Problem Analysis:

We are given two pipes with different flow rates:

1. Larger inlet pipe: Fills the tank in 3 days.
2. Smaller inlet pipe: Fills the tank in 6 days.

We need to determine how long it will take to fill the tank if both pipes are used simultaneously.

Step 1: Determine the rate of each pipe.

• The larger inlet pipe fills the tank in 3 days, so its rate is $\frac{1}{3}$ of the tank per day.
• The smaller inlet pipe fills the tank in 6 days, so its rate is $\frac{1}{6}$ of the tank per day.

Step 2: Combine the rates.

When both pipes are used together, their combined rate is the sum of their individual rates:

$\text{Combined Rate} = \frac{1}{3} + \frac{1}{6}$

Step 3: Simplify the combined rate.

To add the fractions, find a common denominator:

$\frac{1}{3} = \frac{2}{6}$

So,

$\text{Combined Rate} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$

Step 4: Determine the time required to fill the tank.

The combined rate $\frac{1}{2}$ means that together, the two pipes fill $\frac{1}{2}$ of the tank in 1 day. To fill the entire tank:

$\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{2}} = 2 \text{ days}$

Final Answer:

It will take 2 days to fill the tank if both pipes are used simultaneously.

---

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How would the time change if the smaller pipe could fill the tank in 4 days instead of 6 days?
2. What if the larger pipe were twice as fast as the smaller one? How would that affect the filling time?
3. How much of the tank would be filled after 1 day if both pipes were used?
4. What is the effect on the filling time if the smaller pipe is closed halfway through?
5. How would the solution change if a third pipe, filling the tank in 12 days, is also opened?

Tip: When solving problems involving rates, it’s often helpful to think of the task as a “work problem,” where each component contributes a fraction of the task per unit of time.