One custodian cleans a suite of offices in 3 hours. When a second worker is asked to join the regular custodian, the job takes only 1(1)/(2) hours. How long does It take the second worker to do the same job alone? The second worker can do the same job alone in hours. square (Simplify your answer. Type an integer, fraction, or mixed number.)

Let's denote the time it takes for the second worker to clean the suite of offices alone as \( x \) hours.<br /><br />First, we determine the cleaning rates of both workers. The regular custodian cleans the suite in 3 hours, so their rate is:<br />\[ \frac{1}{3} \text{ of the job per hour} \]<br /><br />When both workers work together, they complete the job in \( 1 \frac{1}{2} \) hours, which is equivalent to \( \frac{3}{2} \) hours. Therefore, their combined rate is:<br />\[ \frac{1}{\frac{3}{2}} = \frac{2}{3} \text{ of the job per hour} \]<br /><br />Let the rate of the second worker be \( \frac{1}{x} \text{ of the job per hour} \).<br /><br />Since the combined rate of both workers is the sum of their individual rates, we have:<br />\[ \frac{1}{3} + \frac{1}{x} = \frac{2}{3} \]<br /><br />To solve for \( \frac{1}{x} \), we subtract \( \frac{1}{3} \) from both sides:<br />\[ \frac{1}{x} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \]<br /><br />Thus,<br />\[ \frac{1}{x} = \frac{1}{3} \]<br /><br />Taking the reciprocal of both sides gives us:<br />\[ x = 3 \]<br /><br />Therefore, the second worker can do the same job alone in \( 3 \) hours.<br /><br />The second worker can do the same job alone in \( 3 \) hours.