To divide the mixed numbers \(4 \frac{2}{1}\) (which is 4.0 in improper fraction form) by \(7 \frac{5}{1}\) (which is 7.0), we first convert them into improper fractions:
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Convert \(4 \frac{2}{1}\) to an improper fraction: \[ 4 \frac{2}{1} = \frac{4 \times 1 + 2}{1} = \frac{4 + 2}{1} = \frac{6}{1} \]
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Convert \(7 \frac{5}{1}\) to an improper fraction: \[ 7 \frac{5}{1} = \frac{7 \times 1 + 5}{1} = \frac{7 + 5}{1} = \frac{12}{1} \]
Now, our problem becomes: \[ \frac{6}{1} \div \frac{12}{1} \]
To divide fractions, we multiply by the reciprocal: \[ \frac{6}{1} \times \frac{1}{12} = \frac{6 \times 1}{1 \times 12} = \frac{6}{12} \]
Now simplify \(\frac{6}{12}\): \[ \frac{6}{12} = \frac{1}{2} \]
Since \(\frac{1}{2}\) is already in simplest form, we can express it as a mixed number: \[ 0 \frac{1}{2} \]
Thus, the final answer is: \[ \frac{1}{2} \]