In the following data sufficiency question on family relations, use the given statements to decide whether the data are sufficient to answer the question. Question: H is the mother of T. How is T related to W? Statement I: W is the only daughter of H. Statement II: W is the sister of T.

Introduction / Context:
This is a data sufficiency question involving family relationships. We know that H is the mother of T, and we are given two additional statements about W. The goal is not simply to find the exact relationship but to determine which statements provide enough information to answer the question, “How is T related to W?” This evaluates your ability to use relational clues and gender information in logical reasoning.

Given Data / Assumptions:

• Base fact: H is the mother of T.
• Statement I: W is the only daughter of H.
• Statement II: W is the sister of T.
• All relationships are standard: mother, daughter, sister, brother, and so on.
• The answer required is T's relation to W (for example, brother, sister, sibling).

Concept / Approach:
For family relation questions, we track parent child links and gender information. A statement is sufficient if it allows a single unambiguous relationship to be inferred. We must consider each statement separately first, and then together, to decide whether the data are sufficient. Note that the given base fact, “H is the mother of T,” already tells us that T is a child of H and that H is female. Additional statements may reveal T's gender and sibling relationships.

Step-by-Step Solution:
Step 1: Use Statement I with the base fact. H is the mother of T, so T is a child of H. Statement I says W is the only daughter of H. Since W is the only daughter, there cannot be any other daughter of H. Step 2: T is already known to be a child of H. If W is the only daughter of H, then T cannot be a daughter, otherwise W would not be the only daughter. That means T must be a son of H. Step 3: From the base fact and Statement I together, H is the mother of both W (daughter) and T (son). This makes W and T siblings, with W as the sister and T as the brother. The question asks, “How is T related to W?” The answer is: T is W's brother. Step 4: Therefore, Statement I alone allows us to uniquely determine T's relation to W. So Statement I is sufficient. Step 5: Now use Statement II alone with the base fact. We know H is the mother of T, and Statement II says W is the sister of T. Knowing that W is T's sister tells us that W is female and that W and T share at least one parent. Step 6: However, Statement II alone does not tell us whether H is also W's mother or only T's mother. The question is still, “How is T related to W?” Since we do not know T's gender from Statement II alone (T could be male or female), T could be W's brother or W's sister. Thus T's exact relation to W is ambiguous with Statement II alone.

Verification / Alternative check:
To confirm, construct two possible scenarios consistent with Statement II:

• Scenario 1: T is male and H is mother of both T and W. Then W is T's sister and T is W's brother.
• Scenario 2: T is female and H is mother of both T and W. Then W is T's sister and T is also W's sister.

Since both scenarios satisfy the base statement and Statement II but give different answers to how T is related to W, Statement II alone is not sufficient.

Why Other Options Are Wrong:
Option b: Claiming that Statement II alone is sufficient is wrong because T's gender remains unknown and two different relationships are possible. Option c: Saying that both statements together are required is unnecessary, because Statement I already gives a complete answer by itself. Option d: Suggesting that even both statements together are not sufficient is incorrect, as with Statement I we already pinpoint T's relationship as brother. Option e: Claiming that either statement alone is sufficient is false because Statement II alone is not sufficient.

Common Pitfalls:
A frequent mistake is to assume that when one person is a sister, the other must be a brother, which is not logically necessary. Another common error is to ignore the phrase “only daughter” in Statement I, which is crucial for deducing that T must be male. Always carefully track both gender and parent child links when dealing with such questions, and check whether more than one relationship remains possible.

Final Answer:
The data in statement I alone are sufficient to answer the question, but the data in statement II alone are not sufficient.