The sum of ages of two brothers will be doubled after 10 yr, if the difference between their ages is 8 yr. Then, what is the ratio of age of younger brother to age of order brother ?

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the age of the younger brother be \( y \) and the age of the older brother be \( x \).

### Step 2: Set Up the Equations
From the problem, we know two things:
1. The difference between their ages is 8 years:
\[ x - y = 8 \quad \text{(1)} \]
2. The sum of their ages will be doubled after 10 years:
\[ (x + 10) + (y + 10) = 2(x + y) \quad \text{(2)} \]

### Step 3: Simplify Equation (2)
Expanding equation (2):
\[ x + 10 + y + 10 = 2(x + y) \]
This simplifies to:
\[ x + y + 20 = 2x + 2y \]
Rearranging gives us:
\[ 20 = 2x + 2y - x - y \]
\[ 20 = x + y \quad \text{(3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x - y = 8 \) (from equation (1))
2. \( x + y = 20 \) (from equation (3))

We can solve these equations simultaneously.

Adding equations (1) and (3):
\[ (x - y) + (x + y) = 8 + 20 \]
This simplifies to:
\[ 2x = 28 \]
Thus, we find:
\[ x = 14 \]

Now substituting \( x = 14 \) back into equation (3):
\[ 14 + y = 20 \]
This gives:
\[ y = 20 - 14 = 6 \]

### Step 5: Calculate the Ratio
Now we have the ages:
- Older brother's age \( x = 14 \)
- Younger brother's age \( y = 6 \)

The ratio of the age of the younger brother to the age of the older brother is:
\[ \text{Ratio} = \frac{y}{x} = \frac{6}{14} = \frac{3}{7} \]

### Final Answer
Thus, the ratio of the age of the younger brother to the age of the older brother is \( \frac{3}{7} \).

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