Question

Euclid 高中最后一场篮球赛后，统计得知全队总分的 $\dfrac{1}{4}$ 由 Alexa 得到， $\dfrac{2}{7}$ 由 Brittany 得到。Chelsea 得了 $15$ 分。其余 $7$ 名队员中没有人得分超过 $2$ 分。其余 $7$ 名队员一共得了多少分？

After Euclid High School's last basketball game, it was determined that $\dfrac{1}{4}$ of the team's points were scored by Alexa and $\dfrac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members?

$10$

$11$

$12$

$13$

$14$

答案：B

知识点：整除性一次方程

难度评级：1490

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设总得分为 $x$ ，所求答案为 $y$ 。

由题意可得 $\dfrac{x}{4} + \dfrac{2x}{7} + 15 + y = x.$

化简得 $y + 15 = \dfrac{13x}{28}.$

因为其余 $7$ 名队员每人的得分不超过 $2$ 分，所以 $y \leq 14$ 。

此外， $x$ 必须是 $28.$ 的倍数。若 $x = 28,$ 则 $y + 15 = 13 \Rightarrow y = -2,$ 不合要求。

若 $x = 2 \cdot 28 = 56,$ 则 $y + 15 = 26 \Rightarrow y = 11,$ 符合要求。

下一个可能的总分是 $x=84,$ 这会得到 $y=24>14.$ 更大的 $28$ 的倍数会使 $y$ 更大，所以 $y=11$ 是唯一可能值。

所以正确答案是 B。

Let $x$ be the total number of points and $y$ be the desired answer.

Then from the problem statement we get that $\dfrac{x}{4} + \dfrac{2x}{7} + 15 + y = x.$

Simplifying yields $y + 15 = \dfrac{13x}{28}.$

We know that $y \leq 14$ since none of the $7$ team members scored more than $2$ points.

We also know that $x$ must be a multiple of $28.$ If $x = 28,$ then we get that $y + 15 = 13 \Rightarrow y = -2,$ which is not allowed.

If $x = 2 \cdot 28 = 56,$ then we have that $y + 15 = 26 \Rightarrow y = 11,$ which works.

The next possible total is $x=84,$ which would give $y=24>14.$ Every larger multiple of $28$ makes $y$ still larger, so $y=11$ is the only possible value.

Thus, the correct answer is B.

2019 AMC 8 第 23 题

答案：B

难度评级：1490

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