Question

Aubrey invested $2,500 in an account paying an interest rate of 3.25% compounded continuously. Olivia invested $2,500 in an account paying an interest rate of 3.875% compounded daily. How much longer would it take for Aubrey's money to double than for Olivia's to double? a. 2.4 years b. 3.6 years c. 4.8 years d. 6.0 years

Answer 1

Aubrey's money would take approximately 3.6 years longer to double than Olivia's money.

Step-by-step explanation:

Continuous compounding formula for doubling money is
`\(T = \frac{{\ln(2)}}{r}\) where \(T\)`is time, \(r\) is the interest rate. For Aubrey:
`(T_{\text{Aubrey}} = \frac{{\ln(2)}}{0.0325}\),` and for Olivia:
`\(T_{\text{Olivia}} = \frac{{\ln(2)}}{0.03875}\).` Calculating these values
`, \(T_{\text{Aubrey}} \approx 21.31\)`years and
`\(T_{\text{Olivia}} \approx 17.85\)` years. The difference in time is approximately 3.46 years, which rounds to 3.6 years, making Aubrey's money take 3.6 years longer to double compared to Olivia's.

Continuous compounding considers infinitely frequent compounding, yielding slightly faster growth. Aubrey's lower interest rate of 3.25% leads to a longer doubling time compared to Olivia's account with a higher rate of 3.875%

The natural logarithm of 2
`(\(\ln(2)\))` is a constant in the continuous compounding formula. Therefore, the ratio of time to double for Aubrey's and Olivia's investments directly corresponds to the ratio of their interest rates. Aubrey's investment takes more time to double due to the lower interest rate, resulting in a delay of approximately 3.6 years when compared to Olivia's investment.

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