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2012 AMC 12A 第 13 题

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所有题目均经美国数学协会（MAA）官方合法授权使用。

13.

油漆工 Paula 和她的两名助手各自以恒定但不同的速度刷漆。他们总是在上午 8:008{:}00 开始工作，并且三人每天午餐都花同样长的时间。星期一，三人一起粉刷了房子的 50%50\%，在下午 4:004{:}00 停工。星期二 Paula 不在，两名助手只粉刷了房子的 24%24\%，并在下午 2:122{:}12 停工。星期三 Paula 独自工作，一直到晚上 7:127{:}12 才完成整栋房子。每天的午餐休息时间是多少分钟？

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:008{:}00 AM and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50%50\% of a house, quitting at 4:004{:}00 PM. On Tuesday, when Paula wasn’t there, the two helpers painted only 24%24\% of the house and quit at 2:122{:}12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:127{:}12 PM. How long, in minutes, was each day’s lunch break?

3030

3636

4242

4848

6060

答案：D
知识点：速率方程组
难度评级：1810
小提示：

设午餐休息为 mm 分钟，设 pp 和 hh 分别为 Paula 与两名助手合计的刷漆速度（每分钟百分比）

Let the lunch break be mm minutes and let pp and hh be the painting rates (percent per minute)

大提示：

工作分钟数为 480−m480 - m、372−m372 - m 和 672−m672 - m；列方程 (p+h)(480−m)=50(p+h)(480-m) = 50，h(372−m)=24h(372-m) = 24，p(672−m)=26p(672-m) = 26

Working minutes are 480−m,480 - m, 372−m,372 - m, and 672−m672 - m; set up (p+h)(480−m)=50,(p+h)(480-m) = 50, h(372−m)=24,h(372-m) = 24, p(672−m)=26p(672-m) = 26

解答：

设午餐时长为 mm 分钟。星期一三人工作了 480−m480 - m 分钟，星期二助手工作了 372−m372 - m 分钟，星期三 Paula 工作了 672−m672 - m 分钟。

若 Paula 每分钟刷 p%p\%，两名助手合计每分钟刷 h%h\%，则 (p+h)(480−m)=50,h(372−m)=24,p(672−m)=26。 \begin{aligned} (p+h)(480-m) &= 50, \\ h(372-m) &= 24, \\ p(672-m) &= 26 \end{aligned}\text{。}

将后两个方程相加，再从第一个方程中减去，得到 108h−192p=0108h - 192p = 0，所以 h=169ph = \tfrac{16}{9}p。解这个方程组得到 p=124p = \tfrac{1}{24} 且 m=48m = 48。

因此，正确答案是 D。

Let mm be the lunch length in minutes. The three worked 480−m480 - m minutes Monday, the helpers 372−m372 - m minutes Tuesday, and Paula 672−m672 - m minutes Wednesday.

If Paula paints p%p\% per minute and the helpers together paint h%h\% per minute, then (p+h)(480−m)=50,h(372−m)=24,p(672−m)=26. \begin{aligned} (p+h)(480-m) &= 50, \\ h(372-m) &= 24, \\ p(672-m) &= 26. \end{aligned}

Adding the last two equations and subtracting from the first gives 108h−192p=0,108h - 192p = 0, so h=169p.h = \tfrac{16}{9}p. Solving the system gives p=124p = \tfrac{1}{24} and m=48.m = 48.

Thus, the correct answer is D.

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