Question

A farmer wants to mix milk containing 3% butterfat with cream containing 30% butterfat to obtain 900 gallons of milk which is 8% butterfat. How many gallons of the 30% butterfat is used?

Answer 1

To achieve 900 gallons of milk with an 8% butterfat mixture, the farmer needs to use 240 gallons of 30% butterfat cream and the rest with 3% butterfat milk.

Step-by-step explanation:

Calculating the Amount of Cream for Butterfat Percentage

The problem at hand involves a farmer who wants to mix milk with a low butterfat percentage (3%) with cream that has a high butterfat percentage (30%) to create a mixture of 900 gallons with a desired butterfat percentage of 8%. To calculate the number of gallons containing 30% butterfat needed in this mixture, we will use a system of equations based on the concept of mixtures in algebra.

Let x be the number of gallons of the 3% butterfat milk, and y be the number of gallons of the 30% butterfat cream. We can set up the following equations based on the total volume and the total amount of butterfat required:

x + y = 900 (total volume)

0.03x + 0.30y = 0.08(900) (total butterfat)

Now we solve these equations simultaneously. Multiplying the second equation by 100 to remove decimals gives us:

3x + 30y = 72 × 900

Then we can multiply the first equation by 3 to substitute in:

3x + 3y = 3 × 900

Subtracting this from the second equation:

27y = 72 × 900 - 3 × 900

y = (72 × 900 - 3 × 900) / 27

y = 240 gallons

Therefore, the farmer needs 240 gallons of the 30% butterfat cream to obtain 900 gallons of milk that is 8% butterfat.