`19` divisions on the main scale of a vernier calipers coincide with `20` divisions on the vernier scale. If each division on the main scale is of `1 cm`, determine the least count of instrument.

Explanation

To determine the least count of the vernier calipers based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Main Scale Division (MSD)**:
- We are given that each division on the main scale is \(1 \, \text{cm}\).
- Therefore, \( \text{MSD} = 1 \, \text{cm} \).

2. **Understand the Coincidence of Scales**:
- The problem states that 19 divisions on the main scale coincide with 20 divisions on the vernier scale.
- This means that the length of 19 main scale divisions is equal to the length of 20 vernier scale divisions.

3. **Calculate the Length of 20 Vernier Scale Divisions**:
- The total length of 19 main scale divisions is:
\[ 19 \times \text{MSD} = 19 \times 1 \, \text{cm} = 19 \, \text{cm} \]
- Since this length is equal to 20 vernier scale divisions, we can express the length of one vernier scale division (VSD) as:
\[ \text{VSD} = \frac{19 \, \text{cm}}{20} = \frac{19}{20} \, \text{cm} \]

4. **Calculate the Least Count (LC)**:
- The least count of the vernier calipers is given by the formula:
\[ \text{LC} = \text{MSD} - \text{VSD} \]
- Substituting the values we have:
\[ \text{LC} = 1 \, \text{cm} - \frac{19}{20} \, \text{cm} \]
- To perform the subtraction, convert \(1 \, \text{cm}\) to a fraction:
\[ 1 \, \text{cm} = \frac{20}{20} \, \text{cm} \]
- Now, we can subtract:
\[ \text{LC} = \frac{20}{20} \, \text{cm} - \frac{19}{20} \, \text{cm} = \frac{1}{20} \, \text{cm} \]

5. **Convert to Decimal**:
- Converting \(\frac{1}{20} \, \text{cm}\) to decimal gives:
\[ \text{LC} = 0.05 \, \text{cm} \]

### Final Answer:
The least count of the instrument is \(0.05 \, \text{cm}\).

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