📑 Topics on this page (Jump to Topic):
📌 1. Basics of Ratio
- a : b = a/b (same units); b ≠ 0.
- Compound ratio of a:b and c:d = ac : bd. Duplicate ratio = a² : b²; sub-duplicate = √a : √b.
- Combining: A:B = 2:3, B:C = 4:5 → equalise B: A:B:C = 8:12:15.
📘 Formulas:
- Dividing S in a : b : c: A = S × a/(a + b + c)
- Fourth proportional of (a, b, c) = bc/a
- Third proportional of (a, b) = b²/a
- Mean proportional of (a, b) = √(ab)
📌 2. Applications
- Coins: ₹1, 50 p, 25 p in number ratio a:b:c → value ratio = a : b/2 : c/4.
- Income-expenditure: incomes a:b, expenses c:d, each saves S → (ax − cy) = (bx − dy) = S.
- Adding: adding x to a:b gives c:d → (a·k + x)/(b·k + x) = c/d.
⚠️ Exam Traps:
- Ratio terms are not actual values — always use a multiplier k.
📌 3. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. The third proportional to 2 and 8 is: Practice
Correct answer: (B)
Solution: b²/a = 8²/2 = 32.
Solution: b²/a = 8²/2 = 32.
Q2. A bag has ₹1, 50 p and 25 p coins in the ratio 1 : 3 : 4, total value ₹140. The number of 50 p coins is: Practice
Correct answer: (C)
Solution: Value ratio = 1 : 1.5 : 1 (sum 3.5) → k = 140/3.5 = 40; 50 p coins = 3 × 40 = 120.
Solution: Value ratio = 1 : 1.5 : 1 (sum 3.5) → k = 140/3.5 = 40; 50 p coins = 3 × 40 = 120.
Q3. Two numbers are in the ratio 2 : 3. When 6 is added to each, the ratio becomes 16 : 21. The larger number is: Practice
Correct answer: (D)
Solution: (2k + 6)/(3k + 6) = 16/21 → k = 5; larger = 3k = 15.
Solution: (2k + 6)/(3k + 6) = 16/21 → k = 5; larger = 3k = 15.
Q4. The duplicate ratio of 4 : 5 is: Practice
Correct answer: (A)
Solution: Duplicate ratio = a² : b² = 16 : 25.
Solution: Duplicate ratio = a² : b² = 16 : 25.
Q5. y varies directly as the square of x. When x = 2, y = 8. Find y when x = 9. Practice
Correct answer: (B)
Solution: y = kx² → 8 = k × 4 → k = 2; y = 2 × 9² = 162.
Solution: y = kx² → 8 = k × 4 → k = 2; y = 2 × 9² = 162.
Q6. ₹2800 is divided among A, B, C in the ratio 2 : 5 : 7. C's share is: Practice
Correct answer: (C)
Solution: C = 2800 × 7/14 = ₹1400.
Solution: C = 2800 × 7/14 = ₹1400.
Q7. The third proportional to 4 and 6 is: Practice
Correct answer: (D)
Solution: b²/a = 6²/4 = 9.
Solution: b²/a = 6²/4 = 9.
Q8. A bag has ₹1, 50 p and 25 p coins in the ratio 1 : 3 : 4, total value ₹70. The number of 50 p coins is: Practice
Correct answer: (A)
Solution: Value ratio = 1 : 1.5 : 1 (sum 3.5) → k = 70/3.5 = 20; 50 p coins = 3 × 20 = 60.
Solution: Value ratio = 1 : 1.5 : 1 (sum 3.5) → k = 70/3.5 = 20; 50 p coins = 3 × 20 = 60.
Q9. Two numbers are in the ratio 5 : 8. When 4 is added to each, the ratio becomes 9 : 14. The larger number is: Practice
Correct answer: (B)
Solution: (5k + 4)/(8k + 4) = 9/14 → k = 10; larger = 8k = 80.
Solution: (5k + 4)/(8k + 4) = 9/14 → k = 10; larger = 8k = 80.
Q10. The duplicate ratio of 5 : 6 is: Practice
Correct answer: (C)
Solution: Duplicate ratio = a² : b² = 25 : 36.
Solution: Duplicate ratio = a² : b² = 25 : 36.
Q11. y varies directly as the square of x. When x = 4, y = 48. Find y when x = 12. Practice
Correct answer: (D)
Solution: y = kx² → 48 = k × 16 → k = 3; y = 3 × 12² = 432.
Solution: y = kx² → 48 = k × 16 → k = 3; y = 3 × 12² = 432.
Q12. In 80 litres of a mixture, milk and water are in the ratio 4 : 1. How much water must be added to make the ratio 2 : 3? Practice
Correct answer: (A)
Solution: Milk = 64 L, water = 16 L. 64/(16 + x) = 2/3 → x = 80 L.
Solution: Milk = 64 L, water = 16 L. 64/(16 + x) = 2/3 → x = 80 L.
Q13. ₹1200 is divided among A, B, C in the ratio 1 : 2 : 3. C's share is: Practice
Correct answer: (B)
Solution: C = 1200 × 3/6 = ₹600.
Solution: C = 1200 × 3/6 = ₹600.
Q14. The mean proportional between 16 and 25 is: Practice
Correct answer: (C)
Solution: √(16 × 25) = √400 = 20.
Solution: √(16 × 25) = √400 = 20.
Q15. A bag has ₹1, 50 p and 25 p coins in the ratio 1 : 2 : 4, total value ₹30. The number of 50 p coins is: Practice
Correct answer: (D)
Solution: Value ratio = 1 : 1 : 1 (sum 3) → k = 30/3 = 10; 50 p coins = 2 × 10 = 20.
Solution: Value ratio = 1 : 1 : 1 (sum 3) → k = 30/3 = 10; 50 p coins = 2 × 10 = 20.
Q16. The incomes of A and B are in the ratio 4 : 5 and their expenses in the ratio 4 : 7. If each saves ₹12000, A's income is: Practice
Correct answer: (A)
Solution: 4x − 4y = 5x − 7y = 12000 → x = 4500, y = 1500. A's income = 4x = ₹18000.
Solution: 4x − 4y = 5x − 7y = 12000 → x = 4500, y = 1500. A's income = 4x = ₹18000.
Q17. Two numbers are in the ratio 5 : 8. When 9 is added to each, the ratio becomes 29 : 41. The larger number is: Practice
Correct answer: (B)
Solution: (5k + 9)/(8k + 9) = 29/41 → k = 4; larger = 8k = 32.
Solution: (5k + 9)/(8k + 9) = 29/41 → k = 4; larger = 8k = 32.
Q18. The duplicate ratio of 3 : 7 is: Practice
Correct answer: (C)
Solution: Duplicate ratio = a² : b² = 9 : 49.
Solution: Duplicate ratio = a² : b² = 9 : 49.
Q19. y varies directly as the square of x. When x = 3, y = 18. Find y when x = 12. Practice
Correct answer: (D)
Solution: y = kx² → 18 = k × 9 → k = 2; y = 2 × 12² = 288.
Solution: y = kx² → 18 = k × 9 → k = 2; y = 2 × 12² = 288.
Q20. ₹2800 is divided among A, B, C in the ratio 2 : 5 : 7. B's share is: Practice
Correct answer: (A)
Solution: B = 2800 × 5/14 = ₹1000.
Solution: B = 2800 × 5/14 = ₹1000.