Section Scores and Weighted Averages
A scholarship test has 100 questions: Section A has 40 questions and Section B has 60. Candidate X answered 75% of Section A correctly and 60% of the whole test correctly. Candidate Y answered 50% of Section A and 70% of Section B correctly. Candidate W answered 70% of Section A and 55% of Section B correctly. The scholarship committee first ranks candidates by total correct answers. It then considers a proposal to weight Section A at 60% and Section B at 40% of the final score.
1. What percentage of Section B did X answer correctly? Compare X and Y on total correct answers.
Read the worked answer
X answered 60% of 100 questions correctly, so 60 questions. In Section A, X answered 75% of 40 = 30 correctly. The remaining correct answers come from Section B: 60 − 30 = 30. Section B has 60 questions, so X's Section B score is 30/60 = 50%.
Y: Section A = 50% of 40 = 20; Section B = 70% of 60 = 42. Total = 62 correct, or 62%.
X has 60 correct and Y has 62, so Y leads by 2 questions (2 percentage points). Notice that X is much stronger in Section A, but the larger Section B carries more weight under equal marks per question.
2. Compute W's score under the current equal-per-question scheme and under the proposed 60/40 weighting. Explain why the two differ.
Read the worked answer
Current scheme: each question counts equally, so the sections are weighted 40/60. W = 0.70 × 40 + 0.55 × 60 = 28 + 33 = 61 correct out of 100, or 61%.
Proposed scheme: W = 0.60 × 70 + 0.40 × 55 = 42 + 22 = 64%.
The difference arises because the proposal shifts weight toward Section A, where W scores 70% rather than 55%. The current scheme is not a 50/50 weighting; it is a 40/60 weighting that favours Section B. W gains 3 points under the proposal, so the choice of weights is a substantive judgement, not a neutral technical detail.
3. The committee considers ranking by each candidate's better section percentage instead of total correct answers. Under this rule, who ranks higher, X or Y? Does this rule preserve the order of total correct answers in general? Explain.
Read the worked answer
X's better section is Section A at 75%; Y's better section is Section B at 70%. Under the better-section rule, X ranks higher - the opposite of the total-correct order, where Y leads 62 to 60.
The rule does not preserve total order. It discards information about the weaker section, so a candidate with one very strong section and one weak section can overtake a more balanced candidate with more total correct answers. That may be defensible if the minor requires a single strength, but it is not a general-purpose ranking rule. It also creates an incentive to neglect one section entirely.