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The best thing your brother can do is read IRS Publication 523, which he can find online here:

I don't know the terminology, but they made about 27% on selling the house. And I don't know what that will come to after closing costs, but still, they stand to increase their wealth.

When you sell property (like a house) for a profit, you have what's called a capital gain. Generally, you pay taxes on such gains - a lot of taxes if you owned the property a short while, rather less in taxes if you owned it a long time.

None of us really know how it works, but we think that if you make money on your house, and you don't re-invest it in equity, you get taxed on that amount as income. Is this right?

But you'll see in Pub 523 that sometimes capital gains on a home sale are forgiven. See the section "Excluding the Gain" on p 12 of Pub 523. In particular, if your brother owned and lived in the house for at least two of the last five years, then he can exclude up to $250,000 of gain; if he and his wife file jointly, they can exclude up to $500,000 of gain...

But, say they decided to rent, instead of buy, and used the money they make on the sale to pay off credit card debt, what would that do to them?

Many years ago, if you sold a house, you could avoid paying capital gains tax if you purchased a new house of equal or greater value within some period of time. That's not true any more - now you have that much more generous 2 of 5 years provision. It doesn't matter what he does with the sale proceeds.

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