Question

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?
\\(\frac{1}{4}\\) cup
\\(\frac{4}{5}\\) cup
\\(\frac{3}{4}\\) cup
\\(\frac{1}{5}\\) cup

Explanation:

Step1: Define variables

Let \( x \) be the volume of cream (in cups), so the volume of milk is \( 1 - x \) cups.

Step2: Set up fat equation

The fat from milk plus fat from cream equals fat in mixture. So \( 0.02(1 - x) + 0.18x = 0.06(1) \).

Step3: Simplify equation

\( 0.02 - 0.02x + 0.18x = 0.06 \) → \( 0.02 + 0.16x = 0.06 \).

Step4: Solve for x

\( 0.16x = 0.06 - 0.02 \) → \( 0.16x = 0.04 \) → \( x = \frac{0.04}{0.16} = \frac{1}{4} \)? Wait, no, wait: \( 0.04\div0.16 = 0.25 \)? Wait, no, \( 0.04/0.16 = 1/4 \)? Wait, no, 0.04 divided by 0.16: 0.04 / 0.16 = (4/100)/(16/100) = 4/16 = 1/4? Wait, but let's re - check. Wait, the equation:

Fat from milk: \( 2\%(1 - x) \), fat from cream: \( 18\%x \), total fat: \( 6\%\times1 \).

So \( 0.02(1 - x)+0.18x = 0.06 \)

Expand: \( 0.02-0.02x + 0.18x=0.06 \)

Combine like terms: \( 0.02 + 0.16x = 0.06 \)

Subtract 0.02: \( 0.16x=0.04 \)

Then \( x=\frac{0.04}{0.16}=\frac{4}{16}=\frac{1}{4} \)? Wait, but \( \frac{1}{4}=0.25 \), and \( 1 - \frac{1}{4}=\frac{3}{4} \). Let's check the fat:

Milk fat: \( 0.02\times\frac{3}{4}=0.015 \)

Cream fat: \( 0.18\times\frac{1}{4}=0.045 \)

Total fat: \( 0.015 + 0.045 = 0.06 \), which is \( 6\% \) of 1 cup. So it works. Wait, but the options are \( \frac{1}{4} \) cup, \( \frac{4}{5} \) cup, \( \frac{3}{4} \) cup, \( \frac{1}{5} \) cup. So the volume of cream is \( \frac{1}{4} \) cup? Wait, no, wait: Wait, \( x \) is the volume of cream. So from the calculation, \( x = \frac{1}{4} \) cup.

Wait, but let's re - do the equation. Let me define variables again. Let \( c \) be the volume of cream (in cups), \( m \) be the volume of milk. So \( m + c=1 \), so \( m = 1 - c \).

Fat equation: \( 0.02m+0.18c = 0.06\times1 \)

Substitute \( m = 1 - c \):

\( 0.02(1 - c)+0.18c=0.06 \)

\( 0.02-0.02c + 0.18c = 0.06 \)

\( 0.02 + 0.16c=0.06 \)

\( 0.16c=0.04 \)

\( c=\frac{0.04}{0.16}=\frac{1}{4} \) cup.

Answer:

\(\frac{1}{4}\) cup