Simplify : `(( - 18 (1)/(3) xx 2 (8)/(11)) - (4 (5)/(7) xx 2 (1)/(3)))/([ (3)/(5) + ((-9)/(10)) ] + [- ((-3)/( 5)) ])`

Options

A. `63 (4)/(81)`
B. `-23 (7)/(9)`
C. `-203 (1)/(3)`
D. `12 (6)/(17)`

Answer

To simplify the expression \(\frac{(-18 \frac{1}{3} \times 2 \frac{8}{11}) - (4 \frac{5}{7} \times 2 \frac{1}{3})}{\left(\frac{3}{5} + \left(-\frac{9}{10}\right)\right) + \left[-\left(-\frac{3}{5}\right)\right]}\), we will follow the steps outlined below:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert the mixed numbers in the expression to improper fractions.

- For \(-18 \frac{1}{3}\):
\[ -18 \frac{1}{3} = -\frac{54 + 1}{3} = -\frac{55}{3} \]

- For \(2 \frac{8}{11}\):
\[ 2 \frac{8}{11} = \frac{2 \times 11 + 8}{11} = \frac{22 + 8}{11} = \frac{30}{11} \]

- For \(4 \frac{5}{7}\):
\[ 4 \frac{5}{7} = \frac{4 \times 7 + 5}{7} = \frac{28 + 5}{7} = \frac{33}{7} \]

- For \(2 \frac{1}{3}\):
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

### Step 2: Rewrite the Expression
Now, substitute these improper fractions back into the expression:
\[ \frac{\left(-\frac{55}{3} \times \frac{30}{11}\right) - \left(\frac{33}{7} \times \frac{7}{3}\right)}{\left(\frac{3}{5} - \frac{9}{10}\right) + \frac{3}{5}} \]

### Step 3: Simplify the Numerator
Calculate the two products in the numerator.

1. For \(-\frac{55}{3} \times \frac{30}{11}\):
\[ -\frac{55 \times 30}{3 \times 11} = -\frac{1650}{33} = -50 \]

2. For \(\frac{33}{7} \times \frac{7}{3}\):
\[ \frac{33 \times 7}{7 \times 3} = \frac{33}{3} = 11 \]

Now substitute these results back into the numerator:
\[ -50 - 11 = -61 \]

### Step 4: Simplify the Denominator
Calculate the denominator:
\[ \left(\frac{3}{5} - \frac{9}{10}\right) + \frac{3}{5} \]

1. Find a common denominator for \(\frac{3}{5}\) and \(-\frac{9}{10}\):
\[ \frac{3}{5} = \frac{6}{10} \quad \text{so} \quad \frac{6}{10} - \frac{9}{10} = -\frac{3}{10} \]

2. Now add \(-\frac{3}{10}\) and \(\frac{3}{5}\):
\[ \frac{3}{5} = \frac{6}{10} \quad \text{so} \quad -\frac{3}{10} + \frac{6}{10} = \frac{3}{10} \]

### Step 5: Combine the Results
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{-61}{\frac{3}{10}} = -61 \times \frac{10}{3} = -\frac{610}{3} \]

### Final Answer
The simplified form of the expression is:
\[ -\frac{610}{3} \]