6. $$ \quad S=13 \frac{4}{9}-\left(8 \frac{1}{3}+1 \frac{6}{7}\right) $$ 7. $$ \quad K=3 \frac{1}{5}+\left(20 \frac{2}{15}-8 \frac{1}{3}\right) $$ 8. $$ \quad I=\left(15 \frac{1}{2}+15 \frac{3}{6}\right)-15 \frac{3}{10} $$ 9. $$ \quad N=\left(21 \frac{3}{6}+20 \frac{7}{9}\right)-10 \frac{1}{6} $$

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Sure, let's solve each of the given problems step by step.

---

### **Problem 6**
$$
S = 13 \frac{4}{9} - \left(8 \frac{1}{3} + 1 \frac{6}{7}\right)
$$

**Step 1: Add the numbers inside the parentheses.**

Convert mixed numbers to improper fractions:
$$
8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{25}{3}
$$
$$
1 \frac{6}{7} = \frac{1 \times 7 + 6}{7} = \frac{13}{7}
$$

Add the two fractions:
$$
\frac{25}{3} + \frac{13}{7}
$$

Find a common denominator (21):
$$
\frac{25}{3} = \frac{175}{21}, \quad \frac{13}{7} = \frac{39}{21}
$$
$$
\frac{175}{21} + \frac{39}{21} = \frac{214}{21} = 10 \frac{4}{21}
$$

**Step 2: Subtract the result from $$13 \frac{4}{9}$$.**

Convert $$13 \frac{4}{9}$$ to an improper fraction:
$$
13 \frac{4}{9} = \frac{13 \times 9 + 4}{9} = \frac{121}{9}
$$

Subtract $$10 \frac{4}{21}$$ from $$13 \frac{4}{9}$$:
$$
\frac{121}{9} - \frac{214}{21}
$$

Find a common denominator (63):
$$
\frac{121}{9} = \frac{847}{63}, \quad \frac{214}{21} = \frac{642}{63}
$$
$$
\frac{847}{63} - \frac{642}{63} = \frac{205}{63} = 3 \frac{16}{63}
$$

**Final Answer:**
$$
S = 3 \frac{16}{63}
$$

---

### **Problem 7**
$$
K = 3 \frac{1}{5} + \left(20 \frac{2}{15} - 8 \frac{1}{3}\right)
$$

**Step 1: Subtract the numbers inside the parentheses.**

Convert mixed numbers to improper fractions:
$$
20 \frac{2}{15} = \frac{20 \times 15 + 2}{15} = \frac{302}{15}
$$
$$
8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{25}{3}
$$

Subtract the fractions:
$$
\frac{302}{15} - \frac{25}{3}
$$

Convert to a common denominator (15):
$$
\frac{25}{3} = \frac{125}{15}
$$
$$
\frac{302}{15} - \frac{125}{15} = \frac{177}{15} = 11 \frac{12}{15} = 11 \frac{4}{5}
$$

**Step 2: Add the result to $$3 \frac{1}{5}$$.**

Convert mixed numbers to improper fractions:
$$
3 \frac{1}{5} = \frac{16}{5}, \quad 11 \frac{4}{5} = \frac{59}{5}
$$

Add the fractions:
$$
\frac{16}{5} + \frac{59}{5} = \frac{75}{5} = 15
$$

**Final Answer:**
$$
K = 15
$$

---

### **Problem 8**
$$
I = \left(15 \frac{1}{2} + 15 \frac{3}{6}\right) - 15 \frac{3}{10}
$$

**Step 1: Simplify and add the numbers inside the parentheses.**

Notice that $$15 \frac{3}{6}$$ simplifies to $$15 \frac{1}{2}$$:
$$
15 \frac{1}{2} + 15 \frac{1}{2} = 31 \times \frac{1}{2} = 32 \times \frac{1}{2} = 30
$$

**Step 2: Subtract $$15 \frac{3}{10}$$ from the result.**

Convert to improper fractions with a common denominator (10):
$$
30 = \frac{300}{10}, \quad 15 \frac{3}{10} = \frac{153}{10}
$$
$$
\frac{300}{10} - \frac{153}{10} = \frac{147}{10} = 14 \frac{7}{10}
$$

**Final Answer:**
$$
I = 14 \frac{7}{10}
$$

---

### **Problem 9**
$$
N = \left(21 \frac{3}{6} + 20 \frac{7}{9}\right) - 10 \frac{1}{6}
$$

**Step 1: Simplify and add the numbers inside the parentheses.**

Simplify $$21 \frac{3}{6}$$ to $$21 \frac{1}{2}$$:
$$
21 \frac{1}{2} + 20 \frac{7}{9}
$$

Convert to improper fractions:
$$
21 \frac{1}{2} = \frac{43}{2}, \quad 20 \frac{7}{9} = \frac{187}{9}
$$

Add the fractions:
$$
\frac{43}{2} + \frac{187}{9}
$$

Find a common denominator (18):
$$
\frac{43}{2} = \frac{387}{18}, \quad \frac{187}{9} = \frac{374}{18}
$$
$$
\frac{387}{18} + \frac{374}{18} = \frac{761}{18} = 42 \frac{5}{18}
$$

**Step 2: Subtract $$10 \frac{1}{6}$$.**

Convert to improper fractions with a common denominator (18):
$$
42 \frac{5}{18} = \frac{761}{18}, \quad 10 \frac{1}{6} = \frac{61}{6} = \frac{183}{18}
$$
$$
\frac{761}{18} - \frac{183}{18} = \frac{578}{18} = 32 \frac{2}{18} = 32 \frac{1}{9}
$$

**Final Answer:**
$$
N = 32 \frac{1}{9}
$$

---
Solutions: 1. $$ S = 3 \frac{16}{63} $$ 2. $$ K = 15 $$ 3. $$ I = 14 \frac{7}{10} $$ 4. $$ N = 32 \frac{1}{9} $$

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