In a vernier callipers `n` divisions of its main scale match with `(n+5)` divisions on its vernier scale. Each division of the main scale is of `x` unit. Using the vernier principle, find the least count.

Options

A. `x/(n+5)`
B. `(2x)/(n+5)`
C. `(5x)/(n+5)`
D. `(7x)/(n+5)`

Answer

To find the least count of the vernier calipers, we can follow these steps:

### Step 1: Understand the relationship between the main scale and the vernier scale
According to the problem, \( n \) divisions of the main scale match with \( n + 5 \) divisions of the vernier scale. We can express this relationship mathematically:
\[ n \text{ (Main Scale Divisions)} = (n + 5) \text{ (Vernier Scale Divisions)} \]
This can be labeled as Equation (1).

### Step 2: Define the value of one main scale division
Each division of the main scale is given as \( x \) units. We can express this as:
\[ \text{1 Main Scale Division} = x \text{ units} \]
This can be labeled as Equation (2).

### Step 3: Calculate the value of one vernier scale division
From the relationship established in Step 1, we can derive the value of one vernier scale division. Since \( n \) divisions of the main scale equal \( n + 5 \) divisions of the vernier scale, we can express one vernier scale division as:
\[ \text{1 Vernier Scale Division} = \frac{n \cdot x}{n + 5} \text{ units} \]

### Step 4: Use the least count formula
The least count (LC) of a vernier caliper is defined as:
\[ \text{Least Count} = \text{1 Main Scale Division} - \text{1 Vernier Scale Division} \]
Substituting the values from Equations (2) and the derived value of the vernier scale division:
\[ \text{Least Count} = x - \frac{n \cdot x}{n + 5} \]

### Step 5: Simplify the expression
To simplify the expression for the least count:
\[ \text{Least Count} = \frac{x(n + 5) - n \cdot x}{n + 5} \]
This simplifies to:
\[ \text{Least Count} = \frac{xn + 5x - nx}{n + 5} \]
The \( nx \) terms cancel out:
\[ \text{Least Count} = \frac{5x}{n + 5} \]

### Final Answer
Thus, the least count of the vernier calipers is:
\[ \text{Least Count} = \frac{5x}{n + 5} \]