Question

Two pipes can fill a tank in 20 hours and 30 hours respectively. A third pipe empties the tank in 15 hours. If all three pipes work together, how long will it take to fill the tank?

• A. 12 hours
• B. 15 hours
• C. 60 hours
• D. 20 hours

Hint:

Use LCM to find tank capacity and compute net rate by adding filling rates and subtracting emptying rates. Verify calculations with given options.

Answer 1

The Correct Option is C

To solve the problem, we need to find out how long it will take to fill the tank when two pipes are filling it and one pipe is emptying it at the same time.

1. Understanding the Concepts:

- There are three pipes:
    - Two pipes fill the tank (add water).
    - One pipe empties the tank (removes water).
- Each pipe has a rate of work, which is how much of the tank it can fill or empty in one hour.
- When pipes work together, their rates combine: the filling rates add up, and the emptying rate subtracts from the total.
- Total time to fill the tank = \( \frac{1}{\text{net rate}} \) where net rate is the combined filling rate.

2. Given Values:

- Pipe 1 fills the tank in 20 hours.
- Pipe 2 fills the tank in 30 hours.
- Pipe 3 empties the tank in 15 hours.

3. Calculating Individual Rates:

The rate of a pipe is the part of the tank it fills or empties in one hour.

• Pipe 1 fills \( \frac{1}{20} \) of the tank per hour.
• Pipe 2 fills \( \frac{1}{30} \) of the tank per hour.
• Pipe 3 empties \( \frac{1}{15} \) of the tank per hour.

4. Calculating the Net Rate:

When all three pipes work together, the net rate of filling the tank is:

\[ \text{Net rate} = \text{(Pipe 1 rate)} + \text{(Pipe 2 rate)} - \text{(Pipe 3 rate)} \]

\[ = \frac{1}{20} + \frac{1}{30} - \frac{1}{15} \]

To add and subtract these fractions, find a common denominator:

The least common denominator of 20, 30, and 15 is 60.

\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{15} = \frac{4}{60} \]

Now substitute back:

\[ \text{Net rate} = \frac{3}{60} + \frac{2}{60} - \frac{4}{60} = \frac{3 + 2 - 4}{60} = \frac{1}{60} \]

5. Calculating the Time to Fill the Tank:

The net rate \( \frac{1}{60} \) means that together, the pipes fill \( \frac{1}{60} \) of the tank in one hour.

So, the total time to fill the tank is the reciprocal of this rate:

\[ \text{Time} = \frac{1}{\text{Net rate}} = \frac{1}{\frac{1}{60}} = 60 \text{ hours} \]

Final Answer:

When all three pipes work together, the tank will be filled in 60 hours.