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 first 8% simple interest per year
and then 8% interest compounded annually.
To determine the ending balance using simple interest,
we will use the formula A equals P sub zero
times the quantity one plus rt
where A is the balance after t years,
P sub zero is the beginning balance,
r is the annual simple interest rate as a decimal,
and t is time in years.
Which means for the simple interest formula,
P sub zero is $21,000,
r is 8%, which must be expressed as a decimal.
8% is equal to 0.08,
and t, the time in years, is 20,
which means the amount after 20 years, A,
is equal to P sub zero, which is $21,000,
times the quantity one plus r times t,
which is 0.08 times 20.
And now let's evaluate this on the calculator.
We have 21,000
times the quantity one
plus 0.08 times 20.
And Enter.
The balance will be exactly $54,600.
Remember, for simple interest,
the interest payments are based
upon initial investment of $21,000.
It is not based upon the increasing balance
of the account each year.
Now let's determine the balance
using 8% interest compounded annually.
To determine the ending balance,
we will use the compounded interest formula shown here
where A is the amount after t years,
P sub zero is the beginning balance or initial investment,
r is the annual interest rate expressed as a decimal,
n is the number of compounds per year,
and t is the time in years.
So for the compounded interest formula,
P sub zero is still $21,000.
The annual interest rate is still 8%,
which as a decimal is 0.08.
T, the time in years, is still 20.
But because we have compounded interest annually,
n, the number of compounds per year, is one,
which means the amount A after 20 years
using 8% interest compounded annually
is equal to 21,000
times the quantity one plus r divided by n
is going to be 0.08 divided by one.
Because we're dividing by one, though,
we could leave that off.
Raised to the power of n times t
where n is one and t is 20,
giving us an exponent of one times 20.
And now let's go back to the calculator.
We have 21,000
times the quantity one plus,
of course 0.08 divided by one is just 0.08
and one times 20 is 20,
so we raise this to the 20th power and press Enter.
We're told to round to the nearest cent.
Because we have a zero in the third decimal place,
the value is approximately $97,880.10.
Now, you might be surprised
by how much larger the balance is with compounded interest,
but when we have interest compounded annually,
the interest is paid on the increasing balance,
not just the beginning balance.
Our interest is paid on the interest earned
year after year for 20 years.
I hope you found this helpful.