Final answer:
The mixture problem of maintaining a 4% butterfat concentration, we use a system of linear equations with x being the quarts of 6% milk and y being the quarts of 3% milk. The correct mixture to add to 30 quarts of 4% milk is 15 quarts of 6% milk and 15 quarts of 3% milk.
Step-by-step explanation:
The question involves solving a mixture problem using percentages and volumes. We want to create a combination of 6% butterfat milk and 3% butterfat milk to mix with 30 quarts of 4% butterfat milk. To maintain the 4% butterfat concentration, the amounts of 6% and 3% milk added must result in an average butterfat concentration of 4%. Let us denote x quarts to be the amount of 6% milk and y quarts to be the amount of 3% milk.
Using the concept that the total amount of butterfat before mixing equals the total amount of butterfat after mixing, we can set up the following equation:
0.06x + 0.03y + 0.04(30) = 0.04(x + y + 30). Additionally, since we are mixing only two kinds of milk, we also have the equation x + y = 30. Solving this system of equations will give us the answer.
Mathematically, the problem looks like this:
- 0.06x + 0.03y + 1.2 = 0.04x + 0.04y + 1.2
- x + y = 30
Solving these equations, we can manipulate them to deduce the individual values for x and y. In this case, x represents the amount of 6% milk and y represents the amount of 3% milk needed. Through substitution or elimination, we can find that the correct answer is option (b) 15 quarts of 6%, 15 quarts of 3%, maintaining the overall 4% butterfat content.