A young’s double-slit experiment is performed using monochromatic light of wavelength $\lambda$ . The intensity of light at a point on the screen, where the path difference is $\lambda$  , is K units. The intensity of light at a point where the path difference is $\frac{\lambda }{6}$ is given by $\frac{nK}{12}$  , where n is an integer. The value of n is ________

In Young’s double slip experiment, intensity at a point is given by
 $I=I_0{\cos }^2\frac{\varphi }{2}$
Where, $\varphi$ =phase difference,
Using phase difference , $\varphi =\frac{2\pi }{\lambda }\times$ path difference 
For path difference $\lambda$  ,  $phasedifference{\varphi }_1=2\pi$
For path difference $\frac{\lambda }{6}$  ,  $phasedifference{\varphi }_2=\frac{\pi }{3}$
Using equation (i)
 $$\begin{gathered} \frac{I_1}{I_2}=\frac{{\cos }^2\left(\frac{{\varphi }_1}{2}\right)}{{\cos }^2\left(\frac{{\varphi }_1}{2}\right)}=\frac{{\cos }^2\left(\frac{2\pi }{2}\right)}{{\cos }^2\left(\frac{\pi }{6}\right)} \\ =\frac{K}{I_2}=\frac{1}{3}=\frac{4}{3}\Rightarrow I_2=\frac{3K}{4}=\frac{9K}{12}\therefore n=9 \end{gathered}$$