The sum of the present ages of a father and a daughter is 56 year. 10 year ago, the ratio of their ages was 5 : 1. Find the current age of the father.

Options

A. 55 year
B. 40 year
C. 45 year
D. 35 year

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

### Step 1: Define the variables
Let the present age of the father be \( F \) years and the present age of the daughter be \( D \) years.

### Step 2: Set up the equations
From the problem, we know:
1. The sum of their present ages is 56 years:
\[ F + D = 56 \quad \text{(Equation 1)} \]

2. Ten years ago, the ratio of their ages was 5:1. Therefore, ten years ago, the father's age was \( F - 10 \) and the daughter's age was \( D - 10 \). This gives us the second equation:
\[ \frac{F - 10}{D - 10} = 5 \quad \text{(Equation 2)} \]

### Step 3: Cross-multiply to eliminate the fraction
From Equation 2:
\[ F - 10 = 5(D - 10) \]
Expanding this gives:
\[ F - 10 = 5D - 50 \]
Rearranging it, we get:
\[ F = 5D - 40 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 1
Now, we will substitute Equation 3 into Equation 1:
\[ (5D - 40) + D = 56 \]
Combining like terms:
\[ 6D - 40 = 56 \]
Adding 40 to both sides:
\[ 6D = 96 \]
Dividing both sides by 6:
\[ D = 16 \]

### Step 5: Find the father's age using Equation 1
Now that we have the daughter's age, we can find the father's age using Equation 1:
\[ F + 16 = 56 \]
Subtracting 16 from both sides gives:
\[ F = 40 \]

### Conclusion
The current age of the father is \( \boxed{40} \) years.

---