>Try this. It finds out which direction to take and exits as soon as there is a sign reversal
>
>

>		static void Main(string[] args)
>		{
>			double totalTax, fedTax, provTax;
>
>
>			totalTax = 500.00;
>
>			if (TrySolveTaxes(totalTax, out fedTax, out provTax))
>				Console.WriteLine("TotalTax= {0} fedTax= {1} provTax= {2}", totalTax, fedTax, provTax);
>			else
>				Console.WriteLine("cannot solve");
>
>
>			Console.ReadLine();
>
>		}
>		public static bool TrySolveTaxes(double totalTax, out double fedTax, out double provTax)
>		{
>			bool success = false;
>
>			double fedPortion = .05;
>			double provPortion = .095;
>
>			// first shot
>			double amount = Math.Round(totalTax / (fedPortion * provPortion + fedPortion + provPortion), 2);
>			fedTax = Math.Round(amount  * fedPortion, 2) ;
>			provTax = Math.Round((amount  + fedTax) * provPortion, 2) ;
>
>
>			int deltaSign = Math.Sign(totalTax - (fedTax + provTax));
>
>			if (deltaSign == 0)
>				success = true;
>			else
>			{
>				double direction = .01 * deltaSign;
>				int oldDeltaSign = deltaSign;
>
>				do
>				{
>					amount += direction;
>					fedTax = Math.Round(amount * fedPortion, 2);
>					provTax = Math.Round((amount + fedTax) * provPortion, 2);
>
>					deltaSign = Math.Sign(totalTax - (fedTax + provTax));
>					provTax = totalTax - fedTax;
>
>				} while (deltaSign == oldDeltaSign);
>				success = true;
>				
>			}
>
>			return success;
>		}
>

With this approach, for the totalTax of 74.88$ (correction from the 500.00$ mentioned above BTW), it falls right away in the delta and returns a base amount of 500.03$.

The code converted to VB.NET I used is as follow:

        TrySolveTaxes(74.88)

    Private Function TrySolveTaxes(totalTax As Double) As Boolean
        Dim fedtax As Double = 0
        Dim provtax As Double = 0
        Dim success As Boolean = False

        Dim fedPortion As Double = 0.05
        Dim provPortion As Double = 0.095

        ' first shot
        Dim amount As Double = Math.Round(totalTax / (fedPortion * provPortion + fedPortion + provPortion), 2)
        fedTax = Math.Round(amount * fedPortion, 2)
        provTax = Math.Round((amount + fedTax) * provPortion, 2)


        Dim deltaSign As Integer = Math.Sign(totalTax - (fedTax + provTax))

        If deltaSign = 0 Then
            success = True
        Else
            Dim direction As Double = 0.01 * deltaSign
            Dim oldDeltaSign As Integer = deltaSign

            Do
                amount += direction
                fedtax = Math.Round(amount * fedPortion, 2)
                provtax = Math.Round((amount + fedtax) * provPortion, 2)

                deltaSign = Math.Sign(totalTax - (fedtax + provtax))

                provtax = totalTax - fedtax
            Loop While deltaSign = oldDeltaSign

            success = True
        End If

        Return success
    End Function