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 ratio between the ages of the father and the son ? I. The sum of their ages is 50 years. II. 3 times the sum of their ages is equal to 5 times the father's age.

### Concept & Logic To find a ratio between two variables, you do not necessarily need their exact individual values. Any single linear equation that relates *only* those two variables directly to each other without constant terms can be rearranged to yield their ratio. ### Step-by-Step Solution Let the father's age be $F$ and the son's age be $S$. * **Goal:** Find $\frac{F}{S}$. * **Evaluating Statement I alone:** "The sum of their ages is 50 years." $$F + S = 50$$ This equation has a constant term (50). We cannot isolate $\frac{F}{S}$ because there are infinite combinations (e.g., 40 and 10, 30 and 20) that yield different ratios. *Statement I alone is NOT sufficient.* * **Evaluating Statement II alone:** "3 times the sum of their ages is equal to 5 times the father's age." $$3(F + S) = 5F$$ Expand the equation: $$3F + 3S = 5F$$ Subtract $3F$ from both sides: $$3S = 2F$$ Rearrange to find the ratio $\frac{F}{S}$: $$\frac{F}{S} = \frac{3}{2}$$ We have found the exact ratio without needing their individual ages. *Statement II alone is sufficient.* ### Exam Strategy & Shortcut Scan the equations for constant numbers. Statement I has "50", tying the variables to specific values but not fixing them. Statement II only contains multipliers (3 and 5) relating the variables to each other. An equation with only variables and coefficients (no loose constants) can *always* be rearranged into a ratio. Spotting this saves you from doing the algebra. ### Common Pitfall Students often assume that because they don't know the exact ages (like 30 and 20), they cannot answer the question, mistakenly combining I and II to solve for $F=30$ and $S=20$. Data Sufficiency asks *if* you can answer the question, not *what* the final numerical ages are. ### Final Answer **Therefore, the correct answer is 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.**