CHICAGO — Cook County, Ill., faces a nearly 16% shortfall in its general fund next year as revenues continue to fall, officials said yesterday as they unveiled preliminary 2012 budget numbers.

The county’s deficit could reach $315 million in 2012, out of a proposed $2 billion general fund budget, said county board president Toni Preckwinkle. The preliminary all-funds budget totals $3.2 billion.

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