In the United States, welfare (financial aid) refers more specifically to money paid by the government to those who are in need of financial assistance -- people who are unable to support themselves, perhaps due to unemployment, disability or some other reason.
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The amounts paid are typically very low, and may fall below the poverty line. Recipients must usually demonstrate a low level of income (eg. by way of "means testing") or financial hardship, or that they satisfy some other requirement (eg. childcare responsibilities or disability), or regularly demonstrate that they are periodically searching for employment.
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In the United States, personal welfare is normally given to households with children (often headed by single mothers) and even these households have only been able to access benefits for a maximum of five years per lifetime of the adult recipient since 1996.
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With regard to personal welfare for the individuals without children, most American states had been providing welfare benefits to single adults and childless married couples since the Great Depression, but the number of states doing so declined steeply during the 1990s, and many of the states still doling out such benefits use methods other than cash payments to render the assistance. Today only two states - New Jersey and Utah - still provide cash benefits to poverty-stricken adults who do not have child dependents. These programs were often known officially by such names as Home Relief and General Assistance.