Dona Benta divided the Sítio do Picapau Amarelo among six characters, keeping a part of the Sítio as a forest reserve. The division is indicated in the figure, where the area of each character is given in hectares and the shaded area is the forest reserve. The Sítio has a rectangular shape and $AB$ is a diagonal.

![](https://cdn.mathpix.com/cropped/2024_05_01_1c2853cb1112cfb2ee2dg-11.jpg?height=457&width=762&top_left_y=888&top_left_x=750)

a) What is the area of the forest reserve?

b) To prepare the lands for planting, each of the six characters spent an amount proportional to the area of their land. Quindim and Cuca together spent $R \$ 2.420,00$. How much did the Saci spend?

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Solution

a) A rectangle is divided into two regions of the same area by its diagonal. Therefore, the lands of Quindim, Visconde de Sabugosa, and Cuca together have an area equal to half the area of the Sítio. The area of these lands, in hectares, totals $4+7+12=23$. The other half of the Sítio has the same area and is equal to the sum of the areas of the lands of Saci, Narizinho, Rabicó, and the forest reserve. Therefore, $6+5+10+$ (area of the reserve) $=23$, which means the area of the reserve is equal to 23ha $-21 \mathrm{ha}=2$ ha.

b) Quindim and Cuca together own 4ha +7 ha $=11 \mathrm{ha}$. Thus, they spent $\frac{2420}{11}=220$ reais per hectare. Since Saci's land is 6ha, he spent $6 \times 220=1320$ reais.