2019 AMC 10B Problem 3

Two of the three percentages are really about students who do not play, so count non-players.

1. Total non-players: $46.8\%$ of $500$ is $0.468 \cdot 500 = 234$.
2. $40\%$ of seniors play, so $60\%$ of seniors do not play.
3. $30\%$ of non-seniors do not play (this is given directly).
4. Let $s$ be the number of seniors. Then there are $500 - s$ non-seniors, and the non-players add up to
$$0.6s + 0.3(500 - s) = 234.$$
5. Simplify: $0.6s + 150 - 0.3s = 234$, so $0.3s = 84$ and $s = 280$.
6. Non-seniors: $500 - 280 = 220$.
7. The question asks for non-seniors who play. Since $30\%$ of them do not play, $70\%$ do: $0.7 \cdot 220 = 154$.

Check: non-players are $0.6 \cdot 280 + 0.3 \cdot 220 = 168 + 66 = 234$. It matches.

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

When a problem gives percentages of different groups, first put them all in the same category ("do not play" here). Then the pieces add up to the total, and one equation finds the group sizes. Be careful at the end: the question asks for players, so one more step turns $30\%$ into $70\%$. Two wrong choices, $66$ and $220$, are numbers you meet along the way.

Work backward from the choices. Try (B): if $154$ non-seniors play and that is $70\%$ of them, there are $154 \div 0.7 = 220$ non-seniors and $280$ seniors. Non-players: $0.3 \cdot 220 + 0.6 \cdot 280 = 66 + 168 = 234$, which is $46.8\%$ of $500$. It works.

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 $\boxed{\textbf{(B)}\ 154}$.