Two pipes A and B can fill an empty cistern when they are opened alternately for one hour each, starting with pipe A. Additional information: (A) Working alone, pipe A can fill the cistern in 40 hours. (B) Pipe B is one third as efficient as pipe A. Based on these statements, which option correctly describes the sufficiency of the information for finding the total time taken to fill the cistern?

Introduction / Context:
This is a data sufficiency problem involving rates of work. Two pipes fill a cistern alternately for one hour each. We need to know whether the given statements provide enough information to find the total time required. Instead of focusing only on numerical computation, we ask whether each statement alone or both together give us a unique answer for the filling time.

Given Data / Assumptions:

• The cistern is initially empty.
• Pipes A and B are opened alternately, one hour at a time, starting with pipe A.
• Statement (A): Pipe A alone can fill the cistern in 40 hours.
• Statement (B): Pipe B is one third as efficient as pipe A.
• Time taken to fill the cistern is measured in hours.

Concept / Approach:
When dealing with work rate problems, each pipe has a certain fraction of the cistern it can fill per hour. If A alone fills the cistern in 40 hours, its rate is 1/40 of the cistern per hour. If B is one third as efficient as A, its rate is (1/3) * (1/40). When the pipes work alternately, their combined effect over a two hour cycle is the sum of their individual hourly contributions. We must see which combination of statements allows us to compute these rates and then the total time.

Step-by-Step Solution:
Step 1: From statement (A), the rate of pipe A alone is 1/40 of the cistern per hour.Step 2: Statement (A) tells us nothing about pipe B, so we cannot determine the combined effect when the pipes alternate. Therefore statement (A) alone is not sufficient.Step 3: From statement (B), pipe B is one third as efficient as A. This compares B to A but does not tell us the actual rate of A. Without knowing A's rate in absolute terms, we cannot determine how long either pipe takes to fill the tank.Step 4: Hence statement (B) alone is also not sufficient.Step 5: Now combine statements (A) and (B). From (A), A's rate is 1/40 per hour. From (B), B's rate is one third of that, that is (1/3) * (1/40) = 1/120 per hour.Step 6: When the pipes work alternately starting with A, in the first hour A fills 1/40 of the cistern and in the second hour B fills 1/120 of the cistern.Step 7: Over a complete 2 hour cycle, the total fraction filled is 1/40 + 1/120 = (3/120 + 1/120) = 4/120 = 1/30 of the cistern.Step 8: Thus, in each 2 hour block, 1/30 of the cistern is filled. So, to fill the entire cistern, we need 30 such cycles.Step 9: Since each cycle is 2 hours, total time = 30 * 2 = 60 hours.Step 10: This calculation shows that with both statements we can uniquely determine the total time. With either statement alone we cannot.

Verification / Alternative check:
If we plug the rates back, in 60 hours there are 30 hours of operation for A and 30 hours for B.Total fraction filled by A = 30 * (1/40) = 30/40 = 3/4.Total fraction filled by B = 30 * (1/120) = 30/120 = 1/4.Total fraction filled = 3/4 + 1/4 = 1 full cistern, confirming our computation.

Why Other Options Are Wrong:
Option a is wrong because statement (A) alone does not give any information about B, which is essential for alternate working.Option b is wrong because statement (B) alone does not provide an absolute rate for either pipe.Option d is wrong because together the statements are clearly sufficient and lead to a unique answer.Option e is wrong because neither individual statement gives complete information, only their combination does.

Common Pitfalls:
Thinking that knowing the time taken by one pipe alone automatically determines the combined time with another pipe.Misinterpreting one third as efficient as as if it were one third slower in time, rather than one third in rate.Trying to solve numerically without first checking whether the information is sufficient in principle.

Final Answer:
The correct evaluation is that Both statements (A) and (B) together are sufficient, but neither alone is sufficient to determine the time taken to fill the cistern when the pipes work alternately.