Aubree is saving money and plans on making monthly contributions into an account earning a monthly interest rate of $0.675 \%$ . If Aubree would like to end up with $\$ 7,000$ after $2$ years, how much does she 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:

Full solution

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

0.675 \%

. If Aubree would like to end up with

\$ 7,000

after

2

years, how much does she 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. $\newline$ $A = \$7,000$ (future value of the account) $\newline$ $i = 0.675\%$ per month (interest rate per period) $\newline$ $n = 2$ years $* 12$ months/year $= 24$ months (number of periods) $\newline$ We need to find the value of $d$ (the amount invested at the end of each period).
Convert Interest Rate: Convert the interest rate from a percentage to a decimal. $i = 0.675\% = \frac{0.675}{100} = 0.00675$
Plug Values into Formula: Plug the values into the formula to solve for $d$ . $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$ $\$7,000 = d \times \left(\frac{(1 + 0.00675)^{24} - 1}{0.00675}\right)$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ $(1 + 0.00675)^{24} - 1$ $\newline$ = $(1.00675)^{24} - 1$ $\newline$ = $1.00675^{24} - 1$ $\newline$ = $1.177978 - 1$ $\newline$ = $0.177978$
Divide by Interest Rate: Divide the result by the interest rate $i$ . $\frac{0.177978}{0.00675}$ $= 26.3691852$
Divide Future Value by Result: Divide the future value $A$ by the result from Step $5$ to solve for $d$ . $\$7,000 / 26.3691852 = 265.449$
Round Monthly Contribution: Round the monthly contribution to the nearest dollar. $\newline$ $d \approx \$265$ (to the nearest dollar)