To divide the mixed number \(4 \frac{2}{1}\) by the mixed number \(7 \frac{5}{1}\), we first convert the mixed numbers into improper fractions.
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Convert \(4 \frac{2}{1}\): \[ 4 \frac{2}{1} = 4 + \frac{2}{1} = \frac{4 \times 1 + 2}{1} = \frac{4 + 2}{1} = \frac{6}{1} \]
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Convert \(7 \frac{5}{1}\): \[ 7 \frac{5}{1} = 7 + \frac{5}{1} = \frac{7 \times 1 + 5}{1} = \frac{7 + 5}{1} = \frac{12}{1} \]
Now, we have the problem:
\[ \frac{6}{1} \div \frac{12}{1} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{6}{1} \div \frac{12}{1} = \frac{6}{1} \times \frac{1}{12} = \frac{6 \times 1}{1 \times 12} = \frac{6}{12} \]
Now simplify \(\frac{6}{12}\):
\[ \frac{6}{12} = \frac{1}{2} \]
Thus, the answer is:
\[ \frac{1}{2} \] or as a mixed number, it remains \(\frac{1}{2}\).