AMC 8 Daily Practice Round 8
Complete problem set with solutions and individual problem pages
In a factory production team working on a batch of parts, if each worker stays at their original position, the task can be completed in hours. If workers and exchange positions while the efficiency of the other workers remains unchanged, the task can be completed hour earlier. If workers and exchange positions while the efficiency of the other workers remains unchanged, the task can also be completed hour earlier. If and exchange positions and and also exchange positions at the same time, with the efficiency of the other workers unchanged, how many minutes earlier can the task be completed?
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Divide the total task into parts.
Originally, the team completed parts per hour, so each part took minutes.
After exchanging and , the team completed parts per hour, which is 1 more part per hour than before.
Since the efficiency of the other workers remains unchanged, this extra part is entirely due to the improved combined efficiency of and .
Similarly, after exchanging and , they also completed 1 more part per hour than before, due to the improved combined efficiency of and .
If both exchanges are made at the same time, and and and each contribute an extra part per hour.
Therefore, the team completes parts per hour, meaning each part takes minutes to complete.
Thus, for each part, the time saved is minutes, and for all parts, the total time saved is minutes.
Answer: The task can be completed minutes earlier.
