How many gallons of 45% butterfat cream must be mixed with 500 gallons of 3% butterfat milk to obtain a 5% butterfat milk?

fat = f

.45 f + .03 (500) = .05 (500+f)
or
.40 f = .02(500)

Let x = gallons 45%

45x + (500*3) = (500+x)*5

To calculate the number of gallons of 45% butterfat cream that must be mixed with 500 gallons of 3% butterfat milk to obtain a 5% butterfat milk, we can follow these steps:

Step 1: Determine the amount of butterfat needed in the final milk mixture.
The desired butterfat percentage in the final milk mixture is 5%. This means that for every 100 gallons of the mixture, 5 gallons should be butterfat.

Step 2: Calculate the amount of butterfat in the 500 gallons of 3% butterfat milk.
Since the 500 gallons of milk have a butterfat percentage of 3%, the amount of butterfat in the milk is 3% of 500 gallons, which is (3/100) * 500 = 15 gallons.

Step 3: Calculate the butterfat needed from the 45% butterfat cream.
To achieve the desired 5% butterfat in the final mixture, we need an additional 5 - 3 = 2 gallons of butterfat for every 100 gallons of mixture.
The 45% butterfat cream has 45 gallons of butterfat for every 100 gallons.
Therefore, the proportion can be set up as follows:
45/100 = 2/x, where x is the number of gallons of 45% butterfat cream needed.

Step 4: Solve for x to find the number of gallons of 45% butterfat cream.
By cross-multiplying the above proportion, we get:
45x = 2 * 100
45x = 200
x = 200 / 45
x ≈ 4.44

So, approximately 4.44 gallons of the 45% butterfat cream should be mixed with 500 gallons of the 3% butterfat milk to obtain a 5% butterfat milk.