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2008 AMC 12A Problem 10

Problem 10 of 25EasierAlgebraArithmetic

Doug can paint a room in 55 hours. Dave can paint the same room in 77 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let tt be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t?t?

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Solution

In one hour Doug paints 15\tfrac{1}{5} of the room and Dave paints 17,\tfrac{1}{7}, so together they paint 15+17\tfrac{1}{5} + \tfrac{1}{7} of the room per hour. Of the total time t,t, one hour is spent at lunch, so they work for t−1t - 1 hours. The fraction painted must equal 1,1, giving (15+17)(t−1)=1. \left(\dfrac{1}{5} + \dfrac{1}{7}\right)(t - 1) = 1. Thus, D is the correct answer.
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