Madeline deposits $\$ 5,800$ every year into an account earning an annual interest rate of $5.2 \%$ compounded annually. How much would she have in the account after $15$ years, to the nearest dollar? Use the following formula to determine your answer. $\newline$ $A=d\left(\frac{(1+i)^{n}-1}{i}\right)$ $\newline$ $A=$ the future value of the account after $n$ periods $\newline$ $d=$ the amount invested at the end of each period $\newline$ $i=$ the interest rate per period $\newline$ $n=$ the number of periods $\newline$ Answer:

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Math Problems

Grade 7

Compound interest

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Q. Madeline deposits

\$ 5,800

every year into an account earning an annual interest rate of

5.2 \%

compounded annually. How much would she have in the account after

15

years, to the nearest dollar? Use the following formula to determine your answer.

\newline

A=d\left(\frac{(1+i)^{n}-1}{i}\right)

\newline

A=

the future value of the account after

n

periods

\newline

d=

the amount invested at the end of each period

\newline

i=

the interest rate per period

\newline

n=

the number of periods

\newline

Answer:

Identify variables: Identify the variables from the problem. $\newline$ We are given: $\newline$ $d = \$5,800$ (the amount invested at the end of each period) $\newline$ $i = 5.2\%$ or $0.052$ (the interest rate per period) $\newline$ $n = 15$ (the number of periods) $\newline$ We need to find $A$ , the future value of the account after $n$ periods.
Convert interest rate: Convert the percentage interest rate to a decimal. $i = 5.2\% = \frac{5.2}{100} = 0.052$
Plug values into formula: Plug the values into the compound interest formula. $\newline$ $A = d \times \left(\left(1 + i\right)^n - 1\right) / i$ $\newline$ $A = \$(5,800) \times \left(\left(1 + 0.052\right)^{15} - 1\right) / 0.052$
Calculate compound factor: Calculate the compound factor $(1 + i)^n$ . $(1 + i)^n = (1 + 0.052)^{15}$ $(1 + i)^n \approx 1.052^{15}$ $(1 + i)^n \approx 2.1133137$ (rounded to $7$ decimal places for precision)
Calculate numerator: Calculate the numerator of the formula: $((1 + i)^n - 1)$ . $((1 + i)^n - 1) \approx 2.1133137 - 1$ $((1 + i)^n - 1) \approx 1.1133137$
Divide by interest rate: Divide the result from Step $5$ by the interest rate $i$ . $\newline$ $\frac{1.1133137}{0.052} \approx 21.4094952$ (rounded to $7$ decimal places for precision)
Multiply by amount: Multiply the result from Step $6$ by the amount invested at the end of each period $d$ . $A \approx \$(5,800 \times 21.4094952)$ $A \approx \$124,174.8716$
Round future value: Round the future value of the account to the nearest dollar. $A \approx \$124,175$