B2. Rectangle $A B C D$ is divided into four rectangles as shown in the figure. The area of rectangle $A E I G$ is 3, the area of rectangle $E B H I$ is 5, and the area of rectangle $I H C F$ is 12. What is the area of the parallelogram $A H J F$?

![](https://cdn.mathpix.com/cropped/2024_04_17_6ebf2d3f59be4ec6e677g-1.jpg?height=311&width=374&top_left_y=1358&top_left_x=1429)
B2. $24 \frac{1}{5}$

Instead of looking at the entire parallelogram $A H J F$, we first consider half of it: triangle $A F H$. We can divide triangle $A F H$ into three smaller triangles: $A F I, F I H$, and $H I A$. If we calculate the area of triangle $A F I$ using the formula $\frac{1}{2} \cdot$ base $\cdot$ height and take $F I$ as the base, we see that it is exactly half the area of rectangle $D F I G$. The area of triangle $F I H$ is half the area of rectangle $C H I F$, and the area of triangle $H I A$ is half the area of rectangle $B H I E$.

![](https://cdn.mathpix.com/cropped/2024_04_17_014c53579b995e6f1c4fg-1.jpg?height=337&width=396&top_left_y=1708&top_left_x=1521)

The total area of the parallelogram, which is twice the area of triangle $A H F$, is thus the sum of the areas of rectangles $D F I G$, $C H I F$, and $B H I E$. The last two areas are 12 and 5. We only need to find the area of $D F I G$. This is equal to

$$
|G I| \cdot|F I|=\frac{(|G I| \cdot|I E|) \cdot(|F I| \cdot|I H|)}{|I H| \cdot|I E|}=\frac{3 \cdot 12}{5}=7 \frac{1}{5}
$$

The total area of $A H J F$ is thus $5+12+7 \frac{1}{5}=24 \frac{1}{5}$.