Correct Option
Let the present age of the son be x years.
According to the first statement, "n years back I (father) was as old as you (son) are now."
- Son's age n years back = x - n years.
- Father's age n years back = x years.
- Therefore, the father's present age = x + n years.
According to the second statement, "My (father's) present age is four times your (son's) age n years back."
- Father's present age = x + n
- Son's age n years back = x - n
- Thus, x + n = 4(x - n)
- x + n = 4x - 4n
- 3x - 5n = 0 (Equation 1)
According to the third statement, "the sum of the present ages of the father and the son is 130 years."
- Father's present age + Son's present age = 130
- (x + n) + x = 130
- 2x + n = 130 (Equation 2)
Now, we solve the system of linear equations:
- 3x - 5n = 0
- 2x + n = 130
Multiply Equation 2 by 5:
- 5(2x + n) = 5(130)
- 10x + 5n = 650 (Equation 3)
Add Equation 1 and Equation 3:
- (3x - 5n) + (10x + 5n) = 0 + 650
- 13x = 650
- x = 650 / 13
- x = 50
Substitute the value of x into Equation 2:
- 2(50) + n = 130
- 100 + n = 130
- n = 130 - 100
- n = 30
The difference of their ages is the father's present age minus the son's present age:
- Difference = (x + n) - x
- Difference = n
- Difference = 30 years.
Incorrect Options
Options 2 (32 years), 3 (34 years), and 4 (36 years) are incorrect as they do not align with the derived solution from the given conditions and calculations.