There are four vernier scales, whose observations are given in column - I and the correcponding calculations of least count are given in column - II. Match column - I with column - II appropriately. (Here S = value of IMSD : N = Value of number of divisions on vernier) (Assume ideal conditions)
`{:(,"Column - I",,"Column - II"),((A),S=1mm : N=10,(P),0.05 mm),((B),S=0.5mm : N=10,(Q),0.01mm),((C ),S=0.5 mm : N=20,(R ),0.1 mm),((D),S=1mm : N=100,(S),0.025 mm):}`

Answer

To solve the problem of matching the vernier scales in Column I with their respective least counts in Column II, we will use the formula for calculating the least count of a vernier scale:

**Least Count (LC) = (Value of 1 Main Scale Division (MSD)) / (Number of Vernier Scale Divisions (N))**

Now, let's go through each option step by step.

### Step 1: Calculate the Least Count for each Vernier Scale

**(A) S = 1 mm, N = 10**
- LC = 1 mm / 10 = 0.1 mm

**(B) S = 0.5 mm, N = 10**
- LC = 0.5 mm / 10 = 0.05 mm

**(C) S = 0.5 mm, N = 20**
- LC = 0.5 mm / 20 = 0.025 mm

**(D) S = 1 mm, N = 100**
- LC = 1 mm / 100 = 0.01 mm

### Step 2: Match the Calculated Least Counts with Column II

Now that we have calculated the least counts, we can match them with the options provided in Column II:

- For (A), the least count is **0.1 mm**, which corresponds to (R).
- For (B), the least count is **0.05 mm**, which corresponds to (P).
- For (C), the least count is **0.025 mm**, which corresponds to (S).
- For (D), the least count is **0.01 mm**, which corresponds to (Q).

### Final Matching

- A → R (0.1 mm)
- B → P (0.05 mm)
- C → S (0.025 mm)
- D → Q (0.01 mm)

### Summary of the Matches:
- A matches with R
- B matches with P
- C matches with S
- D matches with Q