Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statements is/are necessary to answer the question. What is the difference between the ages of Y and X? I. The ratio between the ages of X and Y is 2 : 3. II. Y's age is 50% more than X's age. III. One-fourth of X's age is equal to one-sixth of Y's age.

### Concept & Logic To find an absolute numerical value (like the exact difference in ages), at least one piece of given data must provide an absolute number (e.g., a sum, a specific age, or a difference). If all statements only provide relative data (ratios, fractions, or percentages), the exact values cannot be determined. ### Step-by-Step Solution Let's analyze the mathematical meaning of each statement: * **Statement I:** $\frac{X}{Y} = \frac{2}{3}$ $\Rightarrow 3X = 2Y$ * **Statement II:** $Y = X + 50\% \text{ of } X \Rightarrow Y = 1.5X \Rightarrow \frac{X}{Y} = \frac{1}{1.5} = \frac{2}{3} \Rightarrow 3X = 2Y$ * **Statement III:** $\frac{X}{4} = \frac{Y}{6} \Rightarrow 6X = 4Y \Rightarrow 3X = 2Y$ **Deduction:** All three statements provide the exact same relative relationship between $X$ and $Y$. They all simplify to the ratio $X:Y = 2:3$. Because we have zero absolute values provided (no total sum, no future age, no actual difference given in years), we have infinitely many solutions. For example, $X$ and $Y$ could be 20 and 30 (difference 10), or 40 and 60 (difference 20). ### Exam Strategy & Shortcut **The Absolute Value Rule:** Scan the data sufficiency statements immediately for absolute numbers. If the question asks for a specific value (difference, sum, age) and all statements contain ONLY ratios, percentages, or fractions relative to each other, the answer is always "Cannot be determined". ### Common Pitfall Students often get tricked by the different phrasing (ratio, percentage, fraction) and assume they represent independent pieces of information. This leads them to mistakenly believe combining them will yield a solution, wasting valuable time trying to solve a redundant system of equations. ### Final Answer **Therefore, the correct answer is Question cannot be answered even with information in all three statements.**