>> This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos were organized
by topic.
We're going
to do the following three
problems on this video.
They all involved
in investment
of 20,000 dollars
and a 12 percent interest
account and we're trying
to find out how much many
there is in the account
after 20 years,
but the conditions
are different.
The first one is just if it's
in the simple interest
account, the second problem is
if it's compounded yearly
so we'll be using the compound
interest formula,
and the third one will be
if it's compounded daily also
using the compound
interest formula.
So, here's the first problem.
You're investing 20,000
at 12 percent for 20 years.
So, you've got this formula,
I equals PRT
so the principal is 20,000,
the rate is 12 percent
which you could write
as 12/100 or 0.12
and the time is 20 years.
So, we simply put
that in the formula,
I equals 20,000 times
and I'm going to do it
as 12/100 so if I can do this
in my head just
as well times 20.
So, I can cancel that hundred
with two of those zeros
and what does that give me
in interest?
Well, I have 200 dollars,
right, times 12 times 20,
so I'm just going
to do 2 times 2 is 4 times 12
that's 48,
and then how many zeros do I
have here, 1, 2, 3 zeros,
so that's what you make an
interest, pretty amazing
isn't it?
In other words,
way more than you're
original investment.
Now, how much money is
in the account?
It says, how much money is
in the account,
that's a different question.
So, I have my 20,000 dollars
then I get to add on to that.
That's was already
in the account to begin with
and so my answer is
68,000 dollars.
So if you--
now if you got an extra 20,000
dollars around which I don't,
but if you do
and you just let it sit
in an account earning simple
interest, that's how much
money would be in the account
after 20 years, of course,
20 years is a long time
to wait.
So, we're going
to remember this.
We're going
to compare all these later.
Let's go on to the
next problem.
So, if 20,000 is invested
in an account earning 12
percent interest compounded
yearly, how much money is
in the account after 20 years?
We need this formula.
This is the compound interest
formula and remember,
these are what these variables
stand for in the compound
interest formula.
All right,
so my principal here is 20,000
and my rate is still 12
percent or 0.12 or 12/100.
My variable N,
okay that's how often--
how many times per year
it's compounded.
Well, it's only yearly,
so N is just one in this case,
once a year, and the time is
in years, it's 20 years.
So, T equals 20.
So, if we put
that in this formula,
we've got 20,000 times 1 plus,
now what's the rate,
0.12 over 1 times N times T,
1 times 20.
That's the exponent,
so I've got 20,000 times,
all right,
what's this going to be?
Well, 0.12 by 1 is just 0.12
and 1 plus that is 1.12
to the 20th.
So, of course
if you have a lot of time,
you could take 1.12
and multiply it,
finds itself 20 times
to get the answer,
that I'm going to suggest,
you put that on your
calculator
and there's different
calculators
on how you're going
to enter this, so go ahead
and try it.
I showed how I did it
in my calculator and I'm going
to just show the series
of keystrokes I use.
So, these are the keystrokes I
use on my calculator now
that's because I'm doing the
order of operation myself.
I'm just thinking well,
in order of operations,
I first have to do 1.12
and raise it to the power,
that's what this is,
1.12 and I use the Y
to the X key
or you might have a little
caret, this is called a caret,
and then the number
for the exponent would be 20
then I put equals
and there will be a number
that comes up
and then I'll use
the multiplication.
I'm going to take that answer
and multiply it
by 20,000 dollars
and then I'm going
to put equal sign
and when I do that,
I get this number rounded
to the nearest cent,
192,925.86 cents,
okay that is a huge difference
than when we did the simple
interest formula,
I'm going to remind you,
this is how much money is
in the account after 20 years.
So, you put in 20,000,
you let it sit there
for 20 years,
you've got a 192,925 dollars
in your account,
simple interest you'd only
had 68,000.
It's really amazing.
So, compounding interest is
great if you're putting money
an account and you want
to earn a lot of interest.
But if you're borrowing many,
you're hoping somebody is
going to give it to you
as simple interest
because you would pay a
lot less.
Now, let's see what would
happen if we actually
compounded it daily instead
of yearly.
How much more
of a difference could that be
and most banks do
compound daily.
All right,
so here's the next part.
What about if you invested it
and you compounded it daily?
So, we're going
to have the same
variables here.
At the beginning,
you've got the principal is
still 20,000, right?
And the rate is still 12%,
but N is different.
How many times per year is
that if it's done daily.
And I know
that we don't have the same
number of days per year,
but usually its 365
and that's what for years.
So N is 365 and then
for how many years, 20.
So, the only thing different
from the previous problem is
that N is 365 as opposed to 1.
So, we're going
to plug those numbers in,
A equals 20,000 times 1 plus
0.12, right?
That's the rate over,
right now, what was the N
in this case,
365 and then you're going
to raise that to the N times
T, so, 365 times 20.
[ Pause ]
Now again,
the trick is entering this.
You could, you know,
enter it just like it--
you see it right here
if you've got a calculator
that allow--
I would say
if you had a graphing
calculator
and for sure can make--
make sure you enter
everything correctly.
I tend to like
to simplify it just a little
bit and there is an easy way
to always simplify this,
that's inside the parenthesis,
because whatever this
denominator is, 365,
you just think of--
that would be the whole
number, the number 1
over here, I could rewrite
that always as this number
over itself,
365 over 365 right,
which means this ends
up being the whole number part
in front of the 12
and that ends
up being the denominator.
So, imagine if you change
that to 365
over 365 then you'd a common
denominator
and you'd have 365 plus 0.12.
So, I do that
and then I also do the 365
times 20, but
it's unnecessary.
You could leave it like this
and use parenthesis and,
you know, work it
in your calculator however
you like.
So, I'm going
to do just a little bit
of simplification before I put
it in the calculator,
and so like I said,
this will be 365.12
over 365 so, it's a--
and so I'm going to do--
I'm not going to actually do
that division.
I'm going to enter it just
like that in my calculator
and then 365 times 20,
let's see what is that,
that's 7,300, all right.
So, again,
you could just enter it
like it is at this point or at
that point
and my keystrokes again,
I'm going to start with what's
in the parenthesis here
so I'm actually going
to use a parenthesis.
So, I would do 365.12 divided
by 365 first
and then I would raised it
so you'd use the Y
to the X button
or the little caret
to the 7,300
and then you would equal
and then you would times
up by 20,000.
So, again here are
my keystrokes.
So, these are my keystrokes
down here.
I do what's in parenthesis,
365.12 divided by 365,
I write what
that equals then I'm going
to raise that to the 7,300.
So, on my calculator,
that's the Y to the X button,
7,300 then I'm going
to say what that equal so far
and I'm going to multiply
that answer by 20,000
and write equals.
And when you do that,
this is the number you
should get.
You see, I got 220,
376 dollars and 58 cents
so that's how much many she
gets an account
and I didn't double
check this.
I probably will do that
but I always suggest you enter
your numbers more than once.
Okay, so now let's compare
this 3.
All right,
here are our results
if we invest 20,000 dollars
at 12% for 20 years,
you can see the amount
of the account at the end
of the 20 years
if you simply use
simple interest.
You'd only have 68,000 dollars
in the account.
If you compounded interest
yearly, you would have a
192,925 huge difference
and if you compounded daily,
you have more but it's not
as huge of a difference
220,376 dollars still I'll
take 28,000 dollars more
approximately,
that would be fine
and I prefer the
compounded daily.
The reason that com--
when you compound interest
by so much more money is your
getting interest
on your interest
and it just grows
very quickly.
Now of course
if this was 6 percent instead
of 12 percent it wouldn't be
as much of a difference.
Of course the higher
percentage is going
to make the bigger difference,
but it's compounding it or not
that makes a very
big difference.
So, I think that's a pretty
interesting to think about,
especially if you're borrowing
on your credit cards
and it's being compounded
like crazy
and you have a high
percentage rate.
So, think about that.
This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos were organized
by topic.