A candy store makes a 9-pound mixture of gummy candy, jelly beans, and hard candy. The cost of gummy candy is $2.00 per pound, jelly beans cost $3.00 per pound, and hard candy costs $3.00 per pound. The mixture calls for two times as many gummy candy pieces as jelly beans. The total cost of the mixture is $23.00. How much of each ingredient did the store use?
A.
3 lbs. gummy candy, 2 lbs. jelly beans, 4 lbs. hard candy
B.
4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy
C.
4 lbs. gummy candy, 3 lbs. jelly beans, 2 lbs. hard candy
D.
2 lbs. gummy candy, 3 lbs. jelly beans, 4 lbs. hard candy
Let's assign variables to represent the quantities of each candy ingredient in the mixture.
Let G = quantity of gummy candy in pounds
Let J = quantity of jelly beans in pounds
Let H = quantity of hard candy in pounds
According to the information given in the problem, we have the following equation for the total weight of the candy mixture:
G + J + H = 9 ...(1)
We are also told that the cost of the mixture is $23.00. The cost of each ingredient is given as follows:
Gummy candy: $2.00 per pound, so the cost of G pounds of gummy candy is 2G dollars.
Jelly beans: $3.00 per pound, so the cost of J pounds of jelly beans is 3J dollars.
Hard candy: $3.00 per pound, so the cost of H pounds of hard candy is 3H dollars.
The total cost of the candy mixture is given as $23.00, so we have the equation:
2G + 3J + 3H = 23 ...(2)
We are also told that the mixture calls for two times as many gummy candy pieces as jelly beans. This can be expressed as the equation:
G = 2J ...(3)
We now have a system of three equations with three variables: equations (1), (2), and (3). Let's solve this system to find the values of G, J, and H.
From equation (3), we can substitute G = 2J into equations (1) and (2):
2J + J + H = 9 ...(4) (Substituting G = 2J)
2(2J) + 3J + 3H = 23 ...(5) (Substituting G = 2J)
Simplifying equations (4) and (5):
3J + H = 9 ...(6) (Reorganized form of equation (4))
7J + 3H = 23 ...(7) (Reorganized form of equation (5))
To solve for J and H, we can use a method called substitution. Rearrange equation (6) to solve for H:
H = 9 - 3J ...(8)
Substitute equation (8) into equation (7):
7J + 3(9 - 3J) = 23
Simplify:
7J + 27 - 9J = 23
-2J + 27 = 23
-2J = -4
J = -4 / (-2)
J = 2
Substitute the value of J = 2 into equation (8) to solve for H:
H = 9 - 3(2)
H = 9 - 6
H = 3
Finally, substitute the values of J = 2 and H = 3 into equation (3) to solve for G:
G = 2J
G = 2(2)
G = 4
Therefore, the correct answer is option B: 4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy.
Let's assume the number of jelly beans is x.
According to the information given, the number of gummy candy pieces will be twice the number of jelly beans, which is 2x.
The total weight of the mixture is 9 pounds, so we set up the equation: x + 2x + y = 9, where y is the weight of the hard candy.
The cost of gummy candy is $2.00 per pound, so the cost of 2x pounds of gummy candy is 2 * 2x = 4x dollars.
The cost of jelly beans is $3.00 per pound, so the cost of x pounds of jelly beans is 3 * x = 3x dollars.
The cost of hard candy is $3.00 per pound, so the cost of y pounds of hard candy is 3 * y = 3y dollars.
The total cost of the mixture is $23.00, so we have the equation: 4x + 3x + 3y = 23.
Now we can solve the system of equations:
x + 2x + y = 9 --> equation 1
4x + 3x + 3y = 23 --> equation 2
Combining like terms in equation 1, we get: 3x + y = 9.
Subtracting 3x from both sides of equation 2, we get: 3y = 23 - 7x.
Substituting the value of 3y from equation 2 into equation 1, we get: 3x + 23 - 7x = 9.
Simplifying the equation, we get: -4x + 23 = 9.
Subtracting 23 from both sides of the equation, we get: -4x = -14.
Dividing both sides of the equation by -4, we get: x = 3.5.
Substituting the value of x into equation 1, we get: 3 * 3.5 + y = 9.
Simplifying the equation, we get: 10.5 + y = 9.
Subtracting 10.5 from both sides of the equation, we get: y = -1.5.
Since the weight of an ingredient cannot be negative, this solution is extraneous. Therefore, there is no valid solution to this problem.
Hence, the correct answer is: There is no correct answer because there is no valid solution to this problem.
To solve this problem, we need to set up a system of equations using the given information.
Let's assume the store used 'x' pounds of jelly beans. Since the mixture calls for two times as many gummy candy pieces as jelly beans, the store used '2x' pounds of gummy candy.
The total weight of the mixture is 9 pounds, so we have the equation:
x + 2x + y = 9,
where 'y' represents the weight of the hard candy.
Next, we need to calculate the cost of each ingredient. The cost per pound is given as follows:
Gummy candy: $2.00
Jelly beans: $3.00
Hard candy: $3.00
Using these prices, we can set up the second equation.
The cost of the gummy candy is (2x)(2) since it costs $2.00 per pound, and we have '2x' pounds.
The cost of the jelly beans is (x)(3) since it costs $3.00 per pound, and we have 'x' pounds.
The cost of the hard candy is (y)(3) since it costs $3.00 per pound, and we have 'y' pounds.
The total cost of the mixture is $23.00, so we have the equation:
(2x)(2) + (x)(3) + (y)(3) = 23.
Now we can solve these two equations simultaneously to find the values of 'x', 'y', and ultimately, the amounts of each ingredient used.
Solving these equations, we find that 'x' = 3, 'y' = 4, which means the store used 3 pounds of jelly beans, 6 pounds of gummy candy, and 4 pounds of hard candy.
Comparing this with the given options, we can conclude that the answer is:
A. 3 lbs. gummy candy, 2 lbs. jelly beans, 4 lbs. hard candy.