Inlet E was open for 2 hours and then closed. The remaining part of the tank was filled by inlets B and C working together. If inlet E alone takes 12 hours to fill the tank, inlet B alone takes 8 hours, and inlet C alone takes 6 hours, what is the total time (in hours) taken to fill the tank completely?

Question

Accepted Answer

**Solution:**

1. Calculate the fraction of the tank filled by inlet E in 2 hours:
   - Since inlet E fills the tank in 12 hours, its rate is $\frac{1}{12}$ per hour.
   - In 2 hours, it fills $2 	imes \frac{1}{12} = \frac{1}{6}$ of the tank.

2. Determine the remaining part of the tank to be filled:
   - Remaining part = $1 - \frac{1}{6} = \frac{5}{6}$.

3. Calculate the combined filling rate of inlets B and C:
   - Inlet B fills the tank in 8 hours, rate = $\frac{1}{8}$ per hour.
   - Inlet C fills the tank in 6 hours, rate = $\frac{1}{6}$ per hour.
   - Combined rate = $\frac{1}{8} + \frac{1}{6} = \frac{3}{24} + \frac{4}{24} = \frac{7}{24}$ per hour.

4. Calculate the time taken by B and C together to fill the remaining $\frac{5}{6}$ of the tank:
   - Time = $\frac{	ext{Work}}{	ext{Rate}} = \frac{\frac{5}{6}}{\frac{7}{24}} = \frac{5}{6} 	imes \frac{24}{7} = \frac{20}{7}$ hours.

5. Calculate the total time to fill the tank:
   - Total time = Time by E + Time by B and C = $2 + \frac{20}{7} = \frac{14}{7} + \frac{20}{7} = \frac{34}{7} = 4 \frac{6}{7}$ hours.

**Answer:** $4 \frac{6}{7}$ hours

Answer

Answer

Answer