To solve the given fraction division problem, we need to follow these steps:
1. Understand the division of fractions:
When dividing two fractions, we multiply the first fraction by the reciprocal of the second fraction. In this case, we are given:
[tex]\[
\frac{2}{3} \div -\frac{7}{9}
\][/tex]
The division of [tex]\(\frac{2}{3}\)[/tex] by [tex]\(-\frac{7}{9}\)[/tex] can be rewritten as:
[tex]\[
\frac{2}{3} \times \left( -\frac{9}{7} \right)
\][/tex]
2. Multiply the fractions:
To multiply two fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{2}{3} \times \left( -\frac{9}{7} \right) = \frac{2 \times -9}{3 \times 7} = \frac{-18}{21}
\][/tex]
3. Simplify the fraction:
To simplify the fraction [tex]\(\frac{-18}{21}\)[/tex], we need to find the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 18 and 21 is 3. So, we divide both the numerator and the denominator by 3:
[tex]\[
\frac{-18 \div 3}{21 \div 3} = \frac{-6}{7}
\][/tex]
4. Result:
The simplified quotient is:
[tex]\[
-\frac{6}{7}
\][/tex]
Therefore, the correct number that belongs in the green box is:
[tex]\[
\boxed{6}
\][/tex]
