The Fairfax County Board of Supervisors early yesterday approved a 9.9 percent increase in the county's annual operating budget and levied a real estate tax rate that will raise typical homeowner tax bills by about 4 percent.

The total county operating budget for the year beginning July 1 is about $520 million. Of this total, about $335 million, a 7 percent increase over last year, will be paid out of count tax revenues. The real estate levy is the principal county tax. The balance of the operating budget is financed by state and U.S. funds.

The supervisors actually reduced the property tax rate by 10 cents from $1.74 per $100 of assessed valuation to $1.64. However, the new rate will be applied to assessed house values that average 8.7 percent more than last year. The result is a tax increase of about 4 percent for typical homeowners.

For instance, the owner of a $69,000 home who paid $1,200 in taxes last year will pay $1,246 this year.

The Alexandria City Council on Monday night also raised the real estate tax by about 7 percent for the typical homeowner to fund a city budget of $88.3 million, 7.4 percent more than current spending.

The Alexandria tax increase results from the adoption of a tax rate of $1.54 per $100 of assessed value. Although the new rate is 16 cents below the current levy of $1.70, it will be applied to houses whose assessed values have increased an average of 17 percent.

The tax bill for the owner of a $60,000 home in Alexandria will rise $61 from $1,020 to $1081.

Fairfax County Manager Leonard Whorton said that increased government spending in Fairfax is required by inflation and rapid growth. The budget provides for the hiring of 121 new police officers and firefighters and 54 other new employes.

The supervisors cut about $9.5 million from Whorton's proposed budget. The spending total adopted by the board provides for a pay increase of 5.5 percent for county employees, including school teachers.

The school board has authority to grant teachers the 7.1 percent pay raise recommended by School Superintendent S. John Davis, but will have to cut spending in other areas to do so.

In addition to the $520 million operating budget, the county adopted a capital budget of about $57 million for construction of schools and other public facilities.