Gift box

A gift box has dimensions of 8 1/2 inches, 5 1/2 inches, and 2 1/2 inches, respectively. How many cubes with side lengths of 1/2 inches would be needed to fill the gift box?

Correct answer:

n =  935

Step-by-step explanation:

$$\begin{gathered} x=8\frac{1}{2}=8+\frac{1}{2}=\frac{8\cdot 2+1}{2}=\frac{16+1}{2}=\frac{17}{2}=8\frac{1}{2}=8.5\text{in} \\ y=5\frac{1}{2}=5+\frac{1}{2}=\frac{5\cdot 2+1}{2}=\frac{10+1}{2}=\frac{11}{2}=5\frac{1}{2}=5.5\text{in} \\ z=2\frac{1}{2}=2+\frac{1}{2}=\frac{2\cdot 2+1}{2}=\frac{4+1}{2}=\frac{5}{2}=2\frac{1}{2}=2.5\text{in} \\ a=\frac{1}{2}=0.5\text{in} \\ {n}_1=x/a=\frac{17}{2}/\frac{1}{2}=\frac{17}{2}:\frac{1}{2}=\frac{17}{2}\cdot \frac{2}{1}=\frac{17\cdot 2}{2\cdot 1}=\frac{34}{2}=17 \\ {n}_2=y/a=\frac{11}{2}/\frac{1}{2}=\frac{11}{2}:\frac{1}{2}=\frac{11}{2}\cdot \frac{2}{1}=\frac{11\cdot 2}{2\cdot 1}=\frac{22}{2}=11 \\ {n}_3=z/a=\frac{5}{2}/\frac{1}{2}=\frac{5}{2}:\frac{1}{2}=\frac{5}{2}\cdot \frac{2}{1}=\frac{5\cdot 2}{2\cdot 1}=\frac{10}{2}=5 \\ n=n_1\cdot n_2\cdot n_3=17\cdot 11\cdot 5=935 \end{gathered}$$

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