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?

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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.
Your friend would take approximately 13.33 hours to paint the room alone.

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?

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