Let's use "r" to represent the number of rectangular pieces and "t" to represent the number of triangular pieces.
From the problem, we know that for every 8 rectangular pieces, there are 12 triangular pieces. This can be written as:
12/8 = 3/2
So for every 2/3 of a triangular piece, there is 1 rectangular piece.
We also know that there are a total of 80 pieces. So we can set up an equation:
r + t = 80
We can use the ratio of 2/3 to set up another equation:
r = (2/3)t
Now we can substitute the second equation into the first equation and solve for t:
(2/3)t + t = 80
(5/3)t = 80
t = (3/5) * 80
t = 48
So there are 48 triangular pieces. To find the number of rectangular pieces, we can use the second equation:
r = (2/3)t
r = (2/3) * 48
r = 32
So there are 32 rectangular pieces.
Therefore, there are 48 triangular pieces and 32 rectangular pieces used in the mosaic table top