Directions: Each of the questions below consists of a statement and/or a question and two statements labelled I and II given below it. You have to decide whether the data provided in the statements 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; and Give answer (e) if the data in both Statements I and II are necessary to answer the question. What is the value of $a$? I. Ratio of $a$ and $b$ is $3 : 5$, where $b$ is positive. II. Ratio of $2a$ and $b$ is $12 : 10$, where $a$ is positive.

### Concept & Redundant Information To find a specific numerical value for a variable, you need an equation that connects it to a known constant. Ratios only provide relationships between variables, not their absolute values. ### Step-by-Step Solution * **Analyzing Statement I:** * We are given $\frac{a}{b} = \frac{3}{5}$. This means $a = \frac{3}{5}b$. * Because $b$ is an unknown positive number, $a$ can be any positive number. * Statement I is not sufficient. * **Analyzing Statement II:** * We are given $\frac{2a}{b} = \frac{12}{10}$. * Simplifying the right side: $\frac{2a}{b} = \frac{6}{5}$. * Dividing by 2: $\frac{a}{b} = \frac{3}{5}$. * This is the exact same ratio provided in Statement I. It still does not yield an absolute numerical value for $a$. * Statement II is not sufficient. * **Combining Both Statements:** * Since both statements provide identical mathematical information ($\frac{a}{b} = \frac{3}{5}$), combining them does not give us any new equations to solve for the exact value of $a$. ### Exam Strategy & Shortcut Recognize when statements are just different expressions of the exact same fact. If statement I doesn't work and statement II is mathematically identical, combining them will never work either. ### Common Pitfall A common error is assuming that $a$ strictly equals 3 just because the ratio is $3:5$. A ratio defines proportions, so $a$ could be 3, 6, 9, 30, etc. ### Final Answer Therefore, the correct answer is **if the data even in both Statements I and II together are not sufficient to answer the question**.