I $8$ 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 $5.2 \%$ unded quarterly. If she is planning to pay off the loan in $10$ years, what quarterly payment be? \$.$\newline$arterly payment would be $ \$ $ (Round to $2$ decimal

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Q. I

8

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 \%

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

10

years, what quarterly payment be? \$.$\newline$arterly payment would be $ \$ $ (Round to $2$ decimal

Identify loan details: Identify the loan details. $\newline$ We have: $\newline$ Principal amount $P$ = $\$18,000$ $\newline$ Annual interest rate $r$ = $5.2\%$ or $0.052$ (as a decimal) $\newline$ Compounding frequency per year $n$ = $4$ (quarterly) $\newline$ Total number of years $t$ = $10$ $\newline$ We need to calculate the quarterly payment $R$ .
Convert annual interest rate: Convert the annual interest rate to the quarterly interest rate. $\newline$ Quarterly interest rate = Annual interest rate / Number of quarters in a 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 payments $=$ Number of years $\times$ Number of quarters in a year $\newline$ Total number of payments $= 10 \times 4$ $\newline$ Total number of payments $= 40$
Use annuity payment formula: Use the formula for the annuity payment for a loan compounded at different periods. $\newline$ The formula for the annuity payment is: $\newline$ $R = P \times \left(\frac{r}{n}\right) / \left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right]$ $\newline$ Where $R$ is the quarterly payment.
Calculate quarterly payment: Plug the values into the formula and calculate the quarterly payment. $\newline$ $R = 18000 \times (0.013) / [1 - (1 + 0.013)^{-40}]$ $\newline$ $R = 234 / [1 - (1.013)^{-40}]$ $\newline$ First, calculate the value inside the brackets: $(1.013)^{-40}$
Calculate value in brackets: Calculate the value inside the brackets. $\newline$ $(1.013)^{(-40)} \approx 0.607523$ $\newline$ Now, subtract this value from $1$ . $\newline$ $1 - 0.607523 \approx 0.392477$
Complete payment calculation: Complete the calculation for the quarterly payment. $\newline$ $R = \frac{234}{0.392477}$ $\newline$ $R \approx 596.253$ $\newline$ Round to two decimal places. $\newline$ $R \approx \$(596.25)$