A father said to his son, “n years back I was as old as you are now. My present age is four times your age n years back”. If the sum of the present ages of the father and the son is 130 years, what is the difference of their ages?

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:

1. 3x - 5n = 0
2. 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.