Milk and cream are combined to make a 60text{-liter} mixture that is 50% butterfat. If the milk is 5% butterfat...

1. Translate the problem requirements: We need to find how many liters of milk (5% butterfat) when combined with cream (75% butterfat) creates 60 liters of mixture with 50% butterfat concentration
2. Set up the quantity relationship: Express the total volume constraint using milk and cream quantities
3. Apply the butterfat concentration balance: Calculate total butterfat contributed by each component to match the target concentration
4. Solve for milk quantity: Use the concentration equation to find the amount of milk in the mixture

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in everyday terms. Imagine you're mixing two different liquids with different concentrations of butterfat to create a final mixture with a specific target concentration.

We have:

- Milk: 5% butterfat (relatively low fat content)
- Cream: 75% butterfat (very high fat content)
- Target mixture: 60 liters total, 50% butterfat (somewhere in between)

The question is asking: how much of the low-fat milk do we need?

Think of it this way: since 50% is much closer to 75% than to 5%, we'd expect to use more milk than cream to "pull down" that high cream concentration to our target of 50%.

Let's define our variables:

- Let M = liters of milk in the mixture
- Let C = liters of cream in the mixture

Process Skill: TRANSLATE - Converting the real-world mixture scenario into mathematical terms

2. Set up the quantity relationship

The first constraint is straightforward - the total volume must equal 60 liters.

In plain English: All the milk plus all the cream must add up to 60 liters total.

\(\mathrm{M + C = 60}\)

This means: \(\mathrm{C = 60 - M}\)

So if we can find M (the amount of milk), we automatically know C (the amount of cream).

3. Apply the butterfat concentration balance

Now here's the key insight: the total amount of butterfat in the final mixture must equal the sum of butterfat contributed by the milk and the cream.

Let's think about this step by step:

- The final mixture has 60 liters at 50% butterfat
- So total butterfat = \(\mathrm{60 \times 0.50 = 30}\) liters of pure butterfat

Where does this 30 liters of butterfat come from?

- From the milk: \(\mathrm{M \times 0.05}\) liters of butterfat
- From the cream: \(\mathrm{C \times 0.75}\) liters of butterfat

So our butterfat balance equation is:
\(\mathrm{M \times 0.05 + C \times 0.75 = 30}\)

Process Skill: INFER - Recognizing that total butterfat must be conserved in the mixture

4. Solve for milk quantity

Now we substitute \(\mathrm{C = 60 - M}\) into our butterfat equation:

\(\mathrm{M \times 0.05 + (60 - M) \times 0.75 = 30}\)

Let's expand this:
\(\mathrm{M \times 0.05 + 60 \times 0.75 - M \times 0.75 = 30}\)
\(\mathrm{0.05M + 45 - 0.75M = 30}\)

Combining the M terms:
\(\mathrm{-0.70M + 45 = 30}\)
\(\mathrm{-0.70M = 30 - 45}\)
\(\mathrm{-0.70M = -15}\)
\(\mathrm{M = -15 \div (-0.70) = 15 \div 0.70}\)

To make this division easier:
\(\mathrm{M = 15 \div 0.70 = 15 \div (7/10) = 15 \times (10/7) = 150/7}\)

\(\mathrm{M = 150 \div 7 \approx 21.43}\) liters

Let's verify: If \(\mathrm{M = 21.43}\), then \(\mathrm{C = 60 - 21.43 = 38.57}\)
Butterfat check: \(\mathrm{21.43 \times 0.05 + 38.57 \times 0.75 = 1.07 + 28.93 = 30.00}\) ✓

5. Final Answer

The amount of milk in the mixture is approximately 21.43 liters.

This matches answer choice B. 21.43.

Notice how our intuition was correct - we needed more milk (21.43 L) than cream (38.57 L) to bring down that high cream concentration to our target of 50%.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the concentration calculation

Students often confuse how percentages work in mixtures. They might think that mixing 5% and 75% butterfat automatically gives 40% (simple average), forgetting that the final concentration depends on the relative quantities of each component. This leads to setting up incorrect equations or expecting equal amounts of milk and cream.

2. Setting up the wrong constraint equations

Some students focus only on the volume constraint (\(\mathrm{M + C = 60}\)) and forget about the butterfat constraint entirely, or they might set up the butterfat equation incorrectly by using total volume instead of the actual butterfat content. For example, they might write: \(\mathrm{M \times 0.05 + C \times 0.75 = 60}\) instead of \(\mathrm{M \times 0.05 + C \times 0.75 = 30}\).

3. Confusing what the question is asking for

Students sometimes lose track of whether they need to find the amount of milk, cream, or total butterfat. They might set up the problem correctly but solve for the wrong variable, especially since both milk and cream quantities are reasonable-looking numbers.

Errors while executing the approach

1. Sign errors during algebraic manipulation

When expanding \(\mathrm{M \times 0.05 + (60 - M) \times 0.75 = 30}\), students frequently make sign errors, particularly with the term \(\mathrm{-M \times 0.75}\). They might write \(\mathrm{+0.75M}\) instead of \(\mathrm{-0.75M}\), leading to completely wrong answers.

2. Arithmetic mistakes in decimal calculations

The problem involves multiple decimal calculations (0.05, 0.75, 0.70) and division by 0.70. Students often make errors when computing \(\mathrm{-0.05M - 0.75M = -0.70M}\) or when dividing \(\mathrm{15 \div 0.70}\), especially if they don't convert to fractions (\(\mathrm{15 \div 7/10 = 150/7}\)).

3. Incorrect substitution

When substituting \(\mathrm{C = 60 - M}\) into the butterfat equation, students sometimes substitute into the wrong equation or make errors in the substitution process, such as writing \(\mathrm{(60 + M)}\) instead of \(\mathrm{(60 - M)}\).

Errors while selecting the answer

1. Selecting the cream amount instead of milk amount

After correctly calculating \(\mathrm{M \approx 21.43}\) and \(\mathrm{C \approx 38.57}\), students might accidentally select 38.57 (answer choice D) because they computed both values and picked the wrong one. This is especially likely since both numbers appear in the answer choices.

2. Rounding errors or precision issues

Students might get 21.4 or 21.5 instead of 21.43 due to premature rounding in their calculations, then select 20.00 (choice A) or 21.43 (choice B) based on which seems closer, without being confident in their precision.