In a slide callipers, (m +1) number of vernier divisions is equal to m number of smallest main scale divisions. If d unit is the magnitude of the smallest main scale division, then the magnitude of the vernier constant is .

Options

A. `(d)/((m+1)` unit
B. `(d)/(m)` unit
C. `(md)/((m+1))` unit
D. `((m+1)d)/(m)` unit

Explanation

To find the magnitude of the vernier constant in a slide caliper, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that (m + 1) number of vernier divisions is equal to m number of smallest main scale divisions. We need to find the vernier constant.

2. **Define the Terms**:
- Let \( d \) be the magnitude of the smallest main scale division (MSD).
- Let \( VSD \) be the magnitude of one vernier scale division.

3. **Set Up the Equation**:
According to the problem, we can write the relationship between the vernier divisions and the main scale divisions as:
\[ (m + 1) \times VSD = m \times MSD \]
Since \( MSD = d \), we can rewrite this as:
\[ (m + 1) \times VSD = m \times d \]

4. **Solve for VSD**:
Rearranging the equation to find \( VSD \):
\[ VSD = \frac{m \times d}{m + 1} \]

5. **Define the Vernier Constant**:
The vernier constant (VC) is defined as:
\[ VC = MSD - VSD \]

6. **Substituting for MSD and VSD**:
Substitute \( MSD = d \) and \( VSD = \frac{m \times d}{m + 1} \) into the equation for VC:
\[ VC = d - \frac{m \times d}{m + 1} \]

7. **Simplifying the Expression**:
Factor out \( d \):
\[ VC = d \left(1 - \frac{m}{m + 1}\right) \]
Now, simplify the expression inside the parentheses:
\[ 1 - \frac{m}{m + 1} = \frac{(m + 1) - m}{m + 1} = \frac{1}{m + 1} \]

8. **Final Expression for VC**:
Thus, we have:
\[ VC = d \times \frac{1}{m + 1} = \frac{d}{m + 1} \]

### Conclusion:
The magnitude of the vernier constant is:
\[ \text{Vernier Constant} = \frac{d}{m + 1} \]