The age of father 10 years ago was thrice the age of his son. Ten years hence, father's age will be twice that of his son. The ratio of their present ages is:

Options

A. `5:2`
B. `7:3`
C. `9:2`
D. `13:4`

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( F \) = present age of the father
- \( S \) = present age of the son

### Step 2: Set Up the First Equation
According to the problem, 10 years ago, the father's age was thrice the son's age. This can be expressed as:
\[ F - 10 = 3(S - 10) \]
Expanding this equation gives:
\[ F - 10 = 3S - 30 \]
Rearranging it, we have:
\[ F = 3S - 20 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that 10 years hence, the father's age will be twice the son's age. This can be expressed as:
\[ F + 10 = 2(S + 10) \]
Expanding this equation gives:
\[ F + 10 = 2S + 20 \]
Rearranging it, we have:
\[ F = 2S + 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( F = 3S - 20 \)
2. \( F = 2S + 10 \)

Since both equations equal \( F \), we can set them equal to each other:
\[ 3S - 20 = 2S + 10 \]
Subtract \( 2S \) from both sides:
\[ S - 20 = 10 \]
Adding 20 to both sides gives:
\[ S = 30 \]

### Step 5: Find the Father's Age
Now that we have the son's age, we can substitute \( S = 30 \) back into either equation to find \( F \). Using Equation 2:
\[ F = 2(30) + 10 = 60 + 10 = 70 \]

### Step 6: Present Ages and Ratio
Now we have:
- Father's present age \( F = 70 \)
- Son's present age \( S = 30 \)

The ratio of their present ages is:
\[ \frac{F}{S} = \frac{70}{30} = \frac{7}{3} \]

### Final Answer
The ratio of their present ages is \( 7:3 \).

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