Question

You have $46,030.32 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $230,000. 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.

Answer 1

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 $46030.32=$46030.32*(1.1)^n

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

=$4000[(1.1)^n-1]/0.1

Hence

230,000=$46030.32*(1.1)^n+$4000[(1.1)^n-1]/0.1

230,000=$46030.32*(1.1)^n+$40000[(1.1)^n-1]

230,000=$46030.32*(1.1)^n+$40000(1.1)^n-$40000

(230000+40000)=(1.1)^n[$46030.32+$40000]

270000=(1.1)^n*86030.32

(270000/86030.32)=(1.1)^n

(1.1)^n=3.138428405

Taking log on both sides;

n*log 1.1=log3.138428405

n=log 3.138428405/log 1.1

which is equal to

=12 years(Approx).