Paper cutting

Susanna is cutting construction paper into rectangles for a project. She needs to cut one rectangle 10 inches × 14 1/2 inches. She needs to cut another rectangle 10 1/3 inches by 10 1/2 inches. How many total square inches of construction paper does Susanna need for her project?

Correct answer:

S =  253 ¹⁄₂ = 507/2 in²

Step-by-step explanation:

$$\begin{gathered} a=10\text{in} \\ b=14\frac{1}{2}=14+\frac{1}{2}=\frac{14\cdot 2+1}{2}=\frac{28+1}{2}=\frac{29}{2}=14.5\text{in} \\ x=10\frac{1}{3}=10+\frac{1}{3}=\frac{10\cdot 3+1}{3}=\frac{30+1}{3}=\frac{31}{3}\doteq 10.3333\text{in} \\ y=10\frac{1}{2}=10+\frac{1}{2}=\frac{10\cdot 2+1}{2}=\frac{20+1}{2}=\frac{21}{2}=10.5\text{in} \\ {S}_1=a\cdot b=10\cdot \frac{29}{2}=\frac{10\cdot 29}{2}=\frac{290}{2}=145{\text{in}}^2 \\ {S}_2=x\cdot y=\frac{31}{3}\cdot \frac{21}{2}=\frac{31\cdot 21}{3\cdot 2}=\frac{651}{6}=\frac{217}{2}=108.5{\text{in}}^2 \\ S=S_1+S_2=145+\frac{217}{2}=\frac{145\cdot 2}{2}+\frac{217}{2}=\frac{290}{2}+\frac{217}{2}=\frac{290+217}{2}=\frac{507}{2}=\frac{507}{2}{\text{in}}^2=253\frac{1}{2}{\text{in}}^2=253.5{\text{in}}^2 \end{gathered}$$

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