The value of `(4 1/7 - 2 1/4)/(3 1/2 + 1 1/7) div 1/(2+1/(2 + 1/(5 - 1/5)))`.

Options

A. `7//29`
B. `5//6`
C. `1`
D. `9//13`

Answer

To solve the expression \((4 \frac{1}{7} - 2 \frac{1}{4}) / (3 \frac{1}{2} + 1 \frac{1}{7}) \div \frac{1}{(2 + \frac{1}{(2 + \frac{1}{(5 - \frac{1}{5})})})}\), we will break it down into manageable steps.

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(4 \frac{1}{7}\) to an improper fraction:
\[ 4 \frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{28 + 1}{7} = \frac{29}{7} \]
2. Convert \(2 \frac{1}{4}\) to an improper fraction:
\[ 2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4} \]
3. Convert \(3 \frac{1}{2}\) to an improper fraction:
\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]
4. Convert \(1 \frac{1}{7}\) to an improper fraction:
\[ 1 \frac{1}{7} = \frac{1 \times 7 + 1}{7} = \frac{7 + 1}{7} = \frac{8}{7} \]

### Step 2: Substitute the Improper Fractions into the Expression
Now the expression becomes:
\[ \left(\frac{29}{7} - \frac{9}{4}\right) / \left(\frac{7}{2} + \frac{8}{7}\right) \div \frac{1}{(2 + \frac{1}{(2 + \frac{1}{(5 - \frac{1}{5})})})} \]

### Step 3: Calculate the Numerator
1. Find a common denominator for \(\frac{29}{7}\) and \(\frac{9}{4}\):
- The least common multiple of 7 and 4 is 28.
- Convert \(\frac{29}{7}\) and \(\frac{9}{4}\):
\[ \frac{29}{7} = \frac{29 \times 4}{28} = \frac{116}{28}, \quad \frac{9}{4} = \frac{9 \times 7}{28} = \frac{63}{28} \]
2. Subtract the fractions:
\[ \frac{116}{28} - \frac{63}{28} = \frac{116 - 63}{28} = \frac{53}{28} \]

### Step 4: Calculate the Denominator
1. Find a common denominator for \(\frac{7}{2}\) and \(\frac{8}{7}\):
- The least common multiple of 2 and 7 is 14.
- Convert \(\frac{7}{2}\) and \(\frac{8}{7}\):
\[ \frac{7}{2} = \frac{7 \times 7}{14} = \frac{49}{14}, \quad \frac{8}{7} = \frac{8 \times 2}{14} = \frac{16}{14} \]
2. Add the fractions:
\[ \frac{49}{14} + \frac{16}{14} = \frac{49 + 16}{14} = \frac{65}{14} \]

### Step 5: Divide the Two Results
Now we have:
\[ \frac{\frac{53}{28}}{\frac{65}{14}} = \frac{53}{28} \times \frac{14}{65} = \frac{53 \times 14}{28 \times 65} = \frac{742}{1820} \]

### Step 6: Simplify the Fraction
1. Find the greatest common divisor (GCD) of 742 and 1820:
- The GCD is 2.
2. Divide both the numerator and the denominator by the GCD:
\[ \frac{742 \div 2}{1820 \div 2} = \frac{371}{910} \]

### Step 7: Calculate the Division with the Last Part
Now we need to calculate:
\[ \frac{1}{(2 + \frac{1}{(2 + \frac{1}{(5 - \frac{1}{5})})})} \]
1. Calculate \(5 - \frac{1}{5} = \frac{25 - 1}{5} = \frac{24}{5}\).
2. Calculate \(2 + \frac{24}{5} = \frac{10 + 24}{5} = \frac{34}{5}\).
3. Calculate \(2 + \frac{34}{5} = \frac{10 + 34}{5} = \frac{44}{5}\).
4. Finally, the reciprocal is \(\frac{5}{44}\).

### Step 8: Final Calculation
Now we need to multiply:
\[ \frac{371}{910} \div \frac{5}{44} = \frac{371}{910} \times \frac{44}{5} = \frac{371 \times 44}{910 \times 5} = \frac{16324}{4550} \]

### Step 9: Simplify the Final Fraction
1. The GCD of 16324 and 4550 is 2.
2. Divide both by the GCD:
\[ \frac{16324 \div 2}{4550 \div 2} = \frac{8162}{2275} \]

### Final Answer
The final simplified answer is:
\[ \frac{8162}{2275} \]