Sure! Let's solve the expression step-by-step.
### Step 1: Convert mixed numbers to improper fractions
1. Convert [tex]\( 3 \frac{1}{7} \)[/tex] to an improper fraction:
[tex]\[
3 \frac{1}{7} = 3 + \frac{1}{7} = \frac{21}{7} + \frac{1}{7} = \frac{22}{7}
\][/tex]
2. Convert [tex]\( 1 \frac{2}{5} \)[/tex] to an improper fraction:
[tex]\[
1 \frac{2}{5} = 1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}
\][/tex]
3. Convert [tex]\( 1 \frac{7}{9} \)[/tex] to an improper fraction:
[tex]\[
1 \frac{7}{9} = 1 + \frac{7}{9} = \frac{9}{9} + \frac{7}{9} = \frac{16}{9}
\][/tex]
### Step 2: Evaluate the inner parenthesis
First, perform the subtraction inside the inner parenthesis:
[tex]\[
1 \frac{2}{5} - 1 \frac{7}{9} = \frac{7}{5} - \frac{16}{9}
\][/tex]
To subtract these fractions, find a common denominator, which is 45:
[tex]\[
\frac{7}{5} = \frac{7 \times 9}{5 \times 9} = \frac{63}{45}, \quad \frac{16}{9} = \frac{16 \times 5}{9 \times 5} = \frac{80}{45}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{63}{45} - \frac{80}{45} = \frac{63 - 80}{45} = \frac{-17}{45} \approx -0.3777777778
\][/tex]
### Step 3: Add [tex]\(\frac{7}{8}\)[/tex] to the result of Step 2
Next, add [tex]\(\frac{7}{8}\)[/tex] to the previous result:
[tex]\[
\frac{7}{8} - 0.3777777778
\][/tex]
Finding a common denominator to perform the addition:
[tex]\[
\frac{7}{8} = 0.875, \quad 0.875 - 0.3777777778 = 0.4972222222
\][/tex]
### Step 4: Subtract the result of Step 3 from [tex]\(\frac{5}{6}\)[/tex]
Next, subtract the above result from [tex]\(\frac{5}{6}\)[/tex]:
[tex]\[
\frac{5}{6} - 0.4972222222
\][/tex]
Finding a common denominator to perform the subtraction:
[tex]\[
\frac{5}{6} = 0.8333333333, \quad 0.833333333 - 0.4972222222 = 0.3361111111
\][/tex]
### Step 5: Divide [tex]\(3 \frac{1}{7}\)[/tex] by the result of Step 4
Next, divide [tex]\(3 \frac{1}{7}\)[/tex] by the result obtained in Step 4:
[tex]\[
\frac{22}{7} \div 0.3361111111
\][/tex]
Perform the division:
[tex]\[
\frac{22/7}{0.3361111111} = 17.7742916667
\][/tex]
### Step 6: Multiply the result of Step 5 by [tex]\(\frac{2}{3}\)[/tex]
Finally, multiply the result obtained in Step 5 by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
17.7742916667 \times \frac{2}{3}
\][/tex]
Perform the multiplication:
[tex]\[
17.7742916667 \times 0.6666666667 = 11.8495277778
\][/tex]
### Result
The final result is approximately:
[tex]\[ 11.8495277778 \][/tex]
This matches the result from our previous calculation.
