According to the directions on a can of frozen orange juice concentrate, 1 can of concentrate is to be mixed...

1. Translate the problem requirements: Understand that \(1\) can concentrate \(+ 3\) cans water \(= 4\) cans total juice mixture. We need to find how many 12-ounce concentrate cans are required to make enough juice for 200 servings of 6 ounces each.
2. Calculate total juice volume needed: Determine the total ounces of orange juice required for all servings.
3. Apply the concentration ratio: Use the \(1:3\) concentrate-to-water ratio to find what fraction of the total juice volume must be concentrate.
4. Convert to number of concentrate cans: Divide the required concentrate volume by the size of each concentrate can to get the final answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in everyday language:

• The recipe calls for mixing 1 can of concentrate with 3 cans of water
• This means for every 1 part concentrate, we add 3 parts water, giving us 4 parts total juice
• We need to make 200 servings, where each serving is 6 ounces
• Each can of concentrate contains 12 ounces
• We need to find how many of these 12-ounce concentrate cans we need

Think of it like making lemonade - if your recipe says "1 cup lemon juice + 3 cups water = 4 cups lemonade," then you know that \(\frac{1}{4}\) of your final lemonade is pure lemon juice.

Process Skill: TRANSLATE - Converting the mixing directions into a usable ratio

2. Calculate total juice volume needed

This step is straightforward multiplication:

• We need 200 servings
• Each serving is 6 ounces
• Total juice needed = \(200 \times 6 = 1200\) ounces

So we need to make 1,200 ounces of orange juice total.

3. Apply the concentration ratio

Now we use our understanding from step 1. Since the recipe mixes 1 can concentrate with 3 cans water:

• Total parts in the mixture = 1 part concentrate + 3 parts water = 4 parts total
• This means concentrate makes up \(\frac{1}{4}\) of the final juice volume
• Therefore: concentrate needed = \(\frac{1}{4} \times 1200 = 300\) ounces

We can verify this makes sense: if we have 300 ounces of concentrate, we'll mix it with 900 ounces of water (3 times as much), giving us \(300 + 900 = 1200\) ounces total. Perfect!

Process Skill: INFER - Recognizing that concentrate represents \(\frac{1}{4}\) of the total mixture

4. Convert to number of concentrate cans

Finally, we convert our concentrate volume needs into actual cans:

• We need 300 ounces of concentrate total
• Each can contains 12 ounces of concentrate
• Number of cans needed = \(300 \div 12 = 25\) cans

Mathematically: \(300 \div 12 = 25\)

Final Answer

We need 25 cans of 12-ounce concentrate to prepare 200 6-ounce servings of orange juice.

The answer is A. 25.

Verification: \(25 \times 12 = 300\) ounces concentrate. Mix with 900 ounces water = 1,200 ounces total juice. \(1200 \div 6 = 200\) servings. ✓

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the mixing ratio
Students often misinterpret "1 can of concentrate mixed with 3 cans of water" as meaning concentrate makes up \(\frac{1}{3}\) of the final mixture instead of \(\frac{1}{4}\). They forget that the total mixture includes both the concentrate AND the water, so 1 part concentrate + 3 parts water = 4 total parts, making concentrate \(\frac{1}{4}\) of the total.

2. Confusing what needs to be calculated
Some students get distracted by the different container sizes (12-ounce cans vs 6-ounce servings) and try to directly convert between them without first determining the total volume needed. They might attempt to find relationships between 12 and 6 without considering the actual juice preparation requirements.

3. Setting up the wrong equation
Students may incorrectly assume they need to work backwards from the concentrate cans rather than forward from the serving requirements. Instead of calculating total juice needed first, they might try to guess how many concentrate cans to start with and see if it works out.

Errors while executing the approach

1. Arithmetic errors in basic multiplication
When calculating the total juice volume (\(200 \times 6\) ounces), students may make simple multiplication errors, getting 1,000 or 1,400 ounces instead of 1,200 ounces. This error cascades through the entire solution.

2. Incorrectly applying the concentration ratio
Even if students understand that concentrate is \(\frac{1}{4}\) of the mixture, they might apply this fraction incorrectly. For example, calculating \(\frac{1}{4} \times 1200\) as 400 instead of 300, or mistakenly using \(\frac{1}{3} \times 1200 = 400\) if they misunderstood the ratio from the approach phase.

3. Division errors in the final step
When dividing 300 ounces by 12 ounces per can, students might make calculation mistakes, getting 24, 26, or other numbers close to 25, especially if they're working under time pressure.

Errors while selecting the answer

No likely faltering points - the calculation directly gives a whole number (25) that matches one of the answer choices exactly, making selection straightforward once the math is completed correctly.