A corporation that had $115.19 billion in profits for the year paid out $230.10 million in employee benefits. Approximately what...

1. Translate the problem requirements: We need to find what percentage the employee benefits ($230.10 million) represent of the total profits ($115.19 billion). This means we're looking for \((\mathrm{benefits} \div \mathrm{profits}) \times 100\%\).
2. Convert units for comparison: Since we're comparing million to billion, we need to express both amounts in the same units to make the calculation manageable.
3. Set up the percentage ratio: Create the fraction (employee benefits ÷ total profits) and recognize that we can simplify this calculation significantly given the answer choices are far apart.
4. Approximate strategically: Since the answer choices range from 50% down to 0.2%, we can round the numbers to make mental math easier while still arriving at the correct answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to find. We have a company that made $115.19 billion in profits, and they paid out $230.10 million in employee benefits. We want to know: "What percentage of the profits were the employee benefits?"

Think of it this way - if you earned $100 this month and spent $2 on coffee, you'd want to know what percentage of your earnings went to coffee. That's exactly what we're doing here, just with much bigger numbers.

In mathematical terms, we need to calculate: \((\mathrm{employee\ benefits} \div \mathrm{total\ profits}) \times 100\%\)

So we need: \((\$230.10\ \mathrm{million} \div \$115.19\ \mathrm{billion}) \times 100\%\)

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

2. Convert units for comparison

Right now we have millions and billions mixed together, which makes comparison tricky. It's like trying to compare 5 feet to 200 inches - we need the same units.

Since \(1\ \mathrm{billion} = 10^9\) and \(1\ \mathrm{million} = 10^6\), let's convert everything to millions to make our math easier:

• Profits: \(\$115.19\ \mathrm{billion} = \$115.19 \times 1,000\ \mathrm{million} = \$115,190\ \mathrm{million}\)
• Benefits: $230.10 million (already in millions)

Now we can see the relationship more clearly: we're comparing $230.10 million to $115,190 million.

3. Set up the percentage ratio

Now we can set up our percentage calculation with the same units:

Percentage = \((\mathrm{Benefits} \div \mathrm{Profits}) \times 100\%\)
Percentage = \((\$230.10\ \mathrm{million} \div \$115,190\ \mathrm{million}) \times 100\%\)

Looking at this fraction, notice that $230.10 is much, much smaller than $115,190. This tells us right away that our percentage will be quite small - definitely less than 1%.

4. Approximate strategically

Since our answer choices range from 50% all the way down to 0.2%, and they're spread far apart, we can round our numbers to make the calculation much simpler.

Let's approximate:
• Benefits: $230.10 million ≈ $230 million
• Profits: $115,190 million ≈ $115,000 million

Now our calculation becomes:
Percentage ≈ \((230 \div 115,000) \times 100\%\)

To make this even simpler, let's think of it as:
Percentage ≈ \((230 \div 115,000) \times 100\% = (23 \div 11,500) \times 100\%\)

Now, \(23 \div 11,500\) is approximately \(2 \div 1,000 = 0.002\)

So: \(0.002 \times 100\% = 0.2\%\)

Process Skill: SIMPLIFY - Using strategic rounding to make complex calculations manageable

4. Final Answer

Our calculation shows that employee benefits represent approximately 0.2% of the company's profits.

This makes intuitive sense - $230 million sounds like a lot, but compared to $115 billion in profits, it's actually a very small percentage.

The answer is (E) 0.2%

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the percentage relationship
Students often confuse which number should be in the numerator vs. denominator when calculating percentages. In this problem, we want "employee benefits as a percentage of profits," which means benefits ÷ profits × 100%. Some students might incorrectly set up the calculation as profits ÷ benefits, especially when the profit number ($115.19 billion) is much larger than the benefits ($230.10 million).

Faltering Point 2: Not recognizing the need for unit conversion
The question presents profits in billions and benefits in millions. Students may attempt to calculate the percentage directly without converting to the same units first, leading to incorrect results. They might plug in \(230.10 \div 115.19\) without accounting for the billion vs. million difference.

Errors while executing the approach

Faltering Point 1: Incorrect unit conversion
Even when students recognize they need to convert units, they may make errors in the conversion process. Common mistakes include converting billions to millions incorrectly (using 100 or 10,000 instead of 1,000 as the conversion factor) or converting in the wrong direction (trying to convert millions to billions when billions to millions would be easier).

Faltering Point 2: Calculation errors during approximation
When simplifying 230 ÷ 115,000, students might make arithmetic errors. For example, they might approximate this as \(2 \div 1,000 = 0.02\%\) instead of the correct \(0.002 = 0.2\%\). This often happens when students lose track of decimal places during the simplification process.

Errors while selecting the answer

Faltering Point 1: Decimal place confusion in final answer
Students may calculate the decimal correctly as 0.002 but then convert it incorrectly to a percentage. They might select 2% instead of 0.2% because they forget that \(0.002 \times 100\% = 0.2\%\), not 2%. This is especially common when dealing with very small percentages where students aren't used to seeing answers less than 1%.