Milk and cream are mixed together for a recipe. The total volume of the mixture is 1 cup. If the milk contains $2 \%$ fat, the cream contains $18 \%$ fat, and the mixture contains $6 \%$ fat, how much cream is in the mixture?
A. $\frac{1}{5}$ cup
B. $\frac{1}{4}$ cup
C. $\frac{3}{4}$ cup
D. $\frac{4}{5}$ cup
Answer (1)
Define x as the volume of milk and y as the volume of cream.
Write the volume equation: x + y = 1 .
Write the fat content equation: 0.02 x + 0.18 y = 0.06 .
Solve the system of equations to find the volume of cream: y = 4 1 .
Explanation
Problem Analysis Let's analyze the problem. We are given the fat percentages of milk, cream, and a mixture of the two. We also know the total volume of the mixture. Our goal is to find the volume of cream in the mixture.
Setting up the Equations (Volume) Let x be the volume of milk in cups and y be the volume of cream in cups. We can set up a system of equations based on the given information. The total volume of the mixture is 1 cup, so we have: x + y = 1
Setting up the Equations (Fat Content) The milk contains 2% fat, the cream contains 18% fat, and the mixture contains 6% fat. This gives us the equation: 0.02 x + 0.18 y = 0.06 ( 1 ) 0.02 x + 0.18 y = 0.06
Solving for Cream Volume Now we have a system of two equations with two variables: x + y = 1 0.02 x + 0.18 y = 0.06 We can solve for x in the first equation: x = 1 − y . Substitute this into the second equation: 0.02 ( 1 − y ) + 0.18 y = 0.06 0.02 − 0.02 y + 0.18 y = 0.06 0.16 y = 0.04 y = 0.16 0.04 = 16 4 = 4 1
Final Answer So, the volume of cream in the mixture is 4 1 cup.
Examples
Imagine you're a chef creating a signature sauce that requires a specific fat content. By understanding how to mix ingredients with different fat percentages, like milk and cream, you can precisely control the final fat content of your sauce. This ensures the perfect taste and texture every time. This problem demonstrates how to calculate the exact amount of each ingredient needed to achieve the desired fat percentage in the mixture, which is crucial for consistent and high-quality culinary creations.
