Charlotte is saving money and plans on making monthly contributions into an account earning an annual interest rate of $3 \%$ compounded monthly. If Charlotte would like to end up with $\$ 61,000$ after $9$ 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:

View step-by-step help

Home

Math Problems

Grade 7

Compound interest

Full solution

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

3 \%

compounded monthly. If Charlotte would like to end up with

\$ 61,000

after

9

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) = $\$61,000$ $\newline$ $n$ (number of periods) = $9$ years $*$ $12$ months/year = $108$ months $\newline$ $i$ (interest rate per period) = $3\%$ per year / $12$ months = $\$61,000$ $0$
Substitute Values into Formula: Substitute the given values into the formula. $\newline$ $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$ $\newline$ $\$61,000 = d \times \left(\frac{(1 + 0.03/12)^{108} - 1}{0.03/12}\right)$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses $(1 + i)^{n}$ . $(1 + \frac{0.03}{12})^{108} = (1 + 0.0025)^{108}$
Calculate Value of $(1 + i)^{n}$ : Calculate the value of $(1 + i)^{n}$ . $(1 + 0.0025)^{108} \approx 1.349858807576003$
Calculate Numerator of Formula: Calculate the numerator of the formula $(((1 + i)^{n} - 1)$ . $1.349858807576003 - 1 \approx 0.349858807576003$
Calculate Denominator of Formula: Calculate the denominator of the formula (i). $\frac{0.03}{12} = 0.0025$
Calculate Entire Fraction: Calculate the entire fraction of the formula $\left(\frac{(1 + i)^{n} - 1}{i}\right)$ . $\frac{0.349858807576003}{0.0025} \approx 139.9435230304012$
Solve for d: Solve for d (the amount invested at the end of each period). $\newline$ $\$61,000 = d \times 139.9435230304012$ $\newline$ $d = \frac{\$61,000}{139.9435230304012}$
Calculate Value of d: Calculate the value of d. $\newline$ $d \approx \$61,000 / 139.9435230304012 \approx \$435.62$