`(-(1)/(2) - (2)/(3) + (4)/(5) - (1)/(3) + (1)/(5) + (3)/(4))/((1)/(2) + (2)/(3) - (4)/(3) + (1)/(3) - (1)/(5) - (4)/(5))` is simplified to

Options

A. `- (10)/(3)`
B. `- (3)/(10)`
C. 1
D. -2

Answer

To simplify the expression

\[ \frac{-(\frac{1}{2}) - (\frac{2}{3}) + (\frac{4}{5}) - (\frac{1}{3}) + (\frac{1}{5}) + (\frac{3}{4})}{(\frac{1}{2}) + (\frac{2}{3}) - (\frac{4}{3}) + (\frac{1}{3}) - (\frac{1}{5}) - (\frac{4}{5})} \]

we will break it down step by step.

### Step 1: Simplify the Numerator

The numerator is:

\[ -(\frac{1}{2}) - (\frac{2}{3}) + (\frac{4}{5}) - (\frac{1}{3}) + (\frac{1}{5}) + (\frac{3}{4}) \]

First, combine the negative fractions:

\[ -(\frac{1}{2}) - (\frac{2}{3}) - (\frac{1}{3}) = -(\frac{1}{2}) - (\frac{3}{3}) = -(\frac{1}{2}) - 1 = -\frac{3}{2} \]

Now, combine the positive fractions:

\[ (\frac{4}{5}) + (\frac{1}{5}) + (\frac{3}{4}) = (\frac{5}{5}) + (\frac{3}{4}) = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \]

Now, combine the results:

\[ -\frac{3}{2} + \frac{7}{4} \]

To combine these, convert \(-\frac{3}{2}\) to a fraction with a denominator of 4:

\[ -\frac{3}{2} = -\frac{6}{4} \]

Now we can add:

\[ -\frac{6}{4} + \frac{7}{4} = \frac{1}{4} \]

### Step 2: Simplify the Denominator

The denominator is:

\[ (\frac{1}{2}) + (\frac{2}{3}) - (\frac{4}{3}) + (\frac{1}{3}) - (\frac{1}{5}) - (\frac{4}{5}) \]

Combine the fractions:

\[ (\frac{2}{3}) - (\frac{4}{3}) + (\frac{1}{3}) = -\frac{2}{3} + \frac{1}{3} = -\frac{1}{3} \]

Now, combine the remaining fractions:

\[ (\frac{1}{2}) - (\frac{1}{5}) - (\frac{4}{5}) = \frac{1}{2} - 1 = -\frac{1}{2} \]

Now combine the results:

\[ -\frac{1}{3} - \frac{1}{2} \]

To combine these, convert to a common denominator of 6:

\[ -\frac{1}{3} = -\frac{2}{6}, \quad -\frac{1}{2} = -\frac{3}{6} \]

Now we can add:

\[ -\frac{2}{6} - \frac{3}{6} = -\frac{5}{6} \]

### Step 3: Combine Numerator and Denominator

Now we have:

\[ \frac{\frac{1}{4}}{-\frac{5}{6}} = \frac{1}{4} \times -\frac{6}{5} = -\frac{6}{20} = -\frac{3}{10} \]

### Final Answer

Thus, the simplified value of the expression is:

\[ -\frac{3}{10} \]