The sum of the ages of two brothers, having a difference of 8 years between them, will double after 10 years. What is the ratio of the age of the younger brother to that of the elder brother?
आयु में 8 वर्ष का अंतर होने पर दो भाइयों की आयु का योगफल 10 वर्ष बाद दुगुना हो जाएगा। छोटे भाई और बड़े भाई की आयु का अनुपात क्या है?

Options

A. A)`10:13`
B. B)`7:11`
C. C)3:7`
D. D)`8:9`

Explanation

To solve the problem step by step, let's denote the age of the younger brother as \( K \) years. Since the elder brother is 8 years older, his age will be \( K + 8 \) years.

### Step 1: Write the current ages
- Younger brother's age = \( K \)
- Elder brother's age = \( K + 8 \)

### Step 2: Write the sum of their current ages
The sum of their current ages is:
\[ K + (K + 8) = 2K + 8 \]

### Step 3: Write their ages after 10 years
After 10 years, the ages will be:
- Younger brother's age = \( K + 10 \)
- Elder brother's age = \( (K + 8) + 10 = K + 18 \)

### Step 4: Write the sum of their ages after 10 years
The sum of their ages after 10 years will be:
\[ (K + 10) + (K + 18) = 2K + 28 \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the sum of their ages after 10 years is double the sum of their current ages:
\[ 2(2K + 8) = 2K + 28 \]

### Step 6: Simplify the equation
Expanding the left side:
\[ 4K + 16 = 2K + 28 \]

### Step 7: Rearranging the equation
Now, move \( 2K \) to the left side:
\[ 4K - 2K + 16 = 28 \]
This simplifies to:
\[ 2K + 16 = 28 \]

### Step 8: Solve for \( K \)
Subtract 16 from both sides:
\[ 2K = 12 \]
Now divide by 2:
\[ K = 6 \]

### Step 9: Find the ages of both brothers
- Younger brother's age = \( K = 6 \) years
- Elder brother's age = \( K + 8 = 6 + 8 = 14 \) years

### Step 10: Find the ratio of their ages
The ratio of the age of the younger brother to that of the elder brother is:
\[ \text{Ratio} = \frac{K}{K + 8} = \frac{6}{14} = \frac{3}{7} \]

Thus, the ratio of the age of the younger brother to that of the elder brother is \( 3:7 \).

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