To solve the given mathematical problem, we'll go through it step by step, making sure to address each operation carefully.
### Step-by-Step Solution:
1. Convert the mixed number to an improper fraction:
We start with the mixed number [tex]\(8 \frac{5}{9}\)[/tex].
[tex]\[ 8 \frac{5}{9} = 8 + \frac{5}{9} \][/tex]
[tex]\[ = \frac{72}{9} + \frac{5}{9} \][/tex]
[tex]\[ = \frac{72 + 5}{9} \][/tex]
[tex]\[ = \frac{77}{9} \][/tex]
2. Division Step:
We need to divide 1 by [tex]\(\frac{77}{9}\)[/tex].
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[ 1 \div \frac{77}{9} = 1 \times \frac{9}{77} \][/tex]
[tex]\[ = \frac{9}{77} \][/tex]
So, the first intermediate result is:
[tex]\[ \frac{9}{77} \approx 0.11688311688311688 \][/tex]
3. First Multiplication Step:
Now, we need to multiply the result from the division step by [tex]\(\frac{4}{13}\)[/tex].
[tex]\[ \frac{9}{77} \times \frac{4}{13} = \frac{9 \times 4}{77 \times 13} \][/tex]
[tex]\[ = \frac{36}{1001} \][/tex]
So, the second intermediate result is:
[tex]\[ \frac{36}{1001} \approx 0.03596403596403597 \][/tex]
4. Second Multiplication Step:
Finally, we need to multiply the result from the first multiplication step by [tex]\(\frac{13}{39}\)[/tex].
[tex]\[ \frac{36}{1001} \times \frac{13}{39} = \frac{36 \times 13}{1001 \times 39} \][/tex]
[tex]\[ = \frac{468}{39039} \][/tex]
Simplify the fraction if possible:
[tex]\[ \frac{468}{39039} \approx 0.011988011988011988 \][/tex]
### Summary:
1. The result of [tex]\(1 \div 8 \frac{5}{9}\)[/tex] is:
[tex]\[ \approx 0.11688311688311688 \][/tex]
2. The result after multiplying by [tex]\(\frac{4}{13}\)[/tex] is:
[tex]\[ \approx 0.03596403596403597 \][/tex]
3. The final result after multiplying by [tex]\(\frac{13}{39}\)[/tex] is:
[tex]\[ \approx 0.011988011988011988 \][/tex]
Thus, the final result for the complete operation is:
[tex]\[ \approx 0.011988011988011988 \][/tex]
