The Arlington County Board approved a $198 million budget for fiscal 1982 yesterday, trimming the county's personal property tax rate slightly and performing microsurgery on an already reduced school funding package.

The new property tax rate -- reduced from $5.45 to $5.30 per $100 of assessed value -- is expected to lower motor vehicle taxes for most county residents when coupled with a planned 10 percent reduction in vehicle assessments, according to county officials.

After colliding once again yesterday with school board members, the county board split 3 to 2 along party lines over a $45 million education spending plan. The action pointed up the sharp philosophical differences between the board's liberal Democratic and conservative Republican factions and showed the conservative majority's detemination to strengthen their control over school administrators.

The budget will force the school system to cut nearly $1 million from its instruction programs next year, a cut that some officials said could cost more than 100 school employes their jobs.

The new budget allows for a 7 percent hike in school worker salaries and a 9 percent raise for county employes, increases significantly lower than workers had sought and school board members had proposed.

The budget is based largely on a real estate tax rate the board cut by 14 percent last month. Under the previously-approved rate of 96 cents per $100 of assessed value, the average Arlington homeowner's tax bill will drop only $13, because of sharply higher assessments which have boosted residential assessments by 15.1 percent.

The tightened purse strings have raised fears among some board members and administrators that the county government could become mired in financial troubles later this year if the Reagan administration's proposed reductions in federal assistance become effective.

The budget session again opened the dispute between those sympathetic to the school board's funding requests and Republican board members who argue that Arlington's declining enrollments should mean a halt to increases in education spending.

At ojne point yesterday, board member Walter Frankland, the majority's most outspoken conservative, accused former school superintendent Larry Cuban of deliberately attempting to undermine the board's wishes by ignoring its budget guidelines.

"Either they're not giving us the correct budget figures or they're changing this thing like a moving target," Frankland said, his voice rising to the hissing and booing from teachers in the audience. "The crunch in instruction is not as serious as everyone is making it out to be."

After an occassionally acrimonious exchange with school board member Torill B. Floyd, Frankland added: "The more I hear people complaining, the more I think we must be on the right track."

"I don't think you really understand the consequences of what you're doing," Floyd responded.

Republican board member Dorothy Grotos, who earlier this week admonished teachers for acting political and unprofessional, joined the debate. "I believe there should be some cuts in personnel in that white palace over there," Grotos said, referring to the schools administration building. "That was exactly the intent," she said, of the board's initial $43.5 million school budget guideline.

The approved $45 million education plan had been offered as a compromise by Republican board chairman Stephen Detwiler. Both Democratic board members John Milliken and Ellen Bozman said that the amount was insufficient and voted against it. They offered plans for transferring funds from elsewhere in the county budget to boost the schools budget.

Finally, Grotos agreed to transfer $19,000 from the Human Resources budget, a proposal Milliken called "the meagerest of gestures. It's an indication of what I consider to be a wrongheaded attitude here."

According to state law, county boards generally are not supposed to consider school budgets line by line. The Arlington board, however, refused school board member requests to deliver the $45 million in a lump sum.

Floyd said the "setting the budget in concrete" would tie the hands of newly appointed schools superintendent Charles E. Nunley, a noted fiscal conservative from Lorain, Ohio.

"They're just using this new superintendent as a smokescreen," Frankland responded.