Let's simplify the expression:

$\frac{5}{6} \div \frac{2}{1} \quad + \quad \frac{5}{12} + \frac{2}{3}$

Step 1: Simplifying the division

When dividing by a fraction, multiply by its reciprocal. So, we need to multiply $\frac{5}{6}$ by the reciprocal of $\frac{2}{1}$, which is $\frac{1}{2}$:

$\frac{5}{6} \div \frac{2}{1} = \frac{5}{6} \times \frac{1}{2} = \frac{5 \times 1}{6 \times 2} = \frac{5}{12}$

Step 2: Simplifying the addition

Now, add $\frac{5}{12}$ (from the division result) to $\frac{5}{12} + \frac{2}{3}$.

To add fractions, we need a common denominator. The least common denominator between 12 and 3 is 12. Convert $\frac{2}{3}$ into an equivalent fraction with denominator 12:

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

Now, add the fractions:

$\frac{5}{12} + \frac{8}{12} = \frac{5 + 8}{12} = \frac{13}{12}$

Final Answer:

$\frac{13}{12}$

This is an improper fraction, and it can also be written as a mixed number:

$1 \frac{1}{12}$

Would you like more details or have any questions on this? Here are a few related questions:

1. How do you simplify a fraction when dividing by a fraction?
2. How do you find the least common denominator for adding fractions?
3. What is the process to convert improper fractions to mixed numbers?
4. How do you handle addition or subtraction of fractions with different denominators?
5. What are some strategies for simplifying complex fractions?

Tip: Always check for the least common denominator (LCD) when adding or subtracting fractions to make the process smoother!