## Elementary Technical Mathematics

Each 17-ft duct must be cut into 4 pieces. 3 of the 4 pieces will be $4\frac{1}{2}$ ft long while the fourth piece is $3\frac{1}{2}$ft long
Divide $17$ ft by $4\frac{1}{2}$ ft to obtain: $=17 \div 4\frac{1}{2} \\=\frac{17(4)}{4} \div \frac{2(4)+1}{2} \\=\frac{68}{4} \div \frac{9}{2}$ Use the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ to obtain: $\require{cancel} =\frac{68}{4} \times \frac{2}{9} \\=\frac{68}{\cancel{4}2} \times \frac{\cancel{2}}{9} \\=\frac{68}{2} \times \frac{1}{9} \\=34 \times \frac{1}{9} \\=\frac{34(1)}{9} \\=\frac{34}{9} \\=3\frac{7}{9}$ Thus, 3 pieces of $4\frac{1}{2}$ ft long ducts can be cut from the 17-ft duct. The length of the 4th piece is: $=17 - (3 \times 4\frac{1}{2}) \\=17 - (3 \times \frac{2(4)+1}{2}) \\=17-(3 \times \frac{9}{2}) \\=17-\frac{27}{2} \\=17-13\frac{1}{2} \\=3\frac{1}{2}$ Therefore, each 17-ft duct must be cut into 4 pieces. 3 of the 4 pieces will be $4\frac{1}{2}$ ft long while the fourth piece is $3\frac{1}{2}$ ft long.