AMC10 2023 B

AMC10 2023 B · Q5

AMC10 2023 B · Q5. It mainly tests Systems of equations, Word problems (algebra).

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?

麦迪和劳拉看到黑板上写着一列数字。麦迪将列表中每个数字加 $3$，发现新数字的和是 $45$。劳拉将列表中每个数字乘以 $3$，发现新数字的和也是 $45$。黑板上写了多少个数字？

(A) 10 10

(B) 5 5

(D) 8 8

(E) 9 9

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let there be $n$ numbers in the list of numbers, and let their sum be $S$. Then we have the following \[S+3n=45\] \[3S=45\] From the second equation, $S=15$. So, $15+3n=45$ $\Rightarrow$ $n=\boxed{\textbf{(A) }10}.$

设列表中有 $n$ 个数字，它们的和为 $S$。则有以下方程 \[S+3n=45\] \[3S=45\] 由第二个方程，$S=15$。所以，$15+3n=45$ $\Rightarrow$ $n=\boxed{\textbf{(A) }10}$。

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