AMC12 2000 A

AMC12 2000 A · Q13

AMC12 2000 A · Q13. It mainly tests Word problems (algebra), Mixture / concentration.

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?

一天早上，Angela 家的每个成员都喝了一杯 8 盎司的咖啡加牛奶混合饮料。每杯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。这个家庭有多少人？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $c$ be the total amount of coffee, $m$ of milk, and $p$ the number of people in the family. Then each person drinks the same total amount of coffee and milk (8 ounces), so \[\left(\frac{c}{6} + \frac{m}{4}\right)p = c + m\] Regrouping, we get $2c(6-p)=3m(p-4)$. Since both $c,m$ are positive, it follows that $6-p$ and $p-4$ are also positive, which is only possible when $p = 5\ \mathrm{(C)}$.

设咖啡总量为 $c$，牛奶总量为 $m$，家庭人数为 $p$。则每个人喝的咖啡与牛奶总量相同（8 盎司），所以 \[\left(\frac{c}{6} + \frac{m}{4}\right)p = c + m\] 整理得 $2c(6-p)=3m(p-4)$。由于 $c,m$ 都为正，因此 $6-p$ 与 $p-4$ 也都为正，这只可能在 $p = 5\ \mathrm{(C)}$ 时成立。

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