I watched a video on Vernier Calipers which gave me an understanding of how it works and how to read a Vernier Caliper. I also tried the GeoGebra simulation. But I am not able to understand the formula (least count and zero error formula) required to do numerical problems. Please explain the formula with an example numerical. For instance, I am not able to do the following problem which was asked in IIT JEE 2016: IIT JEE 2016

From https://en.wikipedia.org/wiki/Vernier_scale#Least_count_or_vernier_constant:

The difference between the value of one main scale division and the value of one vernier scale division is known as the least count of the vernier, also known as the vernier constant.

$$\text{Least Count of Vernier Calipers} = {1\text{MSD (Main Scale Division)}\over1\text{VSD (Vernier Scale Division)}}$$ or $$\text{Least Count of Vernier Calipers} = {1\text{MSD (Main Scale Division)}\over\text{Number of divisions in Vernier Scale}}$$

I understood least count but I am not able to understand the length of (n − 1) main-scale divisions = the length of n vernier-scale division (IIT JEE 2003), or $$(n − 1)S = nV$$, or $$nS − S = nV$$.

I am also not able to understand the Zero error formula $\text{Zero error} (ZE) = ±n × \text{least count} (LC)$ and method to use a vernier scale or caliper with zero error is to use the formula $\text{actual reading} = \text{main scale} + \text{vernier scale} − (\text{zero error})$.

VernierscaleHow a vernier scale works.gif Vernier scale zero error +0.10.gif

(Images uploaded by User:Lookang, Wikipedia)

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    $\begingroup$ I am solving the numerical problems from scribd.com/document/416066050/… If you are not able to access the document (if you don't have Scribd subscription) I can send you the document as an attachment. $\endgroup$
    – user279106
    Commented Jul 4, 2021 at 13:28
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    $\begingroup$ Nobody in their right mind would ever make a real Vernier caliper with a scale like your C2 example. And the drawings in the question are so inaccurate they are ridiculous IMO. $\endgroup$
    – alephzero
    Commented Jul 4, 2021 at 17:00