Dealing with nonresident tax liability
The first step is to figure out exactly how much income you have in each state in question. Sometimes, the partnership or LLC itself will break down all of your income by state, sending you multiple schedules that allocate appropriate income and deductions by the place where they were generated. In other situations, you might receive information on the percentage of total income and deductions allocable to each state. That will require you to do the math yourself, taking the federal figures on the K-1 and allocating them by hand to the various states in question.
Once you've done that, the next step is looking at each state's filing requirements rules. In some states, if you earn less than a certain minimum amount, then the state doesn't require you to file a tax return. Above that amount, though, you may have to file as a nonresident and owe money to that state for income tax on the income that's allocable to business activities there. If you don't, you risk having the state pursue you and charge interest and penalties for failing to meet the requirements.
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