The sum of the present ages of father and son is 90 years. 10 years earlier the ratio of their ages was `5:2`. The present age of the father is :

Options

A. 65 year
B. 68 years
C. 60 years
D. 15 years

Answer

To solve the problem, we will denote the present age of the father as \( F \) and the present age of the son as \( S \).

### Step 1: Set up the equations based on the information given.

1. The sum of their present ages is 90 years:
\[ F + S = 90 \quad \text{(Equation 1)} \]

2. Ten years ago, the ratio of their ages was \( 5:2 \). Therefore, we can express their ages 10 years ago as \( F - 10 \) for the father and \( S - 10 \) for the son. The ratio can be written as:
\[ \frac{F - 10}{S - 10} = \frac{5}{2} \quad \text{(Equation 2)} \]

### Step 2: Cross-multiply to eliminate the fraction in Equation 2.

Cross-multiplying gives us:
\[ 2(F - 10) = 5(S - 10) \]
Expanding both sides:
\[ 2F - 20 = 5S - 50 \]

### Step 3: Rearrange the equation.

Rearranging the equation gives us:
\[ 2F - 5S = -30 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations.

Now we have a system of equations:
1. \( F + S = 90 \) (Equation 1)
2. \( 2F - 5S = -30 \) (Equation 3)

From Equation 1, we can express \( S \) in terms of \( F \):
\[ S = 90 - F \]

### Step 5: Substitute \( S \) in Equation 3.

Substituting \( S \) into Equation 3:
\[ 2F - 5(90 - F) = -30 \]
Expanding this gives:
\[ 2F - 450 + 5F = -30 \]
Combining like terms:
\[ 7F - 450 = -30 \]
Adding 450 to both sides:
\[ 7F = 420 \]

### Step 6: Solve for \( F \).

Dividing both sides by 7:
\[ F = 60 \]

### Step 7: Find \( S \).

Now, substituting \( F \) back into Equation 1 to find \( S \):
\[ S = 90 - F = 90 - 60 = 30 \]

### Conclusion

The present age of the father is \( \boxed{60} \) years.