Miguel deposits $\$ 470$ every month into an account earning an annual interest rate of $8.1 \%$ compounded monthly. How much would he have in the account after $30$ months, 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. Miguel deposits

\$ 470

every month into an account earning an annual interest rate of

8.1 \%

compounded monthly. How much would he have in the account after

30

months, 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. $\newline$ $d = \$470$ (the amount invested at the end of each period) $\newline$ $i = 8.1\%$ annual interest rate, which needs to be converted to a monthly rate by dividing by $12$ . $\newline$ $n = 30$ (the number of periods, which are months in this case)
Convert interest rate: Convert the annual interest rate to a monthly interest rate. $\newline$ $i = \frac{8.1\% \text{ per year}}{12 \text{ months per year}}$ $\newline$ $i = \frac{0.081}{12}$ $\newline$ $i = 0.00675$ (monthly interest rate)
Calculate future value: Use the formula to calculate the future value of the account after $n$ periods. $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$
Plug values and calculate: Plug the values into the formula and calculate the future value. $\newline$ $A = 470 \times \left(\left(1 + 0.00675\right)^{30} - 1\right) / 0.00675$
Calculate exponent: Calculate the value inside the parentheses and the exponent. $\newline$ $(1 + 0.00675)^{30} = 1.00675^{30}$
Continue formula calculation: Calculate the exponent. $\newline$ $1.00675^{30} \approx 1.22139$
Calculate numerator: Continue with the formula calculation. $\newline$ $A = 470 \times ((1.22139 - 1) / 0.00675)$
Calculate future value: Calculate the numerator of the fraction. $\newline$ $1.22139 - 1 = 0.22139$
Round to nearest dollar: Calculate the future value $A$ . $A = 470 \times (0.22139 / 0.00675)$ $A \approx 470 \times 32.80593$ $A \approx 15418.7851$
Round to nearest dollar: Calculate the future value $A$ . $A = 470 \times (0.22139 / 0.00675)$ $A \approx 470 \times 32.80593$ $A \approx 15418.7851$ Round the future value to the nearest dollar. $A \approx \$(15419)$