To divide the mixed numbers \(-4 \frac{2}{3}\) and \(3 \frac{1}{5}\), we first need to convert them into improper fractions.
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Convert the mixed numbers to improper fractions:
For \(-4 \frac{2}{3}\):
\[ -4 \frac{2}{3} = -\left(4 \times 3 + 2\right)/3 = -\left(12 + 2\right)/3 = -14/3 \]
For \(3 \frac{1}{5}\):
\[ 3 \frac{1}{5} = 3 \times 5 + 1/5 = (15 + 1)/5 = 16/5 \]
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Rewrite the division of the fractions as multiplication by the reciprocal:
\[ -\frac{14}{3} \div \frac{16}{5} = -\frac{14}{3} \times \frac{5}{16} \]
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Multiply the fractions:
\[ = -\frac{14 \times 5}{3 \times 16} = -\frac{70}{48} \]
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Simplify the fraction:
We need to find the greatest common divisor (GCD) of 70 and 48. The GCD is 2, so we divide both the numerator and denominator by 2:
\[ -\frac{70 \div 2}{48 \div 2} = -\frac{35}{24} \]
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Convert to a mixed number:
To convert \(-\frac{35}{24}\) to a mixed number, we divide 35 by 24:
\[ 35 \div 24 = 1 \quad \text{(with a remainder of 11)} \]
So, we can express it as:
\[ -\frac{35}{24} = -1 \frac{11}{24} \]
Thus, the final simplified mixed number is:
\[ \boxed{-1 \frac{11}{24}} \]
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