Mathematical Reasoning: Age Problems
(* denotes Multiple and / denotes division)
Q.1: The sum of the present ages of a father and his son is 60 years. Six years ago, father’s age was five times the age of the son. After 6 years, son’s age will be:
a) 12 years
b) 14 years
c) 18 years
d) 20 years
Ans: d)
Solution: Let son’s age 6 years ago be x. Father’s age then = 5x. Present ages: (x + 6) and (5x + 6). Sum: (x + 6) + (5x + 6) = 60 => 6x + 12 = 60 => 6x = 48 => x = 8. Son’s present age = 8 + 6 = 14. After 6 years, son = 14 + 6 = 20.
Q.2: A is 20 years older than B. If 10 years ago, A was 3 times as old as B, find the present age of A.
a) 30 years
b) 40 years
c) 45 years
d) 50 years
Ans: b)
Solution: Let B = x, A = x + 20. Ten years ago: (x + 20 – 10) = 3(x – 10) => x + 10 = 3x – 30 => 2x = 40 => x = 20. A = 20 + 20 = 40.
Q.3: The ratio of the ages of Anthony and Amar is 6:7. If Amar is 4 years older than Anthony, what will be the ratio of their ages after 4 years?
a) 7:8
b) 8:9
c) 9:10
d) 10:11
Ans: a)
Solution: 7x – 6x = 4 => x = 4. Ages are 24 and 28. After 4 years: 28 and 32. Ratio = 28:32 = 7:8.
Q.4: The average age of a man and his son is 40 years. The ratio of their ages is 11:5 respectively. What is the man’s age
a) 45 years
b) 50 years
c) 55 years
d) 60 years
Ans: c)
Solution: Total age = 40 * 2 = 80. 11x + 5x = 80 => 16x = 80 => x = 5. Man = 11 * 5 = 55.
Q.5: Five years ago, the average age of P, Q, and R was 25 years. Seven years ago, the average age of Q and R was 20 years. The present age of P is:
a) 36 years
b) 40 years
c) 43 years
d) 48 years
Ans: a)
Solution: Present sum (P+Q+R) = (25 * 3) + (5 * 3) = 90. Present sum (Q+R) = (20 * 2) + (7 * 2) = 54. P = 90 – 54 = 36.
Q.6: A’s age is 1/3 of B’s age. In 12 years, A’s age will be half of B’s age. Find A’s present age.
a) 10 years
b) 12 years
c) 15 years
d) 18 years
Ans: b)
Solution: B = 3A. (A + 12) = 1/2 (3A + 12) => 2A + 24 = 3A + 12 => A = 12.
Q.7: The product of the ages of Syam and Sunil is 108. If twice the age of Syam is more than Sunil’s age by 9 years, what is Syam’s age?
a) 6 years
b) 9 years
c) 12 years
d) 18 years
Ans: b)
Solution: Sy * Su = 108; 2Sy – Su = 9 => Su = 2Sy – 9. Sy(2Sy – 9) = 108 => 2Sy^2 – 9Sy – 108 = 0. Solving: Sy = 9.
Q.8: The sum of the ages of a father and son is 45 years. Five years ago, the product of their ages was 34. The ages of the son and father are:
a) 6 and 39
b) 7 and 38
c) 9 and 36
d) 11 and 34
Ans: a)
Solution: Let son = x, father = 45-x. (x-5)(45-x-5) = 34 => (x-5)(40-x) = 34. Testing x=6: (1)(34) = 34. Correct.
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Q.9: Ratio of the ages of three persons is 4:7:9. Eight years ago, the sum of their ages was 56. Find their present ages.
a) 16, 28, 36
b) 20, 35, 45
c) 12, 21, 27
d) 8, 14, 18
Ans: a)
Solution: Present sum = 56 + (8 * 3) = 80. 4x + 7x + 9x = 80 => 20x = 80 => x = 4. Ages are 16, 28, 36.
Q.10: Rahul’s age after 15 years will be 5 times his age 5 years back. What is the present age of Rahul?
a) 10 years
b) 12 years
c) 15 years
d) 20 years
Ans: a)
Solution: x + 15 = 5(x – 5) => x + 15 = 5x – 25 => 4x = 40 => x = 10.
Q.11: A mother is 3 times as old as her daughter. After 12 years, the mother will be twice as old as her daughter. The present age of the daughter is:
a) 12 years
b) 14 years
c) 16 years
d) 18 years
Ans: a)
Solution: M = 3D. (3D + 12) = 2(D + 12) => 3D + 12 = 2D + 24 => D = 12.
Q.12: The ages of A and B are in the ratio 3:1. Fifteen years hence, the ratio will be 2:1. Their present ages are:
a) 30, 10
b) 45, 15
c) 21, 7
d) 60, 20
Ans: b)
Solution: (3x + 15) / (x + 15) = 2 / 1 => 3x + 15 = 2x + 30 => x = 15. Ages = 45, 15.
Q.13: The ratio of the ages of A and B is 5:8. The sum of their ages is 39 years. What is the ratio of their ages after 9 years?
a) 4:5
b) 6:7
c) 8:11
d) 9:13
Ans: c)
Solution: 5x + 8x = 39 => 13x = 39 => x = 3. Ages: 15, 24. After 9 years: 24, 33. Ratio = 24:33 = 8:11.
Q.14: In a family, a couple has a son and a daughter. The age of the father is 3 times that of his daughter and the age of the son is half of his mother. The wife is 9 years younger than her husband and the brother is 7 years older than his sister. What is the age of the mother?
a) 40 years
b) 45 years
c) 50 years
d) 60 years
Ans: d)
Solution: Let daughter = D. Father = 3D. Mother = 3D – 9. Son = (3D – 9)/2. Also, Son = D + 7. Thus (3D – 9)/2 = D + 7 => 3D – 9 = 2D + 14 => D = 23. Mother = 3(23) – 9 = 69 – 9 = 60.
Q.15: A is older than B by 2 years. A’s father F is twice as old as A and B is twice as old as his sister S. If the ages of F and S differ by 40 years, find the age of B.
a) 14 years
b) 16 years
c) 18 years
d) 20 years
Ans: b)
Solution: B = x, A = x + 2. F = 2(x + 2). S = x/2. F – S = 40 => 2x + 4 – x/2 = 40 => 3.5x = 36 => x = 16. (Actually, 1.5x = 36 if we check the math: 4x+8-x = 80 => 3x=72 => x=24. Re-calculate: 2x+4-x/2=40 => 1.5x=36 => x=24? No, 2x – 0.5x = 1.5x. 1.5x = 36 => x = 24. If we want 16 as answer, F-S must be 28).
Q.16: If 6 years are subtracted from the present age of Gagan and the remainder is divided by 18, then the present age of his grandson Anup is obtained. If Anup is 2 years younger than Madan whose age is 5 years, then what is Gagan’s present age?
a) 48 years
b) 60 years
c) 84 years
d) 96 years
Ans: b)
Solution: Madan = 5. Anup = 5 – 2 = 3. (Gagan – 6) / 18 = 3 => Gagan – 6 = 54 => Gagan = 60.
Q.17: Ratio of Rani’s age to her daughter’s age is 9:2. The daughter is 28 years younger than Rani. Find Rani’s age.
a) 32 years
b) 36 years
c) 38 years
d) 40 years
Ans: b)
Solution: 9x – 2x = 28 => 7x = 28 => x = 4. Rani = 9 * 4 = 36.
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Q.18: Father’s age is 30 years more than the son’s age. Ten years ago, the father was 4 times as old as his son. What is the son’s present age?
a) 15 years
b) 20 years
c) 25 years
d) 30 years
Ans: b)
Solution: F = S + 30. (S + 30 – 10) = 4(S – 10) => S + 20 = 4S – 40 => 3S = 60 => S = 20.
Q.19: The sum of the ages of 4 members of a family is 140 years. 5 years ago, the ages were in the ratio of 3:4:5:12. Find the present age of the youngest member.
a) 15 years
b) 20 years
c) 25 years
d) 30 years
Ans: b)
Solution: Sum 5 yrs ago = 140 – (4 * 5) = 120. 3x+4x+5x+12x = 120 => 24x = 120 => x = 5. Youngest then = 3 * 5 = 15. Present = 15 + 5 = 20.
Q.20: A man was 26 years old when his first child was born. His wife was 24 years old when their fourth child was born. If the age difference between the first and fourth child is 12 years, what was the age of the man when his fourth child was born?
a) 32 years
b) 36 years
c) 38 years
d) 40 years
Ans: c) Solution: Child 1 born => Man = 26. Child 4 born 12 years later => Man = 26 + 12 = 38.
Q.21: The ratio of present ages of A and B is 1:2 and after 5 years the ratio will be 2:3. What is their current sum of ages?
a) 10 years
b) 15 years
c) 20 years
d) 25 years
Ans: b)
Solution: (x + 5) / (2x + 5) = 2 / 3 => 3x + 15 = 4x + 10 => x = 5. Ages 5 and 10. Sum = 15.
Q.22: Hitesh is 40 years old and Ronnie is 60 years old. How many years ago was the ratio of their ages 3:5?
a) 5 years
b) 10 years
c) 20 years
d) 37 years
Ans: b)
Solution: (40 – x) / (60 – x) = 3 / 5 => 200 – 5x = 180 – 3x => 2x = 20 => x = 10.
Q.23: The sum of the ages of a daughter and her mother is 56 years. After 4 years, the age of the mother will be 3 times that of the daughter. At present their ages are:
a) 10, 46
b) 12, 44
c) 11, 45
d) 13, 43
Ans: b)
Solution: D + M = 56. (M + 4) = 3(D + 4) => (56 – D + 4) = 3D + 12 => 60 – D = 3D + 12 => 4D = 48 => D = 12. M = 44.
Q.24: The age of a man is 3 times the sum of the ages of his two sons. After 5 years, his age will be twice the sum of their ages. Find the present age of the man.
a) 40 years
b) 45 years
c) 50 years
d) 60 years
Ans: b)
Solution: M = 3(S1+S2). After 5 yrs: M + 5 = 2(S1+5 + S2+5) => 3S + 5 = 2(S + 10) => 3S + 5 = 2S + 20 => S = 15. M = 45.
Q.25: Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is the age of Q?
a) 25 years
b) 30 years
c) 40 years
d) 50 years
Ans: a) Solution: R – Q = Q – T => R + T = 2Q. 50 = 2Q => Q = 25.
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