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2008 AMC 10A Problem 13

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2008 AMC 10A Problem 13AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 2 of 5 · about 1 min · position P11-15

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 tt ?

  1. A)

    (15+17)(t+1)=1\left(\frac{1}{5}+\frac{1}{7}\right)\left(t+1\right)=1

  2. B)

    (15+17)t+1=1\left(\frac{1}{5}+\frac{1}{7}\right)t+1=1

  3. C)

    (15+17)t=1\left(\frac{1}{5}+\frac{1}{7}\right)t=1

  4. D)

    (15+17)(t−1)=1\left(\frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1

  5. E)

    (5+7)t=1\left(5+7\right)t=1

Answer

D

Key insight

Their combined rate is 1/5 + 1/7 rooms per hour, and they paint for only t - 1 of the t hours because lunch takes one hour.

Solution

Doug paints 15\frac15 of the room per hour and Dave paints 17\frac17 per hour, so together they paint 15+17\frac15 + \frac17 of the room per hour.

The total time tt includes a one-hour lunch during which no painting happens, so the actual painting time is t−1t - 1 hours. Painting one whole room means
(15+17)(t−1)=1.\left(\frac{1}{5}+\frac{1}{7}\right)(t - 1) = 1.

The answer is (D) (15+17)(t−1)=1\boxed{\textbf{(D)}\ \left(\frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1}.

Why this works

Work problems run on rate ×\times time == work, with rates in "rooms per hour" so that they add. The only subtlety is which time the rate multiplies: the hours actually spent working, which is the total minus the break. Sanity check with a quick estimate: the working time is 3512≈2.9\frac{35}{12} \approx 2.9 hours, so t≈3.9t \approx 3.9, and only (D) produces that.

The trap

Using t + 1 as the working time (choice A), reversing the direction of the lunch adjustment.

Common mistakes

  • Using t + 1 as the working time (choice A), reversing the direction of the lunch adjustment.
  • Adding the hour to the finished work (choice B) or adding times 5+75 + 7 instead of rates (choice E).

Techniques

Set up the equation/formula and compute; no special trick needed