Z6–I–5 MO 2019

The shape in the picture was created by cutting a small cross out of a large cross. Each of these crosses can be composed of five identical squares, with the sides of the small squares being half the sides of the large squares. The area of ​​the gray shape in the picture is 45 cm². What is the area of ​​the large cross?

Correct answer:

S =  60 cm²

Step-by-step explanation:

$$\begin{gathered} S_1=45{\text{cm}}^2 \\ {S}_1=S-S_2 \\ S=5\cdot a^2 \\ {S}_2=5\cdot (a/2{)}^2 \\ {S}_1=5a^2-5\cdot (a/2{)}^2 \\ {S}_1=5a^2-5\cdot a^2/4 \\ {S}_1=15/4\cdot a^2 \\ {a}^2=4\cdot S_1/15 \\ S=5\cdot a^2 \\ S=5\cdot 4\cdot S_1/15 \\ S=4\cdot S_1/3 \\ S=\frac{4}{3}\cdot S_1=\frac{4}{3}\cdot 45=\frac{4\cdot 45}{3}=\frac{180}{3}=60{\text{cm}}^2 \end{gathered}$$

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