2008 AMC 12A Problem 10

Question

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?

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

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

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

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

(5 + 7)t = 1

Answer: D

Concepts:rate linear equation

Difficulty rating: 1380

Small hint:

Together they paint

\dfrac{1}{5} + \dfrac{1}{7}

of the room each hour

Big hint:

Of the

t

hours, one hour is lunch, so they actually work

t - 1

hours

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.