3.11 Exam Pass/Fail Marks Problems

These questions revolve around one core idea: the difference between two percentage-of-maximum-marks figures, when it corresponds to a known number of actual marks, lets you find the maximum marks itself.

Pass Marks = Pass% × Maximum Marks

If a candidate scores x% and falls short of the pass mark by d marks, then: Pass marks = (x% of Max Marks) + d.

Worked Example 1: A student secures 35% marks and fails by 10 marks. If the passing requirement is 40%, find the maximum marks. The gap between 40% and 35%, i.e., 5% of the maximum marks, equals the 10-mark shortfall. 5% of M = 10 → M = 10 × (100/5) = 200.

Worked Example 2: The pass percentage in an exam is 33%. A candidate scores 25% and fails by 30 marks. Find the maximum marks. (33 − 25)% of M = 30 → 8% of M = 30 → M = 30 × (100/8) = 375.

Worked Example 3: In an examination, A scores 20% of the maximum marks and fails by 5 marks, while B, appearing for the same exam, scores 30% of the maximum marks and gets 20 marks more than the passing marks. Find the maximum marks and the pass percentage. Let maximum marks = M and pass marks = P. From A: 0.20M = P − 5 → P = 0.20M + 5 From B: 0.30M = P + 20 Substitute: 0.30M = 0.20M + 5 + 20 → 0.10M = 25 → M = 250. P = 0.20(250) + 5 = 55. Pass % = (55/250) × 100 = 22%.

Worked Example 4: A candidate who scores 40% of the total marks in an examination fails by 40 marks, while another candidate who scores 45% of the total marks gets 10 marks more than the minimum passing marks. Find the maximum marks and the passing marks. Let maximum marks = M, pass marks = P. 0.40M = P − 40 → P = 0.40M + 40 0.45M = P + 10 Substituting: 0.45M = 0.40M + 40 + 10 → 0.05M = 50 → M = 1,000. P = 0.40(1000) + 40 = 440.

Worked Example 5: A scores 36% of the maximum marks and gets 18 marks more than the passing marks, while B scores 28% of the maximum marks and falls short of the passing marks by 30 marks. Find the maximum marks and the passing percentage. Let maximum marks = M, pass marks = P. 0.36M = P + 18 → P = 0.36M − 18 0.28M = P − 30 → P = 0.28M + 30 Setting equal: 0.36M − 18 = 0.28M + 30 → 0.08M = 48 → M = 600. P = 0.36(600) − 18 = 216 − 18 = 198. Pass % = (198/600) × 100 = 33%.

Worked Example 6: In an exam, the passing marks are 40% of the maximum marks. A student scores 130 marks and fails by 50 marks. A second student scores 20% more marks than the first student. Does the second student pass, and if not, by how many marks does he fall short? Pass marks = 130 + 50 = 180 = 40% of M → M = 180/0.40 = 450. Second student’s marks = 130 × 1.20 = 156. Since 156 < 180 (the passing marks), the second student fails, falling short by 180 − 156 = 24 marks.