To find the quotient of the given fractions and simplify it completely, we'll follow these steps:
1. Identify the fractions involved:
- First fraction: [tex]\(\frac{8}{11}\)[/tex]
- Second fraction: [tex]\(-\frac{4}{5}\)[/tex]
2. Remember the rule for dividing fractions:
Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we convert the division problem into a multiplication one:
[tex]\[
\frac{8}{11} \div -\frac{4}{5} = \frac{8}{11} \times -\frac{5}{4}
\][/tex]
3. Multiply the fractions:
- Multiply the numerators: [tex]\(8 \times -5 = -40\)[/tex]
- Multiply the denominators: [tex]\(11 \times 4 = 44\)[/tex]
Thus, we get:
[tex]\[
\frac{8}{11} \div -\frac{4}{5} = \frac{-40}{44}
\][/tex]
4. Simplify the fraction:
- Find the greatest common divisor (GCD) of 40 and 44, which is 4.
- Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{-40}{44} = \frac{-40 \div 4}{44 \div 4} = \frac{-10}{11}
\][/tex]
Therefore, when you simplify the fraction [tex]\(\frac{\frac{8}{11}}{-\frac{4}{5}}\)[/tex], the simplified form is [tex]\(-\frac{10}{11}\)[/tex].
Now, we need to fill in the blank in the expression [tex]\(-\frac{[?]}{\square}\)[/tex]:
- The numerator in the simplified fraction is [tex]\(-10\)[/tex].
So, the number that belongs in the green box is [tex]\(\boxed{10}\)[/tex].
