Question

油漆工 Paula 和她的两名助手各自以固定但不同的速度刷漆。他们总是上午 $8:00$ 开始工作，并且三人每天午餐休息的时间相同。

星期一，三人一起刷完了一栋房子的 $50\%$ ，并在下午 $4:00$ 停工。星期二，Paula 不在，两名助手只刷完了房子的 $24\%$ ，并在下午 $2:12$ 停工。星期三，Paula 独自工作，到晚上 $7:12$ 完成整栋房子。

每天的午餐休息是多少分钟？

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00$ AM, and all three always take the same amount of time to eat lunch.

On Monday the three of them painted $50\%$ of a house, quitting at $4:00$ PM. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $2:12$ PM. On Wednesday Paula worked by herself and finished the house by working until $7:12$ P.M.

How long, in minutes, was each day's lunch break?

$30$

$36$

$42$

$48$

$60$

答案：D

知识点：速率方程组

难度评级：2060

解答：

设午餐休息为 $m$ 分钟，Paula 的工作速度为每分钟 $p$ 个百分点，两名助手的合计速度为每分钟 $h$ 个百分点。

星期一给出 $(p+h)(480-m)=50$ 。星期二给出 $h(372-m)=24$ 。星期三剩余工作量为百分之 $26$ ，所以 $p(672-m)=26$ 。

由后两式相加再减去第一式，得到 $108h-192p=0$ ，所以 $h=\dfrac{16}{9}p$ 。

代入星期一的方程得 $\dfrac{25}{9}p(480-m)=50$ ，而星期三方程为 $p(672-m)=26$ 。联立解得 $m=48$ 。

所以正确答案是 D。

Let the lunch break be $m$ minutes, Paula's rate be $p$ percent per minute, and the helpers' combined rate be $h$ percent per minute.

Monday gives $(p+h)(480-m)=50$ . Tuesday gives $h(372-m)=24$ . Since the remaining work on Wednesday was $26$ percent, Wednesday gives $p(672-m)=26$ .

Adding the Tuesday and Wednesday equations and subtracting the Monday equation gives $108h-192p=0$ , so $h=\dfrac{16}{9}p$ .

Substituting into Monday gives $\dfrac{25}{9}p(480-m)=50$ , while Wednesday gives $p(672-m)=26$ . Solving these two equations gives $m=48$ .

Thus, D is the correct answer.

2012 AMC 10A 第 19 题

答案：D

难度评级：2060

解答：

其他年份的第 19 题