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. How much is four-fifths of the number? I. Three-fourths of the number is 2.5 less than its four-fifths. II. Half of the number added to it is 75.

Concept & Logic Any single linear equation with one variable can be solved for that variable. If a statement provides a complete algebraic relationship involving only the unknown number, that statement alone is mathematically sufficient to find the number, and consequently, four-fifths of it. Step-by-Step Solution * **Goal:** Find $\frac{4}{5}x$, where $x$ is the unknown number. * **Evaluating Statement I:** Three-fourths of the number is $2.5$ less than its four-fifths. Translating this into algebra: $$\frac{4}{5}x - \frac{3}{4}x = 2.5$$ This is a linear equation with one variable. Solving this will yield a unique value for $x$, which means we can definitively find $\frac{4}{5}x$. *(Self-proof: $\frac{16x - 15x}{20} = 2.5 \implies \frac{x}{20} = 2.5 \implies x = 50$. So $\frac{4}{5} \times 50 = 40$.)* Thus, Statement I is sufficient alone. * **Evaluating Statement II:** Half of the number added to it is $75$. Translating this into algebra: $$x + \frac{x}{2} = 75$$ $$\frac{3x}{2} = 75$$ This is also a linear equation with one variable. Solving this yields a unique value for $x$, allowing us to find $\frac{4}{5}x$. *(Self-proof: $3x = 150 \implies x = 50$. So $\frac{4}{5} \times 50 = 40$.)* Thus, Statement II is sufficient alone. Exam Strategy & Shortcut (MANDATORY) In Data Sufficiency, you do NOT need to calculate the final answer. The moment you translate a word problem statement into a solvable single-variable linear equation (e.g., $ax + b = cx + d$), you can instantly declare that statement sufficient and move on. This saves massive amounts of time during the exam. Common Pitfall (MANDATORY) A common mistake is actually taking the time to solve for $x$ and then calculating $\frac{4}{5}x$ for both statements to "make sure" they match. In actual exam conditions, calculating the final numeric answer wastes precious seconds. Trust the algebraic rules of sufficiency. Final Answer **Therefore, the correct answer is that the data either in Statement I or in Statement II alone are sufficient to answer the question.**