Ratio-constrained scores with a cap: Marks of A, B, and C are in the ratio 10 : 12 : 15. If the maximum possible mark is 100, identify the range in which B’s mark cannot lie.

Introduction / Context: When scores maintain a fixed ratio, all three are scaled by a single positive factor k. Imposing a maximum score cap restricts k, which in turn restricts the allowable ranges for each student’s marks. We test intervals for feasibility under this cap.

Given Data / Assumptions:

• A : B : C = 10 : 12 : 15 ⇒ A = 10k, B = 12k, C = 15k.
• Maximum per paper = 100 marks.

Concept / Approach: Since C = 15k ≤ 100, we must have k ≤ 100/15 = 6.666…. Therefore B = 12k ≤ 12 * 6.666… = 80. Thus B cannot exceed 80 under the common scaling factor constraint.

Step-by-Step Solution:

Verification / Alternative check: Feasible examples: k = 2 ⇒ B = 24 (20–30 possible); k = 4 ⇒ B = 48 (40–50 possible); k ≈ 6 ⇒ B = 72 (70–80 possible). No k produces B within 80–90 strictly.

Why Other Options Are Wrong: The intervals 20–30, 40–50, 70–80 each contain at least one attainable multiple of 12k with k ≤ 6.666…, but 80–90 does not.

Common Pitfalls: Allowing k to vary separately for each student or forgetting that all three marks must share the same scaling factor.

Final Answer: 80 - 90