Ratio and proportion
- ₹7,800 is divided among A, B and C in the ratio \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\). What is B's share? Answer: B. ₹2,400 Answer: (B) ₹2,400 Multiply by LCM 12: ratio = 6 : 4 : 3, total 13 parts. B's share = 4/13 × 7800 = 2400 . 1
- The ratio of boys to girls in a class is 4 : 5. When 8 more boys join the class, the numbers of boys and girls become equal. How many girls are there in the class? Answer: B. 40 Answer: (B) 40 4k + 8 = 5k gives k = 8. Girls = 5 × 8 = 40. 1
- What is the mean proportional between 9 and 64? Answer: C. 24 Answer: (C) 24 Mean proportional = √(9 × 64) = √(576) = 24 . 1
- Rs. 7,200 is divided among A, B and C in the ratio 3 : 4 : 5. By how much does C's share exceed A's share (in Rs.)? Answer: B. 1200 Answer: (B) 1200 One part = 7200 ÷ 12 = 600. C − A = (5 − 3) × 600 = Rs. 1,200. 1
- ₹1,800 is divided between two friends in the ratio 4 : 5. What is the larger share? Answer: D. ₹1000 Answer: (D) ₹1000 Shortcut: 4 + 5 = 9 parts, 1 part = 1800 ÷ 9 = 200. Larger share = 5 × 200 = ₹1000. 1
- Find the fourth proportional to 6, 8 and 15. Answer: C. 20 Answer: (C) 20 If 6 : 8 = 15 : x, then x = 8 × 15/6 = 20 . 1
- What is the mean proportional between 9 and 16? Answer: C. 12 Answer: (C) 12 Mean proportional = √(9 × 16) = √(144) = 12 . Trap: 12.5 is the average, not the mean proportional. 1
- If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c. Answer: A. 8 : 12 : 15 Answer: (A) 8 : 12 : 15 Shortcut: make b equal. Multiply the first ratio by 4 and the second by 3: a : b = 8 : 12 and... 1
- What is the third proportional to 4 and 12? Answer: A. 36 Answer: (A) 36 If 4 : 12 = 12 : x, then x = 12²/4 = 36 . 1
- Two numbers are in the ratio 5 : 7 and their sum is 168. What is their difference? Answer: B. 28 Answer: (B) 28 Shortcut: 12 parts = 168, so 1 part = 14. Difference = 7 − 5 = 2 parts = 28. 1
- What is the compounded ratio of 2 : 3, 6 : 11 and 11 : 2? Answer: B. 2 : 1 Answer: (B) 2 : 1 Multiply the antecedents and the consequents: 2 × 6 × 11/3 × 11 × 2 = 132/66 = 2/1 . Compounded ratio = 2... 1
- In a class the ratio of boys to girls is 7 : 5. If there are 84 boys, how many students are there in all? Answer: C. 144 Answer: (C) 144 1 part = 84 ÷ 7 = 12. Total = 12 parts = 12 × 12 = 144. 1
- The incomes of A and B are in the ratio 5 : 4 and their expenditures are in the ratio 3 : 2. If each of them saves ₹1,600, what is A's income? Answer: D. ₹4000 Answer: (D) ₹4000 Let incomes be 5x, 4x and expenditures 3y, 2y. Then 5x − 3y = 1600 and 4x − 2y = 1600. From the second, y... 1
- A bag contains ₹1, 50-paise and 25-paise coins in the ratio 5 : 6 : 8. The total value is ₹210. How many 50-paise coins are there? Answer: B. 126 Answer: (B) 126 Shortcut: convert the number ratio into a value ratio: 5 × 1 : 6 × 0.5 : 8 × 0.25 = 5 : 3 : 2, i.e. 10 value... 1
- What number must be added to each of 6, 10, 14 and 22 so that the resulting numbers are in proportion? Answer: A. 2 Answer: (A) 2 Let x be added. Then (6 + x)(22 + x) = (10 + x)(14 + x). 132 + 28x = 140 + 24x, so 4x = 8 and x = 2. Check: 8 : 1... 1
- Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. What is the smaller number? Answer: A. 15 Answer: (A) 15 Let the numbers be 3k and 5k. Then 7(3k + 10) = 5(5k + 10), so 21k + 70 = 25k + 50 and k = 5. Smaller number =... 1
- A 40-litre mixture contains milk and water in the ratio 3 : 1. How much water must be added to make the ratio 3 : 2? Answer: C. 10 litres Answer: (C) 10 litres Milk = 30 L, water = 10 L. Milk does not change, so for 3 : 2 water must be 30 × 2/3 = 20 L. Wate... 1
- The salaries of A, B and C are in the ratio 2 : 3 : 5. If their salaries are increased by 15%, 10% and 20% respectively, what is the new ratio? Answer: B. 23 : 33 : 60 Answer: (B) 23 : 33 : 60 New salaries: 2 × 1.15 : 3 × 1.10 : 5 × 1.20 = 2.3 : 3.3 : 6. Multiply by 10: 23 : 33 : 60. 1
- If x : y = 3 : 4, what is (4x + 5y) : (5x − 2y)? Answer: D. 32 : 7 Answer: (D) 32 : 7 Shortcut: put x = 3, y = 4. 4x + 5y = 12 + 20 = 32 and 5x − 2y = 15 − 8 = 7. Ratio = 32 : 7. 1
- ₹3,400 is divided among A, B and C such that A gets \(\frac{2}{3}\) of what B gets and B gets \(\frac{1}{4}\) of what C gets. What is C's share? Answer: C. ₹2400 Answer: (C) ₹2400 Take C = 12 (LCM-friendly). Then B = 12 ÷ 4 = 3 and A = 2/3 × 3 = 2. So A : B : C = 2 : 3 : 12 (17 parts)... 1
- Two numbers are in the ratio 3 : 5. If 9 is subtracted from each, the ratio becomes 12 : 23. What is the smaller number? Answer: B. 33 Answer: (B) 33 Let the numbers be 3k and 5k. Then 23(3k − 9) = 12(5k − 9). 69k − 207 = 60k − 108, so 9k = 99 and k = 11. Small... 1
- ₹6,500 is divided among A, B and C in the ratio \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\). What is B's share? Answer: D. ₹2000 Answer: (D) ₹2000 Multiply by the LCM 12: the ratio becomes 6 : 4 : 3, a total of 13 parts. 1 part = 6500 ÷ 13 = 500. B's s... 1
- The numbers of students in three sections are in the ratio 3 : 5 : 7. If 15 students are added to each section, the ratio becomes 3 : 4 : 5. How many students were there in all at first? Answer: D. 75 Answer: (D) 75 Let the sections have 3k, 5k, 7k. From the first two: 4(3k + 15) = 3(5k + 15), so 12k + 60 = 15k + 45 and k = 5... 1
- If (a + b) : (a − b) = 7 : 3, what is a : b? Answer: C. 5 : 2 Answer: (C) 5 : 2 Shortcut (componendo–dividendo): a : b = (7 + 3) : (7 − 3) = 10 : 4. So a : b = 5 : 2. 1