lyle needs 200g of chocolate that is 86% cocoa for a cake recipe he has one kind that is 99% cocoa and another kind that is 70% cocoa. How much of each kind does he need to make the cake
g = grams of 99% cocoa
0.99g = value of 99% cocoa
200 - g = grams of 70% cocoa
0.70(200 - g) = value of 70% cocoa
0.86(200) = value of mixture
0.99g + 0.70(200 - g) = 0.86(200)
Solve for g, grams of 99% cocoa
200 - g = grams of 70% cocoa
110 for 99% and 90 for 70%
To determine how much of each kind of chocolate Lyle needs to make the cake, we can set up a system of equations based on the cocoa percentage and the total weight of chocolate required.
Let's assume Lyle needs x grams of the 99% cocoa chocolate and y grams of the 70% cocoa chocolate.
Based on the percentage of cocoa in the chocolate, we can write the equations:
Equation 1: (0.99x + 0.70y) / (x + y) = 0.86 (86% cocoa)
Equation 2: x + y = 200g (total weight required)
To solve this system of equations, we can use the substitution or elimination method. Let's solve it using the elimination method:
Rearranging Equation 1:
0.99x + 0.70y = 0.86(x + y)
0.99x + 0.70y = 0.86x + 0.86y
0.13x - 0.16y = 0
Multiplying Equation 2 by 0.13 to eliminate x:
0.13x + 0.13y = 26
Now we have the following system of equations:
0.13x - 0.16y = 0
0.13x + 0.13y = 26
Adding the two equations, we get:
0.13x - 0.16y + 0.13x + 0.13y = 0 + 26
0.26x - 0.03y = 26
Now, rearranging this equation:
0.26x = 0.03y + 26
26x = 3y + 2600
y = (26x - 2600) / 3
Substituting this value of y into Equation 2:
x + (26x - 2600) / 3 = 200
3x + 26x - 2600 = 600
29x = 3200
x = 3200 / 29
x ≈ 110.34
Now, substituting the value of x back into Equation 2:
110.34 + y = 200
y = 89.66
Therefore, Lyle needs approximately 110.34g of the chocolate that is 99% cocoa and approximately 89.66g of the chocolate that is 70% cocoa to make the cake.
To find out how much of each type of chocolate Lyle needs to make the cake, we can set up a system of equations. Let's assume Lyle needs x grams of the 99% cocoa chocolate and y grams of the 70% cocoa chocolate.
Since Lyle needs a total of 200g of chocolate, we have the equation:
x + y = 200 -- (equation 1)
In addition, we know that the percentage of cocoa in the 99% cocoa chocolate is 99% or 0.99, while the percentage of cocoa in the 70% cocoa chocolate is 70% or 0.70.
The amount of cocoa in the 99% cocoa chocolate is calculated as:
0.99x
And the amount of cocoa in the 70% cocoa chocolate is calculated as:
0.70y
Since the total amount of cocoa needed is 86% or 0.86 of the total chocolate (200g), we have:
0.99x + 0.70y = 0.86 * 200 -- (equation 2)
Now, we can solve this system of equations to find the values of x and y.
First, let's simplify equation 2:
0.99x + 0.70y = 172
To eliminate decimal values, we can multiply equation 1 by 0.99:
0.99x + 0.99y = 198
Now, subtract equation 2 from the modified equation 1:
0.99x + 0.99y - (0.99x + 0.70y) = 198 - 172
0.99x + 0.99y - 0.99x - 0.70y = 26
0.29y = 26
y = 26 / 0.29
y ≈ 89.655
From equation 1, we can find the value of x:
x + y = 200
x + 89.655 ≈ 200
x ≈ 200 - 89.655
x ≈ 110.345
Therefore, Lyle needs approximately 110.345 grams of the 99% cocoa chocolate and approximately 89.655 grams of the 70% cocoa chocolate to make the cake.