Hari Ram’s present age is three times his son’s present age and two fifth of his father’s present age. The average of the present age of all of them is 46 years. What is the difference between Hari Ram’s son’s present age and Hari Ram’s father’s present age?

Options

A. A. 44 yrs
B. B. 56 yrs
C. C. 67 yrs
D. D. 78 yrs

Answer

To solve the problem, we will define the variables for the ages of Hari Ram, his son, and his father, and then use the information given in the question to set up equations.

### Step 1: Define the variables
Let:
- \( S \) = Hari Ram's son's present age
- \( H \) = Hari Ram's present age
- \( F \) = Hari Ram's father's present age

### Step 2: Set up the equations based on the problem statement
From the problem, we know:
1. Hari Ram’s present age is three times his son’s present age:
\[ H = 3S \]
2. Hari Ram’s present age is two-fifths of his father’s present age:
\[ H = \frac{2}{5}F \]
3. The average of their present ages is 46 years:
\[ \frac{H + S + F}{3} = 46 \]

### Step 3: Rearranging the average equation
Multiplying both sides of the average equation by 3 gives:
\[ H + S + F = 138 \]

### Step 4: Substitute \( H \) in terms of \( S \) and \( F \)
From the first equation \( H = 3S \), we can substitute \( H \) in the average equation:
\[ 3S + S + F = 138 \]
This simplifies to:
\[ 4S + F = 138 \]

### Step 5: Substitute \( F \) in terms of \( H \)
From the second equation \( H = \frac{2}{5}F \), we can express \( F \) in terms of \( H \):
\[ F = \frac{5}{2}H \]

### Step 6: Substitute \( H \) in terms of \( S \) into the equation for \( F \)
Substituting \( H = 3S \) into \( F = \frac{5}{2}H \):
\[ F = \frac{5}{2}(3S) = \frac{15}{2}S \]

### Step 7: Substitute \( F \) back into the equation \( 4S + F = 138 \)
Now substituting \( F \):
\[ 4S + \frac{15}{2}S = 138 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 8S + 15S = 276 \]
This simplifies to:
\[ 23S = 276 \]

### Step 8: Solve for \( S \)
Now, divide both sides by 23:
\[ S = \frac{276}{23} = 12 \]

### Step 9: Find \( H \) and \( F \)
Now that we have \( S \):
1. Calculate \( H \):
\[ H = 3S = 3 \times 12 = 36 \]
2. Calculate \( F \):
\[ F = \frac{5}{2}H = \frac{5}{2} \times 36 = 90 \]

### Step 10: Find the difference between Hari Ram’s son’s age and his father’s age
Now we need to find the difference between \( F \) and \( S \):
\[ F - S = 90 - 12 = 78 \]

### Final Answer
The difference between Hari Ram’s son’s present age and Hari Ram’s father’s present age is **78 years**.

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