Alter given values to achieve desired result

Sorry - you have 4 unknown values (cells D3:G3 - which you want to change), but only two constraints (or equations): that the sum of the 4 variables be equal to A3, and that the Sum of the Values divided by the constants in D2:G2 be another specific value (A3/B3).

They will not give a unique solution - in fact, they will give an almost infinite number of solutions - the first two solutions from my brute force solving are, for example:

1 934 2055 10
1 944 2015 40

You can only solve those for 2 of the unknowns as expressions of the other 2 values.

Here is code that does the brute force solution, for just one value of the first cell (D3 = 1):

Sub FirstSolution()
    Dim i As Integer
    Dim j As Integer
    Dim k As Integer
    Dim l As Integer
    
    Dim i1 As Integer
    Dim j1 As Integer
    Dim k1 As Integer
    Dim l1 As Integer
    
    Dim iH As Integer
    Dim iL As Integer
    
    Dim lRow As Long
    Dim Sh1 As Worksheet
    Dim Sh2 As Worksheet
    
    Set Sh1 = Worksheets("Sheet1")
    On Error Resume Next
    Application.DisplayAlerts = False
    Worksheets("Solutions").Delete
    On Error GoTo 0
    
    Set Sh2 = Worksheets.Add(After:=Sh1)
    Sh2.Name = "Solutions"
    
    iH = Sh1.Range("A3").Value
    iL = Sh1.Range("A3").Value / Sh1.Range("B3").Value
    
    i1 = Sh1.Range("D2").Value
    j1 = Sh1.Range("E2").Value
    k1 = Sh1.Range("F2").Value
    l1 = Sh1.Range("G2").Value
    
    lRow = 0
    
    i = i1
    For j = j1 To iH - i Step j1
        For k = k1 To iH - i - j Step k1
            l = iH - i - j - k
            If l Mod l1 = 0 Then
                If i / i1 + j / j1 + k / k1 + l / l1 = iL Then
                    lRow = lRow + 1
                    With Sh2
                        .Cells(lRow, "A") = i
                        .Cells(lRow, "B") = j
                        .Cells(lRow, "C") = k
                        .Cells(lRow, "D") = l
                    End With
                End If
            End If
        Next k
    Next j
    
End Sub

Hi juriemagic, the answer in red/orange is 3000 divided by the blue total, so if you want a different answer and you need to keep the 3000 then the blue has to change. When you have 3 parts of an equation and 2 of them are fixed then only 1 is left.
The blue is the relationship between green and the numbers above green. Can either of those change or only the green?
I did notice that your desired answer is exactly double the incorrect result. Can you just add a /2 into the blue cells?
e.g. Cell I3 =D3/D2/2 (copy over to L3)

Edit: Sorry Bernie Deitrick, I started typing this ages ago while I was playing with the numbers. I didn't realize that you had answered this (and much better then me too). I should have refreshed the page before I posted.

For your stated problem, there are 30,352 solutions, all given by this code - all give the 3.409 ratio as desired, since that is part of the check.

Option Explicit

Sub AllSolutions()
    Dim i As Integer
    Dim j As Integer
    Dim k As Integer
    Dim l As Integer
    
    Dim i1 As Integer
    Dim j1 As Integer
    Dim k1 As Integer
    Dim l1 As Integer
    
    Dim iH As Integer
    Dim iL As Integer
    Dim iC As Integer
    
    Dim lRow As Long
    Dim Sh1 As Worksheet
    Dim Sh2 As Worksheet
    
    Set Sh1 = Worksheets("Sheet1")
    On Error Resume Next
    Application.DisplayAlerts = False
    Worksheets("Solutions").Delete
    On Error GoTo 0
    
    Set Sh2 = Worksheets.Add(After:=Sh1)
    Sh2.Name = "Solutions"
    
    iH = Sh1.Range("A3").Value
    iL = Sh1.Range("A3").Value / Sh1.Range("B3").Value
    
    i1 = Sh1.Range("D2").Value
    j1 = Sh1.Range("E2").Value
    k1 = Sh1.Range("F2").Value
    l1 = Sh1.Range("G2").Value
    
    lRow = 0
    
    For i = i1 To iH Step i1
        For j = j1 To iH - i Step j1
            For k = k1 To iH - i - j Step k1
                l = iH - i - j - k
                If l Mod l1 = 0 Then
                    If i / i1 + j / j1 + k / k1 + l / l1 = iL Then
                        lRow = lRow + 1
                        With Sh2
                            .Cells(lRow, "A") = i
                            .Cells(lRow, "B") = j
                            .Cells(lRow, "C") = k
                            .Cells(lRow, "D") = l
                        End With
                    End If
                End If
            Next k
        Next j
    Next i
    
End Sub

Play with numbers.xlsm

Click the button in the attached.