liza arranged 66 roses in seven vases. some vases had 6 red roses and the others had 12 pink roses. how many vases had red roses?
p+r = 7
6p+12r = 66
since p = 7-r,
6(7-r)+12r = 66
42+6r = 66
r = 4
Well, I must confess that Liza certainly knows how to spread some flower power! Let's do some math, shall we? We know that each vase had either 6 red roses or 12 pink roses. Since there were 7 vases in total, let's assume x vases had red roses. This means the remaining (7 - x) vases had pink roses.
Now, when we multiply the number of vases with red roses (x) by the number of red roses in each vase (6), and the number of vases with pink roses (7 - x) by the number of pink roses in each vase (12), the total number of roses should add up to 66.
So, we can set up the equation:
6x + 12(7 - x) = 66
Now, let's solve it and see how many vases had those lovely red roses!
Let's start by finding the total number of roses. We know that there are 66 roses in total.
Now, let's assume the number of vases with red roses is "x" and the number of vases with pink roses is "y".
Since some vases had 6 red roses and the others had 12 pink roses, we can write two equations based on this information.
Equation 1: x + y = 7 (Because there are a total of seven vases)
Equation 2: 6x + 12y = 66 (Because there are 6 red roses in each vase with red roses, and 12 pink roses in each vase with pink roses)
Now, we can solve these equations simultaneously.
Multiplying Equation 1 by 6, we get:
6x + 6y = 42
Subtracting Equation 2 from the above equation:
6x + 6y - (6x + 12y) = 42 - 66
6x + 6y - 6x - 12y = -24
-6y = -24
Dividing both sides by -6:
y = 4
Substituting the value of y in Equation 1, we get:
x + 4 = 7
x = 7 - 4
x = 3
Therefore, there are 3 vases with red roses.
To determine how many vases had red roses, we can use the information provided in the question:
1. Start by finding the total number of roses Liza arranged: 66 roses.
2. Next, determine the number of roses in each type of vase: 6 red roses and 12 pink roses.
3. Let's suppose there were "x" vases with red roses. The remaining "7 - x" vases would have pink roses.
4. Now we can set up an equation to solve for "x" by calculating the total number of roses:
Number of red roses + Number of pink roses = Total number of roses
(6 red roses * x vases) + (12 pink roses * (7 - x) vases) = 66 roses
5. Simplify the equation:
6x + 12(7 - x) = 66
6x + 84 - 12x = 66
-6x = 66 - 84
-6x = -18
6. Solve for "x":
x = (-18) / (-6)
x = 3
Therefore, the answer is that 3 vases had red roses.