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。这个家庭有多少人？

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：分数一次方程极限情形界定

难度评级：1710

小提示：

设 Angela 自己的一杯中有 $c$ 杯咖啡和 $m$ 杯牛奶，且 $c + m = 1$ 。

Suppose Angela’s own cup contains $c$ units of coffee and $m$ units of milk with $c + m = 1$

大提示：

则全家共有 $6c$ 杯咖啡和 $4m$ 杯牛奶，所以人数为 $6c + 4m$ 。

Then the family total is $6c$ coffee and $4m$ milk, so the number of people is $6c + 4m$

解答：

以 $8$ -盎司的一杯为单位计量，所以 Angela 的杯中有 $c$ 杯咖啡和 $m$ 杯牛奶，且 $c + m = 1$ 。

因为 Angela 喝掉了咖啡总量的六分之一，所以咖啡总量为 $6c$ ；因为她喝掉了牛奶总量的四分之一，所以牛奶总量为 $4m$ 。人数等于总杯数： $\begin{aligned} 6c + 4m &= 6c + 4(1 - c) \\ &= 4 + 2c \end{aligned}\text{。}$

这必须是整数，所以 $2c$ 是整数。又因 $0 \lt c \lt 1$ ，只能有 $c = \tfrac12$ ，从而人数为 $4 + 1 = 5$ 。

因此，正确答案是 C。

Measure amounts in $8$ -ounce cups, so Angela’s cup holds $c$ coffee and $m$ milk with $c + m = 1.$

Since Angela drank a sixth of the coffee, the total coffee is $6c$ ; since she drank a quarter of the milk, the total milk is $4m.$ The number of people equals the total number of cups, $\begin{aligned} 6c + 4m &= 6c + 4(1 - c) \\ &= 4 + 2c. \end{aligned}$

This is an integer only when $2c$ is an integer, and since $0 \lt c \lt 1$ this forces $c = \tfrac12,$ giving $4 + 1 = 5$ people.

Thus, the correct answer is C.

2000 AMC 12 第 13 题

答案：C

难度评级：1710

小提示：

大提示：

解答：

其他年份的第 13 题