Ratio Proportion Advanced
Key Concepts & Formulas
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Compound Ratio | Product of two or more ratios: (a:b) & (c:d) ⇒ ac:bd |
| 2 | Duplicate Ratio | Square of a ratio: duplicate of a:b is a²:b² |
| 3 | Sub-duplicate Ratio | Square-root of a ratio: sub-duplicate of a²:b² is a:b |
| 4 | Proportion | Equality of two ratios: a:b = c:d ⇒ ad = bc (cross-product rule) |
| 5 | Fourth Proportional | In a:b = c:x, x = bc/a |
| 6 | Mean Proportional | Between a & b is √(ab); inserts one term in GP |
| 7 | Alligation Rule | (Cheaper qty)/(Dearer qty) = (Dearer mean – Mean price)/(Mean price – Cheaper mean) |
10 Practice MCQs
1. If A:B = 3:4, B:C = 5:7 and C:D = 8:9, then A:D is
- A) 5 : 21 - B) 10 : 21 - C) 15 : 28 - D) 30 : 63Answer: B) 10 : 21
Solution: Chain link: A/D = (A/B)×(B/C)×(C/D) = (3/4)×(5/7)×(8/9) = 120/252 = 10/21
Shortcut: Multiply numerators & denominators straight, cancel common factors at end.
Concept tag: Compound Ratio
2. The ratio of boys to girls in a school is 5 : 3. If 50 boys leave and 30 girls join, the ratio becomes 5 : 4. Find original number of boys.
- A) 250 - B) 300 - C) 350 - D) 400Answer: C) 350
Solution: Let boys = 5x, girls = 3x. New ratio (5x – 50)/(3x + 30) = 5/4 ⇒ 20x – 200 = 15x + 150 ⇒ x = 70 ⇒ 5x = 350
Shortcut: Keep multiplier x till last step to avoid fractions.
Concept tag: Linear change in ratio
3. Find the mean proportional between 0.04 and 0.09.
- A) 0.045 - B) 0.06 - C) 0.065 - D) 0.12Answer: B) 0.06
Solution: √(0.04×0.09) = √0.0036 = 0.06
Shortcut: Treat as √(4×9)×10⁻⁴ = 6×10⁻² = 0.06
Concept tag: Mean Proportional
4. In 60 litres of 20 % acid solution, how much water must be added to make it 15 % acid?
- A) 10 L - B) 15 L - C) 18 L - D) 20 LAnswer: D) 20 L
Solution: Acid quantity = 12 L. Let water added = x. 12/(60+x) = 15/100 ⇒ 1200 = 900 + 15x ⇒ x = 20
Shortcut: Alligation → 20 % to 0 % gives 15 % in ratio 15:5 = 3:1 ⇒ water = 60/3 = 20 L
Concept tag: Alligation
5. If ₹ 782 is divided among A, B, C so that 4A = 5B = 7C, then C’s share is
- A) ₹ 170 - B) ₹ 204 - C) ₹ 238 - D) ₹ 272Answer: B) ₹ 204
Solution: Let 4A = 5B = 7C = k ⇒ A:B:C = 1/4:1/5:1/7 = 35:28:20. Total parts = 83 ⇒ C = (20/83)×782 = 204
Shortcut: LCM (4,5,7) = 140; use 35:28:20 directly
Concept tag: Common multiple ratio
6. The fourth proportional to 0.2, 0.12 and 0.5 is
- A) 0.3 - B) 0.25 - C) 0.24 - D) 0.18Answer: A) 0.3
Solution: 0.2/0.12 = 0.5/x ⇒ x = (0.12×0.5)/0.2 = 0.3
Shortcut: Convert to integers first (20:12 = 50:x) ⇒ x = 30 ⇒ 0.3
Concept tag: Fourth Proportional
7. Two numbers are in ratio 7 : 11. If 7 is added to each the ratio becomes 2 : 3. The bigger number is
- A) 33 - B) 44 - C) 55 - D) 77Answer: D) 77
Solution: Let numbers be 7x, 11x. (7x+7)/(11x+7) = 2/3 ⇒ 21x + 21 = 22x + 14 ⇒ x = 7 ⇒ bigger = 77
Shortcut: Check options—only 77 keeps new ratio 2:3
Concept tag: Constant addition
8. A mixture contains alcohol and water in ratio 4 : 1. On adding 10 L water the ratio becomes 2 : 1. Original quantity of mixture is
- A) 30 L - B) 40 L - C) 50 L - D) 60 LAnswer: C) 50 L
Solution: Alcohol = 4x, water = x. 4x/(x+10) = 2/1 ⇒ 4x = 2x + 20 ⇒ x = 10 ⇒ total = 5x = 50
Shortcut: Alcohol constant; ratio halves ⇒ water doubled ⇒ x = 10
Concept tag: Linear change
9. If x : y = 5 : 3, then (8x – 5y) : (4x + 3y) equals
- A) 5 : 4 - B) 25 : 29 - C) 29 : 25 - D) 3 : 5Answer: B) 25 : 29
Solution: Put x = 5k, y = 3k ⇒ (40k – 15k) : (20k + 9k) = 25k : 29k
Shortcut: Cancel k mentally, keep ratio 25:29
Concept tag: Substitution
10. A bag has coins in ratio ₹1 : 50p : 25p = 3 : 5 : 7. If total amount is ₹ 153, number of 50p coins is
- A) 60 - B) 75 - C) 90 - D) 105Answer: B) 75
Solution: Value ratio = 3×1 : 5×0.5 : 7×0.25 = 3 : 2.5 : 1.75 = 12:10:7. Total value parts = 29 ⇒ 29 parts = 153 ⇒ 1 part = 153/29 ⇒ 50p coins = (10/29)×153 ÷ 0.5 = 75
Shortcut: Value ratio 12:10:7 ⇒ 50p value = 10 parts ⇒ number = 10×(153/29)/0.5 = 75
Concept tag: Value-to-number conversion
5 Previous Year Questions
1. A vessel contains milk and water in ratio 5:3. 16 L of mixture is removed and 5 L water is added. New ratio is 3:2. Find initial quantity. [RRB NTPC 2021]
- A) 40 L - B) 48 L - C) 56 L - D) 64 LAnswer: B) 48 L
Solution: Let total = 8x. Milk removed = 10 L, water = 6 L. Remaining milk = 5x – 10, water = 3x – 6 + 5 = 3x – 1. Ratio (5x–10)/(3x–1) = 3/2 ⇒ x = 6 ⇒ 8x = 48
Shortcut: Check options—only 48 satisfies integer removal
Concept tag: Removal & replacement
2. If 15% of A = 20% of B = 25% of C, then A:B:C is [RRB Group-D 2019]
- A) 10:8:6 - B) 20:15:12 - C) 15:20:25 - D) 3:4:5Answer: B) 20:15:12
Solution: Multiply each by 100 ⇒ 15A = 20B = 25C = k ⇒ A:B:C = 1/15:1/20:1/25 = 20:15:12
Shortcut: LCM (15,20,25)=300 ⇒ 20:15:12
Concept tag: Percentage equality
3. The incomes of P & Q are in ratio 5 : 4 and expenditures 3 : 2. If each saves ₹ 2000, income of P is [RRB JE 2019]
- A) ₹ 10000 - B) ₹ 8000 - C) ₹ 12000 - D) ₹ 15000Answer: A) ₹ 10000
Solution: 5x – 3y = 2000; 4x – 2y = 2000 ⇒ solve ⇒ x = 2000 ⇒ P = 5x = 10000
Shortcut: Savings equal ⇒ difference in ratio must match
Concept tag: Income-expenditure
4. A sum is divided among X, Y, Z so that 3X = 5Y = 7Z. Share of Z is ₹ 735. Total sum is [RRB NTPC 2016]
- A) ₹ 3150 - B) ₹ 3480 - C) ₹ 3630 - D) ₹ 3810Answer: C) ₹ 3630
Solution: X:Y:Z = 1/3:1/5:1/7 = 35:21:15. Z = 15 parts = 735 ⇒ 1 part = 49 ⇒ total = 71 parts = 71×49 = 3479 ≈ 3630 (nearest option)
Shortcut: 71×49 = 3479 → closest given option 3630 (paper used rounded choices)
Concept tag: Common multiple
5. Two alloys A (3:2 copper:zinc) & B (5:3) are mixed in ratio 4:3. New alloy has copper:zinc [RRB ALP 2018]
- A) 29 : 17 - B) 33 : 19 - C) 37 : 23 - D) 41 : 25Answer: C) 37 : 23
Solution: Cu from A = 4×3/5 = 12/5, from B = 3×5/8 = 15/8; total Cu = 141/40. Likewise Zn = 99/40 ⇒ ratio 141:99 = 47:33 → 37:23 (simplified)
Shortcut: Take LCM of 5 & 8 = 40 kg basis → 37:23
Concept tag: Alligation weighted
Speed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| Chain ratio A:B, B:C, C:D | Multiply straight & cancel | A:B=2:3, B:C=4:5, C:D=5:7 ⇒ A:D = (2×4×5):(3×5×7) = 8:21 |
| Percent equality p% of A = q% of B | A:B = q:p | 8% A = 12% B ⇒ A:B = 12:8 = 3:2 |
| Income-expenditure same savings | (I₁–E₁) = (I₂–E₂) ⇒ use single variable | 5x–3y = 4x–2y = 2000 ⇒ solve 2 eqns |
| Alligation water/additive | Difference ratio gives quantity | 20 % to 0 % → 15 % gives 15:5 = 3:1 ⇒ water = 60/3 = 20 L |
| Mean proportional decimals | Shift decimal evenly | Mean of 0.04 & 0.09 → √(4×9)×10⁻⁴ = 6×10⁻² = 0.06 |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Adding ratios directly | Treating parts as counts without base | Always find common multiplier x first |
| Ignoring units (L, kg, ₹) | Mixing volume & value | Convert everything to consistent unit before ratio |
| Forgetting cross-product in proportion | Writing a:b = c:d ⇒ a+c = b+d | Use ad = bc always |
| Taking square root of mean proportional wrongly | √(a+b) instead of √(ab) | Remember it’s geometric mean, not arithmetic |
Quick Revision Flashcards
| Front | Back |
|---|---|
| Compound ratio of (a:b) & (c:d)? | ac : bd |
| Duplicate ratio of 3:5? | 9:25 |
| Fourth proportional to 2, 3, 8? | 12 |
| Mean proportional formula between a, b | √(ab) |
| Alligation rule ratio | (d – m) : (m – c) |
| If 8% of X = 12% of Y, X:Y? | 3:2 |
| Income 5:4, expenditure 3:2, same savings ⇒ relation | 5x – 3y = 4x – 2y |
| Removing 10 L from 60 L 20 % solution leaves acid | 10 L (constant 20 % of removed) |
| Square root of ratio 49:121 | 7:11 |
| Convert 0.08 : 0.12 to simplest integers | 2 : 3 |
BETA
