2008 AMC 12A Problem 10

Problem 10 of 25EasierAlgebra

Doug can paint a room in

5

hours. Dave can paint the same room in

7

hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let

t

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?

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Solution

In one hour Doug paints

\tfrac{1}{5}

of the room and Dave paints

\tfrac{1}{7},

so together they paint

\tfrac{1}{5} + \tfrac{1}{7}

of the room per hour. Of the total time

t,

one hour is spent at lunch, so they work for

t - 1

hours. The fraction painted must equal

1,

giving

\left(\dfrac{1}{5} + \dfrac{1}{7}\right)(t - 1) = 1.

Thus, D is the correct answer.

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