Problems on Ages

4 years ago, H+W = 2×27 = 54. Now, H+W = 54+8 = 62. With child, average is 21, so H+W+C = 3×21 = 63. Child's age = 63-62 = 1 year.

Let A's present age be a years. Then B's present age is 2a years. After 4 years: B's age = 2a + 4, and twice A's age = 2(a + 4) = 2a + 8. The condition states 2a + 4 = (2a + 8) - 4, which simplifies to 2a + 4 = 2a + 4. This is an identity that holds for all values of a, meaning the information given is insufficient to determine a unique numerical answer for the age difference.

Let Rahul's age = 8x + 9, Suresh's age = 7x + 9 (9 years ago). After 6 + 9 = 15 years: (8x + 24)/(7x + 24) = 5/6. This gives 48x + 144 = 35x + 120, or 13x = -24, giving negative x. The system is inconsistent, making the data inadequate to determine Suresh's present age.

Let uncle's present age = U. Ram is 30 years younger, so Ram's age = U - 30. Five years ago: (U - 30 - 5) = (1/4)(U - 5). Solving: U - 35 = (U - 5)/4, giving 4U - 140 = U - 5, so 3U = 135 and U = 45. Uncle's present age is 45, so after 5 years it will be 50. Option C is correct.

Working backwards: Rehana's age after 7 years = 85, so her present age = 85-7 = 78. Vinod is 12 years younger than Rehana, so Vinod's present age = 78-12 = 66. Manoj:Vinod = 3:11, so 3x + 11x = 66 gives x = 4.411 (not integer). Wait - the ratio is of their ages, not sum. If Vinod = 11x = 66, then x = 6, so Manoj = 3×6 = 18. Manoj's father is 25 years older than Manoj, so father's age = 18+25 = 43. Options B, C, D, and E are incorrect calculations.

Let woman's age = 7x, daughter's age = x. Average age = (7x + x)/2 = 4x = 16, so x = 4. Woman's age = 7x = 7(4) = 28 years. Verify: Daughter is 4, woman is 28, ratio 7:1 ✓, average (28+4)/2 = 16 ✓. Option B is correct.

Average × 2 gives the sum of ages: 52 × 2 = 104 total. If younger is 40, elder is 104 - 40 = 64. The difference is 64 - 40 = 24 years.

Let Rekha's present age be x. Then Ajay's age is 1.5x. After 8 years, the ratio becomes (1.5x + 8)/(x + 8) = 25/18. Cross-multiplying gives 27x + 144 = 25x + 200, so 2x = 56 and x = 28. Rekha is currently 28 years old. Option A (50) would satisfy the ratio but is the present age, not future. Options C and D don't satisfy the equation.

Let A's and B's ages 10 years ago be 2x and 3x. Their present ages are (2x+10) and (3x+10). Given (2x+10)+(3x+10)=60, we get 5x=40, so x=8. Present ages: A=26, B=34. After 5 years: A=31, B=39. The ratio is 31:39. Option D matches this calculation.

Let present ages be Meena = M, Meeta = T. 5 years ago: T = (5/7)(M - 5). In 5 years: T + 5 = (7/9)(M + 5). Solving these equations yields M:T = 4:3. This matches option E.

Let Pramila's current age = P, Swati's = S. One year ago: P-1 = 4(S-1), so P-1 = 4S-4, thus P = 4S-3. Six years hence: P+6 = (S+6) + 24, so P+6 = S+30, thus P = S+24. Setting equal: 4S-3 = S+24, so 3S = 27, S = 9. Then P = 9+24 = 33. Ratio P:S = 33:9 = 11:3. Verification: One year ago, Pramila was 32, Swati was 8, and 32 = 4×8.

Let John's present age be x. Then (x-7)(x+2) = 52. Expanding: x² - 5x - 14 = 52, so x² - 5x - 66 = 0. Factoring: (x-11)(x+6) = 0. Since age cannot be negative, x = 11 years. Verification: (11-7) × (11+2) = 4 × 13 = 52 ✓

Let ages be 5x and 6x. After 6 years: (5x+6)/(6x+6) = 8/9. Cross-multiplying: 45x+54 = 48x+48, so 3x = 6 and x = 2. Current ages: 10 and 12 years, difference = 2 years. This difference remains constant throughout their lives.

Let the son's present age be x years. Then the man's present age is 2x². Eight years hence: man's age = 2x² + 8, son's age = x + 8. The condition gives 2x² + 8 = 3(x + 8) + 4 = 3x + 28. Solving 2x² - 3x - 20 = 0 gives x = 4 or x = -2.5. Age cannot be negative, so x = 4.

Let ages 10 years ago be 7x and 8x. Present ages: (7x + 10) + (8x + 10) = 50, so 15x + 20 = 50, thus 15x = 30 and x = 2. Shyam's present age = 8x + 10 = 8(2) + 10 = 26 years. Always add the time elapsed to get present age from a past ratio.