To find the quotient of the given fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(-\frac{7}{12}\)[/tex], follow these steps:
1. Rewrite the Problem as a Division of Fractions:
The expression [tex]\(\frac{\frac{2}{3}}{-\frac{7}{12}}\)[/tex] needs to be read as a division of fractions. In fraction form, this can be written as:
[tex]\( \frac{2}{3} \div -\frac{7}{12} \)[/tex]
2. Invert the Divisor and Multiply:
When dividing by a fraction, you multiply by its reciprocal. The reciprocal of [tex]\(-\frac{7}{12}\)[/tex] is [tex]\(-\frac{12}{7}\)[/tex]. So, the problem changes to:
[tex]\( \frac{2}{3} \times -\frac{12}{7} \)[/tex]
3. Multiply the Numerators and Denominators:
Now, we multiply the fractions directly:
[tex]\[
\frac{2 \times -12}{3 \times 7} = \frac{-24}{21}
\][/tex]
4. Simplify the Result:
To simplify [tex]\(\frac{-24}{21}\)[/tex], we need to find the greatest common divisor (GCD) of 24 and 21. The GCD of 24 and 21 is 3. Therefore, we divide the numerator and the denominator by 3:
[tex]\[
\frac{-24 \div 3}{21 \div 3} = \frac{-8}{7}
\][/tex]
So, the completely simplified quotient of [tex]\(\frac{2}{3}\)[/tex] divided by [tex]\(-\frac{7}{12}\)[/tex] is [tex]\(\frac{-8}{7}\)[/tex].
By evaluating [tex]\(\frac{-8}{7}\)[/tex] as a decimal, we get approximately:
[tex]\[
-1.1428571428571428
\][/tex]
Thus, the quotient and the simplified fraction is [tex]\( \boxed{\frac{-8}{7}} \)[/tex] which is [tex]\(-1.1428571428571428\)[/tex].
