Do you both live in Illinois? If so, your wife's income would be taxable in Illinois as well as in Missouri. If that's the case you have to include her MO income on your IL return, but you'll be able to take an "other state credit" on the IL return for the taxes paid to MO.
Your home state can tax all your income, regardless of where it was earned.
Were you both residents of IL all year? If so, you file a resident tax return for IL and a non-resident tax return for MO. How do I file a nonresident state return?
If you moved during the year and lived in both states, you will file2 part-year resident returns. You might think that part-year taxes are proportional to the amount of time spent in the state, but that's often not the case. Most states divide the income earned in that state by the total annual income to come up with a percentage. That percentage is then applied to the state tax on the entire year's income—no matter where it was earned—to prorate the tax liability. TurboTax follows the rules for each state.
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