Data Sufficiency Analysis

The question asks for the time it takes for a leak to empty a full cistern. To answer this, we need to determine the rate of the leak. Let's analyze the sufficiency of the given statements.

Let F be the rate of filling the cistern (in cisterns per hour).
Let L be the rate of the leak (in cisterns per hour).

The question is asking for the value of 1/L.

Statement (I): "The cistern is normally filled in 9 hours."

• "Normally" implies without a leak. This statement allows us to calculate the filling rate, F.
• Time = 1 / Rate → 9 = 1 / F → F = 1/9 cisterns/hour.
• However, this statement provides no information about the leak rate, L.
• Conclusion: Statement (I) alone is not sufficient.

Statement (II): "It takes one hour more than the usual time to fill the cistern because of a leak in the bottom."

• Let T_usual be the normal time to fill. Then T_with_leak = T_usual + 1.
• In terms of rates, this translates to: 1/F + 1 = 1 / (F - L).
• This statement provides a relationship between F and L, but since we don't know the value of F, we cannot find a specific value for L.
• Conclusion: Statement (II) alone is not sufficient.

Combining Statements (I) and (II)

• From Statement (I), we know F = 1/9.
• We can substitute this value into the equation from Statement (II):
• 1 / (1/9) + 1 = 1 / (1/9 - L)
• 9 + 1 = 1 / (1/9 - L)
• 10 = 1 / (1/9 - L)
• 10 × (1/9 - L) = 1
• 10/9 - 10L = 1
• 10L = 10/9 - 1 = 1/9
• L = 1/90 cisterns/hour.
• The time for the leak to empty the cistern is 1/L, which is 90 hours.
• Since we can find a unique answer by combining both statements, they are sufficient together.

Final Answer

The data in both statements I and II together are necessary to answer the question. This corresponds to the answer option Y.