Let the denominator of Eric's fraction be \( x \). According to the riddle, the numerator is 18 less than the denominator. Therefore, we can express the numerator as \( x - 18 \).
So, the fraction can be written as:
\[ \frac{x - 18}{x} \]
According to the riddle, this fraction is equivalent to \( \frac{2}{5} \). Therefore, we can set up the equation:
\[ \frac{x - 18}{x} = \frac{2}{5} \]
To solve this equation, we can cross-multiply:
\[ 5(x - 18) = 2x \]
Expanding the left side:
\[ 5x - 90 = 2x \]
Now, we can rearrange the equation by subtracting \( 2x \) from both sides:
\[ 5x - 2x - 90 = 0 \]
This simplifies to:
\[ 3x - 90 = 0 \]
Adding 90 to both sides gives:
\[ 3x = 90 \]
Now, divide both sides by 3:
\[ x = 30 \]
Now that we have \( x \), we can find the numerator:
\[ \text{Numerator} = x - 18 = 30 - 18 = 12 \]
Thus, the fraction is:
\[ \frac{12}{30} \]
To verify that this fraction is equivalent to \( \frac{2}{5} \), we can simplify \( \frac{12}{30} \):
\[ \frac{12 \div 6}{30 \div 6} = \frac{2}{5} \]
Since the simplification confirms the equivalence, Eric's fraction is:
\[ \frac{12}{30} \]
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