Question

6) Alberto invests $8,534 in a retirement

account with a fixed annual interest rate of
4% compounded 12 times per year. How
long will it take for the account balance to
reach $16,826.03?

Answer 1

It will take 17.2 years to reach account balance of $16826.03

Explanation:

Given

Principal amount P = $ 8534

rate of interest r = 4% = 4/100 = 0.04

No of times interest is compounded, n = 12

Final amount A = $ 16826.03

To find: time in years, t = ?

We know that final amount of compound interest A is given by formula:

A = P
`(1 + (r)/(n) )^(nt)`

Substituting known values,

16826.03 = 8534 *
`(1 + (0.04)/(12) )^(12t)`

`(1 + (0.04)/(12) )^(12t)` = 1.972

`(1.0033)^(12t)` = 1.972

take log on both sides,

ln
`(1.0033)^(12t)` = ln (1.972)

12t * ln (1.0033) = ln (1.972) (since ln
`x^(n)` = n * ln x)

12t = 206.112

t = 17.2 years

Answer 2

17 years

Explanation:

The compound interest formula is ...

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

where P is the principal invested at annual rate r, compounded n times per year for t years.

Filling in the numbers and solving for t, we find ...

16826.03 = 8534(1 +.04/12)^(12t)

16826.03/8534 = 1.0033333...^(12t)

Taking logs, we have ...

log(16826.03/8534) = 12t·log(1.0333333...)

Dividing by the coefficient of t gives ...

log(16826.03/8534)/(12·log(301/300)) = t ≈ 17.000

It will take 17 years for the account balance to reach $16,826.03.