To multiply the fractions \(-5 \frac{2}{3}\) and \(-1 \frac{1}{2}\), we first need to convert these mixed numbers into improper fractions.
Step 1: Convert Mixed Numbers to Improper Fractions
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Convert \(-5 \frac{2}{3}\): \[ -5 \frac{2}{3} = -\left(5 \times 3 + 2\right)/3 = -\left(15 + 2\right)/3 = -\frac{17}{3} \]
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Convert \(-1 \frac{1}{2}\): \[ -1 \frac{1}{2} = -\left(1 \times 2 + 1\right)/2 = -\left(2 + 1\right)/2 = -\frac{3}{2} \]
Step 2: Multiply the Improper Fractions
Now, we will multiply \(-\frac{17}{3}\) and \(-\frac{3}{2}\):
\[ -\frac{17}{3} \times -\frac{3}{2} = \frac{17 \times 3}{3 \times 2} = \frac{51}{6} \]
Step 3: Simplify the Result
We can simplify \(\frac{51}{6}\):
\[ \frac{51}{6} = \frac{17}{2} \quad (\text{by dividing the numerator and the denominator by 3}) \]
Step 4: Convert Back to Mixed Number (if needed)
To convert \(\frac{17}{2}\) back to a mixed number:
\[ 17 \div 2 = 8 \quad \text{remainder 1} \] So, \(\frac{17}{2} = 8 \frac{1}{2}\).
Final Answer
Therefore, the result of \(-5 \frac{2}{3} \times -1 \frac{1}{2}\) is \(\frac{17}{2}\) or \(8 \frac{1}{2}\).