WASHINGTON — President Obama had Senate Republicans nodding in agreement during a recent ice-breaking dinner as he described a basic problem for the nation’s fiscal future: For each dollar that Americans pay for Medicare, they ultimately draw about $3 in benefits. What’s more, he added, most people do not understand that.

By his point that evening, the president was referring to the widespread and incorrect view, especially among older Americans, that Medicare recipients get only what they have paid for through taxes, premiums and medical co-payments. Now that misperception is making it all the harder for politicians to consider trimming those benefits or raising out-of-pocket expenses as they seek to restrain Medicare spending that is rising unsustainably while baby boomers age and medical prices increase.

There has long been similar confusion about Social Security, although the current generation of new beneficiaries is roughly breaking even.

Late in 2012, for a sixth straight year, Medicare trustees issued a warning required by law whenever more than 45 percent of the health program’s costs must be covered by general revenues from all taxpayers.