What is the value of `((3)/(4) div (9)/(32) +(4)/(3) xx (2)/(3) " of "(27)/(16))/((1)/(2) xx ((8)/(2) -2) div (4)/(9) +((1)/(3)+(1)/(6)))`? (a) `(10)/(3)` (b) `(13)/(2)` (c) `(25)/(2)` (d) `(31)/(2)`

Options

A. `(50)/(33)`
B. `(53)/(33)`
C. `(25)/(2)`
D. `(31)/(2)`

Answer

To solve the expression

\[ \frac{\left(\frac{3}{4} \div \frac{9}{32}\right) + \left(\frac{4}{3} \times \frac{2}{3} \text{ of } \frac{27}{16}\right)}{\left(\frac{1}{2} \times \left(\frac{8}{2} - 2\right) \div \frac{4}{9}\right) + \left(\frac{1}{3} + \frac{1}{6}\right)} \]

we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Solve the numerator

1. **Calculate \(\frac{3}{4} \div \frac{9}{32}\)**:
\[ \frac{3}{4} \div \frac{9}{32} = \frac{3}{4} \times \frac{32}{9} = \frac{3 \times 32}{4 \times 9} = \frac{96}{36} = \frac{8}{3} \]

**Hint**: To divide fractions, multiply by the reciprocal of the divisor.

2. **Calculate \(\frac{4}{3} \times \frac{2}{3} \text{ of } \frac{27}{16}\)**:
\[ \frac{4}{3} \times \frac{2}{3} \text{ of } \frac{27}{16} = \frac{4}{3} \times \frac{2}{3} \times \frac{27}{16} \]
First calculate \(\frac{4}{3} \times \frac{2}{3} = \frac{8}{9}\):
\[ \frac{8}{9} \times \frac{27}{16} = \frac{8 \times 27}{9 \times 16} = \frac{216}{144} = \frac{3}{2} \]

**Hint**: "Of" in mathematics often means multiplication.

3. **Combine the results**:
\[ \frac{8}{3} + \frac{3}{2} \]
To add these fractions, find a common denominator (which is 6):
\[ \frac{8}{3} = \frac{16}{6}, \quad \frac{3}{2} = \frac{9}{6} \quad \Rightarrow \quad \frac{16}{6} + \frac{9}{6} = \frac{25}{6} \]

**Hint**: When adding fractions, convert them to have a common denominator.

### Step 2: Solve the denominator

1. **Calculate \(\frac{1}{2} \times \left(\frac{8}{2} - 2\right)\)**:
\[ \frac{8}{2} - 2 = 4 - 2 = 2 \quad \Rightarrow \quad \frac{1}{2} \times 2 = 1 \]

**Hint**: Simplify inside the parentheses first.

2. **Calculate \(1 \div \frac{4}{9}\)**:
\[ 1 \div \frac{4}{9} = 1 \times \frac{9}{4} = \frac{9}{4} \]

**Hint**: Division by a fraction is the same as multiplying by its reciprocal.

3. **Calculate \(\frac{1}{3} + \frac{1}{6}\)**:
Find a common denominator (which is 6):
\[ \frac{1}{3} = \frac{2}{6} \quad \Rightarrow \quad \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

**Hint**: Again, find a common denominator for addition.

4. **Combine the results**:
\[ \frac{9}{4} + \frac{1}{2} \]
Convert \(\frac{1}{2}\) to have a denominator of 4:
\[ \frac{1}{2} = \frac{2}{4} \quad \Rightarrow \quad \frac{9}{4} + \frac{2}{4} = \frac{11}{4} \]

**Hint**: Always convert fractions to a common denominator before adding.

### Step 3: Final calculation

Now we have:

\[ \frac{\frac{25}{6}}{\frac{11}{4}} = \frac{25}{6} \times \frac{4}{11} = \frac{100}{66} = \frac{50}{33} \]

### Conclusion

The final result is:

\[ \frac{50}{33} \]

### Answer

The answer is \((a) \frac{50}{33}\).