Understanding Rank and Data Sufficiency Problems

This question is a classic example of a data sufficiency problem where we are given a question and two statements, and we need to determine if either statement alone, both statements together, or neither statement is sufficient to answer the question. The core concept here is understanding ranks in a sequence or list, like a class of students.

The question asks for X's rank from the top in a class of 50 students.

We have a total of 50 students in the class.

A fundamental formula used in ranking problems is:

\(\text{Total number of students} = \text{Rank from top} + \text{Rank from bottom} - 1\)

We will analyze each statement independently to see if it provides enough information to find X's rank from the top.

Analyzing Statement 1 for Rank Calculation

Statement 1 says:

1. Y's rank is 4 ranks below that of X.
2. Y's rank is 30th from the bottom.

Let X's rank from the top be \(R_X\).

According to point 1, Y's rank from the top (\(R_Y\)) is 4 ranks below X's rank from the top. This means:

\(R_Y = R_X + 4\)

According to point 2, Y's rank from the bottom is 30.

Now we can use the rank formula for Y:

\(\text{Total students} = \text{Y's rank from top} + \text{Y's rank from bottom} - 1\)

\(50 = R_Y + 30 - 1\)

\(50 = R_Y + 29\)

Subtract 29 from both sides to find \(R_Y\):

\(R_Y = 50 - 29\)

\(R_Y = 21\)

So, Y's rank from the top is 21. Now substitute this value back into the relationship between X's and Y's ranks:

\(R_Y = R_X + 4\)

\(21 = R_X + 4\)

Subtract 4 from both sides to find \(R_X\):

\(R_X = 21 - 4\)

\(R_X = 17\)

From Statement 1 alone, we can determine that X's rank from the top is 17. Thus, Statement 1 alone is sufficient to answer the question.

Analyzing Statement 2 for Rank Calculation

Statement 2 says:

1. Z's rank is 2 ranks above that of X.
2. Z's rank is 32nd from the bottom.

Let X's rank from the top be \(R_X\).

According to point 1, Z's rank from the top (\(R_Z\)) is 2 ranks above X's rank from the top. This means:

\(R_Z = R_X - 2\)

According to point 2, Z's rank from the bottom is 32.

Now we can use the rank formula for Z:

\(\text{Total students} = \text{Z's rank from top} + \text{Z's rank from bottom} - 1\)

\(50 = R_Z + 32 - 1\)

\(50 = R_Z + 31\)

Subtract 31 from both sides to find \(R_Z\):

\(R_Z = 50 - 31\)

\(R_Z = 19\)

So, Z's rank from the top is 19. Now substitute this value back into the relationship between X's and Z's ranks:

\(R_Z = R_X - 2\)

\(19 = R_X - 2\)

Add 2 to both sides to find \(R_X\):

\(R_X = 19 + 2\)

\(R_X = 21\)

From Statement 2 alone, we can determine that X's rank from the top is 21. Thus, Statement 2 alone is sufficient to answer the question.

Conclusion: Evaluating Sufficiency

We found that:

• Statement 1 alone is sufficient to determine X's rank from the top (it's 17).
• Statement 2 alone is sufficient to determine X's rank from the top (it's 21).

Since either statement by itself gives a unique value for X's rank, either statement alone is sufficient to answer the question.

Matching the Analysis to the Options

Let's look at the given options based on our analysis:

• Option 1: 2 alone is sufficient while 1 alone is not sufficient to answer the given question. (Incorrect, as 1 is sufficient)
• Option 2: Either 1 or 2 is sufficient to answer the given question. (Correct, as both 1 alone and 2 alone are sufficient)
• Option 3: Both 1 and 2 are sufficient to answer the given question. (While true, option 2 is more precise in indicating individual sufficiency)
• Option 4: 1 alone is sufficient while 2 alone is not sufficient to answer the given question. (Incorrect, as 2 is sufficient)

The most appropriate answer is that either Statement 1 or Statement 2 is sufficient.

Revision Table: Key Concepts in Data Sufficiency

Additional Information: Understanding Rank Problems

Rank problems in logical reasoning or quantitative aptitude tests often involve determining the position of a person or object in a sequence from either end (like top/bottom, left/right, front/back). The total number of items in the sequence is usually provided or can be derived.

Key points to remember:

• The formula Total = Rank from one end + Rank from other end - 1 is crucial. The '-1' is because the person/object is counted twice (once from each end).
• If a person's rank from one end and the total number of people are known, their rank from the other end can be found.
• Problems often involve relationships between the ranks of two or more individuals. For example, "A is 5 ranks below B" means A's rank number is 5 greater than B's rank number (assuming ranking from the top).
• In data sufficiency, evaluate each statement independently first. If a statement is sufficient alone, that eliminates options requiring both statements or neither. If neither statement is sufficient alone, then evaluate if both statements together are sufficient.
• In this specific problem, it is important to note that Statement 1 leads to X having rank 17, while Statement 2 leads to X having rank 21. This doesn't mean the statements contradict each other in the context of the original problem; it means that under the conditions of Statement 1, X's rank would be 17, and under the conditions of Statement 2, X's rank would be 21. The question asks if *either* set of conditions is enough to *determine* X's rank, and indeed, each set of conditions allows us to determine a specific rank for X based on that statement's premise.