A participant in a bicycle race completes the race with an average speed of 24,816 yards per hour. What is...

1. TRANSLATE the problem information

• Given information:
  • Speed: \(\mathrm{24,816\text{ yards per hour}}\)
  • Conversion factor: \(\mathrm{1\text{ mile} = 1,760\text{ yards}}\)
  • Need: speed in miles per hour

2. TRANSLATE the conversion setup

• To convert yards per hour to miles per hour, we need to multiply by a conversion factor
• The conversion factor must cancel out "yards" and leave us with "miles"
• Correct setup: \(\mathrm{24,816\text{ yards/hour} \times \frac{1\text{ mile}}{1,760\text{ yards}}}\)

3. SIMPLIFY the calculation

• The "yards" units cancel out:

\(\mathrm{24,816\text{ yards/hour} \times \frac{1\text{ mile}}{1,760\text{ yards}} = \frac{24,816}{1,760}\text{ miles/hour}}\)

• Calculate: \(\mathrm{24,816 \div 1,760 = 14.1}\) (use calculator)

Answer: \(\mathrm{14.1\text{ miles per hour}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students set up the conversion factor upside down

Students might write: \(\mathrm{24,816 \times \frac{1,760}{1} = 43,721,280}\), thinking they need to multiply by 1,760 since "there are more yards than miles." This completely misses the concept of dimensional analysis where units must cancel properly.

This leads to confusion when they realize their answer is impossibly large, causing them to guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students try to divide by the wrong number

Some students remember they need to divide but think: "Since \(\mathrm{1\text{ mile} = 1,760\text{ yards}}\), I should divide 24,816 by 1 to get miles." They don't properly set up the conversion factor and instead just divide by 1, getting 24,816 as their final answer.

This leads to confusion and guessing when they realize this can't be right.

The Bottom Line:

This problem tests whether students understand dimensional analysis for unit conversion. The key insight is that conversion factors must be set up so unwanted units cancel out, leaving only the desired units.