REUTERS/Gary Cameron Construction on new single-family homes fell 7% to a seasonally adjusted annual rate of 935,000 units. This is a turnaround from May's surprising 18.7% surge. Multifamily housing starts fell 11.6% to 482,000 units, the lowest since December. Housing starts fell in June as higher costs and mortgage rates slowed activity in a key area of the market that has been adding supply.Construction on new single-family homes fell 7% to a seasonally adjusted annual rate of 935,000 units, breaking a four-month streak of gains and following an 18.7% jump in May.Meanwhile, multifamily housing starts dropped to 11.6% to 482,000 units, hitting the lowest rate since December.The decline is bad news for prospective buyers in the housing market, which increasingly relies on construction of new homes for inventory while the existing-home market is frozen.In fact, new construction made up almost a third of active listings in May, while pending sales of existing homes fell 2.7% month to month.