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Q. Can I claim exclusion on capital gains tax for the larger profit generating home?

A.  Yes, probably. 

It appears that you already know, you can only claim one sale.  You may choose either one, depending on the details. See Mike9241's reply. The maximum excludable amount is reduced for the time you actually lived in it, since you don't meet the 2 year rule.  For example, you lived in it 4 months; your maximum exclusion is 4/24 x $250,000* = $41,667.  You may exclude the full amount of your gain, but not more than $41,667.

*$500,000 Married filing jointly

principal residence reg 1.121-1(b) 

(2) Principal residence. In the case of a taxpayer using more than one property as a residence, whether property is used by the taxpayer as the taxpayer's principal residence depends upon all the facts and circumstances. If a taxpayer alternates between 2 properties, using each as a residence for successive periods of time, the property that the taxpayer uses a majority of the time during the year ordinarily will be considered the taxpayer's principal residence. In addition to the taxpayer's use of the property, relevant factors in determining a taxpayer's principal residence, include, but are not limited to -

(i) The taxpayer's place of employment;

(ii) The principal place of abode of the taxpayer's family members;

(iii) The address listed on the taxpayer's federal and state tax returns, driver's license, automobile registration, and voter registration card;

(iv) The taxpayer's mailing address for bills and correspondence;

(v) The location of the taxpayer's banks; and

(vi) The location of religious organizations and recreational clubs with which the taxpayer is affiliated.

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@jtdeyoung1 wrote:
 Both sales were required to move for work outside of 50 miles from each respective location.

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If both sales qualify, you probably can exclude the gain from BOTH sales.