WASHINGTON — Mitt Romney got the math about right. But when he said 47 percent of Americans pay no income taxes and are “dependent on government,” he blurred together half or more of the entire country, ranging from the nation’s neediest to its middle class, and even some of its richest families.

Forty-six percent of the country’s households — some 76 million — paid no federal income taxes last year, according to a study by the Tax Policy Center.

While it’s true most of those families are poor, the numbers include many others who got tax breaks because they are old, have children in college or didn’t owe taxes on interest from state and local bonds. And of those who didn’t write checks to the IRS, six in 10 still paid Social Security and Medicare payroll taxes, and more than that paid federal excise taxes on items such as gasoline, alcohol and cigarettes, said Roberton Williams, who analyzes taxes at the center.

On the spending side, 150 million people — just under half the country — received benefits last year from one or more federal programs, according to the Census Bureau.

A look at just the four biggest programs shows how varied their often-overlapping recipients can be. There were 50 million collecting Social Security and 46 million getting Medicare health coverage for the elderly — programs that pay everyone regardless of their income. Eighty million were living in households getting Medicaid medical coverage, and 48 million were in families receiving food stamps, which are limited to the poor.

Combined, those numbers belie a political point that Romney, the Republican presidential candidate, was making when he spoke at a secretly recorded Florida fundraiser in May.

Some 47 percent, he said, “will vote for the president no matter what.” But that’s not so for the elderly, who favored Romney over President Barack Obama 52 percent to 42 percent in an Associated Press-GfK poll last month, and it wouldn’t be true for millions of middle-class or wealthy voters either.

“A significant fraction of government entitlement benefits are neither poor nor necessarily supporters of the Democratic Party, and many pay significant income taxes,” said Robert Reischauer, a fellow at the Urban Institute and former head of the nonpartisan Congressional Budget Office.

According to the Tax Policy Center study, 38 million taxpayers — half of those who owe no federal income taxes — escape owing money to the IRS because their income is too low. For them, merely using the standard deduction, personal exemptions and other basic parts of the tax code allow them to avoid income taxes.

On top of that, about one in five others who don’t owe federal income taxes — nearly 17 million people — benefit from tax breaks for the elderly. These include the exclusion of taxes on part of their Social Security benefits and a higher standard deduction.

Almost 12 million people avoid income taxes because they use the earned income tax credit for low-income workers, the child tax credit and other breaks for low-earning families with children.

Two million take advantage of a tax credit for college expenses, enacted under Obama, and other education tax breaks. Millions of others owe nothing because taxes are not paid on welfare and other income support programs; because of deductions for business costs, alimony, moving and other expenses; and because capital gains tax rates can drop to zero for some assets held for longer than a year.

Overall, 93 percent of those who owe no federal income taxes earn $50,000 annually or less. But 5 percent of them earn $50,000 to $100,000 and the rest — around 430,000 households — earn more than $100,000 annually. That includes 4,000 households earning more than $1 million a year, thanks largely to tax exempt interest, reduced rates on capital gains and dividends and other deductions.

As for the vast numbers of Americans who are recipients of benefit programs, many do so because they have modest incomes. These include 23 million in households where low-income women and children get food support, 20 million in families getting supplemental security income for the low-income disabled, 14 million in public housing or getting federal rental aid and 6 million on welfare.

There are also 3 million getting veterans compensation and 700,000 receiving workers’ compensation.

Census figures also show that there are racial differences in households receiving federal assistance based on income.

Among whites, 22 percent lived in households last year receiving Medicaid, food stamps or other programs for the poor. That compared to 52 percent of blacks and 56 percent of Hispanics.

After the video was revealed, Romney said his remarks were “off the cuff” and were not “elegantly stated,” but he offered no apologies.

White House spokesman Jay Carney said that Obama “deeply believes that we’re in this together.”

Romney’s comment rippled into congressional races from Florida to Connecticut and into Senate contests, too. Throughout, Democrats tried to tie Romney to Republican candidates they said would make policies that favor the rich over the less fortunate.

“When you are running to be president, you normally don’t insult half of America,” said a fundraising letter for former West Palm Beach, Fla., Mayor Lois Frankel, a Democrat trying to unseat Republican Rep. Allen West.