P, Q and R secured 45%, 50% and 60% marks respectively in Biology. M scored 12.5 marks more than P and 4 marks less than R. You are given three statements about their marks. Which statements are actually required to determine the individual marks of all four students?

Introduction / Context:
This is a data sufficiency style question framed in the context of Biology marks. The problem describes the percentages obtained by three students, P, Q and R, and then relates the marks of a fourth student M to the marks of P and R. It then lists three additional statements about totals and specific marks, and asks which of these statements are required to determine the individual marks of all four students. The challenge is to see whether the original information alone is enough to uniquely determine the marks, or whether some of the extra statements are necessary.

Given Data / Assumptions:

• P secures 45% marks in Biology.
• Q secures 50% marks in Biology.
• R secures 60% marks in Biology.
• Let the maximum marks in Biology be Mmax.
• M secures 12.5 more marks than P.
• M secures 4 fewer marks than R.
• Statements given:
  • A: The total of the marks obtained by the four students is 311.5.
  • B: The total of M and P marks is 147.5.
  • C: R has obtained 84 marks.
• We must decide which of A, B, C are actually required to find individual marks.

Concept / Approach:
Let the maximum marks be Mmax. Then P = 0.45 * Mmax, Q = 0.50 * Mmax and R = 0.60 * Mmax. M is defined both as P + 12.5 and as R - 4. These relationships already form a system of equations involving P, Q, R, M and Mmax. If this system has a unique solution, then we do not need any of the additional statements A, B or C to determine the marks. So our main task is to check whether the base equations from the question alone are sufficient to uniquely determine all values.

Step-by-Step Solution:
Step 1: Express P, Q and R in terms of Mmax. P = 0.45 * Mmax, Q = 0.50 * Mmax, R = 0.60 * Mmax. Step 2: Use the relations for M. M = P + 12.5 and M = R - 4. Step 3: Equate P + 12.5 and R - 4. P + 12.5 = R - 4. Step 4: Substitute P and R in terms of Mmax: 0.45 * Mmax + 12.5 = 0.60 * Mmax - 4. Step 5: Rearrange: 12.5 + 4 = 0.60 * Mmax - 0.45 * Mmax. Step 6: 16.5 = 0.15 * Mmax, so Mmax = 16.5 / 0.15 = 110. Step 7: Now compute P, Q and R: P = 0.45 * 110 = 49.5, Q = 0.50 * 110 = 55, R = 0.60 * 110 = 66. Step 8: From M = P + 12.5, M = 49.5 + 12.5 = 62. From M = R - 4, M = 66 - 4 = 62, which is consistent. Step 9: Thus P, Q, R, M and Mmax are uniquely determined without using statements A, B or C.

Verification / Alternative check:
We can quickly check statement C: it says R has 84 marks, but from our direct derivation, R has 66 marks when maximum marks are 110. This means statement C actually contradicts the values derived from the original data. Therefore, relying on statement C could mislead us. Since the question is about which statements are required, and the core data already gives a unique solution, the correct conclusion is that none of the extra statements are required. The base relationships are sufficient by themselves.

Why Other Options Are Wrong:
Saying “Only C” or “Only A and B” or “All are required” assumes that the base data is not sufficient to compute the marks. However, we have already determined all marks using only the original percentages and the relationships between M, P and R. Therefore, all such choices are incorrect. The additional statements may either be redundant or even inconsistent with the derived values.

Common Pitfalls:
In data sufficiency style questions, it is easy to assume that the extra statements must be used, especially when they look informative. Many learners do not verify whether the base problem is already solvable. Another common mistake is to take any one of the given statements as a fact without checking for consistency with the original data. A systematic approach is to first test whether the original conditions alone yield a unique solution, and only then consider if further statements are necessary.

Final Answer:
None of the additional statements A, B or C is required to determine the individual marks; the original data is sufficient.