EVE/DID=.TALKTALK....

${\color{#D61F06}\text{This problem's question:}}$ Each letter stands for a decimal digit. Each identical letter represents the same decimal digit each time the letter occurs. The period is a decimal point. To the left of the equal sign is a fraction. Write the answer as EVEDID. What is the answer?

Clarification: I lost the four-dot ellipsis in the title meant to indicate that decimal representation was a repeating decimal fraction. It is a coincidence that the ellipsis is four dots long and the repeat interval of the repeating decimal fraction is also four digits long.

$\frac{\text{EVE}}{\text{DID}}=.\overline{\text{TALK}}\text{TALK}\cdot\!\cdot\cdot$ or $\frac{\text{EVE}}{\text{DID}}=.\dot{\text{T}}\text{AL}\dot{\text{K}}\text{TALK}\cdot\!\cdot\cdot$

Second clarification (due to Chris Lewis): $\frac{\text{EVE}}{\text{DID}}$ is a reduced fraction, i. e., the numerator and denominator are co-prime.