Problem 2. Whenever the owner's wish is granted, the length of the rectangular magic carpet decreases by $\frac{1}{2}$ of its current length, and the width decreases by $\frac{1}{3}$ of its current width. After three granted wishes, the carpet had an area of $18 \mathrm{dm}^{2}$. The initial width of the carpet was $1.8 \mathrm{~m}$. Determine the initial length of the carpet. Determine the initial area of the carpet.
Solution. Let $x$ be the initial width of the carpet. The initial length of the carpet is $1.8 \mathrm{~m}=18 \mathrm{dm}$. After fulfilling the first wish, the length of the carpet is $\frac{1}{2} x$, and the width is $18-\frac{1}{3} \cdot 18=12 \mathrm{dm}$. After fulfilling the second wish, the length of the carpet is $\frac{1}{2} \cdot \frac{1}{2} x=\frac{1}{4} x$, and the width is $12-\frac{1}{3} \cdot 12=8 \mathrm{dm}$. After fulfilling the third wish, the length of the carpet is $\frac{1}{2} \cdot \frac{1}{4} x=\frac{1}{8} x$, and the width is $8-\frac{1}{3} \cdot 8=\frac{16}{3} \mathrm{dm}$. Now for the area of the carpet, we get $\frac{1}{8} x \cdot \frac{16}{3}=18$, from which we find $x=27 \mathrm{dm}=2.7 \mathrm{~m}$. Finally, the initial area of the carpet is $P=2.7 \cdot 1.8=4.86 \mathrm{~m}^{2}$.