`(3(1)/(4)-(4)/(5) "of"(5)/(6))/(4(1)/(3)-: (1)/(5)-((3)/(10)+21(1)/(5)))`

Options

A. `(1)/(6)`
B. `2(7)/(12)`
C. `15(1)/(2)`
D. `21(1)/(2)`

Answer

To solve the expression `(3(1)/(4)-(4)/(5) "of"(5)/(6))/(4(1)/(3)-: (1)/(5)-((3)/(10)+21(1)/(5)))`, we will break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions.

- \(3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}\)
- \(4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}\)
- \(21 \frac{1}{5} = \frac{21 \times 5 + 1}{5} = \frac{105 + 1}{5} = \frac{106}{5}\)

So, the expression becomes:
\[ \frac{\left(\frac{13}{4} - \frac{4}{5} \cdot \frac{5}{6}\right)}{\left(\frac{13}{3} \div \frac{1}{5} - \left(\frac{3}{10} + \frac{106}{5}\right)\right)} \]

### Step 2: Simplify the Numerator
Now we simplify the numerator:
\[ \frac{13}{4} - \left(\frac{4}{5} \cdot \frac{5}{6}\right) \]
Calculating \( \frac{4}{5} \cdot \frac{5}{6} = \frac{4 \cdot 5}{5 \cdot 6} = \frac{4}{6} = \frac{2}{3} \).

Now, substituting back:
\[ \frac{13}{4} - \frac{2}{3} \]
To subtract these fractions, we need a common denominator, which is 12:
\[ \frac{13}{4} = \frac{39}{12}, \quad \frac{2}{3} = \frac{8}{12} \]
Thus,
\[ \frac{39}{12} - \frac{8}{12} = \frac{31}{12} \]

### Step 3: Simplify the Denominator
Now we simplify the denominator:
\[ \frac{13}{3} \div \frac{1}{5} - \left(\frac{3}{10} + \frac{106}{5}\right) \]
Calculating \( \frac{13}{3} \div \frac{1}{5} = \frac{13}{3} \cdot 5 = \frac{65}{3} \).

Next, we simplify \( \frac{3}{10} + \frac{106}{5} \):
To add these fractions, we need a common denominator, which is 10:
\[ \frac{3}{10} + \frac{106}{5} = \frac{3}{10} + \frac{212}{10} = \frac{215}{10} \]
Now substituting back into the denominator:
\[ \frac{65}{3} - \frac{215}{10} \]
Finding a common denominator (30):
\[ \frac{65}{3} = \frac{650}{30}, \quad \frac{215}{10} = \frac{645}{30} \]
Thus,
\[ \frac{650}{30} - \frac{645}{30} = \frac{5}{30} = \frac{1}{6} \]

### Step 4: Final Calculation
Now we have:
\[ \frac{\frac{31}{12}}{\frac{1}{6}} = \frac{31}{12} \cdot 6 = \frac{31 \cdot 6}{12} = \frac{186}{12} = \frac{31}{2} \]

### Final Answer
The final answer is:
\[ \frac{31}{2} \]