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 :-

Answer

To find the least count of the vernier callipers, we will follow these steps:

### Step 1: Understand the Definitions
- The least count (LC) of a measuring instrument is the smallest value that can be measured accurately with that instrument.
- In a vernier calliper, the least count can be calculated using the formula:
\[ \text{Least Count (LC)} = \text{One Main Scale Division (MSD)} - \text{One Vernier Scale Division (VSD)} \]

### Step 2: Identify the Given Values
- Let the value of one main scale division (MSD) be \( x \) cm.
- According to the problem, \( n \) divisions of the vernier scale coincide with \( n-1 \) divisions of the main scale.

### Step 3: Calculate the Vernier Scale Division
- Since \( n \) divisions of the vernier scale coincide with \( n-1 \) divisions of the main scale, we can express the vernier scale division (VSD) as:
\[ \text{VSD} = \frac{(n-1) \text{ MSD}}{n} \]
- Substituting the value of MSD:
\[ \text{VSD} = \frac{(n-1) x}{n} \]

### Step 4: Substitute into the Least Count Formula
- Now, substituting the values of MSD and VSD into the least count formula:
\[ \text{LC} = \text{MSD} - \text{VSD} \]
\[ \text{LC} = x - \frac{(n-1) x}{n} \]

### Step 5: Simplify the Expression
- To simplify the expression:
\[ \text{LC} = x - \frac{(n-1)x}{n} = x \left(1 - \frac{(n-1)}{n}\right) \]
- This can be simplified further:
\[ \text{LC} = x \left(\frac{n}{n} - \frac{(n-1)}{n}\right) = x \left(\frac{n - (n-1)}{n}\right) = x \left(\frac{1}{n}\right) \]

### Step 6: Final Result
- Therefore, the least count of the vernier callipers is:
\[ \text{LC} = \frac{x}{n} \text{ cm} \]