Problem 55 Time to reach a financial goal You have 5603032 i

Problem 5-5 Time to reach a financial goal You have $56,030.32 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $240,000. You expect to earn 10% annually on the account. How many years will it take to reach your goal? Round your answer to two decimal places at the end of the calculations. years

Solution

We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period.

Hence future value of $56030.32=56030.32*(1+10/100)^n

=56030.32*(1.1)^n

Also Future value of annuity=Annuity[(1+rate)^time period-1]/rate

=$3000[(1.1)^n-1]/0.1

Hence

240,000=56030.32*(1.1)^n+$3000[(1.1)^n-1]/0.1

240,000=56030.32*(1.1)^n+$30000[(1.1)^n-1]

240,000=56030.32*(1.1)^n+$30000(1.1)^n-$30,000

(240,000+30,000)=(1.1)^n[56030.32+30000]

(1.1)^n=270,000/86030.32

(1.1)^n=3.138428405

Taking log on both sides;

n*log (1.1)=log3.138428405

n=log3.138428405/log 1.1

which is equal to

=12 years(Approx).