Question 1179805: I have $25,000 and I want to have $36,000. If the bank is paying 4.18% compounded monthly, how
long will it take to reach my goal? (Rounded to the nearest day)
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! formula to use is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods..
the interest rate per year = 4.18%
divide that by 100 to get an interest rate of .0418 per year.
divide that by 12 to get an interect rate of .0418/12 per month.
f = 36
p = 25
formula becomes:
36 = 25 * (1 + .0418/12) ^ n
divide both sides by 36 to get:
36/25 = (1 + .0418/12) ^ n
take the log of both sides to get:
log(36/25) = log((1 + .0418/12) ^ n)
since log(x^n) = n * log(x), this becomes:
log(36/25) = n * log(1 + .0418/12)
solve for n to get:
n = log(36/25) / log(1 + .0418/12) = 104.8644495.
that's the number of months for 25000 to grow to 36000 at 4.18% per year compounded monthly.
confirm by solving for f with that value of n to get:
f = 25000 * (1 + .0418/12) ^ 104.8644495 = 36000.
answer is confirmed to be correct.
number of months required is 104.8644495.
number of years required is that divided by 12 = 8.730704122.
in order to confrim that to be correct, you need the effective interest rate per year.
that would be equal to (1 + .0418/12) ^ 12 = 1.04261019 minus 1 = .04261019.
confirm by replacing n with the number of years and r with the effective interest rate per year to get:
f = 25000 * (1 + .04261019) ^ 8.738704122 = 36000.
Answer by ikleyn(52778)  (Show Source):
You can put this solution on YOUR website! .
From solving logarithmic equation, @Theo got
n = 104.8644495 months
and he states that it is the time to get $36000.
This answer is INCORRECT.
The obtained decimal number of months MUST BE ROUNDED to the nearest greater month n = 105
in order for the bank was in position to make the last compounding.
So, the correct answer is THIS:
The goal will be reached in 105 months.
------------
I noticed and repeated it tens times at this forum that in such problems for discretely compounded accounts,
the time, found formally, must be rounded to the end of the closest compounding periods,
but it looks like no one tutor here did not get it.
It is really unclear to me, how these people can teach others, without understanding so simple facts.
It is also an important notice to the COMPOSER of this problem:
It is INCORRECT requirement in the problem's formulation
to request rounding to the nearest day.
The rounding MUST GO to the nearest greater MONTH,
or to the nearest completed compounding period.
///////////
On discretely compounded accounts, see my lesson
- Problems on discretely compound accounts
in this site, and learn the subject from there.
You will find there a variety of similar and different problems, solved with all detailed explanations.
After reading this lesson, you will tackle such problems on your own without asking for help from outside.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic "Logarithms".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
Happy learning (!)
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