2000 AMC 12 第 13 题

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：

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