The value of `(4)/(5) xx 1(1)/(9) div(8)/(15) + (5(1)/(4) div (3)/(7) "of" (1)/(4) xx (2)/(7)) div 5(3)/(5) - (1)/(4) div (3)/(2)` is :

Options

A. 4
B. `(1)/(2)`
C. `(3)/(5)`
D. 6

Answer

To solve the expression \(\frac{4}{5} \times 1\frac{1}{9} \div \frac{8}{15} + \left(\frac{5\frac{1}{4}}{\frac{3}{7}} \text{ of } \left(\frac{1}{4} \times \frac{2}{7}\right)\right) \div 5\frac{3}{5} - \frac{1}{4} \div \frac{3}{2}\), we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert all mixed numbers to improper fractions:
- \(1\frac{1}{9} = \frac{10}{9}\)
- \(5\frac{1}{4} = \frac{21}{4}\)
- \(5\frac{3}{5} = \frac{28}{5}\)

So, the expression becomes:
\[ \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \left(\frac{21}{4} \div \frac{3}{7} \text{ of } \left(\frac{1}{4} \times \frac{2}{7}\right)\right) \div \frac{28}{5} - \frac{1}{4} \div \frac{3}{2} \]

### Step 2: Solve the "of" Operation
Next, we solve the "of" operation:
\[ \frac{1}{4} \times \frac{2}{7} = \frac{2}{28} = \frac{1}{14} \]

Now, substitute this back into the expression:
\[ \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \left(\frac{21}{4} \div \frac{3}{7} \times \frac{1}{14}\right) \div \frac{28}{5} - \frac{1}{4} \div \frac{3}{2} \]

### Step 3: Solve the Division Operations
Now, we will solve the division operations:
1. \(\frac{21}{4} \div \frac{3}{7} = \frac{21}{4} \times \frac{7}{3} = \frac{147}{12} = \frac{49}{4}\)
2. Now, substitute this back into the expression:
\[ \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \left(\frac{49}{4} \times \frac{1}{14}\right) \div \frac{28}{5} - \frac{1}{4} \div \frac{3}{2} \]

### Step 4: Calculate Each Part
1. Calculate \(\frac{4}{5} \times \frac{10}{9} \div \frac{8}{15}\):
- First, \(\frac{4}{5} \times \frac{10}{9} = \frac{40}{45} = \frac{8}{9}\)
- Then, \(\frac{8}{9} \div \frac{8}{15} = \frac{8}{9} \times \frac{15}{8} = \frac{15}{9} = \frac{5}{3}\)

2. Calculate \(\frac{49}{4} \times \frac{1}{14} = \frac{49}{56} = \frac{7}{8}\)
- Now, \(\frac{7}{8} \div \frac{28}{5} = \frac{7}{8} \times \frac{5}{28} = \frac{35}{224} = \frac{5}{32}\)

3. Calculate \(\frac{1}{4} \div \frac{3}{2} = \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}\)

### Step 5: Combine All Parts
Now, we combine all parts:
\[ \frac{5}{3} + \frac{5}{32} - \frac{1}{6} \]

### Step 6: Find a Common Denominator
The common denominator for \(3\), \(32\), and \(6\) is \(96\):
- Convert \(\frac{5}{3} = \frac{160}{96}\)
- Convert \(\frac{5}{32} = \frac{15}{96}\)
- Convert \(\frac{1}{6} = \frac{16}{96}\)

### Step 7: Combine the Fractions
Now, combine:
\[ \frac{160}{96} + \frac{15}{96} - \frac{16}{96} = \frac{159}{96} \]

### Step 8: Simplify the Result
The simplified result is:
\[ \frac{159}{96} \approx 1.65625 \]

### Final Answer
The final answer is approximately \(1.66\).