**Solution**

Given,

Wavelength (λ) = 589.3 nm = 589.3×10^{-9} m,

Angle (2θ) = 27°42' = \(\left(27+\frac{42}{60}\right)^\circ=27.27^\circ\)

⇒θ=27.27/2 = 13.58°

No. of lines per mm (N) =?

We have,

\(d\sin\left(\theta\right)=\lambda\) (for the first order)

Or, \(\frac1N\sin\left(\theta\right)=\lambda\)

Or, \(N=\frac{\sin\left(\theta\right)}\lambda\)

Or, \(N=\frac{\sin\left(13.85\right)}{589.3\times10^{-9}}\)

∴ N = 406212.18 lines/m = 406 lines per mm

Hence the number of rulings per mm of the grating is 406.