Math problem

Question

Kelly wants to accumulate 1 million dollars by investing a lump sum into an IRA (Individual Retirement Account) and letting it grow for the next 45 years.

After researching different IRA's she has narrowed her investment decision to two possibilities.

Option 1 pays an APR of 5.95% compounded continuously, and Option 2 pays an APR of 5.9% compounded annually.
 
 
 
A). How much needs to be invested in Option 1 to accumulate the desired amount?
(Round to the nearest dollar and don't include commas.)
 
 
B).How much needs to be invested in Option 2 to accumulate the desired amount?
(Round to the nearest dollar and don't include commas.)
.

C).Which option is the best (highest return)?
 

Option 1
 
Option 2

Answer 1

Continuously compounded interest

 

A = Pe^rt, where P = principal, r = rate, t = time, A = amount

1000000 = Pe^0.0595(45)

1000000 = Pe^2.6775

ln 1000000 = ln Pe^2.6775

ln 1000000 = ln P + ln e^2.6775

13.8 ≈ ln P + 2.6775 ln e

13.8 ≈ ln P + 2.6775

11.12 ≈ ln P

67507.90 ≈ P

 

For option one, $67,507.90 needs to be invested.

 

Option B

 

1000000 = p(1 + r/n)^nt, where p = principal, r = rate, n = number of calculations per year, t = years

1000000 = p(1 + 0.059)⁴⁵

1000000 = p 1.059⁴⁵

1000000 ≈ 13.19p

75815 ≈ p

 

Option B needs to invest $75,815.