Certainly! Let's work through this problem step-by-step to find the solution to dividing the two mixed numbers, [tex]\(-7 \frac{4}{7} \div -8 \frac{1}{5}\)[/tex].
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert each mixed number into an improper fraction.
#### Converting [tex]\(-7 \frac{4}{7}\)[/tex] to an Improper Fraction
1. The mixed number is [tex]\(-7 \frac{4}{7}\)[/tex].
2. Convert the integer part and the fractional part separately:
- The integer part is [tex]\(-7\)[/tex], which is [tex]\(-7 \cdot 7 = -49\)[/tex].
- The fractional part is [tex]\(\frac{4}{7}\)[/tex].
3. Combine the two parts:
[tex]\[
-49 + \frac{4}{7} \implies \frac{-49 \cdot 7 + 4}{7} = \frac{-49 + 4}{7} = \frac{-45}{7}
\][/tex]
So, [tex]\(-7 \frac{4}{7} = -\frac{53}{7}\)[/tex].
#### Converting [tex]\(-8 \frac{1}{5}\)[/tex] to an Improper Fraction
1. The mixed number is [tex]\(-8 \frac{1}{5}\)[/tex].
2. Convert the integer part and the fractional part separately:
- The integer part is [tex]\(-8\)[/tex], which is [tex]\(-8 \cdot 5 = -40\)[/tex].
- The fractional part is [tex]\(\frac{1}{5}\)[/tex].
3. Combine the two parts:
[tex]\[
-40 + \frac{1}{5} \implies \frac{-40 \cdot 5 + 1}{5} = \frac{-40 + 1}{5} = \frac{-39}{5}
\][/tex]
So, [tex]\(-8 \frac{1}{5} = -\frac{41}{5}\)[/tex].
### Step 2: Perform the Division
Now, we need to divide the two improper fractions:
[tex]\[
-\frac{53}{7} \div -\frac{41}{5}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
-\frac{53}{7} \div -\frac{41}{5} = -\frac{53}{7} \times -\frac{5}{41}
\][/tex]
### Step 3: Simplify the Multiplication
Before multiplying, note that the two negative signs will cancel each other out:
[tex]\[
-\frac{53}{7} \times -\frac{5}{41} = \frac{53}{7} \times \frac{5}{41}
\][/tex]
Now, perform the multiplication:
[tex]\[
\frac{53 \cdot 5}{7 \cdot 41} = \frac{265}{287}
\][/tex]
### Conclusion
Thus, the result of [tex]\(-7 \frac{4}{7} \div -8 \frac{1}{5}\)[/tex] is [tex]\(\frac{265}{287}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{265}{287}}
\][/tex]
