Question 731574
Interest compounded monthly:
The interest for 1 month would be {{{1/12}}} of 8%, meaning {{{(1/12)(8/100) =2/300}}} of the initial balance that month.
So at the end of the first month you would have a balance (in $) of
{{{500+500(2/300)=500+500*2/300=500(1+2/300)}}}
So the new balance would be the initial balance times {{{(1+2/300)}}} .
 
During the second month, you would be earning interest on the whole of that ${{{500(1+2/300)}}} and that balance would be multiplied times {{{(1+2/300)}}} to get the new balance of
{{{500(1+2/300)(1+2/300)=500*(1+2/300)^2}}}
 
After {{{n}}} months you would have a total of
${{{500*(1+2/300)^n}}}
 
a. At the end of 5 years you would have accumulated interest during {{{5*(12months)=60months}}} and would have a total of
${{{500*(1+2/300)^60}}}= ${{{774.92}}} (rounded)
 
b. To get to {{{500*(1+2/300)^n=1000}}} you will need {{{n}}} months, and we can find {{{n}}} like this:
From
{{{500*(1+2/300)^n=1000}}}
taking logarithms on both sides, we get
{{{log((500*(1+2/300)^n))=log((1000))}}} --> {{{log((500))+n*log((1+2/300))=3}}} --> {{{n=(3-log((500)))/log((1+2/300))}}}
That calculates as about {{{104.3}}} (rounded)
If you need $1000, you will have to wait for 105 months because
after 104 months you will have
${{{500*(1+2/300)^104}}}= ${{{997.89}}} (rounded)
but after 105 months you will have
${{{500*(1+2/300)^105}}}= ${{{1004.54}}} (rounded)
 
c. Interest compounded monthly:
If the interest was calculated and added to the balance after shorter and shorter periods, you would gain a little bit more as the periods were shortened, getting as close as you want, but never going over a certain limit.
That limit is what they call continuous compounding.
The balance with continuous compounding is given by the function
{{{balance}}}= ${{{500*e^(0.08*y)}}} with {{{y}}}= number of years.
In that expression,
${{{500}}} is the initial deposit,
{{{0.08}}} is the 8% interest rate expresed as a decimal,
and {{{e}}} is an irrational number, like {{{pi}}}.
For {{{y=5}}} we get the balance after 5 years as
${{{500*e^(0.08*5)}}}= ${{{500*e^0.4}}}= ${{{745.91}}}