`3/4div(2)1/4of 2/3-(1/2-1/3)/(1/2+1/3)xx(3)1/3+5/6=?`

Options

A. A)(7/18)
B. B)(49/54)
C. C)(2/3)
D. D)(1/6)

Answer

To solve the expression \( \frac{3}{4} \div (2 \frac{1}{4} \text{ of } \frac{2}{3}) - \frac{\frac{1}{2} - \frac{1}{3}}{\frac{1}{2} + \frac{1}{3}} \times (3 \frac{1}{3}) + \frac{5}{6} \), we will follow the BODMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- Convert \( 2 \frac{1}{4} \) to an improper fraction:
\[ 2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4} \]
- Convert \( 3 \frac{1}{3} \) to an improper fraction:
\[ 3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \]

2. **Calculate \( 2 \frac{1}{4} \text{ of } \frac{2}{3} \):**
- This means to multiply:
\[ \frac{9}{4} \text{ of } \frac{2}{3} = \frac{9}{4} \times \frac{2}{3} = \frac{9 \times 2}{4 \times 3} = \frac{18}{12} = \frac{3}{2} \]

3. **Perform the Division:**
- Now, calculate \( \frac{3}{4} \div \frac{3}{2} \):
\[ \frac{3}{4} \div \frac{3}{2} = \frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{2}{4} = \frac{1}{2} \]

4. **Calculate the Fraction in the Parentheses:**
- Calculate \( \frac{1}{2} - \frac{1}{3} \):
- Find a common denominator (6):
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
\[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]
- Calculate \( \frac{1}{2} + \frac{1}{3} \):
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

5. **Calculate the Division of Fractions:**
- Now compute \( \frac{\frac{1}{6}}{\frac{5}{6}} \):
\[ \frac{1}{6} \div \frac{5}{6} = \frac{1}{6} \times \frac{6}{5} = \frac{1}{5} \]

6. **Multiply by \( 3 \frac{1}{3} \):**
- Now multiply by \( \frac{10}{3} \):
\[ \frac{1}{5} \times \frac{10}{3} = \frac{10}{15} = \frac{2}{3} \]

7. **Combine All Parts:**
- Now substitute back into the expression:
\[ \frac{1}{2} - \frac{2}{3} + \frac{5}{6} \]
- Convert \( \frac{1}{2} \) to sixths:
\[ \frac{1}{2} = \frac{3}{6} \]
- Now compute:
\[ \frac{3}{6} - \frac{4}{6} + \frac{5}{6} = \frac{3 - 4 + 5}{6} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer:
\[ \frac{2}{3} \]