The age of Shaurya and Ravi is in the ratio 2 : 6. After 5 years, the ratio of their ages will become 6 : 8. Find the average of their ages after 10 years.

Options

A. (a) 12
B. (b) 13
C. (c) 17
D. (d) 24

Answer

To solve the problem, let's denote the current ages of Shaurya and Ravi as follows:

Let Shaurya's age = 2x
Let Ravi's age = 6x

### Step 1: Set up the equations based on the given ratios
According to the problem, the current age ratio of Shaurya and Ravi is 2:6, which can be expressed as:
- Shaurya's age = 2x
- Ravi's age = 6x

### Step 2: Determine their ages after 5 years
After 5 years, their ages will be:
- Shaurya's age after 5 years = 2x + 5
- Ravi's age after 5 years = 6x + 5

### Step 3: Set up the equation for the new ratio after 5 years
According to the problem, after 5 years, the ratio of their ages will be 6:8. This can be simplified to 3:4. Therefore, we can set up the equation:
\[ \frac{2x + 5}{6x + 5} = \frac{3}{4} \]

### Step 4: Cross-multiply to solve for x
Cross-multiplying gives us:
\[ 4(2x + 5) = 3(6x + 5) \]
Expanding both sides:
\[ 8x + 20 = 18x + 15 \]

### Step 5: Rearrange the equation to isolate x
Rearranging gives:
\[ 20 - 15 = 18x - 8x \]
\[ 5 = 10x \]
\[ x = \frac{1}{2} \]

### Step 6: Find the current ages of Shaurya and Ravi
Now substituting x back to find their ages:
- Shaurya's age = 2x = 2 * (1/2) = 1 year
- Ravi's age = 6x = 6 * (1/2) = 3 years

### Step 7: Calculate their ages after 10 years
After 10 years, their ages will be:
- Shaurya's age = 1 + 10 = 11 years
- Ravi's age = 3 + 10 = 13 years

### Step 8: Find the average of their ages after 10 years
The average age after 10 years is calculated as:
\[ \text{Average age} = \frac{\text{Shaurya's age} + \text{Ravi's age}}{2} = \frac{11 + 13}{2} = \frac{24}{2} = 12 \text{ years} \]

### Final Answer
The average of their ages after 10 years is **12 years**.