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Let's solve the problems one by one.

### Question 4:
Multiply \( \frac{7}{9} \) and \( 4 \frac{7}{8} \).

First, convert the mixed number \( 4 \frac{7}{8} \) into an improper fraction:
\[
4 \frac{7}{8} = \frac{4 \times 8 + 7}{8} = \frac{39}{8}
\]

Now, multiply:
\[
\frac{7}{9} \times \frac{39}{8} = \frac{7 \times 39}{9 \times 8} = \frac{273}{72}
\]

Simplify the fraction:
\[
\frac{273}{72} = \frac{273 \div 9}{72 \div 9} = \frac{39}{8}
\]

So, the result is:
\[
\frac{39}{8} \text{ or } 4 \frac{7}{8}
\]

### Question 5:
Divide \( \frac{1}{2} \) by \( \frac{1}{6} \).

When dividing fractions, multiply by the reciprocal of the second fraction:
\[
\frac{1}{2} \div \frac{1}{6} = \frac{1}{2} \times \frac{6}{1} = \frac{6}{2} = 3
\]

So, the result is:
\[
3
\]

### Question 6:
Divide \( 5 \frac{1}{2} \) by \( 4 \frac{4}{7} \).

First, convert both mixed numbers into improper fractions:
\[
5 \frac{1}{2} = \frac{11}{2}, \quad 4 \frac{4}{7} = \frac{32}{7}
\]

Now, divide by multiplying by the reciprocal of the second fraction:
\[
\frac{11}{2} \div \frac{32}{7} = \frac{11}{2} \times \frac{7}{32} = \frac{11 \times 7}{2 \times 32} = \frac{77}{64}
\]

So, the result is:
\[
\frac{77}{64} \text{ or } 1 \frac{13}{64}
\]

### Question 7:
Simplify:
\[
9 \frac{1}{12} - \left( 2 \frac{2}{3} \right)^2 + 3 \frac{5}{8} \div 1 \frac{3}{4}
\]

1. Convert all mixed numbers to improper fractions:
   \[
   9 \frac{1}{12} = \frac{109}{12}, \quad 2 \frac{2}{3} = \frac{8}{3}, \quad 3 \frac{5}{8} = \frac{29}{8}, \quad 1 \frac{3}{4} = \frac{7}{4}
   \]

2. Square \( 2 \frac{2}{3} \):
   \[
   \left( \frac{8}{3} \right)^2 = \frac{64}{9}
   \]

3. Divide \( 3 \frac{5}{8} \) by \( 1 \frac{3}{4} \):
   \[
   \frac{29}{8} \div \frac{7}{4} = \frac{29}{8} \times \frac{4}{7} = \frac{116}{56} = \frac{29}{14}
   \]

4. Now, substitute back into the original expression:
   \[
   \frac{109}{12} - \frac{64}{9} + \frac{29}{14}
   \]

5. Find the least common denominator (LCD) for 12, 9, and 14, which is 252. Convert all fractions:
   \[
   \frac{109}{12} = \frac{2289}{252}, \quad \frac{64}{9} = \frac{1792}{252}, \quad \frac{29}{14} = \frac{522}{252}
   \]

6. Now, simplify the expression:
   \[
   \frac{2289}{252} - \frac{1792}{252} + \frac{522}{252} = \frac{1019}{252}
   \]

So, the simplified result is:
\[
\frac{1019}{252} \text{ or approximately } 4.04
\]

---

Would you like further clarification or details on any of these steps?

Here are some related questions:
1. How do you multiply and divide mixed numbers?
2. How do you simplify improper fractions to mixed numbers?
3. What are the rules for dividing fractions?
4. How do you find the least common denominator in fraction addition and subtraction?
5. What are the steps for squaring a mixed number?

**Tip**: Always convert mixed numbers into improper fractions before performing multiplication or division.

Solve various arithmetic operations with fractions and mixed numbers.

Explore detailed solutions to problems involving multiplication, division, and simplification of fractions and mixed numbers. Learn how to convert mixed numbers to improper fractions, find the least common denominator, and simplify complex expressions.

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