A medication loses 40% of its effectiveness each hour after administration. Initially, the effective dose is 250 milligrams. Which expression...

1. TRANSLATE the problem information

• Given information:
  • Initial effective dose: \(250\text{ mg}\)
  • Loses \(40\%\) effectiveness each hour
  • Need expression for dose \(\mathrm{d}\) after \(\mathrm{t}\) hours

• What this tells us: If \(40\%\) is lost each hour, then \(60\%\) remains each hour

2. INFER the mathematical pattern

• This is exponential decay - the amount decreases by the same percentage each time period
• We need the exponential decay formula: \(\text{Final Amount} = \text{Initial Amount} \times (\text{remaining fraction})^{\text{time}}\)
• The remaining fraction is \(0.6\) (since \(60\%\) remains)

3. APPLY the exponential decay formula

• \(\mathrm{d} = 250 \times (0.6)^{\mathrm{t}}\)
• We can verify: at \(\mathrm{t} = 0\),
  \(\mathrm{d} = 250(0.6)^0\)
  \(= 250 \times 1\)
  \(= 250\text{ mg}\) ✓
• At \(\mathrm{t} = 1\),
  \(\mathrm{d} = 250(0.6)^1\)
  \(= 250 \times 0.6\)
  \(= 150\text{ mg}\) (this is \(250 - 100 = 150\), representing a \(40\%\) loss) ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify this as exponential decay but use \(0.4\) (the amount lost) instead of \(0.6\) (the amount remaining) as the base.

They think: "It loses 40% each hour, so I should use 0.4 in my formula." This leads to \(\mathrm{d} = 250(0.4)^{\mathrm{t}}\).

This may lead them to select Choice C (\(\mathrm{d} = 250(0.4)^{\mathrm{t}}\))

Second Most Common Error:

Poor INFER reasoning about formula structure: Students understand the 60% remaining concept but incorrectly structure the exponential formula by making the initial dose (250) the base instead of the coefficient.

They might think the dose itself grows exponentially rather than shrinks by a factor each hour.

This may lead them to select Choice A (\(\mathrm{d} = 0.4(250)^{\mathrm{t}}\)) or Choice B (\(\mathrm{d} = 0.6(250)^{\mathrm{t}}\))

The Bottom Line:

The key insight is recognizing that "loses 40%" means "retains 60%" and that in exponential decay, we raise the remaining fraction to the power of time, not the initial amount.