Exam Applications — Marks, Population, Salary

🎯 Learning Objective

Master the pass-fail marks pattern, population growth and income-expenditure questions.

💡 Concept

• Pass-marks: if A fails by f marks scoring s → pass mark = s + f; % needed relates to max marks
• Two-student pattern: A gets a%, fails by f; B gets b%, gets e extra → (b−a)% of Max = f + e
• Population after n years at r% → P(1 + r/100)ⁿ; before n years → P/(1 + r/100)ⁿ
• Salary/income: savings = income − expenditure; track each part's % separately

🧮 Key Formulas

(b−a)% of Max = fail-margin + extra

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Future pop = P(1+r/100)ⁿ

✏️ Easy Example

Q. A student needs 40% to pass. He scores 178 and fails by 22 marks. Find the maximum marks.

1. Pass mark = 178 + 22 = 200
2. 40% of Max = 200
3. Max = 200 × 100/40

Answer: 500

🇮🇳 Real-Life Example

Cut-off lists in RRB results work exactly on this maths — pass %, your score, and the gap decide everything.

📝 Exam-Level Example

Q. A scores 30% and fails by 60 marks. B scores 45% and gets 30 more than pass marks. Find maximum marks.

1. Gap between A and B = 45% − 30% = 15% of Max
2. In marks: 60 + 30 = 90
3. 15% of Max = 90 → Max = 90 × 100/15

Answer: 600

📝 Exam-Level Example

Q. Population of a town is 8,000 and grows 5% yearly. Population after 2 years?

1. 8000 × 1.05 = 8400
2. 8400 × 1.05 = 8820

Answer: 8,820

🪄 Memory Trick

In pass-fail: (percent difference) of Max = (marks difference). Write this one equation and solve.

⚠️ Common Mistakes

• ❌ Taking pass % of scored marks instead of maximum marks
• ❌ Simple-growth (P + 2×5%) instead of compound for population

🏆 Exam Tips

• ✅ Underline WHOSE percentage it is before solving
• ✅ Population/depreciation → compound formula always

📌 Summary

• Pass mark = score + fail margin
• (b−a)% of Max = total marks gap
• Population grows compound, not simple
• Income = Expenditure + Savings