jmc

The sum of the first $n$ odd integers is $1+3+\cdots +(2n-1)$. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so this sum is $[1+(2n-1)]/2\cdot n=n^2$.

Then the sum of the odd integers between 0 and 100 is $50^2$, and the sum of the odd integers between 0 and 200 is $100^2$. Therefore, the ratio of the sum of the odd integers between 0 and 100 to the sum of the odd integers between 100 and 200 is $\frac{50^2}{100^2-50^2}=\frac{1}{4-1}=\boxed{\frac{1}{3}}$.