Answer
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
Work Step by Step
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.
