Merit Scholarship Shortlist Under a Weighted Rubric
A specialized master's program awards three merit scholarships. The committee scores each applicant on three components, each on a 0–100 scale: GMAT performance, converted GPA, and interview. The published weights are GMAT 40%, GPA 30%, interview 30%. Scores are reported to one decimal place, but ranking is done on unrounded values. The tie-break rule is higher interview score, then higher GPA. A separate eligibility rule states that any applicant with an interview score below 70 is ineligible for a scholarship, regardless of the weighted score.
Four applicants remain: Applicant A has GMAT 92, GPA 88, interview 70; Applicant B has GMAT 84, GPA 94, interview 80; Applicant C has GMAT 88, GPA 90, interview 76; Applicant D has GMAT 96, GPA 82, interview 68. The committee must publish a ranked shortlist of eligible applicants and explain any change in order if the weights are revised.
1. Compute each applicant's weighted score under the published weights, apply the eligibility rule, and produce the ranked shortlist of eligible applicants.
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Weighted score = 0.40 × GMAT + 0.30 × GPA + 0.30 × interview.
- A: 0.40(92) + 0.30(88) + 0.30(70) = 36.8 + 26.4 + 21.0 = 84.2
- B: 0.40(84) + 0.30(94) + 0.30(80) = 33.6 + 28.2 + 24.0 = 85.8
- C: 0.40(88) + 0.30(90) + 0.30(76) = 35.2 + 27.0 + 22.8 = 85.0
- D: 0.40(96) + 0.30(82) + 0.30(68) = 38.4 + 24.6 + 20.4 = 83.4, but D's interview is 68, below the minimum of 70, so D is ineligible.
Ranking among eligible applicants: B (85.8), C (85.0), A (84.2). D is excluded before ranking, so D's high GMAT cannot rescue the application. The shortlist is B, C, A.
2. Suppose the committee changes the weights to GMAT 30%, GPA 30%, interview 40%. Recompute the eligible applicants' scores and state whether the top-three order changes.
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New score = 0.30 × GMAT + 0.30 × GPA + 0.40 × interview.
- A: 0.30(92) + 0.30(88) + 0.40(70) = 27.6 + 26.4 + 28.0 = 82.0
- B: 0.30(84) + 0.30(94) + 0.40(80) = 25.2 + 28.2 + 32.0 = 85.4
- C: 0.30(88) + 0.30(90) + 0.40(76) = 26.4 + 27.0 + 30.4 = 83.8
D remains ineligible. The order is still B, C, A. The interview weight increase helps A relative to the original weights, but not enough to overtake C, because C also has a stronger interview than A and a competitive GMAT/GPA profile.
3. Under the original weights, what is the minimum integer interview score A would need, holding GMAT and GPA fixed, to overtake B? Show the threshold.
Read the worked answer
A's score as a function of interview score I is:
0.40(92) + 0.30(88) + 0.30I = 36.8 + 26.4 + 0.30I = 63.2 + 0.30I.
B's score is 85.8. To overtake B strictly, A needs:
63.2 + 0.30I > 85.8 0.30I > 22.6 I > 75.333...
Since interview scores are integers, the smallest qualifying score is 76. At I = 75, A's score is 63.2 + 22.5 = 85.7, just below B. At I = 76, A's score is 86.0, so A would rank above B. The eligibility threshold of 70 is already satisfied, so the binding constraint is the ranking threshold.