Directions : 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 question. Read both the statements and Give answer (a) 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 answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. What is the three-digit number? I. Two-fifth of that number is less than half of that number by 20. II. One-fourth of that number is 25% of that number.

Concept & Logic A statement that presents a solvable single-variable equation will yield a distinct numeric answer. A statement that presents a tautology (a universally true fact) provides zero mathematical constraints and is useless for finding a specific variable. Step-by-Step Solution * **Goal:** Find an exact three-digit number, let us call it $x$. * **Evaluating Statement I:** Two-fifth of that number is less than half of that number by 20. Translating this into an equation: $$\frac{1}{2}x - \frac{2}{5}x = 20$$ Find a common denominator (10): $$\frac{5x - 4x}{10} = 20$$ $$\frac{x}{10} = 20 \implies x = 200$$ We found a unique three-digit number (200). Thus, Statement I alone is sufficient. * **Evaluating Statement II:** One-fourth of that number is 25% of that number. Mathematically, one-fourth is $\frac{1}{4}$, and 25% is $\frac{25}{100} = \frac{1}{4}$. $$\frac{1}{4}x = \frac{1}{4}x$$ This is true for absolutely any number in existence. It provides no new information to narrow down the value of $x$. Thus, Statement II alone is not sufficient. Exam Strategy & Shortcut (MANDATORY) Scan the statements for tautologies. Statement II translates to "a quarter of the number is a quarter of the number." You can instantly discard this statement as insufficient in zero seconds, leaving you only needing to verify if Statement I forms a solvable equation. Common Pitfall (MANDATORY) Do not blindly combine statements just because one looks complicated and one looks simple. Students occasionally select option (d) or (e) assuming they must use percentages to verify the fraction, ignoring that Statement II is merely a restatement of basic math definitions. Final Answer **Therefore, the correct answer is that 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.**