Kiran is 15 years older than Rajat. After six years, their ages will be in the ratio 3:2, find their ages.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the age of Kiran be \( X \) and the age of Rajat be \( Y \).

### Step 2: Set Up the First Equation
According to the problem, Kiran is 15 years older than Rajat. This gives us our first equation:
\[ X = Y + 15 \]

### Step 3: Set Up the Second Equation
After 6 years, Kiran's age will be \( X + 6 \) and Rajat's age will be \( Y + 6 \). The problem states that their ages will be in the ratio of 3:2. Therefore, we can write the second equation as:
\[ \frac{X + 6}{Y + 6} = \frac{3}{2} \]

### Step 4: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 2(X + 6) = 3(Y + 6) \]
Expanding both sides:
\[ 2X + 12 = 3Y + 18 \]

### Step 5: Substitute the First Equation into the Second Equation
Now we will substitute \( X \) from the first equation into the second equation:
\[ 2(Y + 15) + 12 = 3Y + 18 \]
Expanding this gives:
\[ 2Y + 30 + 12 = 3Y + 18 \]
This simplifies to:
\[ 2Y + 42 = 3Y + 18 \]

### Step 6: Rearranging the Equation
Now, we will rearrange the equation to solve for \( Y \):
\[ 42 - 18 = 3Y - 2Y \]
This simplifies to:
\[ 24 = Y \]

### Step 7: Find Kiran's Age
Now that we have Rajat's age, we can find Kiran's age using the first equation:
\[ X = Y + 15 = 24 + 15 = 39 \]

### Conclusion
Thus, the ages are:
- Kiran's age: \( 39 \) years
- Rajat's age: \( 24 \) years

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