Question

unit: rational expressions
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
question id: 565295
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
the solution is

Explanation:

Step1: Define variables

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

Step2: Set up the work - rate equation for combined work

When they work together, their combined work rate is \( \frac{1}{t}+\frac{1}{2t} \), and we know that together they can paint the apartment in 4 hours, so their combined work rate is also \( \frac{1}{4} \) (apartments per hour).
First, simplify the left - hand side of the equation:
\( \frac{1}{t}+\frac{1}{2t}=\frac{2 + 1}{2t}=\frac{3}{2t} \)
So we have the equation \( \frac{3}{2t}=\frac{1}{4} \)

Step3: Solve for \( t \)

Cross - multiply: \( 2t\times1=3\times4 \)
\( 2t = 12 \)
Divide both sides by 2: \( t=\frac{12}{2}=6 \)

Step4: Find the time for the new crew

The new crew takes \( 2t \) hours. Substitute \( t = 6 \) into \( 2t \): \( 2\times6 = 12 \) hours.

Answer:

12