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Guests at the Days Inn on New York Avenue NE this fall may have been surprised to find that their neighbors included 51 homeless families. These families all qualify for a program called rapid rehousing. But with D.C. rents skyrocketing, the city can’t find affordable apartments for them. All told, there were 94 homeless families staying in hotels on taxpayers’ dime as of September—at a cost of $100 a night each—because there isn’t enough shelter space or permanent supportive housing for them. The problem is worse during winter, since the city’s required to shelter all homeless people when temperatures with wind chill drop below freezing. The Department of Human Services may have to turn to hotels beyond the District’s borders, with only 284 family shelter units for the 509 homeless families expected to seek shelter this winter.