Welcome, ladies and gentlemen.
So, what I'd like to do is show you
how to determine the time or rate needed
to satisfy our given requirements,
as far as having a given initial investment, as well
as a final investment, or final value.
So we're given a present value.
We're given a final value.
Now we just don't know how long do
we need to compound that interest, or at what rate.
So the first two problems we're going to be finding the time.
And for the second two problems, we're
going to be finding the interest rate.
Now, I'm doing this video in conjunction
with my other two compound interest videos.
But if you have not already gone through the videos
for solving exponential and logarithmic equations,
I would recommend that you stop where you're at now.
Go ahead and go through solving exponential logarithm
equations, and then come back to that.
Because I'm going to use the solving techniques
for exponential logarithmic in this video.
And if you haven't come across that,
or you have no idea what that is,
it's probably going to get a little confusing
on how I'm doing this.
And the reason why I have to use solving
exponential logarithmic equation techniques is because now,
or especially for finding t, you can
see that for continuous interest, our r and our t
are in exponents.
So we've got to figure out how to solve for r and t
when they're in, as in the power.
And we can't use our normal solving techniques.
We're going to have to use solving
logarithmic and exponential equation techniques to help us
with that.
All right.
But anyways, the first step that we're
going to be doing for any of these problems,
even if you have no idea what you're doing,
remember if you're given a word problem instead of like this,
just identify the information like I did.
And now take the formulas that you are provided.
And then plug them in.
And I'll show you what to do next.
So this is a very common type of problem that they say.
Is they say we're given a present value of $3,000.
We want your final value to be doubled the initial amount.
And we are going to invest your money at 3.6% weekly.
How long is it going to take for your money
to double, basically?
So if we know our present value is 3,000, well, for $3,000
to double that just means that's going to be 6,000, right?
6,000.
So what I'm going to do is take 6,000
equals my present value of 3,000 times 1
plus 0.036 divided by weekly, so there's 52 weeks
in a year, raised to 52.
And then t is what we do not know.
So we need to solve for t.
OK.
So, the first thing that we're going to want to do
is I can go ahead and divide out this 3,000.
Let me grab another color here.
So I can divide by 3,000 on both sides.
Because that's just multiplying by that exponent.
And when I do, 6,000 divided by 3,000 I get the number 2.
Actually let's just do it there.
So, that's 2 is equal to-- now, I'm
probably going to want to simplify this.
And again, it gets very, very important when doing this,
is trying to use your calculator to make sure
that we are not approximate in our answers in any form.
So, I'm going to do 1 plus 0.036.
So I'm going to use a my order of operations first.
I'm going to 0.036 divided by 52.
So 0.036 divided by 52.
And then I'm going to add 1 to that.
And that gives me 1.00-- let's write it in there.
Again, this is approximation by the calculator.
So times, let's call that, 52t.
OK.
Now we need to get rid of this exponent.
Right?
So what I basically need to do, to solve for t,
I need to get rid of this base.
So what I'm going to have to do is
I'm going to take the log of both sides.
Now, what's interesting here is when
you take the log of 2 that's going
to give you some decimal answer, I mean and so forth.
But what is going to be our base?
Well, our base has to be that answer.
Right?
Well what's nice about this is the log of that answer--
or let's just even write in, the log
base 1.0 dot, dot, dot, dot, dot, all
that, for instance, log base 3 of 3 is just equal to 1.
Right?
And if that's raised to a power x, then that's equal to x.
So therefore if I take the log base of this,
it's just going to leave me this answer, which is 52t.
But the log base of 1.0 dot, dot, dot, dot, dot of 2,
is going to be some number.
So therefore what I have to use, is
use the change of base formula, since my calculator does not
recognize that as a base.
So it's going to look something crazy,
but it's going to look like this.
It's going to be log base 1.000692308
raised to the second power or of 2, is equal to 52t.
Then let's go ahead and figure out what that number is.
I'm kind of running out of space.
So I'm going to use the change of base formula.
So to calculate that using the change of base formula,
I can basically use ln or log.
I'll just use base 10.
I'll do log of 2, divided by the log base 10 of 1.000692308.
Now remember, I haven't changed this in my calculator.
That answer is still there.
So, the reason why that's important
is I don't want to type in this whole answer.
Because then I'd be approximating the answer.
I want to use the complete answer.
So what I'm going to type in, I'm
going to use my change of base formula.
That equals 52t.
So what I'm going to do is I'm going
to do log of 2, and parentheses, divided
by log of second answer.
And I'm going to get that value.
And that's going to give me-- oh geez, I'm running out of space.
All right.
Well, let's just write it down below.
That gives me 1,001.559.128 equals 52t,
Well, now to solve for t, all I need to do is divide by 52.
And when I just take my answer now,
and I just click divide it by 52,
my final answer is t is equal to 19.26.
I'll just round to the hundredth.
So that means it'll take you almost 20 years, or more
than 19 years to double your money earning the interest
rate of 3.6% compounded weekly.
OK.
So let's try to get to the next one.
So now I want to triple my value.
So that's going to be 750 plus 750
plus 750, which would be 2,250.
Right?
2,250, perfect.
So my final value is 2,250.
Present value is 750.
Now this is compounded continuously.
So therefore, I'm going to be using this formula.
So it's going to be e raised to the r, which
is 0.0425 times t, which we do not know.
OK.
So the first thing I'm going to do is divide by 750.
And that's going to give me 3.
Well, that's money, right?
So that gives me 3 equals e to the 0.0425t.
Now to get rid of a base 1.00, we need to take the log of that
base.
Well, to get rid of e, we need to take the log of base e,
which is just ln.
So I'm going to take the ln, which is base e of both sides.
And what that's going to leave me now is ln of 3 is equal
to 0.0425t.
To solve for t, I just divide by 0.0425.
So therefore, t equals ln of 3 divided by 0.0425.
Now, it was much easier than the first problem.
Now, I can just plug that into my calculator.
So I'll just take in ln divided by-- I'm sorry-- ln of 3.
So I type ln of 3, and then close the parentheses,
and then divide it by 0.0425.
And that gives me 25.85, as I round to the nearest 100th.
So therefore that's going to take you
25 years to triple your investment,
earning an interest rate of 4.25.
OK.
So that's how you figure out t with logarithms
and natural logarithms.
Now let's go ahead and figure out what r is going to be,
or how to find r.
For a regular compound interest, it's
going to be much easier, because we're not
having to solve for our exponent anymore.
But it's still going to be a little bit kind of crazy
as far as our math.
So again, let's plug in our information.
So we have a final value of 1,000,
equals an initial value of $100 times 1
plus my interest rate I do not know,
divided by n which is-- oh wait, this is continuously.
Sorry about that.
I didn't read it.
So e to the r, which we don't know,
and then t is going to be 8.
OK.
So now this one is just going to be very similar to what we
did before in the last example.
Except instead of solving for t, we're solving for r.
So I divide by 100.
OK.
And that's going to leave me with 10 equals e to the 8r.
Then what I'll do is I'll just take the ln of both sides,
the ln of e is equal to 1, which is going to leave me that 8r.
So then I have ln of 10 equals 8r.
Then I'll just divide by 8.
Divide by 8, r equals ln of 10, divided by 8.
So now, again I take my calculator.
And I do ln of 10, close the parenthesis, divide it by 8.
And that's going to give me 0.28-- actually,
I'm going round it to the nearest 100th,
so that's going to be 0.29.
So therefore as percentage, I would
need to earn 29% for me to go from $100 to $1,000
in 8 years compounded continuously.
So, pretty high interest rate, right?
You're not even doubling your money.
I mean you're trying to earn a lot of money
over in a small amount of time.
So compound interest works best over long periods of time.
But over a short period of time, you're
going to have a really high interest rate.
All right.
Last but not least, let's find r in our regular compound
formula.
So I have $18,000, or I want to earn
$18,000 with an initial investment of $500.
I want to know what interest rate
that's going to take if I'm compounding semiannually.
Well, once a year, is 1.
Semiannually is going to be twice a year,
so that's going to be 2, times 2 times 10.
OK.
So again, the first thing you want to do
is eliminate the exponent.
So I'll divide by 500.
And that should give me 36, I believe.
Yep, 36.
OK?
And that's 1 plus r divided by 2 raised to the 20th.
OK.
Now to get rid of raising something to the 20th power,
I need to take the 20th root of both sides, which
is not going to give me a nice number I'm pretty sure.
So I'm going to take the 20th root on both sides.
Now, in my calculator I have an option where if I type in math,
there's a variable in front.
And I can write in what root I want to take.
So I'm going to take the 20th power of 36, which gives me
this crazy number, which again is
going to be approximated by my calculator of 1.196231199.
And that's going to be equal to 1 plus r divided by 2.
Now to solve for r, all I simply need to do
is follow my operations.
So I solve for r, I'm going to subtract the 1 on both sides.
That gives me 0.196231199 equals r divided by 2.
To undo dividing by 2, I'm just going to multiply by 2.
OK.
I'm going to do this in my calculator,
so I don't make a mistake.
And then I get r equals 0.39.
Now again, this is just a percent.
So I'm just going to round it to the nearest 10th.
But 0.39 written as percentage is going to be 39%.
That means if I want to invest $500
and I want to earn $18,000 at the end of 10 years,
I need to earn 39%.
So there you go, ladies and gentlemen.
That is how you determine your time and rate,
if given final value and future value,
as well as compounding in years.
Thanks.