1. **State the problem:** Harry and Joe together can mop a warehouse in 8 hours. Harry alone takes 12 hours. We need to find how long Joe alone would take. 2. **Formula and concept:** The work rate formula is $ \text{Work Rate} = \frac{1}{\text{Time}} $. When two people work together, their rates add up: $$\frac{1}{T_{Harry+Joe}} = \frac{1}{T_{Harry}} + \frac{1}{T_{Joe}}$$ 3. **Given values:** - $T_{Harry+Joe} = 8$ hours - $T_{Harry} = 12$ hours 4. **Set up the equation:** $$\frac{1}{8} = \frac{1}{12} + \frac{1}{T_{Joe}}$$ 5. **Solve for $ \frac{1}{T_{Joe}} $:** $$\frac{1}{T_{Joe}} = \frac{1}{8} - \frac{1}{12}$$ Find common denominator 24: $$\frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24}$$ So, $$\frac{1}{T_{Joe}} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}$$ 6. **Calculate $T_{Joe}$:** $$T_{Joe} = 24 \text{ hours}$$ 7. **Explanation:** Joe alone would take 24 hours to mop the warehouse. **How AI helped:** AI tools helped verify the work-rate formula and confirm the algebraic manipulation steps, ensuring the correct solution for Joe's time.