A distance of 112 furlongs is equivalent to how many feet? (1 furlong = 220 yards and 1 yard =...

1. TRANSLATE the problem information

• Given information:
  • 112 furlongs (what we're converting)
  • 1 furlong = 220 yards (first conversion factor)
  • 1 yard = 3 feet (second conversion factor)
• We need to find: equivalent distance in feet

2. INFER the conversion strategy

• We can't convert directly from furlongs to feet, so we need to chain conversions
• Path: furlongs → yards → feet
• Set up dimensional analysis so units cancel properly

3. TRANSLATE into mathematical setup

Set up the conversion with factors arranged to cancel units:

\(112 \mathrm{furlongs} \times \frac{220 \mathrm{yards}}{1 \mathrm{furlong}} \times \frac{3 \mathrm{feet}}{1 \mathrm{yard}}\)

Notice how:

• "furlongs" cancels between the first term and first fraction
• "yards" cancels between the two fractions
• Only "feet" remains

4. SIMPLIFY through calculation

• First multiply the numbers: \(112 \times 220 \times 3\)
• Calculate: \(112 \times 220 = 24,640\)
• Then: \(24,640 \times 3 = 73,920\) (use calculator)

Answer: 73,920 feet

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students set up conversion factors incorrectly, either using them upside-down or in the wrong order.

For example, they might write: \(112 \mathrm{furlongs} \times \frac{1 \mathrm{furlong}}{220 \mathrm{yards}} \times \frac{1 \mathrm{yard}}{3 \mathrm{feet}}\), which would give them a much smaller number instead of the correct larger number. This leads to confusion about whether their answer makes sense and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the final multiplication step.

They might correctly set up \(112 \times 220 \times 3\) but calculate incorrectly, getting values like 73,290 or 74,820. Since this is a calculation-heavy problem, small arithmetic mistakes compound into significantly wrong final answers.

The Bottom Line:

This problem tests whether students can systematically chain unit conversions without getting lost in the setup or making calculation errors along the way.