1. Write the two populations clearly

Population $A$: $101, 102, 103, \dots, 200$

Population $B$: $151, 152, 153, \dots, 250$

Each population has $100$ observations.

2. Observe the relationship between the two populations

Every term of population $B$ is obtained by adding $50$ to the corresponding term of population $A$: $151=101+50, \quad 152=102+50, \quad \dots, \quad 250=200+50$

So, $B = A + 50$

3. Use the property of variance

If a constant $k$ is added to every observation of a population, then the variance does not change: $\operatorname{Var}(X+k)=\operatorname{Var}(X)$

Hence, $V_B = V_A$

Therefore, $\frac{V_A}{V_B} = 1$

4. Check the options

The correct option is: $\boxed{1}$ which is Option A.

5. Comparison with stored correct answer

Stored correct answer: A

Our derived answer: A

So they agree.