Set B Solutions

B1. Answer: (a) 105 Let numbers = 3x and 4x (in lowest terms, so HCF part =1, meaning LCM = product/HCF = 3x×4x/x =12x). LCM = 12x = 180 → x = 15. Numbers = 45, 60. Sum = 105.

B2. Answer: (a) ₹2700 A:B = 2:3, B:C = 4:5. LCM of B-terms (3,4) = 12. A:B = 2:3 = 8:12. B:C = 4:5 = 12:15. So A:B:C = 8:12:15, total parts = 35. 1 part = 6300/35 = 180. C’s share = 15 × 180 = 2700.

B3. Answer: (a) 19 : 29 A’s capital-months = (35000×4) + (30000×8) = 140000 + 240000 = 380000. B’s capital-months = (45000×4) + (50000×8) = 180000 + 400000 = 580000. Ratio = 380000:580000 = 38:58 = 19:29 (dividing by 2).

B4. Answer: (a) ₹6000 Let incomes = 3x, 2x and expenditures = 5y, 3y. Savings: 3x − 5y = 1000 …(i); 2x − 3y = 1000 …(ii) Multiply (i) by 3: 9x − 15y = 3000. Multiply (ii) by 5: 10x − 15y = 5000. Subtract: (10x − 9x) = 5000 − 3000 → x = 2000. Substitute into (ii): 2(2000) − 3y = 1000 → 4000 − 3y = 1000 → 3y = 3000 → y = 1000. A’s income = 3x = 3 × 2000 = ₹6000. (Check: B’s income = 4000, A’s expenditure = 5000, B’s expenditure = 3000; savings: 6000−5000=1000 ✓, 4000−3000=1000 ✓.)

B5. Answer: (a) 5 : 1 Using reverse componendo-dividendo: if (x+y)/(x−y) = p/q, then x/y = (p+q)/(p−q) = (3+2)/(3−2) = 5/1. Check: x=5,y=1 → (5+1)/(5−1) = 6/4 = 3/2. ✓

B6. Answer: (a) ₹5760 Ratio 1/2 : 1/3 : 1/4 → multiply by LCM 12 → 6 : 4 : 3. A’s investment for first 4 months = 6 units; after increasing by 50%, for remaining 8 months = 6×1.5 = 9 units. A’s capital-months = (6×4) + (9×8) = 24 + 72 = 96. B’s capital-months = 4×12 = 48. C’s capital-months = 3×12 = 36. Total = 96+48+36 = 180. B’s share = 21600 × 48/180 = 21600 × (48/180). 21600/180 = 120. 120×48 = 5760.

B7. Answer: (a) 30 Let ages 6 years ago = 3x, 4x. Present ages = 3x+6, 4x+6. Given present ratio: (3x+6)/(4x+6) = 4/5. Cross-multiply: 5(3x+6) = 4(4x+6) → 15x+30 = 16x+24 → x = 6. Q’s present age = 4x+6 = 24+6 = 30. (Check: P’s present age = 3(6)+6 = 24; 24:30 = 4:5 ✓. Six years ago: 18:24 = 3:4 ✓.)

B8. Answer: (a) 33.33% Boys = 5x, girls = 4x, total = 9x. Scholarship boys = 0.40 × 5x = 2x. Scholarship girls = 0.25 × 4x = 1x. Total scholarship students = 3x. Percentage = (3x/9x) × 100 = 33.33%.

B9. Answer: (a) 40, 48 Let numbers = 5x, 6x. (5x−8)/(6x−8) = 4/5. Cross-multiply: 5(5x−8) = 4(6x−8) → 25x − 40 = 24x − 32 → x = 8. Numbers = 5×8=40, 6×8=48. (Check: after subtracting 8: 32:40 = 4:5 ✓.)

B10. Answer: (a) 3 : 2 Efficiency is inversely proportional to time taken. Efficiency ratio of A:B = 1/12 : 1/18 = 18 : 12 = 3 : 2 (multiplying both by LCM 36 and simplifying: 36/12 : 36/18 = 3:2).

B11. Answer: (a) 5 : 7 a:b = 2:3, b:c = 5:4. LCM of b-terms (3,5) = 15. a:b = 2:3 = 10:15. b:c = 5:4 = 15:12. So a:b:c = 10:15:12. Now c:d = 6:7, and current c = 12. Scale 6:7 so that 6 becomes 12: multiply by 2 → 12:14. So a:b:c:d = 10:15:12:14. a:d = 10:14 = 5:7.

B12. Answer: (a) ₹1680 A’s savings = 25% of A’s income = 2400 → A’s income = 2400/0.25 = 9600. Income ratio A:B = 8:5 → B’s income = 9600 × 5/8 = 6000. A’s expenditure = A’s income − A’s savings = 9600 − 2400 = 7200. Expenditure ratio A:B = 5:3 → B’s expenditure = 7200 × 3/5 = 4320. B’s savings = B’s income − B’s expenditure = 6000 − 4320 = 1680.

B13. Answer: (a) ₹3600 Difference between D’s and A’s parts = 6 − 3 = 3 parts = ₹600 → 1 part = ₹200. Total parts = 3+4+5+6 = 18. Total sum = 18 × 200 = ₹3600.

B14. Answer: (a) 20/29 Let x = 7, y = 3 (using the given ratio directly). x² = 49, y² = 9. (x²−y²)/(x²+y²) = (49−9)/(49+9) = 40/58 = 20/29 (dividing by 2).

B15. Answer: (a) 15 Milk = (4/5)×45 = 36 litres. Water = (1/5)×45 = 9 litres. Let x litres of water be added: 36/(9+x) = 3/2. Cross-multiply: 36×2 = 3×(9+x) → 72 = 27 + 3x → 3x = 45 → x = 15.

B16. Answer: (b) 48 years Let the ages be 2x, 3x, 4x. Given: (2x + 4x) − 3x = 36 → 3x = 36 → x = 12. C’s age = 4x = 4 × 12 = 48 years.

B17. Answer: (a) 8 : 9 A’s capital-months = (42000×5) + (50000×7) = 210000 + 350000 = 560000. (A’s capital becomes 42000+8000 = 50000 for the remaining 12−5 = 7 months.) B’s capital-months = (56000×5) + (50000×7) = 280000 + 350000 = 630000. (B’s capital becomes 56000−6000 = 50000 for the remaining 7 months.) Ratio = 560000 : 630000. Dividing by 70000: 8 : 9.

B18. Answer: (c) 32 Let the other number = x. Mean proportional formula: 24² = 18 × x → 576 = 18x → x = 576/18 = 32.

B19. Answer: (d) 15 : 8 Treat 4x as “a” and 3y as “b”: 4x/3y = (7+3)/(7−3) = 10/4 = 5/2 → 4x × 2 = 5 × 3y → 8x = 15y → x : y = 15 : 8. Check: x=15, y=8 → 4x=60, 3y=24 → (60+24)/(60−24) = 84/36 = 7/3. ✓

B20. Answer: (a) ₹10,000 Income : Expenditure = 7 : 5. Income = 7 parts = 35000 → 1 part = 5000. Expenditure = 5 parts = 5×5000 = 25000. Savings = Income − Expenditure = 35000 − 25000 = ₹10,000.

B21. Answer: (b) 5 : 8 New first number = 3 × 1.25 = 3.75. New second number = 4 × 1.5 = 6. New ratio = 3.75 : 6 = 375 : 600 = 5 : 8 (dividing by 75).

B22. Answer: (c) 20 A’s one-day work = 1/20; B’s one-day work = 1/30. Combined one-day work = 1/20 + 1/30 = 3/60 + 2/60 = 5/60 = 1/12. Work done together in 4 days = 4 × 1/12 = 1/3. Remaining work = 1 − 1/3 = 2/3. Days for B alone to finish 2/3 of the work = (2/3) ÷ (1/30) = (2/3) × 30 = 20 days.

B23. Answer: (a) 12 Third proportional to 6, 12: x = 12²/6 = 144/6 = 24. Fourth proportional to 3, 4, 9: y = (4×9)/3 = 36/3 = 12. x − y = 24 − 12 = 12.

B24. Answer: (d) 10 Milk = 55 × 7/11 = 35 litres. Water = 55 × 4/11 = 20 litres. For the new ratio 7 : 6 with milk unchanged at 35: new water = 35 × 6/7 = 30 litres. Water to be added = 30 − 20 = 10 litres.

B25. Answer: (b) 30 Total parts = 3+5+7 = 15. Value of 1 part = 75/15 = 5. B’s present age = 5 × 5 = 25. B’s age five years hence = 25 + 5 = 30 years.

B26. Answer: (a) 71.67% Let boys = 7x, girls = 5x, total = 12x. Absent boys = 0.20 × 7x = 1.4x. Absent girls = 0.40 × 5x = 2x. Total absent = 3.4x. Present students = 12x − 3.4x = 8.6x. Percentage present = (8.6x/12x) × 100 = 71.67% (approx).

B27. Answer: (c) 8 : 7 Taking the reciprocal of both sides: (7a+2b)/(7a−2b) = 5/3. Applying reverse componendo-dividendo to m = 7a, n = 2b: m/n = (5+3)/(5−3) = 8/2 = 4. So 7a/2b = 4 → 7a = 8b → a : b = 8 : 7. Check: a=8, b=7 → 7a=56, 2b=14 → (56−14)/(56+14) = 42/70 = 3/5. ✓

B28. Answer: (a) ₹16,200 B’s management fee = 10% of 40000 = ₹4000. Remaining profit = 40000 − 4000 = ₹36,000. Investment ratio A:B = 45000:55000 = 9:11, total parts = 20. A’s total earning = 36000 × 9/20 = ₹16,200. (B’s total earning = 4000 + 36000×11/20 = 4000 + 19800 = ₹23,800; check: 16200+23800 = 40000 ✓.)

B29. Answer: (d) 30 First : Second = 2 : 3 = 10 : 15 (multiplying by 5). Second : Third = 5 : 8 = 15 : 24 (multiplying by 3). So First : Second : Third = 10 : 15 : 24, total parts = 49. 1 part = 98/49 = 2. Second number = 15 × 2 = 30.

B30. Answer: (a) 14 years Let the ages 4 years ago = 5x, 2x. Four years from now (i.e., 8 years after the “4 years ago” point): (5x+8)/(2x+8) = 3/2. Cross-multiply: 2(5x+8) = 3(2x+8) → 10x+16 = 6x+24 → 4x = 8 → x = 2. Ages 4 years ago: father = 10, son = 4. Father’s present age = 10 + 4 = 14 years. (Check: son’s present age = 4+4 = 8; four years hence — father = 18, son = 12, ratio = 18:12 = 3:2 ✓.)