A candy store makes an 8-pound mixture of gummy candy (g), jelly beans (j), and hard candy (h). 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 four times as many gummy candy pieces as jelly beans. The total cost of the mixture is $20.00. Question 1 Which system of equations could be used to answer the question, "How many lbs of each ingredient did the store use?"
Use the system of equations you selected above to answer the question, "How many lbs of each ingredient did the store use?"
Let's use the variables g, j, and h to represent the pounds of gummy candy, jelly beans, and hard candy respectively.
From the given information, we can set up the following equations:
1. The total weight of the mixture is 8 pounds: g + j + h = 8.
2. The cost of the mixture is $20.00: 2g + 3j + 3h = 20.
3. The mixture calls for four times as many gummy candy pieces as jelly beans: g = 4j.
We have a system of equations to solve:
g + j + h = 8
2g + 3j + 3h = 20
g = 4j
Let's assign variables to each ingredient:
Let g = pounds of gummy candy
Let j = pounds of jelly beans
Let h = pounds of hard candy
We are given the following information:
1) The total weight of the mixture is 8 pounds:
g + j + h = 8 ---- Equation (1)
2) 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 total cost of the mixture is $20.00:
2g + 3j + 3h = 20 ---- Equation (2)
3) The mixture calls for four times as many gummy candy pieces as jelly beans:
g = 4j ---- Equation (3)
To solve the system of equations, we can use substitution or elimination method. Let's use the substitution method.
From Equation (3), we can substitute g = 4j into Equations (1) and (2):
4j + j + h = 8
5j + h = 8 ---- Equation (4)
2(4j) + 3j + 3h = 20
8j + 3j + 3h = 20
11j + 3h = 20 ---- Equation (5)
Now we have a system of two equations with two variables (Equations 4 and 5). We can solve this system to find the values of j and h, representing the pounds of jelly beans and hard candy used in the mixture.
To solve this problem, we can set up a system of equations based on the given information.
Let's let:
g = pounds of gummy candy
j = pounds of jelly beans
h = pounds of hard candy
1. We know that the total weight of the mixture is 8 pounds:
g + j + h = 8
2. We know that the cost of the mixture is $20.00. We can express this in terms of the cost per pound of each candy:
2g + 3j + 3h = 20
3. We also know that the mixture calls for four times as many gummy candy pieces as jelly beans:
g = 4j
Now we can solve this system of equations using various methods such as substitution or elimination.
Let's use substitution to solve it:
From equation 3, we can express g in terms of j:
g = 4j
Substitute this expression into equation 1:
4j + j + h = 8
5j + h = 8
Now, substitute the values of g and h into equation 2:
2(4j) + 3j + 3h = 20
8j + 3j + 3h = 20
11j + 3h = 20
Now we have two equations:
5j + h = 8
11j + 3h = 20
We can solve this system of equations using any method of solving linear equations, such as substitution, elimination, or matrix methods.
For simplicity, let's solve it using elimination:
Multiply equation 1 by 3 and equation 2 by 5 to eliminate h:
15j + 3h = 24
55j + 15h = 100
Now subtract the two equations:
(55j + 15h) - (15j + 3h) = 100 - 24
40j + 12h = 76
Now we have a new equation:
40j + 12h = 76
Combine this equation with equation 5j + h = 8:
40j + 12h = 76
5j + h = 8
Now we have a system of two equations with two unknowns:
40j + 12h = 76
5j + h = 8
Solve this system of equations to find the values of j, h, and g.