A hardware store mixes paints in a ratio of two parts blue to six parts red to make two gallons of cabbage red. A ration of five parts blue to three parts red makes two gallons of eggplant purple. A gallon of cabbage red sells for $24, and a gallon of eggplant purple sells for $21. Find the cost of 1 quart of blue paint and the cost of 1 quart of red paint.
One quart of blue paint costs $
One quart of red paint costs $
23 4
To find the cost of 1 quart of blue paint and 1 quart of red paint, we need to determine the cost of 1 gallon for each color first.
From the given information, we know that to make 2 gallons of cabbage red, we need a ratio of 2 parts blue to 6 parts red. This means there are 2/8=1/4 gallons of blue paint and 6/8=3/4 gallons of red paint in 2 gallons of cabbage red.
Since 1 gallon of cabbage red sells for $24, 1 quart (1/4 gallon) of blue paint costs 1/4 * $24 = $6.
Similarly, we know that to make 2 gallons of eggplant purple, we need a ratio of 5 parts blue to 3 parts red. This means there are 5/8 gallons of blue paint and 3/8 gallons of red paint in 2 gallons of eggplant purple.
Since 1 gallon of eggplant purple sells for $21, 1 quart (1/4 gallon) of red paint costs 1/4 * $21 = $5.25.
Therefore, the cost of 1 quart of blue paint is $6 and the cost of 1 quart of red paint is $5.25.
To solve this problem, we need to set up a system of equations based on the given information.
Let's use the following variables:
x = cost of 1 quart of blue paint
y = cost of 1 quart of red paint
From the first statement, we know that two parts blue paint and six parts red paint are needed to make two gallons of cabbage red. Since a gallon is equal to four quarts, this means that we need (2 * 4) 8 quarts of cabbage red paint.
Equation 1: 2x + 6y = 8 * 24
From the second statement, we know that five parts blue paint and three parts red paint are needed to make two gallons of eggplant purple. Again, converting gallons to quarts, this means we need (2 * 4) 8 quarts of eggplant purple paint.
Equation 2: 5x + 3y = 8 * 21
Now, we can solve these equations simultaneously to find the values of x and y.
Multiplying Equation 1 by 3 and Equation 2 by 6 will give us a system of equations with coefficients that can be easily eliminated:
6x + 18y = 8 * 3 * 24
30x + 18y = 8 * 6 * 21
To eliminate y, subtract the second equation from the first:
(6x + 18y) - (30x + 18y) = (8 * 3 * 24) - (8 * 6 * 21)
Simplifying:
-24x = 8 * 3 * 24 - 8 * 6 * 21
-24x = 8(3 * 24 - 6 * 21)
Dividing both sides by -24:
x = (8(3 * 24 - 6 * 21)) / -24
Simplifying further:
x = (8 * (72 - 126)) / -24
x = (8 * (-54)) / -24
x = (-432) / -24
x = 18
Now, substitute the value of x back into one of the original equations to find y:
2x + 6y = 8 * 24
2 * 18 + 6y = 8 * 24
36 + 6y = 192
6y = 192 - 36
6y = 156
y = 156 / 6
y = 26
Therefore, the cost of one quart of blue paint is $18, and the cost of one quart of red paint is $26.
2 b + 6 r = 2 * 24
5 b + 3 r = 2 * 21 ... 10 b + 6 r = 84
subtracting equations (to eliminate r) ... 8 b = 36
solve for b , then substitute back to find r