1. **State the problem:** We have 3 hoses filling a swimming pool in 6 hours. We want to find how long 2 hoses take to fill \( \frac{3}{4} \) of the pool. 2. **Formula and rules:** The rate of filling is proportional to the number of hoses and the time taken. The total work (filling the pool) is 1 pool. 3. **Calculate the rate of 3 hoses:** \[ \text{Rate of 3 hoses} = \frac{1 \text{ pool}}{6 \text{ hours}} = \frac{1}{6} \text{ pool/hour} \] 4. **Calculate the rate of 1 hose:** \[ \text{Rate of 1 hose} = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} \text{ pool/hour} \] 5. **Calculate the rate of 2 hoses:** \[ \text{Rate of 2 hoses} = 2 \times \frac{1}{18} = \frac{2}{18} = \frac{1}{9} \text{ pool/hour} \] 6. **Calculate time for 2 hoses to fill \( \frac{3}{4} \) pool:** \[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{\frac{3}{4}}{\frac{1}{9}} = \frac{3}{4} \times 9 = \frac{27}{4} = 6.75 \text{ hours} \] **Final answer:** It will take 2 hoses 6.75 hours (or 6 hours 45 minutes) to fill \( \frac{3}{4} \) of the swimming pool.