Evaluate : `((1+ (2)/(9))- [2 (1)/(2)- (1(1)/(2)- ((-1)/(3)))])/((1+ 3(1)/(5)) + (1+2(2)/(3)))`

Options

A. `25/(354)`
B. `5 (13)/(19)`
C. `3(8)/(33)`
D. `13 (5)/(6)`

Answer

To evaluate the expression

\[ \frac{\left(1 + \frac{2}{9}\right) - \left[2 \cdot \frac{1}{2} - \left(1 \cdot \frac{1}{2} - \left(-\frac{1}{3}\right)\right)\right]}{\left(1 + \frac{3}{5}\right) + \left(1 + \frac{2}{3}\right)} \]

we will follow the order of operations (often remembered by the acronym PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

### Step 1: Simplify the numerator

1. **Calculate \(1 + \frac{2}{9}\)**:
\[ 1 + \frac{2}{9} = \frac{9}{9} + \frac{2}{9} = \frac{11}{9} \]

2. **Calculate the expression inside the brackets**:
- First, calculate \(2 \cdot \frac{1}{2}\):
\[ 2 \cdot \frac{1}{2} = 1 \]
- Now, calculate \(1 \cdot \frac{1}{2}\):
\[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]
- Next, calculate \(\frac{1}{2} - \left(-\frac{1}{3}\right)\):
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
- Now, substitute back:
\[ 1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \]

3. **Combine the results**:
\[ \frac{11}{9} - \frac{1}{6} \]
- To subtract these fractions, find a common denominator (which is 18):
\[ \frac{11}{9} = \frac{22}{18}, \quad \frac{1}{6} = \frac{3}{18} \]
- Now subtract:
\[ \frac{22}{18} - \frac{3}{18} = \frac{19}{18} \]

### Step 2: Simplify the denominator

1. **Calculate \(1 + \frac{3}{5}\)**:
\[ 1 + \frac{3}{5} = \frac{5}{5} + \frac{3}{5} = \frac{8}{5} \]

2. **Calculate \(1 + \frac{2}{3}\)**:
\[ 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

3. **Combine the results**:
\[ \frac{8}{5} + \frac{5}{3} \]
- To add these fractions, find a common denominator (which is 15):
\[ \frac{8}{5} = \frac{24}{15}, \quad \frac{5}{3} = \frac{25}{15} \]
- Now add:
\[ \frac{24}{15} + \frac{25}{15} = \frac{49}{15} \]

### Step 3: Combine the results

Now we have:

\[ \frac{\frac{19}{18}}{\frac{49}{15}} = \frac{19}{18} \cdot \frac{15}{49} = \frac{19 \cdot 15}{18 \cdot 49} \]

### Step 4: Simplify the final expression

1. **Calculate the numerator**:
\[ 19 \cdot 15 = 285 \]

2. **Calculate the denominator**:
\[ 18 \cdot 49 = 882 \]

3. **Final result**:
\[ \frac{285}{882} \]
- Simplifying this fraction (both can be divided by 3):
\[ \frac{285 \div 3}{882 \div 3} = \frac{95}{294} \]

So, the final answer is:

\[ \frac{95}{294} \]