The value of `(2(6)/(7)"of"4(1)/(5)div(2)/(3))xx1(1)/(9)div((3)/(2)xx2(2)/(3)"of"(1)/(2)div(1)/(4))` is :

Options

A. 5
B. `(1)/(8)`
C. 8
D. `(1)/(5)`

Answer

To solve the expression \( \left(2 \frac{6}{7} \text{ of } 4 \frac{1}{5} \div \frac{2}{3}\right) \times 1 \frac{1}{9} \div \left(\frac{3}{2} \times 2 \frac{2}{3} \text{ of } \frac{1}{2} \div \frac{1}{4}\right) \), we will follow the order of operations (BODMAS/BIDMAS).

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

- \( 2 \frac{6}{7} = \frac{2 \times 7 + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7} \)
- \( 4 \frac{1}{5} = \frac{4 \times 5 + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5} \)
- \( 1 \frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{9 + 1}{9} = \frac{10}{9} \)
- \( 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \)

Now, our expression looks like this:
\[ \left(\frac{20}{7} \text{ of } \frac{21}{5} \div \frac{2}{3}\right) \times \frac{10}{9} \div \left(\frac{3}{2} \times \frac{8}{3} \text{ of } \frac{1}{2} \div \frac{1}{4}\right) \]

### Step 2: Solve the "of" Operations
The "of" operation means multiplication.

- \( \frac{20}{7} \text{ of } \frac{21}{5} = \frac{20}{7} \times \frac{21}{5} = \frac{420}{35} = 12 \)
- \( \frac{3}{2} \text{ of } \frac{1}{2} = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4} \)

Now, the expression becomes:
\[ \left(12 \div \frac{2}{3}\right) \times \frac{10}{9} \div \left(\frac{3}{2} \times \frac{8}{3} \div \frac{1}{4}\right) \]

### Step 3: Solve the Division
For \( 12 \div \frac{2}{3} \):
\[ 12 \div \frac{2}{3} = 12 \times \frac{3}{2} = \frac{36}{2} = 18 \]

For \( \frac{3}{2} \times \frac{8}{3} \div \frac{1}{4} \):
\[ \frac{3}{2} \times \frac{8}{3} = 4 \]
Then,
\[ 4 \div \frac{1}{4} = 4 \times 4 = 16 \]

### Step 4: Substitute Back into the Expression
Now, we substitute back into the expression:
\[ 18 \times \frac{10}{9} \div 16 \]

### Step 5: Solve the Remaining Operations
First, calculate \( 18 \times \frac{10}{9} \):
\[ 18 \times \frac{10}{9} = \frac{180}{9} = 20 \]

Now, divide by 16:
\[ 20 \div 16 = \frac{20}{16} = \frac{5}{4} \]

### Final Answer
The value of the expression is \( \frac{5}{4} \).

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