# Magic belt

The magic rectangular belt has the property that whenever its owner wants something, the length of the belt is reduced to 1/2 and the width to 1/3. After three such wishes, the belt had a content area 4 cm2. What was its original length if the original width was 9 cm?

Result

b =  96 cm

#### Solution:

$a=9 \ \\ S_{2}=4 \ \\ a_{2}=a \cdot \ (1/3)^{ 3 }=9 \cdot \ (1/3)^{ 3 } \doteq \dfrac{ 1 }{ 3 } \doteq 0.3333 \ \\ b_{2}=S_{2} / a_{2}=4 / 0.3333=12 \ \\ b=2^{ 3 } \cdot \ b_{2}=2^{ 3 } \cdot \ 12=96 \ \\ b=S_{1}=a \cdot \ b=9 \cdot \ 96=864=96 \ \text{cm}$

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