To solve the division of fractions [tex]\(\frac{-9}{4} \div \frac{27}{-2}\)[/tex], one needs to follow several steps systematically.
### Step-by-Step Solution
1. Understanding Division of Fractions:
Dividing by a fraction is equivalent to multiplying by its reciprocal.
So, [tex]\(\frac{-9}{4} \div \frac{27}{-2}\)[/tex] can be expressed as:
[tex]\[
\frac{-9}{4} \times \text{reciprocal of} \, \frac{27}{-2}
\][/tex]
2. Find the Reciprocal:
The reciprocal of [tex]\(\frac{27}{-2}\)[/tex] is [tex]\(\frac{-2}{27}\)[/tex].
3. Set Up the Multiplication:
Rewriting the expression using the reciprocal, we get:
[tex]\[
\frac{-9}{4} \times \frac{-2}{27}
\][/tex]
4. Multiply the Numerators:
Multiply the numerators of the fractions:
[tex]\[
-9 \times -2 = 18
\][/tex]
5. Multiply the Denominators:
Multiply the denominators of the fractions:
[tex]\[
4 \times 27 = 108
\][/tex]
6. Form the New Fraction:
After performing the multiplications, we have:
[tex]\[
\frac{18}{108}
\][/tex]
7. Simplify the Fraction:
Simplify [tex]\(\frac{18}{108}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 18:
[tex]\[
\frac{18 \div 18}{108 \div 18} = \frac{1}{6}
\][/tex]
### Conclusion
Therefore, after following these steps, the solution to [tex]\(\frac{-9}{4} \div \frac{27}{-2}\)[/tex] is:
[tex]\[
\frac{1}{6}
\][/tex]
Answer:
[tex]\[
\boxed{\frac{1}{6}}
\][/tex]
