As hospitals cut costs to survive ever-increasing financial pressures, nurses argue that inadequate staffing harms patients. Now members of a University of Pennsylvania research team say they have figured out a key reason for that. The study, which appears Monday in the American Journal of Infection Control, is believed to be the first to examine why staffing matters. It also is among the first to measure the potential harm. If hospitals could reduce their proportion of burned-out nurses to 10 percent from the 30 percent that is typical, according to the study, they would prevent 4,160 cases a year of the two most-common hospital-acquired infections statewide and save $41 million.