Josiah is saving money and plans on making monthly contributions into an account earning a monthly interest rate of $0.35 \%$ . If Josiah would like to end up with $\$ 12,000$ after $5$ years, 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. Josiah is saving money and plans on making monthly contributions into an account earning a monthly interest rate of

0.35 \%

. If Josiah would like to end up with

\$ 12,000

after

5

years, 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) = $\$12,000$ $\newline$ $i$ (interest rate per period) = $0.35\%$ per month $\newline$ $n$ (number of periods) = $5$ years $* 12$ months/year = $60$ months $\newline$ We will use the formula $A = d * \left(\frac{(1 + i)^{n} - 1}{i}\right)$ to find $d$ , the amount invested at the end of each period.
Convert Interest Rate: Convert the interest rate from a percentage to a decimal. $i = 0.35\% = \frac{0.35}{100} = 0.0035$
Substitute Values into Formula: Substitute the values into the formula. $\newline$ $A = \$12,000$ $\newline$ $i = 0.0035$ $\newline$ $n = 60$ $\newline$ Now we will plug these values into the formula to solve for $d$ .
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ $(1 + i)^n = (1 + 0.0035)^{60}$ $\newline$ Use a calculator to find the value. $\newline$ $(1 + 0.0035)^{60} \approx 1.229678$
Calculate Numerator: Calculate the numerator of the formula. $((1 + i)^n - 1) = 1.229678 - 1 = 0.229678$
Calculate Denominator: Calculate the denominator of the formula. $\newline$ $i = 0.0035$
Calculate Fraction Value: Calculate the value of the fraction in the formula. $0.229678 / 0.0035 \approx 65.6228571$
Calculate Monthly Contribution: Calculate the monthly contribution $d$ . $A = \$12,000$ $d = \frac{A}{(0.229678 / 0.0035)}$ $d = \frac{\$12,000}{65.6228571}$ $d \approx \$182.88$
Round Monthly Contribution: Round the monthly contribution to the nearest dollar. $d \approx \$(183)$