\(\frac{(39,897)(0.0096)}{198.76}\) is approximately

1. Translate the problem requirements: We need to evaluate a fraction with a large numerator (product of two numbers) divided by a denominator, and find which answer choice it's closest to approximating
2. Analyze answer choice spacing: Notice the answer choices span from 0.02 to 200 (factors of 10 apart), indicating we only need a rough approximation rather than precise calculation
3. Round numbers strategically for estimation: Convert each number to a simpler form that maintains the overall magnitude while making mental math possible
4. Perform simplified calculation: Calculate using the rounded numbers to get an approximate result that clearly falls within one answer choice range

Execution of Strategic Approach

1. Translate the problem requirements

We need to calculate the value of a fraction where the top part (numerator) is the product of 39,897 and 0.0096, and the bottom part (denominator) is 198.76. Think of this like figuring out how many times the denominator fits into the numerator.

The expression is: \((39,897 \times 0.0096) \div 198.76\)

Since this is asking for an approximation, we don't need to be perfectly precise - we just need to get close enough to identify which answer choice is correct.

Process Skill: TRANSLATE

2. Analyze answer choice spacing

Let's look at our answer choices: 0.02, 0.2, 2, 20, and 200.

Notice something important: each choice is exactly 10 times larger than the previous one. This tells us that we only need to figure out the general size or "order of magnitude" of our answer. We don't need to calculate precisely to several decimal places.

This spacing suggests that even rough approximations will clearly point us to the right answer choice.

3. Round numbers strategically for estimation

Let's make these numbers much friendlier to work with by rounding them smartly:

• 39,897 is very close to 40,000 (we're rounding to the nearest ten thousand)
• 0.0096 is very close to 0.01 (we're rounding to the nearest hundredth)
• 198.76 is very close to 200 (we're rounding to the nearest hundred)

So our complicated expression becomes: \((40,000 \times 0.01) \div 200\)

These numbers are much easier to work with mentally!

Process Skill: SIMPLIFY

4. Perform simplified calculation

Now let's calculate step by step with our simplified numbers:

First, let's find \(40,000 \times 0.01\):
• When we multiply by 0.01, we're essentially dividing by 100
• \(40,000 \div 100 = 400\)

Now we need: \(400 \div 200\)
• This is straightforward: \(400 \div 200 = 2\)

So our approximation gives us 2.

Let's verify this makes sense by checking that our rounding didn't drastically change the original numbers:
• 39,897 ≈ 40,000 (less than 1% difference)
• 0.0096 ≈ 0.01 (about 4% difference)
• 198.76 ≈ 200 (less than 1% difference)

These are all very reasonable approximations, so we can be confident in our result.

Final Answer

Our calculation gives us approximately 2, which corresponds to answer choice C.

This makes perfect sense when we look at our answer choices - our result of 2 is clearly much closer to choice C (2) than to the neighboring choices B (0.2) or D (20).

Common Faltering Points

Errors while devising the approach

• Attempting exact calculation instead of strategic approximation: Students may try to calculate the exact value of \((39,897)(0.0096) \div 198.76\) using long multiplication and division, rather than recognizing that the widely spaced answer choices (0.02, 0.2, 2, 20, 200) indicate that strategic rounding and approximation is the intended approach.
• Poor rounding strategy leading to calculation complexity: Students might round numbers arbitrarily (like 39,897 to 40,000 but 0.0096 to 0.009) without considering how their choices affect calculation difficulty, missing the opportunity to create a simple mental math problem like \((40,000 \times 0.01) \div 200\).

Errors while executing the approach

• Decimal placement errors when multiplying by 0.01: When calculating \(40,000 \times 0.01\), students might incorrectly place the decimal point and get 4,000 instead of 400, leading them to calculate \(4,000 \div 200 = 20\) and select answer choice D instead of C.
• Inconsistent rounding direction affecting final result: Students might round all numbers in the same direction (all up or all down) rather than to the nearest convenient values, potentially shifting their approximation significantly away from the true answer and leading to an incorrect order of magnitude.

Errors while selecting the answer

• Lack of reasonableness check on final answer: Students may arrive at their calculated result but fail to verify it makes sense in context - for example, not recognizing that if they get an answer like 0.2 or 20, they should double-check their decimal point placement since the original calculation involves a number around 40,000 in the numerator and around 200 in the denominator.