Ethan deposits $\$ 100$ every quarter into an account earning a quarterly interest rate of $0.825 \%$ . How much would he have in the account after $11$ 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. Ethan deposits

\$ 100

every quarter into an account earning a quarterly interest rate of

0.825 \%

. How much would he have in the account after

11

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: Identify the variables from the problem. $\newline$ We have: $\newline$ $d = \$100$ (the amount invested at the end of each period) $\newline$ $i = 0.825\%$ per quarter (the interest rate per period) $\newline$ $n = 11 \text{ years} \times 4 \text{ quarters/year} = 44 \text{ quarters}$ (the number of periods)
Convert interest rate: Convert the interest rate from a percentage to a decimal. $i = 0.825\% = \frac{0.825}{100} = 0.00825$
Calculate future value: Use the formula to calculate the future value of the account. $A = d \times \left(\frac{(1 + i)^{n} - 1}{i}\right)$
Plug values and calculate: Plug the values into the formula and calculate the future value. $\newline$ $A = 100 \times \left(\left(1 + 0.00825\right)^{44} - 1\right) / 0.00825$
Calculate exponentiation: Calculate the value inside the parentheses and the exponentiation. $\newline$ $(1 + 0.00825)^{44} = 1.00825^{44}$
Calculate future value: Calculate the exponentiation using a calculator. $\newline$ $1.00825^{44} \approx 1.432386$
Calculate numerator: Continue with the formula calculation. $\newline$ $A = 100 \times \left(\frac{1.432386 - 1}{0.00825}\right)$
Calculate future value: Calculate the numerator of the fraction. $\newline$ $1.432386 - 1 = 0.432386$
Finish calculation: Calculate the future value of the account. $\newline$ $A = 100 \times (0.432386 / 0.00825)$
Round final answer: Finish the calculation. $\newline$ $A \approx 100 \times 52.41284848$ $\newline$ $A \approx 5241.284848$
Round final answer: Finish the calculation. $\newline$ $A \approx 100 \times 52.41284848$ $\newline$ $A \approx 5241.284848$ Round the final answer to the nearest dollar. $\newline$ $A \approx \$(5241)$