Question

A dairy needs 296 gallons of milk containing 6% butterfat. How many gallons each of milk containing 7% butterfat and milk containing 3% butterfat must be used to obtain the desired 296 gallons?

Answer 1

To get 296 gallons of milk with 6% butterfat, a dairy must mix 222 gallons of 7% butterfat milk with 74 gallons of 3% butterfat milk by solving a system of linear equations.

Step-by-step explanation:

To solve the problem where a dairy needs 296 gallons of milk containing 6% butterfat and we need to mix milk containing 7% butterfat with milk containing 3% butterfat to achieve this, we can set up a system of equations.

Let x be the amount of 7% butterfat milk and y be the amount of 3% butterfat milk needed. We have two equations based on the total quantity of milk and the total amount of butterfat:

1. x + y = 296 (total milk equation)
2. 0.07x + 0.03y = 0.06 × 296 (butterfat content equation)

Solving the system of equations, we multiply the second equation by 100 to make the numbers easier to work with:

1. 1. 7x + 3y = 1776 (modified butterfat content equation)
2. Subtracting 3 times the first equation from the modified second equation gives us:
3. 7x + 3y - (3x + 3y) = 1776 - (3 × 296)
4. This simplifies to:
5. 4x = 888
6. Dividing by 4, we find:
7. x = 222 gallons of 7% butterfat milk.
8. Substituting x into the first equation:
9. y = 296 - x = 296 - 222
10. y = 74 gallons of 3% butterfat milk.
11. Therefore, to achieve 296 gallons of 6% butterfat milk, the dairy needs to mix 222 gallons of milk containing 7% butterfat and 74 gallons of milk containing 3% butterfat.