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三人合作造管：求 Charlie 单独用时

Builders

专题
Brainteaser / 脑筋急转弯
难度
L4

题目详情

Alice 单独造一根管子需 120 分钟，Bob 单独需 100 分钟。

两人一起干了 40 分钟后，Charlie 加入。再过 10 分钟三人完成。

问：Charlie 单独完成需要多少分钟？

Alice and Bob can each build a pipe on their own in 120120 and 100100 minutes, respectively. Instead, they work together on building a pipe. After 40 minutes of building, Charlie assists them in building the pipe. Together, they finish building the pipe 1010 minutes after Charlie joins. In how many minutes could Charlie build the pipe on his own?

解析

设总工作量为 600 单位，则 Alice 速率 5 单位/分钟，Bob 速率 6 单位/分钟。

40 分钟完成 40⋅11=44040\cdot 11=440 单位，剩余 160 单位由三人 10 分钟完成，三人速率为 16。

因此 Charlie 速率为 16−11=516-11=5 单位/分钟，与 Alice 相同，单独用时 600/5=120600/5=120 分钟。

Original Explanation

Suppose 600 total units need to be built to fully create the pipe. It follows then that Alice builds at 55 units per minute and Bob builds at 66 units per minute (indeed, their respective completion times are 120120 and 100100 minutes). After 40 minutes, Alice and Bob together would have completed 40⋅(5+6)=44040 \cdot (5 + 6) = 440 of the 600600 units. After Charlie joins, they complete the remaining 160160 units in 1010 minutes, meaning that they were building at a rate of 1616 units per minute with all three working. As they were at 1111 per minute before Charlie joined, this means Charlie builds at 55 units per minute, the same rate as Alice, so Charlie also would take 120120 minutes to build the pipe on his own.