A piece of duct measures 48 inches. When pieces are fitted together, 1 72 inches is lost connecting each successive length of duct. On the job 7 and 12 pieces of ducting are used. What, in inches is the total length of the finished duct?

A piece of duct measures 48 inches.

In each connection, we lost 1.72 inches (it also could be 1/2 inches, but i will go with the other option)

Then:

If we have 7 pieces of duct connected we have 5 connections (the ones in the extremes are not connected between them)

So here we have 7 times 48 inches for the pieces, minus 5 times 1.72 inches for the connections, the length is:

L = 7*48in - 5*1.72in = 327.4 inches.

for the 12 pieces duct, we have 12 pieces and 10 connections, so the length is:

L = 12*48in - 10*1.72in = 558.8 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 1.72in of the new connection:

L =  558.8 in +  327.4 in - 1.72in = 884.48 in