What percent of Country M's total budget of 1.3 text{ trillion dollars} is Program K's allocation of 260 text{ million...

1. Translate the problem requirements: We need to find what percentage 260 million dollars represents of 1.3 trillion dollars. This requires converting units to be comparable and then calculating the percentage.
2. Convert units to a common base: Since we're comparing million to trillion, we need to express both amounts in the same unit to make the calculation straightforward.
3. Set up the percentage calculation: Once both amounts are in the same unit, we can directly calculate what fraction Program K represents of the total budget.
4. Convert to percentage and match answer choices: Transform the decimal result into percentage form and identify the correct answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find in plain English. We have Program K, which gets 260 million dollars, and we want to know what slice this represents of Country M's entire budget pie of 1.3 trillion dollars.

Think of it this way: if you had $1,300 and wanted to know what percentage $0.26 represents, you'd divide the smaller amount by the larger amount. That's exactly what we're doing here, just with much larger numbers.

The challenge is that our numbers are given in different units - one in millions and one in trillions. Before we can compare them, we need to express both amounts using the same unit.

Process Skill: TRANSLATE - Converting the percentage question into a clear comparison of two monetary amounts

2. Convert units to a common base

Now let's get both numbers speaking the same language. We're told that:
• \(\mathrm{1,000\,million = 1\,billion}\)
• \(\mathrm{1,000\,billion = 1\,trillion}\)

Let's convert everything to millions since that's easier to work with:
• Program K's allocation: 260 million dollars (already in millions)
• Total budget: 1.3 trillion dollars

To convert 1.3 trillion to millions:
\(\mathrm{1.3\,trillion = 1.3 \times 1,000\,billion = 1,300\,billion}\)
\(\mathrm{1,300\,billion = 1,300 \times 1,000\,million = 1,300,000\,million}\)

So now we're comparing:
• Program K: 260 million
• Total budget: 1,300,000 million

3. Set up the percentage calculation

Now that both amounts are in the same unit (millions), we can find what fraction Program K represents of the total budget.

In everyday terms: out of every 1,300,000 dollars in the budget, how many dollars go to Program K? The answer is 260 dollars.

As a fraction, this is:
\(\mathrm{260\,million \div 1,300,000\,million = \frac{260}{1,300,000}}\)

Let's simplify this fraction by dividing both the numerator and denominator by 260:
\(\mathrm{\frac{260}{1,300,000} = \frac{1}{5,000}}\)

This means Program K gets 1 dollar for every 5,000 dollars in the total budget.

4. Convert to percentage and match answer choices

To convert our fraction to a percentage, we need to express \(\mathrm{\frac{1}{5,000}}\) as a decimal and then multiply by 100.

\(\mathrm{\frac{1}{5,000} = 0.0002}\)

Converting to percentage: \(\mathrm{0.0002 \times 100\% = 0.02\%}\)

Let's verify this makes sense: 0.02% means 2 hundredths of 1 percent, which is indeed a very small slice of the budget - exactly what we'd expect for a 260 million program within a 1.3 trillion budget.

Final Answer

Program K's allocation of 260 million dollars represents 0.02% of Country M's total budget of 1.3 trillion dollars.

The correct answer is C. 0.02%

Common Faltering Points

Errors while devising the approach

1. Unit conversion confusion: Students often struggle to understand that they need to convert both amounts to the same unit before calculating the percentage. They might attempt to directly divide 260 million by 1.3 trillion without recognizing the unit mismatch, leading to an incorrect setup.

2. Misunderstanding the percentage formula: Some students might confuse which number should be the numerator and which should be the denominator. They might incorrectly set up the calculation as (total budget ÷ program allocation) instead of (program allocation ÷ total budget), fundamentally misunderstanding what percentage means.

3. Ignoring the conversion factors: Students might overlook or misread the given conversion information (1,000 million = 1 billion, 1,000 billion = 1 trillion) and either not use these conversions at all or apply them incorrectly.

Errors while executing the approach

1. Incorrect unit conversion calculations: When converting 1.3 trillion to millions, students often make arithmetic errors. They might incorrectly calculate 1.3 trillion as 130,000 million (missing one zero) or 13,000,000 million (adding an extra zero), leading to wrong final percentages.

2. Decimal placement errors in division: When calculating 260 ÷ 1,300,000, students frequently misplace decimal points. They might get 0.002 instead of 0.0002, or other variations, especially when doing long division or using calculators without carefully tracking decimal places.

3. Percentage conversion mistakes: After correctly finding the decimal 0.0002, students sometimes forget to multiply by 100 to convert to percentage form, or they multiply incorrectly, leading them to select 0.0002% instead of the correct 0.02%.

Errors while selecting the answer

1. Misreading answer choices: Given that all answer choices are percentages with different decimal place values (2.0%, 0.2%, 0.02%, 0.002%, 0.0002%), students often miscount the number of zeros and select an adjacent answer choice. This is particularly common when they've calculated 0.02% but accidentally select 0.2% or 0.002%.

2. Scale expectation errors: Students might second-guess their correct answer because they don't have an intuitive sense of how small 260 million should be relative to 1.3 trillion. They might think 0.02% seems "too small" and choose a larger percentage like 0.2% or 2.0% instead.