Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. 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. the solution is

Explanation:

Step1: Define Variables

Let \( x \) be the time (in hours) it takes the experienced crew to paint the apartment. Then the new crew takes \( 2x \) hours. The rate of work of the experienced crew is \( \frac{1}{x} \) (apartments per hour), and the rate of the new crew is \( \frac{1}{2x} \).

Step2: Set Up Work - Rate Equation

When working together, their combined rate is \( \frac{1}{x}+\frac{1}{2x} \), and they can complete 1 apartment in 6 hours, so their combined rate is also \( \frac{1}{6} \).
We set up the equation:

$$ \frac{1}{x}+\frac{1}{2x}=\frac{1}{6} $$

Step3: Combine Fractions on the Left

Find a common denominator for the left - hand side, which is \( 2x \). Then \( \frac{1}{x}+\frac{1}{2x}=\frac{2}{2x}+\frac{1}{2x}=\frac{2 + 1}{2x}=\frac{3}{2x} \)
So our equation becomes \( \frac{3}{2x}=\frac{1}{6} \)

Step4: Cross - Multiply to Solve for \( x \)

Cross - multiplying gives us \( 2x\times1=3\times6 \)

$$ 2x = 18 $$

Divide both sides by 2: \( x=\frac{18}{2}=9 \)

Answer:

9