1. **State the problem:** Calculate the value of $\frac{2}{3} + \frac{5}{6} \times \frac{3}{4}$.\n\n2. **Recall the order of operations:** Multiplication is performed before addition. So first, multiply $\frac{5}{6}$ by $\frac{3}{4}$.\n\n3. **Multiply the fractions:**\n$$\frac{5}{6} \times \frac{3}{4} = \frac{5 \times 3}{6 \times 4} = \frac{15}{24}.$$\n\n4. **Simplify the fraction $\frac{15}{24}$:**\nBoth numerator and denominator can be divided by 3:\n$$\frac{\cancel{15}^{5}}{\cancel{24}^{8}} = \frac{5}{8}.$$\n\n5. **Add the fractions $\frac{2}{3}$ and $\frac{5}{8}$:**\nFind a common denominator, which is 24 (the least common multiple of 3 and 8).\nConvert each fraction:\n$$\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}, \quad \frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}.$$\n\n6. **Add the numerators:**\n$$\frac{16}{24} + \frac{15}{24} = \frac{16 + 15}{24} = \frac{31}{24}.$$\n\n7. **Final answer:** The sum is an improper fraction $\frac{31}{24}$, which can also be written as the mixed number $1 \frac{7}{24}$.