View source for 022 Sample Final A, Problem 10

<span class="exam">Set up the formula to find the amount of money one would have at the end of 8 years if she invests $2100 in an account paying 6% annual interest, compounded quarterly.

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!Foundations: &nbsp;
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|The primary purpose of this problem is to demonstrate that you understand compounding on an interval of time.  When we compound on an interval, say monthly, the value in the account only changes at the end of each interval.  In other words, there is no interest accrued for a week or a day.  As a result, we use the formula
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::<math>A\,=\,P\left(1+\frac{r}{n}\right)^{nt},</math>
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|where <math style="vertical-align: 0%;">A</math> is the value of the account, <math style="vertical-align: 0%;">P</math> is the principal (original amount invested), <math style="vertical-align: 0%;">r</math> is the annual rate and <math style="vertical-align: 0%;">n</math> is the number of compoundings per year.  The value of <math style="vertical-align: 0%;">n</math> is <math style="vertical-align: 0%;">365</math> for compounding daily, <math style="vertical-align: 0%;">52</math> for compounding weekly, and <math style="vertical-align: -5%;">12</math> for compounding monthly.  As a result, the exponent <math style="vertical-align: 0%;">nt</math>, where <math style="vertical-align: 0%;">t</math> is the time in years, is the number of compounding periods where we actually earn interest. Similarly, <math style="vertical-align: -22%;">r/n</math> is the rate per compounding period (the annual rate divided by the number of compoundings per year).
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|For example, if we compound monthly for <math style="vertical-align: 0%;">7</math> years at a <math style="vertical-align: -5%;">6%</math> rate, we would compound <math style="vertical-align: -5%;">nt\,=\,12 \cdot7\,=\,84</math> times, once per month, at a rate of <math style="vertical-align: -22%;">0.06/12\,=\,0.005</math> per monthly period. Notice that we <u>'''always'''</u> use the decimal version for interest rates when using these equations.
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!Solution: &nbsp;
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|We are given all the pieces required.  We begin with <math style="vertical-align: -1px">$2100</math> of principal, and compound quarterly, or <math style="vertical-align: -1.5px">4</math> times per year. Using the formula in 'Foundations', the equation for the account value after 8 years is
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::<math>A\,=\,P\left(1+\frac{r}{n}\right)^{nt}\,=\,2100\left(1+\frac{0.06}{4}\right)^{4\cdot 8}\,=\,2100(1.015)^{32}.</math>
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!Final Answer: &nbsp;
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::<math>A\,=\,2100(1.015)^{32}.</math>
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