1. **State the problem:** A man takes 60 minutes less than his apprentice to paint a room. Together, they take 72 minutes to paint the room. We need to find the time the apprentice takes to paint the room alone. 2. **Define variables:** Let the apprentice's time to paint the room alone be $x$ minutes. 3. **Man's time:** Since the man takes 60 minutes less, his time is $x - 60$ minutes. 4. **Work rates:** The rate of work is the reciprocal of time. So, - Apprentice's rate = $\frac{1}{x}$ rooms per minute - Man's rate = $\frac{1}{x-60}$ rooms per minute 5. **Combined rate:** Together they take 72 minutes, so their 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 $72x(x-60)$ 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 equation:** Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-204$, $c=4320$: $$x = \frac{204 \pm \sqrt{204^2 - 4 \times 1 \times 4320}}{2}$$ Calculate discriminant: $$204^2 = 41616$$ $$4 \times 4320 = 17280$$ $$\sqrt{41616 - 17280} = \sqrt{24336} = 156$$ 12. **Find roots:** $$x = \frac{204 \pm 156}{2}$$ Two possible values: - $$x = \frac{204 + 156}{2} = \frac{360}{2} = 180$$ - $$x = \frac{204 - 156}{2} = \frac{48}{2} = 24$$ 13. **Check validity:** Man's time is $x - 60$. - If $x=24$, man’s time = $24 - 60 = -36$ (not possible) - If $x=180$, man’s time = $180 - 60 = 120$ (valid) 14. **Final answer:** The apprentice takes **180 minutes** to paint the room alone.