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 population of State A? I. Population of State B is $80\%$ of the population of State A. II. Average population of States A and B is 18 lakhs.

### Concept & Logic Data sufficiency problems require evaluating if the provided facts yield a unique numerical solution. We set up algebraic equations for the given conditions to test solvability. ### Step-by-Step Solution * Let the population of State A be $A$ and the population of State B be $B$. * Evaluate Statement I: * $B = 0.8A$ * This gives a ratio but no absolute values. We cannot find $A$. Statement I alone is not sufficient. * Evaluate Statement II: * The average population is 18 lakhs. * $$(A + B) / 2 = 18 \implies A + B = 36$$ * We have one equation with two variables. We cannot find $A$. Statement II alone is not sufficient. * Evaluate Both Statements Together: * Substitute $B = 0.8A$ into the second equation: * $$A + 0.8A = 36$$ * $$1.8A = 36 \implies A = 20$$ * We get a unique value for $A$. Both statements together are required to answer the question. ### Exam Strategy & Shortcut When dealing with linear equations in data sufficiency, you rarely need to solve them completely. Recognize that you need two distinct linear equations to solve for two unknown variables. Statement I gives one equation, Statement II gives the second. Since they are mathematically independent, combining them will definitively yield the answer. ### Common Pitfall Students often waste time calculating the final absolute answer ($A = 20$). In data sufficiency, once you establish that a unique solution exists, stop calculating to save time. ### Final Answer **Therefore, the correct answer is if the data in both statements I and II together are necessary to answer the question.**