13. You can paint a room in 8 hours. Working together, you and your friend can paint the room in just 5 hours. How long would your friend take to paint the room working alone?

1. Let your friend's time to paint the room alone be \( x \) hours. 2. The work rate is defined as "room per hour". - Your rate is \( \frac{1}{8} \) room per hour. - Your friend's rate is \( \frac{1}{x} \) room per hour. 3. When you work together, your combined rate is: \[ \frac{1}{8} + \frac{1}{x} \] According to the problem, together you can paint the room in 5 hours, so the combined rate is: \[ \frac{1}{5} \text{ room per hour.} \] 4. Set up the equation: \[ \frac{1}{8} + \frac{1}{x} = \frac{1}{5} \] 5. Solve for \(\frac{1}{x}\): \[ \frac{1}{x} = \frac{1}{5} - \frac{1}{8} \] 6. Find a common denominator for the fractions: \[ \frac{1}{5} = \frac{8}{40}, \quad \frac{1}{8} = \frac{5}{40} \] Thus: \[ \frac{1}{x} = \frac{8}{40} - \frac{5}{40} = \frac{3}{40} \] 7. Solve for \( x \) by taking the reciprocal: \[ x = \frac{40}{3} \] Therefore, your friend would take \(\frac{40}{3}\) hours, or approximately \(13.33\) hours, to paint the room alone.