Ratio & Proportion — Advanced Exam Problems

🎯 Learning Objective

Combine ratios across stories — partnerships, proportion-preserving shifts and double ratio families.

💡 Concept

• Partnership: profit splits by capital × time — late joiners earn on fewer months
• Subtracting the same x from four numbers in proportion → (a−x)(d−x) = (b−x)(c−x); the x² always cancels
• Fraction ratios like 1/2 : 1/3 : 1/4 → scale by the LCM to get whole numbers first
• Two ratio families (incomes AND spendings) need TWO variables plus one linking condition
• Savings = income − expenditure is the linking line in most hard ratio stories

🧮 Key Formulas

Partnership: share ∝ capital × time

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Proportion after subtracting x: (a−x)(d−x) = (b−x)(c−x)

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Fraction ratio a/p : b/q : c/r → multiply by LCM(p, q, r)

✏️ Easy Example

Q. Divide ₹6,500 among A, B and C in the ratio 1/2 : 1/3 : 1/4.

1. Multiply the whole ratio by LCM(2, 3, 4) = 12 to clear fractions: 6 : 4 : 3 — scaling a ratio changes nothing
2. Total parts = 6 + 4 + 3 = 13
3. One part = 6500/13 = ₹500
4. A = 6 × 500 = 3000, B = 4 × 500 = 2000, C = 3 × 500 = 1500

Answer: A = ₹3,000, B = ₹2,000, C = ₹1,500

🇮🇳 Real-Life Example

Two friends putting money into a business at different times split the year-end profit by capital × time — partnership ratio in action.

📝 Exam-Level Example

Q. A starts a business with ₹15,000. After 4 months B joins with ₹20,000, and after 6 months from the start C joins with ₹30,000. Find B's share in a year-end profit of ₹13,000.

1. Profit divides in the ratio of capital × time — money that stayed invested longer earns more
2. A: 15000 × 12 = 180000; B: 20000 × 8 = 160000 (joined after 4 months, so only 8 months); C: 30000 × 6 = 180000
3. Ratio = 180000 : 160000 : 180000 = 9 : 8 : 9 — divide every term by 20000
4. Total parts = 9 + 8 + 9 = 26; one part = 13000/26 = ₹500
5. B's share = 8 × 500 = ₹4,000

Answer: ₹4,000

📝 Exam-Level Example

Q. What number must be subtracted from each of 23, 30, 57 and 78 so that the remainders are in proportion?

1. Let x be subtracted; proportion means (23−x) : (30−x) :: (57−x) : (78−x)
2. Product of extremes = product of means: (23−x)(78−x) = (30−x)(57−x)
3. Expand both sides: 1794 − 101x + x² = 1710 − 87x + x² — the x² terms cancel, leaving a linear equation
4. 1794 − 1710 = 101x − 87x → 84 = 14x
5. x = 6; check: 17 : 24 :: 51 : 72, and both fractions equal 17/24 ✓

Answer: 6

📝 Exam-Level Example

Q. The incomes of A, B and C are in the ratio 7 : 9 : 12 and their spendings in the ratio 8 : 9 : 15. If A saves 1/4 of his income, find the ratio of their savings.

1. Let incomes be 7x, 9x, 12x and spendings 8y, 9y, 15y — two ratio families need two different variables
2. A saves 1/4 of income, so A spends 3/4 of 7x = 21x/4; but A's spending is also 8y → 8y = 21x/4
3. Solve the link: y = 21x/32 — this one condition ties the two families together
4. Savings: A = 7x/4 = 56x/32; B = 9x − 9y = 9x − 189x/32 = 99x/32; C = 12x − 15y = 12x − 315x/32 = 69x/32
5. The common factor x/32 cancels → savings ratio = 56 : 99 : 69

Answer: 56 : 99 : 69

🪄 Memory Trick

Whenever two ratio families describe the same people, give them different variables (x and y) and hunt for the ONE fact that links them.

⚠️ Common Mistakes

• ❌ Using one variable for both families (incomes 7x and spendings 8x) — they are different scales
• ❌ Missing that the x² terms cancel — the proportion equation is linear, not quadratic
• ❌ Counting B's months as 12 in partnership when he joined after 4 (only 8 months count)

🏆 Exam Tips

• ✅ Fraction ratios → multiply by the LCM before anything else
• ✅ Partnership: write capital × months for each partner in a column, then simplify
• ✅ After solving, plug back — the two proportion fractions must come out equal

📌 Summary

• Capital × time decides partnership shares
• Subtract-x proportion → cross-multiply; x² cancels
• Two ratio families → two variables plus one linking fact
• Clear fractions with the LCM before dividing anything