1. **State the problem:** We have two types of pumps, large and small, filling a swimming pool. Two large and one small pump fill the pool in 4 hours. One large and three small pumps also fill the pool in 4 hours. We want to find how long 4 large and 4 small pumps take to fill the pool. 2. **Define variables:** Let $L$ be the rate of one large pump (pool/hour) and $S$ be the rate of one small pump (pool/hour). 3. **Write equations from the problem:** - Two large and one small pump fill the pool in 4 hours, so their combined rate is $\frac{1}{4}$ pool/hour: $$2L + S = \frac{1}{4}$$ - One large and three small pumps fill the pool in 4 hours, so their combined rate is also $\frac{1}{4}$ pool/hour: $$L + 3S = \frac{1}{4}$$ 4. **Solve the system of equations:** From the first equation: $$S = \frac{1}{4} - 2L$$ Substitute into the second equation: $$L + 3\left(\frac{1}{4} - 2L\right) = \frac{1}{4}$$ Simplify: $$L + \frac{3}{4} - 6L = \frac{1}{4}$$ $$\cancel{L} + \frac{3}{4} - 6\cancel{L} = \frac{1}{4}$$ $$-5L + \frac{3}{4} = \frac{1}{4}$$ Subtract $\frac{3}{4}$ from both sides: $$-5L = \frac{1}{4} - \frac{3}{4} = -\frac{2}{4} = -\frac{1}{2}$$ Divide both sides by $-5$: $$L = \frac{-\frac{1}{2}}{-5} = \frac{1}{10}$$ 5. **Find $S$:** $$S = \frac{1}{4} - 2\times \frac{1}{10} = \frac{1}{4} - \frac{2}{10} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20}$$ 6. **Calculate the rate of 4 large and 4 small pumps:** $$4L + 4S = 4\times \frac{1}{10} + 4\times \frac{1}{20} = \frac{4}{10} + \frac{4}{20} = \frac{4}{10} + \frac{2}{10} = \frac{6}{10} = \frac{3}{5}$$ 7. **Find the time to fill the pool:** Time = $\frac{1}{\text{rate}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$ hours. **Final answer:** It will take $\frac{5}{3}$ hours, or 1 hour and 40 minutes, for 4 large and 4 small pumps to fill the swimming pool.