A grocer sells milk chocolate at $2.90 per pound, dark chocolate at $4.40 per pound, and dark chocolate with almonds at $5.50 per pound. He wants to make a mixture of 50 pounds of mixed chocolates to sell at $4.50 per pound. The mixture is to contain as many pounds of dark chocolate with almonds as milk chocolate and dark chocolate combined. How many pounds of each type must he use in this mixture?
Your mistake was assuming that since the almond chocolate was half the total, the other two were the same amount.
To solve this problem, we can use a system of equations. Let's denote the number of pounds of milk chocolate as 'x', the number of pounds of dark chocolate as 'y', and the number of pounds of dark chocolate with almonds as 'z'.
Given that the grocer wants to make a mixture of 50 pounds of chocolates, we have the equation:
x + y + z = 50 ----(1)
The cost per pound of the mixture is given as $4.50. So, the total cost of the mixture will be:
2.90x + 4.40y + 5.50z = 4.50 * 50 ----(2)
Lastly, the problem states that the amount of dark chocolate with almonds should be equal to the sum of milk chocolate and dark chocolate. So, we have the equation:
z = x + y ----(3)
Now, we can solve this system of equations to find the values of 'x', 'y', and 'z'.
We can begin by rearranging equation (3) to express 'x' in terms of 'z' and 'y':
x = z - y
Substituting this value in equation (1), we get:
(z - y) + y + z = 50
2z = 50
z = 25
Now, using this value of 'z' in equation (3), we find:
25 = x + y ----(4)
Substituting the values of 'x' and 'z' into equation (2), we get:
2.90x + 4.40y + 5.50z = 225
Substituting the value of 'z' and rearranging equation (4), we have:
x = 25 - y ----(5)
Plugging in the above values, the equation becomes:
2.90(25 - y) + 4.40y + 5.50(25) = 225
Simplifying further, we get:
72.50 - 2.90y + 4.40y + 137.50 = 225
1.5y = 15
Dividing both sides by 1.5, we find:
y = 10
Finally, substituting the value of 'y' into equation (5), we can find 'x':
x = 25 - 10
x = 15
Therefore, the grocer must use 15 pounds of milk chocolate, 10 pounds of dark chocolate, and 25 pounds of dark chocolate with almonds in the mixture in order to meet the given conditions.
let x equal amount of milk chocolate
(25 * 5.5) + (x * 2.9) + [(25 - x) * 4.4] = 50 * 4.5
0.5 * 50Lbs. = 25 Lbs. of dark chocolate with almonds.
12.5 Lbs. of milk chocolate.
12.5 Lbs. of dark chocolate.