A radioactive isotope has a half-life of 4 hours, meaning that exactly half of the substance decays every 4 hours....

1. TRANSLATE the problem information

• Given information:
  • Half-life = 4 hours (half the substance decays every 4 hours)
  • Initial amount = 6,400 grams
  • Time period = 20 hours
  • Need to find: Amount remaining after 20 hours

2. INFER the decay pattern

• This is exponential decay - each half-life, exactly half remains
• Key insight: We need to find how many 4-hour periods fit into 20 hours
• Each period reduces the amount by factor of 1/2

3. Calculate the number of half-lives

Number of half-lives = \(\mathrm{20\ hours ÷ 4\ hours = 5\ half-lives}\)

4. INFER the exponential formula

• After 1 half-life: \(\mathrm{6{,}400 × (1/2) = 3{,}200\ grams}\)
• After 2 half-lives: \(\mathrm{6{,}400 × (1/2)^2 = 1{,}600\ grams}\)
• After 5 half-lives: \(\mathrm{6{,}400 × (1/2)^5}\)

5. SIMPLIFY the calculation

• \(\mathrm{(1/2)^5 = 1/32}\)
• \(\mathrm{6{,}400 × (1/32) = 6{,}400 ÷ 32 = 200\ grams}\)

Answer: A (200)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students misinterpret "half-life" as linear decay instead of exponential decay.

They might think: "If half decays every 4 hours, then in 20 hours (5 periods), everything decays except \(\mathrm{6{,}400 - 5(3{,}200) = -9{,}600}\)" which makes no sense, or they calculate \(\mathrm{6{,}400 ÷ 5 = 1{,}280}\), thinking the decay is evenly distributed.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify the exponential pattern but make calculation errors.

They might calculate \(\mathrm{(1/2)^5}\) incorrectly (perhaps as \(\mathrm{1/10}\) instead of \(\mathrm{1/32}\)), or make arithmetic errors in \(\mathrm{6{,}400 ÷ 32}\), potentially getting 320 instead of 200.

This may lead them to select Choice B (400) if they calculated \(\mathrm{(1/2)^5}\) as \(\mathrm{1/16}\) instead of \(\mathrm{1/32}\).

The Bottom Line:

This problem tests whether students truly understand exponential decay versus linear thinking, combined with careful arithmetic with powers and fractions.