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 two-digit number where the digit at the unit's place is smaller? I. The difference between the two digits is 5. II. The sum of the two digits is 7.

Concept & Logic When solving for two unknown variables (the digits of a number), you typically need two distinct linear equations. The question stem provides a crucial asymmetrical constraint: the unit's digit is strictly smaller than the tens digit. Step-by-Step Solution * **Given Constraints:** Let the tens digit be $x$ and the units digit be $y$. The question states that the unit's digit is smaller: $x > y$. * **Evaluating Statement I:** The difference between the two digits is $5$. Since $x > y$, we can securely write: $$x - y = 5$$ Possible $(x,y)$ pairs: $(5,0), (6,1), (7,2), (8,3), (9,4)$. This yields multiple numbers ($50, 61, 72, 83, 94$). Statement I is not sufficient alone. * **Evaluating Statement II:** The sum of the two digits is $7$. $$x + y = 7$$ Possible $(x,y)$ pairs where $x > y$: $(4,3), (5,2), (6,1), (7,0)$. This yields multiple numbers ($43, 52, 61, 70$). Statement II is not sufficient alone. * **Evaluating Both Statements Together:** We have a standard system of linear equations: 1) $x - y = 5$ 2) $x + y = 7$ Adding the equations together: $$2x = 12 \implies x = 6$$ Substituting $x$ back into the second equation: $$6 + y = 7 \implies y = 1$$ The number is exactly $61$. Both statements are required to find this unique solution. Exam Strategy & Shortcut (MANDATORY) Recognize that an initial constraint like "unit's digit is smaller" removes the ambiguity of absolute difference. Once you see you have $x - y = A$ and $x + y = B$, you know immediately from basic algebra that a unique solution exists as long as the results are valid single integers. You can mark (e) without even calculating the final number $61$. Common Pitfall (MANDATORY) Students often ignore the condition in the main question body ("unit's place is smaller"). If you miss this, you might think Statement I means $|x - y| = 5$, and ultimately conclude that even both statements together are not sufficient. Always extract mathematical constraints from the question text itself before reading the statements. Final Answer **Therefore, the correct answer is that the data in both Statements I and II together are necessary to answer the question.**