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?%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A3 lbs. gummy candy, 2 lbs. jelly beans, 4 lbs. hard candy%0D%0A3 lbs. gummy candy, 2 lbs. jelly beans, 4 lbs. hard candy%0D%0A%0D%0A4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy%0D%0A4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy%0D%0A%0D%0A4 lbs. gummy candy, 3 lbs. jelly beans, 2 lbs. hard candy%0D%0A4 lbs. gummy candy, 3 lbs. jelly beans, 2 lbs. hard candy%0D%0A%0D%0A2 lbs. gummy candy, 3 lbs. jelly beans, 4 lbs. hard candy%0D%0A2 lbs. gummy candy, 3 lbs. jelly beans, 4 lbs. hard candy
To solve this problem, let's denote the number of pounds of gummy candy as G, the number of pounds of jelly beans as J, and the number of pounds of hard candy as H.
We are given that the total weight of the mixture is 9 pounds:
G + J + H = 9 (equation 1)
We are also given that the cost of gummy candy is $2.00 per pound, the cost of jelly beans is $3.00 per pound, and the cost of hard candy is $3.00 per pound. The total cost of the mixture is $23.00:
2G + 3J + 3H = 23 (equation 2)
The mixture calls for two times as many gummy candy pieces as jelly beans:
G = 2J (equation 3)
We can solve this system of equations to determine the values of G, J, and H.
From equation 3, we can substitute G = 2J into equation 1 and simplify:
2J + J + H = 9
3J + H = 9 (equation 4)
Now we have a system of two equations with two variables (equations 2 and 4). We can solve this system using substitution or elimination method.
Let's solve it using the substitution method:
From equation 4, we can solve for H:
H = 9 - 3J
Substitute this value of H into equation 2:
2G + 3J + 3(9 - 3J) = 23
2G + 3J + 27 - 9J = 23
2G - 6J = -4
G - 3J = -2 (equation 5)
Now we have a system of two equations with two variables (equations 3 and 5).
Substituting G = 2J from equation 3 into equation 5:
2J - 3J = -2
-J = -2
J = 2
Substitute this value of J into equation 3 to solve for G:
G = 2J
G = 2(2)
G = 4
Now substitute the values of G and J into equation 4 to solve for H:
3J + H = 9
3(2) + H = 9
6 + H = 9
H = 9 - 6
H = 3
So, the solution is G = 4, J = 2, and H = 3.
Therefore, the candy store used 4 pounds of gummy candy, 2 pounds of jelly beans, and 3 pounds of hard candy.
To solve this problem, we can set up a system of equations based on the given information.
Let's denote the amount of gummy candy as "g" in pounds, the amount of jelly beans as "j" in pounds, and the amount of hard candy as "h" in pounds.
1. From the problem, we know that the candy store makes a 9-pound mixture:
g + j + h = 9 (Equation 1)
2. We are also told that the mixture calls for two times as many gummy candy pieces as jelly beans:
g = 2j (Equation 2)
3. Finally, we are given that the total cost of the mixture is $23.00. To determine the cost, we multiply the cost per pound by the amount of each candy and sum them up:
2.00g + 3.00j + 3.00h = 23 (Equation 3)
Now, we can solve this system of equations to find the values of g, j, and h.
To make things easier, we can substitute Equation 2 into Equations 1 and 3, eliminating the variable g:
1. Substitute g = 2j into Equation 1:
2j + j + h = 9
3j + h = 9 (Equation 4)
2. Substitute g = 2j into Equation 3:
2.00(2j) + 3.00j + 3.00h = 23
4j + 3j + 3h = 23
7j + 3h = 23 (Equation 5)
Now, we can solve Equations 4 and 5 simultaneously.
Using Equation 4, we can solve for h in terms of j:
h = 9 - 3j (Equation 6)
Substitute Equation 6 into Equation 5:
7j + 3(9 - 3j) = 23
7j + 27 - 9j = 23
-2j = -4
j = 2
Substitute the value of j back into Equation 6 to find h:
h = 9 - 3(2)
h = 9 - 6
h = 3
Substitute the values of j and h back into Equation 2 to find g:
g = 2j
g = 2(2)
g = 4
Therefore, the store used 4 pounds of gummy candy, 2 pounds of jelly beans, and 3 pounds of hard candy. So the correct answer is:
4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy.
Let's solve this step-by-step:
1. Let's assign variables to the unknown quantities:
- Let's call the number of pounds of gummy candy "g".
- Let's call the number of pounds of jelly beans "j".
- Let's call the number of pounds of hard candy "h".
2. We are given that the total weight of the mixture is 9 pounds, so we can write the first equation:
g + j + h = 9 ...(Equation 1)
3. We are also given that the cost of gummy candy is $2.00 per pound, jelly beans are $3.00 per pound, and hard candy is $3.00 per pound. The total cost of the mixture is $23.00, so we can write the second equation:
2g + 3j + 3h = 23 ...(Equation 2)
4. The mixture calls for two times as many gummy candy pieces as jelly beans, so we can write the third equation:
g = 2j ...(Equation 3)
5. Now we have a system of three equations with three variables. We can solve this system to find the values of g, j, and h.
6. Substituting g = 2j from Equation 3 into Equation 1, we get:
2j + j + h = 9
3j + h = 9 ...(Equation 4)
7. Substituting g = 2j from Equation 3 into Equation 2, we get:
2(2j) + 3j + 3h = 23
4j + 3j + 3h = 23
7j + 3h = 23 ...(Equation 5)
8. Now we have two equations with two variables (Equations 4 and 5). Let's solve this system.
9. Multiply Equation 4 by 7, and Equation 5 by 3 to make the coefficients of "j" in both equations equal:
(7j + h) * 7 becomes 21j + 7h = 63 ...(Equation 6)
(7j + 3h) * 3 becomes 21j + 9h = 69 ...(Equation 7)
10. Subtract Equation 6 from Equation 7 to eliminate "j":
(21j + 9h) - (21j + 7h) = 69 - 63
2h = 6
h = 3 ...(Equation 8)
11. Substitute the value of h = 3 into Equation 4:
3j + 3 = 9
3j = 6
j = 2 ...(Equation 9)
12. Substitute the values of h = 3 and j = 2 into Equation 1 to find g:
g + 2 + 3 = 9
g + 5 = 9
g = 4 ...(Equation 10)
13. Therefore, the solution is:
g = 4 pounds of gummy candy
j = 2 pounds of jelly beans
h = 3 pounds of hard candy
So the correct answer is: 4 lbs. gummy candy, 2 lbs. jelly beans, 3 lbs. hard candy.