Ai Mi deposits $\$ 910$ every month into an account earning an annual interest rate of $3.9 \%$ compounded monthly. How much would she have in the account after $13$ years, 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 8

Compound interest

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Q. Ai Mi deposits

\$ 910

every month into an account earning an annual interest rate of

3.9 \%

compounded monthly. How much would she have in the account after

13

years, 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 Variables: First, let's identify the variables needed to use the formula $A=d\left(\left(1+i\right)^{n}-1\right)/i$ : $d = \$910$ (monthly deposit) $i = 3.9\%$ annual interest rate compounded monthly, which needs to be converted to a monthly rate by dividing by $12$ $n = 13$ years, but since the compounding is monthly, we need to multiply by $12$ to get the number of periods
Calculate Monthly Interest Rate: Now, let's calculate the monthly interest rate ( extit{i}):
extit{i} = annual interest rate / $12$
extit{i} = $3.9\% / 12$
extit{i} = $0.039 / 12$
extit{i} = $0.00325$
Calculate Number of Periods: Next, we calculate the number of periods $n$ : $n = \text{years} \times 12$ (since the deposits are monthly) $n = 13 \times 12$ $n = 156$
Calculate Future Value: Now we can use the formula to calculate the future value of the account $A$ :
$A = d \times \left(\left(1 + i\right)^{n} - 1\right) / i$
$A = 910 \times \left(\left(1 + 0.00325\right)^{156} - 1\right) / 0.00325$
Calculate Inside Parentheses: Let's calculate the part inside the parentheses first: $\newline$ $(1 + i)^{n} - 1$ $\newline$ $(1 + 0.00325)^{156} - 1$
Calculate Exponential Value: Now, we calculate $(1 + 0.00325)^{156}$ : $\newline$ $(1 + 0.00325)^{156} \approx 1.647$
Subtract One: Subtract $1$ from the result: $\newline$ $1.647 - 1 \approx 0.647$
Divide by i: Now we divide this result by i: $0.647 / 0.00325 \approx 199.0769$
Multiply by $d$ : Finally, we multiply this result by $d$ to get $A$ : $A = 910 \times 199.0769$ $A \approx 181155.979$
Round Final Amount: Since we need to round to the nearest dollar, the final amount in the account will be: $A \approx \$(181,156)$