To find the quotient [tex]\(\frac{-\frac{6}{7}}{\frac{8}{5}}\)[/tex] and simplify it completely, follow these steps:
1. Understand the problem:
We need to divide two fractions: [tex]\(\frac{-6}{7}\)[/tex] by [tex]\(\frac{8}{5}\)[/tex].
2. Invert the divisor:
Division by a fraction is the same as multiplying by its reciprocal. Therefore, we need to multiply [tex]\(\frac{-6}{7}\)[/tex] by the reciprocal of [tex]\(\frac{8}{5}\)[/tex], which is [tex]\(\frac{5}{8}\)[/tex].
3. Multiply the fractions:
[tex]\[
\frac{-6}{7} \times \frac{5}{8} = \frac{-6 \times 5}{7 \times 8} = \frac{-30}{56}
\][/tex]
4. Simplify the result:
We need to simplify the fraction [tex]\(\frac{-30}{56}\)[/tex]. To do this, we find the greatest common divisor (GCD) of [tex]\(-30\)[/tex] and [tex]\(56\)[/tex].
5. Find the GCD:
The GCD of [tex]\(-30\)[/tex] and [tex]\(56\)[/tex] is [tex]\(2\)[/tex].
6. Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{-30 \div 2}{56 \div 2} = \frac{-15}{28}
\][/tex]
Therefore, the quotient [tex]\(\frac{-\frac{6}{7}}{\frac{8}{5}} = -\frac{15}{28}\)[/tex].
So, the number that belongs in the green box is:
[tex]\[
\boxed{28}
\][/tex]
