Divide the sum of `(13)/(5) and (-12)/(7)` by the product of `(-31)/(7) and (1)/(-2)`.

Answer

To solve the problem step by step, we will follow the instructions given in the question:

**Question:** Divide the sum of \( \frac{13}{5} \) and \( -\frac{12}{7} \) by the product of \( -\frac{31}{7} \) and \( \frac{1}{-2} \).

### Step 1: Calculate the sum of \( \frac{13}{5} \) and \( -\frac{12}{7} \)

To find the sum, we need a common denominator. The denominators are 5 and 7, and their least common multiple (LCM) is 35.

Convert \( \frac{13}{5} \) to a fraction with a denominator of 35:
\[ \frac{13}{5} = \frac{13 \times 7}{5 \times 7} = \frac{91}{35} \]

Convert \( -\frac{12}{7} \) to a fraction with a denominator of 35:
\[ -\frac{12}{7} = \frac{-12 \times 5}{7 \times 5} = \frac{-60}{35} \]

Now, add the two fractions:
\[ \frac{91}{35} + \frac{-60}{35} = \frac{91 - 60}{35} = \frac{31}{35} \]

### Step 2: Calculate the product of \( -\frac{31}{7} \) and \( \frac{1}{-2} \)

To find the product, multiply the two fractions:
\[ -\frac{31}{7} \times \frac{1}{-2} = \frac{31 \times 1}{7 \times 2} = \frac{31}{14} \]

### Step 3: Divide the sum by the product

Now we need to divide the sum \( \frac{31}{35} \) by the product \( \frac{31}{14} \):
\[ \frac{31}{35} \div \frac{31}{14} \]

Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{31}{35} \times \frac{14}{31} \]

### Step 4: Simplify the expression

The \( 31 \) in the numerator and denominator cancels out:
\[ \frac{14}{35} \]

Now, simplify \( \frac{14}{35} \):
\[ \frac{14 \div 7}{35 \div 7} = \frac{2}{5} \]

### Final Answer
The final answer is:
\[ \frac{2}{5} \]

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