Sophia is saving money and plans on making monthly contributions into an account earning an annual interest rate of $6.3\%$ compounded monthly. If Sophia would like to end up with $\$65,000$ after $11$ 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: ◻

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Grade 7

Compound interest

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

6.3\%

compounded monthly. If Sophia would like to end up with

\$65,000

after

11

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. $\newline$ We are given: $\newline$ Future value of the account, $A = \$(65,000)$ $\newline$ Annual interest rate, $r = 6.3\%$ or $0.063$ (as a decimal) $\newline$ Number of years, $t = 11$ $\newline$ The interest rate per period (monthly), $i = \frac{r}{12}$ $\newline$ The number of periods, $n = t \times 12$ (since the contributions are monthly)
Convert Annual Interest Rate: Convert the annual interest rate to the monthly interest rate. $\newline$ $i = \frac{r}{12}$ $\newline$ $i = \frac{0.063}{12}$ $\newline$ $i = 0.00525$
Calculate Total Periods: Calculate the total number of periods. $\newline$ $n = t \times 12$ $\newline$ $n = 11 \times 12$ $\newline$ $n = 132$
Find Monthly Contribution: Use the formula to find the monthly contribution, $d$ . $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$ $\$65,000 = d \times \left(\frac{(1 + 0.00525)^{132} - 1}{0.00525}\right)$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ $(1 + i)^{n} - 1$ $\newline$ $(1 + 0.00525)^{132} - 1$ $\newline$ $(1.00525)^{132} - 1$
Calculate Value of Expression: Calculate the value of the expression $(1.00525)^{132} - 1$ . This requires a calculator or a computer to compute. $(1.00525)^{132} - 1 \approx 0.9994$
Substitute Calculated Value: Substitute the calculated value back into the formula. $\newline$ $\$65,000 = d \times (0.9994 / 0.00525)$
Solve for Monthly Contribution: Solve for $d$ , the monthly contribution. $d = \frac{\$(65,000)}{(0.9994 / 0.00525)}$ $d \approx \frac{\$(65,000)}{0.1902}$ $d \approx \$(341.75)$
Round Monthly Contribution: Round the monthly contribution to the nearest dollar. $d \approx \$(342)$