David and Ron are ordering food for a business lunch. David thinks that there should be twice as many sandwiches...

1. Translate the problem requirements: David wants twice as many sandwiches as pastries (\(\mathrm{S = 2P}\)), while Ron wants pastries to equal 12 more than one-fourth of sandwiches (\(\mathrm{P = S/4 + 12}\)). We need to find the number of sandwiches where both agree on the pastry count.
2. Set up the agreement condition: Since both David and Ron must agree on the number of pastries, we can set their two expressions for pastries equal to each other.
3. Solve for the number of sandwiches: Use algebraic manipulation to find the sandwich quantity that satisfies both conditions simultaneously.
4. Verify the solution makes practical sense: Check that our answer produces whole numbers for both sandwiches and pastries, and confirm it matches one of the given choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what each person wants in everyday language first:

David's perspective: He thinks there should be twice as many sandwiches as pastries. So if there are 10 pastries, David wants 20 sandwiches. If there are 15 pastries, David wants 30 sandwiches.

Ron's perspective: He thinks the number of pastries should be 12 more than one-fourth of the sandwiches. So if there are 20 sandwiches, Ron wants one-fourth of 20 (which is 5) plus 12 more, giving us 17 pastries.

Now let's write this mathematically:
• David's rule: \(\mathrm{S = 2P}\) (sandwiches equal twice the pastries)
• Ron's rule: \(\mathrm{P = S/4 + 12}\) (pastries equal one-fourth of sandwiches plus 12)

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up the agreement condition

For David and Ron to agree, they need to agree on the number of pastries. This means both of their rules must give the same number of pastries when we use the same number of sandwiches.

From David's rule, we can solve for pastries: If \(\mathrm{S = 2P}\), then \(\mathrm{P = S/2}\)

So we have two expressions for the number of pastries:
• From David's perspective: \(\mathrm{P = S/2}\)
• From Ron's perspective: \(\mathrm{P = S/4 + 12}\)

For them to agree, these must be equal: \(\mathrm{S/2 = S/4 + 12}\)

3. Solve for the number of sandwiches

Now we solve the equation \(\mathrm{S/2 = S/4 + 12}\).

Let's work with this step by step:
\(\mathrm{S/2 = S/4 + 12}\)

Subtract \(\mathrm{S/4}\) from both sides:
\(\mathrm{S/2 - S/4 = 12}\)

To subtract these fractions, let's use a common denominator of 4:
\(\mathrm{2S/4 - S/4 = 12}\)
\(\mathrm{S/4 = 12}\)

Multiply both sides by 4:
\(\mathrm{S = 48}\)

Process Skill: MANIPULATE - Using algebraic techniques to isolate the variable

4. Verify the solution makes practical sense

Let's check our answer \(\mathrm{S = 48}\) sandwiches:

According to David's rule: \(\mathrm{P = S/2 = 48/2 = 24}\) pastries
According to Ron's rule: \(\mathrm{P = S/4 + 12 = 48/4 + 12 = 12 + 12 = 24}\) pastries

Perfect! Both David and Ron agree on 24 pastries when there are 48 sandwiches.

This gives us whole numbers for both items, which makes sense for a catering order.

Final Answer

The answer is E. 48 sandwiches.

This is the number of sandwiches that allows David and Ron to agree on exactly 24 pastries.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting David's constraint: Students often confuse who wants what and may incorrectly write David's rule as \(\mathrm{P = 2S}\) (pastries equal twice the sandwiches) instead of \(\mathrm{S = 2P}\) (sandwiches equal twice the pastries). The phrase "twice as many sandwiches as pastries" can be tricky to translate correctly.

2. Confusion about "agreement" condition: Some students may think they need to find when the total number of items is the same, rather than understanding that David and Ron need to agree specifically on the number of pastries. This leads to setting up incorrect equations.

3. Mixing up the fractions in Ron's rule: Students may misread "12 more than one-fourth of the sandwiches" and write it as \(\mathrm{P = 12 + S/4}\) thinking the addition comes first, or incorrectly write it as \(\mathrm{P = (S + 12)/4}\), putting the entire expression over 4.

Errors while executing the approach

1. Fraction arithmetic errors: When solving \(\mathrm{S/2 = S/4 + 12}\), students commonly make mistakes when subtracting \(\mathrm{S/4}\) from \(\mathrm{S/2}\). They may incorrectly calculate \(\mathrm{S/2 - S/4}\) as \(\mathrm{S/6}\) instead of \(\mathrm{S/4}\), or struggle with finding the common denominator of 4.

2. Sign errors during algebraic manipulation: Students may incorrectly move terms across the equation, such as writing \(\mathrm{S/2 + S/4 = 12}\) instead of \(\mathrm{S/2 - S/4 = 12}\) when trying to isolate terms.

Errors while selecting the answer

1. Verification confusion: Students may correctly solve for \(\mathrm{S = 48}\) but then second-guess themselves because 48 seems "too large" compared to the other answer choices, leading them to pick a smaller number like 24 (which is actually the number of pastries, not sandwiches).

2. Answering the wrong quantity: Some students may solve correctly and find that there should be 48 sandwiches and 24 pastries, but then mistakenly select 24 as their final answer because they confused which quantity the question was asking for.