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Colleen has put $\$26,000$ into a retirement fund. The fund has an estimated annual return of $10\%$ . If Colleen doesn't add any more money, how much can she expect to have in the fund in $7$ years? If necessary, round your answer to the nearest cent. $\newline$ $\$$ ____

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Exponential growth and decay: word problems

Full solution

Q. Colleen has put

\$26,000

into a retirement fund. The fund has an estimated annual return of

10\%

. If Colleen doesn't add any more money, how much can she expect to have in the fund in

7

years? If necessary, round your answer to the nearest cent.

\newline

\$

____

Identify Inputs: Identify the initial amount, rate of return, and time period. $\newline$ Colleen's initial investment (principal) is $\$26,000$ . $\newline$ The annual return rate is $10\%$ , or $0.10$ in decimal form. $\newline$ The time period is $7$ years.
Use Compound Interest Formula: Use the formula for compound interest to calculate the future value of the investment. $\newline$ The formula for compound interest is $A = P(1 + r/n)^{(nt)}$ , where: $\newline$ $A$ = the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ = the principal amount (the initial amount of money). $\newline$ $r$ = the annual interest rate (decimal). $\newline$ $n$ = the number of times that interest is compounded per year. $\newline$ $t$ = the time the money is invested for, in years. $\newline$ Since the problem does not specify how often the interest is compounded, we will assume it is compounded once a year ( $n = 1$ ).
Substitute Values: Substitute the values into the compound interest formula. $\newline$ $P = \$26,000$ $\newline$ $r = 0.10$ $\newline$ $n = 1$ $\newline$ $t = 7$ $\newline$ $A = 26000(1 + 0.10/1)^{1*7}$ $\newline$ $A = 26000(1 + 0.10)^7$ $\newline$ $A = 26000(1.10)^7$
Calculate Future Value: Calculate the future value of the investment. $\newline$ $A = 26000(1.10)^7$ $\newline$ $A = 26000 \times (1.10)^7$ $\newline$ $A = 26000 \times 1.9487171$ (rounded to $7$ decimal places) $\newline$ $A = 50666.2447$ (rounded to $4$ decimal places)
Round to Nearest Cent: Round the answer to the nearest cent if necessary. $\newline$ $A = \$50,666.24$

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