>> In this video we do an application problem
that uses exponents and or logs to solve.
And so we do this problem on this video.
Use the compound interest formula
to find how many years it takes 1300 dollars invested
at 9% compounded monthly to increase to 2000 dollars.
So what this means is if you put 1300 dollars in a bank,
and the interest rate is 9% and it's compounded monthly
which means 12 times a year, when is the value
in that account at 2000 dollars?
So what we want to do is identify what we do know
and use this formula here.
So p stands for the original investment, a is how much it is
after a certain number of years.
So let's fill that in.
We know that p is our original investment of 1300 dollars
and a, we want to find out when it's grown to 2000 dollars.
All right, what about the interest rate, r?
Well that's 9%, we want to write that as a decimal or a fraction.
So it's .09 and we don't know the time, right.
But n is how many times per year the interest is compounded.
Since it says monthly
and there's 12 months in a year, n is 12.
So we want to fill all these numbers in to this formula
and then we will compute and find out what t is.
So let's do that.
So we have a, what we're going to plug in 2000.
p is 1300 and we have 1 plus .09 divided by 12,
we'd have r over n to the nt.
So n is 12, so it's 12t.
So that's the original part of the problem.
Now from here there's different ways you might,
you know, simplify this.
And I'm going to start by simplifying inside parenthesis
and at the same time dividing both sides by 300, I mean 1300.
So if I divide both sides by 1300, okay I'm just going
to leave that as 20/13.
Okay, on the left side 20/13 reduces pretty easily.
20/13 equals, and now I have to simplify inside this parenthesis
because I have 1 in front of here
so I could [inaudible] 1 as 12/12.
Is everybody with me there?
12/12. So I have a common denominator.
So that means in a fraction I would have 12.09
over 12 to the 12t.
So this is what I have so far.
Okay, at this point if you can divide 12.09 divided by 12
and have it not being a repeating decimal,
then you can do that.
If you get something that 12 doesn't go into evenly,
then what you would do is just leave it as is.
But in this case if you do 12, use your calculator,
12.09 divided by 12, you get 1.0075, it does come
out to an exact number here to the 12t.
But notice 20/13.
If I try to divide that out, it doesn't come out exact.
So if something doesn't come out exact,
leave it in fractional form.
All right so here's the problem.
We're now going to try to solve for t and t is in the exponent
and the key is any time you have a variable in the exponent
that you're trying to solve for, you're going to use logs.
You're going to take the log of both sides
or the natural log of both sides.
So we're going to go on to the next page to continue this.
So this is what I have and it's up to you whether you want
to use your log button or your natural log button.
I'm just going to use the log button so I'm going
to write the log of the left hand side is equal to the log
of the right hand side.
Now if you want, you could put brackets around here
but it's certainly not required.
All right, so now if I have this 12t, what I could do is
since it's in the exponent, I could put it
out in front of the log.
So really it's kind of confusing seeing these brackets
but I showed it just that I'm saying
that I'm doing it to both sides.
So what do we have now?
I've got this log of 20/13, it's okay,
don't worry about it, just keep writing it.
And we're going to bring 12t
out to the front times the log of 1.0075.
Now the trick here is you're trying to solve for t.
So remember the log of 1.0075,
that's just some number multiplied by t
and then I also have 12 by t. So what I've got here is the log
of 20/13 and I'm going to write this so you see this
as a coefficient times 12 and I have log,
12 log of 1.0075 times t. So what I want you to note is
that this part here is simply the coefficient of t.
So we're just going to divide both sides by that number.
So I'm going to divide the right side and the left side.
So I don't do any rounding until this very last step.
And keep in mind this is the log of 20/13 divided
by 12 times the log of this.
You can't combine this 20/13 with this 1.0075,
these are two separate computations you're going
to have to make to solve for t. Now if you put
that in the calculator with parenthesis, etcetera,
so in the calculator I would recommend you put like a bracket
around the log here and then parenthesis around the 20/13
and another bracket, then a division sign
and then another bracket, 12 times the log of this.
You want to make sure
that you're very careful how you enter this in your calculator.
So go ahead and try entering this
in your calculator and see what you get.
It might take you a little while to answer all that
and this is what I got in my calculator.
So I'm going to put approximately
because now it's not going to be exact, it does go on and on.
I got 4.8044, you know, and it kind of goes on and on
but it did say round to the nearest tenth of the year.
So I'm going to say that t is approximately 4.8 years
and in fact I'm going to, since it's a word problem it takes
about 4.8 years, which was the question.
It says how long did it take, right?
How many years?
Okay, wow, not an easy problem.
You know at this point I would suggest actually checking to see
if this seems correct.
So let's go back to the original problem.
So here's the original problem and we already identified p, r,
a, n, t but what I want to do is not see
if actually you invested this 1300 dollars at 9% at you know,
compounded monthly for 4.8 years,
is it really 2000 dollars?
So that's what I mean by checking.
So I'm going to try that.
So I'm going to say well what is [inaudible].
And the question is, is it going to come out to 2000 dollars,
that what we would hope.
So what would p be without the 1300?
I'm going to have 1 plus my r is .09 divided by 12 again
to the mt. So it's 12 times, now I'm going to actually plug
in that number 4.8, okay.
So we're just going to simply compute what happens
if you put 1300 dollars in for 4.8 years
at this interest rate compounded monthly.
So again, we want to simplify inside this parenthesis
which was, I think 1.0075 and 12 times 4.8 is 57.6
so that's how many times it actually gets compounded.
And now we're going to use our calculator to do the 1.0075
to the 57.6 and then multiply it by 1300
or if you're careful you could go 1300 times,
make sure there's parenthesis to do this.
So go ahead now and just see what happens
when you simplify that.
Be careful of your order of operations.
I usually go ahead of my calculator just to make sure.
I do this first and so I put my calculator at 1.0075 raised
at 57.6 that you do that according to your calculator
and then I press enter and if you do that,
you should have something about 1.537854 etcetera.
And then I just go ahead and multiply by the 1300 next.
And that gives me 1999.21, okay.
Well that's not exactly 2000 dollars right?
We want to see if it's about 2000 dollars
and of course it's not going to be because 4.8 was rounded.
But it checks because that's pretty darn close, right.
So we must have, when we rounded,
we rounded under a little bit so that's why when we plugged it
in we got a little bit smaller number.
So now we verify that that does make sense, about 4.8 years.
So it takes about 4.8 years was the answer to this problem.