1. **State the problem:** We have three pipes: Pipe A fills the tank in 10 minutes, Pipe B empties the tank in 30 minutes, and Pipe C empties the tank in 35 minutes. The tank starts empty. We want to find the time it takes to fill the tank when all three pipes are open simultaneously. 2. **Calculate the rates:** - Pipe A's filling rate is $\frac{1}{10}$ tank per minute. - Pipe B's emptying rate is $\frac{1}{30}$ tank per minute. - Pipe C's emptying rate is $\frac{1}{35}$ tank per minute. 3. **Find the net rate:** The net rate when all pipes are open is the filling rate minus the sum of emptying rates: $$\text{Net rate} = \frac{1}{10} - \frac{1}{30} - \frac{1}{35}$$ 4. **Calculate the net rate:** Find a common denominator for 10, 30, and 35, which is 210. $$\frac{1}{10} = \frac{21}{210}, \quad \frac{1}{30} = \frac{7}{210}, \quad \frac{1}{35} = \frac{6}{210}$$ So, $$\text{Net rate} = \frac{21}{210} - \frac{7}{210} - \frac{6}{210} = \frac{21 - 7 - 6}{210} = \frac{8}{210} = \frac{4}{105}$$ 5. **Calculate the time to fill the tank:** Since the net rate is $\frac{4}{105}$ tank per minute, the time to fill one tank is the reciprocal: $$\text{Time} = \frac{1}{\frac{4}{105}} = \frac{105}{4} = 26.25 \text{ minutes}$$ **Final answer:** It takes 26.25 minutes (or 26 minutes and 15 seconds) to fill the tank when all three pipes are open simultaneously.