After graduating with master's degree, Claudia combined all of her loans into a single loan of $\$18,000.00$ with an interest rate of $5.2\%$ compounded quarterly. If she is planning to pay off the loan in $10$ years, what quarterly payment would be? $\$$ (Round to $2$ decimal

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Q. After graduating with master's degree, Claudia combined all of her loans into a single loan of

\$18,000.00

with an interest rate of

5.2\%

compounded quarterly. If she is planning to pay off the loan in

10

years, what quarterly payment would be?

\$

(Round to

2

decimal

Given values: We have: $\newline$ Principal amount $P$ = $\$18,000$ $\newline$ Annual interest rate $r$ = $5.2\%$ or $0.052$ $\newline$ Compounding frequency per year $n$ = $4$ (since it's compounded quarterly) $\newline$ Total number of years $t$ = $10$ $\newline$ We need to calculate the quarterly payment $R$ . $\newline$ First, we convert the annual interest rate to the quarterly interest rate by dividing by the number of compounding periods per year. $\newline$ Quarterly interest rate = $\$18,000$ $0$
Calculate quarterly interest rate: Calculate the quarterly interest rate. $\newline$ Quarterly interest rate = $\frac{0.052}{4}$ $\newline$ Quarterly interest rate = $0.013$
Calculate total compounding periods: Next, we calculate the total number of compounding periods $T$ by multiplying the number of years by the compounding frequency. $\newline$ Total number of compounding periods $T$ = $n \times t$
Use annuity payment formula: Calculate the total number of compounding periods. $\newline$ Total number of compounding periods $T$ = $4 \times 10$ $\newline$ Total number of compounding periods $T$ = $40$
Substitute values into formula: Now, we use the formula for the annuity payment for a loan compounded at regular intervals, which is: $\newline$ $R = P \times \left[\frac{i(1+i)^T}{(1+i)^T - 1}\right]$ $\newline$ where $i$ is the quarterly interest rate and $T$ is the total number of compounding periods.
Calculate quarterly payment: Substitute the values into the formula to calculate the quarterly payment. $R = 18000 \times \left[\frac{(0.013(1+0.013)^{40})}{((1+0.013)^{40} - 1)}\right]$
Round quarterly payment: Calculate the quarterly payment using the formula. $R = 18000 \times \left[\frac{(0.013(1.013)^{40})}{((1.013)^{40} - 1)}\right]$
Round quarterly payment: Calculate the quarterly payment using the formula. $\newline$ $R = 18000 \times \left[\frac{(0.013(1.013)^{40})}{((1.013)^{40} - 1)}\right]$ Use a calculator to compute the values. $\newline$ $R \approx 18000 \times \left[\frac{(0.013 \times 1.6771)}{(1.6771 - 1)}\right]$ $\newline$ $R \approx 18000 \times \left[\frac{0.0218023}{0.6771}\right]$ $\newline$ $R \approx 18000 \times 0.032198$ $\newline$ $R \approx 579.564$
Round quarterly payment: Calculate the quarterly payment using the formula. $\newline$ $R = 18000 \times \left[\frac{(0.013(1.013)^{40})}{((1.013)^{40} - 1)}\right]$ Use a calculator to compute the values. $\newline$ $R \approx 18000 \times \left[\frac{(0.013 \times 1.6771)}{(1.6771 - 1)}\right]$ $\newline$ $R \approx 18000 \times \left[\frac{0.0218023}{0.6771}\right]$ $\newline$ $R \approx 18000 \times 0.032198$ $\newline$ $R \approx 579.564$ Round the quarterly payment to two decimal places as requested. $\newline$ Quarterly payment = $\$579.56$