The sum of the present ages of Hari and Mohan is double the difference of their present ages. Four years ago this ratio was one and half times. Find the ratio of their ages after 12 years.

Answer

To solve the problem step by step, let's denote the present ages of Hari and Mohan as \( x \) and \( y \) respectively.

### Step 1: Set Up the Equations
According to the problem, we have two conditions:
1. The sum of their ages is double the difference of their ages.
2. Four years ago, the ratio of their ages was one and a half times.

From the first condition, we can write the equation:
\[ x + y = 2(x - y) \]

### Step 2: Simplify the First Equation
Now, let's simplify the first equation:
\[ x + y = 2x - 2y \]
Rearranging gives:
\[ x + y + 2y = 2x \]
\[ x - 3y = 0 \quad \Rightarrow \quad x = 3y \]

### Step 3: Set Up the Second Equation
Now, let's consider the second condition. Four years ago, their ages would be \( x - 4 \) and \( y - 4 \). According to the problem, the ratio of their ages four years ago was \( \frac{3}{2} \):
\[ \frac{(x - 4) + (y - 4)}{(x - 4) - (y - 4)} = \frac{3}{2} \]

### Step 4: Simplify the Second Equation
Let's simplify this equation:
\[ \frac{x + y - 8}{x - y} = \frac{3}{2} \]
Cross-multiplying gives:
\[ 2(x + y - 8) = 3(x - y) \]
Expanding both sides:
\[ 2x + 2y - 16 = 3x - 3y \]
Rearranging gives:
\[ 2x + 2y + 3y - 3x = 16 \]
\[ -x + 5y = 16 \quad \Rightarrow \quad x = 5y - 16 \]

### Step 5: Substitute the Value of x
Now we have two expressions for \( x \):
1. \( x = 3y \)
2. \( x = 5y - 16 \)

Setting them equal to each other:
\[ 3y = 5y - 16 \]
Rearranging gives:
\[ 16 = 5y - 3y \]
\[ 16 = 2y \quad \Rightarrow \quad y = 8 \]

### Step 6: Find the Value of x
Now substituting \( y \) back into the equation for \( x \):
\[ x = 3y = 3 \times 8 = 24 \]

### Step 7: Calculate Ages After 12 Years
Now we need to find their ages after 12 years:
- Age of Hari after 12 years: \( x + 12 = 24 + 12 = 36 \)
- Age of Mohan after 12 years: \( y + 12 = 8 + 12 = 20 \)

### Step 8: Find the Ratio of Their Ages After 12 Years
Now we find the ratio of their ages after 12 years:
\[ \text{Ratio} = \frac{36}{20} = \frac{9}{5} \]

### Final Answer
Thus, the ratio of their ages after 12 years is \( \frac{9}{5} \).

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