Present age of Ram is three times of his son’s present age and two -third of his father’s present age. At present, the average age of all of them is 34 years. What is the difference between present age of Ram’s son and his father’s age?

The correct answer is 42 yrs (or Option 2: 42 yrs).

Let us solve this problem step-by-step by setting up linear equations based on the relationships given in the question.

Step 1: Define the variables for the present ages.
Let the present age of Ram's son be S years.
Let the present age of Ram be R years.
Let the present age of Ram's father be F years.

Step 2: Express Ram's age and his father's age in terms of his son's age.
1. It is given that the present age of Ram is three times his son's present age:
$R=3S$

2. It is also given that Ram's present age is two-thirds of his father's present age:
$R=\frac{2}{3}F$

Substituting $R=3S$ into the equation above:
$3S=\frac{2}{3}F$

Multiplying both sides by 3 and dividing by 2 gives Ram's father's age in terms of S:
$F=\frac{3\times 3S}{2}=\frac{9S}{2}=4.5S$

Step 3: Use the average age to find the value of S.
The average age of all three people (Ram, his son, and his father) is 34 years. Therefore, the sum of their ages is:
$\text{Sum of ages}=34\times 3=102\text{years}$

Now, write the sum of ages equation in terms of S:
$S+R+F=102$

Substitute $R=3S$ and $F=4.5S$ into the sum equation:
$S+3S+4.5S=102$
$8.5S=102$

Solving for S:
$S=\frac{102}{8.5}=12\text{years}$

Step 4: Find the ages of Ram's son and Ram's father.
Present age of Ram's son ($S$) = 12 years.
Present age of Ram's father ($F$) = $4.5\times 12=54\text{years}$.

Step 5: Calculate the difference between the present age of Ram's father and Ram's son.
$\text{Difference}=F-S=54-12=42\text{years}$

Thus, the difference between the present age of Ram's son and his father's age is 42 yrs.