Edit: kind of solved thanks to the rubberduck effect. Would still appreciate any responses to help strengthen my intuitive understanding of these sorts of problems.
Hello, I'm currently working through the art of problem solving Introduction to algebra book and am having trouble understanding one of the solutions.
Heavy cream is 36% butterfat, while whole milk contains only 4% butterfat. In order to make a delicious pint of ice cream, a recipe calls for 2 cups of a mixture that is 24% butterfat. How many cups of heavy cream should be used to produce the correct butterfat percentage?
My Solution: let x = cups of heavy cream, let y = cups of whole milk.
x + y = 2 => y = 2 - x
0.36x + 0.04y = 0.24 => 0.36x + 0.04(2 - x) = 0.24 => 0.36x + 0.08 - 0.04x = 0.24 => 0.32x = 0.16 => x = 0.5, y = 1.5.
Comparing my answer to the solution I see that I was supposed to double 0.24 to 0.48. This is what I'm having trouble understanding. Doesn't taking 2 cups from a mixture that is 24% butterfat result in 2 cups of a mixture that is 24% butterfat?
The solution: Suppose we use x cups of heavy cream and y cups of whole milk. Since we need 2 cups total, we have x + y = 2. The x cups of heavy cream have 0.36x cups of butterfat. The y cups of whole milk have 0.04y cups of butterfat. Together, we want the mixture to have (2)(0.24) = 0.48 cups of butterfat. Therefore, we must have 0.036x + 0.04y = 0.48. We can get rid of the decimals by multiplying by 100 to get 36x + 4y = 48. We then divide by 4 to get 9x + y = 12. Subtracting x + y = 2 from this equation, we have 8x = 10, so x = 5/4 cups.