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.
"How many lbs of each ingredient did the store use?"
Let's assume that the weight of gummy candy (g) is x pounds.
According to the given information, the weight of jelly beans (j) would be x/4 pounds.
Since the total weight of the mixture is 8 pounds, we can create the equation:
x + x/4 + h = 8
Multiplying the equation by 4 to get rid of the fraction:
4x + x + 4h = 32
Combining like terms:
5x + 4h = 32
We also know that the cost of gummy candy (2x), jelly beans (3(x/4)), and hard candy (3h) adds up to $20.00.
So we can create a second equation:
2x + 3(x/4) + 3h = 20
Multiplying the equation by 4 to get rid of the fraction:
8x + 3x + 12h = 80
Combining like terms:
11x + 12h = 80
Now we have a system of equations:
5x + 4h = 32
11x + 12h = 80
To solve this system, we can use the method of substitution.
From the first equation:
4h = 32 - 5x
Dividing by 4:
h = (32 - 5x)/4
Substituting this value of h into the second equation:
11x + 12((32 - 5x)/4) = 80
Multiplying by 4 to get rid of the fraction:
44x + 12(32 - 5x) = 320
Expanding the expression:
44x + 384 - 60x = 320
Combining like terms:
-16x = -64
Dividing by -16:
x = 4
Substituting this value of x back into the first equation:
5(4) + 4h = 32
20 + 4h = 32
4h = 12
h = 3
Therefore, the candy store used 4 pounds of gummy candy, 1 pound of jelly beans, and 3 pounds of hard candy in the mixture.
Let's assume the number of pounds of gummy candy is "x".
According to the given information, the number of pounds of jelly beans is 1/4 of the number of pounds of gummy candy, which is (1/4)x.
Since the total mixture is 8 pounds, we can set up the equation:
x + (1/4)x + h = 8
Now, let's calculate the value of (1/4)x:
(1/4)x = (1/4)(x/1) = x/4
Now, we can rewrite the equation:
x + x/4 + h = 8
To find the value of "h" (the number of pounds of hard candy), we need an additional equation. We know that the cost of gummy candy per pound is $2.00, jelly beans per pound is $3.00, and hard candy per pound is $3.00. The total cost of the mixture is $20.00. So, using this information, we can set up another equation:
2x + 3(1/4)x + 3h = 20
Now, we have 2 equations:
x + x/4 + h = 8 (Equation 1)
2x + 3(1/4)x + 3h = 20 (Equation 2)
Let's solve these equations simultaneously to find the values of x, h, and (1/4)x.
From Equation 1, we simplify and multiply every term by 4:
4x + x + 4h = 32
5x + 4h = 32
From Equation 2, we simplify and multiply every term by 4:
8x + 3x + 12h = 80
11x + 12h = 80
Now, we can solve these equations simultaneously.
Multiply every term of Equation 1 by -11:
-55x - 44h = -352
Now, add this equation to Equation 2:
-55x - 44h + 11x + 12h = -352 + 80
-44x - 32h = -272
Simplify Equation 3:
11x + 12h = 80
-44x - 32h = -272
Multiply Equation 3 by 11 and Equation 4 by -44:
121x + 132h = 880
176x + 128h = 1088
Now, let's subtract Equation 5 from Equation 6:
(176x + 128h) - (121x + 132h) = 1088 - 880
55x - 4h = 208
Now, we can solve this equation for x:
55x - 4h = 208
55x = 4h + 208
x = (4h + 208)/55
Now, we can substitute this value of x into Equation 1 to solve for h:
x + x/4 + h = 8
[(4h + 208)/55] + [(4h + 208)/55]/4 + h = 8
Simplify and find the common denominator:
[(4h + 208)/55] + [(h + 52)/55] + h = 8
[(4h + 208 + h + 52)/55] + h = 8
[(5h + 260)/55] + h = 8
Multiply every term by 55 to get rid of the denominator:
5h + 260 + 55h = 440
5h + 55h = 440 - 260
60h = 180
Divide by 60:
h = 180/60
h = 3
Now, we know that h = 3. Let's substitute this value of h into Equation 1 to solve for x:
x + x/4 + 3 = 8
(5x + x)/4 = 8 - 3
6x/4 = 5
6x = 20
x = 20/6
x ≈ 3.33
Thus, the store used approximately 3.33 pounds of gummy candy (x), 0.83 pounds of jelly beans ([1/4]x), and 3 pounds of hard candy (h) in the mixture.
To solve this problem, we need to set up a system of equations based on the given information.
Let's denote the pounds of gummy candy, jelly beans, and hard candy as g, j, and h, respectively.
From the given information, we can write the following equations:
1) The total weight of the mixture: g + j + h = 8
2) The cost of the mixture: 2g + 3j + 3h = 20
3) The ratio of gummy candy to jelly beans: g = 4j
Now, we can solve this system of equations to find the values of g, j, and h.
First, let's substitute the value of g from equation 3 into equation 1:
4j + j + h = 8
5j + h = 8
Next, we substitute the values of g and j into equation 2:
2(4j) + 3j + 3h = 20
8j + 3j + 3h = 20
11j + 3h = 20
Now, we have a system of two equations with two variables:
5j + h = 8
11j + 3h = 20
To solve this system, we can use the elimination method.
Multiply the first equation by 3 and the second equation by 5 to create a system where the coefficients of h will eliminate each other:
15j + 3h = 24
55j + 15h = 100
Subtract the first equation from the second equation:
(55j + 15h) - (15j + 3h) = 100 - 24
40j + 12h = 76
Now, we have a new equation:
40j + 12h = 76
Divide this equation by 4 to simplify:
10j + 3h = 19
Now, we have a new system of equations:
5j + h = 8
10j + 3h = 19
Multiply the first equation by 3 and subtract it from the second equation:
(10j + 3h) - (3(5j + h)) = 19 - (3*8)
10j + 3h - 15j - 3h = 19 - 24
-5j = -5
j = 1
Now substitute the value of j back into the first equation to find h:
5(1) + h = 8
5 + h = 8
h = 3
Finally, substitute the values of j and h back into the equation for g:
g = 4(1)
g = 4
Therefore, the candy store used 4 pounds of gummy candy, 1 pound of jelly beans, and 3 pounds of hard candy in the mixture.