1. **State the problem:** A man and his apprentice paint a room. The man takes 1 hour less than the apprentice to paint alone. Together, they take 72 minutes to paint the room. Find the time the apprentice takes alone. 2. **Define variables:** Let the apprentice's time to paint alone be $x$ minutes. 3. **Man's time:** Since the man takes 1 hour (60 minutes) less, his time is $x - 60$ minutes. 4. **Painting rates:** The rate is the reciprocal of time. - Apprentice's rate: $\frac{1}{x}$ rooms per minute. - Man's rate: $\frac{1}{x - 60}$ rooms per minute. 5. **Combined rate:** Together they paint in 72 minutes, so combined rate is $\frac{1}{72}$ rooms per minute. 6. **Equation:** Sum of individual rates equals combined rate: $$\frac{1}{x} + \frac{1}{x - 60} = \frac{1}{72}$$ 7. **Solve the equation:** Multiply both sides by $x(x - 60)72$ to clear denominators: $$72(x - 60) + 72x = x(x - 60)$$ 8. **Expand:** $$72x - 4320 + 72x = x^2 - 60x$$ 9. **Combine like terms:** $$144x - 4320 = x^2 - 60x$$ 10. **Bring all terms to one side:** $$0 = x^2 - 60x - 144x + 4320$$ $$0 = x^2 - 204x + 4320$$ 11. **Solve quadratic:** Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-204$, $c=4320$. 12. **Calculate discriminant:** $$\Delta = (-204)^2 - 4 \times 1 \times 4320 = 41616 - 17280 = 24336$$ 13. **Square root:** $$\sqrt{24336} = 156$$ 14. **Find roots:** $$x = \frac{204 \pm 156}{2}$$ 15. **Two possible values:** - $$x = \frac{204 + 156}{2} = \frac{360}{2} = 180$$ - $$x = \frac{204 - 156}{2} = \frac{48}{2} = 24$$ 16. **Check validity:** Man's time is $x - 60$. - If $x=24$, man’s time is $24 - 60 = -36$ (not possible). - If $x=180$, man’s time is $180 - 60 = 120$ (valid). 17. **Answer:** The apprentice takes **180 minutes** to paint the room alone.