AMC8 2022

AMC8 2022 · Q21

AMC8 2022 · Q21. It mainly tests Linear equations, Percent.

Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

Steph在上半场20次尝试中投中了15个篮，在下半场10次尝试中投中了10个篮。Candace在上半场尝试了12次，下半场尝试了18次。在每半场，Steph的命中率都高于Candace。令人惊讶的是她们最终的总命中率相同。Candace在下半场比上半场多投中了多少个篮？

(A) 7 7

(B) 8 8

(D) 10 10

(E) 11 11

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $x$ be the number of shots that Candace made in the first half, and let $y$ be the number of shots Candace made in the second half. Since Candace and Steph took the same number of attempts, with an equal percentage of baskets scored, we have $x+y=10+15=25.$ In addition, we have the following inequalities: \[\frac{x}{12}<\frac{15}{20} \implies x<9,\] and \[\frac{y}{18}<\frac{10}{10} \implies y<18.\] Pairing this up with $x+y=25$ we see the only possible solution is $(x,y)=(8,17),$ for an answer of $17-8 = \boxed{\textbf{(C) } 9}.$

设$x$为Candace在上半场的命中数，$y$为下半场的命中数。由于总命中率相同，总命中数相等，故$x+y=10+15=25$。此外，有以下不等式： \[\frac{x}{12}<\frac{15}{20} \implies x<9,\] 和 \[\frac{y}{18}<\frac{10}{10} \implies y<18.\] 结合$x+y=25$，唯一可能解是$(x,y)=(8,17)$，故答案是$17-8=\boxed{\textbf{(C) } 9}$。

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