To find out how many minutes Emma spent studying, we need to first convert Sarah's study time into an improper fraction.
Sarah spent \( 35 \frac{1}{2} \) minutes studying. To convert this to an improper fraction:
\[ 35 \frac{1}{2} = \frac{(35 \times 2) + 1}{2} = \frac{70 + 1}{2} = \frac{71}{2} \text{ minutes} \]
Emma studied for \( 1 \frac{1}{2} \) times the amount that Sarah studied. First, we convert \( 1 \frac{1}{2} \) into an improper fraction:
\[ 1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \]
Now we calculate the time Emma spent studying by multiplying Sarah's study time by the factor \( \frac{3}{2} \):
\[ \text{Emma's study time} = \frac{3}{2} \times \frac{71}{2} \]
Multiplying these fractions:
\[ \text{Emma's study time} = \frac{3 \times 71}{2 \times 2} = \frac{213}{4} \]
Next, we'll convert this improper fraction back to a mixed number:
\[ \frac{213}{4} = 53 \frac{1}{4} \]
Therefore, Emma spent \( 53 \frac{1}{4} \) minutes studying, which can also be expressed as:
\[ 53.25 \text{ minutes} \]
Final answer: Emma spent 53 1/4 minutes studying.