## 3. Rectangles

A rectangle is divided by lines parallel to its sides into four smaller rectangles. The areas of three of these four smaller rectangles are $240 \mathrm{~cm}^{2}, 320 \mathrm{~cm}^{2}$, and $480 \mathrm{~cm}^{2}$. What is the area (in $\mathrm{cm}^{2}$) of the largest of the four resulting rectangles?

Result: $\quad 640$
Solution, first method.

![](https://cdn.mathpix.com/cropped/2024_05_30_033072dbc63a78c728f4g-03.jpg?height=368&width=757&top_left_y=1815&top_left_x=684)

With the labels as in the image, we have:

$$
a c=320 \quad a d=240 \quad b d=480 \quad b c=?
$$

By multiplying, we get $(a c)(b d)=320 \cdot 480$,

which is

$$
(a c)(b d)=153600
$$

Rearranging in a different way, we get $(a d)(b c)=153600$, and applying $a d=240$ we get

$$
240(b c)=153600
$$

Therefore, the desired area is $b c=640$.

## Solution, second method.

![](https://cdn.mathpix.com/cropped/2024_05_30_033072dbc63a78c728f4g-04.jpg?height=369&width=757&top_left_y=358&top_left_x=661)

Since the area of the green rectangle is twice the area of the red rectangle, and both rectangles have one side of the same length $d$, it means that the length of the other side of the green rectangle, $b$, is twice the length of the other side of the red rectangle, $a$, i.e., $b=2 a$.

Since the blue and yellow rectangles have one side of length $c$, and the other side of the yellow rectangle is twice as long as the other side of the blue rectangle, it means that the area of the yellow rectangle is twice the area of the blue rectangle.

Therefore, the area of the yellow rectangle is $2 \cdot 320 \mathrm{~cm}^{2}=640 \mathrm{~cm}^{2}$.

Solution, third method.

Draw a line parallel to one side of the large rectangle, dividing the lower rectangles into two equal parts.

![](https://cdn.mathpix.com/cropped/2024_05_30_033072dbc63a78c728f4g-04.jpg?height=319&width=711&top_left_y=1528&top_left_x=678)

This divides the rectangle with an area of $480 \mathrm{~cm}^{2}$ into two rectangles with an area of $240 \mathrm{~cm}^{2}$, so it can be observed that the two parallel lines within the rectangle have divided the rectangle into three equal parts.

![](https://cdn.mathpix.com/cropped/2024_05_30_033072dbc63a78c728f4g-04.jpg?height=326&width=715&top_left_y=2093&top_left_x=679)

Therefore, the area of the fourth rectangle is $2 \cdot 320 \mathrm{~cm}^{2}=640 \mathrm{~cm}^{2}$.