(CEDERJ 2019) - Question Medium of Mathematics

In a civil engineering project, an architect needs to divide a rectangular plot of land into two equal parts for the construction of two semi-detached houses. The coordinates of the vertices of the land on the Cartesian plane are A(0, 0), B(30, 0), C(30, 20), and D(0, 20). Using concepts of analytic geometry, determine the coordinates of the point that represents the midpoint of side CD of the land, which will be the dividing point between the two new properties. Then, verify if the obtained midpoint point is indeed on segment CD, ensuring the accuracy of the project. Explain the reasoning used to determine the coordinates and to verify the positioning of the midpoint on the land.

In Ana's lunchbox, she has 8 green grapes and 6 purple grapes. In João's lunchbox, he has 4 grapes of each color. If they combine all the grapes and divide them equally between them, who will have more grapes, Ana or João? Write two different addition equations that result in the same total number of grapes to help Ana and João divide the grapes fairly.

João is organizing a birthday party and wants to make a larger quantity of his favorite cake recipe. The original recipe yields a cake with 20 slices and uses 3 cups of flour, 2 cups of sugar, and 1 cup of oil. João wants to make a cake with 40 slices. a) How many cups of flour, sugar, and oil should he use to make a cake that yields 40 slices? b) If he wanted to make a cake with 60 slices, what would be the quantity of cups of each ingredient? c) João noticed that the number of cake slices is always half the number of cups of water used in the recipe. How many cups of water are needed for a cake of 80 slices?