"Tammy borrowed money from a bank and she is repaying the loan by making quarterly payments of R6164 at the end of every quarter, each payment includes interest and repayment of the principle. The loan will be paid off in 16 years and interest is calculated at a rate of 8.28% per year, compounded quarterly. How much interest did Tammy pay during year 10 of the loan?" just give me the correct formulas I need for the answer and make sure the values I need to put in are definitely correct

To solve this problem, we need to calculate the interest paid during year 10 of the loan using the following formulas. We’ll use the **Amortization Formula** and the **Loan Balance Formula** to find the interest portion.

### Key Information
- **Quarterly Payment (PMT)**: R6164
- **Annual Interest Rate**: 8.28%
- **Quarterly Interest Rate**: \(\frac{8.28\%}{4} = 2.07\%\)
- **Number of Years**: 16
- **Number of Quarters in 16 years**: \( 16 \times 4 = 64 \)
- **Interest Rate per Quarter (as a decimal)**: \( i = \frac{8.28\%}{4} = 0.0207 \)
- **Loan Term in Quarters**: \( n = 64 \)

### Step 1: Calculate the Initial Loan Amount
To find the loan amount, we use the **Present Value of an Annuity Formula**:

\[
PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}
\]

where:
- \( PMT = 6164 \)
- \( i = 0.0207 \)
- \( n = 64 \)

### Step 2: Calculate the Remaining Loan Balance at the Beginning of Year 10
Determine how many quarters have passed up to year 10. Since there are 4 quarters per year:

- **Quarters Passed** = 9 years \(\times\) 4 = 36 quarters
- **Remaining Quarters at the Beginning of Year 10** = \( 64 - 36 = 28 \)

Now, use the **Loan Balance Formula**:

\[
B_t = PMT \times \frac{1 - (1 + i)^{-(n-t)}}{i}
\]

where:
- \( t = \text{number of payments made} = 36 \)
- \( n = 64 \)
- \( PMT = 6164 \)
- \( i = 0.0207 \)

### Step 3: Calculate the Interest Paid in Year 10
For each payment in year 10, the interest portion is calculated using:

\[
\text{Interest for each payment} = \text{Remaining Balance before the payment} \times i
\]

Since year 10 has 4 payments, repeat the calculation for each quarter in year 10. Sum up the interest portions to get the total interest paid during year 10.

Let me know if you need detailed calculations or any further assistance with implementing these formulas!

Tammy borrowed money from a bank and she is repaying the loan by making quarterly payments of R6164 at the end of every quarter, each payment includes interest and repayment of the principle. The loan will be paid off in 16 years and interest is calculated at a rate of 8.28% per year, compounded quarterly. How much interest did Tammy pay during year 10 of the loan?

Learn how to calculate the interest paid during the 10th year of a loan with quarterly payments of R6164, compounded quarterly at an interest rate of 8.28% per year. Explore the amortization formula, loan balance formula, and compound interest concepts. Suitable for university-level finance or mathematics students.

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