To multiply \( \frac{1}{9} \) by \( 4 \frac{1}{5} \), we first convert \( 4 \frac{1}{5} \) into an improper fraction.
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Convert \( 4 \frac{1}{5} \): \[ 4 \frac{1}{5} = 4 \times 5 + 1 = 20 + 1 = 21 \] Thus, \( 4 \frac{1}{5} = \frac{21}{5} \).
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Now multiply \( \frac{1}{9} \) by \( \frac{21}{5} \): \[ \frac{1}{9} \times \frac{21}{5} = \frac{1 \times 21}{9 \times 5} = \frac{21}{45} \]
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Simplify \( \frac{21}{45} \): Both the numerator and denominator can be divided by their greatest common divisor, which is 3. \[ \frac{21 \div 3}{45 \div 3} = \frac{7}{15} \]
Thus, the answer in simplest form is \( \frac{7}{15} \).