Question

Two pipes supply waters to a cistern whose capacity of 15 cubic feet. How long does it take the two pipes to fill the cistern?
1. The first pipe supplies water at a rate (per minute) that is thrice faster than the second pipe.
2. The pipes fill 8 cubic feet of the tank in ten minute.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

In combined work problems, you don't always need to know the individual rates. Knowing the combined rate is often enough to solve the problem, as demonstrated by Statement (2).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a work-rate problem. The question asks for the total time required for two pipes working together to fill a cistern of a given capacity.
Step 2: Key Formula or Approach:
The fundamental formula for work-rate problems is:
\[ \text{Work} = \text{Rate} \times \text{Time} \] Or, to find the time:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} \] Here, the 'Work' is filling the cistern, which is 15 cubic feet. The 'Rate' is the combined rate of the two pipes. Let \( r_1 \) and \( r_2 \) be the rates of the first and second pipes, respectively. The combined rate is \( R = r_1 + r_2 \).
We need to find \( T = \frac{15}{r_1 + r_2} \).
Step 3: Detailed Explanation:
Analyze Statement (1): "The first pipe supplies water at a rate (per minute) that is thrice faster than the second pipe."
This gives us a relationship between the two rates:
\[ r_1 = 3 \times r_2 \] The combined rate is \( R = r_1 + r_2 = 3r_2 + r_2 = 4r_2 \).
The time to fill the cistern would be \( T = \frac{15}{4r_2} \).
Since we don't know the actual value of \( r_2 \), we cannot calculate a specific time T. Therefore, Statement (1) is not sufficient.
Analyze Statement (2): "The pipes fill 8 cubic feet of the tank in ten minute."
This statement gives us direct information about the combined rate of the two pipes.
Combined Rate, \( R = r_1 + r_2 = \frac{\text{Work}}{\text{Time}} = \frac{8 \text{ cubic feet}}{10 \text{ minutes}} = 0.8 \text{ cubic feet/minute} \).
Now we can calculate the total time to fill the 15 cubic feet cistern:
\[ T = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{15 \text{ cubic feet}}{0.8 \text{ cubic feet/minute}} = 18.75 \text{ minutes} \] This gives a unique value for the time. Therefore, Statement (2) is sufficient.
Step 4: Final Answer:
Statement (2) alone is sufficient to answer the question, but Statement (1) alone is not. The correct option is (B).