If $21,000 is invested in an account for 20 years,
find the value of the investment at the end of 20 years
if the interest is a, 8% simple interest per year,
b, 8% interest compounded annually,
c, 8% interest compounded quarterly,
and d, 8% interest compounded monthly.
For part a, because we have simple interest,
we use the simple interest formula shown here on the right,
where A, the amount after 20 years,
is equal to P sub zero, the initial investment of $21,000,
times the quantity one plus r times t,
where r is the annual interest rate as a decimal
and t is the time in years.
8% as a decimal is 0.08.
t, the time in years, is 20.
And now we go to the calculator to determine
the account balance after 20 years.
We have 21,000
times the quantity one plus 0.08 times 20.
Enter.
After 20 years, if the account pays 8% simple interest,
the balance is $54,600.
For parts b, c, and d, we have compounded interest,
and therefore we use the compounded interest formula
shown here on the right.
Notice for part b, the interest is compounded annually,
and since n is the number of compounds per year,
for part b, n is one.
For part c, we have 8% interest compounded quarterly.
There are four quarters per year,
and therefore n is four.
And for part d, we have 8% interest compounded monthly.
There are 12 months in a year, and therefore n is 12.
So for part b,
where we have 8% interest compounded annually,
A, the amount after 20 years, is equal to P sub zero,
which is $21,000, times the quantity one plus
r divided by n, where r is the annual interest rate
as a decimal and n is the number of compounds per year,
which in our case will give us 0.08,
which is 8% as a decimal, divided by n, which is one,
and this is raised to the power of n times t,
where we know n is one and t is the time in years,
giving us an exponent of one times 20.
Let's set up c and d before we go back to the calculator.
The only difference for part c is that n,
the number of compounds per year, is four rather than one,
which means A, the amount after 20 years,
is equal to $21,000 times the quantity one plus
now r divided by n is going to be 0.08 divided by four,
and the exponent of n times t is now four times 20.
And then for part d, where the interest
is compounded monthly, n is 12,
and therefore the amount A after 20 years is equal to
$21,000 times the quantity one plus 0.08 divided by 12,
and then raised to the power of 12 times 20.
And now let's go to the calculator and determine
these three amounts, rounded to the nearest cent.
So for 8% interest compounded annually,
after 20 years the amount is equal to 21,000
times the quantity one plus 0.08.
We can leave off the divided by one,
since that will still give us 0.08.
And this is raised to the power of one times 20,
which is 20.
So we press the exponent key, 20, and then Enter.
To the nearest cent, after 20 years,
the balance is $97,880.10.
And now when the interest is compounded quarterly,
the account balance after 20 years is going to be 21,000
times the quantity one plus 0.08 divided by four,
and this is raised to the power of four times 20,
which is 80.
So we press the exponent key.
We can enter 80 or we could enter four times 20.
Enter.
If the interest is compounded quarterly, after 20 years,
the balance is $102,384.22.
And then finally, if the interest is compounded monthly,
after 20 years, the balance will be $21,000
times the quantity one plus 0.08 divided by 12,
and this is raised to the power of 12 times 20,
which is 240.
Or again we can enter 12 times 20.
So we press the exponent key and then 12 times 20.
Enter.
After 20 years of interest compounded monthly,
the balance is $103,462.86.
Notice the balance of the account increases
as the number of compounds per year increases.
I hope you found this helpful.