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In a house with shared ownership, if one person moves out and the other assumes mortgage, how do we determine who owns what share in the end?
Both names are on the deed, so the property is jointly owned. You're going to need the second person's signature to be able to sell the property. Ideally the way to know "what happens now" is to consult the written agreement you made before you purchased the house together. The formula for dividing up assets when dissolving your partnership is whatever you agreed to up front. (Your up-front agreement could have said "if you move out, you forfeit any claim to the property".) It sounds like you don't have that, so you'll have to come to some (written) agreement with your partner before you proceed. If you can't come to an agreement, then you'll end up in court, a judge will split up the assets, and the only winners there are the lawyers...