To find the fraction that Eric is thinking of, we can set up a relationship using variables based on the information provided in the riddle.
Let the denominator of the fraction be \( d \). According to the riddle, the numerator is 18 less than the denominator, so we can express the numerator \( n \) as:
\[ n = d - 18 \]
The fraction can then be expressed as:
\[ \frac{n}{d} = \frac{d - 18}{d} \]
We are also told that this fraction is equivalent to \( \frac{2}{5} \). Therefore, we can set up the equation:
\[ \frac{d - 18}{d} = \frac{2}{5} \]
To solve for \( d \), we can cross-multiply:
\[ 5(d - 18) = 2d \]
Expanding the left side gives:
\[ 5d - 90 = 2d \]
Now, we can subtract \( 2d \) from both sides:
\[ 5d - 2d - 90 = 0 \]
This simplifies to:
\[ 3d - 90 = 0 \]
Adding 90 to both sides yields:
\[ 3d = 90 \]
Now, divide by 3:
\[ d = 30 \]
Now that we have the denominator, we can find the numerator:
\[ n = d - 18 = 30 - 18 = 12 \]
So the fraction is:
\[ \frac{n}{d} = \frac{12}{30} \]
To confirm that this fraction is equivalent to \( \frac{2}{5} \), we can simplify \( \frac{12}{30} \):
\[ \frac{12 \div 6}{30 \div 6} = \frac{2}{5} \]
Thus, \( \frac{12}{30} \) is indeed equivalent to \( \frac{2}{5} \).
Therefore, the fraction that Eric is thinking of is \( \frac{12}{30} \).