Determine the value of ` ( x + y)/( x + 2y)` when ` ( 2 x + y)/( x + 4y)` = 3

Options

A. `(3)/(5)`
B. `(7)/(10)`
C. `(4)/(5)`
D. `(10)/(9)`

Answer

To solve the problem, we need to determine the value of \( \frac{x + y}{x + 2y} \) given that \( \frac{2x + y}{x + 4y} = 3 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \frac{2x + y}{x + 4y} = 3 \]

2. **Cross-multiply to eliminate the fraction:**
\[ 2x + y = 3(x + 4y) \]

3. **Expand the right side of the equation:**
\[ 2x + y = 3x + 12y \]

4. **Rearrange the equation to isolate terms involving \( x \) and \( y \):**
\[ 2x - 3x = 12y - y \]
This simplifies to:
\[ -x = 11y \]

5. **Rearranging gives us the ratio of \( x \) to \( y \):**
\[ x = -11y \]

6. **Substitute \( x \) in terms of \( y \) into the expression \( \frac{x + y}{x + 2y} \):**
\[ \frac{-11y + y}{-11y + 2y} \]

7. **Simplify the numerator and the denominator:**
- The numerator becomes:
\[ -11y + y = -10y \]
- The denominator becomes:
\[ -11y + 2y = -9y \]

8. **Now, substitute back into the expression:**
\[ \frac{-10y}{-9y} \]

9. **Cancel out \( y \) (assuming \( y \neq 0 \)):**
\[ \frac{10}{9} \]

Thus, the value of \( \frac{x + y}{x + 2y} \) is \( \frac{10}{9} \).

### Final Answer:
\[ \frac{x + y}{x + 2y} = \frac{10}{9} \]