AMC10 2011 A

AMC10 2011 A · Q12

AMC10 2011 A · Q12. It mainly tests Systems of equations, Arithmetic misc.

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team’s total score was 61 points. How many free throws did they make?

一个篮球队的球员投中了一些三分球、两分球和一分为罚球。他们用两分球所得的分数与三分球相同。成功罚球数比成功两分球数多一个。全队总得分61分。他们投中了多少个罚球？

(A) 13 13

(B) 14 14

(D) 16 16

(E) 17 17

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let $x$, $y$, and $z$ be the number of successful three-point shots, two-point shots, and free throws, respectively. Then the given conditions imply $$ 3x+2y+z=61, $$ $$ 2y=3x, $$ and $$ y+1=z. $$ Solving results in $x=8$, $y=12$, and $z=13$. Hence the team made 13 free throws.

答案（A）：设 $x$、$y$、$z$ 分别为命中的三分球、两分球和罚球的次数。则题目条件推出 $$ 3x+2y+z=61, $$ $$ 2y=3x, $$ 以及 $$ y+1=z. $$ 解得 $x=8$、$y=12$、$z=13$。因此该队罚中了 13 次。

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