2019 AMC 8 第 23 题

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

小提示：

总分必须是

28

的倍数

The total score must be a multiple of

28

大提示：

其他七名队员合计最多得

14

分

The other seven players together scored at most

14

视频讲解：

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文字解答：

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\text{。}

化简得

y + 15 = \dfrac{13x}{28}\text{。}

我们知道

y \leq 14

，因为其余

7

名队员每人的得分不超过

2

分。

此外，

x

必须是

28

的倍数。若

x = 28

，则

y + 15 = 13 \Rightarrow y = -2\text{，}

不合要求。

若

x = 2 \cdot 28 = 56

，则

y + 15 = 26 \Rightarrow y = 11\text{，}

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.