Simplify :
`(((4)/(3) xx (-(25)/(2)))+((-(10)/(3))xx (5)/(2))-((-(16)/(3))xx((-45)/(32))))/((3)/(4) xx ((9)/(14) xx (-(2)/(18))))`

Options

A. `13 (11)/(27)`
B. `606 (2)/(3)`
C. `-133 (7)/(4)`
D. `606 (7)/(3)`

Answer

To simplify the expression

\[ \frac{\left(\frac{4}{3} \times -\frac{25}{2}\right) + \left(-\frac{10}{3} \times \frac{5}{2}\right) - \left(-\frac{16}{3} \times -\frac{45}{32}\right)}{\frac{3}{4} \times \left(\frac{9}{14} \times -\frac{2}{18}\right)}, \]

we will follow these steps:

### Step 1: Simplify the Numerator

1. Calculate \(\frac{4}{3} \times -\frac{25}{2}\):
\[ \frac{4 \times -25}{3 \times 2} = \frac{-100}{6} = -\frac{50}{3}. \]

2. Calculate \(-\frac{10}{3} \times \frac{5}{2}\):
\[ -\frac{10 \times 5}{3 \times 2} = -\frac{50}{6} = -\frac{25}{3}. \]

3. Calculate \(-\frac{16}{3} \times -\frac{45}{32}\):
\[ -\frac{16 \times 45}{3 \times 32} = \frac{720}{96} = \frac{15}{2}. \]

4. Combine the results from the above calculations:
\[ -\frac{50}{3} - \frac{25}{3} + \frac{15}{2}. \]
To combine these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.

Convert each fraction:
\[ -\frac{50}{3} = -\frac{100}{6}, \quad -\frac{25}{3} = -\frac{50}{6}, \quad \frac{15}{2} = \frac{45}{6}. \]

Now combine:
\[ -\frac{100}{6} - \frac{50}{6} + \frac{45}{6} = \frac{-100 - 50 + 45}{6} = \frac{-105}{6} = -\frac{35}{2}. \]

### Step 2: Simplify the Denominator

1. Calculate \(\frac{9}{14} \times -\frac{2}{18}\):
\[ \frac{9 \times -2}{14 \times 18} = \frac{-18}{252} = -\frac{1}{14}. \]

2. Now calculate \(\frac{3}{4} \times -\frac{1}{14}\):
\[ \frac{3 \times -1}{4 \times 14} = \frac{-3}{56}. \]

### Step 3: Combine the Numerator and Denominator

Now we have:
\[ \frac{-\frac{35}{2}}{-\frac{3}{56}}. \]

This can be simplified by multiplying by the reciprocal of the denominator:
\[ -\frac{35}{2} \times -\frac{56}{3} = \frac{35 \times 56}{2 \times 3} = \frac{1960}{6} = \frac{980}{3}. \]

### Final Answer
The simplified expression is:
\[ \frac{980}{3}. \]