1. Basics
- a : b = a/b. Multiply or divide both terms by the same number — ratio unchanged.
- Compound ratio of a:b and c:d = ac : bd. Duplicate a² : b²; triplicate a³ : b³; sub-duplicate √a : √b.
- a : b = c : d ⇒ ad = bc. Fourth proportional to a, b, c = bc/a. Third proportional to a, b = b²/a. Mean proportional = √(ab).
2. Combining ratios
A : B = 2 : 3, B : C = 4 : 5 → make B common (12) → A : B : C = 8 : 12 : 15.
3. Types
| Type | Method |
|---|---|
| Divide an amount in a ratio | Share = amount × part/total parts |
| Add x to both to change ratio | (a + x)/(b + x) = c/d |
| Income-expenditure | Savings = income − expenditure; form two equations |
| Coins | Value of one group of coins → number of groups |
| Componendo-dividendo | a/b = c/d ⇒ (a + b)/(a − b) = (c + d)/(c − d) |
4. Solved examples
(ASO 2021) Fourth proportional to 5, 8, 15 = 24.
(ASO 2021) x/5 = y/9 → (x + 5) : (y + 9) = 5 : 9.
Third proportional to 9 and 12 = 144/9 = 16; mean proportional of 9 and 16 = 12.
₹1,800 among A, B, C in 2 : 3 : 4 → C = 1800 × 4/9 = ₹800.
Incomes 3 : 2, expenses 5 : 3, each saves ₹1,000 → 3x − 5y = 1000, 2x − 3y = 1000 → y = 1000, x = 2000 → incomes ₹6,000 and ₹4,000.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. If A : B = 4 : 5 and B : C = 6 : 7, then A : B : C is:
(a) 24 : 30 : 35 (b) 24 : 35 : 30 (c) 25 : 30 : 35 (d) 4 : 5 : 7
Answer: (a) 24 : 30 : 35
Solution: Make B equal: A : B = 24 : 30, B : C = 30 : 35. Ratio = 24 : 30 : 35.
Example 2. If A : B = 2 : 3 and B : C = 4 : 7, then A : B : C is:
(a) 8 : 12 : 21 (b) 8 : 21 : 12 (c) 9 : 12 : 21 (d) 2 : 3 : 7
Answer: (a) 8 : 12 : 21
Solution: Make B equal: A : B = 8 : 12, B : C = 12 : 21. Ratio = 8 : 12 : 21.
Example 3. The fourth proportional to 6, 10 and 15 is:
(a) 23 (b) 25 (c) 9 (d) 28
Answer: (b) 25
Solution: 6 : 10 = 15 : x → x = 10×15/6 = 25.
Example 4. ₹2,275 is divided among A, B and C in the ratio 8 : 2 : 3. B's share is:
(a) ₹386 (b) ₹1,400 (c) ₹525 (d) ₹350
Answer: (d) ₹350
Solution: Total parts = 13. One part = 2,275/13 = 175. B = 2 × 175 = ₹350.
Example 5. The mean proportional between 180 and 125 is:
(a) 150 (b) 55 (c) 152.5 (d) 155
Answer: (a) 150
Solution: Mean proportional = √(180 × 125) = √22500 = 150.
Example 6. The mean proportional between 2 and 162 is:
(a) 82 (b) 160 (c) 20 (d) 18
Answer: (d) 18
Solution: Mean proportional = √(2 × 162) = √324 = 18.
Example 7. The third proportional to 11 and 22 is:
(a) 33 (b) 22 (c) 44 (d) 484
Answer: (c) 44
Solution: a : b = b : x → x = b²/a = 484/11 = 44.
Example 8. The third proportional to 12 and 60 is:
(a) 300 (b) 3,600 (c) 60 (d) 108
Answer: (a) 300
Solution: a : b = b : x → x = b²/a = 3600/12 = 300.
Example 9. If A : B = 4 : 5, B : C = 6 : 4 and C : D = 2 : 6, then A : D is:
(a) 2 : 5 (b) 5 : 2 (c) 12 : 15 (d) 8 : 30
Answer: (a) 2 : 5
Solution: A/D = (A/B)(B/C)(C/D) = (4×6×2)/(5×4×6) = 48/120 = 2 : 5.
ASO exam level
Example 10. The number of girls in a class is 3 times the number of boys. Which of these cannot be the total number of students?
(a) 40 (b) 24 (c) 20 (d) 25
Answer: (d) 25
Solution: Total = boys × 4, so it must be a multiple of 4. 25 is not.
Example 11. Two numbers are in the ratio 4 : 5. If 5 is added to each, the ratio becomes 5 : 6. The numbers are:
(a) 25 and 30 (b) 21 and 26 (c) 20 and 25 (d) 40 and 50
Answer: (c) 20 and 25
Solution: (4k + 5)/(5k + 5) = 5/6 → k = 5. Numbers 20, 25.
Example 12. Two numbers are in the ratio 4 : 5. If 4 is added to each, the ratio becomes 5 : 6. The numbers are:
(a) 16 and 20 (b) 20 and 24 (c) 32 and 40 (d) 17 and 21
Answer: (a) 16 and 20
Solution: (4k + 4)/(5k + 4) = 5/6 → k = 4. Numbers 16, 20.
Example 13. Two numbers are 50% and 20% more than a third number. The ratio of the first to the second is:
(a) 14 : 11 (b) 50 : 20 (c) 5 : 4 (d) 4 : 5
Answer: (c) 5 : 4
Solution: Third = 100 → first 150, second 120. Ratio 5 : 4.
Example 14. Two numbers are 50% and 20% more than a third number. The ratio of the first to the second is:
(a) 50 : 20 (b) 5 : 4 (c) 4 : 5 (d) 14 : 11
Answer: (b) 5 : 4
Solution: Third = 100 → first 150, second 120. Ratio 5 : 4.
Example 15. A bag has ₹1, 50-paise and 25-paise coins in the ratio 1 : 2 : 4. The total value is ₹60. The number of 25-paise coins is:
(a) 20 (b) 80 (c) 40 (d) 84
Answer: (b) 80
Solution: Value of one group = 1 + 1 + 1 = ₹3. Groups = 60/3 = 20. 25-paise coins = 4 × 20 = 80.
Example 16. If x : y = 7 : 8, then (4x + 2y) : (2x + 1y) is:
(a) 1 : 2 (b) 45 : 22 (c) 3 : 1 (d) 2 : 1
Answer: (d) 2 : 1
Solution: Put x = 7, y = 8: (28 + 16) : (14 + 8) = 44 : 22 = 2 : 1.
Example 17. Salaries of A and B are in the ratio 7 : 9. If each gets an increase of ₹4,500, the ratio becomes 79 : 99. A's present salary is:
(a) ₹45,000 (b) ₹35,000 (c) ₹39,500 (d) ₹35,500
Answer: (b) ₹35,000
Solution: Let salaries be 7k and 9k. (7k + 4,500)/(9k + 4,500) = 79/99 gives k = 5,000. A = ₹35,000.
Example 18. Salaries of A and B are in the ratio 8 : 3. If each gets an increase of ₹500, the ratio becomes 129 : 49. A's present salary is:
(a) ₹64,000 (b) ₹64,500 (c) ₹70,401 (d) ₹24,000
Answer: (a) ₹64,000
Solution: Let salaries be 8k and 3k. (8k + 500)/(3k + 500) = 129/49 gives k = 8,000. A = ₹64,000.
Example 19. Which of these ratios is the greatest: 11 : 7, 11 : 10, 13 : 4, 4 : 6?
(a) 11 : 7 (b) 13 : 4 (c) 11 : 10 (d) 4 : 6
Answer: (b) 13 : 4
Solution: Compare as decimals: 11 : 7 = 1.571, 11 : 10 = 1.100, 13 : 4 = 3.250, 4 : 6 = 0.667. Greatest is 13 : 4.
Example 20. Which of these ratios is the greatest: 10 : 3, 3 : 2, 8 : 6, 4 : 10?
(a) 10 : 3 (b) 4 : 10 (c) 8 : 6 (d) 3 : 2
Answer: (a) 10 : 3
Solution: Compare as decimals: 10 : 3 = 3.333, 3 : 2 = 1.500, 8 : 6 = 1.333, 4 : 10 = 0.400. Greatest is 10 : 3.
Advanced level
Example 21. The incomes of A and B are in the ratio 7 : 5 and their expenditures in the ratio 3 : 2. If each saves ₹3,500, A's income is:
(a) ₹24,500 (b) ₹17,500 (c) ₹21,000 (d) ₹28,000
Answer: (a) ₹24,500
Solution: Incomes 7x, 5x; expenses 3y, 2y. 7x − 3y = 5x − 2y = 3,500 → x = 3,500. A's income = ₹24,500 (check: spends ₹21,000).
Example 22. The incomes of A and B are in the ratio 2 : 5 and their expenditures in the ratio 1 : 4. If each saves ₹3,500, A's income is:
(a) ₹17,500 (b) ₹3,500 (c) ₹10,500 (d) ₹7,000
Answer: (d) ₹7,000
Solution: Incomes 2x, 5x; expenses 1y, 4y. 2x − 1y = 5x − 4y = 3,500 → x = 3,500. A's income = ₹7,000 (check: spends ₹3,500).
Topic-wise practice for Ratio and proportion (Quant): Proportionals, coins and rupees, income-expenditure ratio. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Basic ratio and simplification
- Compound, duplicate and sub-duplicate ratio
- Mean, third and fourth proportional
- Dividing an amount in a ratio
- Adding a number to change the ratio
- Coins and rupees
- Income and expenditure ratio
- Ratio of ages, salaries and students