Simplify `(3/2 + 1/2xx 3/2)/(3/2+1/2 of 3/2)+1/8`

Options

A. 9/8
B. 2
C. 1/6
D. 1/2

Answer

To simplify the expression \((\frac{3}{2} + \frac{1}{2} \times \frac{3}{2}) / (\frac{3}{2} + \frac{1}{2} \text{ of } \frac{3}{2}) + \frac{1}{8}\), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-step Solution:

1. **Identify the expression**:
\[ \frac{\frac{3}{2} + \frac{1}{2} \times \frac{3}{2}}{\frac{3}{2} + \frac{1}{2} \text{ of } \frac{3}{2}} + \frac{1}{8} \]

2. **Calculate the multiplication in the numerator**:
\[ \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \]
So, the numerator becomes:
\[ \frac{3}{2} + \frac{3}{4} \]

3. **Find a common denominator for the numerator**:
The common denominator for \(\frac{3}{2}\) and \(\frac{3}{4}\) is 4.
- Convert \(\frac{3}{2}\) to \(\frac{6}{4}\):
\[ \frac{3}{2} = \frac{6}{4} \]
- Now, add the fractions:
\[ \frac{6}{4} + \frac{3}{4} = \frac{9}{4} \]

4. **Calculate the expression in the denominator**:
The denominator is:
\[ \frac{3}{2} + \frac{1}{2} \text{ of } \frac{3}{2} \]
Again, calculate \(\frac{1}{2} \text{ of } \frac{3}{2}\):
\[ \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \]
So, the denominator becomes:
\[ \frac{3}{2} + \frac{3}{4} \]
Convert \(\frac{3}{2}\) to \(\frac{6}{4}\):
\[ \frac{6}{4} + \frac{3}{4} = \frac{9}{4} \]

5. **Now substitute back into the expression**:
\[ \frac{\frac{9}{4}}{\frac{9}{4}} + \frac{1}{8} \]

6. **Simplify the fraction**:
\[ \frac{9}{4} \div \frac{9}{4} = 1 \]
So, the expression now is:
\[ 1 + \frac{1}{8} \]

7. **Convert 1 to a fraction with a common denominator**:
\[ 1 = \frac{8}{8} \]
Now add:
\[ \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \]

### Final Answer:
\[ \frac{9}{8} \]