To multiply the mixed number \(4 \frac{1}{5}\) by \(\frac{1}{9}\), we first convert the mixed number to an improper fraction.
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Convert \(4 \frac{1}{5}\) to an improper fraction: \[ 4 \frac{1}{5} = \frac{4 \times 5 + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5} \]
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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 \(\frac{21}{45}\):
- Find the greatest common divisor (GCD) of 21 and 45. The GCD is 3.
- Divide both the numerator and denominator by their GCD: \[ \frac{21 \div 3}{45 \div 3} = \frac{7}{15} \]
The answer in simplest form is \(\frac{7}{15}\).
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