Question

topic: solving rational equations and applications
progress:
question id: 90906
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 4 hours. how many hours does it take the new crew to paint the apartment?
it takes ______ hours for the new crew to paint the apartment.
the solution is

Explanation:

Step1: Define variables

Let $x$ = time (hours) for experienced crew to paint the apartment. Then the new crew takes $2x$ hours.

Step2: Set up work rate equation

Work rate of experienced crew: $\frac{1}{x}$ per hour. Work rate of new crew: $\frac{1}{2x}$ per hour. Combined rate: $\frac{1}{4}$ per hour.
$$\frac{1}{x} + \frac{1}{2x} = \frac{1}{4}$$

Step3: Solve for $x$

Find common denominator $2x$:
$$\frac{2}{2x} + \frac{1}{2x} = \frac{1}{4}$$
$$\frac{3}{2x} = \frac{1}{4}$$
Cross-multiply:
$$2x = 3 \times 4$$
$$2x = 12$$
$$x = 6$$

Step4: Find new crew's time

New crew time is $2x$:
$$2x = 2 \times 6 = 12$$

Answer:

12