Calculation Based On % Of Cells

I need a formula that I can place in, lets say, Cell A1 that will reference Cells B1, B2, B3 and C1, C2, C3. These all have numbers & %'s in them. However, I want to "weight" the B cells by the percentages in C cells. The problem I am having is that when I change the %'s nothing is changing in the output...almost like I was doing nothing but averaging B1:B3 without the % to adjust.

This is how I envision it, but my calculation may be wrong of course :
A1 = average(sum(B1,B2,B3)*C1, sum(B1,B2,B3)*C2, sum(B1,B2,B3)*C3)

The Goal:

This would allow me to change the output depending on the percentage. So if I am looking at days to an appointment and the B cells represent areas of a city: B1 = 10, B2 = 5, B3 = 2 the average would be 5.6, however, I know that B1 city has more "weight" to it with our customer base so I want B1 to be closer to what will show as days out (which is obviously important as 10 days is not good). So I would assign C1 = 60%, C2 = 30%, C3 = 10%. This should pull the average more to 7 or 8 as it would lean more towards the B1 value. I hope this makes sense.

I did try going more of a route where I did something like =average(B1*C1,B2*C2,B3*C3) but that lowered all the original values instead and averaged them instead of pushing the number more towards the higher weighted % of B1 cell value. I can send an example spreadsheet if I am losing anyone.