Question

Alyssa opened a retirement account with 7.25% APR in the year 2000. Her initial deposit was $13,500. How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?

Answer 1

Consider the Compound Interest formula;

A(t) = P(1 + r/n)nt

 

Where, A(t) is the account value, “t” is the measured in years, “P” the starting amount or initial value, “r” is the annual percentage rate (APR), “n” is the number of compounding periods in one year.

Consider the initial deposit is P = 13500 dollars.

 

The annual percentage rate is r = 7.25%,

Therefore,

r = 7.25/100

   = 0.0725

 

The interest is compounded monthly n = 12,

 

The investment amount after t = 25 years is determined as follows;

Substitute the known value as follows:

A(25) = 13500(1 + 0.0725/12)12×25

          = 13500(1.006)300

         = 82247.78

 

Therefore, the amount received on 2025 that compounded monthly is 82247.78 dollars.

 

The amount received on 2025 that compounded continuously is determined as follows:

The formula for the continuous growth or decay is:

A(t) = aert

 

Where, the initial deposit is a = 13500 and the continuous growth rate is r;

r = 7.25/100

   = 0.0725

 

The investment amount after t = 25 years is determined as follows;

 

Substitute the known value as follows:

A(25) = 13500(e0.0725×25)

          = 13500(e1.8125)

          = 13500 × 6.1257

         = 82698

 

Therefore, the amount received on 2025 that compounded continuously is 82698 dollars.

Therefore, the amount received on 2025 that compounded continuously is 82698 dollars.