Question

Your goal is to create a college fund for your child. Suppose you find a fund that offers an APR of 4 %. How much should you deposit monthly to accumulate ​$80, 000 in 18 ​years?

Answer 1

Answer:

$701.12

Step-by-step explanation:

Before doing the computing, notice that if we increase an amount D in,say,  r%, it means that the new amount obtained is D+r% of D = D+(r/100)D = D(1+r/100).

That is to say, increasing an amount D in r% is equivalent to multiplying it by (1+r/100)

Taking into account that APR stands for Annual Percentage Rate, the amount we deposit will be increased 4% each year.  

Generally, in this kind of loans the percentage is prorated monthly. That is to say, the money you have in the account will be increased in (4/12)%= 0.3333% each month.

Let D be the amount we are going to deposit each month.

After the month 1 we will have the money increased in 0.3333% plus the new deposit

$D(1+\frac{0.3333}{100})+D=D(1+1.0033)$

After the month 2 we will have the money we already had increased in 0.3333% plus the new deposit D

$D(1+1.0033)(1.0033)+D=D(1.0033+1.0033^2)+D=D(1+1.0033+1.0033^2)$

After the month 3 we will have, for the same reason,

$D(1+1.0033+1.0033^2+1.0033^3)$

It can be noticed then, that after 18 years (96 moths) we will have an amount in the fund of

$D(1+1.0033+1.0033^2+...+1.0033^{96})$

If we call  

$S=1+1.0033+1.0033^2+...+1.0033^{96}$

then

$1.0033S=1.0033+1.0033^2+...+1.0033^{96}+1.0033^{97}$

Subtracting the equations

$S-1.0033S=1-1.0033^{97}\Rightarrow S(1-1.0033)=1-1.0033^{97}$

and we have

$S=\frac{1-1.0033^{97}}{1-1.0033}=114.10298$

So, after 18 years the amount in the fund will be

114.10298D

If we want this amount to be $80,000 then 114.10298D=80,000

$D=\frac{80,000}{114.10298}\approx \ 701.12$

So, the money we would have to deposit each month in a fund with an APR of 4% to accumulate $80,000 in 18 years, is

$\boxed{D=\ 701.12}$