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
To solve this problem, we can set up a system of equations to represent the given information.
Let's assume that the amount of jelly beans in pounds is x. Since the mixture calls for two times as many gummy candy pieces as jelly beans, the amount of gummy candy in pounds would be 2x.
The total weight of the mixture is 9 pounds, so we can write the equation: x + 2x + y = 9, where y represents the weight of the hard candy.
The cost of gummy candy per pound is $2.00, so the cost of the gummy candy would be 2 * (2x) = 4x.
The cost of jelly beans per pound is $3.00, so the cost of the jelly beans would be 3 * x = 3x.
The cost of hard candy per pound is $3.00, so the cost of the hard candy would be 3 * y = 3y.
The total cost of the mixture is $23.00, so we can write the equation: 4x + 3x + 3y = 23.
Now we can solve the system of equations:
x + 2x + y = 9
4x + 3x + 3y = 23
Combining like terms, we get:
3x + y = 9
7x + 3y = 23
We can eliminate y by multiplying the first equation by 3 and the second equation by 1:
9x + 3y = 27
7x + 3y = 23
Subtracting the second equation from the first equation, we get:
(9x + 3y) - (7x + 3y) = 27 - 23
2x = 4
x = 2
Substituting x = 2 into the first equation, we get:
3(2) + y = 9
6 + y = 9
y = 3
Therefore, the solution is x = 2, y = 3.
So, the candy store used 2 pounds of gummy candy, 3 pounds of jelly beans, and 4 pounds of hard candy.
The answer is D. 2 lbs. gummy candy, 3 lbs. jelly beans, 4 lbs. hard candy.
To solve this problem, we need to set up a system of equations.
Let's represent the amount of gummy candy as 'g', the amount of jelly beans as 'j', and the amount of hard candy as 'h'.
From the problem, we know the following information:
1. The total weight of the mixture is 9 pounds.
So, g + j + h = 9 ----(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 $23.00.
So, 2g + 3j + 3h = 23 ----(Equation 2)
3. The mixture calls for two times as many gummy candy pieces as jelly beans.
So, g = 2j ----(Equation 3)
Using these equations, we can solve for the values of g, j, and h.
We can start by solving the system of equations (Equations 1, 2, and 3) to find the values of g, j, and h.
Substituting Equation 3 into Equation 1:
2j + j + h = 9
3j + h = 9 ----(Equation 4)
Substituting Equation 3 into Equation 2:
2(2j) + 3j + 3h = 23
4j + 3j + 3h = 23
7j + 3h = 23 ----(Equation 5)
Now we have two equations (Equations 4 and 5) with two variables (j and h). We can solve these equations simultaneously.
Multiplying Equation 4 by 3:
3j + h = 9
Subtracting this equation from Equation 5:
7j + 3h - (3j + h) = 23 - 9
4j + 2h = 14
2j + h = 7 ----(Equation 6)
Now we have Equation 6 and Equation 4, which is a system of equations with two variables (j and h). We can solve this system of equations simultaneously.
Multiplying Equation 6 by 3:
3(2j + h) = 3(7)
6j + 3h = 21 ----(Equation 7)
Subtracting this equation from Equation 5:
(4j + 2h) - (6j + 3h) = 14 - 21
-2j - h = -7
2j + h = 7 ----(Equation 8)
Equations 7 and 8 are essentially the same equation. This means that there are infinitely many solutions and we can solve for the variables in terms of one of the variables.
Let's solve for h in terms of j:
From Equation 7:
6j + 3h = 21
3h = 21 - 6j
h = (21 - 6j)/3
h = 7 - 2j ----(Equation 9)
Now, we can substitute Equation 9 into Equation 3 to get g in terms of j:
From Equation 3:
g = 2j
So, the solution is:
g = 2j
h = 7 - 2j
To find the specific amounts of each ingredient, we need to substitute these values into Equation 1:
g + j + h = 9
2j + j + (7 - 2j) = 9
2j + j + 7 - 2j = 9
3j + 7 = 9
3j = 2
j = 2/3
Now that we have the value of j, we can find the values of g and h:
g = 2j
g = 2(2/3) = 4/3
h = 7 - 2j
h = 7 - 2(2/3)
h = 7 - 4/3
h = 21/3 - 4/3
h = 17/3
Therefore, the amounts of each ingredient are approximately:
g = 4/3 pounds
j = 2/3 pounds
h = 17/3 pounds
Now let's check which answer option matches these amounts:
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
The only option that matches the amounts we found is option D.
Therefore, the correct answer is:
D. 2 lbs. gummy candy, 3 lbs. jelly beans, 4 lbs. hard candy
The answer is: 6Ibs. gummy candy, 2 Ibs. jelly beans, 5Ibs. hard candy
so, what do you know?
g+j+h = 9
2g+3j+3h = 23
g = 2j
Can you take it from there?