Mohal is saving money and plans on making monthly contributions into an account earning a monthly interest rate of $0.45 \%$ . If Mohal would like to end up with $\$ 19,000$ after $30$ 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:

Full solution

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

0.45 \%

. If Mohal would like to end up with

\$ 19,000

after

30

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) = $\$19,000$ $\newline$ $i$ (interest rate per period) = $0.45\%$ per month, which is $0.0045$ in decimal form $\newline$ $n$ (number of periods) = $30$ months $\newline$ We need to find the value of $d$ (the amount invested at the end of each period).
Plug Values into Formula: Plug the given values into the formula to solve for $d$ . The formula is $A = d \times \left(\left(1 + i\right)^{n} - 1\right) / i$ . We have $A = \$19,000$ , $i = 0.0045$ , and $n = 30$ .
Calculate $(1 + i)^n$ : Calculate the value inside the parentheses $(1 + i)^n$ . $(1 + i)^n = (1 + 0.0045)^{30}$ Use a calculator to find the value. $(1 + 0.0045)^{30} \approx 1.142475$
Calculate $((1 + i)^n - 1)$ : Calculate the numerator of the formula $((1 + i)^n - 1)$ .
$((1 + i)^n - 1) = 1.142475 - 1$
$((1 + i)^n - 1) \approx 0.142475$
Calculate Value of d: Calculate the value of d using the formula. $\newline$ $d = \frac{A}{\left(\left(1 + i\right)^n - 1\right) / i}$ $\newline$ $d = \frac{\$19,000}{\left(0.142475 / 0.0045\right)}$ $\newline$ First, calculate the denominator of the fraction. $\newline$ $0.142475 / 0.0045 \approx 31.6611$
Divide to Find d: Now, divide the future value $A$ by the result from Step $5$ to find $d$ . $d = \frac{\$19,000}{31.6611}$ $d \approx \$600.18$ Since we need to round to the nearest dollar, $d \approx \$600$ .