From a piece of wire 0.5 meter long, a machine cuts off a piece 0.005 meter long. The remaining piece...

1. Translate the problem requirements: We need to find how many times longer the remaining piece is compared to the cut-off piece. This means we're looking for the ratio: \(\mathrm{(remaining\ length)} ÷ \mathrm{(cut-off\ length)}\)
2. Calculate the remaining wire length: Subtract the cut-off piece from the original wire to find what's left
3. Set up the comparison ratio: Divide the remaining length by the cut-off length to determine how many times longer it is
4. Simplify to reach the final answer: Perform the division, noting that the answer choices are whole numbers so we can work efficiently

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in everyday terms. We have a piece of wire that's 0.5 meters long. A machine cuts off a small piece that's 0.005 meters long. Now we want to know: how many times longer is the remaining piece compared to the piece that was cut off?

Think of it this way: if you had a pizza and gave away a small slice, how many times bigger would the remaining pizza be compared to that small slice you gave away? We're looking for this same type of comparison.

The phrase "how many times as long" means we need to find a ratio. Specifically, we want:
\(\mathrm{(length\ of\ remaining\ piece)} ÷ \mathrm{(length\ of\ cut-off\ piece)}\)

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

2. Calculate the remaining wire length

This is straightforward subtraction. We start with 0.5 meters and remove 0.005 meters.

Original length: 0.5 meters
Cut-off piece: 0.005 meters
Remaining length = \(0.5 - 0.005 = 0.495\) meters

So we have:
• Remaining piece: 0.495 meters
• Cut-off piece: 0.005 meters

3. Set up the comparison ratio

Now we need to find how many times longer the remaining piece is compared to the cut-off piece. This means we divide the remaining length by the cut-off length.

Ratio = (remaining length) ÷ (cut-off length)
Ratio = \(0.495 ÷ 0.005\)

Before we calculate this, let's think about what this division is telling us. We're essentially asking: "How many pieces of size 0.005 would fit into a length of 0.495?"

4. Simplify to reach the final answer

Let's calculate \(0.495 ÷ 0.005\). To make this easier, we can work with the division by removing decimals:

\(0.495 ÷ 0.005 = 495 ÷ 5\)

This works because we multiplied both numbers by 1000 to eliminate decimals.

\(495 ÷ 5 = 99\)

Let's verify this makes sense: if we multiply \(99 × 0.005 = 0.495\) ✓

And indeed, \(0.495 + 0.005 = 0.500\), which matches our original wire length ✓

Process Skill: SIMPLIFY - Converting decimal division to simpler whole number division

Final Answer

The remaining piece is 99 times as long as the piece that was cut off.

The answer is B. 99

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "how many times as long"
Students often confuse this phrase and set up the ratio backwards, calculating (cut-off piece) ÷ (remaining piece) instead of (remaining piece) ÷ (cut-off piece). This fundamental misunderstanding of the comparison direction leads to selecting answer choice A (which would be approximately 1/99).

2. Forgetting to subtract the cut-off piece
Some students incorrectly think they need to compare the original wire length (0.5 meters) to the cut-off piece (0.005 meters), setting up \(0.5 ÷ 0.005 = 100\). This leads them to choose answer choice C, missing that the problem specifically asks about the "remaining piece" after cutting.

Errors while executing the approach

1. Decimal division calculation errors
When calculating \(0.495 ÷ 0.005\), students may struggle with decimal division and make arithmetic mistakes. Common errors include incorrectly converting to 495 ÷ 50 instead of 495 ÷ 5, or making errors when eliminating decimals by the same factor from both numerator and denominator.

2. Subtraction errors with decimals
Students may incorrectly calculate \(0.5 - 0.005\), potentially getting 0.45 instead of 0.495. This error often occurs when students don't properly align decimal places or rush through what seems like a "simple" subtraction step.

Errors while selecting the answer

1. Selecting the wrong calculated value
Students who correctly calculate both 0.495 (remaining length) and 99 (the ratio) might mistakenly select answer choice D (495) because they see this number in their work and confuse it with the final answer. They fail to recognize that 495 was just an intermediate step in the decimal division process.