Divide the sum of `(- (2)/(5)) and (9)/((-10))` by the sum of `(3)/(7) and (4)/(5).`

Options

A. `(-35)/(86)`
B. `(-91)/(86)`
C. `(47)/(85)`
D. `(-73)/(85)`

Answer

To solve the problem step by step, we will first calculate the sums required and then perform the division.

### Step 1: Calculate the first sum, S1
We need to find the sum of \(-\frac{2}{5}\) and \(\frac{9}{-10}\).

1. **Convert to a common denominator**: The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10.
- Convert \(-\frac{2}{5}\) to have a denominator of 10:
\[ -\frac{2}{5} = -\frac{2 \times 2}{5 \times 2} = -\frac{4}{10} \]
- \(\frac{9}{-10}\) can be written as \(-\frac{9}{10}\).

2. **Add the fractions**:
\[ S1 = -\frac{4}{10} + -\frac{9}{10} = -\frac{4 + 9}{10} = -\frac{13}{10} \]

### Step 2: Calculate the second sum, S2
Next, we need to find the sum of \(\frac{3}{7}\) and \(\frac{4}{5}\).

1. **Convert to a common denominator**: The denominators are 7 and 5. The LCM of 7 and 5 is 35.
- Convert \(\frac{3}{7}\) to have a denominator of 35:
\[ \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \]
- Convert \(\frac{4}{5}\) to have a denominator of 35:
\[ \frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35} \]

2. **Add the fractions**:
\[ S2 = \frac{15}{35} + \frac{28}{35} = \frac{15 + 28}{35} = \frac{43}{35} \]

### Step 3: Divide S1 by S2
Now we will divide the first sum by the second sum:
\[ \text{Result} = \frac{S1}{S2} = \frac{-\frac{13}{10}}{\frac{43}{35}} \]

1. **Dividing fractions**: Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{-\frac{13}{10}}{\frac{43}{35}} = -\frac{13}{10} \times \frac{35}{43} \]

2. **Multiply the fractions**:
\[ = -\frac{13 \times 35}{10 \times 43} = -\frac{455}{430} \]

3. **Simplify the fraction**:
- The greatest common divisor (GCD) of 455 and 430 is 5.
\[ -\frac{455 \div 5}{430 \div 5} = -\frac{91}{86} \]

### Final Answer
The final result is:
\[ -\frac{91}{86} \]