In a vernier callipers, n divisions of its main scale match with (n+1) divisions on its vernier scale. Each division of the main scale is a units. Using the vernier principle, calculate its least count.

Explanation

To calculate the least count of a vernier caliper given that \( n \) divisions of its main scale match with \( n + 1 \) divisions on its vernier scale, and each division of the main scale is \( a \) units, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Main Scale and Vernier Scale:**
The problem states that \( n \) divisions of the main scale correspond to \( n + 1 \) divisions of the vernier scale. This can be expressed mathematically as:
\[ n \text{ (Main Scale Divisions)} = (n + 1) \text{ (Vernier Scale Divisions)} \]

2. **Define the Value of Each Division:**
Let the value of each division of the main scale be \( a \) units. Therefore, the total length represented by \( n \) divisions of the main scale is:
\[ \text{Length of } n \text{ Main Scale Divisions} = n \times a \]

3. **Calculate the Value of Each Vernier Scale Division:**
Since \( n \) divisions of the main scale equal \( n + 1 \) divisions of the vernier scale, we can express the value of each vernier scale division as:
\[ \text{Value of 1 Vernier Scale Division} = \frac{n \times a}{n + 1} \]

4. **Apply the Formula for Least Count:**
The least count (LC) of the vernier caliper is defined as:
\[ \text{Least Count} = 1 \text{ (Main Scale Division)} - 1 \text{ (Vernier Scale Division)} \]
Substituting the values we have:
\[ \text{Least Count} = a - \frac{n \times a}{n + 1} \]

5. **Simplify the Expression:**
To simplify the expression, we can factor out \( a \):
\[ \text{Least Count} = a \left(1 - \frac{n}{n + 1}\right) \]
Now, simplifying the term in the parentheses:
\[ 1 - \frac{n}{n + 1} = \frac{(n + 1) - n}{n + 1} = \frac{1}{n + 1} \]
Thus, we have:
\[ \text{Least Count} = a \cdot \frac{1}{n + 1} \]

6. **Final Result:**
Therefore, the least count of the vernier caliper is:
\[ \text{Least Count} = \frac{a}{n + 1} \text{ units} \]