Directions: Each of the following questions consists of a question followed by three statements I, II and III. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. A, B and C secured $45\%$, $50\%$ and $60\%$ marks respectively in Biology. D's marks in Biology are 10 more than A's marks and 20 less than C's marks. Find out the total marks of the four students. I. Maximum marks for Biology are 200. II. Total of D's and A's marks is 190. III. C has obtained 120 marks.

### Concept & Logic This is a percentage logic puzzle. To find the exact total marks of the students, we need to convert their percentages into absolute values. To do this, we need the "Maximum Marks" ($M$) for the exam. ### Step-by-Step Solution * Given from Question Stem: * $A = 0.45M$ * $B = 0.50M$ * $C = 0.60M$ * $D = A + 10$ and $D = C - 20$ * *Crucial Deduction*: The question stem mathematically solves itself! * $A + 10 = C - 20 \implies C - A = 30$ * $0.60M - 0.45M = 0.15M = 30 \implies M = 200$ * Since $M$ can be found using *only* the question stem, none of the statements are technically required. However, since "None" is not an option, we evaluate the statements based on the test-setter's intent: providing an explicit shortcut to find $M$. * Evaluate Statement I: Directly gives maximum marks $M = 200$. Sufficient. * Evaluate Statement II: $D + A = 190 \implies (0.45M + 10) + 0.45M = 190 \implies 0.9M = 180 \implies M = 200$. Sufficient. * Evaluate Statement III: $C = 120 \implies 0.60M = 120 \implies M = 200$. Sufficient. * Since each statement independently allows us to find $M$ (ignoring the derivation in the stem), any single statement works. ### Exam Strategy & Shortcut Test setters sometimes accidentally provide self-sufficient question stems. When you notice this, but "None are required" is missing from the options, evaluate each statement to see if it provides the missing linchpin variable (in this case, Maximum Marks) directly. If they all do, the intended answer is "Any one of the three". ### Common Pitfall Students often get confused when the question stem is self-sufficient and waste time second-guessing their math. Trust your algebra. If you derive the maximum marks from the stem, realize the statements are just alternative pathways to the same number. ### Final Answer **Therefore, the correct answer is Any one of the three.**