Zachary is saving money and plans on making monthly contributions into an account earning an annual interest rate of $4.2 \%$ compounded monthly. If Zachary would like to end up with $\$ 8,000$ after $21$ months, how much does he need to contribute to the account every month, 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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Grade 7

Compound interest

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Q. Zachary is saving money and plans on making monthly contributions into an account earning an annual interest rate of

4.2 \%

compounded monthly. If Zachary would like to end up with

\$ 8,000

after

21

months, how much does he need to contribute to the account every month, 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. $\newline$ A (future value of the account) = $\$8,000$ $\newline$ n (number of periods) = $21$ months $\newline$ i (interest rate per period) = $4.2\%$ annual interest rate compounded monthly $\newline$ First, we need to convert the annual interest rate to a monthly interest rate by dividing by $12$ (since there are $12$ months in a year). $\newline$ i = $\frac{4.2\%}{12}$ $\newline$ i = $\frac{0.042}{12}$ $\newline$ i = $0.0035$
Convert Annual Rate: Substitute the values into the formula. $\newline$ We have the formula $A = d \times \left(\left(1 + i\right)^{n} - 1\right) / i$ . $\newline$ Now we substitute the values we have into the formula. $\newline$ $A = \$8,000$ $\newline$ $i = 0.0035$ $\newline$ $n = 21$ $\newline$ We need to find $d$ .
Substitute Values: Solve for $d$ using the formula. $\newline$ $\$8,000 = d \times \left(\left(1 + 0.0035\right)^{21} - 1\right) / 0.0035$ $\newline$ Now we calculate the right side of the equation step by step. $\newline$ First, calculate $(1 + i)^n$ : $\newline$ $(1 + 0.0035)^{21}$
Solve for $d$ : Calculate the compound factor. $(1 + 0.0035)^{21} \approx 1.0759$ Now we subtract $1$ from the compound factor. $1.0759 - 1 \approx 0.0759$
Calculate Compound Factor: Divide by the interest rate per period. $\newline$ $0.0759 / 0.0035 \approx 21.6857$ $\newline$ Now we have the denominator of the formula.
Divide by Interest Rate: Solve for the monthly contribution $d$ .
$\$8,000 = d \times 21.6857$
Now we divide both sides by $21.6857$ to solve for $d$ .
$d \approx \$8,000 / 21.6857$
$d \approx 368.85$
Solve for Monthly Contribution: Round the monthly contribution to the nearest dollar. $\newline$ Zachary needs to contribute approximately $\$369$ every month.