Question

indigo borrowed from a bank at 2.69% compounded quarterly; she settled the loan by repaying $1,069 at the end of every three months for 6.5 years.
a) what was the amount of loan received? $
b) what was the amount of interest charged? $

Explanation:

Step1: Identify the type of annuity

This is an ordinary annuity (payments at the end of each period) with quarterly compounding. The formula for the present value (PV) of an ordinary annuity is $PV = PMT \times \frac{1 - (1 + \frac{r}{n})^{-nt}}{\frac{r}{n}}$, where $PMT$ is the payment per period, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is the number of years.

Given:

• $PMT = \$1,069$
• $r = 2.69\% = 0.0269$
• $n = 4$ (quarterly compounding)
• $t = 6.5$ years

Step2: Calculate the present value (loan amount)

First, calculate the exponent $-nt = -4 \times 6.5 = -26$.

Then, calculate the term $\frac{r}{n} = \frac{0.0269}{4} = 0.006725$.

Next, calculate $(1 + \frac{r}{n})^{-nt} = (1 + 0.006725)^{-26} \approx 0.8432$.

Then, calculate the numerator $1 - (1 + \frac{r}{n})^{-nt} = 1 - 0.8432 = 0.1568$.

Now, calculate the present value factor $\frac{1 - (1 + \frac{r}{n})^{-nt}}{\frac{r}{n}} = \frac{0.1568}{0.006725} \approx 23.32$.

Finally, calculate $PV = 1069 \times 23.32 \approx \$24,930.08$.

Step3: Calculate the total payments

Total payments = $PMT \times nt = 1069 \times 26 = \$27,794$.

Step4: Calculate the interest charged

Interest charged = Total payments - Loan amount = $27,794 - 24,930.08 = \$2,863.92$.

Answer:

a) The amount of loan received is $\$24,930.08$ (rounded to the nearest cent).
b) The amount of interest charged is $\$2,863.92$ (rounded to the nearest cent).