## Task 4 - 020524

The figure shows two different-sized areas. $(a=2 \mathrm{~cm}, b=4 \mathrm{~cm}, c=6 \mathrm{~cm}, d=10 \mathrm{~cm})$
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0020.jpg?height=408&width=988&top_left_y=2023&top_left_x=543)

a) How many times is the smaller area contained in the larger one?

b) Verify the correctness of this claim with a drawing!
a) The area of the small shape is obtained as the difference between the areas of a rectangle with side lengths $b$ and $c$, from which a rectangle with dimensions $a$ and $b$ has been cut out on the top right:

$$
A_{\text{small}}=b c-a b=4 \mathrm{~cm} \cdot 6 \mathrm{~cm}-2 \mathrm{~cm} \cdot 4 \mathrm{~cm}=16 \mathrm{~cm}^{2}
$$

For the large area, a rectangle with sides $c$ and $d$ has a rectangle with sides $a$ and $c$ removed.

$$
A_{\text{large}}=c d-a c=6 \mathrm{~cm} \cdot 10 \mathrm{~cm}-2 \mathrm{~cm} \cdot 6 \mathrm{~cm}=48 \mathrm{~cm}^{2}
$$

Thus, the small area fits into the large area exactly

$$
\frac{A_{\text{large}}}{A_{\text{small}}}=\frac{48 \mathrm{~cm}^{2}}{16 \mathrm{~cm}^{2}}=3
$$

3 times, as the diagram for part b) also shows.
b)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0021.jpg?height=212&width=326&top_left_y=745&top_left_x=888)