If `(a - b) : (a + b) = 1 : 11` find the ratio `(5a + 4b + 15) : (5a - 4b + 3)`.

Answer

To solve the problem, we need to find the ratio \((5a + 4b + 15) : (5a - 4b + 3)\) given that \((a - b) : (a + b) = 1 : 11\).

### Step-by-step Solution:

1. **Set up the ratio**:
Given \((a - b) : (a + b) = 1 : 11\), we can express this as:
\[ \frac{a - b}{a + b} = \frac{1}{11} \]

2. **Cross-multiply**:
Cross-multiplying gives us:
\[ 11(a - b) = 1(a + b) \]
This simplifies to:
\[ 11a - 11b = a + b \]

3. **Rearrange the equation**:
Rearranging the equation, we get:
\[ 11a - a = 11b + b \]
Which simplifies to:
\[ 10a = 12b \]

4. **Simplify the equation**:
Dividing both sides by 2, we find:
\[ 5a = 6b \]

5. **Substitute \(5a\) in the ratio**:
Now, we need to substitute \(5a\) into the expression we are trying to find:
\[ \frac{5a + 4b + 15}{5a - 4b + 3} \]
Replacing \(5a\) with \(6b\):
\[ \frac{6b + 4b + 15}{6b - 4b + 3} \]

6. **Combine like terms**:
This simplifies to:
\[ \frac{10b + 15}{2b + 3} \]

7. **Factor out common terms**:
We can factor out 5 from the numerator:
\[ \frac{5(2b + 3)}{2b + 3} \]

8. **Cancel out the common terms**:
The \(2b + 3\) in the numerator and denominator cancels out:
\[ = 5 \]

9. **Write the final ratio**:
Therefore, the ratio \((5a + 4b + 15) : (5a - 4b + 3)\) is:
\[ 5 : 1 \]

### Final Answer:
The required ratio is \(5 : 1\).