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 difference between the ages of Sumit and Dinesh? (Bank P.O., 2006) I. The ratio of their ages is 7 : 9 respectively. II. Five years hence the sum of their ages will be 58 years.

### Concept & Ratios and Sums To find the exact difference between two unknown quantities (ages), we need both a relative relationship (like a ratio) and an absolute total or difference to ground the values. ### Step-by-Step Solution - **Evaluating Statement I:** The ratio of Sumit's to Dinesh's age is $7:9$. Let their ages be $7x$ and $9x$. The difference is $2x$. We cannot find the exact difference without knowing $x$. Thus, Statement I alone is not sufficient. - **Evaluating Statement II:** Five years from now, the sum of their ages will be $58$. This means their present sum is $58 - (5 + 5) = 48$. This gives us $S + D = 48$. This does not tell us their individual ages or the difference. Thus, Statement II alone is not sufficient. - **Evaluating Both Statements Together:** Combining the two, we know their present sum is $48$ (from II) and their ratio is $7:9$ (from I). - We can set up the equation: $$ 7x + 9x = 48 $$ $$ 16x = 48 \Rightarrow x = 3 $$ - The difference is $9x - 7x = 2x = 2(3) = 6$. - Since we can find the definitive answer using both statements, both are necessary. ### Exam Strategy & Shortcut Recognize that Statement I gives a proportional relationship (ratio) while Statement II gives a discrete, absolute value (sum). In typical age problems, combining one ratio and one absolute sum will always yield the exact ages. You can immediately mark "both necessary" without calculating the final difference. ### Common Pitfall A common mistake when evaluating Statement II is subtracting only $5$ years from the future sum instead of $10$ years (5 years for each of the two people) to find the present sum. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II are necessary to answer the question**.