A father's age is now three times that of his elder daughter. Five years back, his age was eight times that of his younger daughter. If the difference of ages of the two daughters is 5 years, what is the age of the father now?

Options

A. 55
B. 50
C. 60
D. 45

Explanation

To solve the problem step by step, we need to define the variables and set up equations based on the information given.

### Step 1: Define the variables
Let:
- \( E \) = present age of the elder daughter
- \( Y \) = present age of the younger daughter
- \( F \) = present age of the father

### Step 2: Set up the equations based on the problem statement
1. From the first statement, "A father's age is now three times that of his elder daughter":
\[ F = 3E \quad \text{(Equation 1)} \]

2. From the second statement, "Five years back, his age was eight times that of his younger daughter":
\[ F - 5 = 8(Y - 5) \quad \text{(Equation 2)} \]

3. From the third statement, "the difference of ages of the two daughters is 5 years":
\[ E - Y = 5 \quad \text{(Equation 3)} \]

### Step 3: Solve the equations
First, we can express \( Y \) in terms of \( E \) using Equation 3:
\[ Y = E - 5 \quad \text{(Substituting into Equation 3)} \]

Now, substitute \( Y \) into Equation 2:
\[ F - 5 = 8((E - 5) - 5) \]
This simplifies to:
\[ F - 5 = 8(E - 10) \]
Expanding gives:
\[ F - 5 = 8E - 80 \]
Rearranging this gives:
\[ F = 8E - 75 \quad \text{(Equation 4)} \]

### Step 4: Substitute Equation 1 into Equation 4
Now we can substitute \( F \) from Equation 1 into Equation 4:
\[ 3E = 8E - 75 \]
Rearranging this gives:
\[ 75 = 8E - 3E \]
\[ 75 = 5E \]
Dividing both sides by 5:
\[ E = 15 \]

### Step 5: Find the age of the elder daughter and then the younger daughter
Now that we have \( E \):
\[ Y = E - 5 = 15 - 5 = 10 \]

### Step 6: Find the father's age using Equation 1
Now substitute \( E \) back into Equation 1 to find \( F \):
\[ F = 3E = 3 \times 15 = 45 \]

### Conclusion
The present age of the father is \( \boxed{45} \) years.