The interference pattern is obtained with two coherent light sources of intensity ratio $4:1$. And the ratio $\frac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }}$ is $\frac{5}{x}$. Then, the value of $x$ will be equal to :

The ratio of the intensities is $\frac{I_1}{I_2}=\frac{4}{1}$

After taking square root of above equation on both sides, we get, $\frac{\sqrt{I_1}}{\sqrt{I_2}}=\frac{2}{1}$

Apply componendo and dividendo,
$\Rightarrow \frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}=\frac{2+1}{2-1}=\frac{3}{1}$
$$\begin{gathered} \Rightarrow {\left(\frac{\sqrt{I_1}+\sqrt{I_2}}{\sqrt{I_1}-\sqrt{I_2}}\right)}^2=\frac{I_{\max }}{I_{\min }}=\frac{9}{1} \\ \Rightarrow \frac{I_{max}+I_{min}}{I_{max}-I_{min}}=\frac{10}{8}=\frac{5}{4} \end{gathered}$$

Therefore, the answer is $4$.