AMC10 2000 A

AMC10 2000 A · Q22

AMC10 2000 A · Q22. It mainly tests Linear equations, Fractions.

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?

一个早晨，Angela 全家每个人都喝了一杯 8 盎司的咖啡与牛奶混合物。每个杯子中咖啡和牛奶的量各不相同，但都不为零。Angela 喝了总牛奶量的四分之一和总咖啡量的六分之一。全家有多少人？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Suppose that the whole family drank $x$ cups of milk and $y$ cups of coffee. Let $n$ denote the number of people in the family. The information given implies that $x/4 + y/6 = (x + y)/n$. This leads to $3x(n-4)=2y(6-n).$ Since $x$ and $y$ are positive, the only positive integer $n$ for which both sides have the same sign is $n=5.$

答案（C）：设全家一共喝了 $x$ 杯牛奶和 $y$ 杯咖啡。令 $n$ 表示家庭人数。由已知信息可得 $x/4 + y/6 = (x + y)/n$。于是推出 $3x(n-4)=2y(6-n)。$ 由于 $x$ 和 $y$ 均为正数，使得等式两边同号的唯一正整数 $n$ 是 $n=5。$

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