2019 AMC 10B Problem 3

Below is the professionally curated solution for Problem 3 of the 2019 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 10B solutions, or check the answer key.

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Concepts:percentagelinear equation

Difficulty rating: 1120

3.

In a high school with 500500 students, 40%40\% of the seniors play a musical instrument, while 30%30\% of the non-seniors do not play a musical instrument. In all, 46.8%46.8\% of the students do not play a musical instrument. How many non-seniors play a musical instrument?

66 66

154 154

186 186

220 220

266 266

Solution:

Let the number of seniors be s.s. Then, 500s500-s people aren't seniors. We know 60%60\% of seniors don't play an instrument. Then, the number of students who don't play an instrument can be represented as 0.3(500s)+0.6(s)=0.3s+1500.3(500-s)+0.6(s) = 0.3s+150 and 0.468500=234.0.468\cdot 500=234. Thus, 0.3s+150=2340.3s+150=234 s=280.s=280. This makes the number of non-seniors equal to 220.220. Since 70%70\% of non-seniors play instruments, we have the total number as 2200.7=154.220\cdot 0.7=154.

Thus, the answer is B .

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