Simplify : `(4(1)/(7)-2(1)/(4))/(3+(1)/(2)+1(1)/(7))+(1)/(2+(1)/(2+(1)/(5-(1)/(5))))`

Options

A. 0
B. `-1`
C. `3(1)/(24)`
D. `53/65`

Answer

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

### Step 1: Simplify the first part \((4\frac{1}{7} - 2\frac{1}{4})\)

1. Convert the mixed numbers to improper fractions:
- \(4\frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{29}{7}\)
- \(2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}\)

2. Now we have:
\[ \frac{29}{7} - \frac{9}{4} \]

3. Find the least common multiple (LCM) of the denominators (7 and 4), which is 28.

4. Convert both fractions:
- \(\frac{29}{7} = \frac{29 \times 4}{7 \times 4} = \frac{116}{28}\)
- \(\frac{9}{4} = \frac{9 \times 7}{4 \times 7} = \frac{63}{28}\)

5. Now subtract:
\[ \frac{116}{28} - \frac{63}{28} = \frac{116 - 63}{28} = \frac{53}{28} \]

### Step 2: Simplify the denominator \((3 + \frac{1}{2} + 1\frac{1}{7})\)

1. Convert \(1\frac{1}{7}\) to an improper fraction:
- \(1\frac{1}{7} = \frac{8}{7}\)

2. Now we have:
\[ 3 + \frac{1}{2} + \frac{8}{7} \]

3. Convert 3 to a fraction:
- \(3 = \frac{3 \times 14}{14} = \frac{42}{14}\)

4. Find the LCM of the denominators (2 and 7), which is 14.

5. Convert the fractions:
- \(\frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}\)
- \(\frac{8}{7} = \frac{8 \times 2}{7 \times 2} = \frac{16}{14}\)

6. Now add:
\[ \frac{42}{14} + \frac{7}{14} + \frac{16}{14} = \frac{42 + 7 + 16}{14} = \frac{65}{14} \]

### Step 3: Combine the results

Now we have:
\[ \frac{\frac{53}{28}}{\frac{65}{14}} = \frac{53}{28} \times \frac{14}{65} = \frac{53 \times 14}{28 \times 65} \]

### Step 4: Simplify the fraction

1. Simplify \(\frac{14}{28} = \frac{1}{2}\):
\[ \frac{53 \times 1}{2 \times 65} = \frac{53}{130} \]

### Step 5: Simplify the second part \(\frac{1}{2 + \frac{1}{2 + \frac{1}{5 - \frac{1}{5}}}}\)

1. Simplify \(5 - \frac{1}{5} = \frac{25 - 1}{5} = \frac{24}{5}\).

2. Now simplify:
\[ 2 + \frac{1}{2 + \frac{24}{5}} \]

3. Convert \(2\) to a fraction:
- \(2 = \frac{10}{5}\)

4. Combine:
\[ \frac{10}{5} + \frac{24}{5} = \frac{34}{5} \]

5. Now simplify:
\[ 2 + \frac{1}{\frac{34}{5}} = 2 + \frac{5}{34} \]

6. Convert \(2\) to a fraction:
- \(2 = \frac{68}{34}\)

7. Combine:
\[ \frac{68}{34} + \frac{5}{34} = \frac{73}{34} \]

8. Now we have:
\[ \frac{1}{\frac{73}{34}} = \frac{34}{73} \]

### Step 6: Combine both parts

Now we have:
\[ \frac{53}{130} + \frac{34}{73} \]

1. Find the LCM of 130 and 73, which is \(9490\).

2. Convert both fractions:
- \(\frac{53}{130} = \frac{53 \times 73}{9490} = \frac{3869}{9490}\)
- \(\frac{34}{73} = \frac{34 \times 130}{9490} = \frac{4420}{9490}\)

3. Add:
\[ \frac{3869 + 4420}{9490} = \frac{8289}{9490} \]

### Final Answer

Thus, the simplified expression is:
\[ \frac{8289}{9490} \]