For a person m miles from a flash of lightning, the length of the time interval from the moment the...

1. TRANSLATE the problem information

• Given information:
  • Distance from person to lightning: \(\mathrm{m}\) miles
  • Time interval from seeing lightning to hearing thunder: \(\mathrm{k}\) seconds
  • Ratio of \(\mathrm{m}\) to \(\mathrm{k}\) is \(\mathrm{1\ to\ 5}\)
  • Actual time interval: \(\mathrm{k = 25}\) seconds
  • Find: \(\mathrm{m}\) (distance in miles)

• What "ratio of \(\mathrm{m}\) to \(\mathrm{k}\) is \(\mathrm{1\ to\ 5}\)" means:
  • This translates to \(\mathrm{\frac{m}{k} = \frac{1}{5}}\)

2. INFER the solution approach

• Since we know the ratio \(\mathrm{\frac{m}{k} = \frac{1}{5}}\) and we know \(\mathrm{k = 25}\), we can set up a proportion to find \(\mathrm{m}\)
• We need to substitute the known value of \(\mathrm{k}\) into our ratio equation

3. SIMPLIFY to find the answer

• Set up the equation: \(\mathrm{\frac{m}{25} = \frac{1}{5}}\)
• Multiply both sides by 25: \(\mathrm{m = 25 \times \frac{1}{5}}\)
• Calculate: \(\mathrm{m = \frac{25}{5} = 5}\)

Answer: D. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "ratio of \(\mathrm{m}\) to \(\mathrm{k}\) is \(\mathrm{1\ to\ 5}\)" means, thinking it means \(\mathrm{m = 5k}\) instead of \(\mathrm{\frac{m}{k} = \frac{1}{5}}\).

If they think the ratio means \(\mathrm{m}\) is 5 times larger than \(\mathrm{k}\), they would calculate \(\mathrm{m = 5 \times 25 = 125}\). Since 125 isn't among the answer choices, this leads to confusion and guessing among the available options.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the proportion correctly as \(\mathrm{\frac{m}{25} = \frac{1}{5}}\) but make arithmetic errors in the final calculation steps.

For example, they might misapply the multiplication and think \(\mathrm{m = 25 \times 5}\) instead of \(\mathrm{m = 25 \times \frac{1}{5}}\), leading to incorrect calculations. This computational confusion may lead them to select Choice A (10) if they somehow get their arithmetic tangled up.

The Bottom Line:

This problem tests whether students can correctly interpret ratio language and convert it into mathematical notation. The key insight is recognizing that "\(\mathrm{m}\) to \(\mathrm{k}\) is \(\mathrm{1\ to\ 5}\)" means \(\mathrm{\frac{m}{k} = \frac{1}{5}}\), not that \(\mathrm{m}\) is 5 times something.