1. **State the problem:** Calculate $\left(1 \frac{3}{4} + 2 \frac{1}{3}\right) \div \frac{5}{6}$.\n\n2. **Convert mixed numbers to improper fractions:**\n$1 \frac{3}{4} = \frac{7}{4}$ and $2 \frac{1}{3} = \frac{7}{3}$.\n\n3. **Add the fractions:**\n$$\frac{7}{4} + \frac{7}{3} = \frac{7 \times 3}{4 \times 3} + \frac{7 \times 4}{3 \times 4} = \frac{21}{12} + \frac{28}{12} = \frac{49}{12}.$$\n\n4. **Divide by $\frac{5}{6}$:**\nDividing by a fraction is the same as multiplying by its reciprocal. So,\n$$\frac{49}{12} \div \frac{5}{6} = \frac{49}{12} \times \frac{6}{5}.$$\n\n5. **Multiply the fractions:**\n$$\frac{49}{12} \times \frac{6}{5} = \frac{49 \times 6}{12 \times 5} = \frac{294}{60}.$$\n\n6. **Simplify the fraction:**\nFind the greatest common divisor (GCD) of 294 and 60, which is 6.\n$$\frac{\cancel{294}^{49} \times 6}{\cancel{60}^{10} \times 6} = \frac{49}{10}.$$\n\n7. **Convert to mixed number:**\n$$\frac{49}{10} = 4 \frac{9}{10}.$$\n\n**Final answer:** $4 \frac{9}{10}$.