Amira deposits $\$ 560$ every quarter into an account earning an annual interest rate of $8 \%$ compounded quarterly. How much would she have in the account after $12$ 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:

Full solution

Q. Amira deposits

\$ 560

every quarter into an account earning an annual interest rate of

8 \%

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

12

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 Given Values: Identify the given values from the problem. $\newline$ Amira deposits $\$560$ every quarter, so $d = \$560$ . $\newline$ The annual interest rate is $8\%$ , so the quarterly interest rate $i = \frac{8\%}{4} = 2\%$ per quarter. $\newline$ Since the interest is compounded quarterly, we need to find the number of quarters in $12$ years. There are $4$ quarters in a year, so $n = 12 \text{ years} \times 4 \text{ quarters/year} = 48$ quarters.
Convert Interest Rate: Convert the quarterly interest rate into decimal form. $i = 2\%$ per quarter $= \frac{2}{100} = 0.02$
Calculate $(1+i)^n$ : Use the formula $A = d\left(\frac{(1+i)^{n}-1}{i}\right)$ to calculate the future value of the account. $\newline$ First, calculate $(1+i)^n$ . $\newline$ $(1+i)^n = (1+0.02)^{48}$
Calculate $(1+0.02)^{48}$ : Calculate $(1+0.02)^{48}$ using a calculator. $\newline$ $(1+0.02)^{48} \approx 2.20804$
Calculate Numerator: Calculate the numerator of the formula: $((1+i)^{n}-1)$ . $((1+0.02)^{48} - 1) \approx (2.20804 - 1) \approx 1.20804$
Calculate Future Value: Calculate the future value $A$ using the formula. $A = 560 \times \left(\frac{1.20804}{0.02}\right)$
Division and Multiplication: Calculate the division and multiplication to find $A$ . $A \approx 560 \times (60.402) \approx 33825.12$
Round Future Value: Round the future value to the nearest dollar. $\newline$ $A \approx \$33,825$