A Steel Rod $45$ Inches Long Has Twelve Pieces Cut to the Following Lengths

Essay

A steel rod $45$ inches long has twelve pieces cut to the following lengths:
3 pieces, $2 \frac { 1 } { 8 }$ inches in length
2 pieces, $5 \frac { 3 } { 4 }$ inches in length
6 pieces, $\frac { 7 } { 8 }$ inches in length
1 piece, $3 \frac { 1 } { 2 }$ inches in length
Assume $\frac { 1 } { 16 }$ of an inch is wasted with each cut. Find the length of the remaining rod.