AMC10 2023 A

AMC10 2023 A · Q10

AMC10 2023 A · Q10. It mainly tests Systems of equations, Averages (mean).

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently?

Maureen正在跟踪她这学期测验成绩的平均分。如果她在下一次测验中得$11$分，她的平均分将增加$1$。如果她在接下来三次测验中各得$11$分，她的平均分将增加$2$。她当前测验成绩的平均分是多少？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously. We can write the following equations: \[\frac{ax+11}{a+1}=x+1\qquad (1)\] \[\frac{ax+33}{a+3}=x+2\qquad (2)\] Multiplying equation $(1)$ by $(a+1)$ and solving, we get: \[ax+11=ax+a+x+1\] \[11=a+x+1\] \[a+x=10\qquad (3)\] Multiplying equation $(2)$ by $(a+3)$ and solving, we get: \[ax+33=ax+2a+3x+6\] \[33=2a+3x+6\] \[2a+3x=27\qquad (4)\] Solving the system of equations for $(3)$ and $(4)$, we find that $a=3$ and $x=\boxed{\textbf{(D) }7}$.

令$a$表示之前参加的测试次数，$x$表示之前成绩的平均分。我们可以写出以下方程： \[\frac{ax+11}{a+1}=x+1\qquad (1)\] \[\frac{ax+33}{a+3}=x+2\qquad (2)\] 将方程$(1)$乘以$(a+1)$并解得： \[ax+11=ax+a+x+1\] \[11=a+x+1\] \[a+x=10\qquad (3)\] 将方程$(2)$乘以$(a+3)$并解得： \[ax+33=ax+2a+3x+6\] \[33=2a+3x+6\] \[2a+3x=27\qquad (4)\] 解方程组$(3)$和$(4)$，我们发现$a=3$且$x=\boxed{\textbf{(D) }7}$。

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