A distance of 0.51 kilometer is measured along a trail. What is this distance, in centimeters?(1 kilometer = 1{,000 meters};...

1. TRANSLATE the problem information

• Given information:
  • Distance: 0.51 kilometers
  • Need to find: distance in centimeters
  • Conversion factors: \(\mathrm{1\ km = 1{,}000\ m}\); \(\mathrm{1\ m = 100\ cm}\)

2. INFER the conversion strategy

• Since we need to go from km → cm, and we have km → m and m → cm conversions
• Strategy: Convert km → m first, then m → cm (two-step approach)
• Alternative: Calculate direct km → cm conversion factor

3. SIMPLIFY the first conversion (km to m)

• \(\mathrm{0.51\ km \times 1{,}000\ m/km = 510\ m}\)
• The km units cancel, leaving meters

4. SIMPLIFY the second conversion (m to cm)

• \(\mathrm{510\ m \times 100\ cm/m = 51{,}000\ cm}\) (use calculator)
• The m units cancel, leaving centimeters

Answer: C. 51,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make decimal multiplication errors or lose track of place values during the multi-step conversion.

For example, they might calculate \(\mathrm{0.51 \times 1{,}000 = 5{,}100}\) and then forget the second conversion step, selecting Choice B (5,100). Or they might make an arithmetic error and get confused about decimal placement.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students recognize they need to convert but don't properly set up both conversion steps or convert in the wrong direction.

Some students might think "smaller unit means divide" and incorrectly divide by the conversion factors instead of multiply, leading to very small answers. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically apply unit conversions with decimal numbers. The key is staying organized through the two-step process and being careful with decimal arithmetic.