Questions: A new crew of painters takes two times as long to paint a small apartment as an experienced crew. Together, both crews can paint the apartment in 6 hours. How many hours does it take the experienced crew to paint the apartment? It takes hours for the experienced crew to paint the apartment.

A new crew of painters takes two times as long to paint a small apartment as an experienced crew. Together, both crews can paint the apartment in 6 hours. How many hours does it take the experienced crew to paint the apartment?

It takes hours for the experienced crew to paint the apartment.

Transcript text: A new crew of painters takes two times as long to paint a small apartment as an experienced crew. Together, both crews can paint the apartment in 6 hours. How many hours does it take the experienced crew to paint the apartment? It takes $\qquad$ hours for the experienced crew to paint the apartment. The solution is $\square$

Solution

Solution Steps

Step 1: Define Variables

Let $ x $ be the number of hours it takes the experienced crew to paint the apartment. Therefore, the new crew takes $ 2x $ hours to paint the apartment.

Step 2: Set Up the Equation

The rate of work for the experienced crew is $ \frac{1}{x} $ of the apartment per hour, and the rate for the new crew is $ \frac{1}{2x} $ of the apartment per hour. Together, their combined rate is:

\[ \frac{1}{x} + \frac{1}{2x} = \frac{1}{6} \]

Step 3: Solve the Equation

Combine the fractions on the left side:

\[ \frac{1}{x} + \frac{1}{2x} = \frac{2}{2x} + \frac{1}{2x} = \frac{3}{2x} \]

Set the equation equal to the combined rate:

\[ \frac{3}{2x} = \frac{1}{6} \]

Cross-multiply to solve for $ x $:

\[ 3 \cdot 6 = 2x \cdot 1 \]

\[ 18 = 2x \]

Divide both sides by 2:

\[ x = 9 \]