Let the number Amina thinks of be \( x \).
According to the problem, she first subtracts \( \frac{5}{2} \) from \( x \):
\[ x - \frac{5}{2} \]
Next, she multiplies the result by 8:
\[ 8 \left( x - \frac{5}{2} \right) \]
According to the problem, this result is equal to three times the number she thought of, which is \( 3x \). Therefore, we can set up the equation:
\[ 8 \left( x - \frac{5}{2} \right) = 3x \]
Now, we simplify the left side of the equation:
\[ 8x - 8 \cdot \frac{5}{2} = 3x \] \[ 8x - 20 = 3x \]
Next, we can isolate \( x \) by moving \( 3x \) to the left side of the equation:
\[ 8x - 3x - 20 = 0 \] \[ 5x - 20 = 0 \]
Adding 20 to both sides gives:
\[ 5x = 20 \]
Now, divide both sides by 5:
\[ x = 4 \]
Thus, the number Amina thought of is \( \boxed{4} \).