Percentage — Advanced Exam Problems

🎯 Learning Objective

Master the hardest RRB/SSC percentage patterns — elections, two-subject pass-fail and chained changes — with base-first thinking.

💡 Concept

• Election pattern: valid votes = votes cast − invalid; keep 'of the list' and 'of valid votes' as separate bases
• Two-subject pass-fail: failed at least one = fail(A) + fail(B) − fail(both); pass both = 100 − that
• 'A is x% more than B' flips to 'B is x/(100+x) × 100 % less than A' — never the same x
• Three or more successive changes: chain multiplying factors, or apply a + b + ab/100 twice
• Hard questions mix bases on purpose — write the base beside every % before solving

🧮 Key Formulas

Pass both % = 100 − (fA + fB − fAB)

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x% more ↔ x/(100 + x) × 100 % less

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Chained changes: MF = (1 ± a/100)(1 ± b/100)(1 ± c/100)

✏️ Easy Example

Q. Ravi's salary is 25% more than Suresh's. By what per cent is Suresh's salary less than Ravi's?

1. Let Suresh = 100, so Ravi = 125 — assuming 100 turns every percentage into a plain number
2. Gap = 125 − 100 = 25, but the base is now RAVI (125), because the question says 'less than Ravi'
3. Less % = (25/125) × 100 = 20

Answer: 20%

🇮🇳 Real-Life Example

Every election result on TV — turnout %, invalid votes, victory margin — is exactly this maths running live.

📝 Exam-Level Example

Q. In an election between two candidates, 10% of the listed voters did not vote and 60 votes were invalid. The winner got 47% of the listed voters and won by 308 votes. Find the number of voters on the list.

1. Let the list have N voters. 10% did not vote, so votes cast = 90% of N = 0.9N — always anchor every % to its own base first
2. 60 votes were invalid, so valid votes = 0.9N − 60
3. Winner got 47% of N (of the LIST, not of valid votes — read the base carefully) = 0.47N, so the loser got (0.9N − 60) − 0.47N = 0.43N − 60
4. Winning margin = winner − loser = 0.47N − (0.43N − 60) = 0.04N + 60
5. Given margin = 308, so 0.04N + 60 = 308 → 0.04N = 248 → N = 248/0.04 = 6200

Answer: 6,200 voters

📝 Exam-Level Example

Q. In an exam, 42% students failed in Maths and 52% failed in English. If 17% failed in both, find the percentage of students who passed in both subjects.

1. Failed in at least one subject = fail(M) + fail(E) − fail(both) — subtract the overlap once because those students were counted twice
2. = 42 + 52 − 17 = 77
3. Passed in both = 100 − failed in at least one, because 'passed both' is the exact opposite of 'failed at least one'
4. = 100 − 77 = 23
5. Check: only-Maths fail = 42 − 17 = 25, only-English = 52 − 17 = 35, both = 17 → 25 + 35 + 17 = 77 ✓

Answer: 23%

📝 Exam-Level Example

Q. The price of petrol is increased by 25% and then again by 12%. By what per cent (approx) must a driver cut his consumption to keep his petrol budget unchanged?

1. Two successive hikes combine by a + b + ab/100 — never add them straight, because the second hike acts on an already-raised price
2. Net hike = 25 + 12 + (25 × 12)/100 = 37 + 3 = 40%
3. Expenditure = price × consumption; to keep it fixed, consumption must fall by x/(100 + x) × 100 where x is the net price rise
4. Cut = 40/(100 + 40) × 100 = 4000/140 = 200/7
5. 200/7 = 28.57 approx — in fractions, price became ×7/5, so consumption must become ×5/7, a drop of 2/7

Answer: 28.57% (i.e. 2/7 of consumption)

🪄 Memory Trick

Turn the net change into a fraction MF: price ×7/5 means consumption ×5/7, so cut = 2/7 = 28.57%. Reciprocal thinking beats formulas on hard chains.

⚠️ Common Mistakes

• ❌ Taking the winner's % on valid votes when the question says % of total listed voters (or vice versa)
• ❌ Adding 42% + 52% = 94% failed — forgetting to subtract the both-failed overlap
• ❌ Adding successive changes (25 + 12 = 37%) instead of chaining multiplying factors

🏆 Exam Tips

• ✅ Assume 100 (or call the list N) the moment multiple bases appear in one question
• ✅ Draw a two-circle Venn for any pass-fail-both question — 10 seconds, zero confusion
• ✅ Cross-check chained percentages by multiplying MFs: 1.25 × 1.12 = 1.40 exactly

📌 Summary

• Anchor every % to its stated base — list, valid votes, or previous price
• Pass both = 100 − (fA + fB − fboth)
• Chain changes with MFs; never add raw percentages
• x% more ↔ x/(100+x) less — the reciprocal pair