I listened to one call from a woman who sounded as if her world were collapsing. She and her husband operated a business, which seemed to be teetering near collapse, and its finances were intertwined with theirs. They were behind in payments on their mortgage.

Under the administration’s mortgage modification program, the new payment, including escrow payments for taxes and insurance, is to be 31 percent of the borrower’s gross monthly income. The woman first said their income was $6,000 a month, the amount they had taken out of the business when times were good.

That number, it turned out, was too high to qualify for a modification. When told that, the woman said she thought that for at least the next couple of months, they might be able to take only $2,000. That number was too low. She got no modification that day. Had she come up with a number somewhere in between, she might have qualified.

The arithmetic of “Obama mods,” as some call them, is laid out by the government. The 31 percent number is fixed in stone, which provides some simplicity but also can be arbitrary. A family with a lot of other obligations might not be able to afford 31 percent, while one with few other debts could afford much more.

To get the payment down to the 31 percent figure, the bank first cuts the interest rate, to as low as 2 percent, while leaving the other terms of the mortgage unchanged. For the vast majority of mortgages being modified, that is enough. If not, the term of the mortgage is stretched out to as long as 40 years.

Finally, if that is not enough, part of the principal can be deferred. That deferred amount is still owed, but no interest accrues and the lump sum is due at the end of the 40 years, or when the house is sold.

Calculating those numbers is only the first step. After determining the present value of that projected series of payments, the bank then compares it with what it could get by foreclosing. If the bank would be better off by foreclosing, then there is no modification.