Question $8$ $\newline$ Cichers $\newline$ After graduating with a master's degree, Claudia combined all of her student loat a single loan of $\newline$ $\$18,000.00$ with an interest rate of $\newline$ $5.2\%$ compounded quarterly. planning to pay off the loan in $10$ years, what will her quarterly payment be? $\newline$ The quarterly payment would be $\newline$ $\$$ - (Round to $2$ decimal placesi) Question Help: OMessage inctructor

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Q. Question

8

\newline

Cichers

\newline

After graduating with a master's degree, Claudia combined all of her student loat a single loan of

\newline

\$18,000.00

with an interest rate of

\newline

5.2\%

compounded quarterly. planning to pay off the loan in

10

years, what will her quarterly payment be?

\newline

The quarterly payment would be

\newline

\$

- (Round to

2

decimal placesi) Question Help: OMessage inctructor

Identify variables: Identify the variables for the loan payment calculation. $\newline$ Principal amount $P$ = $\$18,000.00$ $\newline$ Annual interest rate $r$ = $5.2\%$ or $0.052$ (as a decimal) $\newline$ Compounding frequency per year $n$ = $4$ (quarterly) $\newline$ Total number of payments $t$ = $10$ years
Convert interest rate: Convert the annual interest rate to the quarterly interest rate. $\newline$ Quarterly interest rate = Annual interest rate / Number of compounding periods per year $\newline$ Quarterly interest rate = $\frac{0.052}{4}$ $\newline$ Quarterly interest rate = $0.013$
Calculate total payments: Calculate the total number of quarterly payments. $\newline$ Total number of quarterly payments = Total number of years $\times$ Number of compounding periods per year $\newline$ Total number of quarterly payments = $10 \times 4$ $\newline$ Total number of quarterly payments = $40$
Use annuity payment formula: Use the formula for the annuity payment with compound interest to calculate the quarterly payment. $\newline$ The formula is: $\newline$ Payment (PMT) = $P \times \left[\frac{r/n}{1 - (1 + r/n)^{-nt}}\right]$ $\newline$ Where: $\newline$ $P$ = Principal amount $\newline$ $r$ = Annual interest rate (as a decimal) $\newline$ $n$ = Number of compounding periods per year $\newline$ $t$ = Total number of years
Calculate quarterly payment: Plug the values into the formula and calculate the quarterly payment. $\newline$ $PMT = 18000 \times \left[\frac{0.013}{1 - (1 + 0.013)^{-40}}\right]$ $\newline$ $PMT = 18000 \times \left[\frac{0.013}{1 - (1.013)^{-40}}\right]$
Calculate denominator: Calculate the denominator of the formula. $\newline$ $(1 + 0.013)^{-40} = 1.013^{-40}$ $\newline$ Use a calculator to find the value. $\newline$ $(1 + 0.013)^{-40} \approx 0.608731$ $\newline$ $1 - 0.608731 = 0.391269$
Finalize quarterly payment: Calculate the quarterly payment using the values from the previous steps. $\newline$ $PMT = 18000 \times \left[\frac{0.013}{0.391269}\right]$ $\newline$ $PMT = 18000 \times \left[\frac{0.033218}{0.391269}\right]$ $\newline$ $PMT = 18000 \times 0.084883$ $\newline$ $PMT \approx 1527.894$
Round quarterly payment: Round the quarterly payment to two decimal places as requested. $\newline$ Quarterly payment $\approx$ $\$1527.89$