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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Math Problems

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. $\newline$ $A$ (future value of the account) = $\$65,000$ $\newline$ $i$ (monthly interest rate) = $6.3\%$ annual interest rate / $12$ months = $0.063 / 12$ $\newline$ $n$ (total number of periods) = $11$ years * $12$ months/year
Convert Annual Interest Rate: Convert the annual interest rate to a monthly interest rate. $\newline$ $i = \frac{0.063}{12}$ $\newline$ $i = 0.00525$ (monthly interest rate)
Calculate Total Periods: Calculate the total number of periods (months) over $11$ years. $\newline$ $n = 11 \text{ years} \times 12 \text{ months/year}$ $\newline$ $n = 132 \text{ months}$
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 and the exponent. $\newline$ $(1 + 0.00525)^{132} - 1$
Perform Exponentiation: Perform the exponentiation. $\newline$ $(1 + 0.00525)^{132} \approx 1.99624$ $\newline$ $1.99624 - 1 \approx 0.99624$
Divide by Monthly Interest Rate: Divide the result by the monthly interest rate. $\newline$ $0.99624 / 0.00525 \approx 189.73619$
Solve for 'd': Solve for 'd' by dividing the future value of the account by the result from the previous step. $\newline$ $\$65,000 / 189.73619 \approx \$342.58$
Round Monthly Contribution: Round the monthly contribution to the nearest dollar. $\newline$ $d \approx \$343$ per month