A ten foot long board is cut into 3 pieces. The second piece is half as long as the first. The third piece is 4 1/2 feet longer than the second piece. How long is the first piece?

say "a" is the length of the first piece, we'll first off, change the mixed fraction to "improper" and add the pieces, keeping in mind that, all 3 pieces add up to 10 feet.

[tex]\bf \stackrel{mixed}{4\frac{1}{2}}\implies \cfrac{4\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{9}{2}}\\\\ -------------------------------[/tex]

[tex]\bf \stackrel{\textit{first piece}}{a}+\stackrel{\textit{second piece}}{\cfrac{a}{2}}+\left(\stackrel{\textit{third piece}}{\cfrac{a}{2}~+~\cfrac{9}{2}}\right)=10\impliedby \begin{array}{llll} \textit{multiplying both sides by }\\ \textit{the LCD of 2} \end{array} \\\\\\ 2\left( a+\cfrac{a}{2}+\cfrac{a}{2}+\cfrac{9}{2} \right)=2(10)\implies 2a+a+a+9=20 \\\\\\ 4a=11\implies a=\cfrac{11}{4}\implies a=2\frac{3}{4}[/tex]