A loan of $\$ 6300$ is repaid by equal payments at the end of each quarter for $3$ years. If interest is $6 \%$ compounded quarterly, how much interest was paid in the first year $\newline$ $\begin{array}{l} P / y= \\ c / y= \end{array}$

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Q. A loan of

\$ 6300

is repaid by equal payments at the end of each quarter for

3

years. If interest is

6 \%

compounded quarterly, how much interest was paid in the first year

\newline

\begin{array}{l} P / y= \\ c / y= \end{array}

Calculate Quarterly Interest Rate: Calculate the quarterly interest rate. $\newline$ Quarterly interest rate = Annual interest rate / Number of quarters in a year $\newline$ Quarterly interest rate = $\frac{6\%}{4}$ $\newline$ Quarterly interest rate = $0.015$ or $1.5\%$
Determine Number of Quarters: Determine the number of quarters in $3$ years. $\newline$ Number of quarters $=$ Years $*$ Quarters per year $\newline$ Number of quarters $=$ $3 * 4$ $\newline$ Number of quarters $=$ $12$
Use Annuity Payment Formula: Use the formula for the annuity payment $A$ for a loan with compound interest: $\newline$ $A = P \times \left(\frac{r(1+r)^n}{((1+r)^n - 1)}\right)$ $\newline$ Where $P$ is the principal, $r$ is the quarterly interest rate, and $n$ is the total number of payments. $\newline$ $A = 6300 \times \left(\frac{0.015(1+0.015)^{12}}{((1+0.015)^{12} - 1)}\right)$
Calculate Annuity Payment: Calculate the annuity payment $A$ .
$A = 6300 \times \left(0.015(1.015)^{12}\right) / \left((1.015)^{12} - 1\right)$
$A = 6300 \times \left(0.015 \times 1.195618\right) / \left(1.195618 - 1\right)$
$A = 6300 \times \left(0.01793427\right) / \left(0.195618\right)$
$A = 112.985901 / 0.195618$
$A = 577.45$ approximately
Calculate Total Amount Paid: Calculate the total amount paid in the first year. $\newline$ Total paid first year = $A \times$ Number of quarters in a year $\newline$ Total paid first year = $577.45 \times 4$ $\newline$ Total paid first year = $2309.8$
Calculate Interest Paid: Calculate the interest paid in the first year. $\newline$ Interest paid $= \text{Total amount paid} - \text{Principal repayment}$ $\newline$ But we need to find the principal repayment first, which is the total amount paid minus the interest for each quarter.
Calculate Interest First Quarter: Calculate the interest for the first quarter. $\newline$ Interest for first quarter = $Principal \times Quarterly \, interest \, rate$ $\newline$ Interest for first quarter = $6300 \times 0.015$ $\newline$ Interest for first quarter = $94.5$
Calculate Principal Repayment First Quarter: Calculate the principal repayment for the first quarter. $\newline$ Principal repayment first quarter = $A - \text{Interest for first quarter}$ $\newline$ Principal repayment first quarter = $577.45 - 94.5$ $\newline$ Principal repayment first quarter = $482.95$
Update Principal Next Quarter: Update the principal for the next quarter. $\newline$ New principal after first quarter = Original principal - Principal repayment first quarter $\newline$ New principal after first quarter = $6300 - 482.95$ $\newline$ New principal after first quarter = $5817.05$
Calculate Interest Second Quarter: Calculate the interest for the second quarter. $\newline$ Interest for second quarter = New principal after first quarter $\times$ Quarterly interest rate $\newline$ Interest for second quarter = $5817.05 \times 0.015$ $\newline$ Interest for second quarter = $87.26$ approximately
Calculate Principal Repayment Second Quarter: Calculate the principal repayment for the second quarter. $\newline$ Principal repayment second quarter = $A - \text{Interest for second quarter}$ $\newline$ Principal repayment second quarter = $577.45 - 87.26$ $\newline$ Principal repayment second quarter = $490.19$
Update Principal Third Quarter: Update the principal for the third quarter. $\newline$ New principal after second quarter = New principal after first quarter - Principal repayment second quarter $\newline$ New principal after second quarter = $5817.05 - 490.19$ $\newline$ New principal after second quarter = $5326.86$
Calculate Interest Third Quarter: Calculate the interest for the third quarter. $\newline$ Interest for third quarter = New principal after second quarter $\times$ Quarterly interest rate $\newline$ Interest for third quarter = $5326.86 \times 0.015$ $\newline$ Interest for third quarter = Oops, made a mistake here, should have used the updated principal.