Direction for the Questions Each... - Solved

Question 64

Logical Reasoning Statement and Conclusions Hard

Direction for the Questions  Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the given question. Read both statements and Give the answer (U) if the data in statement I alone are sufficient to answer the question, while the data in statement II alone are not sufficient to answer the question. Give the answer (V) if the data in statement I| alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question. Give the answer (W) if the data either in Statement I or in Statement II alone are sufficient to the answer of the question. Give the answer (X) if the data in both statements I and II together are not sufficient to answer the question. Give the answer (Y) if the data in both statements I and II together are necessary to answer the question. How much time will the leak take to empty the full cistern? (i) The cistern is normally filled in 9 hours. (ii) It takes one hour more than the usual time to fill the cistern because of a leak in the bottom.

(A) V

(B) U

(C) X

(D) Y

✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

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.