How much cream that is 30% butterfat must be added to milk that is 3% butterfat to have 270 quarts that are at least 4.5% butterfat?

Let the amount of 30% butterfat cream to be added be \( x \) quarts. The remaining amount, \( 270 - x \), will be milk that contains 3% butterfat. We want the final mixture of 270 quarts to have at least 4.5% butterfat.

We can set up the following equation based on the amount of butterfat in each component:

\[
0.30x + 0.03(270 - x) = 0.045 \times 270
\]

### Step-by-Step Solution:
1. **Calculate the total amount of butterfat desired**:
   \[
   0.045 \times 270 = 12.15 \text{ quarts of butterfat}
   \]

2. **Expand and simplify the equation**:
   \[
   0.30x + 0.03(270 - x) = 12.15
   \]
   Expanding both sides:
   \[
   0.30x + 0.03 \times 270 - 0.03x = 12.15
   \]
   Simplifying further:
   \[
   0.30x + 8.1 - 0.03x = 12.15
   \]
   Combine like terms:
   \[
   0.27x + 8.1 = 12.15
   \]

3. **Solve for \( x \)**:
   Subtract 8.1 from both sides:
   \[
   0.27x = 4.05
   \]
   Divide by 0.27:
   \[
   x = \frac{4.05}{0.27} = 15
   \]

So, you need to add **15 quarts** of 30% butterfat cream to the milk.

### Final Answer:
You must add **15 quarts** of 30% butterfat cream to the milk.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the solution change if the final mixture needed to be exactly 5% butterfat?
2. What if the milk had 2% butterfat instead of 3%?
3. How would you calculate the amount if the total volume of the mixture was 300 quarts?
4. What is the percentage of butterfat if 30 quarts of 30% cream is added?
5. How does the percentage of butterfat in cream affect the amount required in mixtures?

**Tip**: When dealing with mixtures, set up equations by equating the total amount of the desired component in each substance to the total amount required in the mixture.

Learn how to calculate the amount of 30% butterfat cream to add to 3% butterfat milk to create 270 quarts with at least 4.5% butterfat. This step-by-step solution covers algebraic concepts like mixture problems and linear equations, suitable for Grades 8-10.

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