The Arlington County Board, splitting along party lines, voted last night to advertise the proposed fiscal 1984 personal property tax rate at its current level of $5.10 per $100 assessed valuation. The tax is applied to cars, boats, airplanes and some pieces of machinery and tools.

The board scheduled a public hearing for April 23. It can cut the rate below the advertised $5.10 figure but may not increase it.

The board's two Republican-backed independents, Walter L. Frankland Jr. and Dorothy T. Grotos, argued for lowering the rate to $4.90 both last night and on Tuesday night, when the board deadlocked 2 to 2. Board member John G. Milliken, a Democrat, was absent Tuesday night, but joined with Democrat Mary Margaret Whipple and Board Chairman Ellen M. Bozman, a Democrat-backed independent, to break the tie last night.

Frankland and Grotos said they wanted the tax rolled back to offset the one-cent increase approved last month in the real estate tax rate, now set at 99 cents per $100 of assessed valuation. The two also opposed the real estate-rate increase.

Bozman, Whipple and Milliken said that, given budget uncertainties, it was preferable to advertise the tax at its current rate. "There are an awful lot of things unresolved," Whipple said last night. "We can always lower the rate. Until we can put all the pieces together and decide priorities for funding and how much we can allocate within the existing revenues, I'm not prepared to make that decision."