Question

You currently have $5,200. First United Bank will pay you an annual interest rate of 8.7, while Second National Bank will pay you an annual interest rate of 9.8. How many fewer years must you wait for your account value to grow to $12,100 at Second National Bank?

Answer 1

Time require for the investment to grow as per both the bank can be computed using formula for compound interest.

A = P x (1+r/n) nxt

A = Accumulated future value = $ 12,100

P = Principal = $ 5,200

n = Annual interest compounding frequency = 1

r = Rate of interest

t = Number of years

First Case (United Bank):

r = 0.087

$ 12,100 = $ 5,200 x (1+ 0.087) n

$ 12,100 = $ 5,200 x (1.087) n

(1.087) n = $ 12,100/$ 5,200

(1.087) n = 2.32692307692308

Taking log of both sides and solving we get n as,

n x log 1.087 = log 2.32692307692308   

n x 0.036229544086 = 0.36678202668

n = 0.36678202668/0.036229544086

    = 10.1238377664745 or 10.12 years

Second Case (National Bank):

r = 0.098

$ 12,100 = $ 5,200 x (1+ 0.098) n

$ 12,100 = $ 5,200 x (1.098) n

(1.098) n = $ 12,100/$ 5,200

(1.098) n = 2.32692307692308

Taking log of both sides and solving we get n as,

n x log 1.098 = log 2.32692307692308   

n x 0.040602340114 = 0.36678202668

n = 0.36678202668/ 0.040602340114

    = 9.03351938952727 or 9.03 years

Number of fewer years to wait in Second National Bank to grow the investment

                              = 10.12 – 9.03 = 1.09 years