1. **Stating the problem:** A and B together can complete a work in 12 days. B and C together can complete the same work in 15 days. C and A together can complete the work in 20 days. We need to find how long A, B, and C working together will take to complete twice the work. 2. **Define variables:** Let the total work be 1 unit. Let the rates of A, B, and C be $a$, $b$, and $c$ units per day respectively. 3. **Write equations from given data:** Since A and B together complete 1 work in 12 days, their combined rate is: $$a + b = \frac{1}{12}$$ Similarly, for B and C: $$b + c = \frac{1}{15}$$ For C and A: $$c + a = \frac{1}{20}$$ 4. **Add all three equations:** $$ (a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} $$ Simplify left side: $$ 2(a + b + c) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} $$ 5. **Calculate the right side sum:** Find common denominator 60: $$ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} $$ 6. **Solve for $a + b + c$:** $$ 2(a + b + c) = \frac{1}{5} \implies a + b + c = \frac{1}{10} $$ 7. **Interpretation:** Together, A, B, and C complete $\frac{1}{10}$ of the work per day. 8. **Find time to complete twice the work:** Twice the work is 2 units. Time taken = $\frac{\text{work}}{\text{rate}} = \frac{2}{\frac{1}{10}} = 20$ days. **Final answer:** A, B, and C working together will complete twice the work in **20 days**.