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. Divya is twice as old as Shruti. What is the difference in their ages ? I. Five years hence, the ratio of their ages would be 9 : 5. II. Ten years back, the ratio of their ages was 3 : 1.

### Concept & Logic The problem already provides one foundational linear equation connecting the two variables. Therefore, providing *any* other distinct linear equation will create a solvable system of two equations, allowing us to find both ages and their difference. ### Step-by-Step Solution Let Divya's present age be $D$ and Shruti's present age be $S$. * **Given:** Divya is twice as old as Shruti. $$D = 2S$$ * **Goal:** Find the difference $D - S$. (Since $D = 2S$, the difference is just $2S - S = S$. So, we simply need to find Shruti's age). * **Evaluating Statement I alone:** "Five years hence, the ratio of their ages would be 9 : 5." $$\frac{D + 5}{S + 5} = \frac{9}{5}$$ We can substitute $D = 2S$ into this equation: $$\frac{2S + 5}{S + 5} = \frac{9}{5}$$ This is one equation with one variable ($S$). It can be solved to find $S$. *Statement I alone is sufficient.* * **Evaluating Statement II alone:** "Ten years back, the ratio of their ages was 3 : 1." $$\frac{D - 10}{S - 10} = \frac{3}{1}$$ Again, substitute $D = 2S$ into this equation: $$\frac{2S - 10}{S - 10} = 3$$ This is also one equation with one variable ($S$). It can be solved to find $S$. *Statement II alone is sufficient.* ### Exam Strategy & Shortcut Don't waste time solving the equations! The prompt gives you Equation 1 ($D=2S$). Statement I gives you Equation 2. Statement II gives you Equation 3. Basic algebra dictates that 2 distinct equations are enough to solve for 2 variables. Therefore, Equation 1 + Equation 2 works. Alternatively, Equation 1 + Equation 3 works. Both statements independently provide the missing link. Mark "Either" immediately. ### Common Pitfall A common time-wasting pitfall is actually doing the cross-multiplication (e.g., $10S + 25 = 9S + 45 \Rightarrow S = 20$). In Data Sufficiency, you just need to determine *solvability*, not the actual solution. Stop calculating as soon as you reduce the problem to one equation with one unknown. ### Final Answer **Therefore, the correct answer is The data either in Statement I or in Statement II alone are sufficient to answer the question.**