1. **State the problem:** We are given points A, B, C, and D on segment AC such that $AD : DC = 2 : 3$. We know $\overrightarrow{AB} = 10a$ and $\overrightarrow{DB} = 2a - 4b$. We need to express $\overrightarrow{CB}$ in terms of $a$ and $b$ and simplify fully. 2. **Understand the vectors and ratio:** Since $D$ lies on $AC$ with ratio $AD : DC = 2 : 3$, point $D$ divides $AC$ in the ratio 2 to 3. 3. **Express $\overrightarrow{AC}$ in terms of $a$:** We know $\overrightarrow{AB} = 10a$ and $\overrightarrow{DB} = 2a - 4b$. We want $\overrightarrow{CB}$. 4. **Express $\overrightarrow{CB}$ using $\overrightarrow{AB}$ and $\overrightarrow{AC}$:** $$\overrightarrow{CB} = \overrightarrow{CA} + \overrightarrow{AB} = -\overrightarrow{AC} + \overrightarrow{AB}$$ 5. **Find $\overrightarrow{AC}$ using the ratio:** Since $D$ divides $AC$ in ratio 2:3, the position vector of $D$ relative to $A$ is $$\overrightarrow{AD} = \frac{2}{2+3} \overrightarrow{AC} = \frac{2}{5} \overrightarrow{AC}$$ 6. **Express $\overrightarrow{DB}$ in terms of $\overrightarrow{AB}$ and $\overrightarrow{AD}$:** $$\overrightarrow{DB} = \overrightarrow{AB} - \overrightarrow{AD}$$ Substitute known vectors: $$2a - 4b = 10a - \overrightarrow{AD}$$ Rearranged: $$\overrightarrow{AD} = 10a - (2a - 4b) = 8a + 4b$$ 7. **Recall from step 5 that $\overrightarrow{AD} = \frac{2}{5} \overrightarrow{AC}$, so:** $$\frac{2}{5} \overrightarrow{AC} = 8a + 4b$$ Multiply both sides by $\frac{5}{2}$: $$\overrightarrow{AC} = \frac{5}{2} (8a + 4b) = 20a + 10b$$ 8. **Now find $\overrightarrow{CB}$ from step 4:** $$\overrightarrow{CB} = -\overrightarrow{AC} + \overrightarrow{AB} = -(20a + 10b) + 10a = -20a - 10b + 10a = -10a - 10b$$ 9. **Final simplified expression:** $$\boxed{\overrightarrow{CB} = -10a - 10b}$$