In Vernier calipers, m divisions of main scale coincide with (m + 1) divisions of vernier scale. If each division of main scale is d units, the least count of the instrument is

Since m divisions of main scale is equivalent to (m + 1) divisions of vernier scale, one division of vernier scale is equivalent to $\frac{\mathrm{m}}{\mathrm{m}+1}$ divisions of main scale.
So, L.C. = 1 division of M.S. - 1 division of V.S.

L.C = $d-\frac{m}{m+1}d = \frac{1}{m+1}d$