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Q.  Can I legally claim children for the time they were in my care ?

A. Probably not*.  They must have lived with you for more than half the year for you to be able to claim them, as a "Qualifying Child",  no matter how much support you provided.

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit. They are interrelated but the rules are different for each.

The support test is different for each type. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.

.A child closely related (a foster child counts as related) to a taxpayer can be a “Qualifying Child (QC)” dependent, regardless of the child's income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or  is totally & permanently disabled
2. He did not provide more than 1/2 his own support
3. He lived with the relative (including temporary absences) for more than half the year
4. He is younger than the relative (not applicable for a disabled child)
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child (this essentially means that you have the parent’s permission to claim the child, if the child also lived with the parent more than half the year)
6. If the parents of a child can claim the child as a qualifying child but no parent so claims the child, no one else can claim the child as a qualifying child unless that person's adjusted gross income (AGI) is higher than the highest AGI of any of the child's parents who can claim the child.

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Ret...

*If the children  also did not live with the mother (or any other close relative) for more than half the year** (that is, they are not the qualifying children of another taxpayer), then you may be able to claim them under the qualifying relative rules if your five months of support amounts to more than half their support for the year (probably unlikely). But, then you can't claim the child tax credit or earned income credit, based on those children.  A Qualifying Relative dependent (or a QC over age 16) gets the taxpayer the $500 Other Dependent Credit (ODC), instead of the Child tax credit.  The ODC is non refundable.  It can only be used to reduce your actual tax. 

**The mother MAY be able to claim that the time with you was only a temporary absence from her home.