A distance of 61 furlongs is equivalent to how many feet? (1 furlong = 220 yards and 1 yard =...

1. TRANSLATE the problem information

• Given information:
  • Need to convert: 61 furlongs to feet
  • Conversion facts: \(\mathrm{1\text{ furlong} = 220\text{ yards}}\), \(\mathrm{1\text{ yard} = 3\text{ feet}}\)
  • Find: equivalent distance in feet

2. INFER the conversion strategy

• We can't convert directly from furlongs to feet
• We need to chain conversions: furlongs → yards → feet
• Use dimensional analysis with conversion factors to ensure units cancel properly

3. TRANSLATE conversion facts into mathematical ratios

• From "1 furlong = 220 yards" we get: \(\mathrm{\frac{220\text{ yards}}{1\text{ furlong}}}\)
• From "1 yard = 3 feet" we get: \(\mathrm{\frac{3\text{ feet}}{1\text{ yard}}}\)
• Set up the calculation: \(\mathrm{61\text{ furlongs} \times \frac{220\text{ yards}}{1\text{ furlong}} \times \frac{3\text{ feet}}{1\text{ yard}}}\)

4. SIMPLIFY by canceling units and calculating

• Notice how units cancel: furlongs cancel first, then yards cancel
• This leaves: \(\mathrm{61 \times 220 \times 3\text{ feet}}\)
• Calculate step by step: \(\mathrm{61 \times 220 = 13{,}420}\)
• Then: \(\mathrm{13{,}420 \times 3 = 40{,}260\text{ feet}}\)

Answer: 40,260 feet

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to chain multiple conversions and instead try to convert directly from furlongs to feet, or they set up the conversions incorrectly.

For example, they might try: \(\mathrm{61 \times 220 + 3 = 13{,}423}\), or \(\mathrm{61 \times (220 + 3) = 13{,}603}\), mixing up how to combine the conversion factors. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the dimensional analysis correctly but make arithmetic errors when calculating \(\mathrm{61 \times 220 \times 3}\).

Common calculation mistakes include: \(\mathrm{61 \times 220 = 1{,}342}\) (missing a zero) leading to \(\mathrm{1{,}342 \times 3 = 4{,}026}\), or calculating \(\mathrm{61 \times 220}\) correctly but then computing \(\mathrm{13{,}420 \times 3 = 4{,}026}\) (decimal point error). This causes them to arrive at incorrect numerical answers.

The Bottom Line:

This problem requires systematic thinking about unit conversion chains. Students who try to shortcut the process or aren't careful with multi-step arithmetic will struggle to get the correct answer.