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 statement(s) is/are necessary to answer the given question. What will be the sum of two numbers? I. Among the two numbers, the bigger number is greater than the smaller number by 6. II. 40% of the smaller number is equal to 30% of the bigger number. III. The ratio between half of the bigger number and one-third of the smaller number is 2 : 1.

Concept & Logic To find the sum of two numbers, we need to determine the exact values of both numbers. This requires two distinct linear equations representing their relationship. Let the bigger number be $x$ and the smaller number be $y$. We need to find $x + y$. Step-by-Step Solution * **Analyze Statement I:** The bigger number is greater by 6. Equation: $x - y = 6$. * **Analyze Statement II:** $40\%$ of smaller $= 30\%$ of bigger. * Equation for II: $0.40y = 0.30x \implies 4y = 3x \implies x/y = 4/3$. * **Analyze Statement III:** Ratio between half of bigger and one-third of smaller is 2:1. * Equation for III: $(x/2) / (y/3) = 2/1 \implies (3x)/(2y) = 2 \implies 3x = 4y \implies x/y = 4/3$. * Notice that Statement II and Statement III yield the exact same ratio ($x/y = 4/3$). They are mathematically identical. * To find the exact values, we must combine the difference from Statement I with the ratio from either Statement II or Statement III. * Using I and II (or I and III): $x = 4k, y = 3k \implies 4k - 3k = 6 \implies k=6$. The numbers are 24 and 18, and their sum is 42. * Thus, we need "Statement I and either Statement II or Statement III" to answer the question. Since this option is not available, we must select "None of these". Exam Strategy & Shortcut Ratios, percentages, and fractions often express the exact same relationship disguised in different formats. Instantly simplify percentage relationships ($40\%$ and $30\%$) into simple ratios ($4:3$) and do the same for fraction relationships. If they match, you know the statements are redundant, saving you from calculating the final answer. Common Pitfall Many test-takers assume "Any two of the three" is correct because they see three distinct text strings. However, if you combine Statement II and Statement III, you only get the ratio $4:3$ without any absolute value (like the difference of 6), leaving the exact numbers unknown. Final Answer **Therefore, the correct answer is None of these.**