Paige decided to paint 4/5 of the rooms in her home last year. By the end of the year, Paige completed 2/3 of the rooms. How much of Paige’s home was painted last year?
Since there are 3 * 5 = <<3*5=15>>15 rooms in her home, 4/5 * 15 = <<4/5*15=12>>12 rooms were painted initially.
By the end of the year, she painted 2/3 * 15 = <<2/3*15=10>>10 rooms.
Thus, last year, 12 + 10 = <<12+10=22>>22 rooms were painted. Answer: \boxed{22}.
To find out how much of Paige's home was painted last year, we need to subtract the fraction of rooms she didn't paint from the total number of rooms in her home.
Let's denote the total number of rooms in Paige's home as "r".
Since Paige painted 4/5 of the rooms last year, the fraction of rooms she did not paint is 1 - 4/5 = 5/5 - 4/5 = 1/5.
By the end of the year, Paige completed 2/3 of the rooms, so the fraction of rooms she did not complete is 1 - 2/3 = 3/3 - 2/3 = 1/3.
To find out how much of Paige's home was painted last year, we subtract the fraction of rooms she did not paint from the fraction of rooms she did not complete: 1/5 - 1/3.
To subtract fractions with different denominators, we need to find a common denominator.
The least common multiple (LCM) of 5 and 3 is 15.
Multiply the numerator and denominator of 1/5 by 3: (1/5) * (3/3) = 3/15.
Multiply the numerator and denominator of 1/3 by 5: (1/3) * (5/5) = 5/15.
Now we can subtract the fractions: 3/15 - 5/15 = (3 - 5)/15 = -2/15.
Therefore, Paige painted -2/15 of her home last year.
To find out how much of Paige's home was painted last year, we need to calculate the total proportion of rooms painted.
First, let's find the proportion of rooms Paige painted last year. Paige painted 4/5 of the rooms, which can be written as 4/5.
Next, let's calculate how much of the rooms Paige completed by the end of the year. Paige completed 2/3 of the rooms, which can be written as 2/3.
To find the proportion of Paige's home that was painted last year, we can multiply the proportion of rooms painted by the proportion of rooms completed: 4/5 * 2/3.
To multiply fractions, we multiply the numerators together and the denominators together: (4 * 2) / (5 * 3) = 8/15.
Therefore, 8/15 of Paige's home was painted last year.