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?
2/3
8/15
6/15
1/3
2/9
The fraction of the rooms painted last year is 2/3.
To find out how much of Paige's home was painted last year, we need to find the fraction of the total rooms that were painted.
First, we determine the fraction of rooms painted by the end of the year, which is 2/3.
Next, we can set up a proportion to find out how much 2/3 represents in terms of the total rooms:
2/3 = x/(total rooms)
Now, we need to solve for x by cross-multiplying:
2(total rooms) = 3x
Dividing both sides by 3:
(total rooms) = (3x)/2
So, 2/3 represents (3x)/2 of the total rooms.
Now, we know that Paige decided to paint 4/5 of the rooms in her home last year. So, we can set up another proportion to find out how much 4/5 represents in terms of the total rooms:
4/5 = y/(total rooms)
Now, we need to solve for y by cross-multiplying:
4(total rooms) = 5y
Dividing both sides by 5:
(total rooms) = (5y)/4
Therefore, 4/5 represents (5y)/4 of the total rooms.
Since we know that 2/3 represents (3x)/2 of the total rooms, and 4/5 represents (5y)/4 of the total rooms, the fraction of Paige's home that was painted last year is equal to the sum of these two fractions:
2/3 + 4/5 = (3x)/2 + (5y)/4
Finding a common denominator:
(8/12) + (6/12) = (6x+5y)/4
Simplifying:
14/12 = (6x+5y)/4
Now, we can cross-multiply to solve for (6x+5y):
4(14) = 12(6x+5y)
56 = 72x + 60y
Rearranging the equation:
12y = 72x - 56
Dividing both sides by 12:
y = (72x - 56)/12
Now, we can substitute the given value of x = 2/3:
y = (72(2/3) - 56)/12
Simplifying:
y = (48 - 56)/12
y = -8/12
Therefore, the fraction of Paige's home that was painted last year is 2/3 + (-8/12) = -2/12.
Since fractions cannot be negative, we know that there is an error in our calculations.
Reviewing the problem, it seems that there is a contradiction between the information given. The statement says that Paige decided to paint 4/5 of the rooms, but is also stated that she completed only 2/3 of the rooms. However, it is not possible to complete more rooms than initially decided to paint.
Therefore, there is no valid solution to the problem.
To solve this problem, we need to determine the fraction of Paige's home that was painted last year.
First, we are given that Paige painted 4/5 of the rooms initially. This means that if her home has a total of x rooms, she painted (4/5)x rooms.
Next, we are told that by the end of the year, Paige completed 2/3 of the rooms. This means that she painted (2/3)x rooms.
To find the fraction of Paige's home that was painted, we need to divide the number of rooms she painted by the total number of rooms in her home.
The fraction is given by:
[(2/3)x] / x
Since x/x is equal to 1, we can simplify the expression to:
2/3
Therefore, 2/3 of Paige's home was painted last year.