About 46 percent of Americans owed no federal income tax in 2011, although many of them paid other forms of taxes. More than 16 million elderly Americans avoid federal income taxes solely because of tax breaks that apply only to seniors, according to the nonpartisan Tax Policy Center.

The videos were the latest troubles for Romney’s campaign, which has tried to focus attention on a weak economic recovery and make the case that the Republican’s business background would help spur the economy. In recent weeks, the campaign has dealt with the fallout from Clint Eastwood’s rambling conversation with a chair at the Republican convention and Romney’s omission of the war in Afghanistan or thanks to the troops in his prime-time convention speech.

The eruption of violence in Egypt and Libya last week prompted Romney to issue a statement assailing the Obama administration before it was known that an American ambassador and three other U.S. citizens had died in Libya, a move that generated criticism from Democrats and Republicans alike.

A series of polls have shown Obama with an edge nationally and in key battleground states, leading Republicans to implore Romney to give voters more specifics on how he would govern. The new approach aims to improve Romney’s standing in the lead-up to the first presidential debate on Oct. 3.

Obama campaign manager Jim Messina quickly issued a fundraising appeal based on the initial video, telling supporters: ‘‘If we don’t come through for President Obama right now, this will be the guy making big decisions that affect us and our families every single day.’’

An Obama adviser said the Democratic campaign might use Romney’s comments from the fundraising video in television advertisements. The official wasn’t authorized to discuss campaign strategy publicly and requested anonymity.

Romney’s campaign released a separate television ad Tuesday, arguing that the president’s policies ‘‘are making it harder on women.’’ It cited unemployment and poverty statistics for women in an attempt to close the gender gap that has shown women favoring Obama.

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Associated Press writers Steve Peoples and Kasie Hunt in Washington and Amy Teibel in Jerusalem contributed to this report.

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