2023 AMC 10A Problem 10

Below is the professionally curated solution for Problem 10 of the 2023 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10A solutions, or check the answer key.

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Concepts:mean system of equations

Difficulty rating: 1270

10.

If Maureen scores an $11$ on her next quiz, her mean score will go up by $1.$ If she gets three $11$ s in a row, her mean score will increase by $2.$ What is her current mean quiz score?

$4$

$5$

$6$

$7$

$8$

Solution:

Let $m$ be the current mean over $n$ quizzes. One more $11$ makes the mean $m + 1:$ $\frac{mn + 11}{n + 1} = m + 1,$ which tidies up to $m + n = 10.$ Three more $11$ s make it $m + 2:$ $\frac{mn + 33}{n + 3} = m + 2,$ i.e. $3m + 2n = 27.$ Solve the pair and $m = 7.$ Therefore, the answer is D.

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