`(2)/(3) -((1)/(2) -(1)/(3))/((1)/(2)+(1)/(3))xx3(1)/(3)+(5)/(6) =?`

Options

A. `(7)/(18)`
B. `(49)/(54)`
C. `(2)/(3)`
D. `(5)/(6)`

Answer

To solve the expression \(\frac{2}{3} - \frac{\left(\frac{1}{2} - \frac{1}{3}\right)}{\left(\frac{1}{2} + \frac{1}{3}\right)} \times 3 \frac{1}{3} + \frac{5}{6}\), we will follow the order of operations (BODMAS/BIDMAS rules: Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step-by-Step Solution:

1. **Simplify the inner fractions**:
- Calculate \(\frac{1}{2} - \frac{1}{3}\):
\[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]
- Calculate \(\frac{1}{2} + \frac{1}{3}\):
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

2. **Substitute back into the expression**:
- Now substitute these results into the expression:
\[ \frac{2}{3} - \frac{\frac{1}{6}}{\frac{5}{6}} \times 3 \frac{1}{3} + \frac{5}{6} \]

3. **Simplify the division of fractions**:
- \(\frac{\frac{1}{6}}{\frac{5}{6}} = \frac{1}{6} \times \frac{6}{5} = \frac{1}{5}\)

4. **Convert \(3 \frac{1}{3}\) to an improper fraction**:
- \(3 \frac{1}{3} = \frac{10}{3}\)

5. **Multiply the fractions**:
- Now calculate \(\frac{1}{5} \times \frac{10}{3}\):
\[ \frac{1 \times 10}{5 \times 3} = \frac{10}{15} = \frac{2}{3} \]

6. **Substitute back into the expression**:
- The expression now looks like:
\[ \frac{2}{3} - \frac{2}{3} + \frac{5}{6} \]

7. **Combine the terms**:
- The \(\frac{2}{3} - \frac{2}{3} = 0\), so we are left with:
\[ 0 + \frac{5}{6} = \frac{5}{6} \]

### Final Answer:
\[ \frac{5}{6} \]