The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n – 1) divisions of main scale. The least count of the vernier callipers is,

Given, ‘n’ division of main scale = 1 cm 

1 division of main scale = $=\frac{1}{n}\mathrm{cm}$ 

$1\mathrm{msd}=\frac{1}{\mathrm{n}}\mathrm{cm}$

Also given (n - 1) division of main scale = n vernier scale division
So, n vernier scale division = (n - 1) main scale division

$n vsd=(n-1) m s d.$

$\begin{array}{rl} & \text{l vsd }=\left(\frac{n-1}{n}\right)\mathrm{msd}.\\ & \text{i.e.,}1\mathrm{vsd}=\left(\frac{\mathrm{n}-1}{\mathrm{n}}\right)\frac{1}{\mathrm{n}}\mathrm{cm}.\\ & \text{Least count}=1\mathrm{msd},-1\mathrm{vsd}\\ & =\frac{1}{\mathrm{n}}-\frac{\mathrm{n}-1}{{\mathrm{n}}^2}=\frac{\mathrm{n}-\mathrm{n}+1}{{\mathrm{n}}^2}=\frac{1}{{\mathrm{n}}^2}\mathrm{cm}\end{array}$