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

To solve the given problem, we need to follow these steps:

### Step 1: Find the sum of \(-\frac{13}{5}\) and \(\frac{12}{7}\)
We need a common denominator to add these two fractions.

\[
\text{Common denominator of 5 and 7 is } 35.
\]

Rewrite the fractions with denominator 35:

\[
-\frac{13}{5} = -\frac{13 \times 7}{5 \times 7} = -\frac{91}{35}
\]
\[
\frac{12}{7} = \frac{12 \times 5}{7 \times 5} = \frac{60}{35}
\]

Now, add the two fractions:

\[
-\frac{91}{35} + \frac{60}{35} = \frac{-91 + 60}{35} = \frac{-31}{35}
\]

### Step 2: Find the product of \(-\frac{31}{7}\) and \(-\frac{1}{2}\)
Multiply the two fractions:

\[
-\frac{31}{7} \times -\frac{1}{2} = \frac{31}{14}
\]

### Step 3: Divide the sum from Step 1 by the product from Step 2
Now, divide \(\frac{-31}{35}\) by \(\frac{31}{14}\). To divide fractions, multiply by the reciprocal of the second fraction:

\[
\frac{-31}{35} \div \frac{31}{14} = \frac{-31}{35} \times \frac{14}{31} = \frac{-31 \times 14}{35 \times 31} = \frac{-434}{1085} = -\frac{2}{5}
\]

### Final Answer:
The result is \(-\frac{2}{5}\).

Would you like more details on any of these steps or have further questions?

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Here are 5 related questions for practice:
1. What is the sum of \(\frac{7}{9}\) and \(\frac{2}{3}\)?
2. How do you divide two fractions?
3. What is the product of \(-\frac{5}{6}\) and \(\frac{3}{4}\)?
4. If the sum of \(\frac{-15}{4}\) and \(\frac{8}{4}\) is divided by 2, what is the result?
5. How do you find the least common denominator for two fractions?

**Tip:** When dividing fractions, remember to multiply by the reciprocal of the divisor!

Learn how to divide the sum of fractions -13/5 and 12/7 by the product of -31/7 and -1/2 with a detailed step-by-step solution. This problem involves fraction operations like addition, multiplication, and division. Perfect for students in Grades 6-8.

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