The price of lunch for 15 people was $207 including a 15% gratuity for service. What was the average price...

1. Translate the problem requirements: We need to find the average price per person for the meal only, which means we must first remove the 15% gratuity from the total \(\$207\), then divide by 15 people
2. Remove the gratuity to find the base meal cost: Since \(\$207\) includes a 15% tip, we need to work backwards to find what the original meal cost was before the tip was added
3. Calculate the average cost per person: Divide the base meal cost by 15 people to get the average price per person excluding gratuity
4. Verify against answer choices: Check our calculated result matches one of the given options

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have 15 people who had lunch together, and their total bill was \(\$207\). However, this \(\$207\) includes a 15% gratuity (tip) that was added on top of the actual meal cost.

The question asks for the average price per person EXCLUDING the gratuity. This means we need to:
- First, remove the 15% tip from the \(\$207\) total to find what the meal itself cost
- Then, divide that meal cost by 15 people to get the average per person

Process Skill: TRANSLATE - Converting the problem language into clear mathematical steps

2. Remove the gratuity to find the base meal cost

Here's the key insight: when we say the total bill of \(\$207\) "includes a 15% gratuity," this means the \(\$207\) is 115% of the original meal cost (100% for the meal + 15% for the tip).

Think of it this way: if the original meal cost was some amount (let's call it x), then after adding a 15% tip, we get:
- Original meal cost: \(\mathrm{x}\)
- 15% tip: \(0.15\mathrm{x}\)
- Total bill: \(\mathrm{x} + 0.15\mathrm{x} = 1.15\mathrm{x}\)

We know this total bill equals \(\$207\), so:
\(1.15\mathrm{x} = \$207\)

To find the original meal cost (x), we divide both sides by 1.15:
\(\mathrm{x} = \$207 \div 1.15 = \$180\)

So the meal itself, before the tip, cost \(\$180\).

3. Calculate the average cost per person

Now that we know the meal cost was \(\$180\) (excluding the gratuity), we can find the average cost per person by dividing by the number of people.

Average cost per person = Total meal cost ÷ Number of people
Average cost per person = \(\$180 \div 15 = \$12\)

Each person's share of the meal (excluding the tip) was \(\$12\).

4. Verify against answer choices

Looking at our calculated result of \(\$12\), we can see this matches answer choice B exactly.

Let's do a quick verification: If each person paid \(\$12\) for their meal, then 15 people would pay \(15 \times \$12 = \$180\) for the meal. Adding a 15% tip: \(\$180 \times 1.15 = \$207\). ✓

Final Answer: B. 12

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "including 15% gratuity" means
Students often think that if the bill is \(\$207\) including 15% gratuity, then the gratuity is simply 15% of \(\$207\) (which would be \(\$31.05\)). They fail to recognize that \(\$207\) represents 115% of the original meal cost, not 100%. This leads them to incorrectly subtract \(\$31.05\) from \(\$207\) instead of dividing \(\$207\) by 1.15.

2. Forgetting to exclude the gratuity from the final answer
Some students correctly calculate the total cost excluding gratuity (\(\$180\)) and the average cost per person including gratuity (\(\$13.80\)), but then select the wrong value. They miss the key phrase "EXCLUDING the gratuity" in the question and choose \(\$13.80\) instead of \(\$12\).

Errors while executing the approach

1. Arithmetic errors in division
Students may set up the equation correctly (\(1.15\mathrm{x} = 207\)) but make calculation mistakes when dividing 207 by 1.15. Common errors include getting \(\$179\) instead of \(\$180\), or making decimal placement errors that lead to incorrect intermediate results.

2. Using incorrect percentage calculations
Some students might try to remove 15% by multiplying \(\$207\) by 0.85 instead of dividing by 1.15. While this approach seems logical, it's mathematically incorrect because 15% was added to the original amount, not calculated as 15% of the final amount.

Errors while selecting the answer

1. Selecting the per-person cost that includes gratuity
After calculating both the meal cost excluding gratuity (\(\$180\)) and the total cost per person including gratuity (\(\$13.80\)), students might select answer choice C (\(\$13.80\)) because they lose track of what the question is specifically asking for. The question asks for the average price EXCLUDING gratuity, which should be \(\$12\).