2021 Fall AMC 10B Problem 7

Call a fraction \frac{a}{b}, not necessarily in the simplest form, special if a and b are positive integers whose sum is 15. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.