Ratio of sides 2

The ratio of the side lengths of one square to another is 1:2. Find the ratio of the area of the two squares.

Result

r =  1:4

Solution:

$a_{1}=\dfrac{ 1 }{ 1+2 }=\dfrac{ 1 }{ 3 } \doteq 0.3333 \ \\ a_{2}=\dfrac{ 2 }{ 1+2 }=\dfrac{ 2 }{ 3 } \doteq 0.6667 \ \\ \ \\ k=a_{1}/a_{2}=0.3333/0.6667=\dfrac{ 1 }{ 2 }=0.5 \ \\ \ \\ S_{1}=a_{1}^2 \ \\ S_{2}=a_{2}^2 \ \\ r=S_{1}/S_{2}=a_{1}^2:a_{2}^2=a_{1}^2:(k a_{1})^2 \ \\ \ \\ r=k^2=0.5^2=\dfrac{ 1 }{ 4 }=0.25=1:4$

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