Simplify the following expression :
`((3)/(11)xx(33)/(6))-((9)/(4)xx(12)/(3))+((5)/(11)xx(22)/(10))`

Options

A. `(-9)/(2)`
B. `(-13)/(2)`
C. `(13)/(2)`
D. `9/2`

Answer

To simplify the expression \(\frac{3}{11} \times \frac{33}{6} - \frac{9}{4} \times \frac{12}{3} + \frac{5}{11} \times \frac{22}{10}\), we will follow these steps:

### Step 1: Simplify each term individually
1. **First term**: \(\frac{3}{11} \times \frac{33}{6}\)
- Multiply the fractions:
\[ = \frac{3 \times 33}{11 \times 6} \]
- Simplify:
\[ = \frac{99}{66} = \frac{3}{2} \]

2. **Second term**: \(\frac{9}{4} \times \frac{12}{3}\)
- Multiply the fractions:
\[ = \frac{9 \times 12}{4 \times 3} \]
- Simplify:
\[ = \frac{108}{12} = 9 \]

3. **Third term**: \(\frac{5}{11} \times \frac{22}{10}\)
- Multiply the fractions:
\[ = \frac{5 \times 22}{11 \times 10} \]
- Simplify:
\[ = \frac{110}{110} = 1 \]

### Step 2: Combine the simplified terms
Now, we can substitute the simplified terms back into the expression:
\[ \frac{3}{2} - 9 + 1 \]

### Step 3: Perform the operations
1. Combine \(1\) and \(-9\):
\[ 1 - 9 = -8 \]

2. Now, combine \(\frac{3}{2}\) and \(-8\):
\[ \frac{3}{2} - 8 = \frac{3}{2} - \frac{16}{2} = \frac{3 - 16}{2} = \frac{-13}{2} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{-13}{2} \]