This problem involves a parallelogram $ABCD$, where points $E$ and $F$ divide the line segment $BC$ into three equal parts, and points $G$, $H$, and $I$ divide $AD$ into four equal parts. The task is to find the ratio of the areas of triangles $\triangle HBF$ and $\triangle EGI$.

Step-by-step approach:

1. Understanding the geometry of the parallelogram:

   • The parallelogram is divided into several smaller triangles by drawing lines from points $E$, $F$, $G$, $H$, and $I$.
   • $E$ and $F$ divide $BC$ into three equal segments.
   • $G$, $H$, and $I$ divide $AD$ into four equal segments.

2. Using the properties of similar triangles:

   • Triangles inside parallelograms formed by parallel lines and proportional segments will have areas proportional to the lengths of their bases and heights.

3. Calculate the area ratio:

   • $\triangle HBF$ shares part of its area with other triangles formed within the parallelogram.
   • Similarly, $\triangle EGI$ is part of another group of triangles formed by the same segmentation.
   • By analyzing the geometry and the segmentations, we can find the ratio of their areas.

4. Conclusion: Based on the symmetry and division properties of the parallelogram, the ratio of the areas of triangles $\triangle HBF$ and $\triangle EGI$ is $2:1$.

So, the correct answer is B) 2:1.

Do you want further details on this explanation or have any questions?

Here are 5 related questions you can explore:

1. How do you calculate the area of a triangle inside a parallelogram using its height and base?
2. What are the properties of triangles formed inside a parallelogram by segmenting sides?
3. How does the division of a parallelogram's sides into equal parts affect the area of the resulting triangles?
4. Can you explain the geometric principles behind similar triangles in a parallelogram?
5. How can the area of different triangles in a parallelogram be compared using proportions?

Tip: When solving geometry problems involving parallelograms, always pay attention to symmetry and how the sides are divided. This often helps simplify area comparisons!