Randy and Trey take turns cleaning offices on the weekends. It takes Randy at most 4 hours to clean the offices. It takes Trey at most 6 hours to clean the offices. What is the greatest amount of time it would take them to clean the offices together?

The greatest time occurs when Randy takes the full 4 hours and Trey takes the full 6 hours. Their cleaning rates are 1/4 and 1/6 of the offices per hour, so together they clean 1/4 + 1/6 = 5/12 of the offices each hour. The time for one full job is 1 ÷ (5/12) = 12/5 = 2.4 hours, so the answer is 2.4 hours.

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