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 choose the appropriate option. What is the average age of A and B ? I. The ratio between one-fifth of A's age and one-fourth of B's age is 1 : 2. II. The product of their ages is 20 times B's age.

### Concept & Logic To find the average of two variables, you need either the sum of those variables or the distinct values of both variables. In Data Sufficiency, we evaluate if the given algebraic statements can resolve to these requirements. ### Step-by-Step Solution * **Goal:** Find $\frac{A + B}{2}$. * **Evaluating Statement I alone:** "The ratio between one-fifth of A's age and one-fourth of B's age is 1 : 2." $$\frac{\frac{A}{5}}{\frac{B}{4}} = \frac{1}{2}$$ $$\frac{4A}{5B} = \frac{1}{2}$$ $$8A = 5B \Rightarrow A = \frac{5}{8}B$$ This gives us a ratio between A and B, but no concrete values or sum. *Statement I alone is NOT sufficient.* * **Evaluating Statement II alone:** "The product of their ages is 20 times B's age." $$A \times B = 20 \times B$$ Assuming B's age is not zero, we can divide both sides by $B$: $$A = 20$$ We know A's age, but we have no information about B. We cannot find the average. *Statement II alone is NOT sufficient.* * **Evaluating Statements I and II together:** From Statement II, we know $A = 20$. From Statement I, we know $8A = 5B$. Substituting the value of A into the equation from Statement I: $$8(20) = 5B$$ $$160 = 5B \Rightarrow B = 32$$ Since we can find distinct values for both A and B, we can calculate their average. ### Exam Strategy & Shortcut In Data Sufficiency, once you see that Statement II isolates a single variable ($A = 20$) and Statement I provides a proportional relationship ($8A = 5B$), you immediately know that substituting the known variable into the ratio will yield the second variable. You don't need to calculate $B = 32$ or the final average. You have enough proof to mark "Both are necessary" and move on. ### Common Pitfall A common mistake is glancing at Statement II ($AB = 20B$) and assuming it represents an equation with two unsolvable variables. Remembering that you can divide out a non-zero variable (age cannot be zero) is crucial to unlocking the value of $A$. ### Final Answer **Therefore, the correct answer is The data in both Statements I and II together are necessary to answer the question.**