To find out how much of Paige's home was painted last year, we need to calculate \( \frac{4}{5} \) of the total rooms she started with and then determine how much that equals compared to the \( \frac{2}{3} \) of the rooms that were completed.
Let’s assume the total number of rooms is represented by a common denominator, such as 15, to make the calculations clearer.
- Total rooms (using a common denominator of 15) can be arbitrarily set as 15.
- The size of the home can be seen as \( 1 \) unit, which equals 15 rooms.
Now, let’s compute \( \frac{4}{5} \) of 15:
\[ \frac{4}{5} \times 15 = 12 \text{ rooms} \]
Next, let's calculate \( \frac{2}{3} \) of the total rooms (still using 15):
\[ \frac{2}{3} \times 15 = 10 \text{ rooms} \]
Now, we are trying to establish how much of her home was painted.
Since she intended to paint \( 12 \) rooms but only painted \( 10 \), we need to find what fraction of the total rooms (15) that represents:
\[ \text{Fraction painted} = \frac{10}{15} = \frac{2}{3} \]
Overall, we see that out of the planned \( \frac{4}{5} \) of the home, Paige painted \( \frac{2}{3} \) of the total rooms.
Since the answer choices include \( \frac{2}{3} \), that would be the correct answer.
Thus, the amount of Paige's home that was painted last year is:
2/3.