The ages of Samir and Tanuj are in the ratio of ` 8 : 15` years respectively. After 9 years the ratio of their ages will be `11 : 18`.
What is the difference in years between their ages ?

Options

A. 24 years
B. 20 years
C. 33 years
D. 21 years

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the ages of Samir and Tanuj be represented as:
- Samir's age = \( 8x \)
- Tanuj's age = \( 15x \)

### Step 2: Set Up the Equation for Future Ages
After 9 years, their ages will be:
- Samir's age after 9 years = \( 8x + 9 \)
- Tanuj's age after 9 years = \( 15x + 9 \)

According to the problem, after 9 years, the ratio of their ages will be \( 11:18 \). Therefore, we can set up the equation:
\[ \frac{8x + 9}{15x + 9} = \frac{11}{18} \]

### Step 3: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 18(8x + 9) = 11(15x + 9) \]

### Step 4: Expand Both Sides
Expanding both sides:
\[ 144x + 162 = 165x + 99 \]

### Step 5: Rearrange the Equation
Rearranging the equation to isolate \( x \):
\[ 144x + 162 - 99 = 165x \]
\[ 63 = 165x - 144x \]
\[ 63 = 21x \]

### Step 6: Solve for \( x \)
Dividing both sides by 21:
\[ x = \frac{63}{21} = 3 \]

### Step 7: Calculate Their Ages
Now that we have \( x \), we can find their actual ages:
- Samir's age = \( 8x = 8 \times 3 = 24 \) years
- Tanuj's age = \( 15x = 15 \times 3 = 45 \) years

### Step 8: Find the Difference in Their Ages
The difference in their ages is:
\[ 45 - 24 = 21 \text{ years} \]

Thus, the difference in years between their ages is **21 years**.

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