To find out how long Patricia studied for her science test, we first need to determine how long she studied for her Social Studies test, which is given as \( 4 \frac{1}{2} \) hours.
First, convert \( 4 \frac{1}{2} \) hours into an improper fraction or a decimal:
\[ 4 \frac{1}{2} = 4 + \frac{1}{2} = 4 + 0.5 = 4.5 \text{ hours} \]
Next, we know that Patricia studied half as long for her science test as she did for the Social Studies test. Therefore, the time she spent studying for her science test is:
\[ \text{Science study time} = \frac{1}{2} \times 4.5 \]
Calculating this gives:
\[ \text{Science study time} = \frac{4.5}{2} = 2.25 \text{ hours} \]
To express \( 2.25 \) hours as a mixed number, we can rewrite it:
\[ 2.25 = 2 + 0.25 = 2 + \frac{1}{4} = 2 \frac{1}{4} \text{ hours} \]
Thus, Patricia studied for her science test for \( 2.25 \) hours or \( 2 \frac{1}{4} \) hours.
In conclusion, Patricia studied for her science test for 2.25 hours or 2 hours and 15 minutes.