A model predicts that the amount of a radioactive substance was 500 grams at time t=0. The model also predicts...
GMAT Advanced Math : (Adv_Math) Questions
A model predicts that the amount of a radioactive substance was \(\mathrm{500\ grams}\) at time \(\mathrm{t=0}\). The model also predicts that each minute for the next \(\mathrm{10\ minutes}\), the amount decreases by \(\mathrm{5\%}\) of the previous minute's amount. Which equation best represents this model, where \(\mathrm{t}\) is the number of minutes after \(\mathrm{t=0}\), for \(\mathrm{t \leq 10}\)?
\(\mathrm{m = 0.95(500)^t}\)
\(\mathrm{m = 1.05(500)^t}\)
\(\mathrm{m = 500(1.05)^t}\)
\(\mathrm{m = 500(0.95)^t}\)
1. TRANSLATE the problem information
- Given information:
- Initial amount: 500 grams at \(\mathrm{t=0}\)
- Each minute: amount decreases by 5% of previous amount
- Time range: \(\mathrm{t \leq 10}\) minutes
- What this tells us: We need an equation showing how 500 grams changes over t minutes
2. TRANSLATE the percentage decrease
- "decreases by 5%" means it loses 5% and keeps 95%
- Keeping 95% = multiplying by 95% = multiplying by 0.95
- So each minute, the amount gets multiplied by 0.95
3. INFER the exponential pattern
- After 1 minute: \(\mathrm{500 \times 0.95}\)
- After 2 minutes: \(\mathrm{500 \times 0.95 \times 0.95 = 500 \times (0.95)^2}\)
- After t minutes: \(\mathrm{500 \times (0.95)^t}\)
- This follows the exponential decay model: \(\mathrm{m = \text{initial amount} \times (\text{decay factor})^{\text{time}}}\)
4. APPLY CONSTRAINTS to select the correct equation
- Our equation: \(\mathrm{m = 500(0.95)^t}\)
- Checking against choices: Only choice (D) matches exactly
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "decreases by 5%" as multiplying by 0.05 instead of 0.95.
They think: "5% decrease → multiply by 0.05" rather than "5% decrease → keep 95% → multiply by 0.95". This fundamental misunderstanding of percentage decrease leads them to look for an equation with 0.05, but since none exists, they get confused and may select Choice (A) (\(\mathrm{m = 0.95(500)^t}\)) because it has 0.95 in it, even though the structure is wrong.
Second Most Common Error:
Conceptual confusion about percentage relationships: Students confuse 5% decrease with 5% increase and think the multiplier should be 1.05.
They reason: "5% change → 1.05" without considering the direction of change. This leads them to select Choice (C) (\(\mathrm{m = 500(1.05)^t}\)) because it has the right structure but wrong multiplier.
The Bottom Line:
Success on this problem requires accurately translating percentage language into mathematical multipliers, then recognizing the exponential pattern that emerges from repeated multiplication.
\(\mathrm{m = 0.95(500)^t}\)
\(\mathrm{m = 1.05(500)^t}\)
\(\mathrm{m = 500(1.05)^t}\)
\(\mathrm{m = 500(0.95)^t}\)