Age Problems

X + Y = 45 (1)
Y + Z = 55 (2)
X + Z = 65 (3)
(X + Z) – (X + Y) = 65 – 45 [Equation(2) – (1)]
= Z – Y = 20 (4)
(Z – Y) + (Z + Y) = 20 + 55 [Equation(4) + (2)]
= 2Z = 75
= Z = 37.5
Y + 37.5 = 55 (Substituting the value of Z in equation 2)
= Y = 17.5

Let son be x years old.
Father = 2x
20 years ago,
son = x - 20
father = 2x - 20
2x - 20 = 12*(x - 20)
2x - 20 = 12x - 240
Add 20 to both sides.
2x = 12x - 240 + 20
2x = 12x - 220
Add - 12x to both sides.
2x - 12x = 12x - 12x - 220
-12x + 2x = -240 + 20
-10x = -220
Divide by -10.
x = -220/-10
x = 22 (son's age)
Father's age = 2*22 = 44 years

Let the present ages of A and B be 4x and 5x, respectively.
= 4x - 5x = 8
= x = 8
Now, present age of A is (4x) 32 and B is (5x) is 40.
= 40 + 32 = 72.

"the man is 3 times as old as his son" ___ f = 3s
"15 years ago, the man was 9 times as old as his son was then" ___ f - 15 = 9(s - 15)
substituting ___ 3s - 15 = 9(s - 15)
distributing ___ 3s - 15 = 9s - 135
adding 135 - 3s ___ 120 = 6s
dividing by 6 ___ 20 = s
substituting ___ f = 3(20) = 60
Therefore, after 15 years, the age of the man will be = 60 + 15 = 75 years.

Let the present ages of Jhadhav and Vijay be 3x and 7x, respectively.
= 7x – 3x = 24
= x = 24/4 = 6
Therefore, the present ages of Jhadhav and Vijay are 42 and 18, respectively.
= Sum of their ages = 42 + 18 = 60

Let the three ages be 4x, 7x, and 9x. Eight years ago, their ages were (4x-8), (7x-8), and (9x-8). The sum was (4x-8) + (7x-8) + (9x-8) = 20x - 24 = 56. Solving: 20x = 80, so x = 4. Present ages are 44=16, 74=28, and 9*4=36.

Let the present ages of P and Q be 5x and 8x, respectively.
(5x + 4)/(8x + 4) = 2/3
= 3(5x + 4) = 2(8x + 4)
= 15x + 12 = 16x + 8
= x = 4
Therefore, the present age of Q = 8x = 32.

Let present age of son = x
Age of son before 5 years = x - 5
So, Present age of father = 7(x - 5) + 5 ……… (i)
Age of son after 5 years = x + 5
Age of father after 5 years = 3(x + 5)
So, present age of father = 3(x + 5) - 5 ……….(ii)
Equating age of father, i.e. (i) and (ii): 
7(x - 5) + 5 = 3(x + 5) - 5
7x – 35 + 5 = 3x + 15 - 5
7x - 3x = 15 - 5 + 35 - 5
4x = 40
x = 40/4. x (son) = 10, Father (3(10 + 5) - 5) =  40 years

Let A's age be 4x and B's age be 5x. Four years hence, their ages will be (4x+4) and (5x+4). The ratio becomes (4x+4)/(5x+4) = 14/17. Cross-multiplying: 17(4x+4) = 14(5x+4), which gives 68x + 68 = 70x + 56. Solving: 2x = 12, so x = 6. B's present age is 5*6 = 30.

Let Shivani's current age be x and her mother's age be y. Five years ago: (y-5) = 3(x-5), meaning mother was 3 times as old as Shivani. After 5 years: (y+5) = 2(x+5), meaning mother will be twice as old. From second equation: y = 2x + 5. Substituting into first: 2x = 3x - 15, giving x = 15. Shivani is 15 years old today. Option B is correct.

Let the son's present age be x. The father's present age is 2x. Twenty years ago, their ages were (2x-20) and (x-20). At that time, the father was 12 times as old: (2x-20) = 12(x-20). Solving: 2x - 20 = 12x - 240, giving 10x = 220, so x = 22. Son is 22 years old, father is 44 years old.

Let the son's present age be x. Father's age is 46-x. Five years ago: (46-x-5) = 11(x-5), meaning father was 11 times as old as the son. Solving: 41 - x = 11x - 55, giving 12x = 96, so x = 8. Son is 8 years old now. After 5 years, son will be 8 + 5 = 13 years old.

Let the son's present age be x. Father's age is 60-x. Six years ago: (60-x-6) = 5(x-6), meaning father was 5 times as old as the son. Solving: 54 - x = 5x - 30, giving 6x = 84, so x = 14. Son is 14 years old now. After 6 years, son will be 14 + 6 = 20 years old.

Let Nutan's age be 5x and Vidhya's age be 4x (ratio 5:4). Their sum is 5x + 4x = 9x = 36. Solving: x = 4. Vidhya's present age is 44 = 16 years. Nutan's present age is 54 = 20 years.

Let Sonu's present age be x. Father's age is 5x. After 8 years, Sonu will be (x+8) and father will be (5x+8). At that time, father will be thrice as old: (5x+8) = 3(x+8). Solving: 5x + 8 = 3x + 24, giving 2x = 16, so x = 8. Sonu is 8 years old and father is 40 years old.

Let the present age of the son be x. Therefore, the present age of the father is 4x.
5 years ago, 7(x – 5) = (4x – 5)
= 7x – 35 = 4x - 5
= 3x = 30
= x = 10
Therefore, the present age of the father is 4x = 40 years.

Let A's present age be 3x and B's present age be 5x (ratio 3:5). Six years hence, their ages will be (3x+6) and (5x+6). The new ratio is (3x+6)/(5x+6) = 2/3. Cross-multiplying: 3(3x+6) = 2(5x+6), giving 9x + 18 = 10x + 12. Solving: x = 6. B's present age is 5*6 = 30 years.

Let the father's age be F and the son's age be S. We know F + S = 45. Five years ago, their ages were (F-5) and (S-5). The product was four times the father's age: (F-5)(S-5) = 4(F-5). Assuming F ≠ 5, we can divide: S-5 = 4, so S = 9. Then F = 45 - 9 = 36. Checking: 5 years ago, ages were 31 and 4, product is 124, which is indeed 4*31 = 124. Present ages are 36 and 9.

Let the ages of A and Q be 5x and 6x, respectively.
Now, (5x + 4) + (6x + 4) = 52
5x + 4 + 6x + 4 = 52
11x + 8 = 52
x = 4
Therefore, present age of A (5x) = 24 years and Q (6x) = 28
= Ratio of their present ages is 24 : 28 = 6 : 7

Let the daughter's present age be x. Mother's age is 46-x. After 4 years, daughter will be (x+4) and mother will be (46-x+4) = (50-x). At that time, mother will be twice as old: (50-x) = 2(x+4). Solving: 50 - x = 2x + 8, giving 3x = 42, so x = 14. Daughter is 14 years old.

Let P's age be 2x and Q's age be 3x (ratio 2:3). We're told that P is 12 years less than Q: 2x = 3x - 12. Solving: x = 12. Q's present age is 312 = 36 years. P's present age is 212 = 24 years. The difference is indeed 36 - 24 = 12 years.

Let present ages be 2x and x. After 5 years: (2x+5)/(x+5) = 3/2. Cross-multiplying: 4x+10 = 3x+15, so x = 5. Present ages are 10 and 5 years. Option A (40, 20) and D (20, 10) are proportional but don't satisfy the 5-year condition. Option C's values are not in 2:1 ratio.

Daughter is 10 years old now. Mother's age = 3 × 10 = 30 years. Father is 5 years older than mother = 35 years now. When daughter was born (10 years ago), father was 35 - 10 = 25 years old. Option A (20 years) would make the current age gap too small.

P + Q = 55 (1)
-P + Q = 5 (2)
(P + Q) + (-P + Q) [Adding (1) and (2)]
= 2Q = 60
= Q = 30