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?

Question

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

Answer 1

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.