1. **Problem statement:** Two hoses together fill a pool in 2 hours. Hose A alone fills the pool in 3 hours. We need to find how long hose B alone takes to fill the pool. 2. **Formula and concept:** The rate of filling is the reciprocal of the time taken. If hose A fills the pool in 3 hours, its rate is $\frac{1}{3}$ pool per hour. If hose B fills the pool in $t$ hours, its rate is $\frac{1}{t}$ pool per hour. When both hoses work together, their rates add up. 3. **Set up the equation:** $$\frac{1}{3} + \frac{1}{t} = \frac{1}{2}$$ 4. **Solve for $t$:** Multiply both sides by $6t$ (the least common multiple of denominators 3, $t$, and 2) to clear fractions: $$6t \times \left(\frac{1}{3} + \frac{1}{t}\right) = 6t \times \frac{1}{2}$$ $$2t + 6 = 3t$$ 5. **Isolate $t$:** $$2t + 6 = 3t$$ Subtract $2t$ from both sides: $$\cancel{2t} + 6 = \cancel{2t} + 3t$$ $$6 = t$$ 6. **Answer:** Hose B alone takes 6 hours to fill the pool.