how much whipping cream and 2% milk should be mixed to obtain 35 gallons of milk with 4% butterfat.
if the milk has 2% butterfat and the whipping cream has 9% butterfat
To calculate the amount of whipping cream and 2% milk needed, we need to consider the butterfat content in each ingredient and the desired butterfat content of the final mixture.
Let's denote the amount of whipping cream needed as WC and the amount of 2% milk needed as MC.
First, let's determine the amount of butterfat in the final mixture. We know that the desired butterfat content is 4% and the volume of milk is 35 gallons. Thus, the total amount of butterfat required is:
(4/100) * 35 gallons = 1.4 gallons
Next, let's determine the amount of butterfat contributed by the 2% milk. We know that the 2% milk has a butterfat content of 2%. Therefore, the amount of butterfat contributed by the 2% milk is:
(2/100) * MC
Similarly, for the whipping cream with a butterfat content of 9%, the amount of butterfat contributed is:
(9/100) * WC
Since we want to obtain 35 gallons of milk in total, the sum of whipping cream and 2% milk would be:
WC + MC = 35 gallons
Now, equating the butterfat contributed by the 2% milk and whipping cream to the desired butterfat content (1.4 gallons), we have the equation:
(2/100) * MC + (9/100) * WC = 1.4 gallons
We have two equations now:
WC + MC = 35 gallons
(2/100) * MC + (9/100) * WC = 1.4 gallons
Solving these two equations simultaneously will give us the amounts of whipping cream and 2% milk needed.
To determine the quantities of whipping cream and 2% milk needed, we need to consider the final volume and butterfat content of the mixture.
Let's assume x represents the volume of whipping cream in gallons that needs to be added, and y represents the volume of 2% milk in gallons required to make 35 gallons of milk with 4% butterfat.
The first step is to set up two equations based on the butterfat and volume information:
Equation 1: x + y = 35 (since the total volume of the mixture is 35 gallons)
Equation 2: (0.09x + 0.02y)/35 = 0.04 (since the butterfat content of the final mixture is 4%)
Let's solve these two equations simultaneously to find the values of x and y.
From Equation 1, we can rearrange it to express x in terms of y: x = 35 - y.
Substituting this value of x into Equation 2, we get:
(0.09*(35 - y) + 0.02y)/35 = 0.04
Multiplying through by 35 to eliminate the fractions:
0.09*(35 - y) + 0.02y = 0.04*35
3.15 - 0.09y + 0.02y = 1.4
Combine like terms:
-0.07y = -1.75
Solving for y:
y = (-1.75)/(-0.07) = 25
Now, substitute the value of y back into Equation 1 to find x:
x + 25 = 35
x = 35 - 25 = 10
Therefore, you would need to mix 10 gallons of whipping cream with 25 gallons of 2% milk to obtain 35 gallons of milk with 4% butterfat.
M = amount of milk (gallons)
C = amount of cream (gallons)
M + C = 35
0.02M + C = 0.04 (M + C)
Solve those two equations for M and C.