To find the length of the plot of corn, we start with the width and the given relationship between the length and the width.
The width of the plot is given as \( 5 \frac{3}{5} \) yards. We can first convert this mixed number into an improper fraction:
\[ 5 \frac{3}{5} = 5 + \frac{3}{5} = \frac{5 \times 5}{5} + \frac{3}{5} = \frac{25}{5} + \frac{3}{5} = \frac{25 + 3}{5} = \frac{28}{5} \]
Next, we know that the length of the plot is \( 9 \frac{1}{2} \) times the width. We'll also convert this mixed number into an improper fraction:
\[ 9 \frac{1}{2} = 9 + \frac{1}{2} = \frac{9 \times 2}{2} + \frac{1}{2} = \frac{18}{2} + \frac{1}{2} = \frac{18 + 1}{2} = \frac{19}{2} \]
Now we can use these fractions to find the length:
Let the width \( w = \frac{28}{5} \) and the factor for length \( f = \frac{19}{2} \).
The length \( l \) can be calculated as:
\[ l = f \times w = \frac{19}{2} \times \frac{28}{5} \]
To multiply these fractions, we multiply the numerators together and the denominators together:
\[ l = \frac{19 \times 28}{2 \times 5} = \frac{532}{10} \]
Now, we can simplify \( \frac{532}{10} \). Both the numerator and the denominator can be divided by 2:
\[ l = \frac{532 \div 2}{10 \div 2} = \frac{266}{5} \]
Next, we convert \( \frac{266}{5} \) into a mixed number:
To do this, we divide \( 266 \) by \( 5 \):
\[ 266 \div 5 = 53 \quad \text{(remainder: 1)} \]
Thus, \( 266 = 5 \times 53 + 1\), and we can write:
\[ \frac{266}{5} = 53 \frac{1}{5} \]
Therefore, the length of the plot of corn is
\[ \boxed{53 \frac{1}{5}} \text{ yards.} \]
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