 220 = 154$ .

The answer is $\boxed{\textbf{(B)}\ 154}$ .

Why this works

Two of the three percentages describe non-players, so converting everything to the "do not play" category gives a single linear equation with no translation errors. Once the group sizes are known, the requested count is one more percentage step. Choose the category that makes the most given data line up.

Alternative approach

Test choice (B) directly: $154$ playing non-seniors at a $70\%$ rate means $220$ non-seniors and $280$ seniors. Non-players: $0.3 \cdot 220 + 0.6 \cdot 280 = 66 + 168 = 234 = 0.468 \cdot 500$ . It checks.

The trap

Mixing 'play' and 'do not play' percentages, for example using 40% of seniors as non-players, and then answering 66 or 220.

Common mistakes

Mixing 'play' and 'do not play' percentages, for example using 40% of seniors as non-players, and then answering 66 or 220.
Stopping at $220$ , the number of non-seniors, instead of taking $70\%$ of it.

Techniques

Set up the equation/formula and compute; no special trick needed