If (4m + 3n)/(4m - 3n) = (7)/(4) use properties of proportion to find :
(i) `m : n`
(ii) `(2m^(2) - 11n^(2))/(2m^(2) + 11n^(2))`.

Answer

To solve the equation \((4m + 3n)/(4m - 3n) = 7/4\) using properties of proportion, we can follow these steps:

### Step-by-Step Solution:

1. **Set up the equation:**
\[ \frac{4m + 3n}{4m - 3n} = \frac{7}{4} \]

2. **Apply the Component and Dividendo Rule:**
According to this rule, we can rewrite the left-hand side as:
\[ \frac{(4m + 3n) + (4m - 3n)}{(4m + 3n) - (4m - 3n)} = \frac{7 + 4}{7 - 4} \]

3. **Simplify the numerator and denominator:**
- The numerator becomes:
\[ (4m + 3n) + (4m - 3n) = 4m + 3n + 4m - 3n = 8m \]
- The denominator becomes:
\[ (4m + 3n) - (4m - 3n) = 4m + 3n - 4m + 3n = 6n \]
Thus, we have:
\[ \frac{8m}{6n} = \frac{11}{3} \]

4. **Set the two fractions equal:**
\[ \frac{8m}{6n} = \frac{11}{3} \]

5. **Cross-multiply to solve for \(m/n\):**
\[ 8m \cdot 3 = 11 \cdot 6n \]
\[ 24m = 66n \]
\[ \frac{m}{n} = \frac{66}{24} = \frac{11}{4} \]

6. **Express \(m:n\):**
Therefore, we have:
\[ m:n = 11:4 \]

### For the second part of the question:

7. **Substitute \(m/n\) into the expression:**
We need to find:
\[ \frac{2m^2 - 11n^2}{2m^2 + 11n^2} \]
Divide the numerator and denominator by \(n^2\):
\[ \frac{2\left(\frac{m}{n}\right)^2 - 11}{2\left(\frac{m}{n}\right)^2 + 11} \]

8. **Substituting \(\frac{m}{n} = \frac{11}{4}\):**
\[ \frac{2\left(\frac{11}{4}\right)^2 - 11}{2\left(\frac{11}{4}\right)^2 + 11} \]
Calculate \(\left(\frac{11}{4}\right)^2 = \frac{121}{16}\):
\[ \frac{2 \cdot \frac{121}{16} - 11}{2 \cdot \frac{121}{16} + 11} \]

9. **Simplify the expression:**
- The numerator:
\[ 2 \cdot \frac{121}{16} - 11 = \frac{242}{16} - \frac{176}{16} = \frac{66}{16} \]
- The denominator:
\[ 2 \cdot \frac{121}{16} + 11 = \frac{242}{16} + \frac{176}{16} = \frac{418}{16} \]

10. **Final simplification:**
\[ \frac{\frac{66}{16}}{\frac{418}{16}} = \frac{66}{418} = \frac{33}{209} \]

### Final Answers:
(i) \(m:n = 11:4\)
(ii) \(\frac{2m^2 - 11n^2}{2m^2 + 11n^2} = \frac{33}{209}\)