Answer:
A piece of duct measures 48 inches.
In each connection, we lost 1 1/2 inches (i guess that it is what you wanted to write)
Then:
If we have 7 pieces of duct connected we have 5 connections (the ones in the extremes are not connected between them, so the number of connections is equal to the number of ducts minus two.)
So here we have 7 times 48 inches for the pieces, minus 5 times 1 1/2 in (i will write 1 1/2 = 1 + 0.5 = 1.5 in, so the math is easier) for the connections, the length is:
L = 7*48in - 5*1.5in = 328.5 inches.
for the 12 pieces duct, we have 12 pieces and 10 connections, so the length is:
L = 12*48in - 10*1.5in = 561 in
Now, if we want to make only one duct with those two, then we must add their lengths, but if we connect them, we also need to subtract the 1.5in of the new connection:
L = 561in + 328.5in - 1.5in = 888in