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“Not true, Governor Romney,” the president interrupted.

“So how much did you cut them by?” Mr. Romney asked.

“It’s not true,” Mr. Obama replied.

Fact-checkers have rated the 14 percent drop in 2011 as an accurate claim — but the details of the evening’s exchanges were mostly lost amid the theater of the two men vying for the most powerful job in the world, talking straight to voters.

One woman pressed Mr. Romney to explain his tax plan, saying she was worried he would take away some of the tax breaks such as mortgage and education loan interest deductions she relies on.

Mr. Romney said he wants to cut tax rates across the board, but would also limit some of those special tax breaks — though he again declined to give specifics, saying it should all be open for debate in the next Congress. He did vow, however, that the wealthiest 5 percent will continue to pay 60 percent of the income tax burden.

“I’m going to eliminate tax credits and deductions on the high end,” he said. “I’m not going to have the high end pay less that they pay now.”

For his part, Mr. Obama said he has a record of lowering tax cuts for the middle class and small businesses, and he wants to continue that, while raising taxes on the wealthy “a little more.”

He also said Mr. Romney’s tax plan math doesn’t add up and that he can’t boost defense spending, balance the budget and keep the share of taxes paid by the wealthy at the same high level while following through on his tax rate cuts, which total $5 trillion.

“It costs about $5 trillion,” Mr. Obama said. “Governor Romney then also wants to spend $2 trillion on additional military programs, even though the military’s not asking for them. That’s $7 trillion. He also wants to continue the Bush tax cuts for the wealthiest Americans. That’s another trillion dollars. That’s $8 trillion.”

When asked which deductions and loopholes Mr. Romney would close in order to ensure the tax cuts don’t add to the deficit, Mr. Obama said “he can’t tell you” and blasted his GOP rival for paying just 14 percent on his taxes.

“If somebody came to you, Governor, with a plan that said, here, I want to spend 7 [trillion dollars] or $8 trillion, and then we’re going to pay for it, but we can’t tell you until maybe after the election how we’re going to do it, you wouldn’t have taken such a sketchy deal,” he said. “And neither should you, the American people, because the math doesn’t add up.”

The debate began with a 20-year-old college student, Jeremy, asking whether he’d be able to get a job when he graduates.

Mr. Romney promised him he would.

“It’s not going to be like the last four years. The middle class has been crushed over the last four years, and jobs have been too scarce,” the former governor said. “When you come out in 2014, I presume I’m going to be president — I’m going to make sure you get a job.”

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