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Rachel deposited $10$ $\$$ in an account earning $5\%$ interest compounded annually. To the nearest cent, how much interest will she earn in $3$ years? $\$$ ____

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Exponential growth and decay: word problems

Full solution

Q. Rachel deposited

10

\$

in an account earning

5\%

interest compounded annually. To the nearest cent, how much interest will she earn in

3

years?

\$

____

Identify values: Identify the principal amount, interest rate, and time period. $\newline$ Rachel deposited $\$10$ at an interest rate of $5\%$ per annum, compounded annually, for a period of $3$ years. $\newline$ Principal ( $P$ ) = $\$10$ $\newline$ Interest rate ( $r$ ) = $5\%$ or $0.05$ (as a decimal) $\newline$ Time ( $t$ ) = $3$ years
Use compound interest formula: Use the formula for compound interest to find the total amount after $3$ years. $\newline$ The formula for compound interest is $A = P(1 + r/n)^{(nt)}$ , where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount, $r$ is the annual interest rate (decimal), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. $\newline$ Since the interest is compounded annually, $n = 1$ . $\newline$ $A = P(1 + r/n)^{(nt)}$
Substitute values into formula: Substitute the values into the formula. $\newline$ $A = 10(1 + 0.05/1)^{(1*3)}$ $\newline$ $A = 10(1 + 0.05)^3$ $\newline$ $A = 10(1.05)^3$
Calculate total amount: Calculate the total amount after $3$ years. $\newline$ $A = 10(1.05)^3$ $\newline$ $A = 10(1.157625)$ $\newline$ $A = 11.57625$
Calculate interest earned: Calculate the interest earned by subtracting the principal from the total amount. $\newline$ Interest earned = $A - P$ $\newline$ Interest earned = $11.57625 - 10$ $\newline$ Interest earned = $1.57625$
Round to nearest cent: Round the interest earned to the nearest cent. $\newline$ Interest earned = $\$1.58$ (rounded to the nearest cent)

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