The Scholarship Average
A student needs a weighted average of at least 75% across four assessment components to keep a scholarship. The components and weights are: coursework 20%, midterm 25%, project 25%, final exam 30%. So far the student has scored 82% on coursework, 68% on the midterm, and 71% on the project. The final exam has not been sat. The student tells a friend, 'I only need 75 overall, and I'm already above 75 on one component, so I'm basically safe.' The scholarship rule also states that no single component may score below 60%, or the scholarship is lost regardless of the average.
1. Calculate the minimum final exam score the student needs to reach a weighted average of exactly 75%. Show each step.
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Weighted contribution so far:
- Coursework: 82 × 0.20 = 16.40
- Midterm: 68 × 0.25 = 17.00
- Project: 71 × 0.25 = 17.75
Subtotal = 16.40 + 17.00 + 17.75 = 51.15
Final exam weight = 0.30, so let F be the final score: 51.15 + 0.30F = 75 0.30F = 23.85 F = 79.5%
So the student needs 79.5% on the final to hit exactly 75 overall - well above their current component scores. The friend's reasoning ('above 75 on one component') is irrelevant: a single high component cannot carry a weighted average when it is only 20% of the grade.
2. The student's friend says, 'Don't worry, the 60% minimum rule can't hurt you because your lowest score is 68.' Evaluate this claim and identify the real risk.
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The claim is correct as stated for the components already completed: 82, 68, and 71 are all above 60, so the floor rule is currently satisfied.
The real risk is the final exam, which has not been sat. If the student scores below 60 on the final:
- the scholarship is lost regardless of the average, even if the weighted average is above 75; and
- the required 79.5% target is already close to the 60% floor, so a weak final is a double failure.
So the friend has answered the wrong question. The binding constraint is not the average alone but the conjunction of (average ≥ 75) AND (every component ≥ 60). The student must treat the final exam as the single point of failure.
3. The student actually scores 85% on the final. Compute the final weighted average and state whether the scholarship is retained.
Read the worked answer
Subtotal from earlier = 51.15. Final contribution = 85 × 0.30 = 25.50.
Weighted average = 51.15 + 25.50 = 76.65%.
Check the floor rule: 82, 68, 71, 85 - all ≥ 60. ✔ Check the average: 76.65 ≥ 75. ✔
The scholarship is retained, but only by 1.65 percentage points. This confirms the earlier analysis: the student's margin was thin and depended entirely on a strong final exam, not on the one high coursework score.