Let's resolve each of the given mixed number division problems step by step:
### 1. [tex]\( 17 \frac{11}{9} \div \frac{3}{5} \)[/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 17 \frac{11}{9} = 17 + \frac{11}{9} = \frac{17 \cdot 9 + 11}{9} = \frac{153 + 11}{9} = \frac{164}{9} \][/tex]
Next, perform the division by multiplying by the reciprocal of [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[ \frac{164}{9} \div \frac{3}{5} = \frac{164}{9} \cdot \frac{5}{3} = \frac{164 \cdot 5}{9 \cdot 3} = \frac{820}{27} \approx 30.37037037037037 \][/tex]
### 2. [tex]\( 2 \frac{6}{9} \div \frac{3}{4} \)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 2 \frac{6}{9} = 2 + \frac{6}{9} = \frac{2 \cdot 9 + 6}{9} = \frac{18 + 6}{9} = \frac{24}{9} \][/tex]
Next, perform the division:
[tex]\[ \frac{24}{9} \div \frac{3}{4} = \frac{24}{9} \cdot \frac{4}{3} = \frac{24 \cdot 4}{9 \cdot 3} = \frac{96}{27} = \frac{32}{9} \approx 3.5555555555555554 \][/tex]
### 3. [tex]\( 3 \frac{8}{9} \div \frac{6}{3} \)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 3 \frac{8}{9} = 3 + \frac{8}{9} = \frac{3 \cdot 9 + 8}{9} = \frac{27 + 8}{9} = \frac{35}{9} \][/tex]
Next, perform the division:
[tex]\[ \frac{35}{9} \div \frac{6}{3} = \frac{35}{9} \cdot \frac{3}{6} = \frac{35 \cdot 3}{9 \cdot 6} = \frac{105}{54} = \frac{35}{18} \approx 1.9444444444444444 \][/tex]
### 4. [tex]\( 4 \frac{5}{12} \div \frac{10}{7} \)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 4 \frac{5}{12} = 4 + \frac{5}{12} = \frac{4 \cdot 12 + 5}{12} = \frac{48 + 5}{12} = \frac{53}{12} \][/tex]
Next, perform the division:
[tex]\[ \frac{53}{12} \div \frac{10}{7} = \frac{53}{12} \cdot \frac{7}{10} = \frac{53 \cdot 7}{12 \cdot 10} = \frac{371}{120} \approx 3.091666666666667 \][/tex]
### 5. [tex]\( 5 \frac{9}{14} \div \frac{3}{7} \)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 5 \frac{9}{14} = 5 + \frac{9}{14} = \frac{5 \cdot 14 + 9}{14} = \frac{70 + 9}{14} = \frac{79}{14} \][/tex]
Next, perform the division:
[tex]\[ \frac{79}{14} \div \frac{3}{7} = \frac{79}{14} \cdot \frac{7}{3} = \frac{79 \cdot 7}{14 \cdot 3} = \frac{553}{42} \approx 13.166666666666668 \][/tex]
### 6. [tex]\( 6 \frac{2}{15} \div \frac{8}{5} \)[/tex]
Convert the mixed number to an improper fraction:
[tex]\[ 6 \frac{2}{15} = 6 + \frac{2}{15} = \frac{6 \cdot 15 + 2}{15} = \frac{90 + 2}{15} = \frac{92}{15} \][/tex]
Next, perform the division:
[tex]\[ \frac{92}{15} \div \frac{8}{5} = \frac{92}{15} \cdot \frac{5}{8} = \frac{92 \cdot 5}{15 \cdot 8} = \frac{460}{120} = \frac{23}{6} \approx 3.8333333333333335 \][/tex]
These are the results of the given division problems.
