Problems on Ages

Let Alice = x, John = 2x, Father = 10x. In two years: (x+2) + (2x+2) + (10x+2) = 58. 13x + 6 = 58 -> 13x = 52 -> x = 4. John is 2x, so 2 * 4 = 8. The logic confirms 8 is correct.

Let Al's age be x. The equation is 2x + 15 = 45. Subtracting 15 gives 2x = 30, so x = 15. Thus, Al is 15 years old. The other options do not satisfy the mathematical condition provided.

The age difference is constant: 31 - 8 = 23 years. Let Tom's current age be T, so father is T + 23. Given father = 2×Tom, we have T + 23 = 2T, giving T = 23. If Tom is 23, father is 46, which is exactly double. Options 21, 22, and 24 don't satisfy the doubling condition.

Let son's present age be x. The father was x when the son was born, so the father is now 2x. 2x = 38, so x = 19. Five years ago, the son was 19 - 5 = 14.

Let son be S and man be M. M = S + 24. In 2 years: M + 2 = 2(S + 2). Substituting M: (S + 24) + 2 = 2S + 4. S + 26 = 2S + 4. S = 22.

Let mother be M and son be S. S = 2/5M. In 8 years: (S+8) = 1/2(M+8). Substituting S: 2/5M + 8 = 1/2M + 4. Solving gives 4 = 1/10M, so M = 40.

Arun's current age is 26-6=20 years. If Arun:Deepak = 4:3, then Deepak = (3/4)×20 = 15 years. The ratio is maintained, and 15 is the only option that satisfies this.

Let ages be 4x, 7x, 9x. Eight years ago, sum was (4x-8)+(7x-8)+(9x-8) = 56. 20x - 24 = 56 => 20x = 80 => x = 4. Present ages: 4(4)=16, 7(4)=28, 9(4)=36.

Let Sachin's age be 7x and Rahul's be 9x. The difference is 9x - 7x = 2x. Given 2x = 7, we find x = 3.5. Sachin's age is 7 * 3.5 = 24.5 years. Other options fail to satisfy both the 7:9 ratio and the 7-year difference simultaneously.

Let x be years ago when mother was 3 times Mary's age: (80-x) = 3(54-x). Solving: 80-x = 162-3x, so 2x = 82, x = 41. 41 years ago, Mary was 13 and her mother was 39, and 39 = 3×13. The other options don't satisfy the 3:1 age ratio condition.

Let 'my age' be X. Ram = X - 5. Shyam = X + 10. The problem states Shyam = 4 * Ram. So, X + 10 = 4(X - 5). Solving this: X + 10 = 4X - 20; 30 = 3X; X = 10. My present age is 10.

Let Raj's age = R, Son's age = S. 5 years ago: R-5 = 5(S-5) → R = 5S - 20. 5 years later: R+5 = 3(S+5) - 8 → R = 3S + 2. Equating: 5S - 20 = 3S + 2 → 2S = 22 → S = 11. Then R = 5(11) - 20 = 35. Checking: 5 years ago Raj was 30, Son was 6, and 30 = 5×6. 5 years later Raj will be 40, Son will be 16, and 40 = 3×16 - 8.

Let the father's age be F and son's age be S. 3 years ago: F-3 = 7(S-3), so F = 7S - 18. Present: F = 5S. Equating: 5S = 7S - 18, giving 2S = 18, so S = 9. Then F = 5×9 = 45. Checking: 3 years ago, father was 42 and son was 6, and 42 = 7×6. Option C (45 & 9) is correct.

Let son be x, father 3x. In 15 years: 3x + 15 = 2(x + 15). Solving gives x = 15. Father's present age is 3x = 45. Distractors like 36 or 42 do not satisfy the 'twice as old' condition after 15 years.