1. Stating the problem: Simplify the expression $\frac{4}{7} + \frac{2}{8} \times \frac{3}{4}$.\n\n2. Recall the order of operations: multiplication comes before addition. So first, multiply $\frac{2}{8}$ and $\frac{3}{4}$.\n\n3. Multiply the numerators and denominators:\n$$\frac{2}{8} \times \frac{3}{4} = \frac{2 \times 3}{8 \times 4} = \frac{6}{32}.$$\n\n4. Simplify $\frac{6}{32}$ by dividing numerator and denominator by their greatest common divisor 2:\n$$\frac{\cancel{6}^{3}}{\cancel{32}^{16}} = \frac{3}{16}.$$\n\n5. Now add $\frac{4}{7}$ and $\frac{3}{16}$. Find a common denominator, which is $7 \times 16 = 112$.\n\n6. Convert each fraction:\n$$\frac{4}{7} = \frac{4 \times 16}{7 \times 16} = \frac{64}{112}, \quad \frac{3}{16} = \frac{3 \times 7}{16 \times 7} = \frac{21}{112}.$$\n\n7. Add the fractions:\n$$\frac{64}{112} + \frac{21}{112} = \frac{64 + 21}{112} = \frac{85}{112}.$$\n\n8. The fraction $\frac{85}{112}$ cannot be simplified further since 85 and 112 share no common divisors other than 1.\n\nFinal answer: $\boxed{\frac{85}{112}}$.