To solve the given expression step by step, we will follow the order of operations (BODMAS/BIDMAS rules) which stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
### Step 1: Simplify the first bracket
The first part of the expression is:
\[
\left(\frac{5}{7} \text{ of } 1 \frac{3}{8} \text{ of } \frac{6}{7}\right)
\]
First, convert \(1 \frac{3}{8}\) into an improper fraction:
\[
1 \frac{3}{8} = \frac{8 + 3}{8} = \frac{11}{8}
\]
Now, replace "of" with multiplication:
\[
\frac{5}{7} \times \frac{11}{8} \times \frac{6}{7}
\]
### Step 2: Calculate the multiplication
Now, multiply the fractions:
\[
\frac{5 \times 11 \times 6}{7 \times 8 \times 7} = \frac{330}{392}
\]
### Step 3: Simplify the fraction
Next, simplify \(\frac{330}{392}\):
The GCD of 330 and 392 is 2.
\[
\frac{330 \div 2}{392 \div 2} = \frac{165}{196}
\]
### Step 4: Simplify the second bracket
Now, simplify the second part of the expression:
\[
1 - \left(\frac{1}{7} \times \left(\frac{5}{12} + \frac{1}{3}\right)\right)
\]
First, calculate \(\frac{5}{12} + \frac{1}{3}\):
Convert \(\frac{1}{3}\) to have a common denominator of 12:
\[
\frac{1}{3} = \frac{4}{12}
\]
So,
\[
\frac{5}{12} + \frac{4}{12} = \frac{9}{12} = \frac{3}{4}
\]
Now, substitute back:
\[
1 - \left(\frac{1}{7} \times \frac{3}{4}\right) = 1 - \frac{3}{28}
\]
Convert 1 to a fraction with a denominator of 28:
\[
1 = \frac{28}{28}
\]
So,
\[
\frac{28}{28} - \frac{3}{28} = \frac{25}{28}
\]
### Step 5: Calculate the last part
Now, calculate:
\[
\frac{\frac{1}{7} - \frac{1}{9}}{\frac{1}{7} + \frac{1}{9}}
\]
Find a common denominator for both the numerator and denominator:
For the numerator:
\[
\frac{1}{7} - \frac{1}{9} = \frac{9 - 7}{63} = \frac{2}{63}
\]
For the denominator:
\[
\frac{1}{7} + \frac{1}{9} = \frac{9 + 7}{63} = \frac{16}{63}
\]
So, we have:
\[
\frac{\frac{2}{63}}{\frac{16}{63}} = \frac{2}{16} = \frac{1}{8}
\]
### Step 6: Combine everything
Now, we can combine everything:
\[
\frac{165}{196} \div \left(\frac{25}{28} \times \frac{1}{8}\right)
\]
Calculate the multiplication in the denominator:
\[
\frac{25}{28} \times \frac{1}{8} = \frac{25}{224}
\]
Now, perform the division:
\[
\frac{165}{196} \div \frac{25}{224} = \frac{165}{196} \times \frac{224}{25}
\]
This simplifies to:
\[
\frac{165 \times 224}{196 \times 25}
\]
### Step 7: Simplify the final fraction
Now, we can simplify:
1. \(165 = 5 \times 33\)
2. \(196 = 7 \times 28\)
3. \(224 = 7 \times 32\)
4. \(25 = 5 \times 5\)
Thus, we can cancel the common factors:
\[
\frac{33 \times 32}{28 \times 5} = \frac{1056}{140}
\]
Now, simplify \(\frac{1056}{140}\):
The GCD is 28:
\[
\frac{1056 \div 28}{140 \div 28} = \frac{38}{5}
\]
### Final Answer
The value of the expression is:
\[
\frac{38}{5}
\]
