The value of `3/7+1/14xx2frac(4)(5)-(1/4 div 2/7 of 2frac(1)(3))xx(5frac(1)(5)div 3frac(1)(2)xx5/13)`

Options

A. `22/35`
B. `3/14`
C. `14/21`
D. `29/70`

Answer

To solve the expression \( \frac{3}{7} + \frac{1}{14} \times 2\frac{4}{5} - \left( \frac{1}{4} \div \frac{2}{7} \text{ of } 2\frac{1}{3} \right) \times \left( 5\frac{1}{5} \div 3\frac{1}{2} \times \frac{5}{13} \right) \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-step Solution

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 2\frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{14}{5} \)
- \( 2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} \)
- \( 5\frac{1}{5} = \frac{5 \times 5 + 1}{5} = \frac{26}{5} \)
- \( 3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} \)

The expression now looks like:
\[ \frac{3}{7} + \frac{1}{14} \times \frac{14}{5} - \left( \frac{1}{4} \div \frac{2}{7} \text{ of } \frac{7}{3} \right) \times \left( \frac{26}{5} \div \frac{7}{2} \times \frac{5}{13} \right) \]

2. **Calculate the Multiplication**:
- \( \frac{1}{14} \times \frac{14}{5} = \frac{1}{5} \)

The expression now looks like:
\[ \frac{3}{7} + \frac{1}{5} - \left( \frac{1}{4} \div \frac{2}{7} \text{ of } \frac{7}{3} \right) \times \left( \frac{26}{5} \div \frac{7}{2} \times \frac{5}{13} \right) \]

3. **Calculate the Division and 'of'**:
- \( \frac{1}{4} \div \frac{2}{7} = \frac{1}{4} \times \frac{7}{2} = \frac{7}{8} \)
- \( \frac{7}{8} \text{ of } \frac{7}{3} = \frac{7}{8} \times \frac{7}{3} = \frac{49}{24} \)

The expression now looks like:
\[ \frac{3}{7} + \frac{1}{5} - \frac{49}{24} \times \left( \frac{26}{5} \div \frac{7}{2} \times \frac{5}{13} \right) \]

4. **Calculate the Second Part**:
- \( \frac{26}{5} \div \frac{7}{2} = \frac{26}{5} \times \frac{2}{7} = \frac{52}{35} \)
- \( \frac{52}{35} \times \frac{5}{13} = \frac{52 \times 5}{35 \times 13} = \frac{260}{455} = \frac{52}{91} \)

The expression now looks like:
\[ \frac{3}{7} + \frac{1}{5} - \frac{49}{24} \times \frac{52}{91} \]

5. **Calculate the Multiplication**:
- \( \frac{49}{24} \times \frac{52}{91} = \frac{2548}{2184} = \frac{1274}{1092} \)

The expression now looks like:
\[ \frac{3}{7} + \frac{1}{5} - \frac{1274}{1092} \]

6. **Combine the Fractions**:
- Find a common denominator (LCM of 7, 5, and 1092):
- The LCM is 1092.
- Convert each fraction:
- \( \frac{3}{7} = \frac{3 \times 156}{1092} = \frac{468}{1092} \)
- \( \frac{1}{5} = \frac{1 \times 218.4}{1092} = \frac{218.4}{1092} \)

Combine them:
\[ \frac{468 + 218.4 - 1274}{1092} \]

7. **Final Calculation**:
- Calculate the numerator:
\[ 468 + 218.4 - 1274 = -587.6 \]

Thus, the final answer is:
\[ \frac{-587.6}{1092} \]

### Final Answer
The simplified value of the expression is \( \frac{29}{70} \).