Its all about politeness.
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Anonymous "Then why'd you ask?"
"To see if you'd admit it" Although the second line is obviously "would," I think the first one is "did," as you suspect.
In my experience, "Then why would you ask?" would not be contracted.
I seem to remember coming across other examples of " 'd," but I can't think of one at the moment. Of course, you often find i
CalifJim'd
= would when combined with the base form of a verb
= had when combined with the past participle of a verb
= did when attached to a question word Hi, Jim.
Does this mean you reject "To see if you'd admit it" as a contraction for "To see if you would admit it?"
Best regards, - A.