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

The objective is to determine the duration required to fill a tank when two pipes are supplying water simultaneously with one pipe draining it.

1. Foundational Principles:

- Three pipes are involved:
   - Two pipes contribute to filling the tank.
    - One pipe removes water from the tank.
- Each pipe possesses a specific work rate, indicating the fraction of the tank it can fill or empty per hour.
- When multiple pipes operate concurrently, their rates are aggregated. Filling rates are added, while the emptying rate is subtracted from the total.
- The total time to fill the tank is calculated as \( \frac{1}{\text{net rate}} \), where the net rate represents the combined filling capacity.

2. Provided Data:

- 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. Calculation of Individual Rates:

A pipe's rate is defined as the proportion of the tank it fills or empties within a single hour.

• Pipe 1's rate: \( \frac{1}{20} \) of the tank per hour.
• Pipe 2's rate: \( \frac{1}{30} \) of the tank per hour.
• Pipe 3's rate: \( \frac{1}{15} \) of the tank per hour (emptying).

4. Determination of the Net Rate:

When all three pipes function simultaneously, the net rate at which the tank is filled is calculated as:

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

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

To perform these fractional operations, a common denominator is required:

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

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

Substituting these values back:

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

5. Calculation of the Filling Time:

A net rate of \( \frac{1}{60} \) signifies that collectively, the pipes fill \( \frac{1}{60} \) of the tank per hour.

Consequently, the total time required to fill the tank is the inverse of this rate:

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

Conclusion:

With all three pipes operating concurrently, the tank will be fully filled in 60 hours.