1. The problem is to find the value of $A$ in the equation $4 + 8 : 2 \times 3 = 10 : 2 \times A + 1$. 2. We use the order of operations (PEMDAS/BODMAS): division and multiplication are performed from left to right before addition. 3. Simplify the left side: $$4 + \frac{8}{2} \times 3 = 4 + 4 \times 3 = 4 + 12 = 16$$ 4. Simplify the right side: $$\frac{10}{2} \times A + 1 = 5A + 1$$ 5. Set the two sides equal: $$16 = 5A + 1$$ 6. Solve for $A$: $$5A = 16 - 1 = 15$$ $$A = \frac{15}{5} = 3$$ Therefore, the value of $A$ is 3.