[ Pause ]
>>In this video, we use the
compound interest formula.
And the compound
interest formula,
where interest is
compounded continuously
to do the following 2 problems.
We're going to do a
practical problem now.
A equals p parentheses 1
plus r over n to the nt.
That's called the
compound interest formula.
It's the formula for finding
the amount of money, a,
in a bank account if
you deposit p dollars
at an annual interest rate of r,
compounded n times
a year for t years.
Once we put in some numbers,
that will make more sense.
If interest is compounded
continuously, in other words,
not twice a year, 3 times
a year, 4 times a year,
6 times a year, 100
times a year.
Continuously, so happens
that the formula is a
equals p, e to the rt.
It uses this letter e, which
we also use with natural logs.
Let's look at an example.
Here are the 2 formulas.
Let's do this problem.
Finding the amount of
money in an account,
if $3,000 is deposited
at 8 percent interest
rate for 5 years.
If the interest is
compounded quarterly
and then let's also
see what it would be
if it was compounded
continuously.
I want you to put
the video on pause.
First you're going to
figure out quarterly.
You're going to look at the
compound interest formula
up here and you're
going to have to figure
out what p, r, n and t are.
Plug it in, use the order
of operations carefully
with your calculator and
come up with that amount.
And then I want you to figure
out how much money would be
in it if the interest was
compounded continuously using
this other formula, a
equals p e to the rt.
So put the video on pause and
try both of those on your own.
Let's do quarterly first.
What do we know?
P in both cases is 3,000.
And the rate is 8 percent.
Now, rate we can
write as a decimal.
So 8 percent is 0.8.
0.08, I think I said
0.8 when I meant 0.08.
And let's see.
The time is 5 years
for both of them.
Times 5 years for both of them.
That's all we need to do part
b, but to do part a, quarterly,
that means how many
times a year?
Quarterly is 4 times a year.
So that's n, n is 4.
So if we were going to do the
first 1, we're going to plug
in a equals 1 plus
r over n to the nt.
I'm sorry, p. Forgot
the p in the beginning.
Now we plug that in.
3,000. 1 plus 0.08 over 4
and then you're going to have
to the nt, so that's 4 times 5.
So that's going to
be to the 20th.
And then you need to simplify
inside these parentheses.
0.08 over 4 is 0.02.
So that's 1.102 to the 20th.
So when you put this
in your calculator,
you need to first do the
1.02 to the 20th power.
So you have to do exponents
first and then you're going
to multiply it by 3,000.
That should give you $4,457.84,
if you did that correctly.
So you put the $3,000
in for 5 years,
that's how much money
you now have,
if interest is compounded
quarterly.
Now let's do it continuously.
The formula is a
equals p, e to the rt.
So, again, p is still 3,000.
We have e to the rt.
So we have 0.08 times 5.
So that's 3,000 times
e to the 0.4.
Because 0.08 times 5 is 0.4.
So to do this with your
calculator you have to do e
to the 0.4, so you can put
0.4 and then use that e
to the x button, exponents
first then multiply by 3,000.
And this, and I'm doing both
of these to the nearest cent,
will give you 4475 and 47 cents.
Not a huge difference on money,
but there's a difference.
So if it's only compounded
4 times a year, you have $4,
457- actually almost $8.
And if it's compounded
continuously,
it's $4,475 dollars.
So, not a huge amount
over 5 years,
but that is the difference.
So that's how we
would do that problem.
So I'm going to give
1 for you to try.
Here are the 2 formulas again
and here's a problem
for you to try.
Find the amount of money in an
account if $6,000 is deposited
at 6 percent interest
rate for 7 years,
if interest is compounded.
First let's do it
for twice a year,
and then we'll do
it for continuously.
So put the video on pause
and try each of those.
[ Pause ]
>>For both of them,
we've got p is 6,000.
The rate is 6 percent, 0.06.
The number of years, t, is 7.
And for part a, we also have
to know how many times a year.
So n is going to be 2.
So for the first 1.
We would have a equals 6,000.
1 plus 0.06 over 2 to the nt.
So that's 2 times 7.
So we have 6,000.
Now, 0.06 over 2 is 0.03,
so that's 1.03 to 14th
and then we're going to
put that in our calculator
and round it to the
nearest penny.
First you have to do the
1.03 to the 14th power,
and then multiply it by 6,000.
Make sure you never round
until the end of the problem.
So I come up with
$9075.52 when it's rounded.
Now let's do it continuously.
We're going to have
a equals 6,000.
E to the rt.
So 0.06 times 7.
So we have 6,000.
E to the- I think that's 4.2.
I'm sorry, 0.42, right?
0.42. And remember, we're
going to have to do e
to the 0.42 first and
then multiply it by 6,000.
And I've got 9131.77
when you round it.
We're talking dollars and cents.
So you can see the difference,
just having it compounded
twice a year
as opposed to continuously.
So here's where you're
using the formulas
and what you're really
learning how
to do is use your
calculator very carefully.
[ Pause ]
