The value of `(5 (1)/(9) - 7 (7)/(8)div9 (9)/(20)) xx (9)/(11)- (5(1)/(4) div (3)/(7) " of " (1)/(4) xx (2)/(7)) div 4 (2)/(3) + 1 (3)/(4) ` is
(a)`2(1)/(4)`
(b)`2(1)/(3)`
(c)`3(1)/(4)`
(d)`4(1)/(2)`

Options

A. `2(1)/(4)`
B. `2(1)/(3)`
C. `3(1)/(4)`
D. `4(1)/(2)`

Answer

To solve the expression step by step, we will follow the order of operations (BODMAS/BIDMAS) and convert mixed fractions into improper fractions where necessary.

### Step 1: Convert Mixed Fractions to Improper Fractions
The given expression is:
\[ (5 \frac{1}{9} - 7 \frac{7}{8} \div 9 \frac{9}{20}) \times \frac{9}{11} - (5 \frac{1}{4} \div \frac{3}{7} \text{ of } \frac{1}{4} \times \frac{2}{7}) \div 4 \frac{2}{3} + 1 \frac{3}{4} \]

Convert mixed fractions:
- \(5 \frac{1}{9} = \frac{46}{9}\)
- \(7 \frac{7}{8} = \frac{63}{8}\)
- \(9 \frac{9}{20} = \frac{189}{20}\)
- \(5 \frac{1}{4} = \frac{21}{4}\)
- \(4 \frac{2}{3} = \frac{14}{3}\)
- \(1 \frac{3}{4} = \frac{7}{4}\)

Now, rewrite the expression:
\[ \left(\frac{46}{9} - \frac{63}{8} \div \frac{189}{20}\right) \times \frac{9}{11} - \left(\frac{21}{4} \div \frac{3}{7} \text{ of } \frac{1}{4} \times \frac{2}{7}\right) \div \frac{14}{3} + \frac{7}{4} \]

### Step 2: Solve the Division Inside the Parentheses
First, calculate \(\frac{63}{8} \div \frac{189}{20}\):
\[ \frac{63}{8} \div \frac{189}{20} = \frac{63}{8} \times \frac{20}{189} = \frac{63 \times 20}{8 \times 189} \]
Simplifying:
- \(63 = 3 \times 21\)
- \(189 = 3 \times 63\)

Thus:
\[ \frac{63 \times 20}{8 \times 189} = \frac{20}{8} = \frac{5}{2} \]

Now substitute back:
\[ \left(\frac{46}{9} - \frac{5}{2}\right) \times \frac{9}{11} - \left(\frac{21}{4} \div \frac{3}{7} \text{ of } \frac{1}{4} \times \frac{2}{7}\right) \div \frac{14}{3} + \frac{7}{4} \]

### Step 3: Calculate \(\frac{46}{9} - \frac{5}{2}\)
Finding a common denominator (18):
\[ \frac{46}{9} = \frac{92}{18}, \quad \frac{5}{2} = \frac{45}{18} \]
Thus:
\[ \frac{92}{18} - \frac{45}{18} = \frac{47}{18} \]

### Step 4: Multiply by \(\frac{9}{11}\)
\[ \frac{47}{18} \times \frac{9}{11} = \frac{423}{198} \]

### Step 5: Solve the Second Parenthesis
Calculate \(\frac{21}{4} \div \frac{3}{7}\):
\[ \frac{21}{4} \div \frac{3}{7} = \frac{21}{4} \times \frac{7}{3} = \frac{147}{12} \]

Now calculate the "of" operation:
\[ \frac{147}{12} \text{ of } \frac{1}{4} \times \frac{2}{7} = \frac{147}{12} \times \frac{1}{4} \times \frac{2}{7} = \frac{147 \times 2}{12 \times 28} = \frac{294}{336} = \frac{49}{56} \]

### Step 6: Divide by \(\frac{14}{3}\)
\[ \frac{49}{56} \div \frac{14}{3} = \frac{49}{56} \times \frac{3}{14} = \frac{147}{784} \]

### Step 7: Combine All Parts
Now we have:
\[ \frac{423}{198} - \frac{147}{784} + \frac{7}{4} \]

### Step 8: Find a Common Denominator and Combine
The least common multiple of the denominators (198, 784, 4) is 1560. Convert each fraction:
- \(\frac{423}{198} = \frac{423 \times 8}{198 \times 8} = \frac{3384}{1560}\)
- \(\frac{147}{784} = \frac{147 \times 2}{784 \times 2} = \frac{294}{1560}\)
- \(\frac{7}{4} = \frac{7 \times 390}{4 \times 390} = \frac{2730}{1560}\)

Combine:
\[ \frac{3384 - 294 + 2730}{1560} = \frac{4820}{1560} \]

### Step 9: Simplify the Result
Dividing both numerator and denominator by 20:
\[ \frac{241}{78} \]

Convert to mixed fraction:
\[ 3 \frac{7}{78} \]

### Final Answer
The value of the expression is:
\[ \text{(c) } 3 \frac{1}{4} \]