Problem

Compare investments $A(t)=500\cdot~1.04^{t}$ and $B(t)=300\cdot~1.08^{t}$ by giving their starts and yearly growth rates, solving for the crossover time, and identifying the greater value before and after it.

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What this problem is really about

A larger exponential starting value and a larger growth factor are different advantages, so neither parameter alone determines the long-term comparison. We’ll convert both bases to percent rates, equate the models, and isolate the relative growth factor. Logarithms will locate the crossover, after which the higher start and faster growth explain the order on opposite sides.

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starts = A=$500, B=$300; growth = A 4%, B 8%; crossover = none; comparison = A remains greater forever because it starts larger

starts = A=$500, B=$300; growth = A 4%, B 8%; crossover = t=50 years from 200/4; comparison = A before, B after

starts = A=$500, B=$300; growth = A 4%, B 8%; crossover = none; comparison = B>A initially because 8%>4%

starts = A(0)=$500, B(0)=$300; growth = A 4%/year, B 8%/year; crossover = (1.08/1.04)ᵗ=5/3 ⇒ t≈13.535 years; comparison = A>B before crossover, equal there, B>A after

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Compare investments \(A(t)=500\cdot~1.04^{t}\) and \(B(t)=300\cdot~1.08^{t}\) by giving their starts and yearly growth rates, solving for the crossover time, and identifying the greater value before and after it.

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