Which of the following equals 5{,000}?

1. Translate the problem requirements: We need to find which of the five given fractions equals exactly 5,000. Each fraction has decimal numbers in both numerator and denominator.
2. Identify the most efficient calculation approach: Rather than calculating each fraction precisely, we can use decimal point manipulation to quickly eliminate options and identify patterns.
3. Apply strategic decimal manipulation: Convert fractions by moving decimal points in both numerator and denominator equally, transforming them into simpler equivalent fractions.
4. Verify the promising candidate: Once we identify the most likely answer through pattern recognition, perform the final calculation to confirm it equals 5,000.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We have five different fractions, and we need to find which one equals exactly 5,000.

Looking at our answer choices:

• Choice A: \(0.03\) divided by \(0.00006\)
• Choice B: \(0.3\) divided by \(0.06\)
• Choice C: \(3\) divided by \(0.0006\)
• Choice D: \(30\) divided by \(0.6\)
• Choice E: \(300\) divided by \(0.006\)

Notice that all of these involve dividing by decimal numbers. When we divide by a very small decimal (like \(0.0006\)), we get a much larger result than when we divide by a bigger decimal (like \(0.6\)).

Process Skill: TRANSLATE

2. Identify the most efficient calculation approach

Rather than calculating each fraction from scratch, let's think smart about this. We're looking for an answer of 5,000, which is a fairly large number. This means we probably need either:

• A big number divided by a small number, OR
• A moderate number divided by a very small number

Looking at our choices, let's see which ones have very small denominators:

• Choice A: denominator is \(0.00006\) (very small!)
• Choice B: denominator is \(0.06\) (small)
• Choice C: denominator is \(0.0006\) (very small!)
• Choice D: denominator is \(0.6\) (not that small)
• Choice E: denominator is \(0.006\) (small)

Choices A and C look most promising because they have the smallest denominators.

Process Skill: INFER

3. Apply strategic decimal manipulation

Here's where we use a clever trick. Instead of working with messy decimals, let's move decimal points to make the math easier.

Remember: if we multiply both the top and bottom of a fraction by the same number, the fraction's value doesn't change.

Let's check Choice C: \(3 ÷ 0.0006\)

To eliminate the decimal in the denominator, let's multiply both top and bottom by 10,000:

• Top: \(3 × 10,000 = 30,000\)
• Bottom: \(0.0006 × 10,000 = 6\)

So \(3 ÷ 0.0006\) becomes \(30,000 ÷ 6\)

Now this is easy to calculate: \(30,000 ÷ 6 = 5,000\)

Bingo! Choice C gives us exactly 5,000.

Let's quickly check Choice A to be thorough:

\(0.03 ÷ 0.00006\)

Multiplying both by 100,000:

• Top: \(0.03 × 100,000 = 3,000\)
• Bottom: \(0.00006 × 100,000 = 6\)

So this becomes \(3,000 ÷ 6 = 500\) (not 5,000)

Process Skill: MANIPULATE

4. Verify the promising candidate

Let's double-check our work with Choice C:

We found that \(3 ÷ 0.0006 = 30,000 ÷ 6\)

Dividing: \(30,000 ÷ 6 = 5,000\) ✓

This matches exactly what we're looking for!

Final Answer

The answer is C: \(\frac{3}{0.0006} = 5,000\)

Our decimal manipulation strategy helped us avoid messy calculations while quickly identifying the correct answer.

Common Faltering Points

Errors while devising the approach

• Misunderstanding the goal: Students might think they need to find which fraction is closest to 5,000 rather than exactly equals 5,000, leading them to accept approximate answers instead of calculating precisely.
• Choosing inefficient calculation methods: Students may decide to convert all decimals to long division or use calculators for each option instead of recognizing that strategic decimal manipulation can simplify the work significantly.
• Not recognizing patterns: Students might miss that fractions with very small denominators are more likely to produce large results, failing to prioritize checking options A and C first due to their smallest denominators (\(0.00006\) and \(0.0006\)).

Errors while executing the approach

• Decimal point errors during multiplication: When multiplying both numerator and denominator to eliminate decimals, students frequently miscount decimal places. For example, in \(3 ÷ 0.0006\), they might multiply by 1,000 instead of 10,000, or incorrectly convert \(0.0006 × 10,000\) to 60 instead of 6.
• Division calculation mistakes: After successfully converting to simpler forms like \(30,000 ÷ 6\), students may make basic arithmetic errors, perhaps getting 500 or 50,000 instead of the correct 5,000.
• Inconsistent decimal manipulation: Students might multiply the numerator and denominator by different powers of 10, which changes the fraction's value entirely, rather than multiplying both by the same number to maintain equivalence.

Errors while selecting the answer

• Stopping at the first reasonable result: If students check option A first and get 500, they might incorrectly think this is 'close enough' to 5,000 and select it without checking other options that give the exact answer.
• Misreading calculated results: After correctly calculating that option C equals 5,000, students might accidentally mark a different letter choice (like B or D) due to careless transcription or not double-checking their selection against their work.