To find the quotient of [tex]\(-3 \frac{1}{2} \div 2 \frac{1}{3}\)[/tex], follow these steps:
### Step 1: Convert the Mixed Numbers to Improper Fractions
First, convert [tex]\(-3 \frac{1}{2}\)[/tex] and [tex]\(2 \frac{1}{3}\)[/tex] to improper fractions.
For [tex]\(-3 \frac{1}{2}\)[/tex]:
[tex]\[
-3 \frac{1}{2} = -3 - \frac{1}{2} = -\left(3 + \frac{1}{2}\right) = -\left(\frac{6}{2} + \frac{1}{2}\right) = -\frac{7}{2}
\][/tex]
For [tex]\(2 \frac{1}{3}\)[/tex]:
[tex]\[
2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
\][/tex]
### Step 2: Divide by Multiplying by the Reciprocal
To divide by a fraction, multiply by the reciprocal. So, we need to divide [tex]\(\frac{-7}{2}\)[/tex] by [tex]\(\frac{7}{3}\)[/tex].
Multiply [tex]\(\frac{-7}{2}\)[/tex] by the reciprocal of [tex]\(\frac{7}{3}\)[/tex]:
[tex]\[
\frac{-7}{2} \times \frac{3}{7}
\][/tex]
### Step 3: Perform the Multiplication
Multiply the numerators together and the denominators together:
[tex]\[
\left(\frac{-7 \times 3}{2 \times 7}\right) = \frac{-21}{14}
\][/tex]
### Step 4: Simplify the Fraction
To simplify the fraction [tex]\(\frac{-21}{14}\)[/tex], find the greatest common divisor (GCD) of the numerator and the denominator, which is 7.
Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{-21}{14} = \frac{-21 \div 7}{14 \div 7} = \frac{-3}{2}
\][/tex]
### Conclusion
The quotient of [tex]\(-3 \frac{1}{2} \div 2 \frac{1}{3}\)[/tex] is:
[tex]\[
\boxed{-\frac{3}{2}}
\][/tex]
