In Goblet of Fire Hermione is dragged through the mud in Witch Weekly by Rita Skeeter. After reading the article, Ron suggests Skeeter is making Hermione out to be a “scarlet woman”:
“I told you!” Ron hissed at Hermione as she stared down at the article. “I told you not to annoy Rita Skeeter! She’s made you out to be some sort of — of scarlet woman!”
Hermione stopped looking astonished and snorted with laughter. “Scarlet woman?” she repeated, shaking with suppressed giggles as she looked around at Ron.
“It’s what my mum calls them,” Ron muttered, his ears going red.
— Goblet of Fire, chapter 27 (Padfoot Returns)
That phrase clicked in my mind as a reference to the Scarlet Letter, written in 1850, as a term to refer to a tramp, or women who sleeps around.
Is my assumption that the term “scarlet women” is a reference toward that book correct, or is it from something else in history, and which of these things would Mrs. Weasley a witch who does not interact with muggles have pulled from as a basis for her use of this? Or am I just over thinking this and its a highly common phrase to use in British English, and common enough for a muggle phrase to flow into wizard's vocab?