1. **State the problem:** We have three painters: Alice, Bob, and Charlie, with rates $A$, $B$, and $C$ houses per hour respectively. 2. **Given information:** - Alice and Bob together paint the house in 4 hours, so their combined rate is $\frac{1}{4}$ houses per hour. - Bob and Charlie together paint the house in 5 hours, so their combined rate is $\frac{1}{5}$ houses per hour. - Alice and Charlie together paint the house in 3 hours, so their combined rate is $\frac{1}{3}$ houses per hour. 3. **Set up the system of equations:** $$ \begin{cases} A + B = \frac{1}{4} \\ B + C = \frac{1}{5} \\ A + C = \frac{1}{3} \end{cases} $$ 4. **Solve the system:** Add all three equations: $$ (A + B) + (B + C) + (A + C) = \frac{1}{4} + \frac{1}{5} + \frac{1}{3} $$ Simplify left side: $$ 2A + 2B + 2C = \frac{1}{4} + \frac{1}{5} + \frac{1}{3} $$ Calculate right side: $$ \frac{1}{4} + \frac{1}{5} + \frac{1}{3} = \frac{15}{60} + \frac{12}{60} + \frac{20}{60} = \frac{47}{60} $$ Divide both sides by 2: $$ A + B + C = \frac{47}{120} $$ 5. **Find individual rates:** From $A + B = \frac{1}{4}$, subtract from total: $$ C = \frac{47}{120} - \frac{1}{4} = \frac{47}{120} - \frac{30}{120} = \frac{17}{120} $$ From $B + C = \frac{1}{5}$, subtract $C$: $$ B = \frac{1}{5} - \frac{17}{120} = \frac{24}{120} - \frac{17}{120} = \frac{7}{120} $$ From $A + B = \frac{1}{4}$, subtract $B$: $$ A = \frac{1}{4} - \frac{7}{120} = \frac{30}{120} - \frac{7}{120} = \frac{23}{120} $$ 6. **Interpretation:** - Alice's rate: $A = \frac{23}{120}$ houses/hour - Bob's rate: $B = \frac{7}{120}$ houses/hour - Charlie's rate: $C = \frac{17}{120}$ houses/hour 7. **Time for all three working together:** Combined rate: $$ A + B + C = \frac{47}{120} \text{ houses/hour} $$ Time to paint one house: $$ \text{Time} = \frac{1}{A + B + C} = \frac{1}{\frac{47}{120}} = \frac{120}{47} \approx 2.55 \text{ hours} $$ **Final answer:** It will take approximately 2.55 hours for Alice, Bob, and Charlie to paint the house together.