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 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.

Solution Steps

To solve this problem, we can use the concept of work rates. Let's denote the time it takes for the experienced crew to paint the apartment as \( x \) hours. Therefore, the new crew takes \( 2x \) hours. The work rate of the experienced crew is \( \frac{1}{x} \) of the apartment per hour, and the work rate of the new crew is \( \frac{1}{2x} \) of the apartment per hour. Together, their combined work rate is \( \frac{1}{4} \) of the apartment per hour. We can set up an equation to solve for \( x \) and then find \( 2x \).

Let \( x \) be the time (in hours) it takes for the experienced crew to paint the apartment. Consequently, the new crew takes \( 2x \) hours to complete the same task.

The work rate of the experienced crew is \( \frac{1}{x} \) of the apartment per hour, while the work rate of the new crew is \( \frac{1}{2x} \) of the apartment per hour. Together, their combined work rate is given as \( \frac{1}{4} \) of the apartment per hour. Thus, we can set up the equation: \[ \frac{1}{x} + \frac{1}{2x} = \frac{1}{4} \]

To solve the equation, we first find a common denominator: \[ \frac{2}{2x} + \frac{1}{2x} = \frac{1}{4} \] This simplifies to: \[ \frac{3}{2x} = \frac{1}{4} \] Cross-multiplying gives: \[ 3 \cdot 4 = 2x \implies 12 = 2x \implies x = 6 \]

Now that we have \( x = 6 \), we can find the time it takes for the new crew: \[ 2x = 2 \cdot 6 = 12 \]

Final Answer