If the ratio between 8 and 11 is the same as the ratio of `2x - y` to `x + 2y`, find the value of `(7x)/(9y)`.

Answer

To solve the problem, we need to find the value of \( \frac{7x}{9y} \) given that the ratio between 8 and 11 is the same as the ratio of \( 2x - y \) to \( x + 2y \).

### Step-by-Step Solution:

1. **Set up the equation based on the given ratios:**
\[ \frac{8}{11} = \frac{2x - y}{x + 2y} \]

2. **Cross-multiply to eliminate the fractions:**
\[ 8(x + 2y) = 11(2x - y) \]

3. **Distribute both sides:**
\[ 8x + 16y = 22x - 11y \]

4. **Rearrange the equation to group like terms:**
\[ 8x - 22x = -11y - 16y \]
This simplifies to:
\[ -14x = -27y \]

5. **Divide both sides by -1:**
\[ 14x = 27y \]

6. **Solve for \( x \):**
\[ x = \frac{27y}{14} \]

7. **Substitute \( x \) into \( \frac{7x}{9y} \):**
\[ \frac{7x}{9y} = \frac{7 \left(\frac{27y}{14}\right)}{9y} \]

8. **Simplify the expression:**
\[ = \frac{7 \cdot 27y}{14 \cdot 9y} \]
The \( y \) cancels out:
\[ = \frac{7 \cdot 27}{14 \cdot 9} \]

9. **Further simplify the fraction:**
\[ = \frac{189}{126} \]
Now, divide both numerator and denominator by 63:
\[ = \frac{3}{2} \]

### Final Answer:
\[ \frac{7x}{9y} = \frac{3}{2} \]