In a vernier callipers, one main scale division is x cm and n divisions of the vernier scale coincide with (n-1) divisions of the main scale. The least count (in cm) of the callipers is :-

Options

A. `((n-1)/(n))x`
B. `(nx)/((n-1))`
C. `(x)/(n) `
D. `(x)/((n-1))`

Explanation

To find the least count of a vernier caliper given that one main scale division is \( x \) cm and \( n \) divisions of the vernier scale coincide with \( (n-1) \) divisions of the main scale, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The least count (LC) of a measuring instrument is the smallest value that can be measured accurately using that instrument.
- The formula for least count is given by:
\[ \text{Least Count} = \text{Value of 1 Main Scale Division (MSD)} - \text{Value of 1 Vernier Scale Division (VSD)} \]

2. **Identify the Values**:
- We know that 1 Main Scale Division (MSD) = \( x \) cm.
- We also know that \( n \) divisions of the vernier scale coincide with \( (n-1) \) divisions of the main scale.

3. **Calculate the Value of 1 Vernier Scale Division (VSD)**:
- Since \( n \) divisions of the vernier scale coincide with \( (n-1) \) divisions of the main scale, we can express this relationship mathematically:
\[ n \times \text{VSD} = (n-1) \times \text{MSD} \]
- Substituting the value of MSD:
\[ n \times \text{VSD} = (n-1) \times x \]
- Now, solving for VSD:
\[ \text{VSD} = \frac{(n-1) \times x}{n} \]

4. **Substitute VSD into the Least Count Formula**:
- Now we can substitute the value of VSD into the least count formula:
\[ \text{Least Count} = x - \text{VSD} \]
- Substituting for VSD:
\[ \text{Least Count} = x - \frac{(n-1) \times x}{n} \]

5. **Simplify the Expression**:
- To simplify, we can factor out \( x \):
\[ \text{Least Count} = x \left(1 - \frac{(n-1)}{n}\right) \]
- This simplifies to:
\[ \text{Least Count} = x \left(\frac{n - (n-1)}{n}\right) = x \left(\frac{1}{n}\right) \]
- Therefore, the least count is:
\[ \text{Least Count} = \frac{x}{n} \text{ cm} \]

### Final Answer:
The least count of the vernier calipers is \( \frac{x}{n} \) cm.