6
advanced-math

Which investment grows faster over the long term?

A: $5,000 at 6% compounded annually B: $7,000 at 4% compounded annually

A

Investment A

B

Investment B

C

They grow at the same rate

D

Cannot determine

Correct Answer: A

Choice A is the correct answer. Compare growth rates over time.

  1. Investment A: AA(t)=5000(1.06)tA_A(t) = 5000(1.06)^t.
  2. Investment B: AB(t)=7000(1.04)tA_B(t) = 7000(1.04)^t.
  3. Long-term: Higher growth rate (6% vs 4%) dominates.
  4. Crossover: Eventually, 5000(1.06)t>7000(1.04)t5000(1.06)^t > 7000(1.04)^t as tt increases.
  5. Test t=20t=20:
    • A: 5000(1.06)2016,0365000(1.06)^{20} \approx 16,036
    • B: 7000(1.04)2015,3467000(1.04)^{20} \approx 15,346

💡 Strategic Tip: In exponential functions, the base (growth rate) dominates in the long run.

Choice B is incorrect because despite higher initial value, lower rate loses. Choice C is incorrect because different rates mean different growth. Choice D is incorrect because we can compare by calculating.