A question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements carefully and decide the appropriate answer.
Question: 5 people, P, Q, X, Y and Z, have an exam on different days of the same week between Monday and Friday. Who has an exam on Thursday?
Statements:
I. P has an exam on Wednesday. Only one person has an exam between P and Q. Z has an exam immediately before X.
II. No one has an exam between P and Y. Only two people have an exam between Y and X. Z does not have an exam on Monday.2110
Correct Answer :
Data in statements I and II together (and not statement I alone or statement II alone) is sufficient to answer the question
Solution :
To determine who has an exam on Thursday, let us analyze the information provided in the question and the two statements step-by-step.
The 5 days of the week are Monday, Tuesday, Wednesday, Thursday, and Friday.
Let us represent them as: Mon, Tue, Wed, Thu, Fri.
Analyzing Statement I alone:
1. P has an exam on Wednesday (Wed = P).
2. Only one person has an exam between P and Q. This means Q can have an exam on either Monday (Mon = Q) or Friday (Fri = Q).
3. Z has an exam immediately before X, meaning Z and X must occupy two consecutive days in the order ZX.
Let us consider the two cases for Q:
- Case 1 (Q is on Monday): The arrangement so far is Q (Mon), _ (Tue), P (Wed), _ (Thu), _ (Fri). The only remaining consecutive days for the ZX pair are Thursday and Friday. Thus, Z is on Thursday, X is on Friday, and the remaining person Y is on Tuesday. The final schedule is: Q (Mon), Y (Tue), P (Wed), Z (Thu), X (Fri). In this case, Z has the exam on Thursday.
- Case 2 (Q is on Friday): The arrangement so far is _ (Mon), _ (Tue), P (Wed), _ (Thu), Q (Fri). The only remaining consecutive days for the ZX pair are Monday and Tuesday. Thus, Z is on Monday, X is on Tuesday, and the remaining person Y is on Thursday. The final schedule is: Z (Mon), X (Tue), P (Wed), Y (Thu), Q (Fri). In this case, Y has the exam on Thursday.
Since we have two different possible answers (Z or Y), Statement I alone is not sufficient.
Analyzing Statement II alone:
1. No one has an exam between P and Y, which means P and Y are scheduled on adjacent days.
2. Only two people have an exam between Y and X.
3. Z does not have an exam on Monday.
Without knowing the exact day for any person, multiple arrangements can be formed. For example, if Y is on Thursday and X is on Monday, or Y is on Tuesday and X is on Friday, we get different positions for the exam on Thursday. Thus, Statement II alone is not sufficient.
Analyzing Statements I and II together:
From Statement I, we know P is on Wednesday (Wed = P) and Z and X are consecutive (ZX).
From Statement II, P and Y are adjacent, so Y must be on Tuesday or Thursday.
Let us evaluate these two sub-cases:
- Sub-case A (Y is on Tuesday):
Since Y is on Tuesday, and there are only two people between Y and X, X must be on Friday (with Wed and Thu in between).
Since Z is immediately before X, Z must be on Thursday.
This leaves Monday for Q. The schedule becomes: Q (Mon), Y (Tue), P (Wed), Z (Thu), X (Fri).
Checking all conditions: Z is not on Monday (satisfied). This is a valid arrangement.
- Sub-case B (Y is on Thursday):
Since Y is on Thursday, and there are only two people between Y and X, X must be on Monday (with Tue and Wed in between).
But Statement I requires Z to have an exam immediately before X. Since X is on Monday, there is no day before Monday, making it impossible to place Z. Thus, this sub-case is invalid.
Therefore, only the arrangement from Sub-case A is valid, where Z has the exam on Thursday.
Since we can uniquely determine that Z has the exam on Thursday only by combining both statements, the data in statements I and II together is sufficient, but neither statement alone is sufficient.
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