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

\$ 18,000.00

with an interest rate of

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

\$

(Round to

2

decimal places.) Question Help: OMessage inctructor

Question Prompt: Question prompt: What will Claudia's quarterly payment be for her student loan of $\$18,000.00$ at a $5.2\%$ interest rate compounded quarterly, if she plans to pay it off in $10$ years?
Given Values: We have: $\newline$ Principal amount $P$ = $\$18,000.00$ $\newline$ Annual interest rate $r$ = $5$ . $2$ \% or $0.052$ (in decimal) $\newline$ Compounding frequency per year $n$ = $4$ (quarterly) $\newline$ Total number of years $t$ = $10$ $\newline$ We need to find the quarterly payment $R$ .
Calculate Quarterly Interest Rate: First, we convert the annual interest rate to the quarterly interest rate by dividing it by the number of compounding periods per year. $\newline$ Quarterly interest rate $i$ = $r / n$ = $0.052 / 4$ = $0.013$
Calculate Total Compounding Periods: Next, we calculate the total number of compounding periods $N$ by multiplying the number of years by the compounding frequency. $N = n \times t = 4 \times 10 = 40$
Formula for Quarterly Payment: Now, we use the formula for the quarterly payment of a loan with compound interest, which is: $\newline$ $R = P \times (i \times (1 + i)^N) / ((1 + i)^N - 1)$
Plug in Values: We plug in the values into the formula to calculate the quarterly payment. $\newline$ $R = 18000 \times (0.013 \times (1 + 0.013)^{40}) / ((1 + 0.013)^{40} - 1)$
Calculate Numerator: We calculate the numerator of the formula: $\newline$ $\text{numerator} = 0.013 \times (1 + 0.013)^{40}$
Calculate Denominator: We calculate the denominator of the formula: $\newline$ denominator = $(1 + 0.013)^{40} - 1$
Compute Values: We compute the values for both the numerator and the denominator and then divide them to find $R$ . $\text{numerator\_value} = 0.013 \times (1.013)^{40}$ $\text{denominator\_value} = (1.013)^{40} - 1$ $R = 18000 \times \left(\frac{\text{numerator\_value}}{\text{denominator\_value}}\right)$
Perform Calculations: We perform the calculations using a calculator or a computer to ensure accuracy. $\newline$ $\text{numerator\_value} \approx 0.013 \times 1.677097$ $\newline$ $\text{denominator\_value} \approx 1.677097 - 1$ $\newline$ $R \approx 18000 \times (0.013 \times 1.677097 / 0.677097)$
Simplify Calculation: We simplify the calculation to find the quarterly payment. $\newline$ $R \approx 18000 \times (0.021801261 / 0.677097)$ $\newline$ $R \approx 18000 \times 0.032204$ $\newline$ $R \approx 579.672$
Find Quarterly Payment: Finally, we round the quarterly payment to two decimal places as requested. $\newline$ Rounded quarterly payment = $\$579.67$