# How to calculate compound interest find rate compunded k n times a year algebra 2 al2 U4L7

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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
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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 applications to give us a thumbs-up your
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