If `I_1/I_2` =16 then find the value of `(I_max - I_min)/ (I_max+I_min)`

Explanation

To solve the problem, we need to find the value of \((I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}})\) given that the ratio \(I_1/I_2 = 16\).

### Step-by-Step Solution:

1. **Identify Given Data**:
- We have the ratio \(I_1/I_2 = 16/1\). This means we can set \(I_1 = 16\) and \(I_2 = 1\).

2. **Calculate Maximum Intensity (\(I_{\text{max}}\))**:
- The formula for maximum intensity is given by:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
- Substitute \(I_1 = 16\) and \(I_2 = 1\):
\[ I_{\text{max}} = (\sqrt{16} + \sqrt{1})^2 = (4 + 1)^2 = 5^2 = 25 \]

3. **Calculate Minimum Intensity (\(I_{\text{min}}\))**:
- The formula for minimum intensity is given by:
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
- Substitute \(I_1 = 16\) and \(I_2 = 1\):
\[ I_{\text{min}} = (\sqrt{16} - \sqrt{1})^2 = (4 - 1)^2 = 3^2 = 9 \]

4. **Calculate \((I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}})\)**:
- Now we can substitute the values of \(I_{\text{max}}\) and \(I_{\text{min}}\) into the expression:
\[ \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} = \frac{25 - 9}{25 + 9} = \frac{16}{34} \]

5. **Simplify the Result**:
- The fraction \(\frac{16}{34}\) can be simplified:
\[ \frac{16}{34} = \frac{8}{17} \]

### Final Answer:
The value of \((I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}})\) is \(\frac{8}{17}\).