WEBVTT
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Compound interest is an application of solving exponential equations.
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There are two formulas that must be memorized
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in order to solve these applications.
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Plus you will need your calculator in order to evaluate the final answers.
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So keep it handy. Let's look at the first example.
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It reads: Find the amount that results from 100 dollars
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invested at 10 percent compounded monthly, and then for the b part,
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we'll look at componded continuously for 2 and 1/4 years.
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Okay, so let's look at the a part.
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The problem: We have to decide which of the two formulas to use.
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We look at the way the money is being compounded.
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It's monthly, so we want to use this formula.
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Okay it reads, A equals P times in parentheses 1 plus r over n to the nt power.
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It's very important besides knowing the formula
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to know what the different variables stand for. So let's go over that.
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A is your final amount. P is the principal, which in this case,
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we know is 100 dollars, or the initial amount that you're investing.
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r of course is the rate, which in this problem is 10 percent.
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We'll look at how to put that into the formula.
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n, both of these spots, is the number of times that the money is compounded per year,
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okay, and t of course is time. So looking at the problem,
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we want to find the amount. Okay, so that is the unknown.
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We've already talked about that the principal is 100 dollars.
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Okay, r is the rate that the money is being compounded
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and must be written as a decimal.
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Percent means per hundred, so it's going to be 0.10.
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n is 12, and we'll put the 12 here again and 2 and 1/4.
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Now, the next step is just to enter all of these numbers in the proper order in our calculator.
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If you would like, it may be easier in the calculator
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to enter 2 and 1/4 as 2.25. So we're going to do that,
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and we're going to round to the nearest penny.
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Turn your calculator on. I'm going to enter 100
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parentheses 1 plus 0.1 divided by 12. Close the parentheses.
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Now at this point, depending on your calculator,
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you can use your carat button or the y to the x button
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parentheses to multiply 12 times 2.25. Close the parentheses, equals.
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Now, the answer should be more than 100 dollars.
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Okay so when look at your answer if you get something that's less than 100 dollars,
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you know that you've entered something incorrectly.
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To the nearest penny, we have 125 dollars and 12 cents.
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Now that finishes the a part, so let's go to the b part.
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Okay this time the money is being compounded continuously,
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so we use a different formula. You want to make sure that
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if the money is compounded yearly, semiannually, quarterly, monthly, daily, or whatever,
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you use this formula. If it's continuously, you're going to use this formula.
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Okay, some of the same variables: final amount equals
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the principal times e, and we know what e is, to the rt power, the rate times the time.
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Again, we're looking for the amount. P is still 100,
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r is still 0.10, and the time is 2 and 1/4 years.
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I'll write it as 2.25 this time so that you can see it.
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Okay, let's put an equal here, and then because we're rounding,
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we're going to get an approximate answer.
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In the calculator, 100, and recall how you enter e:
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Second natural log, 0.10 times 2.25.
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Now, my calculator put the parentheses around this quantity.
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If yours does not, you need to add those.
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Okay equals and because we're compounding the interest more times than monthly,
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the answer we get here should be greater than what we got in the a part of the problem.
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It is. It's 125 dollars and 23 cents.
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That should solve this type of problem when we have all of the quantities,
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P, r, n, or t, and we're looking for the final amount.
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In our next example, we want to look at what happens
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when we're given the final amount, and we have to find the time or the rate.
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In this example, we will see how many years will it take an investment
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an investment of 25,000 dollars to grow to 80,000 dollars.
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Assume an interest rate of 7 percent compounded quarterly and then continuously.
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We will look at the a part first.
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From the first example, remember the key is
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to begin the problem, you must decide which formula to use.
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What determines that is the way that the money is being compounded.
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It's quarterly, so we're going to use our first formula.
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A equals P times 1 plus r over n to the nt power.
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Notice that this time, we're looking for how many years,
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so we want to solve for t. t is in the exponent,
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so that creates a little more difficult problem.
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A, final amount, is going to be 80,000 dollars.
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P, the amount that we start with is 25,000.
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Interest rate in this problem is 7 percent. Now 7 percent as a demical is 0.07.
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A common mistake that a lot of algebra students make
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is they write this as 0.7. That would be 70 percent,
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and I don't think any bank is going to give us a 70 percent rate of interest
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compounded quarterly or continuously, so be very careful there.
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Also, the second mistake that a lot of students make is n.
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Quarterly, you do think of it as 1/4 of a year, but n has to be a whole number.
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Since this is the number of times the money is compounded per year
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every 3 months, so n is 4, and of course t is our unknown.
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Now because we're solving for the exponent,
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we must use the procedures that we used in solving exponential equations.
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The first step of which is to isolate the base to the exponent.
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I'm going to divide both sides by 25,000.
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On this side when we reduce we will get 16/5, which you can leave,
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or you can change it to the decimal 3.2.
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Now if the fraction was something like 16/3 or 17/6
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where you would not get a terminating decimal,
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you definitely would need to leave the fraction.
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Also right here, we can enter, because we're dividing by 4,
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we're going to get a terminating decimal, so if you want to divide that,
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you come up with 1.0175. It may make entering your answer
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in the calculator a little simpler.
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Now at this point, we need to bring down that exponent.
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The way we do that is we take the log or the natural log on both sides.
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Let's use log base 10, but natural log would work just as well.
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Now this will allow us to bring down the exponent.
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Finally we can solve for t by dividing by 4
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and by dividing by the log of 1.0175.
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Let's come up here. Move the problem up here.
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We'll have the log of 3.2 divided by 4 log 1.0175.
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Okay finally, we need to enter this in the calculator.
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The one thing you want to be careful of is that we need to put parentheses
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around the denominator. Let's try that.
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Log 3.2 divided by parentheses 4 log 1.0175.
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Okay, be careful. I punched in the wrong number here, so be careful.
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Press equals, and let's see what our answer is.
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We're going to round to two decimal places.
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t is going to be approximately 16.76.
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Now in our money problems, t is always in years,
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so the units here are years. That is our final answer.
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In the b part of example two, we're going to see how many years it will take
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for the money, the 25,000, to grow to 80,000 at the given rate of 7 percent,
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but this time compounded continuously.
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Remember we need to decide which formula we're going to use.
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For compunded continuously, we're going to use this formula,
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which should be very easy to memorize.
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Our final amount equals our principal times e to the rate times the time power.
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Again, 80,000 is our final amount. P, the principal, is 25,000.
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r, we'll just remember that it's 0.07, and t is our unknown.
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t is in years. Again, the procedure is exactly the same.
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We want to divide both sides by 25,000.
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I think I'll leave this as 16/5 this time, so you can see a fraction in action.
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Of course, that will give me e to the 0.07 t power.
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Now in the previous a part when we were compounding quarterly,
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I took log on base 10 on both sides. I mentioned that I could've used the natural log.
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Either one would work. Here I do not want to use log on base 10
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because I have base e. To undo e, I want to use the natural log of 16/5
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equals the natural log of the right side
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because that allows me when I bring down the exponent.
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I multiply by the natural log of e.
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We all know, or we should know, that quantity is 1.
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Of course 0.07 t times 1 is 0.07 t.
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Last step before putting it in the calculator: we want to divide by 0.07,
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and we enter this in the calculator.
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Now my calculator puts parentheses here, but if yours does not,
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you need to put those in because you want to take the natural log
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of the fraction 16/5, not the natural log of 16 divided by 5.
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Here we go. Divide, close the parentheses, divide by 0.07. Enter.
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Again we're going to round to two decimal places,
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and the final answer will be in years. Because we're compounding continuously,
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it should take us less time than it did in the a part. We get 16.62 years.
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If you become proficient at working these problems,
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I think you'll have no trouble with this at all.