Question

某天早上，Angela 家的每个人都喝了一杯 $8$ 盎司的咖啡牛奶混合饮品。每杯中咖啡和牛奶的量可能不同，但都不为零。Angela 喝掉了全家牛奶总量的四分之一和咖啡总量的六分之一。Angela 家有多少人？

One morning each member of Angela's family drank an $8$ -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

$3$

$4$

$5$

$6$

$7$

答案：C

知识点：加权平均数极限情形界定混合问题比与比例

难度评级：1900

解答：

设共有 $n$ 人，总共喝了 $8n$ 盎司，其中牛奶总量为 $M$ ，咖啡总量为 $C$ 。 $\tfrac14 M + \tfrac16 C = \tfrac1n (M + C).$

Angela 的一杯由总牛奶的 $\tfrac14$ 与总咖啡的 $\tfrac16$ 组成，所以 $\tfrac1n$ 严格介于 $\tfrac16$ 和 $\tfrac14$ 之间。这迫使 $4 \lt n \lt 6$ ，即 $n = 5$ 。

所以正确答案是 C。

Let there be $n$ people, drinking $8n$ ounces total, split into milk $M$ and coffee $C.$ Angela drank one cup, so $\tfrac14 M + \tfrac16 C = \tfrac1n (M + C).$

The left side is a weighted average of $\tfrac14$ and $\tfrac16,$ so $\tfrac1n$ lies strictly between $\tfrac16$ and $\tfrac14.$ That forces $4 \lt n \lt 6,$ so $n = 5.$

Thus, the correct answer is C.

2000 AMC 10 第 22 题

答案：C

难度评级：1900

解答：

其他年份的第 22 题