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good day students welcome to
mathgotserved.com in this clip we're
going to be going over how to find the
rate in a compound interest calculation
now let's consider the following problem
Maria invests 5112 dollars in a savings
account with a fixed annual interest
rate that is compounded quarterly after
six years the balance in our account is

00:30

six thousand four hundred ninety dollars
and eighty eight cents question what is
an interest rate on the account of the
account okay all right
so let's start by identifying what we
know and once in this problem so what do
we know in this problem what are the
given values we know that the starting

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amount also known as the principal is
five thousand twelve dollars so let's
write that down the starting amount
which is the principal P is five one one
two what else do we know we know the
balance or the final amount okay the
final amount of six thousand forty six
thousand four hundred ninety dollars and
eighty-eight cents so the final amount
or balance we're going to call this a

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that is six thousand four ninety and
eighty-eight cents what's the investment
period the time period t so the time
that she invested her money and which is
T is six years
now this scenario is a compound interest
that is compounded n times a year so how
many times is it compounded a year its

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quarterly in K so we think about
quarterly things about think about
quarter quarter means four four times
okay
so number of times compounded number of
times compounded per year is the value n
and in this situation is equal to four

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okay so n is four now these are the
information that we know in this problem
now let's shift gears and take a look at
what we want what does the problem asked
us to find well we want to know what the
interest rate is so the interest rate is
given by the variable R and we do not
know what it is okay

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alright so we've identified what we know
once now let's give ourselves a model of
the situation we're gonna use a
graphical representation of what's going
on here so this is an exponential growth
scenario okay so I'm gonna sketch a
graph is definitely not drawn to scale
we have a starting dollar figure of five
thousand one hundred twelve dollars so

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that's how much she starts with let's
put it somewhere here so five one one
two okay the y-axis is basically we're
looking at amount in dollars and this is
time
in years okay now uh she wants to grow
her money from five thousand twelve
dollars to at the end of the period it's
six thousand four hundred ninety dollars
in eighty eight cents so starting with

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this amount she wants to grow it up to
this amount right here the function is
exponential in nature so it has an
upward-opening concavity something like
this the time period that it's going to
take for her to accomplish this growth
is six years okay so that's just a
visual representation of what we're
looking for
of what we have and what we're looking

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for is basically the interest rate okay
all right now let's go ahead and take a
look at the equations that will help us
solve this problem so we need to know
the formula for computing the compound
interest of an investment scenario okay
so the formula is as follows the formula

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for compound interest is the balance or
a final amount equals to the principal
times 1 plus the interest rate over N
raised to the n T okay so this is an
equation that we're going to be using to
solve this problem now next we're going
to carry out a substitution okay we're
going to substitute our nose and one
into the equation so a is six four nine

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zero point eight eight equals the
principal which is five one one two
times one plus the interest rate we do
not know what that is so we look at as R
over N because its quarterly raised to
the four
times four multiplied by T which is six
okay alright so now that we substituted
our knows I want into our equation we're

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not gonna proceed to solve okay we're
gonna solve the equation for R okay so
let's solve this let's start by
multiplying the exponent we have a
product there so we have 6,000 4 9 0 8 8
equals 5 1 1 2 times 1 plus the interest
rate over 4 raised to the 4 times 6
which is 24 now next we need to move get

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rid of 512 from the little right side of
the equation so we'll just simply divide
both sides by 5 1 1 2 so divided by 5 1
1 2 divided by 5 1 1 2 these divide out
so let's go ahead and bring out our
calculators and computes the quotient on
the left side so we have six thousand

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four hundred ninety dollars 88 cents
divided by five one one two okay so we
will just keep it to four decimal places
for our intermediate calculations and
then we'll round up at the end okay so
let's say we have one thousand I mean
I'm sorry one point
two six nine seven one point two six

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nine seven equals one plus the interest
rate over four raised to the 24th power
now the next thing we want to do is get
rid of that 24 up there that power that
this expression is raised to in order to
get rid of this 24 we need to power it
down we need to bring it down okay
we can't just bring down exponents the
function that enables us to power down

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exponents is known as the logarithmic
function so what we'll do is we'll
proceed to take the natural logarithm of
both sides you could take the natural
log or you can take the common log log
base 10 um the reason why we're taking
the natural log is that one of the
properties enables you to power down
exponents okay so our to the our over
four to the 24 now after taking the

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natural logarithm of both sides we can
now power down this 24 okay that's one
of the properties of exponents power
that down now let's compute the natural
logarithm of one point two six nine
seven okay so take the natural logarithm
we can just grab our previous answers so
let's use that second function answer

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enter we have two point three eight
eight okay so on the left side we have
two wrong color we have two points three
eight eight zero points two three eight
eight
equals now will power down to 24 and we
have 24 times the natural logarithm of 1
plus R over 4 ok now what's do we do

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next
remember the goal is to get our isolated
with brought down 24 why did we bring it
down so we could take it out of there
okay so to get rid of 24 we're gonna go
ahead and divide both sides of the
equation by 24 okay the ultimate goal is
to have our isolated on the right side
so divide both sides by 24 we're gonna

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bring out our calculators again we're
going to compute 24 divided by as
divided by the previous answer okay 20
we're gonna have our previous answer
divided by 24 okay and that gives us
point 0 0 9 9 5 if we round it up to 4
decimal places is going to be point zero

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one okay so we have zero point zero one
equals the natural logarithm of one plus
R over four now how do we get rid of
natural logarithm we eliminate that
using the natural exponents okay which
is the exponent with base e so what
we're going to do is put a big key on
both sides of the equation we're going
to exponentiate both sides with E as a

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base of our exponents because we have
the natural log so we have to use a
natural base on the left we have a
little 0.01 and on the right side we
have a little natural logarithm of 1
plus R over 4
now let's evaluate this term on the left
side so we're just going to compute
second function Ln is e to the point 0 1

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close that enter we'll get one point
zero one zero one okay if we take it to
four decimal places so 0 1 0 1 so this
becomes one point zero one zero one to
four decimal places on the right side a
and LNR inverses so they cancel each
other out and we're left with 1 plus R

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over 4 now we're going to proceed to
subtract 1 from both sides of the
equation if we carry out that
subtraction this is easy we don't need
to use a calculator to subtract 1 so we
are going to end up with zero point zero
one zero one equals r over four and then
the final step involves just simply
multiplying both sides of the equation

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by four and then let's multiply that out
we're gonna have this multiplied by four
zero point zero four zero four equals R
okay so this is the interest rate in
decimal form if the interest rate in
decimal form is 0.04 zero four what is
the interest rate in percent all we'll

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just do is move the decimal point two
places to the right so it's going to be
the interest rate is going to be four
point zero four if we want to round it
to the nearest percent the interest
rates will be approximately four percent
okay now let's interpret what this
answer means within the context of the
problem so what we're going to do to

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state what our answer means is we're
going to take a look at the original
problem and just restate it in answer
form okay so if you take a look at the
problem it's asks for what the inch what
is the interest rate of the account okay
so that's going to help us formulate the
interpretation or the explanation of our
answer so we're just going to write the
different ways to write it the interest

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rate of the account that will cause that
will cause Maria's Maria to grow her
money to grow her money from the
starting amount of five twelve to six
thousand four hundred ninety dollars and

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eighty-eight cents in six years is what
the interest rate that will enable this
to happen is four percent of course the
its compounded quarterly okay so this is
how you find the interest rate in
compound interest calculations thanks so
much for taking the time to watch this
presentation is this tutorial help you
in your studies of logarithms and their
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16:22

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