1. **Stating the problem:** We have triangle ABC with points M on BC such that $BM:MC=2:3$, and point K on segment AM such that $AK:KM=2:1$. We want to find the ratios of areas $\frac{S_{AKC}}{S_{ABC}}$ and $\frac{S_{AABC}}{S_{ABC}}$ (assuming $S_{AABC}$ means the area of triangle formed by points A, A, B, C which is likely a typo or means the whole triangle ABC, so we focus on $S_{AKC}$ and $S_{ABC}$). 2. **Understanding the problem:** - Point M divides BC in ratio 2:3. - Point K divides AM in ratio 2:1. - We want the ratio of areas of smaller triangles inside ABC. 3. **Set coordinates for calculation:** Let’s place points for convenience: - Let $B=(0,0)$ - Let $C=(5,0)$ (since $BM:MC=2:3$, total length BC=5 units) - Let $A=(0,h)$ for some height $h$. 4. **Find coordinates of M:** Since $BM:MC=2:3$, M divides BC at 2/5 from B: $$M=\left(\frac{2}{5} \times 5, 0\right) = (2,0)$$ 5. **Find coordinates of K on AM:** Segment AM connects $A=(0,h)$ and $M=(2,0)$. Ratio $AK:KM=2:1$ means K divides AM in ratio 2:1 starting from A. Using section formula: $$K=\left(\frac{2 \times 2 + 1 \times 0}{2+1}, \frac{2 \times 0 + 1 \times h}{2+1}\right) = \left(\frac{4}{3}, \frac{h}{3}\right)$$ 6. **Calculate areas:** - Area of $\triangle ABC$: $$S_{ABC} = \frac{1}{2} \times BC \times height = \frac{1}{2} \times 5 \times h = \frac{5h}{2}$$ - Area of $\triangle AKC$ with points $A=(0,h)$, $K=\left(\frac{4}{3}, \frac{h}{3}\right)$, $C=(5,0)$: Use determinant formula for area: $$S_{AKC} = \frac{1}{2} \left| x_A(y_K - y_C) + x_K(y_C - y_A) + x_C(y_A - y_K) \right|$$ Substitute: $$= \frac{1}{2} |0 \times (\frac{h}{3} - 0) + \frac{4}{3} \times (0 - h) + 5 \times (h - \frac{h}{3})|$$ $$= \frac{1}{2} |0 - \frac{4h}{3} + 5 \times \frac{2h}{3}| = \frac{1}{2} | - \frac{4h}{3} + \frac{10h}{3}| = \frac{1}{2} \times \frac{6h}{3} = \frac{1}{2} \times 2h = h$$ 7. **Find ratio:** $$\frac{S_{AKC}}{S_{ABC}} = \frac{h}{\frac{5h}{2}} = \frac{h}{1} \times \frac{2}{5h} = \frac{2}{5}$$ **Final answers:** - $\boxed{\frac{S_{AKC}}{S_{ABC}} = \frac{2}{5}}$ Since $S_{AABC}$ is unclear or possibly a typo, we only provide the ratio for $S_{AKC}$.