4. Each time it fulfills the owner's wish, the magic carpet in the shape of a rectangle shrinks: its length decreases by $\frac{1}{2}$ of its length, and its width by $\frac{1}{3}$ of its width. After three fulfilled wishes, the magic carpet has an area of $18 \mathrm{dm}^{2}$. The initial width of the magic carpet was $1.8 \mathrm{~m}$. What was its initial length? What was its initial area?
First method:

Let $x$ be the initial length of the magic carpet.

The initial width of the carpet was $1.8 \mathrm{~m}=18 \mathrm{dm}$.

1 POINT

After the first wish was granted, the sides were:

length $\frac{1}{2} x$, and width $18-\frac{1}{3} \cdot 18=18-6=12 \mathrm{dm}$.

2 POINTS

After the second wish was granted, the sides were:

length $\frac{1}{2} \cdot \frac{1}{2} x=\frac{1}{4} x$, and width $12-\frac{1}{3} \cdot 12=12-4=8 \mathrm{dm}$.

2 POINTS

After the third wish was granted, the sides were:

length $\frac{1}{2} \cdot \frac{1}{4} x=\frac{1}{8} x$, and width $8-\frac{1}{3} \cdot 8=8-\frac{8}{3}=\frac{16}{3} \mathrm{dm}$.

2 POINTS

The area of a rectangle is calculated using the formula $P=a \cdot b$, where $a$ and $b$ are the lengths of adjacent sides.

From the condition of the problem, it follows that

$\frac{1}{8} x \cdot \frac{16}{3}=18$, or $\frac{2}{3} x=18$, hence $x=27 \mathrm{dm}=2.7 \mathrm{~m}$.

2 POINTS

The initial area of the magic carpet was $P=1.8 \mathrm{~m} \cdot 2.7 \mathrm{~m}=4.86 \mathrm{~m}^{2}$.

1 POINT

TOTAL 10 POINTS