Christopher deposits $\$ 420$ every month into an account earning an annual interest rate of $3.9 \%$ compounded monthly. How much would he have in the account after $19$ 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. Christopher deposits

\$ 420

every month into an account earning an annual interest rate of

3.9 \%

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

19

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 Given Values: Identify the given values from the problem. $\newline$ Christopher deposits $\$420$ every month $d = \$420$ . $\newline$ The annual interest rate is $3.9\%$ (annual interest rate = $3.9\%$ ). $\newline$ The interest is compounded monthly, so we need to find the monthly interest rate $i$ . $\newline$ The number of periods is $19$ months $n = 19$ .
Convert Annual Interest Rate: Convert the annual interest rate to a monthly interest rate. $\newline$ The annual interest rate is $3.9\%$ , which as a decimal is $0.039$ . $\newline$ To find the monthly interest rate, divide the annual rate by $12$ (the number of months in a year). $\newline$ $i = \frac{0.039}{12}$ $\newline$ $i = 0.00325$
Calculate Future Value: Use the formula to calculate the future value of the account after $19$ months. $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$ Substitute the given values into the formula. $A = 420 \times \left(\frac{(1 + 0.00325)^{19} - 1}{0.00325}\right)$
Calculate Future Value: Calculate the future value using the values substituted into the formula. $\newline$ $A = 420 \times \left(\left(1 + 0.00325\right)^{19} - 1\right) / 0.00325$ $\newline$ $A = 420 \times \left(\left(1.00325\right)^{19} - 1\right) / 0.00325$ $\newline$ First, calculate $\left(1.00325\right)^{19}$ . $\newline$ $\left(1.00325\right)^{19} \approx 1.063383$ $\newline$ Now, subtract $1$ from this result. $\newline$ $1.063383 - 1 \approx 0.063383$ $\newline$ Finally, divide by $0.00325$ . $\newline$ $0.063383 / 0.00325 \approx 19.502462$
Multiply by Deposit Amount: Multiply the result by the monthly deposit amount to find the future value. $\newline$ $A = 420 \times 19.502462$ $\newline$ $A \approx 8190.03324$
Round to Nearest Dollar: Round the future value to the nearest dollar as the problem asks for the answer to the nearest dollar. $\newline$ $A \approx \$(8190)$