You look at the year as a whole, not broken down by dates or segments.
Add up all the expenses for the kids for the whole year. (Things such as rent or mortgage interest, groceries, medical care, etc.) Call this amount "A"
Figure out the amount you paid or contributed to their expenses. (This amount can come from whatever source, whether it's wages, unemployment, business income, savings, etc.) Call this amount "B".
Divide B / A and if the answer is more than 0.5 then you can file as head of household.
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