A rectangular plot was divided into four plots, all rectangular. The areas of three of them are given in the figure, in $\mathrm{km}^{2}$. What is the area of the plot?

| 27 | 18 |
| :--- | :--- |
|  | 72 |
Solution 1: First, observe that the ratio between the areas of two rectangles that have the same base is equal to the ratio between their heights. Indeed, in the figure on the left,

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-082.jpg?height=246&width=582&top_left_y=2210&top_left_x=340)

| 27 | 18 |
| :--- | :--- |
| $S$ | 72 |

two rectangles are represented that have the same base $b$ and heights $h_{1}$ and $h_{2}$. Their areas $S_{1}$ and $S_{2}$ are given by $S_{1}=b h_{1}$ and $S_{2}=b h_{2}$, therefore,

$$
\frac{S_{1}}{S_{2}}=\frac{b h_{1}}{b h_{2}}=\frac{h_{1}}{h_{2}}
$$

Applying this observation to the two pairs of rectangles given (see the figure on the right) and denoting by $S$ the area of the fourth rectangle, we have

$$
\frac{27}{S}=\frac{h_{1}}{h_{2}}=\frac{18}{72}=\frac{1}{4}
$$

so that $S=27 \times 4=108$. Thus, the area of the plot that was divided measures $27+18+72+108=225 \mathrm{~km}^{2}$.

Solution 2: Let $x, y, z$ and $w$ be the measures of the smaller rectangles, as shown in the figure. The area sought is

$$
(x+w)(y+z)=x y+x z+w y+w z
$$

It is sufficient to determine $w y$, since it is known that

| $x$ | 18 |
| :--- | :---: | :---: |
|  | 72 |

$x y=27, x z=18$ and $w z=72$. But,

$$
\frac{w}{x}=\frac{w z}{x z}=\frac{72}{18}=4
$$

and therefore, $w=4 x$. Since $x y=27$, it follows that $w y=4 x \times 27 / x=4 \times 27=108$. Thus, the area of the plot that was divided measures

$$
27+18+72+108=225 \mathrm{~km}^{2}
$$