A father said to his son, “ n years back l 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?

The correct option is 30 years.

Let us break down the problem step-by-step to find the relation between the ages of the father and the son.

Let the present age of the father be represented by $F$ and the present age of the son be represented by $S$.

According to the first statement: "n years back I was as old as you are now".
The father's age $n$ years back was $F-n$, and the son's present age is $S$.
So, we can write the equation:
$F-n=S$
This can be rewritten to find the difference between their ages:
$F-S=n$ (Equation 1)

According to the second statement: "My present age is four times your age n years back".
The son's age $n$ years back was $S-n$.
So, the father's present age $F$ is:
$F=4(S-n)$ (Equation 2)

Now, let us substitute $n=F-S$ from Equation 1 into Equation 2:
$F=4(S-(F-S\left)\right)$
Simplify the terms inside the parentheses:
$F=4(S-F+S)$
$F=4(2S-F)$
Expand the right side:
$F=8S-4F$
Add $4F$ to both sides:
$5F=8S$
Therefore, we get:
$F=\frac{8}{5}S$ (Equation 3)

We are also given that the sum of their present ages is 130 years:
$F+S=130$ (Equation 4)

Substitute Equation 3 into Equation 4:
$\frac{8}{5}S+S=130$
Taking the common denominator:
$\frac{8S+5S}{5}=130$
$\frac{13S}{5}=130$
Multiply both sides by 5:
$13S=650$
Divide by 13:
$S=50$

So, the present age of the son is 50 years.
Now, find the father's present age using Equation 4:
$F=130-S$
$F=130-50=80$

The father's present age is 80 years.
The difference between their ages is:
$F-S=80-50=30\text{years}$