4000$ invested at bank A with a normal rate of 3.23 compounded quarterly. 3000$ invested at bank B with a normal rate of 4.83 compunded annually.

Question

How long would it take for both values to be equal? 

Answer 1

 

Let A = account balance with bank A

Let B = account balance with bank B

 

A = 4000(1+r/n)^nt    

 

where

         r=annual rate as a decimal

         n=number of payments per year

         t=number of years

 

So   A = 4000(1+0.0323/4)^4t

 

and  B = 3000(1+0.0483)^t

 

 

Set A=B:    4000(1+0.0323/4)^4t  = 3000(1+0.0483)^t

 

Combine where possible:                

       1.3333(1.008075)^4t = (1.0483)^t

 

Here's the key step, take log's of both sides, the scientist in me like to use ln:

 

       ln(1.3333) + 4t(ln(1.008075)) = t ln(1.0483)

 

Taking the logs and solving for t yields

                   

                 t = 19.18 years

 

 

 

Check:  

      A = 4000(1+0.0323/4)^4(19.18)  = 7413.63

      B = 3000(1+0.0483)^19.18         = 7413.69   (need to carry more digits of

                                                                       precision throughout to get

                                                                       agreement to the penny)