How many yards are equivalent to 77 rods? (5.5 text{ yards} = 1 text{ rod})

1. TRANSLATE the problem information

• Given information:
  • 77 rods (what we're converting from)
  • \(\mathrm{5.5\,yards = 1\,rod}\) (conversion factor)
  • Need to find equivalent yards

• We need to convert rods to yards using the given relationship.

2. INFER the conversion approach

• Since \(\mathrm{5.5\,yards = 1\,rod}\), each rod is worth 5.5 yards
• To convert 77 rods to yards, we multiply by the conversion factor
• Set up: \(\mathrm{77\,rods \times \frac{5.5\,yards}{1\,rod}}\)

3. SIMPLIFY the calculation

• The "rods" units cancel out:
  \(\mathrm{77\,rods \times \frac{5.5\,yards}{1\,rod} = 77 \times 5.5\,yards}\)
• Calculate:
  \(\mathrm{77 \times 5.5 = 423.5}\)
• Final result: \(\mathrm{423.5\,yards}\)

Answer: \(\mathrm{423.5\,yards}\) (or \(\mathrm{\frac{847}{2}\,yards}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the direction of conversion and divide instead of multiply.

They might think: "If \(\mathrm{5.5\,yards = 1\,rod}\), then to convert 77 rods, I should divide \(\mathrm{77 \div 5.5}\)." This backwards reasoning comes from not clearly understanding which unit they're converting FROM and which unit they're converting TO. This leads to an answer of 14 yards instead of 423.5 yards, causing confusion and likely guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{77 \times 5.5}\).

Common mistakes include treating it as \(\mathrm{77 \times 5 = 385}\) (forgetting the 0.5) or making errors in decimal multiplication. This leads to answers like 385 yards or other incorrect values, causing them to second-guess their approach and potentially abandon the systematic solution.

The Bottom Line:

Unit conversion problems require careful attention to the direction of conversion. The key insight is recognizing that when you want MORE of something (rods to yards), and each rod gives you 5.5 yards, you multiply rather than divide.