Why is it "sind" and not "haben" in this case? Can it go both ways, and Duolingo is just teaching us both methods?
In an earlier lesson, "fit" was translated to "healthy" in the suggested answer ...
"Fit" is, in general, "healthy", but more specifically "physically suitable" or "good shape". It's not precisely identical, but should be an acceptable in this usage, although you could be healthy without quite being fit.
Yes, I believe the distinction is kind of the same as in English - I just think Duolingo could be more consistent in its translations
"Ihre" does not mean "your".] See this table of possessive pronouns. [EDIT: As mizinamo points out below, the table clearly shows ihre is the proper form for "their" when concerning Nominativ und Akkusativ feminine or plural nouns. One must remember--as I did not--that the forms in the column for sie are also used for the polite/formal second person, Sie (so long as they are capitalized, as Ihre is here).] Additionally, geworden is a past tense form of werden. See this conjugation table.
Um, what? Ihre Kinder (with capitalised Ihre) does mean "your".
The table you linked to is missing a column for Sie (the polite "you"), but it is inflected exactly like sie which means "they" except for capitalisation.
Thus just as ihre Kinder means "their children", so Ihre Kinder means "your children".
Um, right. That explains the nagging feeling I had that I was missing something.
So this sentence could be correctly translated as 'Your children have become fit?'
Can "Ihre Kinder" translate to "her kids", "your (formal) kids" and "their kids"? How would you tell which one is intended?
Can "Ihre Kinder" translate to "her kids", "your (formal) kids" and "their kids"?
At the beginning of a sentence, yes.
How would you tell which one is intended?
Pronouns usually refer back to something that had been mentioned earlier.
Without context, all three are possible (and should be accepted as translations).
As noted, sind . . . geworden forms the past tense: "have become", not "are becoming".
Auf Deutsch, "Her children are becoming fit" would be "Ihre Kinder werden fit." One does not use sein as a modal verb to form the present continuous.
It should be emphasized that "Ihre Kinder" is also "Your Children" using the formal mode to address a person and by using the capital I in Ihre at the beginning of the sentence you don't know if you are referring to "their" or "your" in formal mode.
Too late, but I will get in the habit of doing this for future questions...thanks!