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 an area of 4 cm². What was its original length if the original width was 9 cm?

Correct answer:

b =  96 cm

Step-by-step explanation:

$$\begin{gathered} a=9\text{cm} \\ {S}_2=4{\text{cm}}^2 \\ {a}_2=a\cdot (1/3{)}^3=9\cdot (1/3{)}^3=\frac{1}{3}\doteq 0.3333\text{cm} \\ {b}_2=S_2/a_2=4/0.3333=12\text{cm} \\ b=2^3\cdot b_2=2^3\cdot 12=96\text{cm} \\ \text{ Verifying Solution: } \\ {S}_1=a\cdot b=9\cdot 96=864{\text{cm}}^2 \end{gathered}$$

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