Problems on Ages

Set up equations: let S = son's age, M = mother's age. From the problem: 3S + M = 75 and S + 2M = 80. Solve this system by substituting M = 75 - 3S into the second equation: S + 2(75 - 3S) = 80, which simplifies to S + 150 - 6S = 80, giving -5S = -70, so S = 14.

Let Aastha's age be x. Then mother's age is 4x. After 5 years: (4x+5) = 3(x+5). Solving: 4x+5 = 3x+15, so x = 10. Mother's present age = 4x = 40 years.

Let present ages be E (elder) and Y (youngier). E + Y = 46. Eight years ago: (E-8) = 2(Y-8). Substituting Y = 46-E: E-8 = 2(46-E-8) = 2(38-E) = 76-2E. So 3E = 84, E = 28. Verification: Y = 18, eight years ago ages were 20 and 10, and 20 = 2(10). Present age of elder is 28 years.

When father was 52, there's a 2-year age gap between sisters. Father is 2 years older than mother (so mother was 50 then). Elder sister = half of 50 = 25. Younger sister = 25 - 2 = 23.

Let Vikash's age 4 years ago = 3x, Rahul's age 4 years ago = 5x. Present ages: Vikash = 3x+4, Rahul = 5x+4. After 6 years: Vikash = 3x+10, Rahul = 5x+10. Ratio is (3x+10)/(5x+10) = 4/5. Cross multiply: 5(3x+10) = 4(5x+10), giving 15x+50 = 20x+40, so 5x = 10, x = 2. Rahul's present age = 5(2)+4 = 14.

Let the father's current age be F and the sum of children's current ages be S. Given F = S. After 15 years, sum of children's ages will be S + 5*15 = S + 75, and father's age will be F + 15. Given S + 75 = 2(F + 15). Substituting F = S: S + 75 = 2S + 30, so 45 = S. Therefore F = 45 years.

Let son's age 7 years ago = s. Then father's age 7 years ago = 7s. Current ages: son = s + 7, father = 7s + 7. In 3 years: son = s + 10, father = 7s + 10. Given: 7s + 10 = 3(s + 10). Solving: 7s + 10 = 3s + 30, so 4s = 20, s = 5. Current ages: son = 12, father = 42. Option A matches.

5 years ago, average age of 3 people was 35, so sum of ages was 105. Present sum of ages = 105 + 15 = 120 (each aged 5 years). 3 years ago, average of father and mother was 46, so sum was 92. Present sum of parents' ages = 92 + 6 = 98. Son's present age = 120 - 98 = 22 years.

Sangram's age is derived from Avinash's age of 7 years plus 2 years, giving Sangram's age as 9 years. Shyam's age satisfies the equation (S-10)/7 = 9, which gives S-10 = 63, so Shyam's present age is 73 years. This is verified by option D.

Let present ages of A and B be 6x and 7x. B is 8 years older than A, so 7x - 6x = 8, which gives x = 8. Present ages: A = 48, B = 56. After 18 years: A = 66, B = 74. Ratio = 66:74 = 33:37. Option C is correct. Option A (33:35) is incorrect - this would require B to be only 2 years older than A.

B is equally distant from A and C in age. Let ages be A, B, C. Then B - A = C - B, so 2B = A + C. Given A:C = 5:9, let A = 5k, C = 9k. Then 2B = 5k + 9k = 14k, so B = 7k. B after 10 years = 7k + 10 = 24, so 7k = 14, k = 2. Present ages: A = 10, B = 14, C = 18. Average = (10 + 14 + 18)/3 = 42/3 = 14.

Let daughter's age = d. Then Ashutosh = 3d and mother = (13/9) × 3d = (13/3)d. Sum = d + 3d + (13/3)d = 125. This gives (3d + 9d + 13d)/3 = 125, so 25d = 375 and d = 15. Daughter = 15, mother = (13/3) × 15 = 65. Difference = 65 - 15 = 50.

Raman celebrated his 6th birthday 6 years ago, so Raman's present age = 6 + 6 = 12 years. After 8 years, Raman's age = 12 + 8 = 20 years. After 8 years, father's age will be twice Raman's age, so father's age after 8 years = 2 × 20 = 40 years. Therefore, father's present age = 40 - 8 = 32 years. Sumit's present age = 1/8 × father's present age = 1/8 × 32 = 4 years.

Let R, M, S, Rm be present ages. Three years ago: (R-3)+(M-3)+(S-3)+(Rm-3) = 4×32 = 128, so R+M+S+Rm = 140. Also M+S = 2×36 = 72. Then R+Rm = 140-72 = 68. Given Rm = R+4, so R+(R+4) = 68, 2R = 64, R = 32. But 32 isn't in options A-D, so E (None of these) is correct.

Let Ram's present age be R and his son's present age be S. Given R = 4S and R - 5 = 9(S - 5). Substituting R = 4S in the second equation: 4S - 5 = 9S - 45, which gives 5S = 40, so S = 8. Therefore, R = 4 × 8 = 32. Their present ages are 32 and 8 years.