Question

Water is pumped into the completely empty tank at a constant rate through an inlet pipe. At the same time, there is a leak at the bottom of the tank which leaks water at a constant rate. How long it will take the tank get filled completely?
I. Total capacity of water the tank can hold is 120 gallons.
II. Inlet pipe can completely fill the empty tank in 10 hours if there is no leak in the tank, and also the leak at the bottom of the tank can completely empty the filled tank in 15 hours if there is no water pumped into the tank.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In work-rate problems, it's often easiest to work with fractional rates (e.g., jobs per hour, tanks per hour). The total volume (like 120 gallons) is often extra information if the rates can be determined relative to the whole job.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a Data Sufficiency question involving rates of work (filling and leaking a tank). To find the total time to fill the tank, we need to determine the net filling rate. The net rate is the filling rate minus the leaking rate. The time to fill is the reciprocal of the net rate.
Let \(R_{in}\) be the rate of the inlet pipe (in tanks per hour).
Let \(R_{out}\) be the rate of the leak (in tanks per hour).
Net Rate = \(R_{in} - R_{out}\).
Time to fill = \(\frac{1}{\text{Net Rate}}\).
Step 2: Detailed Explanation:
Analyze Statement I: "Total capacity of water the tank can hold is 120 gallons."
This statement provides the volume of the tank but gives no information about the rate of filling or leaking. We cannot determine the time it takes to fill the tank. Therefore, Statement I alone is not sufficient.
Analyze Statement II: "Inlet pipe can completely fill the empty tank in 10 hours if there is no leak in the tank, and also the leak at the bottom of the tank can completely empty the filled tank in 15 hours if there is no water pumped into the tank."
This statement gives us the individual rates:

The inlet pipe fills the tank in 10 hours, so its rate is \(R_{in} = \frac{1}{10}\) tank per hour.
The leak empties the tank in 15 hours, so its rate is \(R_{out} = \frac{1}{15}\) tank per hour.
We can now calculate the net filling rate: \[ \text{Net Rate} = R_{in} - R_{out} = \frac{1}{10} - \frac{1}{15} \] To subtract the fractions, find a common denominator, which is 30: \[ \text{Net Rate} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \text{ tank per hour} \] The time required to fill the tank completely is the reciprocal of the net rate: \[ \text{Time to fill} = \frac{1}{1/30} = 30 \text{ hours} \] Since we found a unique numerical answer for the time, Statement II alone is sufficient.
Step 3: Final Answer:
Statement II alone is sufficient, but Statement I alone is not. This corresponds to option (B).