Solving Age Comparison Puzzles with Data Sufficiency

This question asks us to determine who is the oldest among Pankaj, Sonali, Rupali, and Tinu based on initial information and additional statements. We need to figure out which statement or combination of statements is sufficient to definitively identify the oldest person.

Initial Age Information

We are given two initial relationships:

• Pankaj is younger than Sonali. We can write this as: \( \text{Pankaj} < \text{Sonali} \)
• Rupali is older than Tinu. We can write this as: \( \text{Rupali} > \text{Tinu} \)

From this initial information alone, we cannot determine the oldest person. We don't know how Rupali and Tinu relate to Pankaj and Sonali.

Analyzing Statements for Sufficiency

Now, let's examine each statement and see if it helps us find the oldest person when combined with the initial information.

Statement A: Rupali is older than Pankaj

Statement A gives us: \( \text{Rupali} > \text{Pankaj} \)

Combining with initial information:

• \( \text{Pankaj} < \text{Sonali} \)
• \( \text{Rupali} > \text{Tinu} \)
• \( \text{Rupali} > \text{Pankaj} \)

Let's try to form a chain or check possibilities:

\( \text{Pankaj} < \text{Sonali} \)

\( \text{Tinu} < \text{Rupali} > \text{Pankaj} \)

We know Rupali is older than Tinu and Pankaj. But we don't know the relationship between Sonali and Rupali, or Sonali and Tinu. Sonali could be the oldest, Rupali could be the oldest. For example:

• Case 1: Sonali is oldest. \( \text{Tinu} < \text{Pankaj} < \text{Rupali} < \text{Sonali} \) (Satisfies P<S, R>T, R>P)
• Case 2: Rupali is oldest. \( \text{Tinu} < \text{Pankaj} < \text{Sonali} < \text{Rupali} \) (Satisfies P<S, R>T, R>P)
• Case 3: Sonali is oldest. \( \text{Pankaj} < \text{Tinu} < \text{Rupali} < \text{Sonali} \) (Satisfies P<S, R>T, R>P)

Since we cannot uniquely determine the oldest person using Statement A and the initial information, Statement A alone is not sufficient.

Statement B: Sonali is older than Rupali

Statement B gives us: \( \text{Sonali} > \text{Rupali} \)

Combining with initial information:

• \( \text{Pankaj} < \text{Sonali} \)
• \( \text{Rupali} > \text{Tinu} \)
• \( \text{Sonali} > \text{Rupali} \)

Let's combine these inequalities:

From \( \text{Sonali} > \text{Rupali} \) and \( \text{Rupali} > \text{Tinu} \), we get \( \text{Sonali} > \text{Rupali} > \text{Tinu} \).

So, Sonali is older than Rupali, and Rupali is older than Tinu. This means Sonali is also older than Tinu (\( \text{Sonali} > \text{Tinu} \)).

We also know \( \text{Pankaj} < \text{Sonali} \).

Let's put all known relations together:

• \( \text{Sonali} > \text{Rupali} \)
• \( \text{Rupali} > \text{Tinu} \)
• \( \text{Pankaj} < \text{Sonali} \)

From \( \text{Sonali} > \text{Rupali} > \text{Tinu} \), we know that Sonali is older than Rupali and Tinu. We also know Pankaj is younger than Sonali. This means Sonali is older than everyone else mentioned (Rupali, Tinu, and Pankaj).

Therefore, Sonali is the oldest.

Since we can definitively determine the oldest person using Statement B and the initial information, Statement B alone is sufficient.

Statement C: Tinu is youngest among all

Statement C gives us: \( \text{Tinu} \) is the youngest.

This means: \( \text{Tinu} < \text{Pankaj} \), \( \text{Tinu} < \text{Sonali} \), and \( \text{Tinu} < \text{Rupali} \).

Combining with initial information:

• \( \text{Pankaj} < \text{Sonali} \)
• \( \text{Rupali} > \text{Tinu} \) (consistent with Tinu being youngest)
• \( \text{Tinu} \) is youngest

From Tinu being the youngest, we know everyone else is older than Tinu. But we still only know \( \text{Pankaj} < \text{Sonali} \). We don't know the relationship between Sonali and Rupali. Sonali could be older than Rupali, or Rupali could be older than Sonali.

• Case 1: Sonali is oldest. \( \text{Tinu} < \text{Rupali} < \text{Pankaj} < \text{Sonali} \) (Satisfies P<S, R>T, T youngest). This case is invalid as P < S is satisfied, R > T is satisfied, T is youngest satisfied, but we also need to check if Sonali is the oldest here - yes.
• Let's take another case: Tinu < Pankaj < Rupali < Sonali (P<S, R>T, T youngest is satisfied, S is oldest).
• Let's take another case: Tinu < Pankaj < Sonali < Rupali (P<S, R>T, T youngest is satisfied, R is oldest).

Since we cannot uniquely determine the oldest person using Statement C and the initial information, Statement C alone is not sufficient.

Evaluating Combinations of Statements

We found that Statement B alone is sufficient. According to the options, if a single statement is sufficient, we should choose that option. Option 1 says "B only".

Let's quickly check why A and B together would also be sufficient but maybe not the *minimal* requirement if B alone works:

Initial: \( \text{Pankaj} < \text{Sonali} \), \( \text{Rupali} > \text{Tinu} \)

Statements A & B: \( \text{Rupali} > \text{Pankaj} \), \( \text{Sonali} > \text{Rupali} \)

From \( \text{Sonali} > \text{Rupali} \) and \( \text{Rupali} > \text{Pankaj} \), we get \( \text{Sonali} > \text{Rupali} > \text{Pankaj} \).

We also know \( \text{Rupali} > \text{Tinu} \). Combining \( \text{Sonali} > \text{Rupali} \) and \( \text{Rupali} > \text{Tinu} \) gives \( \text{Sonali} > \text{Rupali} > \text{Tinu} \).

So, we have \( \text{Sonali} > \text{Rupali} > \text{Pankaj} \) and \( \text{Sonali} > \text{Rupali} > \text{Tinu} \).

Both sets of inequalities show that Sonali is older than Rupali, and Rupali is older than Pankaj and Tinu. Therefore, Sonali is the oldest. Statements A and B together are sufficient. However, since Statement B alone was sufficient, "B only" is the correct option in a data sufficiency question format.

Summary of Sufficiency Analysis

Statement B alone is sufficient to answer the question "Who among them is oldest?".

Conclusion on Age Comparison Sufficiency

Based on our analysis, Statement B provides enough information when combined with the initial conditions (\( \text{Pankaj} < \text{Sonali} \) and \( \text{Rupali} > \text{Tinu} \)) to conclude that Sonali is the oldest. Neither Statement A nor Statement C is sufficient on its own.

Therefore, only Statement B is required to answer the question.

Revision Table: Understanding Age Comparisons

Additional Information on Logical Reasoning Puzzles

Logical reasoning puzzles, especially those involving comparisons like age or height, often require careful step-by-step analysis of given conditions and statements. It's helpful to:

• List all given facts clearly.
• Translate facts into symbolic representations (like inequalities).
• Analyze each piece of additional information (statements) independently, combined with the initial facts, to see if it solves the problem.
• If single statements aren't sufficient, analyze combinations of statements.
• Remember that in data sufficiency, the goal is often to find the *least* amount of information needed to answer the question. If Statement B alone works, then Statements A and B together might also work but "B only" is usually the intended answer if presented as an option.
• Draw diagrams or chains of relationships (like \( \text{A} > \text{B} > \text{C} \)) to visualize the information, especially when dealing with multiple people or items.

Practicing different types of comparison puzzles helps build skill in quickly identifying crucial relationships and determining sufficiency.