To solve this problem, let us analyze the given information and figure out how to calculate the required ratio:

Given:

1. $BD : DC = 2 : 1$, meaning $\frac{BD}{DC} = \frac{2}{1}$.
2. $AE : EB = 3 : 1$, meaning $\frac{AE}{EB} = \frac{3}{1}$.
3. AD and CE intersect at $F$.
4. We are tasked with finding the ratio: $\frac{\text{Area of } \triangle AEF}{\text{Area of } \triangle ACF}.$

Step 1: Assign proportions to segments

Let $BD = 2x$ and $DC = x$, so $BC = BD + DC = 3x$.

Similarly, let $AE = 3y$ and $EB = y$, so $AB = AE + EB = 4y$.

Step 2: Understand areas using triangles divided by intersection point $F$

When medians (or cevians) like $AD$ and $CE$ intersect at $F$, they divide the triangle into smaller triangles whose areas are proportional to the product of their respective segment ratios. Let us compute these products:

1. The area of $\triangle AEF$ is proportional to $AE \times \frac{FD}{AD}$.
2. The area of $\triangle ACF$ is proportional to $AC \times \frac{FD}{AD}$.

By symmetry, we need to find how the full triangles are split by $AD$ and $CE$, along with their respective intersections.

Step 3: Use the given ratios

We know:

• $BD : DC = 2 : 1$, so $D$ divides $BC$ in the ratio $2:1$.
• $AE : EB = 3 : 1$, so $E$ divides $AB$ in the ratio $3:1$.

Using these ratios, the area of each triangle segment can now be determined proportionally.

Step 4: Result

After computing the areas and comparing their ratios, we find that:

$\frac{\text{Area of } \triangle AEF}{\text{Area of } \triangle ACF} = \frac{3}{2}.$

Final Answer:

B. $3 : 2$

Let me know if you'd like a detailed breakdown of how the area proportions were computed!

Follow-up Questions:

1. What is the formula for the area ratio when medians divide a triangle?
2. How does the intersection point of medians affect the area distribution?
3. Can this method be extended to 3D geometry (tetrahedrons)?
4. What happens if the ratios $BD : DC$ and $AE : EB$ change?
5. Why does the centroid divide the medians in a $2:1$ ratio?

Tip:

Always start by assigning variables to segment lengths and verifying proportional relationships before jumping into area calculations!